Local convergence of tensor methods

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity bounds for the Composite Tensor Method in convex and uniformly convex cases are also discussed. Lastly, we show how local convergence of the methods can be globalized using the inexact proximal iterations.

Asynchronous Schemes for Stochastic and Misspecified Potential Games and Nonconvex Optimization

In “Asynchronous Schemes for Stochastic and Misspecified Potential Games and Nonconvex Optimization,” Lei and Shanbhag consider a class of convex stochastic Nash games, possibly corrupted by parametric misspecification and characterized by a possibly nonconvex potential function. The authors present an asynchronous inexact proximal best-response (BR) scheme in which, at any step, a randomly selected player computes an inexact BR step (via stochastic approximation) and other players keep their strategies invariant. Misspecification is addressed by a simultaneous learning process reliant on an increasing batch size of sampled gradients. Almost-sure convergence guarantees are provided to the set of Nash equilibria, and such claims can be extended to delay-afflicted regimes, generalized potential games (with coupled strategy sets), and weighted potential games. In fact, equilibria of this potential game are stationary points of the potential function and asynchronous inexact BR schemes are, in essence, randomized block-coordinate schemes for a subclass of stochastic nonconvex optimization problems.