On Graduated Optimization for Stochastic Non-Convex Problems
This provides theoretical guarantees for a popular heuristic in non-convex optimization, addressing a gap in the literature for researchers and practitioners in machine learning and optimization.
The paper tackles the lack of theoretical convergence analysis for graduated optimization in non-convex problems by proposing a new first-order algorithm that provably converges to a global optimum for a parameterized family of functions, achieving an ε-approximate solution within O(1/ε^2) gradient steps, and extends this to stochastic and zero-order settings with rates of O(1/ε^2) and O(d^2/ε^4), respectively.
The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite its popularity, very little is known in terms of theoretical convergence analysis. In this paper we describe a new first-order algorithm based on graduated optimiza- tion and analyze its performance. We characterize a parameterized family of non- convex functions for which this algorithm provably converges to a global optimum. In particular, we prove that the algorithm converges to an ε-approximate solution within O(1/ε^2) gradient-based steps. We extend our algorithm and analysis to the setting of stochastic non-convex optimization with noisy gradient feedback, attaining the same convergence rate. Additionally, we discuss the setting of zero-order optimization, and devise a a variant of our algorithm which converges at rate of O(d^2/ε^4).