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 these potential functions relies heavily on guessing. To streamline this workflow, we present a framework that generates new potential functions by solving a Partial Differential Equation (PDE). Specifically, when losses are 1-Lipschitz, our framework produces a novel algorithm with anytime regret 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 unknown and unbounded a priori. Such a bound attains an optimal loss-regret trade-off without the impractical doubling trick. Moreover, a matching lower bound shows that the leading order term, including the constant multiplier $\sqrt{2}$, is tight. To our knowledge, the proposed algorithm is the first to achieve such optimalities.

翻译：不受限制的在线线性优化( OLO) 是研究机器学习模型培训的一个实际问题。 现有的工程提出了若干基于潜在算法, 但一般而言, 这些潜在函数的设计在很大程度上依赖于猜测。 为了简化这一工作流程, 我们提出了一个框架, 通过解决部分差异化( PDE) 来产生新的潜在功能。 具体地说, 当损失为1- Lipschitz 时, 我们的框架产生了一种新颖的算法, 随时会后悔的$C\ sqrt{$( 1 ⁇ u ⁇ / C) } [ sqrt $[ sqrt t $(1 ⁇ ⁇ { { { { { { { { { { { { { { { }} [ sqrt}}} 。 据我们所知, 拟议的算法是第一个实现这种最佳性。