Sparse Online Learning via Truncated Gradient: Appendix

In the setting of standard online learning, we are interested in sequential prediction problems where for i = 1, 2, . . .: 1. An unlabeled example xi = [xi , . . . , x d i ] ∈ R arrives. 2. We make a prediction ŷi based on the current weights wi = [w i , . . . , w d i ] ∈ R. 3. We observe yi, let zi = (xi, yi), and incur some known loss L(wi, zi) convex in parameter wi. 4. We update weights according to some rule: wi+1 ← f(wi). We want an update rule f that allows us to bound the sum of losses, ∑t i=1 L(wi, zi), as well as achieving sparsity. For this purpose, we start with the standard stochastic gradient descent (SGD) rule, which is of the form: