Motivated by broad applications in reinforcement learning and machine learning, this paper considers the popular stochastic gradient descent (SGD) when the gradients of the underlying objective function are sampled from Markov processes. This Markov sampling leads to the gradient samples being biased and not independent. The existing results for the convergence of SGD under Markov randomness are often established under the assumptions on the boundedness of either the iterates or the gradient samples. Our main focus is to study the finite-time convergence of SGD for different types of objective functions, without requiring these assumptions. We show that SGD converges nearly at the same rate with Markovian gradient samples as with independent gradient samples. The only difference is a logarithmic factor that accounts for the mixing time of the Markov chain.
University of Alberta（阿尔伯塔大学）；Alberta Machine Intelligence Institute (Amii)（阿尔伯塔机器智能研究所）；Deepmind（深度思维）
The Backprop algorithm for learning in neural networks utilizes two mechanisms: first, stochastic gradient descent and second, initialization with small random weights, where the latter is essential to the effectiveness of the former. We show that in continual learning setups, Backprop performs well initially, but over time its performance degrades. Stochastic gradient descent alone is insufficient to learn continually; the initial randomness enables only initial learning but not continual learning. To the best of our knowledge, ours is the first result showing this degradation in Backprop's a…
Georgia Institute of Technology（佐治亚理工学院）；Virginia Tech（弗吉尼亚理工大学）
We study the problem of finding the Nash equilibrium in a two-player zero-sum Markov game. Due to its formulation as a minimax optimization program, a natural approach to solve the problem is to perform gradient descent/ascent with respect to each player in an alternating fashion. However, due to the non-convexity/non-concavity of the underlying objective function, theoretical understandings of this method are limited. In our paper, we consider solving an entropy-regularized variant of the Markov game. The regularization introduces structure into the optimization landscape that make the soluti…
The massive size of modern neural networks has motivated substantial recent interest in neural network quantization. We introduce Stochastic Markov Gradient Descent (SMGD), a discrete optimization method applicable to training quantized neural networks. The SMGD algorithm is designed for settings where memory is highly constrained during training. We provide theoretical guarantees of algorithm performance as well as encouraging numerical results.
National University of Defense Technology（国防科技大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
The method of block coordinate gradient descent (BCD) has been a powerful method for large-scale optimization. This paper considers the BCD method that successively updates a series of blocks selected according to a Markov chain. This kind of block selection is neither i.i.d. random nor cyclic. On the other hand, it is a natural choice for some applications in distributed optimization and Markov decision process, where i.i.d. random and cyclic selections are either infeasible or very expensive. By applying mixing-time properties of a Markov chain, we prove convergence of Markov chain BCD for m…
Microsoft Research（微软研究院）；University of Washington（华盛顿大学）；Stanford University（斯坦福大学）
This work provides a simplified proof of the statistical minimax optimality of (iterate averaged) stochastic gradient descent (SGD), for the special case of least squares. This result is obtained by analyzing SGD as a stochastic process and by sharply characterizing the stationary covariance matrix of this process. The finite rate optimality characterization captures the constant factors and addresses model mis-specification.
Decentralized stochastic gradient method emerges as a promising solution for solving large-scale machine learning problems. This paper studies the decentralized Markov chain gradient descent (DMGD) algorithm - a variant of the decentralized stochastic gradient methods where the random samples are taken along the trajectory of a Markov chain. This setting is well-motivated when obtaining independent samples is costly or impossible, which excludes the use of the traditional stochastic gradient algorithms. Specifically, we consider the first- and zeroth-order versions of decentralized Markov chai…
Institute for Mathematical Methods in Medicine（医学数学方法研究所）；Data-Based Modeling Johannes Kepler University Linz, Altenberger Strasse 69, A-4020 Linz, Austria（基于数据的建模约翰· Kepler大学林茨分校）
Stochastic Gradient Descent Langevin Dynamics (SGLD) algorithms, which add noise to the classic gradient descent, are known to improve the training of neural networks in some cases where the neural network is very deep. In this paper we study the possibilities of training acceleration for the numerical resolution of stochastic control problems through gradient descent, where the control is parametrized by a neural network. If the control is applied at many discretization times then solving the stochastic control problem reduces to minimizing the loss of a very deep neural network. We numerical…
We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical systems perspective in which appropriate instantiations of the Stable Manifold Theorem allow for a global stability analysis. Thus, neither access to second-order derivative information nor randomness beyond initialization is necessary to provably avoid saddle points.
We study infinite-horizon discounted two-player zero-sum Markov games, and develop a decentralized algorithm that provably converges to the set of Nash equilibria under self-play. Our algorithm is based on running an Optimistic Gradient Descent Ascent algorithm on each state to learn the policies, with a critic that slowly learns the value of each state. To the best of our knowledge, this is the first algorithm in this setting that is simultaneously rational (converging to the opponent's best response when it uses a stationary policy), convergent (converging to the set of Nash equilibria under…
California Institute of Technology（加州理工学院）；Georgia Institute of Technology（佐治亚理工学院）；University of California San Diego（加利福尼亚大学圣迭戈分校）；Nvidia（英伟达）
Many economic games and machine learning approaches can be cast as competitive optimization problems where multiple agents are minimizing their respective objective function, which depends on all agents' actions. While gradient descent is a reliable basic workhorse for single-agent optimization, it often leads to oscillation in competitive optimization. In this work we propose polymatrix competitive gradient descent (PCGD) as a method for solving general sum competitive optimization involving arbitrary numbers of agents. The updates of our method are obtained as the Nash equilibria of a local …
Classical learning theory suggests that the optimal generalization performance of a machine learning model should occur at an intermediate model complexity, with simpler models exhibiting high bias and more complex models exhibiting high variance of the predictive function. However, such a simple trade-off does not adequately describe deep learning models that simultaneously attain low bias and variance in the heavily overparameterized regime. A primary obstacle in explaining this behavior is that deep learning algorithms typically involve multiple sources of randomness whose individual contri…
North Carolina State University（北卡罗来纳州立大学）；IQVIA（艾昆纬）
We consider the stochastic gradient descent (SGD) algorithm driven by a general stochastic sequence, including i.i.d noise and random walk on an arbitrary graph, among others; and analyze it in the asymptotic sense. Specifically, we employ the notion of `efficiency ordering', a well-analyzed tool for comparing the performance of Markov Chain Monte Carlo (MCMC) samplers, for SGD algorithms in the form of Loewner ordering of covariance matrices associated with the scaled iterate errors in the long term. Using this ordering, we show that input sequences that are more efficient for MCMC sampling a…
We study generalization properties of random features (RF) regression in high dimensions optimized by stochastic gradient descent (SGD) in under-/over-parameterized regime. In this work, we derive precise non-asymptotic error bounds of RF regression under both constant and polynomial-decay step-size SGD setting, and observe the double descent phenomenon both theoretically and empirically. Our analysis shows how to cope with multiple randomness sources of initialization, label noise, and data sampling (as well as stochastic gradients) with no closed-form solution, and also goes beyond the commo…
Sogang University（西江大学）；University of Pennsylvania（宾夕法尼亚大学）；Princeton University（普林斯顿大学）
Minimizing the inclusive Kullback-Leibler (KL) divergence with stochastic gradient descent (SGD) is challenging since its gradient is defined as an integral over the posterior. Recently, multiple methods have been proposed to run SGD with biased gradient estimates obtained from a Markov chain. This paper provides the first non-asymptotic convergence analysis of these methods by establishing their mixing rate and gradient variance. To do this, we demonstrate that these methods-which we collectively refer to as Markov chain score ascent (MCSA) methods-can be cast as special cases of the Markov c…
University of Chicago（芝加哥大学）；New York University（纽约大学）
The stochastic gradient descent (SGD) algorithm is widely used for parameter estimation, especially for huge data sets and online learning. While this recursive algorithm is popular for computation and memory efficiency, quantifying variability and randomness of the solutions has been rarely studied. This paper aims at conducting statistical inference of SGD-based estimates in an online setting. In particular, we propose a fully online estimator for the covariance matrix of averaged SGD iterates (ASGD) only using the iterates from SGD. We formally establish our online estimator's consistency a…
We develop two fundamental stochastic sketching techniques; Penalty Sketching (PS) and Augmented Lagrangian Sketching (ALS) for solving consistent linear systems. The proposed PS and ALS techniques extend and generalize the scope of Sketch & Project (SP) method by introducing Lagrangian penalty sketches. In doing so, we recover SP methods as special cases and furthermore develop a family of new stochastic iterative methods. By varying sketch parameters in the proposed PS method, we recover novel stochastic methods such as Penalty Newton Descent, Penalty Kaczmarz, Penalty Stochastic Descent, Pe…
