Accelerated Multiplicative Weights Update Avoids Saddle Points almost always
This addresses a theoretical gap in optimization algorithms for researchers in game theory, machine learning, and multi-agent systems, but it is incremental as it builds on existing MWU methods.
The paper tackles the problem of whether an accelerated version of Multiplicative Weights Update (MWU) can provably avoid saddle points in non-convex optimization on product-of-simplices constraints, and it provides a positive answer by developing an accelerated MWU based on Riemannian Accelerated Gradient Descent and proving it almost always avoids saddle points.
We consider non-convex optimization problems with constraint that is a product of simplices. A commonly used algorithm in solving this type of problem is the Multiplicative Weights Update (MWU), an algorithm that is widely used in game theory, machine learning and multi-agent systems. Despite it has been known that MWU avoids saddle points, there is a question that remains unaddressed:"Is there an accelerated version of MWU that avoids saddle points provably?" In this paper we provide a positive answer to above question. We provide an accelerated MWU based on Riemannian Accelerated Gradient Descent, and prove that the Riemannian Accelerated Gradient Descent, thus the accelerated MWU, almost always avoid saddle points.