Daily Papers

Domain adversarial training has been ubiquitous for achieving invariant representations and is used widely for various domain adaptation tasks. In recent times, methods converging to smooth optima have shown improved generalization for supervised learning tasks like classification. In this work, we analyze the effect of smoothness enhancing formulations on domain adversarial training, the objective of which is a combination of task loss (eg. classification, regression, etc.) and adversarial terms. We find that converging to a smooth minima with respect to (w.r.t.) task loss stabilizes the adversarial training leading to better performance on target domain. In contrast to task loss, our analysis shows that converging to smooth minima w.r.t. adversarial loss leads to sub-optimal generalization on the target domain. Based on the analysis, we introduce the Smooth Domain Adversarial Training (SDAT) procedure, which effectively enhances the performance of existing domain adversarial methods for both classification and object detection tasks. Our analysis also provides insight into the extensive usage of SGD over Adam in the community for domain adversarial training.

We present new algorithms for optimizing non-smooth, non-convex stochastic objectives based on a novel analysis technique. This improves the current best-known complexity for finding a (delta,epsilon)-stationary point from O(epsilon^{-4}delta^{-1}) stochastic gradient queries to O(epsilon^{-3}delta^{-1}), which we also show to be optimal. Our primary technique is a reduction from non-smooth non-convex optimization to online learning, after which our results follow from standard regret bounds in online learning. For deterministic and second-order smooth objectives, applying more advanced optimistic online learning techniques enables a new complexity of O(epsilon^{-1.5}delta^{-0.5}). Our techniques also recover all optimal or best-known results for finding epsilon stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are sub-optimal. Instead, these problems have been shown to satisfy certain generalized-smooth conditions, which have not been well understood in the existing literature. In this paper, we propose a notion of alpha-symmetric generalized-smoothness that extends the existing notions and covers many important functions such as high-order polynomials and exponential functions. We study the fundamental properties and establish descent lemmas for the functions in this class. Then, to solve such a large class of nonconvex problems, we design a special deterministic normalized gradient descent algorithm that achieves the optimal iteration complexity O(epsilon^{-2}), and also prove that the popular SPIDER variance reduction algorithm achieves the optimal sample complexity O(epsilon^{-3}) in the stochastic setting. Our results show that solving generalized-smooth nonconvex problems is as efficient as solving smooth nonconvex problems.

We consider online learning in multi-player smooth monotone games. Existing algorithms have limitations such as (1) being only applicable to strongly monotone games; (2) lacking the no-regret guarantee; (3) having only asymptotic or slow O(1{T}) last-iterate convergence rate to a Nash equilibrium. While the O(1{T}) rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms. We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no-regret learning algorithm for smooth monotone games. Namely, our algorithm achieves both (i) the optimal O(T) regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal O(1{T}) last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games. As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an O(log T) individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art O(T) bound.

Regularized optimal transport (OT) is now increasingly used as a loss or as a matching layer in neural networks. Entropy-regularized OT can be computed using the Sinkhorn algorithm but it leads to fully-dense transportation plans, meaning that all sources are (fractionally) matched with all targets. To address this issue, several works have investigated quadratic regularization instead. This regularization preserves sparsity and leads to unconstrained and smooth (semi) dual objectives, that can be solved with off-the-shelf gradient methods. Unfortunately, quadratic regularization does not give direct control over the cardinality (number of nonzeros) of the transportation plan. We propose in this paper a new approach for OT with explicit cardinality constraints on the transportation plan. Our work is motivated by an application to sparse mixture of experts, where OT can be used to match input tokens such as image patches with expert models such as neural networks. Cardinality constraints ensure that at most k tokens are matched with an expert, which is crucial for computational performance reasons. Despite the nonconvexity of cardinality constraints, we show that the corresponding (semi) dual problems are tractable and can be solved with first-order gradient methods. Our method can be thought as a middle ground between unregularized OT (recovered in the limit case k=1) and quadratically-regularized OT (recovered when k is large enough). The smoothness of the objectives increases as k increases, giving rise to a trade-off between convergence speed and sparsity of the optimal plan.

Human motion capture from monocular videos has made significant progress in recent years. However, modern approaches often produce temporal artifacts, e.g. in form of jittery motion and struggle to achieve smooth and physically plausible motions. Explicitly integrating physics, in form of internal forces and exterior torques, helps alleviating these artifacts. Current state-of-the-art approaches make use of an automatic PD controller to predict torques and reaction forces in order to re-simulate the input kinematics, i.e. the joint angles of a predefined skeleton. However, due to imperfect physical models, these methods often require simplifying assumptions and extensive preprocessing of the input kinematics to achieve good performance. To this end, we propose a novel method to selectively incorporate the physics models with the kinematics observations in an online setting, inspired by a neural Kalman-filtering approach. We develop a control loop as a meta-PD controller to predict internal joint torques and external reaction forces, followed by a physics-based motion simulation. A recurrent neural network is introduced to realize a Kalman filter that attentively balances the kinematics input and simulated motion, resulting in an optimal-state dynamics prediction. We show that this filtering step is crucial to provide an online supervision that helps balancing the shortcoming of the respective input motions, thus being important for not only capturing accurate global motion trajectories but also producing physically plausible human poses. The proposed approach excels in the physics-based human pose estimation task and demonstrates the physical plausibility of the predictive dynamics, compared to state of the art. The code is available on https://github.com/cuongle1206/OSDCap

Training neural networks requires optimizing a loss function that may be highly irregular, and in particular neither convex nor smooth. Popular training algorithms are based on stochastic gradient descent with momentum (SGDM), for which classical analysis applies only if the loss is either convex or smooth. We show that a very small modification to SGDM closes this gap: simply scale the update at each time point by an exponentially distributed random scalar. The resulting algorithm achieves optimal convergence guarantees. Intriguingly, this result is not derived by a specific analysis of SGDM: instead, it falls naturally out of a more general framework for converting online convex optimization algorithms to non-convex optimization algorithms.

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-m rate, where m is the number of simulated samples. This can lead to significant computational challenges since a large m is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.

Gradient-free/zeroth-order methods for black-box convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration complexity, and propose a generic approach that, based on optimal first-order methods, allows to obtain in a black-box fashion new zeroth-order algorithms for non-smooth convex optimization problems. Our approach not only leads to optimal oracle complexity, but also allows to obtain iteration complexity similar to first-order methods, which, in turn, allows to exploit parallel computations to accelerate the convergence of our algorithms. We also elaborate on extensions for stochastic optimization problems, saddle-point problems, and distributed optimization.

Multi-agent systems play a central role in area coverage tasks across search-and-rescue, environmental monitoring, and precision agriculture. Achieving non-uniform coverage, where spatial priorities vary across the domain, requires coordinating agents while respecting dynamic and communication constraints. Density-driven approaches can distribute agents according to a prescribed reference density, but existing methods do not ensure connectivity. This limitation often leads to communication loss, reduced coordination, and degraded coverage performance. This letter introduces a connectivity-preserving extension of the Density-Driven Optimal Control (D2OC) framework. The coverage objective, defined using the Wasserstein distance between the agent distribution and the reference density, admits a convex quadratic program formulation. Communication constraints are incorporated through a smooth connectivity penalty, which maintains strict convexity, supports distributed implementation, and preserves inter-agent communication without imposing rigid formations. Simulation studies show that the proposed method consistently maintains connectivity, improves convergence speed, and enhances non-uniform coverage quality compared with density-driven schemes that do not incorporate explicit connectivity considerations.

The problem of optimal sizing and power scheduling in microgrids subject to uncertainties is well known to the control community. Commonly, the optimal control problem is cast as a mixed-integer program to model the logical constraints arising in energy storage systems, and is then solved approximately using numerical methods such as the scenario approach. In this paper, we propose and compare two formulations of a robust microgrid sizing and power scheduling optimal control problem with logical constraints and uncertainties in the user's power demand, solar power generation, grid electricity prices and battery efficiencies. The first formulation uses binary variables and big-M constraints, leading to a mixed-integer linear program. The second formulation casts the problem as a continuous nonlinear program through an exact smooth reformulation of the logical constraints, consisting of additional modelling variables and non-convex constraints. We then propose a novel local reduction algorithm, extending an existing method, to solve both problems. The two formulations are compared by evaluating the solutions returned by local reduction using 100,000-sample Monte Carlo simulations and achieve promising results, with both averaging feasibility rates above 90%.

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of nodes evaluations.

This study addresses the challenge of estimating average treatment effects (ATEs) for advertising campaigns in online marketplaces where complete randomized experimentation is infeasible. We propose two key innovations: (1) a shrinkage estimator that optimally combines observational and experimental data without assuming smooth treatment effects across campaigns, and (2) a Bayesian adaptive experimental design framework that efficiently selects campaigns for randomized evaluation that minimizes cumulative risk. Our shrinkage estimator achieves lower risk compared to existing methods by balancing bias-variance tradeoffs, while our adaptive design significantly reduces the costs of campaign randomization. We establish theoretical guarantees including asymptotic normality and regret bounds. In an application to Amazon Ads data analyzing 2,583 campaigns, our approach achieves equivalent estimation precision while requiring only half of the randomized experiments needed by random sampling, the standard method widely used in practice today. The proposed method serves as a practical solution for marketplace platforms to efficiently measure advertising effectiveness while managing experimentation costs.

Efficient key-value (KV) cache compression is critical for scaling transformer-based Large Language Models (LLMs) in long sequences and resource-limited settings. Existing methods evict tokens based on their positions or importance scores, but position-based strategies can miss crucial information outside predefined regions, while those relying on global importance scores resulting in strong regional biases, limiting the KV cache's overall context retention and potentially impairing the performance of LLMs on complex tasks. Our wavelet analysis reveals that as tokens approach the end of sequence, their contributions to generation gradually increase and tends to diverge more from neighboring tokens, indicating a smooth transition with increasing complexity and variability from distant to nearby context. Motivated by this observation, we propose TreeKV, an intuitive, training-free method that employs a tree structure for smooth cache compression. TreeKV maintains a fixed cache size, allowing LLMs to deliver high-quality output even in long text scenarios. Unlike most compression methods, TreeKV is applicable to both the generation and prefilling stages. TreeKV consistently surpasses all baseline models in language modeling tasks on PG19 and OpenWebText2, allowing LLMs trained with short context window to generalize to longer window with a 16x cache reduction. On the Longbench benchmark, TreeKV achieves the best performance with only 6\% of the budget at optimal efficiency.

The growing safety concerns surrounding large language models raise an urgent need to align them with diverse human preferences to simultaneously enhance their helpfulness and safety. A promising approach is to enforce safety constraints through Reinforcement Learning from Human Feedback (RLHF). For such constrained RLHF, typical Lagrangian-based primal-dual policy optimization methods are computationally expensive and often unstable. This paper presents a perspective of dualization that reduces constrained alignment to an equivalent unconstrained alignment problem. We do so by pre-optimizing a smooth and convex dual function that has a closed form. This shortcut eliminates the need for cumbersome primal-dual policy iterations, greatly reducing the computational burden and improving training stability. Our strategy leads to two practical algorithms in model-based and preference-based settings (MoCAN and PeCAN, respectively). A broad range of experiments demonstrate the effectiveness and merits of our algorithms.

We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret {O} big( varepsilon^{-1} log^{1.5}{d} big) where d is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are {O} big( varepsilon^{-1} minbig{d, T^{1/3}log dbig} big). We also develop an adaptive algorithm for the small-loss setting with regret O(L^starlog d + varepsilon^{-1} log^{1.5}{d}) where L^star is the total loss of the best expert. Additionally, we consider DP online convex optimization in the realizable setting and propose an algorithm with near-optimal regret O big(varepsilon^{-1} d^{1.5} big), as well as an algorithm for the smooth case with regret O big( varepsilon^{-2/3} (dT)^{1/3} big), both significantly improving over existing bounds in the non-realizable regime.

Neural networks increasingly embed non-differentiable components (spiking neurons, quantized layers, discrete routing, blackbox simulators, etc.) where backpropagation is inapplicable and surrogate gradients introduce bias. We present PolyStep, a gradient-free optimizer that updates parameters using only forward passes. Each step evaluates the loss at structured polytope vertices in a compressed subspace, computes softmax-weighted assignments over the resulting cost matrix, and displaces particles toward low-cost vertices via barycentric projection. This update corresponds to the one-sided limit of a regularized optimal-transport problem, inheriting its geometric structure without Sinkhorn iterations. PolyStep trains genuinely non-differentiable models where existing gradient-free methods collapse to near-random accuracy. On hard-LIF spiking networks we reach 93.4% test accuracy, outperforming all gradient-free baselines by over 60~pp and closing to within 4.4~pp of a surrogate-gradient Adam ceiling. Across four additional non-differentiable architectures (int8 quantization, argmax attention, staircase activations, hard MoE routing) we lead every gradient-free competitor. On MAX-SAT scaling from 100 to 1M variables, we sustain above 92% clause satisfaction while evolution strategies drop 8--12~pp. On RL policy search, we match OpenAI-ES on classical control and retain performance under integer and binary quantization that collapses gradient-based methods. We prove convergence to conservative-stationary points at rate O(log T/T) on piecewise-smooth losses, upgraded to Clarke-stationary on the headline architectures and extended to the piecewise-constant regime via a hitting-time bound. These rates match the known zeroth-order query-complexity lower bounds that all forward-only methods inherit. Code is available at https://github.com/anindex/polystep.

