New paper proves tight L∞ sample bounds for Boolean polynomial learning

A team of researchers (van Doornmalen, Molina, Verdugo, Verschae) has published a paper on arXiv (2606.17319) tackling the problem of learning polynomial surrogates over the Boolean hypercube with uniform L∞ error guarantees—a critical requirement for safely optimizing bounded black-box functions. Unlike typical L2-type guarantees, L∞ error ensures the surrogate deviates from the true function at every point, making it reliable for optimization. The study characterizes the minimax sample complexity under subgaussian noise for two polynomial classes.

For polynomials of degree at most d on n variables, the required sample size scales as n^{d+1}, a factor of n more than the noiseless case. For s-sparse Fourier-Walsh polynomials (s ≤ n), it scales as n s^2, compared to n s without noise. These lower bounds hold even for arbitrary adaptive learners, confirming that the extra factors are intrinsic to noise. The proofs overcome traditional Fourier-analysis limitations by using auxiliary norms as proxies for L∞ control. This work provides a tight theoretical foundation for building optimization-safe surrogates, crucial for applications like combinatorial optimization and drug discovery.