DeepAI

# Riemannian Langevin Algorithm for Solving Semidefinite Programs

We propose a Langevin diffusion-based algorithm for non-convex optimization and sampling on a product manifold of spheres. Under a logarithmic Sobolev inequality, we establish a guarantee for finite iteration convergence to the Gibbs distribution in terms of Kullback-Leibler divergence. We show that with an appropriate temperature choice, the suboptimality gap to the global minimum is guaranteed to be arbitrarily small with high probability. As an application, we analyze the proposed Langevin algorithm for solving the Burer-Monteiro relaxation of a semidefinite program (SDP). In particular, we establish a logarithmic Sobolev inequality for the Burer-Monteiro problem when there are no spurious local minima; hence implying a fast escape from saddle points. Combining the results, we then provide a global optimality guarantee for the SDP and the Max-Cut problem. More precisely, we show the Langevin algorithm achieves ϵ-multiplicative accuracy with high probability in Ω( n^2 ϵ^-3 ) iterations, where n is the size of the cost matrix.

• 6 publications
• 29 publications
03/25/2017

### Solving SDPs for synchronization and MaxCut problems via the Grothendieck inequality

A number of statistical estimation problems can be addressed by semidefi...
08/07/2018

### On the integrality gap of the maximum-cut semidefinite programming relaxation in fixed dimension

We describe a factor-revealing convex optimization problem for the integ...
06/10/2019

### Efficiently escaping saddle points on manifolds

Smooth, non-convex optimization problems on Riemannian manifolds occur i...
07/23/2018

### Fisher Information and Logarithmic Sobolev Inequality for Matrix Valued Functions

We prove a version of Talagrand's concentration inequality for subordina...
01/17/2023

### Noisy, Non-Smooth, Non-Convex Estimation of Moment Condition Models

A practical challenge for structural estimation is the requirement to ac...
06/11/2018

### Smoothed analysis of the low-rank approach for smooth semidefinite programs

We consider semidefinite programs (SDPs) of size n with equality constra...
03/20/2023

### A Cheeger Inequality for Size-Specific Conductance

The μ-conductance measure proposed by Lovasz and Simonovits is a size-sp...