Konsep Inti
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.
Abstrak
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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