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function f_{rho}^{*} in [H]^{s}, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) H for some sin (0,1). The existing minimax optimal results require |f_{rho}^{*}|_{L^{infty}}<infty which implicitly requires s > alpha_{0} where alpha_{0}in (0,1) is the embedding index, a constant depending on H. Whether the KRR is optimal for all sin (0,1) is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any sin (0,1) when the H is a Sobolev RKHS.

Current LLM alignment techniques use pairwise human preferences at a sample level, and as such, they do not imply an alignment on the distributional level. We propose in this paper Alignment via Optimal Transport (AOT), a novel method for distributional preference alignment of LLMs. AOT aligns LLMs on unpaired preference data by making the reward distribution of the positive samples stochastically dominant in the first order on the distribution of negative samples. We introduce a convex relaxation of this first-order stochastic dominance and cast it as an optimal transport problem with a smooth and convex cost. Thanks to the one-dimensional nature of the resulting optimal transport problem and the convexity of the cost, it has a closed-form solution via sorting on empirical measures. We fine-tune LLMs with this AOT objective, which enables alignment by penalizing the violation of the stochastic dominance of the reward distribution of the positive samples on the reward distribution of the negative samples. We analyze the sample complexity of AOT by considering the dual of the OT problem and show that it converges at the parametric rate. Empirically, we show on a diverse set of alignment datasets and LLMs that AOT leads to state-of-the-art models in the 7B family of models when evaluated with Open LLM Benchmarks and AlpacaEval.

We compare the (1,lambda)-EA and the (1 + lambda)-EA on the recently introduced benchmark DisOM, which is the OneMax function with randomly planted local optima. Previous work showed that if all local optima have the same relative height, then the plus strategy never loses more than a factor O(nlog n) compared to the comma strategy. Here we show that even small random fluctuations in the heights of the local optima have a devastating effect for the plus strategy and lead to super-polynomial runtimes. On the other hand, due to their ability to escape local optima, comma strategies are unaffected by the height of the local optima and remain efficient. Our results hold for a broad class of possible distortions and show that the plus strategy, but not the comma strategy, is generally deceived by sparse unstructured fluctuations of a smooth landscape.

Transfer learning is an essential technique for many machine learning/AI models of complex structures such as large language models and generative AI. The essence of transfer learning is to leverage knowledge from resolved source tasks for a new target task, especially when the sample size m of the training data for the latter is low. In this work, we rigorously analyze the potential benefit of transfer learning in terms of sample efficiency. Specifically, taking an optimal transport viewpoint of transfer learning, we find that when the data dimension d is higher than 3, the sample complexity for transfer learning is O(m^{-(α+1)/d}), with α indicating the smoothness of the data distribution, as opposed to the O(m^{-p/d}) sample complexity for direct learning with p indicating the smoothness of the optimal target model. Our finding theoretically supports a better sample efficiency for transfer learning, when the target task is optimizing over a family of not-so-smooth models (i.e., highly complex networks with the possible use of non-smooth activation functions). Using image classification as an example, we numerically demonstrate the sample efficiency for transfer learning, that is, in the data hungry regime, the model performance can be significantly improved by transfer learning.

Solving partial differential equations (PDEs) is a central task in scientific computing. Recently, neural network approximation of PDEs has received increasing attention due to its flexible meshless discretization and its potential for high-dimensional problems. One fundamental numerical difficulty is that random samples in the training set introduce statistical errors into the discretization of loss functional which may become the dominant error in the final approximation, and therefore overshadow the modeling capability of the neural network. In this work, we propose a new minmax formulation to optimize simultaneously the approximate solution, given by a neural network model, and the random samples in the training set, provided by a deep generative model. The key idea is to use a deep generative model to adjust random samples in the training set such that the residual induced by the approximate PDE solution can maintain a smooth profile when it is being minimized. Such an idea is achieved by implicitly embedding the Wasserstein distance between the residual-induced distribution and the uniform distribution into the loss, which is then minimized together with the residual. A nearly uniform residual profile means that its variance is small for any normalized weight function such that the Monte Carlo approximation error of the loss functional is reduced significantly for a certain sample size. The adversarial adaptive sampling (AAS) approach proposed in this work is the first attempt to formulate two essential components, minimizing the residual and seeking the optimal training set, into one minmax objective functional for the neural network approximation of PDEs.

In deployment of the VLA models to real-world robotic tasks, execution speed matters. In previous work arXiv:2510.26742 we analyze how to make neural computation of VLAs on GPU fast. However, we leave the question of how to actually deploy the VLA system on the real robots open. In this report we describe a set of practical techniques to achieve the end-to-end result of running a VLA-driven robot at an impressive speed in real world tasks that require both accuracy and dexterity. The stack of technology ranges across calibration, planning & control, and learning based method to identify optimal execution speed. In the tasks we show, the robot even executes in a speed on par with casual human operation and approaching the hardware limit of our lightweight arm. The unaccelerated videos and inference traces are provided in https://dexmal.github.io/realtime-vla-v2/.

We propose SmoothCruiser, a new planning algorithm for estimating the value function in entropy-regularized Markov decision processes and two-player games, given a generative model of the environment. SmoothCruiser makes use of the smoothness of the Bellman operator promoted by the regularization to achieve problem-independent sample complexity of order O~(1/epsilon^4) for a desired accuracy epsilon, whereas for non-regularized settings there are no known algorithms with guaranteed polynomial sample complexity in the worst case.

The deep learning recipe of casting real-world problems as mathematical optimisation and tackling the optimisation by training deep neural networks using gradient-based optimisation has undoubtedly proven to be a fruitful one. The understanding behind why deep learning works, however, has lagged behind its practical significance. We aim to make steps towards an improved understanding of deep learning with a focus on optimisation and model regularisation. We start by investigating gradient descent (GD), a discrete-time algorithm at the basis of most popular deep learning optimisation algorithms. Understanding the dynamics of GD has been hindered by the presence of discretisation drift, the numerical integration error between GD and its often studied continuous-time counterpart, the negative gradient flow (NGF). To add to the toolkit available to study GD, we derive novel continuous-time flows that account for discretisation drift. Unlike the NGF, these new flows can be used to describe learning rate specific behaviours of GD, such as training instabilities observed in supervised learning and two-player games. We then translate insights from continuous time into mitigation strategies for unstable GD dynamics, by constructing novel learning rate schedules and regularisers that do not require additional hyperparameters. Like optimisation, smoothness regularisation is another pillar of deep learning's success with wide use in supervised learning and generative modelling. Despite their individual significance, the interactions between smoothness regularisation and optimisation have yet to be explored. We find that smoothness regularisation affects optimisation across multiple deep learning domains, and that incorporating smoothness regularisation in reinforcement learning leads to a performance boost that can be recovered using adaptions to optimisation methods.

We study the problem of black-box optimization of a function f of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one of its global optima, but this smoothness is unknown. Our contribution is an adaptive optimization algorithm, POO or parallel optimistic optimization, that is able to deal with this setting. POO performs almost as well as the best known algorithms requiring the knowledge of the smoothness. Furthermore, POO works for a larger class of functions than what was previously considered, especially for functions that are difficult to optimize, in a very precise sense. We provide a finite-time analysis of POO's performance, which shows that its error after n evaluations is at most a factor of sqrt(ln n) away from the error of the best known optimization algorithms using the knowledge of the smoothness.

We study the problem of optimizing a function under a budgeted number of evaluations. We only assume that the function is locally smooth around one of its global optima. The difficulty of optimization is measured in terms of 1) the amount of noise b of the function evaluation and 2) the local smoothness, d, of the function. A smaller d results in smaller optimization error. We come with a new, simple, and parameter-free approach. First, for all values of b and d, this approach recovers at least the state-of-the-art regret guarantees. Second, our approach additionally obtains these results while being agnostic to the values of both b and d. This leads to the first algorithm that naturally adapts to an unknown range of noise b and leads to significant improvements in a moderate and low-noise regime. Third, our approach also obtains a remarkable improvement over the state-of-the-art SOO algorithm when the noise is very low which includes the case of optimization under deterministic feedback (b=0). There, under our minimal local smoothness assumption, this improvement is of exponential magnitude and holds for a class of functions that covers the vast majority of functions that practitioners optimize (d=0). We show that our algorithmic improvement is borne out in experiments as we empirically show faster convergence on common benchmarks.

We propose a method that achieves near-optimal rates for smooth stochastic convex optimization and requires essentially no prior knowledge of problem parameters. This improves on prior work which requires knowing at least the initial distance to optimality d0. Our method, U-DoG, combines UniXGrad (Kavis et al., 2019) and DoG (Ivgi et al., 2023) with novel iterate stabilization techniques. It requires only loose bounds on d0 and the noise magnitude, provides high probability guarantees under sub-Gaussian noise, and is also near-optimal in the non-smooth case. Our experiments show consistent, strong performance on convex problems and mixed results on neural network training.

Recently, research on denoising diffusion models has expanded its application to the field of image restoration. Traditional diffusion-based image restoration methods utilize degraded images as conditional input to effectively guide the reverse generation process, without modifying the original denoising diffusion process. However, since the degraded images already include low-frequency information, starting from Gaussian white noise will result in increased sampling steps. We propose Resfusion, a general framework that incorporates the residual term into the diffusion forward process, starting the reverse process directly from the noisy degraded images. The form of our inference process is consistent with the DDPM. We introduced a weighted residual noise, named resnoise, as the prediction target and explicitly provide the quantitative relationship between the residual term and the noise term in resnoise. By leveraging a smooth equivalence transformation, Resfusion determine the optimal acceleration step and maintains the integrity of existing noise schedules, unifying the training and inference processes. The experimental results demonstrate that Resfusion exhibits competitive performance on ISTD dataset, LOL dataset and Raindrop dataset with only five sampling steps. Furthermore, Resfusion can be easily applied to image generation and emerges with strong versatility. Our code and model are available at https://github.com/nkicsl/Resfusion.

We present WUKONG, a novel training-free framework for high-fidelity textured 3D morphing that takes a pair of source and target prompts (image or text) as input. Unlike conventional methods -- which rely on manual correspondence matching and deformation trajectory estimation (limiting generalization and requiring costly preprocessing) -- WUKONG leverages the generative prior of flow-based transformers to produce high-fidelity 3D transitions with rich texture details. To ensure smooth shape transitions, we exploit the inherent continuity of flow-based generative processes and formulate morphing as an optimal transport barycenter problem. We further introduce a sequential initialization strategy to prevent abrupt geometric distortions and preserve identity coherence. For faithful texture preservation, we propose a similarity-guided semantic consistency mechanism that selectively retains high-frequency details and enables precise control over blending dynamics. This avoids common artifacts like oversmoothing while maintaining semantic fidelity. Extensive quantitative and qualitative evaluations demonstrate that WUKONG significantly outperforms state-of-the-art methods, achieving superior results across diverse geometry and texture variations.

Variational inequalities have recently attracted considerable interest in machine learning as a flexible paradigm for models that go beyond ordinary loss function minimization (such as generative adversarial networks and related deep learning systems). In this setting, the optimal O(1/t) convergence rate for solving smooth monotone variational inequalities is achieved by the Extra-Gradient (EG) algorithm and its variants. Aiming to alleviate the cost of an extra gradient step per iteration (which can become quite substantial in deep learning applications), several algorithms have been proposed as surrogates to Extra-Gradient with a single oracle call per iteration. In this paper, we develop a synthetic view of such algorithms, and we complement the existing literature by showing that they retain a O(1/t) ergodic convergence rate in smooth, deterministic problems. Subsequently, beyond the monotone deterministic case, we also show that the last iterate of single-call, stochastic extra-gradient methods still enjoys a O(1/t) local convergence rate to solutions of non-monotone variational inequalities that satisfy a second-order sufficient condition.

