Move from Perturbed scheme to exponential weighting average

In an online decision problem, one makes decisions often with a pool of decisions’ sequence called experts but without knowledge of the future. After each step, one pays a cost based on the decision and observed rate. One reasonal goal would be to perform as well as the best expert in the pool. The modern and well-known way to attain this goal is the algorithm of exponential weighting. However, recently, another algorithm called follow the perturbed leader is developed and achieved about the same performance. In our work, we first show the properties shared in common by the two algorithms which explain the similarities on the performance. Next we will show that for a specific perturbation, the two algorithms are identical. Finally, we show with some examples that follow-the-leader style algorithms extend naturally to a large class of structured online problems for which the exponential algorithms are inefficient. 1 Online problem setting In an online decision problem, one makes decisions often with a pool of decisions’ sequence called experts but without knowledge of the future. After each step, one pays a cost based on the decision and observed state. As there is no prior knowledge on the accuracy of experts in the pool, one reasonable goal for this general problem would be to perform as well as the best expert in the pool after a number of steps. More precisely, we consider the following mathematical problem: • A set S of experts is given. • The algorithm interacts with an adversary in a series of T steps. • In each step j, the algorithm picks an expert xj ∈ S , and the adversary selects a cost function cj: S → R. The adversary could be adaptive, in that cj may depend on {xi : i < j}. • The algorithm incurs cost , and receives as feedback the value of cj(xj). • Minimize the algorithm’s regret which is defined as difference in expected cost between the algorithm’s sequence of choices and that of best fixed expert in S: