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• W809555510 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, whereK 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 1 2 (lnK) p T ) andO(n 3 2K lnT ), respectively, whereT is the horizon of the game. The bounds roughly means that OGD is better whenK is large while ONS is better whenK is small. But how large can K be? We show that K can be as large as ( T n 1 ), which implies that the worst case regret bounds of OGD and ONS areO(n 3 2 p T lnT ) and ~ O(n 3 2 (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 5 2T 1 3 lnT ), which is slightly better than OGD with respect to T . In addition, our algorithm" @default.
• W809555510 created "2016-06-24" @default.
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• W809555510 date "2015-06-26" @default.
• W809555510 modified "2023-09-24" @default.
• W809555510 title "Online Density Estimation of Bradley-Terry Models" @default.
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