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Optimization · Stochastic

Yuhan Ye, Ying Cui, Jingyi Wang

We propose an online adaptive sampling algorithm for solving stochastic nonsmooth difference-of-convex (DC) problems under time-varying distributions. At each iteration, the algorithm relies solely on data generated from the current distribution and employs distinct adaptive sampling rates for the convex and concave components of the DC function, a novel design guided by our theoretical analysis. We show that, under proper conditions on the convergence of distributions, the algorithm converges subsequentially to DC critical points almost surely. Furthermore, the sample size requirement of our proposed algorithm matches the results achieved in the smooth case or when a measurable subgradient selector is available, both under static distributions. A key element of this analysis is the derivation of a novel $O(\sqrt{p/n})$ pointwise convergence rate (modulo logarithmic factors) for the sample average approximation of subdifferential mappings, where $p$ is the dimension of the variable and $n$ is the sample size -- a result of independent interest. Numerical experiments confirm that the proposed algorithm is both efficient and effective for addressing stochastic nonsmooth problems.

Optimization · Stochastic

Xinyu Luo, Cedar Site Bai, Bolian Li, Petros Drineas, Ruqi Zhang, Brian Bullins

While popular optimization methods such as SGD, AdamW, and Lion depend on steepest descent updates in either $\ell_2$ or $\ell_\infty$ norms, there remains a critical gap in handling the non-Euclidean structure observed in modern deep networks training. In this work, we address this need by introducing a new accelerated $\ell_p$ steepest descent algorithm, called Stacey, which uses interpolated primal-dual iterate sequences to effectively navigate non-Euclidean smooth optimization tasks. In addition to providing novel theoretical guarantees for the foundations of our algorithm, we empirically compare our approach against these popular methods on tasks including image classification and language model (LLM) pretraining, demonstrating both faster convergence and higher final accuracy. We further evaluate different values of $p$ across various models and datasets, underscoring the importance and efficiency of non-Euclidean approaches over standard Euclidean methods. Code can be found at https://github.com/xinyuluo8561/Stacey.

Theory · Optimization

Tehila Dahan, Kfir Y Levy

Optimization methods are crucial to the success of machine learning, with Stochastic Gradient Descent (SGD) serving as a foundational algorithm for training models. However, SGD is often sensitive to the choice of the learning rate, which necessitates extensive hyperparameter tuning. In this work, we introduce a new variant of SGD that brings enhanced stability in two key aspects. First, our method allows the use of the same fixed learning rate to attain optimal convergence rates regardless of the noise magnitude, eliminating the need to adjust learning rates between noiseless and noisy settings. Second, our approach achieves these optimal rates over a wide range of learning rates, significantly reducing sensitivity compared to standard SGD, which requires precise learning rate selection.Our key innovation is a novel gradient estimator based on a double-momentum mechanism that combines two recent momentum-based techniques. Utilizing this estimator, we design both standard and accelerated algorithms that are robust to the choice of learning rate. Specifically, our methods attain optimal convergence rates in both noiseless and noisy stochastic convex optimization scenarios without the need for learning rate decay or fine-tuning. We also prove that our approach maintains optimal performance across a wide spectrum of learning rates, underscoring its stability and practicality. Empirical studies further validate the robustness and enhanced stability of our approach.

Optimization · Non-Convex

Sixin Zhang

We study min-max algorithms to solve zero-sum differential games onRiemannian manifold.Based on the notions ofdifferential Stackelberg equilibriumand differential Nash equilibrium on Riemannian manifold,we analyze the local convergence of two representative deterministic simultaneous algorithms $\tau$-GDA and $\tau$-SGAto such equilibria.Sufficient conditions are obtained to establish the linear convergence rateof $\tau$-GDA based on the Ostrowski theorem on manifold and spectral analysis. To avoid strong rotational dynamics in $\tau$-GDA, $\tau$-SGA is extended fromthe symplectic gradient-adjustment method in Euclidean space.We analyze an asymptotic approximation of $\tau$-SGA when the learning rate ratio $\tau$ is big. In some cases, it can achieve a faster convergence rate to differential Stackelberg equilibrium compared to $\tau$-GDA. We show numerically how the insights obtained from theconvergence analysis may improvethe training of orthogonal Wasserstein GANs using stochastic $\tau$-GDA and $\tau$-SGA on simple benchmarks.

