Minimax Adaptive Online Nonparametric Regression over Besov Spaces

• URL: http://arxiv.org/abs/2505.19741v1
• Date: Mon, 26 May 2025 09:23:11 GMT
• Title: Minimax Adaptive Online Nonparametric Regression over Besov Spaces
• Authors: Paul Liautaud, Pierre Gaillard, Olivier Wintenberger,
• Abstract summary: We study online adversarial regression with convex losses against a rich class of continuous yet highly irregular prediction rules.<n>We introduce an adaptive wavelet-based algorithm that performs sequential prediction without prior knowledge of $(s,p,q)$.<n>We also design a locally adaptive extension capable of dynamically tracking spatially inhomogeneous smoothness.
• Score: 10.138723409205497
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study online adversarial regression with convex losses against a rich class of continuous yet highly irregular prediction rules, modeled by Besov spaces $B_{pq}^s$ with general parameters $1 \leq p,q \leq \infty$ and smoothness $s > d/p$. We introduce an adaptive wavelet-based algorithm that performs sequential prediction without prior knowledge of $(s,p,q)$, and establish minimax-optimal regret bounds against any comparator in $B_{pq}^s$. We further design a locally adaptive extension capable of dynamically tracking spatially inhomogeneous smoothness. This adaptive mechanism adjusts the resolution of the predictions over both time and space, yielding refined regret bounds in terms of local regularity. Consequently, in heterogeneous environments, our adaptive guarantees can significantly surpass those obtained by standard global methods.

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