Proper Agnostic Learning of Functions of Halfspaces under Gaussian Marginals

• URL: http://arxiv.org/abs/2605.27594v1
• Date: Tue, 26 May 2026 19:07:06 GMT
• Title: Proper Agnostic Learning of Functions of Halfspaces under Gaussian Marginals
• Abstract summary: Given i.i.d. labeled samples from an unknown distribution over $mathbbRd times pm 1$ whose marginal on $mathbbRd$ is Gaussian, the goal is to output a hypothesis from a target class $mathcalF$ whose 0-1 loss is within $$ of that of the best classifier in $mathcalF$.<n>Our algorithm runs in time $dO(K2 log (1/)/2) + (K/)O(K
• Score: 5.7652356955571085
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of computationally efficient proper agnostic learning of multidimensional concept classes under the Gaussian distribution. In this setting, given i.i.d. labeled samples from an unknown distribution over $\mathbb{R}^d \times \{\pm 1\}$ whose marginal on $\mathbb{R}^d$ is Gaussian, the goal is to output a hypothesis from a target class $\mathcal{F}$ whose 0-1 loss is within $ε$ of that of the best classifier in $\mathcal{F}$. We give the first efficient proper agnostic learning algorithm for arbitrary Boolean functions of $K$ halfspaces under Gaussian marginals. Our algorithm runs in time $d^{O(K^2 \log(1/ε)/ε^2)} + (K/ε)^{O(K^3/ε^{2.5})}$. Prior to our work, the only known algorithm for $K \geq 2$ was brute-force search, with run-time exponential in $d$. Moreover, the dependence of our run-time on the dimension $d$ matches that of the best known improper learning algorithm, namely $d^{\widetilde{O}(K^2/ε^2)}$. For the special case of a single halfspace ($K=1$), the best previous run-time was $d^{O(1/ε^4)} + (1/ε)^{O(1/ε^6)}$. Our algorithm improves this to $d^{O(1/ε^2)} + (1/ε)^{O(1/ε^{2.5})}$. Once again, the dependence on $d$ matches that of the best known improper algorithm, namely $d^{O(1/ε^2)}$. Furthermore, the dependence of our run-time on the dimension $d$ is essentially optimal in the statistical query model.

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