Representing continuous time is a critical and under-explored challenge in modeling temporal event sequences with large language models (LLMs). Various strategies like byte-level representations or calendar tokens have been proposed. However, the optimal approach remains unclear, especially given the diverse statistical distributions of real-world event data, which range from smooth log-normal to discrete, spiky patterns. This paper presents the first empirical study of temporal tokenization for event sequences, comparing distinct encoding strategies: naive numeric strings, high-precision byte-level representations, human-semantic calendar tokens, classic uniform binning, and adaptive residual scalar quantization. We evaluate these strategies by fine-tuning LLMs on real-world datasets that exemplify these diverse distributions. Our analysis reveals that no single strategy is universally superior; instead, prediction performance depends heavily on aligning the tokenizer with the data's statistical properties, with log-based strategies excelling on skewed distributions and human-centric formats proving robust for mixed modalities.

Bilevel optimization has shown its utility across various machine learning settings, yet most algorithms in practice require second-order information, making it challenging to scale them up. Only recently, a paradigm of first-order algorithms has emerged in the theoretical literature, capable of effectively addressing bilevel optimization problems. Nevertheless, the practical efficiency of this paradigm remains unverified, particularly in the context of large language models (LLMs). This paper introduces the first scalable instantiation of this paradigm called ScaleBiO, focusing on bilevel optimization for large-scale LLM data reweighting. By combining with a recently proposed memory-efficient training technique called LISA, our novel algorithm allows the paradigm to scale to sim30B-sized LLMs on 8timesH100 GPUs, marking the first successful application of bilevel optimization under practical scenarios for large-sized LLMs. Empirically, extensive experiments on data reweighting verify the effectiveness of ScaleBiO for different-scaled models, including Llama-3-8B, Gemma-2-9B, Qwen-2-7B, and Qwen-2.5-32B, where bilevel optimization succeeds in instruction-following and math reasoning tasks, outperforming several popular baselines, including uniform sampling, influence-aware data filtering, and reference-model-based sampling methods. Theoretically, ScaleBiO ensures the optimality of the learned data weights, along with a convergence guarantee matching the conventional first-order bilevel optimization paradigm on smooth and strongly convex objectives.

In this paper, we study a class of composite optimization problems whose objective function is given by the summation of a general smooth and nonsmooth component, together with a relatively simple nonsmooth term. While restart strategies are commonly employed in first-order methods to achieve optimal convergence under strong convexity, they introduce structural complexity and practical overhead, making algorithm design and nesting cumbersome. To address this, we propose a restart-free stochastic gradient sliding algorithm that eliminates the need for explicit restart phases when the simple nonsmooth component is strongly convex. Through a novel and carefully designed parameter selection strategy, we prove that the proposed algorithm achieves an ε-solution with only O(log(1ε)) gradient evaluations for the smooth component and O(1ε) stochastic subgradient evaluations for the nonsmooth component, matching the optimal complexity of existing multi-phase (restart-based) methods. Moreover, for the case where the nonsmooth component is structured, allowing the overall problem to be reformulated as a bilinear saddle-point problem, we develop a restart-free accelerated stochastic gradient sliding algorithm. We show that the resulting method requires only O(log(1ε)) gradient computations for the smooth component while preserving an overall iteration complexity of O(1{sqrtε}) for solving the corresponding saddle-point problems. Our work thus provides simpler, restart-f

Adaptive sparse coding methods learn a possibly overcomplete set of basis functions, such that natural image patches can be reconstructed by linearly combining a small subset of these bases. The applicability of these methods to visual object recognition tasks has been limited because of the prohibitive cost of the optimization algorithms required to compute the sparse representation. In this work we propose a simple and efficient algorithm to learn basis functions. After training, this model also provides a fast and smooth approximator to the optimal representation, achieving even better accuracy than exact sparse coding algorithms on visual object recognition tasks.

Transformers have found extensive applications across various domains due to the powerful fitting capabilities. This success can be partially attributed to their inherent nonlinearity. Thus, in addition to the ReLU function employed in the original transformer architecture, researchers have explored alternative modules such as GeLU and SwishGLU to enhance nonlinearity and thereby augment representational capacity. In this paper, we propose a novel category of polynomial composition activations (PolyCom), designed to optimize the dynamics of transformers. Theoretically, we provide a comprehensive mathematical analysis of PolyCom, highlighting its enhanced expressivity and efficacy relative to other activation functions. Notably, we demonstrate that networks incorporating PolyCom achieve the optimal approximation rate, indicating that PolyCom networks require minimal parameters to approximate general smooth functions in Sobolev spaces. We conduct empirical experiments on the pre-training configurations of large language models (LLMs), including both dense and sparse architectures. By substituting conventional activation functions with PolyCom, we enable LLMs to capture higher-order interactions within the data, thus improving performance metrics in terms of accuracy and convergence rates. Extensive experimental results demonstrate the effectiveness of our method, showing substantial improvements over other activation functions. Code is available at https://github.com/BryceZhuo/PolyCom.

Despite significant work on low-bit quantization-aware training (QAT), there is still a large accuracy gap between such techniques and native training. To address this, we introduce CAGE (Curvature-Aware Gradient Estimation), a new QAT method that augments the straight-through estimator (STE) gradient with a curvature-aware correction designed to counteract the loss increase induced by quantization. CAGE is derived from a multi-objective view of QAT that balances loss minimization with adherence to quantization constraints, yielding a principled correction term that depends on local curvature information. On the theoretical side, we introduce the notion of Pareto-optimal solutions for quantized optimization, and establish that CAGE yields strong convergence guarantees in the smooth non-convex setting. In terms of implementation, our approach is optimizer-agnostic, but we provide a highly-efficient implementation that leverages Adam statistics. When pre-training Llama-style models of up to 800M-parameters, CAGE recovers over 10% of the quantization-induced loss increase in the W4A4 regime over outlier-mitigation methods. These results indicate that curvature-aware gradient corrections can bridge the remaining performance gap beyond current outlier-handling methods.

Pre-training Transformer models is resource-intensive, and recent studies have shown that sign momentum is an efficient technique for training large-scale deep learning models, particularly Transformers. However, its application in distributed training remains underexplored. This paper investigates a novel communication-efficient distributed sign momentum method with multiple local steps, to cope with the scenarios where communicating at every step is prohibitive. Our proposed method allows for a broad class of base optimizers for local steps, and uses sign momentum in the global step, where momentum is generated from differences accumulated during local steps. For generic base optimizers, by approximating the sign operator with a randomized version that acts as a continuous analog in expectation, we present a general convergence analysis, which specializes to an O(1/T) rate for a particular instance. When local step is stochastic gradient descent, we show an optimal O(1/T^{1/4}) rate in terms of ell_1 gradient norm for nonconvex smooth cost functions. We extensively evaluate our method on the pre-training of various sized GPT-2 models from scratch, and the empirical results show significant improvement compared to other distributed methods with multiple local steps.

In this paper, by introducing Generalized Bernstein condition, we propose the first Obig(sqrt{p}{nepsilon}big) high probability excess population risk bound for differentially private algorithms under the assumptions G-Lipschitz, L-smooth, and Polyak-{\L}ojasiewicz condition, based on gradient perturbation method. If we replace the properties G-Lipschitz and L-smooth by alpha-H{\"o}lder smoothness (which can be used in non-smooth setting), the high probability bound comes to Obig(n^{-alpha{1+2alpha}}big) w.r.t n, which cannot achieve Oleft(1/nright) when alphain(0,1]. To solve this problem, we propose a variant of gradient perturbation method, max{1,g-Normalized Gradient Perturbation} (m-NGP). We further show that by normalization, the high probability excess population risk bound under assumptions alpha-H{\"o}lder smooth and Polyak-{\L}ojasiewicz condition can achieve Obig(sqrt{p}{nepsilon}big), which is the first Oleft(1/nright) high probability excess population risk bound w.r.t n for differentially private algorithms under non-smooth conditions. Moreover, we evaluate the performance of the new proposed algorithm m-NGP, the experimental results show that m-NGP improves the performance of the differentially private model over real datasets. It demonstrates that m-NGP improves the utility bound and the accuracy of the DP model on real datasets simultaneously.

Posterior sampling, i.e., exponential mechanism to sample from the posterior distribution, provides varepsilon-pure differential privacy (DP) guarantees and does not suffer from potentially unbounded privacy breach introduced by (varepsilon,delta)-approximate DP. In practice, however, one needs to apply approximate sampling methods such as Markov chain Monte Carlo (MCMC), thus re-introducing the unappealing delta-approximation error into the privacy guarantees. To bridge this gap, we propose the Approximate SAample Perturbation (abbr. ASAP) algorithm which perturbs an MCMC sample with noise proportional to its Wasserstein-infinity (W_infty) distance from a reference distribution that satisfies pure DP or pure Gaussian DP (i.e., delta=0). We then leverage a Metropolis-Hastings algorithm to generate the sample and prove that the algorithm converges in W_infty distance. We show that by combining our new techniques with a careful localization step, we obtain the first nearly linear-time algorithm that achieves the optimal rates in the DP-ERM problem with strongly convex and smooth losses.

Our research aims to unify existing works' diverging opinions on how architectural components affect the adversarial robustness of CNNs. To accomplish our goal, we synthesize a suite of three generalizable robust architectural design principles: (a) optimal range for depth and width configurations, (b) preferring convolutional over patchify stem stage, and (c) robust residual block design through adopting squeeze and excitation blocks and non-parametric smooth activation functions. Through extensive experiments across a wide spectrum of dataset scales, adversarial training methods, model parameters, and network design spaces, our principles consistently and markedly improve AutoAttack accuracy: 1-3 percentage points (pp) on CIFAR-10 and CIFAR-100, and 4-9 pp on ImageNet. The code is publicly available at https://github.com/poloclub/robust-principles.

Offline (or batch) reinforcement learning (RL) algorithms seek to learn an optimal policy from a fixed dataset without active data collection. Based on the composition of the offline dataset, two main categories of methods are used: imitation learning which is suitable for expert datasets and vanilla offline RL which often requires uniform coverage datasets. From a practical standpoint, datasets often deviate from these two extremes and the exact data composition is usually unknown a priori. To bridge this gap, we present a new offline RL framework that smoothly interpolates between the two extremes of data composition, hence unifying imitation learning and vanilla offline RL. The new framework is centered around a weak version of the concentrability coefficient that measures the deviation from the behavior policy to the expert policy alone. Under this new framework, we further investigate the question on algorithm design: can one develop an algorithm that achieves a minimax optimal rate and also adapts to unknown data composition? To address this question, we consider a lower confidence bound (LCB) algorithm developed based on pessimism in the face of uncertainty in offline RL. We study finite-sample properties of LCB as well as information-theoretic limits in multi-armed bandits, contextual bandits, and Markov decision processes (MDPs). Our analysis reveals surprising facts about optimality rates. In particular, in all three settings, LCB achieves a faster rate of 1/N for nearly-expert datasets compared to the usual rate of 1/N in offline RL, where N is the number of samples in the batch dataset. In the case of contextual bandits with at least two contexts, we prove that LCB is adaptively optimal for the entire data composition range, achieving a smooth transition from imitation learning to offline RL. We further show that LCB is almost adaptively optimal in MDPs.

We present Dojo, a differentiable physics engine for robotics that prioritizes stable simulation, accurate contact physics, and differentiability with respect to states, actions, and system parameters. Dojo models hard contact and friction with a nonlinear complementarity problem with second-order cone constraints. We introduce a custom primal-dual interior-point method to solve the second order cone program for stable forward simulation over a broad range of sample rates. We obtain smooth gradient approximations with this solver through the implicit function theorem, giving gradients that are useful for downstream trajectory optimization, policy optimization, and system identification applications. Specifically, we propose to use the central path parameter threshold in the interior point solver as a user-tunable design parameter. A high value gives a smooth approximation to contact dynamics with smooth gradients for optimization and learning, while a low value gives precise simulation rollouts with hard contact. We demonstrate Dojo's differentiability in trajectory optimization, policy learning, and system identification examples. We also benchmark Dojo against MuJoCo, PyBullet, Drake, and Brax on a variety of robot models, and study the stability and simulation quality over a range of sample frequencies and accuracy tolerances. Finally, we evaluate the sim-to-real gap in hardware experiments with a Ufactory xArm 6 robot. Dojo is an open source project implemented in Julia with Python bindings, with code available at https://github.com/dojo-sim/Dojo.jl.

Learning-based methods have gained attention as general-purpose solvers due to their ability to automatically learn problem-specific heuristics, reducing the need for manually crafted heuristics. However, these methods often face scalability challenges. To address these issues, the improved Sampling algorithm for Combinatorial Optimization (iSCO), using discrete Langevin dynamics, has been proposed, demonstrating better performance than several learning-based solvers. This study proposes a different approach that integrates gradient-based update through continuous relaxation, combined with Quasi-Quantum Annealing (QQA). QQA smoothly transitions the objective function, starting from a simple convex function, minimized at half-integral values, to the original objective function, where the relaxed variables are minimized only in the discrete space. Furthermore, we incorporate parallel run communication leveraging GPUs to enhance exploration capabilities and accelerate convergence. Numerical experiments demonstrate that our method is a competitive general-purpose solver, achieving performance comparable to iSCO and learning-based solvers across various benchmark problems. Notably, our method exhibits superior speed-quality trade-offs for large-scale instances compared to iSCO, learning-based solvers, commercial solvers, and specialized algorithms.

