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Optimization · Non-Convex

Zhuqing Liu, Xin Zhang, Jia Liu, Zhengyuan Zhu, Songtao Lu

Learning an accurate value function for a given policy is a critical step in solving reinforcement learning (RL) problems. So far, however, the convergence speed and sample complexity performances of most existing policy evaluation algorithms remain unsatisfactory, particularly with non-linear function approximation. This challenge motivates us to develop a new path-integrated primal-dual stochastic gradient (PILOT) method, that is able to achieve a fast convergence speed for RL policy evaluation with nonlinear function approximation. To further alleviate the periodic full gradient evaluation requirement, we further propose an enhanced method with an adaptive-batch adjustment called PILOT$^+$. The main advantages of our methods include: i) PILOT allows the use of {\em{constant}} step sizes and achieves the $\mathcal{O}(1/K)$ convergence rate to first-order stationary points of non-convex policy evaluation problems; ii) PILOT is a generic {\em{single}}-timescale algorithm that is also applicable for solving a large class of non-convex strongly-concave minimax optimization problems; iii) By adaptively adjusting the batch size via historical stochastic gradient information, PILOT$^+$ is more sample-efficient empirically without loss of theoretical convergence rate. Our extensive numerical experiments verify our theoretical findings and showcase the high efficiency of the proposed PILOT and PILOT$^+$ algorithms compared with the state-of-the-art methods.

Jeongyeol Kwon, Dohyun Kwon, Stephen Wright, Robert Nowak

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 {\it 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 $O(\epsilon^{-7})$ accesses to first-order stochastic gradient oracles. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, {\it i.e.,} with $O(1)$ samples per iteration, and achieves the improved oracle-complexity of $O(\epsilon^{-5})$.

Optimization · Zero-order and Black-box Optimization

Feiyang YE, YUEMING LYU, Xuehao Wang, Yu Zhang, Ivor Tsang

Multi-objective optimization (MOO) has become an influential framework for various machine learning problems, including reinforcement learning and multi-task learning. In this paper, we study the black-box multi-objective optimization problem, where we aim to optimize multiple potentially conflicting objectives with function queries only. To address this challenging problem and find a Pareto optimal solution or the Pareto stationary solution, we propose a novel adaptive stochastic gradient algorithm for black-box MOO, called ASMG. Specifically, we use the stochastic gradient approximation method to obtain the gradient for the distribution parameters of the Gaussian smoothed MOO with function queries only. Subsequently, an adaptive weight is employed to aggregate all stochastic gradients to optimize all objective functions effectively. Theoretically, we explicitly provide the connection between the original MOO problem and the corresponding Gaussian smoothed MOO problem and prove the convergence rate for the proposed ASMG algorithm in both convex and non-convex scenarios.Empirically, the proposed ASMG method achieves competitive performance on multiple numerical benchmark problems. Additionally, the state-of-the-art performance on the black-box multi-task learning problem demonstrates the effectiveness of the proposed ASMG method.

Optimization · Large Scale, Parallel and Distributed

Yi-Rui Yang, Chang-Wei Shi, Wu-Jun Li

Byzantine-robust distributed learning (BRDL), in which computing devices are likely to behave abnormally due to accidental failures or malicious attacks, has recently become a hot research topic. However, even in the independent and identically distributed (i.i.d.) case, existing BRDL methods will suffer a significant drop on model accuracy due to the large variance of stochastic gradients. Increasing batch sizes is a simple yet effective way to reduce the variance. However, when the total number of gradient computation is fixed, a too-large batch size will lead to a too-small iteration number (update number), which may also degrade the model accuracy. In view of this challenge, we mainly study the effect of batch size when the total number of gradient computation is fixed in this work. In particular, we show that when the total number of gradient computation is fixed, the optimal batch size corresponding to the tightest theoretical upper bound in BRDL increases with the fraction of Byzantine workers. Therefore, compared to the case without attacks, a larger batch size is preferred when under Byzantine attacks. Motivated by the theoretical finding, we propose a novel method called Byzantine-robust stochastic gradient descent with normalized momentum (ByzSGDnm) in order to further increase model accuracy in BRDL. We theoretically prove the convergence of ByzSGDnm for general non-convex cases under Byzantine attacks. Empirical results show that when under Byzantine attacks, compared to the cases of small batch sizes, setting a relatively large batch size can significantly increase the model accuracy, which is consistent with our theoretical results. Moreover, ByzSGDnm can achieve higher model accuracy than existing BRDL methods when under deliberately crafted attacks. In addition, we empirically show that increasing batch sizes has the bonus of training acceleration.

