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Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise

Shiwei Zeng, Jie Shen

TL;DR

We address the problem of PAC learning $K$-sparse degree-$d$ polynomial threshold functions on $\mathbb{R}^n$ under Gaussian marginals in the presence of nasty noise. The authors leverage a structural sparsity result in the Hermite basis, showing that the Chow vector $\chi_{f^*}$ is $k$-sparse, and develop an attribute-efficient robust Chow-vector estimator (SparseFilter) that uses a restricted Frobenius-norm criterion to certify a good approximation or to filter corrupted samples. The main contribution is an algorithm running in time $(nd/\epsilon)^{O(d)}$ with sample complexity $O\big( K^{4d}(d\log n)^{5d} / \epsilon^{2d+2} \big)$, tolerating up to $\eta \le O(\epsilon^{d+1}/d^{2d})$ fraction of nasty-noise, and enabling PAC learning of $\mathcal{H}_{d,K}$ under Gaussian marginals with dimension-independent noise tolerance. This work significantly generalizes robustness results from sparse homogeneous halfspaces to general sparse low-degree PTFs, providing a practical and theoretically principled pathway for robust, attribute-efficient learning in high dimensions.

Abstract

The concept class of low-degree polynomial threshold functions (PTFs) plays a fundamental role in machine learning. In this paper, we study PAC learning of $K$-sparse degree-$d$ PTFs on $\mathbb{R}^n$, where any such concept depends only on $K$ out of $n$ attributes of the input. Our main contribution is a new algorithm that runs in time $({nd}/ε)^{O(d)}$ and under the Gaussian marginal distribution, PAC learns the class up to error rate $ε$ with $O(\frac{K^{4d}}{ε^{2d}} \cdot \log^{5d} n)$ samples even when an $η\leq O(ε^d)$ fraction of them are corrupted by the nasty noise of Bshouty et al. (2002), possibly the strongest corruption model. Prior to this work, attribute-efficient robust algorithms are established only for the special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a structural result that translates the attribute sparsity to a sparsity pattern of the Chow vector under the basis of Hermite polynomials, and 2) a novel attribute-efficient robust Chow vector estimation algorithm which uses exclusively a restricted Frobenius norm to either certify a good approximation or to validate a sparsity-induced degree-$2d$ polynomial as a filter to detect corrupted samples.

Attribute-Efficient PAC Learning of Low-Degree Polynomial Threshold Functions with Nasty Noise

TL;DR

We address the problem of PAC learning -sparse degree- polynomial threshold functions on under Gaussian marginals in the presence of nasty noise. The authors leverage a structural sparsity result in the Hermite basis, showing that the Chow vector is -sparse, and develop an attribute-efficient robust Chow-vector estimator (SparseFilter) that uses a restricted Frobenius-norm criterion to certify a good approximation or to filter corrupted samples. The main contribution is an algorithm running in time with sample complexity , tolerating up to fraction of nasty-noise, and enabling PAC learning of under Gaussian marginals with dimension-independent noise tolerance. This work significantly generalizes robustness results from sparse homogeneous halfspaces to general sparse low-degree PTFs, providing a practical and theoretically principled pathway for robust, attribute-efficient learning in high dimensions.

Abstract

The concept class of low-degree polynomial threshold functions (PTFs) plays a fundamental role in machine learning. In this paper, we study PAC learning of -sparse degree- PTFs on , where any such concept depends only on out of attributes of the input. Our main contribution is a new algorithm that runs in time and under the Gaussian marginal distribution, PAC learns the class up to error rate with samples even when an fraction of them are corrupted by the nasty noise of Bshouty et al. (2002), possibly the strongest corruption model. Prior to this work, attribute-efficient robust algorithms are established only for the special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a structural result that translates the attribute sparsity to a sparsity pattern of the Chow vector under the basis of Hermite polynomials, and 2) a novel attribute-efficient robust Chow vector estimation algorithm which uses exclusively a restricted Frobenius norm to either certify a good approximation or to validate a sparsity-induced degree- polynomial as a filter to detect corrupted samples.
Paper Structure (36 sections, 36 theorems, 134 equations, 2 algorithms)

This paper contains 36 sections, 36 theorems, 134 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Main results
  3. Overview of main techniques
  4. Related works
  5. Roadmap
  6. Preliminaries
  7. Main Algorithms
  8. Pruning
  9. Filtering
  10. Termination
  11. Performance Guarantees
  12. Analysis of SparseFilter
  13. Analysis of termination
  14. Main results
  15. Conclusion
  16. ...and 21 more sections

Key Result

Theorem 4

Assume that $D$ is the standard Gaussian distribution $\mathcal{N}(0, I_{n\times n})$. There is an algorithm that runs in time $({nd}/{\epsilon})^{O(d)}$ and PAC learns $\mathcal{H}_{d, K}$ by drawing $C \cdot \frac{ K^{4d} (d\log n)^{5d}}{\epsilon^{2d+2}}$ samples from the nasty adversary for some

Theorems & Definitions (77)

  • Definition 1: PAC learning with nasty noise
  • Theorem 4: Theorem \ref{['thm:main']}, informal
  • Remark 5: Sample complexity
  • Remark 6: Noise tolerance
  • Remark 7: Comparison to prior works
  • Remark 8: Running time
  • Definition 9: Sparse polynomials and PTFs
  • Lemma 10
  • Lemma 11
  • Definition 12: Good set
  • ...and 67 more