# Effect of Random Learning Rate: Theoretical Analysis of SGD Dynamics in Non-Convex Optimization via Stationary Distribution

arXiv: https://arxiv.org/abs/2406.16032 · HTML: https://mlsanity.com/paper/2406.16032/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 48 references that are in the map.
- First posted on arXiv 23 Jun 2024.

## Builds on (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Train longer, generalize better: closing the generalization gap in large batch training of neural networks](https://mlsanity.com/paper/1705.08741/) (2017)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Stochastic modified equations and adaptive stochastic gradient algorithms](https://mlsanity.com/paper/1511.06251/) (2015)
- [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017)
- [Shape Matters: Understanding the Implicit Bias of the Noise Covariance](https://mlsanity.com/paper/2006.08680/) (2020)
- [First Exit Time Analysis of Stochastic Gradient Descent Under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/1906.09069/) (2019)
- [Revisiting the Noise Model of Stochastic Gradient Descent](https://mlsanity.com/paper/2303.02749/) (2023)
- [The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method](https://mlsanity.com/paper/1510.02451/) (2015)
- [Analysis of Stochastic Gradient Descent in Continuous Time](https://mlsanity.com/paper/2004.07177/) (2020)
- [Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization](https://mlsanity.com/paper/2111.06328/) (2021)
- [Learning with Random Learning Rates](https://mlsanity.com/paper/1810.01322/) (2018)
- [A Discrete Bouncy Particle Sampler](https://mlsanity.com/paper/1707.05200/) (2017)
- [Stochastic gradient descent with random learning rate](https://mlsanity.com/paper/2003.06926/) (2020)
- [Approximations of Piecewise Deterministic Markov Processes and their convergence properties](https://mlsanity.com/paper/2109.11827/) (2021)

## Abstract (arXiv)

We consider a variant of the stochastic gradient descent (SGD) with a random learning rate and reveal its convergence properties. SGD is a widely used stochastic optimization algorithm in machine learning, especially deep learning. Numerous studies reveal the convergence properties of SGD and its theoretically favorable variants. Among these, the analysis of convergence using a stationary distribution of updated parameters provides generalizable results. However, to obtain a stationary distribution, the update direction of the parameters must not degenerate, which limits the applicable variants of SGD. In this study, we consider a novel SGD variant, Poisson SGD, which has degenerated parameter update directions and instead utilizes a random learning rate. Consequently, we demonstrate that a distribution of a parameter updated by Poisson SGD converges to a stationary distribution under weak assumptions on a loss function. Based on this, we further show that Poisson SGD finds global minima in non-convex optimization problems and also evaluate the generalization error using this method. As a proof technique, we approximate the distribution by Poisson SGD with that of the bouncy particle sampler (BPS) and derive its stationary distribution, using the theoretical advance of the piece-wise deterministic Markov process (PDMP).