This paper addresses the issue of policy evaluation in Markov Decision Processes, using linear function approximation. It provides a unified view of algorithms such as TD(lambda), LSTD(lambda), iLSTD, residual-gradient TD. It is asserted that they all consist in minimizing a gradient function and differ by the form of this function and their means of minimizing it. Two new schemes are introduced in that framework: Full-gradient TD which uses a generalization of the principle introduced in iLSTD, and EGD TD, which reduces the gradient by successive equi-gradient descents. These three algorithms…
Momentum is a popular technique for improving convergence rates during gradient descent. In this research, we experiment with adding momentum to the Baum-Welch expectation-maximization algorithm for training Hidden Markov Models. We compare discrete Hidden Markov Models trained with and without momentum on English text and malware opcode data. The effectiveness of momentum is determined by measuring the changes in model score and classification accuracy due to momentum. Our extensive experiments indicate that adding momentum to Baum-Welch can reduce the number of iterations required for initia…
Massachusetts Institute of Technology（麻省理工学院）；Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
We study the problem of least squares linear regression where the data-points are dependent and are sampled from a Markov chain. We establish sharp information theoretic minimax lower bounds for this problem in terms of $τ_{\mathsf{mix}}$, the mixing time of the underlying Markov chain, under different noise settings. Our results establish that in general, optimization with Markovian data is strictly harder than optimization with independent data and a trivial algorithm (SGD-DD) that works with only one in every $\tildeΘ(τ_{\mathsf{mix}})$ samples, which are approximately independent, is minim…
We consider coordinate descent (CD) methods with exact line search on convex quadratic problems. Our main focus is to study the performance of the CD method that use random permutations in each epoch and compare it to the performance of the CD methods that use deterministic orders and random sampling with replacement. We focus on a class of convex quadratic problems with a diagonally dominant Hessian matrix, for which we show that using random permutations instead of random with-replacement sampling improves the performance of the CD method in the worst-case. Furthermore, we prove that as the …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Department of Mathematics（数学系）；Madison, WI 53706（威斯康星州麦迪逊市 53706（注：该地址非机构，此处仅按要求提取原文中明确出现的机构相关内容））
Sampling from a log-concave distribution function is one core problem that has wide applications in Bayesian statistics and machine learning. While most gradient free methods have slow convergence rate, the Langevin Monte Carlo (LMC) that provides fast convergence requires the computation of gradients. In practice one uses finite-differencing approximations as surrogates, and the method is expensive in high-dimensions. A natural strategy to reduce computational cost in each iteration is to utilize random gradient approximations, such as random coordinate descent (RCD) or simultaneous perturbat…
Boston College（波士顿学院）；Johns Hopkins University（约翰斯·霍普金斯大学）
The time to converge to the steady state of a finite Markov chain can be greatly reduced by a lifting operation, which creates a new Markov chain on an expanded state space. For a class of quadratic objectives, we show an analogous behavior where a distributed ADMM algorithm can be seen as a lifting of Gradient Descent algorithm. This provides a deep insight for its faster convergence rate under optimal parameter tuning. We conjecture that this gain is always present, as opposed to the lifting of a Markov chain which sometimes only provides a marginal speedup.
Department of Statistics and Data Science, Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）；Department of Statistics, Purdue University（普渡大学统计系）
We introduce a novel and efficient algorithm called the stochastic approximate gradient descent (SAGD), as an alternative to the stochastic gradient descent for cases where unbiased stochastic gradients cannot be trivially obtained. Traditional methods for such problems rely on general-purpose sampling techniques such as Markov chain Monte Carlo, which typically requires manual intervention for tuning parameters and does not work efficiently in practice. Instead, SAGD makes use of the Langevin algorithm to construct stochastic gradients that are biased in finite steps but accurate asymptotical…
Central Limit Theorems for Stochastic Gradient Descent Quantile Estimators
Department of Statistics, University of Chicago（芝加哥大学统计系）；Department of Statistics and Data Science, Washington University in St. Louis（圣路易斯华盛顿大学统计与数据科学系）
We study the convergence of a random iterative sequence of a family of operators on infinite dimensional Hilbert spaces, inspired by the Stochastic Gradient Descent (SGD) algorithm in the case of the noiseless regression, as studied in [1]. We identify conditions that are strictly broader than previously known for polynomial convergence rate in various norms, and characterize the roles the randomness plays in determining the best multiplicative constants. Additionally, we prove almost sure convergence of the sequence.