Optimization · Convex

Venkata Sai Surya Subramanyam Duvvuri, Fnu Devvrit, Rohan Anil, Cho-Jui Hsieh, Inderjit Dhillon

Adaptive regularization based optimization methods such as full-matrix Adagrad which use gradient second-moment information hold significant potential for fast convergence in deep neural network (DNN) training, but are memory intensive and computationally demanding for large neural nets. We develop a technique called Combining AxeS PReconditioners (CASPR), which optimizes matrix-shaped DNN parameters by finding different preconditioners for each mode/axis of the parameter and combining them using a Kronecker-sum based approximation. We show tighter convergence guarantees in stochastic optimization compared to a Kronecker product based preconditioner, Shampoo, which arises as a special case of CASPR. Furthermore, our experiments demonstrates that CASPR approximates the gradient second-moment matrix in full-matrix Adagrad more accurately, and shows significant improvement in training and generalization performance compared to existing practical adaptive regularization based methods such as Shampoo and Adam in a variety of tasks including graph neural network on OGBG-molpcba, Transformer on a universal dependencies dataset and auto-regressive large language modeling on C4 dataset.

Theory · Game Theory

Ian Gemp, Luke Marris, Georgios Piliouras

We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

Optimization

Zijian Liu, Zhengyuan Zhou

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 $O(\log(1/\delta)\log T/\sqrt{T})$ or $O(\sqrt{\log(1/\delta)/T})$ 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.

Optimization · First-order

Ioannis Mavrothalassitis, Stratis Skoulakis, Leello Dadi, Volkan Cevher

Given a sequence of functions $f_1,\ldots,f_n$ with $f_i:\mathcal{D}\mapsto \mathbb{R}$, finite-sum minimization seeks a point ${x}^\star \in \mathcal{D}$ minimizing $\sum_{j=1}^nf_j(x)/n$. In this work, we propose a key twist into the finite-sum minimization, dubbed as *continual finite-sum minimization*, that asks for a sequence of points $x_1^\star, \ldots, x_n^\star \in D$ such that each ${x}^\star_i \in D$ minimizes the prefix-sum $\sum_{j=1}^if_j(x)/i$. Assuming that each prefix-sum is strongly convex, we develop a first-order continual stochastic variance reduction gradient method ($\mathrm{CSVRG}$) producing an $\epsilon$-optimal sequence with $\tilde{\mathcal{O}}(n/\epsilon^{1/3} + 1/\sqrt{\epsilon})$ overall *first-order oracles* (FO). An FO corresponds to the computation of a single gradient $\nabla f_j(x)$ at a given $x \in \mathcal{D}$ for some $j \in [n]$. Our approach significantly improves upon the $\mathcal{O}(n/\epsilon)$ FOs that $\mathrm{StochasticGradientDescent}$ requires and the $\mathcal{O}(n^2 \log (1/\epsilon))$ FOs that state-of-the-art variance reduction methods such as $\mathrm{Katyusha}$ require. We also prove that there is no natural first-order method with $\mathcal{O}\left(n/\epsilon^\alpha\right)$ gradient complexity for $\alpha < 1/4$, establishing that the first-order complexity of our method is nearly tight.

Xinyue Liu, Hualin Zhang, Bin Gu, Hong Chen

Zeroth-order optimization algorithms are widely used for black-box optimization problems, such as those in machine learning and prompt engineering, where the gradients are approximated using function evaluations. Recently, a generalization result was provided for zeroth-order stochastic gradient descent (SGD) algorithms through stability analysis. However, this result was limited to the vanilla 2-point zeroth-order estimate of Gaussian distribution used in SGD algorithms. To address these limitations, we propose a general proof framework for stability analysis that applies to convex, strongly convex, and non-convex conditions, and yields results for popular zeroth-order optimization algorithms, including SGD, GD, and SVRG, as well as various zeroth-order estimates, such as 1-point and 2-point with different distributions and coordinate estimates. Our general analysis shows that coordinate estimation can lead to tighter generalization bounds for SGD, GD, and SVRG versions of zeroth-order optimization algorithms, due to the smaller expansion brought by coordinate estimates to stability analysis.

