How to escape sharp minima

Modern machine learning applications have seen a remarkable success of optimization algorithms that are designed to find flat minima. Motivated by this paradigm, this work formulates and studies the algorithmic question of how to find flat minima. As an initial effort, this work adopts the trace of hessian of the cost function as the measure of flatness, and formally defines the notion of approximate flat minima. Under this notion, we then design algorithms that find approximate flat minima efficiently. For general cost functions, we present a gradient-based algorithm that finds an approximate flat local minimum efficiently. The main component of the algorithm is to use gradients computed from randomly perturbed iterates to estimate a direction that leads to flatter minima. For the setting where the cost function is an empirical risk over training data, we present a faster algorithm that is inspired by a recently proposed practical algorithm called sharpness-aware minimization, supporting its success in practice.