Optimization · Non-Convex

Dimitris Oikonomou, Nicolas Loizou

Sharpness-Aware Minimization (SAM) has emerged as a powerful method for improving generalization in machine learning models by minimizing the sharpness of the loss landscape. However, despite its success, several important questions regarding the convergence properties of SAM in non-convex settings are still open, including the benefits of using normalization in the update rule, the dependence of the analysis on the restrictive bounded variance assumption, and the convergence guarantees under different sampling strategies. To address these questions, in this paper, we provide a unified analysis of SAM and its unnormalized variant (USAM) under one single flexible update rule (Unified SAM), and we present convergence results of the new algorithm under a relaxed and more natural assumption on the stochastic noise. Our analysis provides convergence guarantees for SAM under different step size selections for non-convex problems and functions that satisfy the Polyak-Lojasiewicz (PL) condition (a non-convex generalization of strongly convex functions). The proposed theory holds under the arbitrary sampling paradigm, which includes importance sampling as special case, allowing us to analyze variants of SAM that were never explicitly considered in the literature. Experiments validate the theoretical findings and further demonstrate the practical effectiveness of Unified SAM in training deep neural networks for image classification tasks.

Optimization · Non-Convex

Sueda Taner, Ziyi Wang, Christoph Studer

We introduce a novel class of regularization functions, called Cauchy–Schwarz (CS) regularizers, which can be designed to induce a wide range of properties in solution vectors of optimization problems. To demonstrate the versatility of CS regularizers, we derive regularization functions that promote discrete-valued vectors, eigenvectors of a given matrix, and orthogonal matrices. The resulting CS regularizers are simple, differentiable, and can be free of spurious stationary points, making them suitable for gradient-based solvers and large-scale optimization problems. In addition, CS regularizers automatically adapt to the appropriate scale, which is, for example, beneficial when discretizing the weights of neural networks. To demonstrate the efficacy of CS regularizers, we provide results for solving underdetermined systems of linear equations and weight quantization in neural networks. Furthermore, we discuss specializations, variations, and generalizations, which lead to an even broader class of new and possibly more powerful regularizers.

Optimization · Large Scale, Parallel and Distributed

Ziheng Cheng, Margalit Glasgow

We study distributed adaptive algorithms with local updates (intermittent communication). Despite the great empirical success of adaptive methods in distributed training of modern machine learning models, the theoretical benefits of local updates within adaptive methods, particularly in terms of reducing communication complexity, have not been fully understood yet. In this paper, for the first time, we prove that \em Local SGD \em with momentum (\em Local \em SGDM) and \em Local \em Adam can outperform their minibatch counterparts in convex and weakly convex settings in certain regimes, respectively. Our analysis relies on a novel technique to prove contraction during local iterations, which is a crucial yet challenging step to show the advantages of local updates, under generalized smoothness assumption and gradient clipping strategy.

Deep Learning · Theory

Lijia Yu, Yibo Miao, Yifan Zhu, XIAOSHAN GAO, Lijun Zhang

The primary objective of learning methods is generalization. Classic generalization bounds, based on VC-dimension or Rademacher complexity, are uniformly applicable to all networks in the hypothesis space. On the other hand, algorithm-dependent generalization bounds, like stability bounds, address more practical scenarios and provide generalization conditions for neural networks trained using SGD. However, these bounds often rely on strict assumptions, such as the NTK hypothesis or convexity of the empirical loss, which are typically not met by neural networks. In order to establish generalizability under less stringent assumptions, this paper investigates generalizability of neural networks that minimize the empirical risk. A lower bound for population accuracy is established based on the expressiveness of these networks, which indicates that with adequately large training sample and network sizes, these networks can generalize effectively. Additionally, we provide a lower bound necessary for generalization, demonstrating that, for certain data distributions, the quantity of data required to ensure generalization exceeds the network size needed to represent that distribution. Finally, we provide theoretical insights into several phenomena in deep learning, including robust overfitting, importance of over-parameterization networks, and effects of loss functions.