Theory · Learning Theory

Xuheng Li, Yihe Deng, Jingfeng Wu, Dongruo Zhou, Quanquan Gu

Accelerated stochastic gradient descent (ASGD) is a workhorse in deep learning and often achieves better generalization performance than SGD. However, existing optimization theory can only explain the faster convergence of ASGD, but cannot explain its better generalization. In this paper, we study the generalization of ASGD for overparameterized linear regression, which is possibly the simplest setting of learning with overparameterization. We establish an instance-dependent excess risk bound for ASGD within each eigen-subspace of the data covariance matrix. Our analysis shows that (i) ASGD outperforms SGD in the subspace of small eigenvalues, exhibiting a faster rate of exponential decay for bias error, while in the subspace of large eigenvalues, its bias error decays slower than SGD; and (ii) the variance error of ASGD is always larger than that of SGD. Our result suggests that ASGD can outperform SGD when the difference between the initialization and the true weight vector is mostly confined to the subspace of small eigenvalues. Additionally, when our analysis is specialized to linear regression in the strongly convex setting, it yields a tighter bound for bias error than the best-known result.

Optimization · Convex

Rui Pan, Yuxing Liu, Xiaoyu Wang, Tong Zhang

Heavy-ball momentum with decaying learning rates is widely used with SGD for optimizing deep learning models. In contrast to its empirical popularity, the understanding of its theoretical property is still quite limited, especially under the standard anisotropic gradient noise condition for quadratic regression problems. Although it is widely conjectured that heavy-ball momentum method can provide accelerated convergence and should work well in large batch settings, there is no rigorous theoretical analysis. In this paper, we fill this theoretical gap by establishing a non-asymptotic convergence bound for stochastic heavy-ball methods with step decay scheduler on quadratic objectives, under the anisotropic gradient noise condition. As a direct implication, we show that heavy-ball momentum can provide $\tilde{\mathcal{O}}(\sqrt{\kappa})$ accelerated convergence of the bias term of SGD while still achieving near-optimal convergence rate with respect to the stochastic variance term. The combined effect implies an overall convergence rate within log factors from the statistical minimax rate. This means SGD with heavy-ball momentum is useful in the large-batch settings such as distributed machine learning or federated learning, where a smaller number of iterations can significantly reduce the number of communication rounds, leading to acceleration in practice.

Optimization · Non-Convex

Mohammad Pedramfar, Yididiya Nadew, Chris Quinn, Vaneet Aggarwal

This paper introduces unified projection-free Frank-Wolfe type algorithms for adversarial continuous DR-submodular optimization, spanning scenarios such as full information and (semi-)bandit feedback, monotone and non-monotone functions, different constraints, and types of stochastic queries. For every problem considered in the non-monotone setting, the proposed algorithms are either the first with proven sub-linear $\alpha$-regret bounds or have better $\alpha$-regret bounds than the state of the art, where $\alpha$ is a corresponding approximation bound in the offline setting. In the monotone setting, the proposed approach gives state-of-the-art sub-linear $\alpha$-regret bounds among projection-free algorithms in 7 of the 8 considered cases while matching the result of the remaining case. Additionally, this paper addresses semi-bandit and bandit feedback for adversarial DR-submodular optimization, advancing the understanding of this optimization area.

