Online Convex Optimization 3 Proof of Theorem

In this lecture we shall look at a fairly general setting of online convex optimization which, as we shall see, encompasses some of the online learning problems we have seen so far as special cases. A standard (offline) convex optimization problem involves a convex set Ω ⊆ R and a fixed convex cost function c : Ω 7→ R; the goal is to find a point x∗ ∈ Ω that minimizes c(x) over Ω. An online convex optimization problem also involves a convex set Ω ⊆ R, but proceeds in trials: at each trial t, the player/algorithm must choose a point x ∈ Ω, after which a convex cost function ct : Ω→R is revealed by the environment/adversary, and a loss of ct(x ) is incurred for the trial; the goal is to minimize the total loss incurred over a sequence of trials, relative to the best single point in Ω in hindsight.