We present an algorithm for minimizing an objective with hard-to-compute gradients by using a related, easier-to-access function as a proxy. Our algorithm is based on approximate proximal point iterations on the proxy combined with relatively few stochastic gradients from the objective. When the difference between the objective and the proxy is delta-smooth, our algorithm guarantees convergence at a rate matching stochastic gradient descent on a delta-smooth objective, which can lead to substantially better sample efficiency. Our algorithm has many potential applications in machine learning, and provides a principled means of leveraging synthetic data, physics simulators, mixed public and private data, and more.

Deep reinforcement learning agents often exhibit erratic, high-frequency control behaviors that hinder real-world deployment due to excessive energy consumption and mechanical wear. We systematically investigate action smoothness regularization through higher-order derivative penalties, progressing from theoretical understanding in continuous control benchmarks to practical validation in building energy management. Our comprehensive evaluation across four continuous control environments demonstrates that third-order derivative penalties (jerk minimization) consistently achieve superior smoothness while maintaining competitive performance. We extend these findings to HVAC control systems where smooth policies reduce equipment switching by 60%, translating to significant operational benefits. Our work establishes higher-order action regularization as an effective bridge between RL optimization and operational constraints in energy-critical applications.

In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The key innovation of our analysis is a novel notion called Gradient-Normalized Smoothness, which characterizes the maximum radius of a ball around the current point that yields a good relative approximation of the gradient field. Our theory establishes a natural intrinsic connection between Hessian approximation and the linearization of the gradient. Importantly, Gradient-Normalized Smoothness does not depend on the specific problem class of the objective functions, while effectively translating local information about the gradient field and Hessian approximation into the global behavior of the method. This new concept equips approximate second-order algorithms with universal global convergence guarantees, recovering state-of-the-art rates for functions with H\"older-continuous Hessians and third derivatives, quasi-self-concordant functions, as well as smooth classes in first-order optimization. These rates are achieved automatically and extend to broader classes, such as generalized self-concordant functions. We demonstrate direct applications of our results for global linear rates in logistic regression and softmax problems with approximate Hessians, as well as in non-convex optimization using Fisher and Gauss-Newton approximations.

Understanding and analyzing markets is crucial, yet analytical equilibrium solutions remain largely infeasible. Recent breakthroughs in equilibrium computation rely on zeroth-order policy gradient estimation. These approaches commonly suffer from high variance and are computationally expensive. The use of fully differentiable simulators would enable more efficient gradient estimation. However, the discrete allocation of goods in economic simulations is a non-differentiable operation. This renders the first-order Monte Carlo gradient estimator inapplicable and the learning feedback systematically misleading. We propose a novel smoothing technique that creates a surrogate market game, in which first-order methods can be applied. We provide theoretical bounds on the resulting bias which justifies solving the smoothed game instead. These bounds also allow choosing the smoothing strength a priori such that the resulting estimate has low variance. Furthermore, we validate our approach via numerous empirical experiments. Our method theoretically and empirically outperforms zeroth-order methods in approximation quality and computational efficiency.

In this article, we introduce a novel normalization technique for neural network weight matrices, which we term weight conditioning. This approach aims to narrow the gap between the smallest and largest singular values of the weight matrices, resulting in better-conditioned matrices. The inspiration for this technique partially derives from numerical linear algebra, where well-conditioned matrices are known to facilitate stronger convergence results for iterative solvers. We provide a theoretical foundation demonstrating that our normalization technique smoothens the loss landscape, thereby enhancing convergence of stochastic gradient descent algorithms. Empirically, we validate our normalization across various neural network architectures, including Convolutional Neural Networks (CNNs), Vision Transformers (ViT), Neural Radiance Fields (NeRF), and 3D shape modeling. Our findings indicate that our normalization method is not only competitive but also outperforms existing weight normalization techniques from the literature.

Traditional theories of optimization cannot describe the dynamics of optimization in deep learning, even in the simple setting of deterministic training. The challenge is that optimizers typically operate in a complex, oscillatory regime called the "edge of stability." In this paper, we develop theory that can describe the dynamics of optimization in this regime. Our key insight is that while the *exact* trajectory of an oscillatory optimizer may be challenging to analyze, the *time-averaged* (i.e. smoothed) trajectory is often much more tractable. To analyze an optimizer, we derive a differential equation called a "central flow" that characterizes this time-averaged trajectory. We empirically show that these central flows can predict long-term optimization trajectories for generic neural networks with a high degree of numerical accuracy. By interpreting these central flows, we are able to understand how gradient descent makes progress even as the loss sometimes goes up; how adaptive optimizers "adapt" to the local loss landscape; and how adaptive optimizers implicitly navigate towards regions where they can take larger steps. Our results suggest that central flows can be a valuable theoretical tool for reasoning about optimization in deep learning.

Explaining the output of a deep network remains a challenge. In the case of an image classifier, one type of explanation is to identify pixels that strongly influence the final decision. A starting point for this strategy is the gradient of the class score function with respect to the input image. This gradient can be interpreted as a sensitivity map, and there are several techniques that elaborate on this basic idea. This paper makes two contributions: it introduces SmoothGrad, a simple method that can help visually sharpen gradient-based sensitivity maps, and it discusses lessons in the visualization of these maps. We publish the code for our experiments and a website with our results.

In multi-fidelity optimization, biased approximations of varying costs of the target function are available. This paper studies the problem of optimizing a locally smooth function with a limited budget, where the learner has to make a tradeoff between the cost and the bias of these approximations. We first prove lower bounds for the simple regret under different assumptions on the fidelities, based on a cost-to-bias function. We then present the Kometo algorithm which achieves, with additional logarithmic factors, the same rates without any knowledge of the function smoothness and fidelity assumptions, and improves previously proven guarantees. We finally empirically show that our algorithm outperforms previous multi-fidelity optimization methods without the knowledge of problem-dependent parameters.

We study the problem of global maximization of a function f given a finite number of evaluations perturbed by noise. We consider a very weak assumption on the function, namely that it is locally smooth (in some precise sense) with respect to some semi-metric, around one of its global maxima. Compared to previous works on bandits in general spaces (Kleinberg et al., 2008; Bubeck et al., 2011a) our algorithm does not require the knowledge of this semi-metric. Our algorithm, StoSOO, follows an optimistic strategy to iteratively construct upper confidence bounds over the hierarchical partitions of the function domain to decide which point to sample next. A finite-time analysis of StoSOO shows that it performs almost as well as the best specifically-tuned algorithms even though the local smoothness of the function is not known.

Stochastic optimizers are central to deep learning, yet widely used methods such as Adam and Adan can degrade in non-stationary or noisy environments, partly due to their reliance on momentum-based magnitude estimates. We introduce Ano, a novel optimizer that decouples direction and magnitude: momentum is used for directional smoothing, while instantaneous gradient magnitudes determine step size. This design improves robustness to gradient noise while retaining the simplicity and efficiency of first-order methods. We further propose Anolog, which removes sensitivity to the momentum coefficient by expanding its window over time via a logarithmic schedule. We establish non-convex convergence guarantees with a convergence rate similar to other sign-based methods, and empirically show that Ano provides substantial gains in noisy and non-stationary regimes such as reinforcement learning, while remaining competitive on low-noise tasks such as standard computer vision benchmarks.

This paper investigates the global convergence of stepsized Newton methods for convex functions. We propose several simple stepsize schedules with fast global convergence guarantees, up to O (k^{-3}), nearly matching lower complexity bounds Omega (k^{-3.5}) of second-order methods. For cases with multiple plausible smoothness parameterizations or an unknown smoothness constant, we introduce a stepsize backtracking procedure that ensures convergence as if the optimal smoothness parameters were known.

We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate O(k^{-1/4}). As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.

This paper focuses on the problem of constrained stochastic optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient free. Under convexity and smoothness assumption, we show that the proposed algorithm converges to the optimal objective function at a rate Oleft(1/T^{1/3}right), where T denotes the iteration count. In particular, the primal sub-optimality gap is shown to have a dimension dependence of Oleft(d^{1/3}right), which is the best known dimension dependence among all zeroth order optimization algorithms with one directional derivative per iteration. For non-convex functions, we obtain the Frank-Wolfe gap to be Oleft(d^{1/3}T^{-1/4}right). Experiments on black-box optimization setups demonstrate the efficacy of the proposed algorithm.

Recent studies have demonstrated the effectiveness of directly aligning diffusion models with human preferences using differentiable reward. However, they exhibit two primary challenges: (1) they rely on multistep denoising with gradient computation for reward scoring, which is computationally expensive, thus restricting optimization to only a few diffusion steps; (2) they often need continuous offline adaptation of reward models in order to achieve desired aesthetic quality, such as photorealism or precise lighting effects. To address the limitation of multistep denoising, we propose Direct-Align, a method that predefines a noise prior to effectively recover original images from any time steps via interpolation, leveraging the equation that diffusion states are interpolations between noise and target images, which effectively avoids over-optimization in late timesteps. Furthermore, we introduce Semantic Relative Preference Optimization (SRPO), in which rewards are formulated as text-conditioned signals. This approach enables online adjustment of rewards in response to positive and negative prompt augmentation, thereby reducing the reliance on offline reward fine-tuning. By fine-tuning the FLUX.1.dev model with optimized denoising and online reward adjustment, we improve its human-evaluated realism and aesthetic quality by over 3x.

Synthesizing optimal controllers for dynamical systems often involves solving optimization problems with hard real-time constraints. These constraints determine the class of numerical methods that can be applied: computationally expensive but accurate numerical routines are replaced by fast and inaccurate methods, trading inference time for solution accuracy. This paper provides techniques to improve the quality of optimized control policies given a fixed computational budget. We achieve the above via a hypersolvers approach, which hybridizes a differential equation solver and a neural network. The performance is evaluated in direct and receding-horizon optimal control tasks in both low and high dimensions, where the proposed approach shows consistent Pareto improvements in solution accuracy and control performance.

Large Language Model (LLM) based multi-agent systems (MAS) show remarkable potential in collaborative problem-solving, yet they still face critical challenges: low communication efficiency, poor scalability, and a lack of effective parameter-updating optimization methods. We present Optima, a novel framework that addresses these issues by significantly enhancing both communication efficiency and task effectiveness in LLM-based MAS through LLM training. Optima employs an iterative generate, rank, select, and train paradigm with a reward function balancing task performance, token efficiency, and communication readability. We explore various RL algorithms, including Supervised Fine-Tuning, Direct Preference Optimization, and their hybrid approaches, providing insights into their effectiveness-efficiency trade-offs. We integrate Monte Carlo Tree Search-inspired techniques for DPO data generation, treating conversation turns as tree nodes to explore diverse interaction paths. Evaluated on common multi-agent tasks, including information-asymmetric question answering and complex reasoning, Optima shows consistent and substantial improvements over single-agent baselines and vanilla MAS based on Llama 3 8B, achieving up to 2.8x performance gain with less than 10\% tokens on tasks requiring heavy information exchange. Moreover, Optima's efficiency gains open new possibilities for leveraging inference-compute more effectively, leading to improved inference-time scaling laws. By addressing fundamental challenges in LLM-based MAS, Optima shows the potential towards scalable, efficient, and effective MAS (https://chenweize1998.github.io/optima-project-page).

This paper introduces a new learning paradigm termed Neural Metamorphosis (NeuMeta), which aims to build self-morphable neural networks. Contrary to crafting separate models for different architectures or sizes, NeuMeta directly learns the continuous weight manifold of neural networks. Once trained, we can sample weights for any-sized network directly from the manifold, even for previously unseen configurations, without retraining. To achieve this ambitious goal, NeuMeta trains neural implicit functions as hypernetworks. They accept coordinates within the model space as input, and generate corresponding weight values on the manifold. In other words, the implicit function is learned in a way, that the predicted weights is well-performed across various models sizes. In training those models, we notice that, the final performance closely relates on smoothness of the learned manifold. In pursuit of enhancing this smoothness, we employ two strategies. First, we permute weight matrices to achieve intra-model smoothness, by solving the Shortest Hamiltonian Path problem. Besides, we add a noise on the input coordinates when training the implicit function, ensuring models with various sizes shows consistent outputs. As such, NeuMeta shows promising results in synthesizing parameters for various network configurations. Our extensive tests in image classification, semantic segmentation, and image generation reveal that NeuMeta sustains full-size performance even at a 75% compression rate.

First-order optimization (FOO) algorithms are pivotal in numerous computational domains such as machine learning and signal denoising. However, their application to complex tasks like neural network training often entails significant inefficiencies due to the need for many sequential iterations for convergence. In response, we introduce first-order optimization expedited with approximately parallelized iterations (OptEx), the first framework that enhances the efficiency of FOO by leveraging parallel computing to mitigate its iterative bottleneck. OptEx employs kernelized gradient estimation to make use of gradient history for future gradient prediction, enabling parallelization of iterations -- a strategy once considered impractical because of the inherent iterative dependency in FOO. We provide theoretical guarantees for the reliability of our kernelized gradient estimation and the iteration complexity of SGD-based OptEx, confirming that estimation errors diminish to zero as historical gradients accumulate and that SGD-based OptEx enjoys an effective acceleration rate of Omega(N) over standard SGD given parallelism of N. We also use extensive empirical studies, including synthetic functions, reinforcement learning tasks, and neural network training across various datasets, to underscore the substantial efficiency improvements achieved by OptEx.

We present AI-SARAH, a practical variant of SARAH. As a variant of SARAH, this algorithm employs the stochastic recursive gradient yet adjusts step-size based on local geometry. AI-SARAH implicitly computes step-size and efficiently estimates local Lipschitz smoothness of stochastic functions. It is fully adaptive, tune-free, straightforward to implement, and computationally efficient. We provide technical insight and intuitive illustrations on its design and convergence. We conduct extensive empirical analysis and demonstrate its strong performance compared with its classical counterparts and other state-of-the-art first-order methods in solving convex machine learning problems.

We introduce two complementary techniques for efficient adaptive optimization that reduce memory requirements while accelerating training of large-scale neural networks. The first technique, Subset-Norm adaptive step size, generalizes AdaGrad-Norm and AdaGrad(-Coordinate) by reducing the second moment term's memory footprint from O(d) to O(d) through step-size sharing, where d is the model size. For non-convex smooth objectives under coordinate-wise sub-gaussian gradient noise, we prove a noise-adapted high-probability convergence guarantee showing improved dimensional dependence over existing methods. Our second technique, Subspace-Momentum, reduces the momentum state's memory footprint by operating in a low-dimensional subspace while applying standard SGD in the orthogonal complement. We establish high-probability convergence rates under similar relaxed assumptions. Empirical evaluation on LLaMA models from 60M to 1B parameters demonstrates the effectiveness of our methods, where combining subset-norm with subspace-momentum achieves Adam's validation perplexity in approximately half the training tokens (6.8B vs 13.1B) while using only 20% of the Adam's optimizer-states memory footprint and requiring minimal additional hyperparameter tuning.

We present a class of novel optimisers for training neural networks that makes use of the Riemannian metric naturally induced when the loss landscape is embedded in higher-dimensional space. This is the same metric that underlies common visualisations of loss landscapes. By taking this geometric perspective literally and using the induced metric, we develop a new optimiser and compare it to existing methods, namely: SGD, Adam, AdamW, and Muon, across a range of tasks and architectures. Empirically, we conclude that this new class of optimisers is highly effective in low dimensional examples, and provides slight improvement over state-of-the-art methods for training neural networks. These new optimisers have theoretically desirable properties. In particular, the effective learning rate is automatically decreased in regions of high curvature acting as a smoothed out form of gradient clipping. Similarly, one variant of these optimisers can also be viewed as inducing an effective scheduled learning rate and decoupled weight decay is the natural choice from our geometric perspective. The basic method can be used to modify any existing preconditioning method. The new optimiser has a computational complexity comparable to that of Adam.

We introduce Riemannian Lyapunov Optimizers (RLOs), a family of optimization algorithms that unifies classic optimizers within one geometric framework. Unlike heuristic improvements to existing optimizers, RLOs are systematically derived from a novel control-theoretic framework that reinterprets optimization as an extended state discrete-time controlled dynamical system on a Riemannian parameter manifold. Central to this framework is the identification of a Normally Attracting Invariant Manifold (NAIM), which organizes training dynamics into two distinct stages: rapid alignment of the speed state to a target graph, followed by controlled evolution within it. We formalize this by constructing a strict Lyapunov function that certifies convergence to a target manifold. This perspective yields a constructive ``optimizer generator" that not only recovers classic algorithms but enables the principled design of RLOs. We validate our theory via geometric diagnostics and demonstrate that grounding optimizer design in control theory yields state-of-the-art performance in large-scale benchmarks. Overall, RLOs bridge control theory and modern machine learning optimization, providing a unified language and a systematic toolkit for designing stable, effective optimizers.

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the full-batch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones~--~by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

Significant recent work has studied the ability of gradient descent to recover a hidden planted direction θ^star in S^{d-1} in different high-dimensional settings, including tensor PCA and single-index models. The key quantity that governs the ability of gradient descent to traverse these landscapes is the information exponent k^star (Ben Arous et al., (2021)), which corresponds to the order of the saddle at initialization in the population landscape. Ben Arous et al., (2021) showed that n gtrsim d^{max(1, k^star-1)} samples were necessary and sufficient for online SGD to recover θ^star, and Ben Arous et al., (2020) proved a similar lower bound for Langevin dynamics. More recently, Damian et al., (2023) showed it was possible to circumvent these lower bounds by running gradient descent on a smoothed landscape, and that this algorithm succeeds with n gtrsim d^{max(1, k^star/2)} samples, which is optimal in the worst case. This raises the question of whether it is possible to achieve the same rate without explicit smoothing. In this paper, we show that Langevin dynamics can succeed with n gtrsim d^{ k^star/2 } samples if one considers the average iterate, rather than the last iterate. The key idea is that the combination of noise-injection and iterate averaging is able to emulate the effect of landscape smoothing. We apply this result to both the tensor PCA and single-index model settings. Finally, we conjecture that minibatch SGD can also achieve the same rate without adding any additional noise.

Most 3D object generators focus on aesthetic quality, often neglecting physical constraints necessary in applications. One such constraint is that the 3D object should be self-supporting, i.e., remains balanced under gravity. Prior approaches to generating stable 3D objects used differentiable physics simulators to optimize geometry at test-time, which is slow, unstable, and prone to local optima. Inspired by the literature on aligning generative models to external feedback, we propose Direct Simulation Optimization (DSO), a framework to use the feedback from a (non-differentiable) simulator to increase the likelihood that the 3D generator outputs stable 3D objects directly. We construct a dataset of 3D objects labeled with a stability score obtained from the physics simulator. We can then fine-tune the 3D generator using the stability score as the alignment metric, via direct preference optimization (DPO) or direct reward optimization (DRO), a novel objective, which we introduce, to align diffusion models without requiring pairwise preferences. Our experiments show that the fine-tuned feed-forward generator, using either DPO or DRO objective, is much faster and more likely to produce stable objects than test-time optimization. Notably, the DSO framework works even without any ground-truth 3D objects for training, allowing the 3D generator to self-improve by automatically collecting simulation feedback on its own outputs.

We introduce Equilibrium Matching (EqM), a generative modeling framework built from an equilibrium dynamics perspective. EqM discards the non-equilibrium, time-conditional dynamics in traditional diffusion and flow-based generative models and instead learns the equilibrium gradient of an implicit energy landscape. Through this approach, we can adopt an optimization-based sampling process at inference time, where samples are obtained by gradient descent on the learned landscape with adjustable step sizes, adaptive optimizers, and adaptive compute. EqM surpasses the generation performance of diffusion/flow models empirically, achieving an FID of 1.90 on ImageNet 256times256. EqM is also theoretically justified to learn and sample from the data manifold. Beyond generation, EqM is a flexible framework that naturally handles tasks including partially noised image denoising, OOD detection, and image composition. By replacing time-conditional velocities with a unified equilibrium landscape, EqM offers a tighter bridge between flow and energy-based models and a simple route to optimization-driven inference.

We first propose a decentralized proximal stochastic gradient tracking method (DProxSGT) for nonconvex stochastic composite problems, with data heterogeneously distributed on multiple workers in a decentralized connected network. To save communication cost, we then extend DProxSGT to a compressed method by compressing the communicated information. Both methods need only O(1) samples per worker for each proximal update, which is important to achieve good generalization performance on training deep neural networks. With a smoothness condition on the expected loss function (but not on each sample function), the proposed methods can achieve an optimal sample complexity result to produce a near-stationary point. Numerical experiments on training neural networks demonstrate the significantly better generalization performance of our methods over large-batch training methods and momentum variance-reduction methods and also, the ability of handling heterogeneous data by the gradient tracking scheme.

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

Bilevel optimization is an important formulation for many machine learning problems. Current bilevel optimization algorithms assume that the gradient of the upper-level function is Lipschitz. However, recent studies reveal that certain neural networks such as recurrent neural networks (RNNs) and long-short-term memory networks (LSTMs) exhibit potential unbounded smoothness, rendering conventional bilevel optimization algorithms unsuitable. In this paper, we design a new bilevel optimization algorithm, namely BO-REP, to address this challenge. This algorithm updates the upper-level variable using normalized momentum and incorporates two novel techniques for updating the lower-level variable: initialization refinement and periodic updates. Specifically, once the upper-level variable is initialized, a subroutine is invoked to obtain a refined estimate of the corresponding optimal lower-level variable, and the lower-level variable is updated only after every specific period instead of each iteration. When the upper-level problem is nonconvex and unbounded smooth, and the lower-level problem is strongly convex, we prove that our algorithm requires mathcal{O}(1/epsilon^4) iterations to find an epsilon-stationary point in the stochastic setting, where each iteration involves calling a stochastic gradient or Hessian-vector product oracle. Notably, this result matches the state-of-the-art complexity results under the bounded smoothness setting and without mean-squared smoothness of the stochastic gradient, up to logarithmic factors. Our proof relies on novel technical lemmas for the periodically updated lower-level variable, which are of independent interest. Our experiments on hyper-representation learning, hyperparameter optimization, and data hyper-cleaning for text classification tasks demonstrate the effectiveness of our proposed algorithm.

Recently, flat minima are proven to be effective for improving generalization and sharpness-aware minimization (SAM) achieves state-of-the-art performance. Yet the current definition of flatness discussed in SAM and its follow-ups are limited to the zeroth-order flatness (i.e., the worst-case loss within a perturbation radius). We show that the zeroth-order flatness can be insufficient to discriminate minima with low generalization error from those with high generalization error both when there is a single minimum or multiple minima within the given perturbation radius. Thus we present first-order flatness, a stronger measure of flatness focusing on the maximal gradient norm within a perturbation radius which bounds both the maximal eigenvalue of Hessian at local minima and the regularization function of SAM. We also present a novel training procedure named Gradient norm Aware Minimization (GAM) to seek minima with uniformly small curvature across all directions. Experimental results show that GAM improves the generalization of models trained with current optimizers such as SGD and AdamW on various datasets and networks. Furthermore, we show that GAM can help SAM find flatter minima and achieve better generalization.

Machine unlearning has gained increasing attention in recent years, as a promising technique to selectively remove unwanted privacy or copyrighted information from Large Language Models that are trained on a massive scale of human data. However, the emergence of Large Reasoning Models (LRMs), which emphasize long chain-of-thought (CoT) reasoning to address complex questions, presents a dilemma to unlearning: existing methods either struggle to completely eliminate undesired knowledge from the CoT traces or degrade the reasoning performances due to the interference with the reasoning process. To this end, we introduce Counterfactual Unlearning through iterative Preference Optimization (CiPO), a novel framework that redefines unlearning as the targeted intervention of the CoT reasoning in LRMs. More specifically, given a desired unlearning target answer, CiPO instructs LRMs to generate a logically valid counterfactual reasoning trace for preference tuning. As the LRM adjusts to the counterfactual trace, CiPO iteratively updates the preference learning data to increase the discrepancy from the original model. This iterative loop ensures both desirable unlearning and smooth optimization, effectively mitigating the dilemma. Experiments on challenging benchmarks demonstrate that CiPO excels at unlearning, completely removing knowledge from both the intermediate CoT steps and the final answer, while preserving the reasoning abilities of LRMs.

Humans manipulate various kinds of fluids in their everyday life: creating latte art, scooping floating objects from water, rolling an ice cream cone, etc. Using robots to augment or replace human labors in these daily settings remain as a challenging task due to the multifaceted complexities of fluids. Previous research in robotic fluid manipulation mostly consider fluids governed by an ideal, Newtonian model in simple task settings (e.g., pouring). However, the vast majority of real-world fluid systems manifest their complexities in terms of the fluid's complex material behaviors and multi-component interactions, both of which were well beyond the scope of the current literature. To evaluate robot learning algorithms on understanding and interacting with such complex fluid systems, a comprehensive virtual platform with versatile simulation capabilities and well-established tasks is needed. In this work, we introduce FluidLab, a simulation environment with a diverse set of manipulation tasks involving complex fluid dynamics. These tasks address interactions between solid and fluid as well as among multiple fluids. At the heart of our platform is a fully differentiable physics simulator, FluidEngine, providing GPU-accelerated simulations and gradient calculations for various material types and their couplings. We identify several challenges for fluid manipulation learning by evaluating a set of reinforcement learning and trajectory optimization methods on our platform. To address these challenges, we propose several domain-specific optimization schemes coupled with differentiable physics, which are empirically shown to be effective in tackling optimization problems featured by fluid system's non-convex and non-smooth properties. Furthermore, we demonstrate reasonable sim-to-real transfer by deploying optimized trajectories in real-world settings.

