Black-Box Reductions for Parameter-free Online Learning in Banach Spaces

We introduce several new black-box reductions that significantly improve the design of adaptive and parameterfree online learning algorithms by simplifying analysis, improving regret guarantees, and sometimes even improving runtime. We reduce parameter-free online learning to online exp-concave optimization, we reduce optimization in a Banach space to one-dimensional optimization, and we reduce optimization over a constrained domain to unconstrained optimization. All of our reductions run as fast as online gradient descent. We use our new techniques to improve upon the previously best regret bounds for parameter-free learning, and do so for arbitrary norms. 1 Parameter Free Online Learning Online learning is a popular framework for understanding iterative optimization algorithms, including stochastic optimization algorithms or algorithms operating on large data streams. For each of T iterations, an online learning algorithm picks a point wt in some space W , observes a loss function lt : W → R, and suffers loss lt(wt). Performance is measured by the regret, which is the total loss suffered by the algorithm in comparison to some benchmark point ẘ ∈ W : RT (ẘ) = T ∑