Online Density Estimation of Bradley-Terry Models

Abstract

We consider an online density estimation problem for the Bradley-Terry model, where each model parameter defines the probability of a match result between any pair in a set of n teams. The problem is hard because the loss function (i.e., the negative log-likelihood function in our problem setting) is not convex. To avoid the non-convexity, we can change parameters so that the loss function becomes convex with respect to the new parameter. But then the radius K of the reparameterized domain may be infinite, where K depends on the outcome sequence. So we put a mild assumption that guarantees that K is finite. We can thus employ standard online convex optimization algorithms, namely OGD and ONS, over the reparameterized domain, and get regret bounds O(n^\frac12(\ln K)\sqrt{T}) and O(n^\frac32K\ln T), respectively, where T is the horizon of the game. The bounds roughly means that OGD is better when K is large while ONS is better when K is small. But how large can K be? We show that K can be as large as Θ(T^n-1), which implies that the worst case regret bounds of OGD and ONS are O(n^\frac32\sqrt{T}\ln T) and \tildeO(n^\frac32(T)^n-1), respectively. We then propose a version of Follow the Regularized Leader, whose regret bound is close to the minimum of those of OGD and ONS. In other words, our algorithm is competitive with both for a wide range of values of K. In particular, our algorithm achieves the worst case regret bound O(n^\frac52T^\frac13 \ln T), which is slightly better than OGD with respect to T. In addition, our algorithm works without the knowledge K, which is a practical advantage.