Online to Batch Conversions

We have recently been studying the case where have a training set T generated from an underlying distribution and our goal is to find some good hypothesis, with respect to the true underlying distribution, using the training set T . We now examine how to use online learning algorithms (which work on individual, arbitrary sequences) in a stochastic setting. Let us consider the training set T as the ordered sequence: T = {(X1, Y1), . . . , (Xm, Ym)} and let us run an online learning algorithm on this sequence. In particular, let us say that each round t our algorithm chooses some θ ∈ Θ and we suffer loss `(θ; (xi, yi)). Here, the decision space/parameter space Θ could be the space corresponding to the parameterization of our hypothesis class. The regret of our algorithm on the sequence is defined as: RT = m ∑