Convergence Analysis of Stochastic Gradient Descent with MCMC Estimators

• URL: http://arxiv.org/abs/2303.10599v2
• Date: Sat, 23 Mar 2024 13:26:31 GMT
• Title: Convergence Analysis of Stochastic Gradient Descent with MCMC Estimators
• Authors: Tianyou Li, Fan Chen, Huajie Chen, Zaiwen Wen,
• Abstract summary: gradient descent (SGD) and its variants is essential for machine learning. In this paper, we consider the SGD algorithm that employ the Markov Chain Monte Carlo (MCMC) estimator to compute the gradient. It is shown that MCMC-SGD escapes from saddle points and reaches $(epsilon,epsilon1/4)$ approximate second order stationary points.
• Score: 8.493584276672971
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Understanding stochastic gradient descent (SGD) and its variants is essential for machine learning. However, most of the preceding analyses are conducted under amenable conditions such as unbiased gradient estimator and bounded objective functions, which does not encompass many sophisticated applications, such as variational Monte Carlo, entropy-regularized reinforcement learning and variational inference. In this paper, we consider the SGD algorithm that employ the Markov Chain Monte Carlo (MCMC) estimator to compute the gradient, called MCMC-SGD. Since MCMC reduces the sampling complexity significantly, it is an asymptotically convergent biased estimator in practice. Moreover, by incorporating a general class of unbounded functions, it is much more difficult to analyze the MCMC sampling error. Therefore, we assume that the function is sub-exponential and use the Bernstein inequality for non-stationary Markov chains to derive error bounds of the MCMC estimator. Consequently, MCMC-SGD is proven to have a first order convergence rate $O(\log K/\sqrt{n K})$ with $K$ iterations and a sample size $n$. It partially explains how MCMC influences the behavior of SGD. Furthermore, we verify the correlated negative curvature condition under reasonable assumptions. It is shown that MCMC-SGD escapes from saddle points and reaches $(\epsilon,\epsilon^{1/4})$ approximate second order stationary points or $\epsilon^{1/2}$-variance points at least $O(\epsilon^{-11/2}\log^{2}(1/\epsilon) )$ steps with high probability. Our analysis unveils the convergence pattern of MCMC-SGD across a broad class of stochastic optimization problems, and interprets the convergence phenomena observed in practical applications.

Related papers

• Nonasymptotic Analysis of Stochastic Gradient Descent with the Richardson-Romberg Extrapolation [22.652143194356864]
  We address the problem of solving strongly convex and smooth problems using a descent gradient (SGD) algorithm with a constant step size. We provide an expansion of the mean-squared error of the resulting estimator with respect to the number iterations of $n$. We establish that this chain is geometrically ergodic with respect to a defined weighted Wasserstein semimetric.
  arXiv  Detail & Related papers  (2024-10-07T15:02:48Z)
• Stochastic Approximation with Biased MCMC for Expectation Maximization [9.4641739998927]
  An approximation scheme with Markov chain Monte Carlo can be used to create an algorithm known as MCMC-SAEM. MCMC-SAEM is often run withally biased MCMC algorithms, for which the consequences are theoretically less understood. We show that ULA is more stable with respect to the choice of Langevin stepsize and can sometimes result in faster convergence.
  arXiv  Detail & Related papers  (2024-02-27T20:10:03Z)
• Sharper Analysis for Minibatch Stochastic Proximal Point Methods: Stability, Smoothness, and Deviation [41.082982732100696]
  We study a minibatch variant of proximal point (SPP) methods, namely M-SPP, for solving convex composite risk minimization problems. We show that M-SPP with minibatch-size $n$ and quadratic count $T$ enjoys an in-expectation fast rate of convergence. In the small-$n$-large-$T$ setting, this result substantially improves the best known results of SPP-type approaches.
  arXiv  Detail & Related papers  (2023-01-09T00:13:34Z)
• On the Double Descent of Random Features Models Trained with SGD [78.0918823643911]
  We study properties of random features (RF) regression in high dimensions optimized by gradient descent (SGD) We derive precise non-asymptotic error bounds of RF regression under both constant and adaptive step-size SGD setting. We observe the double descent phenomenon both theoretically and empirically.
  arXiv  Detail & Related papers  (2021-10-13T17:47:39Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• Mean-Square Analysis with An Application to Optimal Dimension Dependence of Langevin Monte Carlo [60.785586069299356]
  This work provides a general framework for the non-asymotic analysis of sampling error in 2-Wasserstein distance. Our theoretical analysis is further validated by numerical experiments.
  arXiv  Detail & Related papers  (2021-09-08T18:00:05Z)
• Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity [49.66890309455787]
  We introduce the expected co-coercivity condition, explain its benefits, and provide the first last-iterate convergence guarantees of SGDA and SCO. We prove linear convergence of both methods to a neighborhood of the solution when they use constant step-size. Our convergence guarantees hold under the arbitrary sampling paradigm, and we give insights into the complexity of minibatching.
  arXiv  Detail & Related papers  (2021-06-30T18:32:46Z)
• Estimation in Tensor Ising Models [5.161531917413708]
  We consider the problem of estimating the natural parameter of the $p$-tensor Ising model given a single sample from the distribution on $N$ nodes. In particular, we show the $sqrt N$-consistency of the MPL estimate in the $p$-spin Sherrington-Kirkpatrick (SK) model. We derive the precise fluctuations of the MPL estimate in the special case of the $p$-tensor Curie-Weiss model.
  arXiv  Detail & Related papers  (2020-08-29T00:06:58Z)
• On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems [75.58134963501094]
  This paper analyzes the trajectories of gradient descent (SGD) We show that SGD avoids saddle points/manifolds with $1$ for strict step-size policies.
  arXiv  Detail & Related papers  (2020-06-19T14:11:26Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.