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October 3, 2025Open Access

Minimax Adaptive Online Nonparametric Regression over Besov Spaces

Key Points

An adaptive wavelet-based algorithm achieves minimax-optimal regret bounds against prediction rules in Besov spaces.
The algorithm is effective without prior knowledge of the parameters, enhancing adaptability in various environments.
Local adaptability allows for dynamic adjustments in both time and space, leading to refined predictions of local regularity.
In heterogeneous settings, the proposed method can significantly outperform standard global regression techniques.

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\₏ₐˢ with general parameters 1 p, q and smoothness s > dp. 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\₏ₐˢ. 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.

Minimax Adaptive Online Nonparametric Regression over Besov Spaces

Key Points

Abstract

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