# Convergence Analysis of Stochastic Saddle Point Mirror Descent Algorithm -- A Projected Dynamical View Point

arXiv: https://arxiv.org/abs/2404.04907 · HTML: https://mlsanity.com/paper/2404.04907/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 16 of its 52 references that are in the map.
- First posted on arXiv 7 Apr 2024.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

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

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [A Primer on Zeroth-Order Optimization in Signal Processing and Machine Learning](https://mlsanity.com/paper/2006.06224/) (2020)
- [Stochastic Variance Reduction Methods for Policy Evaluation](https://mlsanity.com/paper/1702.07944/) (2017)
- [The Information Geometry of Mirror Descent](https://mlsanity.com/paper/1310.7780/) (2013)
- [Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity](https://mlsanity.com/paper/1802.01504/) (2018)
- [Saddle-point dynamics: conditions for asymptotic stability of saddle points](https://mlsanity.com/paper/1510.02145/) (2015)
- [Analysis of gradient descent methods with non-diminishing, bounded errors](https://mlsanity.com/paper/1604.00151/) (2016)
- [Convergence of the Iterates in Mirror Descent Methods](https://mlsanity.com/paper/1805.01526/) (2018)
- [k-server via multiscale entropic regularization](https://mlsanity.com/paper/1711.01085/) (2017)
- [The role of convexity on saddle-point dynamics: Lyapunov function and robustness](https://mlsanity.com/paper/1608.08586/) (2016)
- [On Stochastic Subgradient Mirror-Descent Algorithm with Weighted Averaging](https://mlsanity.com/paper/1307.1879/) (2013)
- [Accelerated Stochastic Algorithms for Convex-Concave Saddle-Point Problems](https://mlsanity.com/paper/1903.01687/) (2019)
- [Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization](https://mlsanity.com/paper/1809.04831/) (2018)
- [Robust Analysis of Almost Sure Convergence of Zeroth-Order Mirror Descent Algorithm](https://mlsanity.com/paper/2303.09793/) (2023)

## Built on by (most cited first, 1 of 1)

- [Explainable Learning Rate Regimes for Stochastic Optimization](https://mlsanity.com/paper/2508.13639/) (2025)

## Abstract (arXiv)

Saddle point problems, ubiquitous in optimization, extend beyond game theory to diverse domains like power networks and reinforcement learning. This paper presents novel approaches to tackle saddle point problem, with a focus on continuous-time contexts. In this paper we propose a continuous time dynamics to tackle saddle point problem utilizing projected dynamical system in non-Euclidean domain. This involves computing the (sub/super) gradient of the min-max function within a Riemannian metric. Additionally, we establish viable Caratheodory solutions also prove the Lyapunov stability and asymptotic set stability of the proposed continuous time dynamical system. Next, we present the Stochastic Saddle Point Mirror Descent (SSPMD) algorithm and establish its equivalence with the proposed continuous-time dynamics. Leveraging stability analysis of the continuous-time dynamics, we demonstrate the almost sure convergence of the algorithm's iterates. Furthermore, we introduce the Zeroth-Order Saddle Point Mirror Descent (SZSPMD) algorithm, which approximates gradients using Nesterov's Gaussian Approximation, showcasing convergence to a neighborhood around saddle points. The analysis in this paper provides geometric insights into the mirror descent algorithm and demonstrates how these insights offer theoretical foundations for various practical applications of the mirror descent algorithm in diverse scenarios.