Convergence Analysis of Noisy Distributed Gradient Descent for Non-convex Optimization -- Saddle Point Escape

TL;DR

The paper addresses distributed non-convex optimization over networks by introducing Noisy Distributed Gradient Descent (NDGD), which perturbs local gradients with Gaussian noise to actively escape saddle points. By reformulating the problem in a lifted centralized form via the auxiliary function $Q_{\alpha}$ and applying concentration inequalities, the authors prove that, for sufficiently small fixed step-size $\alpha$ and noise variance, each agent converges with high probability to a neighborhood of a common local minimizer while maintaining consensus. The results include probabilistic convergence guarantees to an approximate second-order stationary point with rates comparable to centralized first-order methods, and a detailed exploration of the trade-offs between convergence accuracy and noise-induced perturbations. Numerical experiments on a non-convex quartic problem and a binary classification task demonstrate that NDGD significantly improves saddle-point escape and accelerates convergence relative to standard DGD. Overall, the work advances distributed non-convex optimization by providing rigorous per-agent convergence guarantees under fixed-step, noise-perturbed dynamics and practical demonstrations of the method’s efficacy.

Abstract

A variant of consensus based distributed gradient descent (\textbf{DGD}) is studied for finite sums of smooth but possibly non-convex functions. In particular, the local gradient term in the fixed step-size iteration of each agent is randomly perturbed to evade saddle points. Under regularity conditions, it is established that for sufficiently small step size and noise variance, each agent converges with high probability to a specified radius neighborhood of a common second-order stationary point, i.e., local minimizer. The rate of convergence is shown to be comparable to centralized first-order algorithms. Numerical experiments are presented to validate the efficacy of the proposed approach over standard \textbf{DGD} in a non-convex setting.

Figures (3)

• Figure 1: Logical structure of the proof of Lemma \ref{['lem: Approximate second order']}.
• Figure 2: Second order properties of NDGD and DGD for the non-convex quartic optimization example, where $\Theta^{\star}$ denotes the set of local minimizers of $J$ (marked by a red $\star$). "bi" and "gi" refer to bad and good initializations, respectively; "1" and "2" refer to agent $1$ and $2$ respectively.
• Figure 3: Second order properties of NDGD and DGD for the binary classification example, where $\mathcal{V}^{\star}$ denotes the set of local minimizers of $\Tilde{L}$ marked by a red $\star$). "bi" and "gi" refer to bad and good initializations, respectively; "1" and "2" refer to agent $1$ and $2$ respectively.

Theorems & Definitions (54)

• Definition II.1: First order stationarity
• Definition II.2: Second order stationarity
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Definition II.3: Consensual first order stationarity
• Definition II.4: Consensual second-order stationarity
• Remark 6