Papers › PDE-Based Optimal Strategy for Unconstrained Online Learning

PDE-Based Optimal Strategy for Unconstrained Online Learning

19 Jan 2022arXiv:2201.07877archive 2025-07-28

ZhiYu Zhang, Ashok Cutkosky, Ioannis Paschalidis

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√(T)+||u||√(2T)[√(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 √(2), is tight. To our knowledge, the proposed algorithm is the first to achieve such optimalities.

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