Unconstrained Online Learning with Unbounded Losses
This work addresses a fundamental limitation in online learning by removing boundedness assumptions, which is crucial for applications with unbounded data or non-smooth losses, though it is incremental in extending existing theory.
The paper tackles the problem of online learning with unbounded domains and non-Lipschitz losses, developing an algorithm that achieves $ ilde O(G\|u\|\sqrt{T}+L\|u\|^{2}\sqrt{T})$ regret, which is shown to be unimprovable, and extends this to saddle-point optimization and dynamic regret with matching lower bounds.
Algorithms for online learning typically require one or more boundedness assumptions: that the domain is bounded, that the losses are Lipschitz, or both. In this paper, we develop a new setting for online learning with unbounded domains and non-Lipschitz losses. For this setting we provide an algorithm which guarantees $R_{T}(u)\le \tilde O(G\|u\|\sqrt{T}+L\|u\|^{2}\sqrt{T})$ regret on any problem where the subgradients satisfy $\|g_{t}\|\le G+L\|w_{t}\|$, and show that this bound is unimprovable without further assumptions. We leverage this algorithm to develop new saddle-point optimization algorithms that converge in duality gap in unbounded domains, even in the absence of meaningful curvature. Finally, we provide the first algorithm achieving non-trivial dynamic regret in an unbounded domain for non-Lipschitz losses, as well as a matching lower bound. The regret of our dynamic regret algorithm automatically improves to a novel $L^{*}$ bound when the losses are smooth.