Definition: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
Input: We propose an online convex optimization algorithm (RESCALEDEXP) that achieves<lb>optimal regret in the unconstrained setting without prior knowledge of any bounds<lb>on the loss functions. We prove a lower bound showing an exponential sep-<lb>aration between the regret of existing algorithms that require a known bound<lb>on the loss functions and any algorithm that does not require such knowledge.<lb>RESCALEDEXP matches this lower bound asymptotically in the number of itera-<lb>tions. RESCALEDEXP is naturally hyperparameter-free and we demonstrate empir-<lb>ically that it matches prior optimization algorithms that require hyperparameter<lb>optimization. 1 Online Convex Optimization Online Convex Optimization (OCO) [1, 2] provides an elegant framework for modeling noisy,<lb>antagonistic or changing environments. The problem can be stated formally with the help of the<lb>following definitions:<lb>Convex Set: A setW is convex ifW is contained in some real vector space and tw+(1− t)w′ ∈W<lb>for all w,w′ ∈W and t ∈ [0, 1].<lb>Convex Function: f :W → R is a convex function if f(tw + (1− t)w′) ≤ tf(w) + (1− t)f(w′)<lb>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<lb>wt in some convex space W , then receives a convex loss function `t, and suffers loss `t(wt). The<lb>regret of the learner with respect to some other u ∈W is defined by

Online Convex Optimization with Unconstrained Domains and Losses