Optimization · Non-Convex

Runzhe Wang, Sadhika Malladi, Tianhao Wang, Kaifeng Lyu, Zhiyuan Li

Momentum is known to accelerate the convergence of gradient descent in strongly convex settings without stochastic gradient noise. In stochastic optimization, such as training neural networks, folklore suggests that momentum may help deep learning optimization by reducing the variance of the stochastic gradient update, but previous theoretical analyses do not find momentum to offer any provable acceleration. Theoretical results in this paper clarify the role of momentum in stochastic settings where the learning rate is small and gradient noise is the dominant source of instability, suggesting that SGD with and without momentum behave similarly in the short and long time horizons. Experiments show that momentum indeed has limited benefits for both optimization and generalization in practical training regimes where the optimal learning rate is not very large, including small- to medium-batch training from scratch on ImageNet and fine-tuning language models on downstream tasks.

Optimization/Stochastic

Lijun Zhang, Peng Zhao, Zhen-Hua Zhuang, Tianbao Yang, Zhi-Hua Zhou

This paper investigates group distributionally robust optimization (GDRO), with the purpose to learn a model that performs well over $m$ different distributions. First, we formulate GDRO as a stochastic convex-concave saddle-point problem, and demonstrate that stochastic mirror descent (SMD), using $m$ samples in each iteration, achieves an $O(m (\log m)/\epsilon^2)$ sample complexity for finding an $\epsilon$-optimal solution, which matches the $\Omega(m/\epsilon^2)$ lower bound up to a logarithmic factor. Then, we make use of techniques from online learning to reduce the number of samples required in each round from $m$ to $1$, keeping the same sample complexity. Specifically, we cast GDRO as a two-players game where one player simply performs SMD and the other executes an online algorithm for non-oblivious multi-armed bandits. Next, we consider a more practical scenario where the number of samples that can be drawn from each distribution is different, and propose a novel formulation of weighted GDRO, which allows us to derive distribution-dependent convergence rates. Denote by $n_i$ the sample budget for the $i$-th distribution, and assume $n_1 \geq n_2 \geq \cdots \geq n_m$. In the first approach, we incorporate non-uniform sampling into SMD such that the sample budget is satisfied in expectation, and prove that the excess risk of the $i$-th distribution decreases at an $O(\sqrt{n_1 \log m}/n_i)$ rate. In the second approach, we use mini-batches to meet the budget exactly and also reduce the variance in stochastic gradients, and then leverage stochastic mirror-prox algorithm, which can exploit small variances, to optimize a carefully designed weighted GDRO problem. Under appropriate conditions, it attains an $O((\log m)/\sqrt{n_i})$ convergence rate, which almost matches the optimal $O(\sqrt{1/n_i})$ rate of only learning from the $i$-th distribution with $n_i$ samples.

Optimization/Zero-order and Black-box Optimization

Nikita Kornilov, Ohad Shamir, Aleksandr Lobanov, Darina Dvinskikh, Alexander Gasnikov, Innokentiy Shibaev, Eduard Gorbunov, Samuel Horváth

In this paper, we consider non-smooth stochastic convex optimization with two function evaluations per round under infinite noise variance. In the classical setting when noise has finite variance, an optimal algorithm, built upon the batched accelerated gradient method, was proposed in (Gasnikov et. al., 2022). This optimality is defined in terms of iteration and oracle complexity, as well as the maximal admissible level of adversarial noise. However, the assumption of finite variance is burdensome and it might not hold in many practical scenarios. To address this, we demonstrate how to adapt a refined clipped version of the accelerated gradient (Stochastic Similar Triangles) method from (Sadiev et al., 2023) for a two-point zero-order oracle. This adaptation entails extending the batching technique to accommodate infinite variance — a non-trivial task that stands as a distinct contribution of this paper.

Theory/Online Learning and Bandits

Hamish Flynn, David Reeb, Melih Kandemir, Jan Peters

We present improved algorithms with worst-case regret guarantees for the stochastic linear bandit problem. The widely used "optimism in the face of uncertainty" principle reduces a stochastic bandit problem to the construction of a confidence sequence for the unknown reward function. The performance of the resulting bandit algorithm depends on the size of the confidence sequence, with smaller confidence sets yielding better empirical performance and stronger regret guarantees. In this work, we use a novel tail bound for adaptive martingale mixtures to construct confidence sequences which are suitable for stochastic bandits. These confidence sequences allow for efficient action selection via convex programming. We prove that a linear bandit algorithm based on our confidence sequences is guaranteed to achieve competitive worst-case regret. We show that our confidence sequences are tighter than competitors, both empirically and theoretically. Finally, we demonstrate that our tighter confidence sequences give improved performance in several hyperparameter tuning tasks.