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. Y is to be predicted upon observing a (possibly high dimensional) random vector X by means of a predictive function f(X) as accurately as possible in the mean-squared sense and study a nearest-neighbour-based pointwise estimate of the gradient of the optimal predictive function, the regression function m(x)=E[Ymid X=x]. Under classic smoothness conditions combined with the assumption that the tails of Y-m(X) are sub-Gaussian, we prove nonasymptotic bounds improving upon those obtained for alternative estimation methods. Beyond the novel theoretical results established, several illustrative numerical experiments have been carried out. The latter provide strong empirical evidence that the estimation method proposed works very well for various statistical problems involving gradient estimation, namely dimensionality reduction, stochastic gradient descent optimization and quantifying disentanglement.

Optimization plays a vital role in scientific research and practical applications. However, formulating a concrete optimization problem described in natural language into a mathematical form and selecting a suitable solver to solve the problem requires substantial domain expertise. We introduce OptimAI, a framework for solving Optimization problems described in natural language by leveraging LLM-powered AI agents, and achieve superior performance over current state-of-the-art methods. Our framework is built upon the following key roles: (1) a formulator that translates natural language problem descriptions into precise mathematical formulations; (2) a planner that constructs a high-level solution strategy prior to execution; and (3) a coder and a code critic capable of interacting with the environment and reflecting on outcomes to refine future actions. Ablation studies confirm that all roles are essential; removing the planner or code critic results in 5.8times and 3.1times drops in productivity, respectively. Furthermore, we introduce UCB-based debug scheduling to dynamically switch between alternative plans, yielding an additional 3.3times productivity gain. Our design emphasizes multi-agent collaboration, and our experiments confirm that combining diverse models leads to performance gains. Our approach attains 88.1% accuracy on the NLP4LP dataset and 82.3% on the Optibench dataset, reducing error rates by 58% and 52%, respectively, over prior best results.

Non-convex optimization plays a key role in a growing number of machine learning applications. This motivates the identification of specialized structure that enables sharper theoretical analysis. One such identified structure is quasar-convexity, a non-convex generalization of convexity that subsumes convex functions. Existing algorithms for minimizing quasar-convex functions in the stochastic setting have either high complexity or slow convergence, which prompts us to derive a new class of stochastic methods for optimizing smooth quasar-convex functions. We demonstrate that our algorithms have fast convergence and outperform existing algorithms on several examples, including the classical problem of learning linear dynamical systems. We also present a unified analysis of our newly proposed algorithms and a previously studied deterministic algorithm.

Direct preference optimization (DPO) has shown success in aligning diffusion models with human preference. Previous approaches typically assume a consistent preference label between final generations and noisy samples at intermediate steps, and directly apply DPO to these noisy samples for fine-tuning. However, we theoretically identify inherent issues in this assumption and its impacts on the effectiveness of preference alignment. We first demonstrate the inherent issues from two perspectives: gradient direction and preference order, and then propose a Tailored Preference Optimization (TailorPO) framework for aligning diffusion models with human preference, underpinned by some theoretical insights. Our approach directly ranks intermediate noisy samples based on their step-wise reward, and effectively resolves the gradient direction issues through a simple yet efficient design. Additionally, we incorporate the gradient guidance of diffusion models into preference alignment to further enhance the optimization effectiveness. Experimental results demonstrate that our method significantly improves the model's ability to generate aesthetically pleasing and human-preferred images.

Training large neural networks (NNs) requires optimizing high-dimensional data-dependent loss functions. The optimization landscape of these functions is often highly complex and textured, even fractal-like, with many spurious local minima, ill-conditioned valleys, degenerate points, and saddle points. Complicating things further is the fact that these landscape characteristics are a function of the data, meaning that noise in the training data can propagate forward and give rise to unrepresentative small-scale geometry. This poses a difficulty for gradient-based optimization methods, which rely on local geometry to compute updates and are, therefore, vulnerable to being derailed by noisy data. In practice,this translates to a strong dependence of the optimization dynamics on the noise in the data, i.e., poor generalization performance. To remediate this problem, we propose a new optimization procedure: Rolling Ball Optimizer (RBO), that breaks this spatial locality by incorporating information from a larger region of the loss landscape in its updates. We achieve this by simulating the motion of a rigid sphere of finite radius rolling on the loss landscape, a straightforward generalization of Gradient Descent (GD) that simplifies into it in the infinitesimal limit. The radius serves as a hyperparameter that determines the scale at which RBO sees the loss landscape, allowing control over the granularity of its interaction therewith. We are motivated by the intuition that the large-scale geometry of the loss landscape is less data-specific than its fine-grained structure, and that it is easier to optimize. We support this intuition by proving that our algorithm has a smoothing effect on the loss function. Evaluation against SGD, SAM, and Entropy-SGD, on MNIST and CIFAR-10/100 demonstrates promising results in terms of convergence speed, training accuracy, and generalization performance.

We consider the problem of designing a smooth trajectory that traverses a sequence of convex sets in minimum time, while satisfying given velocity and acceleration constraints. This problem is naturally formulated as a nonconvex program. To solve it, we propose a biconvex method that quickly produces an initial trajectory and iteratively refines it by solving two convex subproblems in alternation. This method is guaranteed to converge, returns a feasible trajectory even if stopped early, and does not require the selection of any line-search or trust-region parameter. Exhaustive experiments show that our method finds high-quality trajectories in a fraction of the time of state-of-the-art solvers for nonconvex optimization. In addition, it achieves runtimes comparable to industry-standard waypoint-based motion planners, while consistently designing lower-duration trajectories than existing optimization-based planners.

Nesterov's accelerated gradient (AG) is a popular technique to optimize objective functions comprising two components: a convex loss and a penalty function. While AG methods perform well for convex penalties, such as the LASSO, convergence issues may arise when it is applied to nonconvex penalties, such as SCAD. A recent proposal generalizes Nesterov's AG method to the nonconvex setting. The proposed algorithm requires specification of several hyperparameters for its practical application. Aside from some general conditions, there is no explicit rule for selecting the hyperparameters, and how different selection can affect convergence of the algorithm. In this article, we propose a hyperparameter setting based on the complexity upper bound to accelerate convergence, and consider the application of this nonconvex AG algorithm to high-dimensional linear and logistic sparse learning problems. We further establish the rate of convergence and present a simple and useful bound to characterize our proposed optimal damping sequence. Simulation studies show that convergence can be made, on average, considerably faster than that of the conventional proximal gradient algorithm. Our experiments also show that the proposed method generally outperforms the current state-of-the-art methods in terms of signal recovery.

Recent advancements in human preference optimization, initially developed for Language Models (LMs), have shown promise for text-to-image Diffusion Models, enhancing prompt alignment, visual appeal, and user preference. Unlike LMs, Diffusion Models typically optimize in pixel or VAE space, which does not align well with human perception, leading to slower and less efficient training during the preference alignment stage. We propose using a perceptual objective in the U-Net embedding space of the diffusion model to address these issues. Our approach involves fine-tuning Stable Diffusion 1.5 and XL using Direct Preference Optimization (DPO), Contrastive Preference Optimization (CPO), and supervised fine-tuning (SFT) within this embedding space. This method significantly outperforms standard latent-space implementations across various metrics, including quality and computational cost. For SDXL, our approach provides 60.8\% general preference, 62.2\% visual appeal, and 52.1\% prompt following against original open-sourced SDXL-DPO on the PartiPrompts dataset, while significantly reducing compute. Our approach not only improves the efficiency and quality of human preference alignment for diffusion models but is also easily integrable with other optimization techniques. The training code and LoRA weights will be available here: https://huggingface.co/alexgambashidze/SDXL\_NCP-DPO\_v0.1

Dynamical generative models that produce samples through an iterative process, such as Flow Matching and denoising diffusion models, have seen widespread use, but there have not been many theoretically-sound methods for improving these models with reward fine-tuning. In this work, we cast reward fine-tuning as stochastic optimal control (SOC). Critically, we prove that a very specific memoryless noise schedule must be enforced during fine-tuning, in order to account for the dependency between the noise variable and the generated samples. We also propose a new algorithm named Adjoint Matching which outperforms existing SOC algorithms, by casting SOC problems as a regression problem. We find that our approach significantly improves over existing methods for reward fine-tuning, achieving better consistency, realism, and generalization to unseen human preference reward models, while retaining sample diversity.

Fine-tuning foundation models via reinforcement learning (RL) has proven promising for aligning to downstream objectives. In the case of diffusion models (DMs), though RL training improves alignment from early timesteps, critical issues such as training instability and mode collapse arise. We address these drawbacks by exploiting the hierarchical nature of DMs: we train them dynamically at each epoch with a tailored RL method, allowing for continual evaluation and step-by-step refinement of the model performance (or alignment). Furthermore, we find that not every denoising step needs to be fine-tuned to align DMs to downstream tasks. Consequently, in addition to clipping, we regularise model parameters at distinct learning phases via a sliding-window approach. Our approach, termed Hierarchical Reward Fine-tuning (HRF), is validated on the Denoising Diffusion Policy Optimisation method, where we show that models trained with HRF achieve better preservation of diversity in downstream tasks, thus enhancing the fine-tuning robustness and at uncompromising mean rewards.

Minimax problems are notoriously challenging to optimize. However, we demonstrate that the two-timescale extragradient can be a viable solution. By utilizing dynamical systems theory, we show that it converges to points that satisfy the second-order necessary condition of local minimax points, under a mild condition. This work surpasses all previous results as we eliminate a crucial assumption that the Hessian, with respect to the maximization variable, is nondegenerate.

Trustworthy interpretation of deep learning models is critical for neuroimaging applications, yet commonly used Explainable AI (XAI) methods lack rigorous validation, risking misinterpretation. We performed the first large-scale, systematic comparison of XAI methods on ~45,000 structural brain MRIs using a novel XAI validation framework. This framework establishes verifiable ground truth by constructing prediction tasks with known signal sources - from localized anatomical features to subject-specific clinical lesions - without artificially altering input images. Our analysis reveals systematic failures in two of the most widely used methods: GradCAM consistently failed to localize predictive features, while Layer-wise Relevance Propagation generated extensive, artifactual explanations that suggest incompatibility with neuroimaging data characteristics. Our results indicate that these failures stem from a domain mismatch, where methods with design principles tailored to natural images require substantial adaptation for neuroimaging data. In contrast, the simpler, gradient-based method SmoothGrad, which makes fewer assumptions about data structure, proved consistently accurate, suggesting its conceptual simplicity makes it more robust to this domain shift. These findings highlight the need for domain-specific adaptation and validation of XAI methods, suggest that interpretations from prior neuroimaging studies using standard XAI methodology warrant re-evaluation, and provide urgent guidance for practical application of XAI in neuroimaging.

