Mixability of a loss is known to characterise when constant regret bounds are achievable in games of prediction with expert advice through the use of the aggregating algorithm [Vovk, 2001]. We provide a new interpretation of mixability via convex analysis that highlights the role of the Kullback-Leibler divergence in its definition. This naturally generalises to what we call �-mixability where the Bregman divergence D� replaces the KL divergence. We prove that losses that are �-mixable also enjoy constant regret bounds via a generalised aggregating algorithm that is similar to mirror descent.
The analysis of mirror descent by @cite_2 shows that it achieves constant regret when the entropic regulariser is used. However, they do not consider whether similar results extend to other entropies defined on the simplex.
@cite_2
2949921677 2016384870 2150695437 2963868636
We show that for a general class of convex online learning problems, Mirror Descent can always achieve a (nearly) optimal regret guarantee. The mirror descent algorithm (MDA) was introduced by Nemirovsky and Yudin for solving convex optimization problems. This method exhibits an efficiency estimate that is mildly dependent in the decision variables dimension, and thus suitable for solving very large scale optimization problems. We present a new derivation and analysis of this algorithm. We show that the MDA can be viewed as a nonlinear projected-subgradient type method, derived from using a general distance-like function instead of the usual Euclidean squared distance. Within this interpretation, we derive in a simple way convergence and efficiency estimates. We then propose an Entropic mirror descent algorithm for convex minimization over the unit simplex, with a global efficiency estimate proven to be mildly dependent in the dimension of the problem. The mirror descent algorithm (MDA) generalizes gradient descent by using a Bregman divergence to replace squared Euclidean distance. In this paper, we similarly generalize the alternating direction method of multipliers (ADMM) to Bregman ADMM (BADMM), which allows the choice of different Bregman divergences to exploit the structure of problems. BADMM provides a unified framework for ADMM and its variants, including generalized ADMM, inexact ADMM and Bethe ADMM. We establish the global convergence and the O(1 T) iteration complexity for BADMM. In some cases, BADMM can be faster than ADMM by a factor of O(n ln n) where n is the dimensionality. In solving the linear program of mass transportation problem, BADMM leads to massive parallelism and can easily run on GPU. BADMM is several times faster than highly optimized commercial software Gurobi. We study an equivalence of (i) deterministic pathwise statements appearing in the online learning literature (termed regret bounds), (ii) high-probability tail bounds for the supremum of a collection of martingales (of a specific form arising from uniform laws of large numbers for martingales), and (iii) in-expectation bounds for the supremum. By virtue of the equivalence, we prove exponential tail bounds for norms of Banach space valued martingales via deterministic regret bounds for the online mirror descent algorithm with an adaptive step size. We extend these results beyond the linear structure of the Banach space: we define a notion of martingale type for general classes of real-valued functions and show its equivalence (up to a logarithmic factor) to various sequential complexities of the class (in particular, the sequential Rademacher complexity and its offset version). For classes with the general martingale type 2, we exhibit a finer notion of variation that allows partial adaptation to the function indexing the martingale. Our proof technique rests on sequential symmetrization and on certifying the existence of regret minimization strategies for certain online prediction problems.