A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics
This provides theoretical guarantees for escaping suboptimal local minima in non-convex optimization, which is incremental for machine learning practitioners dealing with empirical risk minimization.
The paper tackles the analysis of Stochastic Gradient Langevin Dynamics (SGLD) for non-convex optimization, proving that it achieves an approximate local minimum of the population risk in polynomial time and improves learnability results for linear classifiers under the zero-one loss.
We study the Stochastic Gradient Langevin Dynamics (SGLD) algorithm for non-convex optimization. The algorithm performs stochastic gradient descent, where in each step it injects appropriately scaled Gaussian noise to the update. We analyze the algorithm's hitting time to an arbitrary subset of the parameter space. Two results follow from our general theory: First, we prove that for empirical risk minimization, if the empirical risk is point-wise close to the (smooth) population risk, then the algorithm achieves an approximate local minimum of the population risk in polynomial time, escaping suboptimal local minima that only exist in the empirical risk. Second, we show that SGLD improves on one of the best known learnability results for learning linear classifiers under the zero-one loss.