---
title: "Beyond Multi-Index Models: Learning Intersections of Degree-2 PTFs"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2611.05114v1"
permalink: "https://feed.astralor.com/items/itm_908e2494f86c5a35d25fd6ae"
published: "2026-11-03T16:05:15.000Z"
score: 62
channel: "ai"
tags: ["Beyond Multi-Index Models","Learning Intersections","Degree-2 PTFs","PAC","Gaussian","Our"]
---

# Beyond Multi-Index Models: Learning Intersections of Degree-2 PTFs

## 速览

We give an algorithm for PAC learning intersections of $k$ degree-2 polynomial threshold functions over $R^d$ under a Gaussian distribution. Our algorithm achieves classification error $ε$ using $d^{O(k)}2^{O_k(\log^3(1…

## 关键点

- We give an algorithm for PAC learning intersections of $k$ degree-2 polynomial threshold functions over $R^d$ under a Gaussian distribution.
- Our algorithm achieves classification error $ε$ using $d^{O(k)}2^{O_k(\log^3(1/ε))}$ time and samples, improving over the $d^{O(k^2/ε^2)}$ running time of the polynomial regression approach.
- To our knowledge, our result is the first to obtain a dimension exponent independent of the accuracy parameter, even for $k=2$.

## 全文

We give an algorithm for PAC learning intersections of $k$ degree-2 polynomial threshold functions over $R^d$ under a Gaussian distribution. Our algorithm achieves classification error $ε$ using $d^{O(k)}2^{O_k(\log^3(1/ε))}$ time and samples, improving over the $d^{O(k^2/ε^2)}$ running time of the polynomial regression approach. To our knowledge, our result is the first to obtain a dimension exponent independent of the accuracy parameter, even for $k=2$. We complement our algorithmic result with an SQ lower bound showing that the complexity dependence of $d^{Ω(k)}$ is unavoidable. Lifting the input to polynomial features turns the target into an intersection of $k$ halfspaces. This suggests using the first two labeled moments for subspace recovery. In the lifted space, however, these moments can reveal nothing. Our starting point is a Gaussian comparison theorem showing that, for sufficiently regular defining quadratics, the labeled moments needed for subspace recovery are approximately preserved. To handle arbitrary defining quadratics, we establish a new structural result showing that every direction with significant resampling influence can be detected through degree-$O(k)$ moments. These results yield an initial $d^{O(k)}2^{\mathrm{poly}_k(1/ε)}$-time learner using adaptive restrictions. To improve the exponential dependence on $1/ε$, we develop a two-stage dimension-reduction approach that avoids the costly adaptive partitioning. After identifying a subspace containing all directions with large influence, we use generalized regularity and Gaussian comparison to recover the remaining relevant quadratic features through a new derivative-based moment operator. The recovered features support an intersection close to the target. Combining them with robust hinge-loss minimization gives our quasipolynomial-time learner.

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