Online Convex Optimization with Unconstrained Domains and Losses

We propose an online convex optimization algorithm (RESCALEDEXP) that achieves optimal regret in the unconstrained setting without prior knowledge of any bounds on the loss functions. We prove a lower bound showing an exponential separation between the regret of existing algorithms that require a known bound on the loss functions and any algorithm that does not require such knowledge. RESCALEDEXP matches this lower bound asymptotically in the number of iterations. RESCALEDEXP is naturally hyperparameter-free and we demonstrate empirically that it matches prior optimization algorithms that require hyperparameter optimization. 1 Online Convex Optimization Online Convex Optimization (OCO) [1, 2] provides an elegant framework for modeling noisy, antagonistic or changing environments. The problem can be stated formally with the help of the following definitions: Convex Set: A setW is convex ifW is contained in some real vector space and tw+(1− t)w′ ∈W for all w,w′ ∈W and t ∈ [0, 1]. Convex Function: f :W → R is a convex function if f(tw + (1− t)w′) ≤ tf(w) + (1− t)f(w′) for all w,w′ ∈W and t ∈ [0, 1]. An OCO problem is a game of repeated rounds in which on round t a learner first chooses an element wt in some convex space W , then receives a convex loss function `t, and suffers loss `t(wt). The regret of the learner with respect to some other u ∈W is defined by