Optimization · Everything Else

Shulan Zhu, Chenglong Bao, Defeng Sun, Yancheng Yuan

The \textbf{i}nexact \textbf{s}tochastic \textbf{p}roximal \textbf{p}oint \textbf{a}lgorithm (isPPA) is popular for solving stochastic composite optimization problems with many applications in machine learning. While the convergence theory of the (inexact) PPA has been well established, the known convergence guarantees of isPPA require restrictive assumptions. In this paper, we establish the stability and almost sure convergence of isPPA under mild assumptions, where smoothness and (restrictive) strong convexity of the objective function are not required. Imposing a local Lipschitz condition on component functions and a quadratic growth condition on the objective function, we establish last-iterate iteration complexity bounds of isPPA regarding the distance to the solution set and the Karush–Kuhn–Tucker (KKT) residual. Moreover, we show that the established iteration complexity bounds are tight up to a constant by explicitly analyzing the bounds for the regularized Fr\'echet mean problem. We further validate the established convergence guarantees of isPPA by numerical experiments.

Theory · Optimization

Zhen Qin, Zhuqing Liu, Songtao Lu, Yingbin Liang, Jia (Kevin) Liu

Decentralized bilevel optimization (DBO) provides a powerful framework for multi-agent systems to solve local bilevel tasks in a decentralized fashion without the need for a central server. However, most existing DBO methods rely on lower-level strong convexity (LLSC) to guarantee unique solutions and a well-defined hypergradient for stationarity measure, hindering their applicability in many practical scenarios not satisfying LLSC. To overcome this limitation, we introduce a new single-loop DBO algorithm called diminishing quadratically-regularized bilevel decentralized optimization (DUET), which eliminates the need for LLSC by introducing a diminishing quadratic regularization to the lower-level (LL) objective. We show that DUET achieves an iteration complexity of $O(1/T^{1-5p-\frac{11}{4}\tau})$ for approximate KKT-stationary point convergence under relaxed assumptions, where $p$ and $\tau $ are control parameters for LL learning rate and averaging, respectively.In addition, our DUET algorithm incorporates gradient tracking to address data heterogeneity, a key challenge in DBO settings. To the best of our knowledge, this is the first work to tackle DBO without LLSC under decentralized settings with data heterogeneity.Numerical experiments validate the theoretical findings and demonstrate the practical effectiveness of our proposed algorithms.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Tianqu Zhuang, Hongyao Yu, Yixiang Qiu, Hao Fang, Bin Chen, Shu-Tao Xia

Model inversion attacks (MIAs) aim to reconstruct the private training data by accessing a public model, raising concerns about privacy leakage. Black-box MIAs, where attackers can only query the model and obtain outputs, are closer to real-world scenarios. The latest black-box attacks have outperformed the state-of-the-art white-box attacks, and existing defenses cannot resist them effectively. To fill this gap, we propose Stealthy Shield Defense (SSD), a post-processing algorithm against black-box MIAs. Our idea is to modify the model's outputs to minimize the conditional mutual information (CMI). We mathematically prove that CMI is a special case of information bottlenecks (IB), and thus inherits the advantages of IB---making predictions less dependent on inputs and more dependent on ground truths. This theoretically guarantees our effectiveness, both in resisting MIAs and preserving utility. For minimizing CMI, we formulate a convex optimization problem and solve it via the water-filling method. Adaptive rate-distortion is introduced to constrain the modification to the outputs, and the water-filling is implemented on GPUs to address computation cost. Without the need to retrain the model, our algorithm is plug-and-play and easy to deploy. Experimental results indicate that SSD outperforms existing defenses, in terms of MIA resistance and model's utility, across various attack algorithms, training datasets, and model architectures. Our code is available at https://github.com/ZhuangQu/Stealthy-Shield-Defense.

