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Unconstrained Online Linear Optimization (OLO) is a practical problem setting to study the training of machine learning models. Existing works proposed a number of potential-based algorithms, but in general the design of such potential functions is ad hoc and heavily relies on guessing. In this paper, we present a framework that generates time-varying potential functions by solving a Partial Differential Equation (PDE). Our framework recovers some classical potentials, and more importantly provides a systematic approach to design new ones. The power of our framework is demonstrated through a concrete example. When losses are 1-Lipschitz, we design a novel OLO algorithm with anytime regret upper bound $C\sqrt{T}+||u||\sqrt{2T}[\sqrt{\log(1+||u||/C)}+2]$, where $C$ is a user-specified constant and $u$ is any comparator whose norm is unknown and unbounded a priori. By constructing a matching lower bound, we further show that the leading order term, including the constant multiplier $\sqrt{2}$, is tight. To our knowledge, this is the first parameter-free algorithm with optimal leading constant. The obtained theoretical benefits are validated by experiments.
