Efficient Robust Learning of Low-Degree Polynomial Threshold Functions under Adversarial Corruptions

Core Concepts

We present a polynomial-time algorithm that robustly learns the class of degree-d polynomial threshold functions under the Gaussian distribution in the presence of a constant fraction of adversarial corruptions, achieving error Oc,d(1) opt^(1-c) for any constant c > 0, where opt is the fraction of corruptions.

Abstract

The paper studies the efficient learnability of low-degree polynomial threshold functions (PTFs) in the presence of a constant fraction of adversarial corruptions. The main algorithmic result is a polynomial-time PAC learning algorithm for this concept class in the strong contamination model under the Gaussian distribution, with an error guarantee of Od,c(opt^(1-c)), for any desired constant c > 0, where opt is the fraction of corruptions.
The algorithm employs an iterative approach inspired by localization techniques previously used in the context of learning linear threshold functions. Specifically, it uses a robust perceptron algorithm to compute a good partial classifier and then iterates on the unclassified points. To achieve this, the paper develops new polynomial decomposition techniques, introducing the notion of "super non-singular" polynomial decompositions.
The key technical contributions are:

An efficient algorithm for constructing super non-singular polynomial decompositions (Theorem 4.1).
A structural result demonstrating the (anti)-concentration properties of Gaussian distributions conditioned on sets defined by super non-singular polynomial transformations (Theorem 5.1).
A localization sub-routine that partitions sets defined by polynomial inequalities into subsets with good (anti-)concentration properties (Proposition 6.3).
A robust margin-perceptron algorithm that can learn under the weaker (anti-)concentration assumptions provided by the partitioning routine (Proposition 7.3).

By combining these components, the paper presents the first polynomial-time algorithm that can robustly learn degree-d PTFs with error Oc,d(1) opt^(1-c) under the Gaussian distribution, for any constant c > 0.

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Key Insights Distilled From

Super Non-singular Decompositions of Polynomials and their Application to Robustly Learning Low-degree PTFs

by Ilias Diakon... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00529.pdf

Deeper Inquiries

Can the error guarantee be further improved to Od(opt) or Od(opt^c) for some constant c < 1, without sacrificing the polynomial runtime

The error guarantee can potentially be improved to Od(opt) or Od(opt^c) for some constant c < 1 without sacrificing the polynomial runtime. One approach to achieve this improvement could involve refining the partitioning technique used in the algorithm. By enhancing the partitioning process to create subsets with even better (anti)-concentration properties, it may be possible to reduce the error further while maintaining the polynomial-time complexity. Additionally, exploring advanced localization methods or incorporating more sophisticated learning algorithms could also contribute to enhancing the error guarantee without compromising efficiency.

Are there other natural distribution families, beyond the Gaussian, for which a similar robust learning algorithm for low-degree PTFs can be developed

There are several other natural distribution families beyond the Gaussian distribution for which a similar robust learning algorithm for low-degree Polynomial Threshold Functions (PTFs) could be developed. One potential candidate is the Laplace distribution, which is commonly used in robust statistics due to its heavy tails and robustness to outliers. By adapting the algorithm to work with the Laplace distribution, it may be possible to achieve robust learning of low-degree PTFs under this distribution. Furthermore, exploring distributions with specific properties that align with the characteristics of PTFs could lead to the development of tailored robust learning algorithms for these distributions.

What are the broader implications of the super non-singular polynomial decomposition techniques developed in this work, and how might they find applications in other areas of theoretical computer science

The super non-singular polynomial decomposition techniques developed in this work have significant implications beyond the specific application to robust learning of low-degree PTFs. These techniques can find applications in various areas of theoretical computer science, particularly in the fields of machine learning, optimization, and signal processing.

Machine Learning: The super non-singular decomposition techniques can be utilized in the development of robust learning algorithms for a wide range of concept classes beyond PTFs. By applying these decomposition methods to other types of functions or models, researchers can enhance the robustness and efficiency of learning algorithms in the presence of adversarial noise or corruptions.

Optimization: The decomposition techniques can be leveraged in optimization problems where the objective function is a polynomial. By decomposing the polynomial into super non-singular components, optimization algorithms can exploit the structure of the function to improve convergence rates and solution quality.

Signal Processing: In signal processing applications, the decomposition techniques can be used for feature extraction, denoising, and signal representation. By decomposing signals into non-singular components, it becomes easier to analyze and process the signals effectively, leading to improved performance in various signal processing tasks.