Optimization · Non-Convex

Dan Greenstein, Elazar Gershuni, Ilan Ben-Bassat, Yaroslav Fyodorov, Ran Moshe, Fiana Raiber, Alex Shtoff, Oren Somekh, Nadav Hallak

The subset selection problem is fundamental in machine learning and other fields of computer science.We introduce a stochastic formulation for the minimum cost subset selection problem in a black box setting, in which only the subset metric value is available.Subsequently, we can handle two-stage schemes, with an outer subset-selection component and an inner subset cost evaluation component. We propose formulating the subset selection problem in a stochastic manner by choosing subsets at random from a distribution whose parameters are learned. Two stochastic formulations are proposed.The first explicitly restricts the subset's cardinality, and the second yields the desired cardinality in expectation.The distribution is parameterized by a decision variable, which we optimize using Stochastic Mirror Descent.Our choice of distributions yields constructive closed-form unbiased stochastic gradient formulas and convergence guarantees, including a rate with favorable dependency on the problem parameters.Empirical evaluation of selecting a subset of layers in transfer learning complements our theoretical findings and demonstrates the potential benefits of our approach.

Deep Learning · Generative Models and Autoencoders

Ye Yuan, Can Chen, Christopher Pal, Xue Liu

In offline multi-objective optimization (MOO), we leverage an offline dataset of designs and their associated labels to simultaneously minimize multiple objectives. This setting more closely mirrors complex real-world problems compared to single-objective optimization. Recent works mainly employ evolutionary algorithms and Bayesian optimization, with limited attention given to the generative modeling capabilities inherent in such data. In this study, we explore generative modeling in offline MOO through flow matching, noted for its effectiveness and efficiency. We introduce \textit{ParetoFlow}, specifically designed to guide flow sampling to approximate the Pareto front. Traditional predictor~(classifier) guidance is inadequate for this purpose because it models only a single objective. In response, we propose a \textit{multi-objective predictor guidance} module that assigns each sample a weight vector, representing a weighted distribution across multiple objective predictions. A local filtering scheme is introduced to address non-convex Pareto fronts. These weights uniformly cover the entire objective space, effectively directing sample generation towards the Pareto front. Since distributions with similar weights tend to generate similar samples, we introduce a \textit{neighboring evolution} module to foster knowledge sharing among neighboring distributions. This module generates offspring from these distributions, and selects the most promising one for the next iteration. Our method achieves state-of-the-art performance across various tasks. Our code is available.

Reinforcement Learning · Planning

Yining Li, Peizhong Ju, Ness Shroff

Multi-Objective Markov Decision Processes (MO-MDPs) are receiving increasing attention, as real-world decision-making problems often involve conflicting objectives that cannot be addressed by a single-objective MDP. The Pareto front identifies the set of policies that cannot be dominated, providing a foundation for finding Pareto optimal solutions that can efficiently adapt to various preferences.However, finding the Pareto front is a highly challenging problem. Most existing methods either (i) rely on traversing the *continuous preference space*, which is impractical and results in approximations that are difficult to evaluate against the true Pareto front, or (ii) focus solely on deterministic Pareto optimal policies, from which there are no known techniques to characterize the full Pareto front. Moreover, finding the structure of the Pareto front itself remains unclear even in the context of dynamic programming, where the MDP is fully known in advance.In this work, we address the challenge of efficiently discovering the Pareto front, involving both deterministic and stochastic Pareto optimal policies.By investigating the geometric structure of the Pareto front in MO-MDPs, we uncover a key property: the Pareto front is on the boundary of a convex polytope whose vertices all correspond to deterministic policies, and neighboring vertices of the Pareto front differ by only one state-action pair of the deterministic policy, almost surely.This insight transforms the global comparison across all policies into a localized search among deterministic policies that differ by only one state-action pair, drastically reducing the complexity of searching for the exact Pareto front. We develop an efficient algorithm that identifies the vertices of the Pareto front by solving a single-objective MDP only once and then traversing the edges of the Pareto front, making it more efficient than existing methods. Furthermore, the entire Pareto front can be found in $V$ iterations, where $V$ represents the number of vertices on the Pareto front.Our empirical studies demonstrate the effectiveness of our theoretical strategy in discovering the Pareto front efficiently.

