Tight $L_\infty$ Sample Complexity for Low-Degree and Sparse Boolean Polynomials
Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube. To ensure that optimizing the surrogate yields good solutions for the underlying objective, we require uniform $L_\infty$-error guarantees rather than the usual $L_2$-type guarantees. We characterize the minimax sample complexity of uniform estimation under subgaussian noise for two classes of bounded polynomials. First, for polynomials of degree at most $d$ on $n$ variables, the sample complexity scales as $n^{d+1}$. Second, for $s$-sparse Fourier-Walsh polynomials with $s \leq n$, it scales as $ns^2$. These rates differ structurally from...
| Authors | Jasper van Doornmalen, Mathieu Molina, Victor Verdugo et al. |
|---|---|
| Primary category | stat.ML |
| Published | 15 June 2026 |
| Last updated at source | 15 June 2026 |
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