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Unconstrained Online Linear Optimization (OLO) is a practical problem settingto study the training of machine learning models. Existing works proposed anumber of potential-based algorithms, but in general the design of thesepotential functions relies heavily on guessing. To streamline this workflow, wepresent a framework that generates new potential functions by solving a PartialDifferential Equation (PDE). Specifically, when losses are 1-Lipschitz, ourframework produces a novel algorithm with anytime regret bound$C\sqrt{T}+||u||\sqrt{2T}[\sqrt{\log(1+||u||/C)}+2]$, where $C$ is auser-specified constant and $u$ is any comparator unknown and unbounded apriori. Such a bound attains an optimal loss-regret trade-off without theimpractical doubling trick. Moreover, a matching lower bound shows that theleading order term, including the constant multiplier $\sqrt{2}$, is tight. Toour knowledge, the proposed algorithm is the first to achieve suchoptimalities.

PDE-Based Optimal Strategy for Unconstrained Online Learning