Tight $L_\infty$ Sample Complexity for Low-Degree and Sparse Boolean Polynomials

2014-06-22 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Data Science & Analytics · Depth: Expert, extended

Summary

The paper characterizes the minimax sample complexity for learning polynomial surrogates over the Boolean hypercube, focusing on uniform L-infinity error guarantees under subgaussian noise. It identifies distinct scaling behaviors for two function classes: low-degree polynomials (degree at most d on n variables) and s-sparse Fourier-Walsh polynomials (s ≤ n). For low-degree polynomials, the sample complexity scales as n^(d+1), while for s-sparse polynomials, it scales as ns^2. These rates significantly differ from the noiseless setting, where complexities are n^d and ns, respectively. The authors demonstrate that these additional factors are intrinsic to noisy scenarios, holding even for adaptive learners. The proofs overcome challenges in extending L2-norm Fourier analysis to L-infinity by employing auxiliary norms, providing a tight characterization for optimization-safe polynomial surrogates.

Key takeaway

For AI Scientists or Research Scientists developing or deploying polynomial surrogates for black-box Boolean optimization, you must account for substantially higher sample complexity when uniform L-infinity error is required in noisy settings. Plan for n^(d+1) samples for low-degree or ns^2 for sparse models, as adaptive learning offers no escape from these intrinsic statistical barriers. This ensures optimization-safe surrogates.

Key insights

Noisy L-infinity learning of Boolean polynomial surrogates requires significantly higher sample complexity than noiseless or L2-norm settings.

Principles

• L-infinity error guarantees are crucial for optimization-safe surrogates.
• Adaptive learners cannot avoid increased sample complexity in noisy L-infinity settings.
• Standard L2-norm Fourier analysis tools are insufficient for L-infinity guarantees.

Method

The paper uses Bayes-risk arguments with Gaussian priors on block-multilinear polynomial subclasses, analyzing posterior distribution covariance and injective norms to derive lower bounds. Upper bounds use fixed-design least-squares or empirical Walsh coefficients with hard thresholding.

In practice

• When optimizing black-box Boolean functions, prioritize L-infinity error for surrogate reliability.
• Be aware that noisy environments drastically increase sample needs for uniform surrogate accuracy.
• Consider specialized techniques beyond standard L2-norm methods for L-infinity guarantees.

Topics

• Boolean Hypercube
• L-infinity Norm
• Sample Complexity
• Fourier-Walsh Expansion
• Polynomial Surrogates
• Subgaussian Noise
• Minimax Estimation

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.