Optimization · Non-Convex

Weiyuan Gong, Chenyi Zhang, Tongyang Li

In this paper, we systematically study quantum algorithms for finding an $\epsilon$-approximate second-order stationary point ($\epsilon$-SOSP) of a $d$-dimensional nonconvex function, a fundamental problem in nonconvex optimization, with noisy zeroth- or first-order oracles as inputs. We first prove that, up to noise of $O(\epsilon^{10}/d^5)$, perturbed accelerated gradient descent equipped with quantum gradient estimation takes $O(\log d/\epsilon^{1.75})$ quantum queries to find an $\epsilon$-SOSP. We then prove that standard perturbed gradient descent is robust to the noise of $O(\epsilon^6/d^4)$ and $O(\epsilon/d^{0.5+\zeta})$ for any $\zeta>0$ on the zeroth- and first-order oracles, respectively, which provides a quantum algorithm with poly-logarithmic query complexity. Furthermore, we propose a stochastic gradient descent algorithm using quantum mean estimation on the Gaussian smoothing of noisy oracles, which is robust to $O(\epsilon^{1.5}/d)$ and $O(\epsilon/\sqrt{d})$ noise on the zeroth- and first-order oracles, respectively. The quantum algorithm takes $O(d^{2.5}/\epsilon^{3.5})$ and $O(d^2/\epsilon^3)$ queries to the two oracles, giving a polynomial speedup over the classical counterparts. As a complement, we characterize the domains where quantum algorithms can find an $\epsilon$-SOSP with poly-logarithmic, polynomial, or exponential number of queries in $d$, or the problem is information-theoretically unsolvable even with an infinite number of queries. In addition, we prove an $\Omega(\epsilon^{-12/7})$ lower bound on $\epsilon$ for any randomized classical and quantum algorithm to find an $\epsilon$-SOSP using either noisy zeroth- or first-order oracles.

Optimization · Non-Convex

Xianliang Li, Jun Luo, Zhiwei Zheng, Hanxiao Wang, Li Luo, Lingkun Wen, Linlong Wu, Sheng Xu

Momentum-based optimizers are widely adopted for training neural networks. However, the optimal selection of momentum coefficients remains elusive. This uncertainty impedes a clear understanding of the role of momentum in stochastic gradient methods. In this paper, we present a frequency domain analysis framework that interprets the momentum method as a time-variant filter for gradients, where adjustments to momentum coefficients modify the filter characteristics. Our experiments support this perspective and provide a deeper understanding of the mechanism involved. Moreover, our analysis reveals the following significant findings: high-frequency gradient components are undesired in the late stages of training; preserving the original gradient in the early stages, and gradually amplifying low-frequency gradient components during training both enhance performance. Based on these insights, we propose Frequency Stochastic Gradient Descent with Momentum (FSGDM), a heuristic optimizer that dynamically adjusts the momentum filtering characteristic with an empirically effective dynamic magnitude response. Experimental results demonstrate the superiority of FSGDM over conventional momentum optimizers.

