Efficient Tracking of Large Classes of Experts

Abstract

In the framework of prediction of individual sequences, sequential prediction methods are to be constructed that perform nearly as well as the best expert from a given class. We consider prediction strategies that compete with the class of switching strategies that can segment a given sequence into several blocks, and follow the advice of a different "base" expert in each block. As usual, the performance of the algorithm is measured by the regret defined as the excess loss relative to the best switching strategy selected in hindsight for the particular sequence to be predicted. In this paper, we construct prediction strategies of low computational cost for the case where the set of base experts is large. In particular, we provide a method that can transform any prediction algorithm that is designed for the base class into a tracking algorithm. The resulting tracking algorithm can take advantage of the prediction performance and potential computational efficiency of in the sense that it can be implemented with time and space complexity only times larger than that of , where is the time horizon and is a parameter of the algorithm. With properly chosen, our algorithm achieves a regret bound of optimal order for , and only times larger than the optimal order for for all typical regret bound types we examined. For example, for predicting binary sequences with switching parameters under the logarithmic loss, our method achieves the optimal regret rate with time complexity for any .

Learning to Bid Without Knowing your Value

“…Also, at timestep t = 0, we incur no regret. We also apply the following slight modifications in Algorithm 2 in [27] so as to match the nature of our problem. First, instead of computing the expected loss at each timestep t, we will now compute the expected outcome-based utility, i.e.…”

“…is the cumulative outcome-based expected utility gained from our WIN-EXP algorithm in the time interval [t c , t c+1 ) 12 with respect toū s for s ∈ [t c , t c+1 ). Now, we are computing the regret components of [27] so as to achieve the desired result.…”

“…Before we show the specifics of the computation, we note here that g > 0 is a parameter of the Tracking Regret algorithm presented by [27] and can be set a priori from the designer of the algorithm. The complexity of g affects the computational complexity of the algorithm and there is a tradeoff between the computational complexity and the regret of the algorithm.…”

Efficient tracking of large classes of experts

“…Due to space constraints the main results will only be presented in their simplest form and for exp-concave loss function only (i.e., for loss functions for which there exists an η > 0 such that F (p) = e −η (p,y) is concave for fixed y ∈ Y); proofs and more general statements can be found in [17].…”

“…There the best linear-complexity scheme achieves O((C + 1) ln n/n 1/6 ) distortion redundancy when an upper bound C on the number of switches in the reference class is known in advance. On the other hand, applying a modified version of our scheme for randomized prediction [17] with g = O (1) in the method of [8], one can show that a linear-complexity version of this algorithm can achieve distortion redundancy O((C + 1) 1/2 ln 3/4 (n)/n 1/4 ) + O((C + 1) ln(n)/n 1/2 ) distortion redundancy for any (a priori unknown) C. When g = 2n γ − 1, the distortion redundancy for linear complexity becomes somewhat worse, proportional to n − 1 2(2+γ) up to logarithmic factors.…”