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. We propose an end-to-end optimization framework, Trace, which treats the computational workflow of an AI system as a graph akin to neural networks, based on a generalization of back-propagation. Optimization of computational workflows often involves rich feedback (e.g. console output or user's responses), heterogeneous parameters (e.g. prompts, hyper-parameters, codes), and intricate objectives (beyond maximizing a score). Moreover, its computation graph can change dynamically with the inputs and parameters. We frame a new mathematical setup of iterative optimization, Optimization with Trace Oracle (OPTO), to capture and abstract these properties so as to design optimizers that work across many domains. In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. Trace is the tool to implement OPTO in practice. Trace has a Python interface that efficiently converts a computational workflow into an OPTO instance using a PyTorch-like interface. Using Trace, we develop a general-purpose LLM-based optimizer called OptoPrime that can effectively solve OPTO problems. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We believe that Trace, OptoPrime and the OPTO framework will enable the next generation of interactive agents that automatically adapt using various kinds of feedback. Website: https://microsoft.github.io/Trace

We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an epsilon-stationary solution of the bilevel problem after epsilon^{-7/2}, epsilon^{-5/2}, and epsilon^{-3/2} iterations (each iteration using O(1) samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to epsilon^{-5/2}, epsilon^{-4/2}, and epsilon^{-3/2}, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

Policy-gradient methods usually optimize expected return, but many real world applications care about distributional properties of returns: tail risk, outlier robustness, or best-of-K discovery. We introduce OrderGrad, a family of likelihood-ratio and reparameterization gradient estimators for order-statistic objectives. OrderGrad optimizes finite-sample L-statistics, i.e., weighted averages of sorted rewards or costs, recovering objectives such as VaR, CVaR, trimmed means, medians, and top-m/best-of-K criteria by changing only the rank weights. For any fixed sample size and rank-weight vector, OrderGrad provides an unbiased gradient estimator for the corresponding order-statistic objective. The method is implemented as a simple reward transformation that can then be used in an otherwise standard policy-gradient or reparameterized update. We study the resulting estimator's variance behavior and evaluate it on tasks where mean optimization is mismatched to the deployment objective, including LLM math post-training and other tasks. OrderGrad provides a unified, plug-and-play route to risk-averse, robust, and exploratory learning. Code: https://github.com/paavo5/ordergrad

Reinforcement learning fine-tuning has become the dominant approach for aligning diffusion models with human preferences. However, assessing images is intrinsically a multi-dimensional task, and multiple evaluation criteria need to be optimized simultaneously. Existing practice deal with multiple rewards by training one specialist model per reward, optimizing a weighted-sum reward R(x)=sum_k w_k R_k(x), or sequentially fine-tuning with a hand-crafted stage schedule. These approaches either fail to produce a unified model that can be jointly trained on all rewards or necessitates heavy manually tuned sequential training. We find that the failure stems from using a naive weighted-sum reward aggregation. This approach suffers from a sample-level mismatch because most rollouts are specialist samples, highly informative for certain reward dimensions but irrelevant for others; consequently, weighted summation dilutes their supervision. To address this issue, we propose MARBLE (Multi-Aspect Reward BaLancE), a gradient-space optimization framework that maintains independent advantage estimators for each reward, computes per-reward policy gradients, and harmonizes them into a single update direction without manually-tuned reward weighting, by solving a Quadratic Programming problem. We further propose an amortized formulation that exploits the affine structure of the loss used in DiffusionNFT, to reduce the per-step cost from K+1 backward passes to near single-reward baseline cost, together with EMA smoothing on the balancing coefficients to stabilize updates against transient single-batch fluctuations. On SD3.5 Medium with five rewards, MARBLE improves all five reward dimensions simultaneously, turns the worst-aligned reward's gradient cosine from negative under weighted summation in 80% of mini-batches to consistently positive, and runs at 0.97X the training speed of baseline training.

Vision Transformers (ViTs) and MLPs signal further efforts on replacing hand-wired features or inductive biases with general-purpose neural architectures. Existing works empower the models by massive data, such as large-scale pre-training and/or repeated strong data augmentations, and still report optimization-related problems (e.g., sensitivity to initialization and learning rates). Hence, this paper investigates ViTs and MLP-Mixers from the lens of loss geometry, intending to improve the models' data efficiency at training and generalization at inference. Visualization and Hessian reveal extremely sharp local minima of converged models. By promoting smoothness with a recently proposed sharpness-aware optimizer, we substantially improve the accuracy and robustness of ViTs and MLP-Mixers on various tasks spanning supervised, adversarial, contrastive, and transfer learning (e.g., +5.3\% and +11.0\% top-1 accuracy on ImageNet for ViT-B/16 and Mixer-B/16, respectively, with the simple Inception-style preprocessing). We show that the improved smoothness attributes to sparser active neurons in the first few layers. The resultant ViTs outperform ResNets of similar size and throughput when trained from scratch on ImageNet without large-scale pre-training or strong data augmentations. Model checkpoints are available at https://github.com/google-research/vision_transformer.

While preference optimization is crucial for improving visual generative models, how to effectively scale this paradigm remains largely unexplored. Current open-source preference datasets contain conflicting preference patterns, where winners excel in some dimensions but underperform in others. Naively optimizing on such noisy datasets fails to learn preferences, hindering effective scaling. To enhance robustness against noise, we propose Poly-DPO, which extends the DPO objective with an additional polynomial term that dynamically adjusts model confidence based on dataset characteristics, enabling effective learning across diverse data distributions. Beyond biased patterns, existing datasets suffer from low resolution, limited prompt diversity, and imbalanced distributions. To facilitate large-scale visual preference optimization by tackling data bottlenecks, we construct ViPO, a massive-scale preference dataset with 1M image pairs at 1024px across five categories and 300K video pairs at 720p+ across three categories. State-of-the-art generative models and diverse prompts ensure reliable preference signals with balanced distributions. Remarkably, when applying Poly-DPO to our high-quality dataset, the optimal configuration converges to standard DPO. This convergence validates dataset quality and Poly-DPO's adaptive nature: sophisticated optimization becomes unnecessary with sufficient data quality, yet remains valuable for imperfect datasets. We validate our approach across visual generation models. On noisy datasets like Pick-a-Pic V2, Poly-DPO achieves 6.87 and 2.32 gains over Diffusion-DPO on GenEval for SD1.5 and SDXL, respectively. For ViPO, models achieve performance far exceeding those trained on existing open-source preference datasets. These results confirm that addressing both algorithmic adaptability and data quality is essential for scaling visual preference optimization.

Robustness remains a paramount concern in deep reinforcement learning (DRL), with randomized smoothing emerging as a key technique for enhancing this attribute. However, a notable gap exists in the performance of current smoothed DRL agents, often characterized by significantly low clean rewards and weak robustness. In response to this challenge, our study introduces innovative algorithms aimed at training effective smoothed robust DRL agents. We propose S-DQN and S-PPO, novel approaches that demonstrate remarkable improvements in clean rewards, empirical robustness, and robustness guarantee across standard RL benchmarks. Notably, our S-DQN and S-PPO agents not only significantly outperform existing smoothed agents by an average factor of 2.16times under the strongest attack, but also surpass previous robustly-trained agents by an average factor of 2.13times. This represents a significant leap forward in the field. Furthermore, we introduce Smoothed Attack, which is 1.89times more effective in decreasing the rewards of smoothed agents than existing adversarial attacks.

AdamW has been the default optimizer for transformer pretraining. For many years, our community searches for faster and more stable optimizers with only constraint positive outcomes. In this work, we propose a single-line modification in Pytorch to any momentum-based optimizer, which we rename Cautious Optimizer, e.g. C-AdamW and C-Lion. Our theoretical result shows that this modification preserves Adam's Hamiltonian function and it does not break the convergence guarantee under the Lyapunov analysis. In addition, a whole new family of optimizers is revealed by our theoretical insight. Among them, we pick the simplest one for empirical experiments, showing speed-up on Llama and MAE pretraining up to 1.47times. Code is available at https://github.com/kyleliang919/C-Optim

We consider the optimization problem of the form min_{x in R^d} f(x) triangleq E_{xi} [F(x; xi)], where the component F(x;xi) is L-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most O( L^4 d^{3/2} epsilon^{-4} + Delta L^3 d^{3/2} delta^{-1} epsilon^{-4}) stochastic zeroth-order oracle complexity to find a (delta,epsilon)-Goldstein stationary point of objective function, where Delta = f(x_0) - inf_{x in R^d} f(x) and x_0 is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to O(L^3 d^{3/2} epsilon^{-3}+ Delta L^2 d^{3/2} delta^{-1} epsilon^{-3}).

Trajectory planning is commonly used as part of a local planner in autonomous driving. This paper considers the problem of planning a continuous-curvature-rate trajectory between fixed start and goal states that minimizes a tunable trade-off between passenger comfort and travel time. The problem is an instance of infinite dimensional optimization over two continuous functions: a path, and a velocity profile. We propose a simplification of this problem that facilitates the discretization of both functions. This paper also proposes a method to quickly generate minimal-length paths between start and goal states based on a single tuning parameter: the second derivative of curvature. Furthermore, we discretize the set of velocity profiles along a given path into a selection of acceleration way-points along the path. Gradient-descent is then employed to minimize cost over feasible choices of the second derivative of curvature, and acceleration way-points, resulting in a method that repeatedly solves the path and velocity profiles in an iterative fashion. Numerical examples are provided to illustrate the benefits of the proposed methods.

Gaining insight into how deep convolutional neural network models perform image classification and how to explain their outputs have been a concern to computer vision researchers and decision makers. These deep models are often referred to as black box due to low comprehension of their internal workings. As an effort to developing explainable deep learning models, several methods have been proposed such as finding gradients of class output with respect to input image (sensitivity maps), class activation map (CAM), and Gradient based Class Activation Maps (Grad-CAM). These methods under perform when localizing multiple occurrences of the same class and do not work for all CNNs. In addition, Grad-CAM does not capture the entire object in completeness when used on single object images, this affect performance on recognition tasks. With the intention to create an enhanced visual explanation in terms of visual sharpness, object localization and explaining multiple occurrences of objects in a single image, we present Smooth Grad-CAM++ Simple demo: http://35.238.22.135:5000/, a technique that combines methods from two other recent techniques---SMOOTHGRAD and Grad-CAM++. Our Smooth Grad-CAM++ technique provides the capability of either visualizing a layer, subset of feature maps, or subset of neurons within a feature map at each instance at the inference level (model prediction process). After experimenting with few images, Smooth Grad-CAM++ produced more visually sharp maps with better localization of objects in the given input images when compared with other methods.

The optimization algorithm and its hyperparameters can significantly affect the training speed and resulting model accuracy in machine learning applications. The wish list for an ideal optimizer includes fast and smooth convergence to low error, low computational demand, and general applicability. Our recently introduced continual resilient (CoRe) optimizer has shown superior performance compared to other state-of-the-art first-order gradient-based optimizers for training lifelong machine learning potentials. In this work we provide an extensive performance comparison of the CoRe optimizer and nine other optimization algorithms including the Adam optimizer and resilient backpropagation (RPROP) for diverse machine learning tasks. We analyze the influence of different hyperparameters and provide generally applicable values. The CoRe optimizer yields best or competitive performance in every investigated application, while only one hyperparameter needs to be changed depending on mini-batch or batch learning.

Diffusion models (DMs) have achieved state-of-the-art generative performance but suffer from high sampling latency due to their sequential denoising nature. Existing solver-based acceleration methods often face significant image quality degradation under a low-latency budget, primarily due to accumulated truncation errors arising from the inability to capture high-curvature trajectory segments. In this paper, we propose the Ensemble Parallel Direction solver (dubbed as EPD-Solver), a novel ODE solver that mitigates these errors by incorporating multiple parallel gradient evaluations in each step. Motivated by the geometric insight that sampling trajectories are largely confined to a low-dimensional manifold, EPD-Solver leverages the Mean Value Theorem for vector-valued functions to approximate the integral solution more accurately. Importantly, since the additional gradient computations are independent, they can be fully parallelized, preserving low-latency sampling nature. We introduce a two-stage optimization framework. Initially, EPD-Solver optimizes a small set of learnable parameters via a distillation-based approach. We further propose a parameter-efficient Reinforcement Learning (RL) fine-tuning scheme that reformulates the solver as a stochastic Dirichlet policy. Unlike traditional methods that fine-tune the massive backbone, our RL approach operates strictly within the low-dimensional solver space, effectively mitigating reward hacking while enhancing performance in complex text-to-image (T2I) generation tasks. In addition, our method is flexible and can serve as a plugin (EPD-Plugin) to improve existing ODE samplers.

We study Stochastic Gradient Descent with AdaGrad stepsizes: a popular adaptive (self-tuning) method for first-order stochastic optimization. Despite being well studied, existing analyses of this method suffer from various shortcomings: they either assume some knowledge of the problem parameters, impose strong global Lipschitz conditions, or fail to give bounds that hold with high probability. We provide a comprehensive analysis of this basic method without any of these limitations, in both the convex and non-convex (smooth) cases, that additionally supports a general ``affine variance'' noise model and provides sharp rates of convergence in both the low-noise and high-noise~regimes.

Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al. [2022] as an interpolation between stochastic and adversarial online convex optimization. Under the smoothness condition, they demonstrate that the expected regret of optimistic follow-the-regularized-leader (FTRL) depends on the cumulative stochastic variance sigma_{1:T}^2 and the cumulative adversarial variation Sigma_{1:T}^2 for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance sigma_{max}^2 and the maximal adversarial variation Sigma_{max}^2 for strongly convex functions. Inspired by their work, we investigate the theoretical guarantees of optimistic online mirror descent (OMD) for the SEA model. For convex and smooth functions, we obtain the same O(sigma_{1:T^2}+Sigma_{1:T^2}) regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an O(min{log (sigma_{1:T}^2+Sigma_{1:T}^2), (sigma_{max}^2 + Sigma_{max}^2) log T}) bound, better than their O((sigma_{max}^2 + Sigma_{max}^2) log T) bound. For exp-concave and smooth functions, we achieve a new O(dlog(sigma_{1:T}^2+Sigma_{1:T}^2)) bound. Owing to the OMD framework, we can further extend our result to obtain dynamic regret guarantees, which are more favorable in non-stationary online scenarios. The attained results allow us to recover excess risk bounds of the stochastic setting and regret bounds of the adversarial setting, and derive new guarantees for many intermediate scenarios.