Theory · Optimization

Yuzhou Gu, Zhao Song, Lichen Zhang

Quadratic programming is a ubiquitous prototype in convex programming. Many machine learning problems can be formulated as quadratic programming, including the famous Support Vector Machines (SVMs). Linear and kernel SVMs have been among the most popular models in machine learning over the past three decades, prior to the deep learning era.Generally, a quadratic program has an input size of $\Theta(n^2)$, where $n$ is the number of variables. Assuming the Strong Exponential Time Hypothesis ($\textsf{SETH}$), it is known that no $O(n^{2-o(1)})$ time algorithm exists when the quadratic objective matrix is positive semidefinite (Backurs, Indyk, and Schmidt, NeurIPS'17). However, problems such as SVMs usually admit much smaller input sizes: one is given $n$ data points, each of dimension $d$, and $d$ is oftentimes much smaller than $n$. Furthermore, the SVM program has only $O(1)$ equality linear constraints. This suggests that faster algorithms are feasible, provided the program exhibits certain structures.In this work, we design the first nearly-linear time algorithm for solving quadratic programs whenever the quadratic objective admits a low-rank factorization, and the number of linear constraints is small. Consequently, we obtain results for SVMs:* For linear SVM when the input data is $d$-dimensional, our algorithm runs in time $\widetilde O(nd^{(\omega+1)/2}\log(1/\epsilon))$ where $\omega\approx 2.37$ is the fast matrix multiplication exponent; * For Gaussian kernel SVM, when the data dimension $d = O(\log n)$ and the squared dataset radius is sub-logarithmic in $n$, our algorithm runs in time $O(n^{1+o(1)}\log(1/\epsilon))$. We also prove that when the squared dataset radius is at least $\Omega(\log^2 n)$, then $\Omega(n^{2-o(1)})$ time is required. This improves upon the prior best lower bound in both the dimension $d$ and the squared dataset radius.

Optimization · Non-Convex

Frank Zhengqing Wu, Berfin Simsek, François Ged

In this paper, we study the loss landscape of one-hidden-layer neural networks with ReLU-like activation functions trained with the empirical squared loss using gradient descent (GD). We identify the stationary points of such networks, which significantly slow down loss decrease during training. To capture such points while accounting for the non-differentiability of the loss, the stationary points that we study are directional stationary points, rather than other notions like Clarke stationary points. We show that, if a stationary point does not contain "escape neurons", which are defined with first-order conditions, it must be a local minimum. Moreover, for the scalar-output case, the presence of an escape neuron guarantees that the stationary point is not a local minimum. Our results refine the description of the *saddle-to-saddle* training process starting from infinitesimally small (vanishing) initialization for shallow ReLU-like networks: By precluding the saddle escape types that previous works did not rule out, we advance one step closer to a complete picture of the entire dynamics. Moreover, we are also able to fully discuss how network embedding, which is to instantiate a narrower network with a wider network, reshapes the stationary points.

Optimization · Large Scale, Parallel and Distributed

Demyan Yarmoshik, Alexander Rogozin, Nikita Kiselev, Daniil Dorin, Alexander Gasnikov, Dmitry Kovalev

We consider the decentralized minimization of a separable objective $\sum_{i=1}^{n} f_i(x_i)$, where the variables are coupled through an affine constraint $\sum_{i=1}^n\left(\mathbf{A}_i x_i - b_i\right) = 0$.We assume that the functions $f_i$, matrices $\mathbf{A}_i$, and vectors $b_i$ are stored locally by the nodes of a computational network, and that the functions $f_i$ are smooth and strongly convex. This problem has significant applications in resource allocation and systems control and can also arise in distributed machine learning.We propose lower complexity bounds for decentralized optimization problems with coupled constraints and a first-order algorithm achieving the lower bounds. To the best of our knowledge, our method is also the first linearly convergent first-order decentralized algorithm for problems with general affine coupled constraints.

Theory · Optimization

Daniil Vankov, Anton Rodomanov, Angelia Nedich, Lalitha Sankar, Sebastian Stich

We study gradient methods for optimizing $(L_0, L_1)$-smooth functions, aclass that generalizes Lipschitz-smooth functions and has gained attention forits relevance in machine learning.We provide new insights into the structure of this function class and developa principled framework for analyzing optimization methods in this setting.While our convergence rate estimates recover existing results for minimizingthe gradient norm in nonconvex problems, our approach significantly improvesthe best-known complexity bounds for convex objectives.Moreover, we show that the gradient method with Polyak stepsizes and thenormalized gradient method achieve nearly the same complexity guarantees asmethods that rely on explicit knowledge of $(L_0, L_1)$.Finally, we demonstrate that a carefully designed accelerated gradientmethod can be applied to $(L_0, L_1)$-smooth functions, further improving allprevious results.