Multiplicative Weights Update as a Distributed Optimization - PowerPoint PPT Presentation

Multiplicative Weights Update as a Distributed Constrained Multiplicative Weights Update as a Distributed Optimization Algorithm: Constrained Optimization Algorithm: Convergence to Second-order Convergence to Second-order Stationary Points Stationary Points Almost Almost Always Always Ioannis Panageas, Georgios Piliouras, Xiao Ioannis Panageas Georgios Piliouras Xiao Wang Wang Singapore University of Technology and Design June, 2019

Motivation Multiplicative Non-concave maximization has been the subject of much Weights Update as a recent study in the optimization and machine learning Distributed Constrained communities. Constrained maximization is of importance Optimization Algorithm: in many applications such as hidden Markov model and Convergence to game theory. Second-order Stationary Results in [Lee et al., 2017] suggest the avoidance of Points Almost Always saddle points for deterministic first-order methods with Ioannis random initialization. How ever, the approach has Panageas, Georgios limitations in certain constrained cases. Piliouras, Xiao Wang Question: Is there a provable convergence for the problems of the form max x ∈ D P ( x ) , where P is a non-concave, twice continuously differentiable function and D is the product of simplices, i.e., D = ∆ 1 × ... × ∆ n ?

Key Techniques Multiplicative A classic algorithm for simplicial constrained maximization Weights Update as a is Baum-Eagon algorithm [Baum and Eagon, 1967]: Distributed Constrained Optimization ∂ P � Algorithm: � ∂ x ij x t Convergence x t +1 = x t (1) . to ij ij Σ s x t ∂ P � Second-order is � x t ∂ x is Stationary Points Almost Always (1) is not a diffeomorphism in general. Ioannis We use MWU as an instance of Baum-Eagon algorithm Panageas, Georgios with learning rates: Piliouras, Xiao Wang 1 + ǫ i ∂ P � � ∂ x ij x t x t +1 = x t (2) . ij ij ∂ P � 1 + ǫ i Σ s x t � is ∂ x is x t (2) is a diffeomorphism with small ǫ i . Use of Center-stable Manifold Theorem [Lee et al., 2017].

Our Results Multiplicative Combining the classification of stationary points with Weights Update as a constraints and the Center-stable Manifold Theorem to MWU, Distributed Constrained we prove the following: Optimization Algorithm: Assume that P is twice continuously differentiable in a set Convergence to containing D . There exists small enough fixed stepsizes ǫ i Second-order such that the set of initial conditions x 0 of which the Stationary Points Almost Always MWU dynamics converges to fixed points that violate Ioannis second order KKT conditions is of measure zero. Panageas, Georgios Assume µ is a measure that is absolutely continuous with Piliouras, Xiao Wang respect to the Lebesgue measure and P is a rational function (fraction of polynomials) that is twice continuously differentiable in a set containing D , with isolated stationary points. It follows that with probability one (randomness induced by µ ), MWU dynamics converges to second order stationary points.

Multiplicative Weights Update as a Distributed Constrained Optimization Algorithm: Convergence to Second-order Stationary Points Almost Always Thank You! Ioannis Panageas, Georgios Piliouras, Xiao Wang