We study accelerated mirror descent dynamics in continuous and discrete time. Combining the original continuous-time motivation of mirror descent with a recent ODE interpretation of Nesterov's accelerated method, we propose a family of continuous-time descent dynamics for convex functions with Lipschitz gradients, such that the solution trajectories converge to the optimum at a O(1/t

We study a general adversarial online learning problem, in which we are given a decision set X in a reflexive Banach space X and a sequence of reward vectors in the dual space of X. At each iteration, we choose an action from X, based on the observed sequence of previous rewards. Our goal is to minimize regret. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging to our setting and obtain upper bounds on the worst-case regret that generalize many previous results. Under the assumption of uniformly continuous rewards, we obtain explicit regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space S. Importantly, we make no convexity assumptions on either S or the reward functions. We also prove a general lower bound on the worst-case regret for any online algorithm. We then apply these results to the problem of learning in repeated two-player zero-sum games on compact metric spaces. In doing so, we first prove that if both players play a Hannan-consistent strategy, then with probability 1 the empirical distributions of play weakly converge to the set of Nash equilibria of the game. We then show that, under mild assumptions, Dual Averaging on the (infinite-dimensional) space of probability distributions indeed achieves Hannan-consistency.

We study accelerated descent dynamics for constrained convex optimization. This dynamics can be described naturally as a coupling of a dual variable accumulating gradients at a given rate η(t), and a primal variable obtained as the weighted average of the mirrored dual trajectory, with weights w(t). Using a Lyapunov argument, we give sufficient conditions on η and w to achieve a desired convergence rate. As an example, we show that the replicator dynamics (an example of mirror descent on the simplex) can be accelerated using a simple averaging scheme. We then propose an adaptive averaging heuristic which adaptively computes the weights to speed up the decrease of the Lyapunov function. We provide guarantees on adaptive averaging in continuous-time, prove that it preserves the quadratic convergence rate of accelerated first-order methods in discrete-time, and give numerical experiments to compare it with existing heuristics, such as adaptive restarting. The experiments indicate that adaptive averaging performs at least as well as adaptive restarting, with significant improvements in some cases.