Imperial College London（帝国理工学院）；University of Oxford（牛津大学）
Stein variational gradient descent (SVGD) is a deterministic particle inference algorithm that provides an efficient alternative to Markov chain Monte Carlo. However, SVGD has been found to suffer from variance underestimation when the dimensionality of the target distribution is high. Recent developments have advocated projecting both the score function and the data onto real lines to sidestep this issue, although this can severely overestimate the epistemic (model) uncertainty. In this work, we propose Grassmann Stein variational gradient descent (GSVGD) as an alternative approach, which per…
Stochastic gradient descent is an optimisation method that combines classical gradient descent with random subsampling within the target functional. In this work, we introduce the stochastic gradient process as a continuous-time representation of stochastic gradient descent. The stochastic gradient process is a dynamical system that is coupled with a continuous-time Markov process living on a finite state space. The dynamical system -- a gradient flow -- represents the gradient descent part, the process on the finite state space represents the random subsampling. Processes of this type are, fo…
Stochastic gradient methods are the workhorse (algorithms) of large-scale optimization problems in machine learning, signal processing, and other computational sciences and engineering. This paper studies Markov chain gradient descent, a variant of stochastic gradient descent where the random samples are taken on the trajectory of a Markov chain. Existing results of this method assume convex objectives and a reversible Markov chain and thus have their limitations. We establish new non-ergodic convergence under wider step sizes, for nonconvex problems, and for non-reversible finite-state Markov…
Motivated by broad applications in machine learning, we study the popular accelerated stochastic gradient descent (ASGD) algorithm for solving (possibly nonconvex) optimization problems. We characterize the finite-time performance of this method when the gradients are sampled from Markov processes, and hence biased and dependent from time step to time step; in contrast, the analysis in existing work relies heavily on the stochastic gradients being independent and sometimes unbiased. Our main contributions show that under certain (standard) assumptions on the underlying Markov chain generating …
We consider the minimization of an objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step-size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged SGD iterates that outlines the dependence on initial conditions, the effect of noise and the step-size, as well as the lack of convergence in the general (non-quadratic) case. For this analysis, we bring tools from Markov chain theory into the analysis of stochastic gradient. We t…
We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, but instead receive samples that are coupled over time. We show that as long as the source of randomness is suitably ergodic---it converges quickly enough to a stationary distribution---the method enjoys strong convergence guarantees, both in expectation and with high probability. This result has implications for stochastic optimization in high-dimensional spaces, peer-to-peer distributed optimization schemes, decision problems with dep…
In this paper, we show that the Quadratic Lyapunov function based Algorithm (QLA, also known as MaxWeight or Backpressure) achieves an exact $[O(1/V), O(V)]$ utility-delay tradeoff in stochastic network optimization problems with Markovian network dynamics. Note that though the QLA algorithm has been extensively studied, most of the performance results are obtained under i.i.d. network radnomness, and it has not been formally proven that QLA achieves the exact $[O(1/V), O(V)]$ utility-delay tradeoff under Markov dynamics. Our analysis uses a combination of duality theory and a variable multi-s…
We consider the problem of learning the optimal policy for infinite-horizon Markov decision processes (MDPs). For this purpose, some variant of Stochastic Mirror Descent is proposed for convex programming problems with Lipschitz-continuous functionals. An important detail is the ability to use inexact values of functional constraints and compute the value of dual variables. We analyze this algorithm in a general case and obtain an estimate of the convergence rate that does not accumulate errors during the operation of the method. Using this algorithm, we get the first parallel algorithm for mi…
For model-free reinforcement learning, one of the main difficulty of stochastic Bellman residual minimization is the double sampling problem, i.e., while only one single sample for the next state is available in the model-free setting, two independent samples for the next state are required in order to perform unbiased stochastic gradient descent. We propose new algorithms for addressing this problem based on the idea of borrowing extra randomness from the future. When the transition kernel varies slowly with respect to the state, it is shown that the training trajectory of new algorithms is c…
