An Elementary Approach to Convergence Guarantees of Optimization Algorithms for Deep Networks
University of Washington（华盛顿大学）
Vincent Roulet, Zaid Harchaoui
备注：
The changes from v1 to v2 include i) slightly more general results; ii) slightly more concise proofs; iii) highway and residual networks; iv) implicitly defined network layers; v) additional algorithm boxes and illustration figures
We present an approach to obtain convergence guarantees of optimization algorithms for deep networks based on elementary arguments and computations. The convergence analysis revolves around the analytical and computational structures of optimization oracles central to the implementation of deep networks in machine learning software. We provide a systematic way to compute estimates of the smoothness constants that govern the convergence behavior of first-order optimization algorithms used to train deep networks. A diverse set of example components and architectures arising in modern deep networks intersperse the exposition to illustrate the approach.
We provide statistical guarantees for Bayesian variational boosting by proposing a novel small bandwidth Gaussian mixture variational family. We employ a functional version of Frank-Wolfe optimization as our variational algorithm and study frequentist properties of the iterative boosting updates. Comparisons are drawn to the recent literature on boosting, describing how the choice of the variational family and the discrepancy measure affect both convergence and finite-sample statistical properties of the optimization routine. Specifically, we first demonstrate stochastic boundedness of the boo…
DPOT: A DeepParticle method for Computation of Optimal Transport with convergence guarantee
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学学院）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学数学科学学院）
Gibbs sampling is a Markov chain Monte Carlo method that is often used for learning and inference on graphical models. Minibatching, in which a small random subset of the graph is used at each iteration, can help make Gibbs sampling scale to large graphical models by reducing its computational cost. In this paper, we propose a new auxiliary-variable minibatched Gibbs sampling method, {\it Poisson-minibatching Gibbs}, which both produces unbiased samples and has a theoretical guarantee on its convergence rate. In comparison to previous minibatched Gibbs algorithms, Poisson-minibatching Gibbs su…
MPI for Intelligent Systems（智能系统马克斯·普朗克研究所）；ETH Zurich（苏黎世联邦理工学院）；EPFL（洛桑联邦理工学院）
Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical perfor…
Université de Montréal（蒙特利尔大学）；Ahmedabad University（艾哈迈达巴德大学）；Pandit Deendayal Petroleum University（潘迪特·迪恩达亚尔石油大学）
In this paper, we perform Lyapunov based analysis of the loss function to derive an a priori upper bound on the settling time of deep neural networks. While previous studies have attempted to understand deep learning using control theory framework, there is limited work on a priori finite time convergence analysis. Drawing from the advances in analysis of finite-time control of non-linear systems, we provide a priori guarantees of finite-time convergence in a deterministic control theoretic setting. We formulate the supervised learning framework as a control problem where weights of the networ…
CyberAgent, Inc.（赛博艾坚特公司）；University of Electro-Communications（电气通信大学）
In this study, we consider a variant of the Follow the Regularized Leader (FTRL) dynamics in two-player zero-sum games. FTRL is guaranteed to converge to a Nash equilibrium when time-averaging the strategies, while a lot of variants suffer from the issue of limit cycling behavior, i.e., lack the last-iterate convergence guarantee. To this end, we propose mutant FTRL (M-FTRL), an algorithm that introduces mutation for the perturbation of action probabilities. We then investigate the continuous-time dynamics of M-FTRL and provide the strong convergence guarantees toward stationary points that ap…
We consider the problem of finding critical points of functions that are non-convex and non-smooth. Studying a fairly broad class of such problems, we analyze the behavior of three gradient-based methods (gradient descent, proximal update, and Frank-Wolfe update). For each of these methods, we establish rates of convergence for general problems, and also prove faster rates for continuous sub-analytic functions. We also show that our algorithms can escape strict saddle points for a class of non-smooth functions, thereby generalizing known results for smooth functions. Our analysis leads to a si…
The fragility of deep neural networks to adversarially-chosen inputs has motivated the need to revisit deep learning algorithms. Including adversarial examples during training is a popular defense mechanism against adversarial attacks. This mechanism can be formulated as a min-max optimization problem, where the adversary seeks to maximize the loss function using an iterative first-order algorithm while the learner attempts to minimize it. However, finding adversarial examples in this way causes excessive computational overhead during training. By interpreting the min-max problem as an optimal…
