# Convergence Analysis of Stochastic Gradient Descent with MCMC Estimators

arXiv: https://arxiv.org/abs/2303.10599 · HTML: https://mlsanity.com/paper/2303.10599/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 50 references that are in the map.
- First posted on arXiv 19 Mar 2023.

## Builds on (most cited first, 16 of 16)

- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Stochastic Gradient Descent, Weighted Sampling, and the Randomized Kaczmarz algorithm](https://mlsanity.com/paper/1310.5715/) (2013)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Filtering Variational Objectives](https://mlsanity.com/paper/1705.09279/) (2017)
- [On the Convergence of SGD with Biased Gradients](https://mlsanity.com/paper/2008.00051/) (2020)
- [Escaping Saddles with Stochastic Gradients](https://mlsanity.com/paper/1803.05999/) (2018)
- [On Markov Chain Gradient Descent](https://mlsanity.com/paper/1809.04216/) (2018)
- [Non-asymptotic Analysis of Biased Stochastic Approximation Scheme](https://mlsanity.com/paper/1902.00629/) (2019)
- [Neural-Network Quantum States, String-Bond States, and Chiral Topological States](https://mlsanity.com/paper/1710.04045/) (2017)
- [Approximate Inference with Amortised MCMC](https://mlsanity.com/paper/1702.08343/) (2017)
- [Learning hard quantum distributions with variational autoencoders](https://mlsanity.com/paper/1710.00725/) (2017)

## Built on by (most cited first, 3 of 3)

- [Accurate Ab-initio Neural-network Solutions to Large-Scale Electronic Structure Problems](https://mlsanity.com/paper/2504.06087/) (2025)
- [DIPLI: Deep Image Prior Lucky Imaging for Blind Astronomical Image Restoration](https://mlsanity.com/paper/2503.15984/) (2025)
- [Is Variational Monte Carlo Robust? Sharp Moment Thresholds and Heavy-tailed Stochastic Optimization](https://mlsanity.com/paper/2606.26009/) (2026)

## Abstract (arXiv)

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/√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 (ε,ε1/4) approximate second order stationary points or ε1/2-variance points at least O(ε-11/2log2(1/ε) ) 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.