We consider the block coordinate descent methods of Gauss-Seidel type with proximal regularization (BCD-PR), which is a classical method of minimizing general nonconvex objectives under constraints that has a wide range of practical applications. We theoretically establish the worst-case complexity bound for this algorithm. Namely, we show that for general nonconvex smooth objectives with block-wise constraints, the classical BCD-PR algorithm converges to an epsilon-stationary point within O(1/epsilon) iterations. Under a mild condition, this result still holds even if the algorithm is executed inexactly in each step. As an application, we propose a provable and efficient algorithm for `Wasserstein CP-dictionary learning', which seeks a set of elementary probability distributions that can well-approximate a given set of d-dimensional joint probability distributions. Our algorithm is a version of BCD-PR that operates in the dual space, where the primal problem is regularized both entropically and proximally.

The Sharpness Aware Minimization (SAM) optimization algorithm has been shown to control large eigenvalues of the loss Hessian and provide generalization benefits in a variety of settings. The original motivation for SAM was a modified loss function which penalized sharp minima; subsequent analyses have also focused on the behavior near minima. However, our work reveals that SAM provides a strong regularization of the eigenvalues throughout the learning trajectory. We show that in a simplified setting, SAM dynamically induces a stabilization related to the edge of stability (EOS) phenomenon observed in large learning rate gradient descent. Our theory predicts the largest eigenvalue as a function of the learning rate and SAM radius parameters. Finally, we show that practical models can also exhibit this EOS stabilization, and that understanding SAM must account for these dynamics far away from any minima.

Autonomous Underwater Vehicles (AUVs) are indispensable for marine exploration; yet, their control is hindered by nonlinear hydrodynamics, time-varying disturbances, and localization uncertainty. Traditional controllers provide only limited adaptability, while Reinforcement Learning (RL), though promising, suffers from sample inefficiency, weak long-term planning, and lacks stability guarantees, leading to unreliable behavior. To address these challenges, we propose a diffusion-prior Lyapunov actor-critic framework that unifies exploration, stability, and semantic adaptability. Specifically, a diffusion model generates smooth, multimodal, and disturbance-resilient candidate actions; a Lyapunov critic further imposes dual constraints that ensure stability; and a Large Language Model (LLM)-driven outer loop adaptively selects and refines Lyapunov functions based on task semantics and training feedback. This "generation-filtering-optimization" mechanism not only enhances sample efficiency and planning capability but also aligns stability guarantees with diverse mission requirements in the multi-objective optimization task. Extensive simulations under complex ocean dynamics demonstrate that the proposed framework achieves more accurate trajectory tracking, higher task completion rates, improved energy efficiency, faster convergence, and improved robustness compared with conventional RL and diffusion-augmented baselines.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

This paper investigates the distinctions between gradient methods applied to non-differentiable functions (NGDMs) and classical gradient descents (GDs) designed for differentiable functions. First, we demonstrate significant differences in the convergence properties of NGDMs compared to GDs, challenging the applicability of the extensive neural network convergence literature based on L-smoothness to non-smooth neural networks. Next, we demonstrate the paradoxical nature of NGDM solutions for L_{1}-regularized problems, showing that increasing the regularization penalty leads to an increase in the L_{1} norm of optimal solutions in NGDMs. Consequently, we show that widely adopted L_{1} penalization-based techniques for network pruning do not yield expected results. Finally, we explore the Edge of Stability phenomenon, indicating its inapplicability even to Lipschitz continuous convex differentiable functions, leaving its relevance to non-convex non-differentiable neural networks inconclusive. Our analysis exposes misguided interpretations of NGDMs in widely referenced papers and texts due to an overreliance on strong smoothness assumptions, emphasizing the necessity for a nuanced understanding of foundational assumptions in the analysis of these systems.

In recent years, large-scale pre-trained diffusion models have demonstrated their outstanding capabilities in image and video generation tasks. However, existing models tend to produce visual objects commonly found in the training dataset, which diverges from user input prompts. The underlying reason behind the inaccurate generated results lies in the model's difficulty in sampling from specific intervals of the initial noise distribution corresponding to the prompt. Moreover, it is challenging to directly optimize the initial distribution, given that the diffusion process involves multiple denoising steps. In this paper, we introduce a Fine-tuning Initial Noise Distribution (FIND) framework with policy optimization, which unleashes the powerful potential of pre-trained diffusion networks by directly optimizing the initial distribution to align the generated contents with user-input prompts. To this end, we first reformulate the diffusion denoising procedure as a one-step Markov decision process and employ policy optimization to directly optimize the initial distribution. In addition, a dynamic reward calibration module is proposed to ensure training stability during optimization. Furthermore, we introduce a ratio clipping algorithm to utilize historical data for network training and prevent the optimized distribution from deviating too far from the original policy to restrain excessive optimization magnitudes. Extensive experiments demonstrate the effectiveness of our method in both text-to-image and text-to-video tasks, surpassing SOTA methods in achieving consistency between prompts and the generated content. Our method achieves 10 times faster than the SOTA approach. Our homepage is available at https://github.com/vpx-ecnu/FIND-website.

Exponential Moving Average (EMA) is a widely used weight averaging (WA) regularization to learn flat optima for better generalizations without extra cost in deep neural network (DNN) optimization. Despite achieving better flatness, existing WA methods might fall into worse final performances or require extra test-time computations. This work unveils the full potential of EMA with a single line of modification, i.e., switching the EMA parameters to the original model after each epoch, dubbed as Switch EMA (SEMA). From both theoretical and empirical aspects, we demonstrate that SEMA can help DNNs to reach generalization optima that better trade-off between flatness and sharpness. To verify the effectiveness of SEMA, we conduct comparison experiments with discriminative, generative, and regression tasks on vision and language datasets, including image classification, self-supervised learning, object detection and segmentation, image generation, video prediction, attribute regression, and language modeling. Comprehensive results with popular optimizers and networks show that SEMA is a free lunch for DNN training by improving performances and boosting convergence speeds.

Tobasco et al. [Physics Letters A, 382:382-386, 2018; see https://doi.org/10.1016/j.physleta.2017.12.023] recently suggested that trajectories of ODE systems that optimize the infinite-time average of a certain observable can be localized using sublevel sets of a function that arise when bounding such averages using so-called auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimization is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localize the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimization. Such points provide good initial conditions for UPO computations. As a proof of concept, we then combine these methods with a single-shooting Newton-Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimizes the mean energy dissipation rate and maximizes the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter sigma > 0. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be O(sigma)-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as O(sigma)-approximation of the original BO, we propose first-order algorithms that find an epsilon-stationary solution by optimizing the penalty formulation with sigma = O(epsilon). When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an epsilon-stationary point of the penalty function, using in total O(epsilon^{-3}) and O(epsilon^{-7}) accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, i.e., with O(1) samples per iteration, and achieves the improved oracle-complexity of O(epsilon^{-3}) and O(epsilon^{-5}), respectively.

In the past several years, the last-iterate convergence of the Stochastic Gradient Descent (SGD) algorithm has triggered people's interest due to its good performance in practice but lack of theoretical understanding. For Lipschitz convex functions, different works have established the optimal O(log(1/delta)log T/T) or O(log(1/delta)/T) high-probability convergence rates for the final iterate, where T is the time horizon and delta is the failure probability. However, to prove these bounds, all the existing works are either limited to compact domains or require almost surely bounded noises. It is natural to ask whether the last iterate of SGD can still guarantee the optimal convergence rate but without these two restrictive assumptions. Besides this important question, there are still lots of theoretical problems lacking an answer. For example, compared with the last-iterate convergence of SGD for non-smooth problems, only few results for smooth optimization have yet been developed. Additionally, the existing results are all limited to a non-composite objective and the standard Euclidean norm. It still remains unclear whether the last-iterate convergence can be provably extended to wider composite optimization and non-Euclidean norms. In this work, to address the issues mentioned above, we revisit the last-iterate convergence of stochastic gradient methods and provide the first unified way to prove the convergence rates both in expectation and in high probability to accommodate general domains, composite objectives, non-Euclidean norms, Lipschitz conditions, smoothness, and (strong) convexity simultaneously. Additionally, we extend our analysis to obtain the last-iterate convergence under heavy-tailed noises.

Optimizing multiple competing black-box objectives is a challenging problem in many fields, including science, engineering, and machine learning. Multi-objective Bayesian optimization (MOBO) is a sample-efficient approach for identifying the optimal trade-offs between the objectives. However, many existing methods perform poorly when the observations are corrupted by noise. We propose a novel acquisition function, NEHVI, that overcomes this important practical limitation by applying a Bayesian treatment to the popular expected hypervolume improvement (EHVI) criterion and integrating over this uncertainty in the Pareto frontier. We argue that, even in the noiseless setting, generating multiple candidates in parallel is an incarnation of EHVI with uncertainty in the Pareto frontier and therefore can be addressed using the same underlying technique. Through this lens, we derive a natural parallel variant, qNEHVI, that reduces computational complexity of parallel EHVI from exponential to polynomial with respect to the batch size. qNEHVI is one-step Bayes-optimal for hypervolume maximization in both noisy and noiseless environments, and we show that it can be optimized effectively with gradient-based methods via sample average approximation. Empirically, we demonstrate not only that qNEHVI is substantially more robust to observation noise than existing MOBO approaches, but also that it achieves state-of-the-art optimization performance and competitive wall-times in large-batch environments.

Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such constraints can be softly imposed via loss function penalties, recent advancements in differentiable physics and optimization improve performance by incorporating PDE-constrained optimization as individual layers in neural networks. This enables a stricter adherence to physical constraints. However, imposing hard constraints significantly increases computational and memory costs, especially for complex dynamical systems. This is because it requires solving an optimization problem over a large number of points in a mesh, representing spatial and temporal discretizations, which greatly increases the complexity of the constraint. To address this challenge, we develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE), which can be used with any neural network architecture. Our approach imposes the constraint over smaller decomposed domains, each of which is solved by an "expert" through differentiable optimization. During training, each expert independently performs a localized backpropagation step by leveraging the implicit function theorem; the independence of each expert allows for parallelization across multiple GPUs. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting for predicting the dynamics of challenging non-linear systems. We also improve training stability and require significantly less computation time during both training and inference stages.

Generalized self-concordance is a key property present in the objective function of many important learning problems. We establish the convergence rate of a simple Frank-Wolfe variant that uses the open-loop step size strategy gamma_t = 2/(t+2), obtaining a O(1/t) convergence rate for this class of functions in terms of primal gap and Frank-Wolfe gap, where t is the iteration count. This avoids the use of second-order information or the need to estimate local smoothness parameters of previous work. We also show improved convergence rates for various common cases, e.g., when the feasible region under consideration is uniformly convex or polyhedral.

Recently, Direct Preference Optimization (DPO) has extended its success from aligning large language models (LLMs) to aligning text-to-image diffusion models with human preferences. Unlike most existing DPO methods that assume all diffusion steps share a consistent preference order with the final generated images, we argue that this assumption neglects step-specific denoising performance and that preference labels should be tailored to each step's contribution. To address this limitation, we propose Step-aware Preference Optimization (SPO), a novel post-training approach that independently evaluates and adjusts the denoising performance at each step, using a step-aware preference model and a step-wise resampler to ensure accurate step-aware supervision. Specifically, at each denoising step, we sample a pool of images, find a suitable win-lose pair, and, most importantly, randomly select a single image from the pool to initialize the next denoising step. This step-wise resampler process ensures the next win-lose image pair comes from the same image, making the win-lose comparison independent of the previous step. To assess the preferences at each step, we train a separate step-aware preference model that can be applied to both noisy and clean images. Our experiments with Stable Diffusion v1.5 and SDXL demonstrate that SPO significantly outperforms the latest Diffusion-DPO in aligning generated images with complex, detailed prompts and enhancing aesthetics, while also achieving more than 20x times faster in training efficiency. Code and model: https://rockeycoss.github.io/spo.github.io/

Large vision-language models are steadily gaining personalization capabilities at the cost of fine-tuning or data augmentation. We present two models for image generation using model-agnostic learning that align semantic priors with generative capabilities. RLDF, or Reinforcement Learning from Diffusion Feedback, is a singular approach for visual imitation through prior-preserving reward function guidance. This employs Q-learning (with standard Q*) for generation and follows a semantic-rewarded trajectory for image search through finite encoding-tailored actions. The second proposed method, noisy diffusion gradient, is optimization driven. At the root of both methods is a special CFG encoding that we propose for continual semantic guidance. Using only a single input image and no text input, RLDF generates high-quality images over varied domains including retail, sports and agriculture showcasing class-consistency and strong visual diversity. Project website is available at https://infernolia.github.io/RLDF.