University of Chicago（芝加哥大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Steklov Mathematical Institute（斯捷克洛夫数学研究所）
We continue a long line of research aimed at proving convergence of depth 2 neural networks, trained via gradient descent, to a global minimum. Like in many previous works, our model has the following features: regression with quadratic loss function, fully connected feedforward architecture, RelU activations, Gaussian data instances and network initialization, adversarial labels. It is more general in the sense that we allow both layers to be trained simultaneously and at {\em different} rates. Our results improve on state-of-the-art [Oymak Soltanolkotabi 20] (training the first layer only) a…
Simple Stepsize for Quasi-Newton Methods with Global Convergence Guarantees
Mohamed bin Zayed University of Artificial Intelligence (MBZUAI)（Mohamed bin Zayed大学人工智能学院）；Moscow Institute of Physics and Technology (MIPT)（莫斯科物理技术研究所）；Innopolis University（Innopolis大学）；Skoltech
Provably Efficient Off-Policy Adversarial Imitation Learning with Convergence Guarantees
Division of Systems Engineering（系统工程系）；Department of Computer Science（计算机科学系）；Amazon Robotics（亚马逊机器人）；Department of Electrical and Computer Engineering（电气与计算机工程系）
Existing global convergence guarantees of (stochastic) gradient descent do not apply to practical deep networks in the practical regime of deep learning beyond the neural tangent kernel (NTK) regime. This paper proposes an algorithm, which is ensured to have global convergence guarantees in the practical regime beyond the NTK regime, under a verifiable condition called the expressivity condition. The expressivity condition is defined to be both data-dependent and architecture-dependent, which is the key property that makes our results applicable for practical settings beyond the NTK regime. On…
It is promising to solve linear inverse problems by unfolding iterative algorithms (e.g., iterative shrinkage thresholding algorithm (ISTA)) as deep neural networks (DNNs) with learnable parameters. However, existing ISTA-based unfolded algorithms restrict the network architectures for iterative updates with the partial weight coupling structure to guarantee convergence. In this paper, we propose hybrid ISTA to unfold ISTA with both pre-computed and learned parameters by incorporating free-form DNNs (i.e., DNNs with arbitrary feasible and reasonable network architectures), while ensuring theor…
Mila and DIRO, Université de Montréal（蒙特利尔大学 Mila 与 DIRO）
Two of the most prominent algorithms for solving unconstrained smooth games are the classical stochastic gradient descent-ascent (SGDA) and the recently introduced stochastic consensus optimization (SCO) [Mescheder et al., 2017]. SGDA is known to converge to a stationary point for specific classes of games, but current convergence analyses require a bounded variance assumption. SCO is used successfully for solving large-scale adversarial problems, but its convergence guarantees are limited to its deterministic variant. In this work, we introduce the expected co-coercivity condition, explain it…
In a recent joint work, the author has developed a modification of Newton's method, named New Q-Newton's method, which can avoid saddle points and has quadratic rate of convergence. While good theoretical convergence guarantee has not been established for this method, experiments on small scale problems show that the method works very competitively against other well known modifications of Newton's method such as Adaptive Cubic Regularization and BFGS, as well as first order methods such as Unbounded Two-way Backtracking Gradient Descent. In this paper, we resolve the convergence guarantee iss…
EURECOM（欧洲电信学院）；University of Tübingen（图宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Adaptive Bayesian quadrature (ABQ) is a powerful approach to numerical integration that empirically compares favorably with Monte Carlo integration on problems of medium dimensionality (where non-adaptive quadrature is not competitive). Its key ingredient is an acquisition function that changes as a function of previously collected values of the integrand. While this adaptivity appears to be empirically powerful, it complicates analysis. Consequently, there are no theoretical guarantees so far for this class of methods. In this work, for a broad class of adaptive Bayesian quadrature methods, w…
In this paper, we consider the problem of distributed pose graph optimization (PGO) that has extensive applications in multi-robot simultaneous localization and mapping (SLAM). We propose majorization minimization methods to distributed PGO and show that our proposed methods are guaranteed to converge to first-order critical points under mild conditions. Furthermore, since our proposed methods rely a proximal operator of distributed PGO, the convergence rate can be significantly accelerated with Nesterov's method, and more importantly, the acceleration induces no compromise of theoretical guar…
