[Paper Review] Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis

Maxim Raginsky, Alexander Rakhlin+1 more|arXiv (Cornell University)|Feb 13, 2017

Markov Chains and Monte Carlo Methods18 references152 citations

TL;DR

This paper provides finite-time, nonasymptotic guarantees for Stochastic Gradient Langevin Dynamics (SGLD) in non-convex learning, linking discrete updates to Langevin diffusion and using Wasserstein-based analysis to bound excess risk and generalization.

ABSTRACT

Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased estimate of the gradient at each iteration. This modest change allows SGLD to escape local minima and suffices to guarantee asymptotic convergence to global minimizers for sufficiently regular non-convex objectives (Gelfand and Mitter, 1991). The present work provides a nonasymptotic analysis in the context of non-convex learning problems, giving finite-time guarantees for SGLD to find approximate minimizers of both empirical and population risks. As in the asymptotic setting, our analysis relates the discrete-time SGLD Markov chain to a continuous-time diffusion process. A new tool that drives the results is the use of weighted transportation cost inequalities to quantify the rate of convergence of SGLD to a stationary distribution in the Euclidean $2$-Wasserstein distance.

Motivation & Objective

• Motivate non-convex optimization problems and study SGLD as a practical algorithm with noise to escape local minima.
• Connect the discrete SGLD update to continuous Langevin diffusion to enable nonasymptotic analysis.
• Provide finite-time bounds on excess risk for both empirical and population risks.
• Decompose excess risk into generalization error and empirical-minimizer gaps and bound each component.

Proposed method

• Study SGLD update W_{k+1} = W_k - eta g_k + sqrt(2 eta / beta) xi_k where g_k is a stochastic gradient estimate.
• Model the updates as a discretization of the Langevin diffusion dW(t) = -grad F_Z(W(t)) dt + sqrt(2/beta) dB(t).
• Use weighted transportation-cost inequalities to bound 2-Wasserstein distances between SGLD iterates and the diffusion.
• Establish a logarithmic Sobolev inequality for the Gibbs distribution to obtain exponential convergence in Wasserstein distance.
• Prove stability of the Gibbs distribution under data perturbations to control generalization via uniform stability arguments.
• Utilize nonasymptotic Laplace approximation to show a Gibbs draw is an approximate empirical minimizer.

Experimental results

Research questions

• RQ1 Can SGLD achieve nonasymptotic convergence guarantees for non-convex objectives?
• RQ2 How close is the SGLD iterate distribution to the Gibbs distribution over time in 2-Wasserstein distance?
• RQ3 What are the finite-time bounds for the excess risk when optimizing empirical and population risks with SGLD?
• RQ4 How does stability of the Gibbs distribution relate to generalization in this non-convex setting?

Key findings

• The expected excess risk bound decomposes into three terms with specific scaling: the first term scales as epsilon * Poly(beta, d, 1/lambda_*) for k >= Poly(beta, d, 1/lambda_*) * 1/epsilon^4 and eta <= (epsilon / log(1/epsilon))^4.
• The second and third terms scale as (beta + d)^2 / (lambda_* n) and d log(beta+1) / beta, respectively.
• The analysis connects discrete SGLD to Langevin diffusion and shows the Gibbs distribution concentrates around empirical minimizers for sufficiently large beta.
• A uniform stability bound for the Gibbs algorithm is established under single-coordinate data perturbations, enabling generalization control.
• The main result (Theorem 2.1) provides a finite-time, nonasymptotic excess-risk bound under assumptions including smoothness, dissipativity, and gradient oracle accuracy.