Efficiently Escaping Saddle Points under Generalized Smoothness via Self-Bounding Regularity

In this paper, we study the problem of non-convex optimization on functions that are not necessarily smooth using first order methods. Smoothness (functions whose gradient and/or Hessian are Lipschitz) is not satisfied by many machine learning problems in both theory and practice, motivating a recent line of work studying the convergence of first order methods to first order stationary points under appropriate generalizations of smoothness. We develop a novel framework to study convergence of first order methods to first and \textit{second} order stationary points under generalized smoothness, under more general smoothness assumptions than the literature. Using our framework, we show appropriate variants of GD and SGD (e.g. with appropriate perturbations) can converge not just to first order but also \textit{second order stationary points} in runtime polylogarithmic in the dimension. To our knowledge, our work contains the first such result, as well as the first 'non-textbook' rate for non-convex optimization under generalized smoothness. We demonstrate that several canonical non-convex optimization problems fall under our setting and framework.

From Optimization to Sampling via Lyapunov Potentials

We study the problem of sampling from high-dimensional distributions using Langevin Dynamics, a natural and popular variant of Gradient Descent where at each step, appropriately scaled Gaussian noise is added. The similarities between Langevin Dynamics and Gradient Descent leads to the natural question: if the distribution's log-density can be optimized from all initializations via Gradient Descent, given oracle access to the gradients, can we sample from the distribution using Langevin Dynamics? We answer this question in the affirmative, at low but appropriate temperature levels natural in the context of both optimization and real-world applications. As a corollary, we show we can sample from several new natural and interesting classes of non-log-concave densities, an important setting where we have relatively few examples.

Langevin Dynamics: A Unified Perspective on Optimization via Lyapunov Potentials

We study the problem of non-convex optimization using Stochastic Gradient Langevin Dynamics (SGLD). SGLD is a natural and popular variation of stochastic gradient descent where at each step, appropriately scaled Gaussian noise is added. To our knowledge, the only strategy for showing global convergence of SGLD on the loss function is to show that SGLD can sample from a stationary distribution which assigns larger mass when the function is small (the Gibbs measure), and then to convert these guarantees to optimization results. We employ a new strategy to analyze the convergence of SGLD to global minima, based on Lyapunov potentials and optimization. We convert the same mild conditions from previous works on SGLD into geometric properties based on Lyapunov potentials. This adapts well to the case with a stochastic gradient oracle, which is natural for machine learning applications where one wants to minimize population loss but only has access to stochastic gradients via minibatch training samples. Here we provide 1) improved rates in the setting of previous works studying SGLD for optimization, 2) the first finite gradient complexity guarantee for SGLD where the function is Lipschitz and the Gibbs measure defined by the function satisfies a Poincar\'e Inequality, and 3) prove if continuous-time Langevin Dynamics succeeds for optimization, then discrete-time SGLD succeeds under mild regularity assumptions.