On the Link between Gaussian Homotopy Continuation and Convex Envelopes

Abstract. The continuation method is a popular heuristic in computer vision for nonconvex optimization. The idea is to start from a simpli-fied problem and gradually deform it to the actual task while tracking the solution. It was first used in computer vision under the name of graduated nonconvexity. Since then, it has been utilized explicitly or im-plicitly in various applications. In fact, state-of-the-art optical flow and shape estimation rely on a form of continuation. Despite its empirical success, there is little theoretical understanding of this method. This work provides some novel insights into this technique. Specifically, there are many ways to choose the initial problem and many ways to progres-sively deform it to the original task. However, here we show that when this process is constructed by Gaussian smoothing, it is optimal in a specific sense. In fact, we prove that Gaussian smoothing emerges from the best affine approximation to Vese’s nonlinear PDE. The latter PDE evolves any function to its convex envelope, hence providing the optimal convexification

On Graduated Optimization for Stochastic Non-Convex Problems

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 {\epsilon}-approximate solution within O(1/\epsilon^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/\epsilon^4).Comment: 17 page