Computational imaging has been revolutionized by compressed sensing algorithms, which offer guaranteed uniqueness, convergence, and stability properties. Model-based deep learning methods that combine imaging physics with learned regularization priors have emerged as more powerful alternatives for image recovery. The main focus of this paper is to introduce a memory efficient model-based algorithm with similar theoretical guarantees as CS methods. The proposed iterative algorithm alternates between a gradient descent involving the score function and a conjugate gradient algorithm to encourage …
Massachusetts Institute of Technology（麻省理工学院）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of California, Berkeley（加州大学伯克利分校）
Min-max optimization problems involving nonconvex-nonconcave objectives have found important applications in adversarial training and other multi-agent learning settings. Yet, no known gradient descent-based method is guaranteed to converge to (even local notions of) min-max equilibrium in the nonconvex-nonconcave setting. For all known methods, there exist relatively simple objectives for which they cycle or exhibit other undesirable behavior different from converging to a point, let alone to some game-theoretically meaningful one~\cite{flokas2019poincare,hsieh2021limits}. The only known conv…
Greedy Low-Rank Gradient Compression for Distributed Learning with Convergence Guarantees
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Center for Data Science, Peking University（北京大学数据科学中心）；School of Software, Beihang University（北航软件学院）；School of Life Sciences, Peking University（北京大学生命科学学院）；Center for Machine Learning Research, Peking University（北京大学机器学习研究中心）
Subspace Optimization for Large Language Models with Convergence Guarantees
Peking University（北京大学）；Zhongguancun Academy（中关村学院）；Beihang University（北航大学）；AI for Science Institute, Beijing, China（人工智能科学研究院，北京，中国）；National Engineering Laboratory for Big Data Analytics（大数据分析与应用国家工程实验室）
Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Many supervised machine learning methods are naturally cast as optimization problems. For prediction models which are linear in their parameters, this often leads to convex problems for which many mathematical guarantees exist. Models which are non-linear in their parameters such as neural networks lead to non-convex optimization problems for which guarantees are harder to obtain. In this review paper, we consider two-layer neural networks with homogeneous activation functions where the number of hidden neurons tends to infinity, and show how qualitative convergence guarantees may be derived.
Backpropagation algorithm is indispensable for the training of feedforward neural networks. It requires propagating error gradients sequentially from the output layer all the way back to the input layer. The backward locking in backpropagation algorithm constrains us from updating network layers in parallel and fully leveraging the computing resources. Recently, several algorithms have been proposed for breaking the backward locking. However, their performances degrade seriously when networks are deep. In this paper, we propose decoupled parallel backpropagation algorithm for deep learning opt…
We investigate online nonlinear regression with continually running recurrent neural network networks (RNNs), i.e., RNN-based online learning. For RNN-based online learning, we introduce an efficient first-order training algorithm that theoretically guarantees to converge to the optimum network parameters. Our algorithm is truly online such that it does not make any assumption on the learning environment to guarantee convergence. Through numerical simulations, we verify our theoretical results and illustrate significant performance improvements achieved by our algorithm with respect to the sta…
We present a matrix factorization algorithm that scales to input matrices that are large in both dimensions (i.e., that contains morethan 1TB of data). The algorithm streams the matrix columns while subsampling them, resulting in low complexity per iteration andreasonable memory footprint. In contrast to previous online matrix factorization methods, our approach relies on low-dimensional statistics from past iterates to control the extra variance introduced by subsampling. We present a convergence analysis that guarantees us to reach a stationary point of the problem. Large speed-ups can be ob…
The Ohio State University（俄亥俄州立大学）；National University of Singapore（新加坡国立大学）；Southern University of Science and Technology (SUSTech)（南方科技大学）
Existing studies indicate that momentum ideas in conventional optimization can be used to improve the performance of Q-learning algorithms. However, the finite-sample analysis for momentum-based Q-learning algorithms is only available for the tabular case without function approximations. This paper analyzes a class of momentum-based Q-learning algorithms with finite-sample guarantee. Specifically, we propose the MomentumQ algorithm, which integrates the Nesterov's and Polyak's momentum schemes, and generalizes the existing momentum-based Q-learning algorithms. For the infinite state-action spa…