Optimization/Bilevel optimization

Yifan Yang, Peiyao Xiao, Kaiyi Ji

In this paper, we revisit the bilevel optimization problem, in which the upper-level objective function is generally nonconvex and the lower-level objective function is strongly convex. Although this type of problem has been studied extensively, it still remains an open question how to achieve an $\mathcal{O}(\epsilon^{-1.5})$ sample complexity in Hessian/Jacobian-free stochastic bilevel optimization without any second-order derivative computation. To fill this gap, we propose a novel Hessian/Jacobian-free bilevel optimizer named FdeHBO, which features a simple fully single-loop structure, a projection-aided finite-difference Hessian/Jacobian-vector approximation, and momentum-based updates. Theoretically, we show that FdeHBO requires $\mathcal{O}(\epsilon^{-1.5})$ iterations (each using $\mathcal{O}(1)$ samples and only first-order gradient information) to find an $\epsilon$-accurate stationary point. As far as we know, this is the first Hessian/Jacobian-free method with an $\mathcal{O}(\epsilon^{-1.5})$ sample complexity for nonconvex-strongly-convex stochastic bilevel optimization.

Optimization/Non-Convex

Quanqi Hu, Dixian Zhu, Tianbao Yang

This paper investigates new families of compositional optimization problems, called non-smooth weakly-convex finite-sum coupled compositional optimization (NSWC FCCO). There has been a growing interest in FCCO due to its wide-ranging applications in machine learning and AI, as well as its ability to address the shortcomings of stochastic algorithms based on empirical risk minimization. However, current research on FCCO presumes that both the inner and outer functions are smooth, limiting their potential to tackle a more diverse set of problems. Our research expands on this area by examining non-smooth weakly-convex FCCO, where the outer function is weakly convex and non-decreasing, and the inner function is weakly-convex. We analyze a single-loop algorithm and establish its complexity for finding an $\epsilon$-stationary point of the Moreau envelop of the objective function. Additionally, we also extend the algorithm for solving novel non-smooth weakly-convex tri-level finite-sum coupled compositional optimization problems, which feature a nested arrangement of three functions. Lastly, we explore the applications of our algorithms in deep learning for two-way partial AUC maximization and multi-instance two-way partial AUC maximization, using empirical studies to showcase the effectiveness of the proposed algorithms.

Optimization/Stochastic

Mher Safaryan, Alexandra Peste, Dan Alistarh

Knowledge distillation is a popular approach for enhancing the performance of "student" models, with lower representational capacity, by taking advantage of more powerful "teacher" models. Despite its apparent simplicity, the underlying mechanics behind knowledge distillation (KD) are not yet fully understood. In this work, we shed new light on the inner workings of this method, by examining it from an optimization perspective. Specifically, we show that, in the context of linear and deep linear models, KD can be interpreted as a novel type of stochastic variance reduction mechanism. We provide a detailed convergence analysis of the resulting dynamics, which hold under standard assumptions for both strongly-convex and non-convex losses, showing that KD acts as a form of \emph{partial variance reduction}, which can reduce the stochastic gradient noise, but may not eliminate it completely, depending on the properties of the ``teacher'' model. Our analysis puts further emphasis on the need for careful parametrization of KD, in particular w.r.t. the weighting of the distillation loss, and is validated empirically on both linear models and deep neural networks.

Deep Learning/Theory

Pierre Marion, Raphaël Berthier

We study the training dynamics of shallow neural networks, in a two-timescale regime in which the stepsizes for the inner layer are much smaller than those for the outer layer. In this regime, we prove convergence of the gradient flow to a global optimum of the non-convex optimization problem in a simple univariate setting. The number of neurons need not be asymptotically large for our result to hold, distinguishing our result from popular recent approaches such as the neural tangent kernel or mean-field regimes. Experimental illustration is provided, showing that the stochastic gradient descent behaves according to our description of the gradient flow and thus converges to a global optimum in the two-timescale regime, but can fail outside of this regime.