A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics

• DOI: 10.48550/arxiv.1702.05575
• Publication Date: 2017-01-01
• Authors: Zhang, Yuchen; Liang, Percy; Charikar, Moses
• Keywords: FOS: Computer and information sciences; Computer Science - Machine Learning; Statistics - Machine Learning; Optimization and Control (math.OC); 0202 electrical engineering, electronic engineering, information engineering; FOS: Mathematics; Machine Learning (stat.ML); 02 engineering and technology; Mathematics - Optimization and Control; Machine Learning (cs.LG)

Correct two mistakes in the proofs of Lemma 3 and Lemma 5<br/>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.<br/>