The objective of this study is to address the problem of background/foreground separation with missing pixels by combining the video acquisition, video recovery, background/foreground separation into a single framework. To achieve this, a smooth robust tensor completion (SRTC) model is proposed to recover the data and decompose it into the static background and smooth foreground, respectively. Specifically, the static background is modeled by the low-rank tucker decomposition and the smooth foreground (moving objects) is modeled by the spatiotemporal continuity, which is enforced by the total …
Adam-family Methods for Nonsmooth Optimization with Convergence Guarantees
Institute of Operations Research and Analytics（运筹学与分析研究所）；National University of Singapore（新加坡国立大学）；School of Computer and Computing Science（计算机与计算科学学院）；Hangzhou City University（杭州城市大学）；State Key Laboratory of Scientific and Engineering Computing（科学与工程计算国家重点实验室）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Department of Mathematics（数学系）
We propose a single time-scale actor-critic algorithm to solve the linear quadratic regulator (LQR) problem. A least squares temporal difference (LSTD) method is applied to the critic and a natural policy gradient method is used for the actor. We give a proof of convergence with sample complexity $\mathcal{O}(\varepsilon^{-1} \log(\varepsilon^{-1})^2)$. The method in the proof is applicable to general single time-scale bilevel optimization problem. We also numerically validate our theoretical results on the convergence.
Washington University in St. Louis（圣路易斯华盛顿大学）；California Institute of Technology（加州理工学院）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Plug-and-play priors (PnP) is a broadly applicable methodology for solving inverse problems by exploiting statistical priors specified as denoisers. Recent work has reported the state-of-the-art performance of PnP algorithms using pre-trained deep neural nets as denoisers in a number of imaging applications. However, current PnP algorithms are impractical in large-scale settings due to their heavy computational and memory requirements. This work addresses this issue by proposing an incremental variant of the widely used PnP-ADMM algorithm, making it scalable to large-scale datasets. We theoret…
Courant Institute of Mathematical Sciences（库朗数学科学研究所）；New York University（纽约大学）；Center for Data Science（数据科学中心）
We study the optimization of wide neural networks (NNs) via gradient flow (GF) in setups that allow feature learning while admitting non-asymptotic global convergence guarantees. First, for wide shallow NNs under the mean-field scaling and with a general class of activation functions, we prove that when the input dimension is no less than the size of the training set, the training loss converges to zero at a linear rate under GF. Building upon this analysis, we study a model of wide multi-layer NNs whose second-to-last layer is trained via GF, for which we also prove a linear-rate convergence …
When designing algorithms for finite-time-horizon episodic reinforcement learning problems, a common approach is to introduce a fictitious discount factor and use stationary policies for approximations. Empirically, it has been shown that the fictitious discount factor helps reduce variance, and stationary policies serve to save the per-iteration computational cost. Theoretically, however, there is no existing work on convergence analysis for algorithms with this fictitious discount recipe. This paper takes the first step towards analyzing these algorithms. It focuses on two vanilla policy gra…
We prove a two-way correspondence between the min-max optimization of general CPE loss function GANs and the minimization of associated $f$-divergences. We then focus on $α$-GAN, defined via the $α$-loss, which interpolates several GANs (Hellinger, vanilla, Total Variation) and corresponds to the minimization of the Arimoto divergence. We show that the Arimoto divergences induced by $α$-GAN equivalently converge, for all $α\in \mathbb{R}_{>0}\cup\{\infty\}$. However, under restricted learning models and finite samples, we provide estimation bounds which indicate diverse GAN behavior as a funct…
Recently, invariant risk minimization (IRM) (Arjovsky et al.) was proposed as a promising solution to address out-of-distribution (OOD) generalization. In Ahuja et al., it was shown that solving for the Nash equilibria of a new class of "ensemble-games" is equivalent to solving IRM. In this work, we extend the framework in Ahuja et al. for linear regressions by projecting the ensemble-game on an $\ell_{\infty}$ ball. We show that such projections help achieve non-trivial OOD guarantees despite not achieving perfect invariance. For linear models with confounders, we prove that Nash equilibria o…
