Hardness of Agnostically Learning Halfspaces from Worst-Case Lattice Problems

TL;DR

This paper establishes the computational hardness of agnostically learning halfspaces and polynomial threshold functions under worst-case lattice problem assumptions, providing the first such results based on worst-case complexity.

Contribution

It introduces the first hardness results for learning halfspaces from worst-case lattice problems, extending beyond average-case assumptions and statistical query models.

Findings

Hardness of learning halfspaces with error better than 1/2 - γ under lattice problem assumptions.

Learning halfspaces with near-optimal error in Gaussian distribution takes super-polynomial time.

Similar hardness results are shown for polynomial threshold functions with respect to their degree.

Abstract

We show hardness of improperly learning halfspaces in the agnostic model, both in the distribution-independent as well as the distribution-specific setting, based on the assumption that worst-case lattice problems, such as GapSVP or SIVP, are hard. In particular, we show that under this assumption there is no efficient algorithm that outputs any binary hypothesis, not necessarily a halfspace, achieving misclassfication error better than $\frac 1 2 - \gamma$ even if the optimal misclassification error is as small is as small as $\delta$. Here, $\gamma$ can be smaller than the inverse of any polynomial in the dimension and $\delta$ as small as $exp(-\Omega(\log^{1-c}(d)))$, where $0 < c < 1$ is an arbitrary constant and $d$ is the dimension. For the distribution-specific setting, we show that if the marginal distribution is standard Gaussian, for any $\beta > 0$ learning halfspaces up to error $OPT_{LTF} + \epsilon$ takes time at least $d^{\tilde{\Omega}(1/\epsilon^{2-\beta})}$ under the same hardness assumptions. Similarly, we show that learning degree-$\ell$ polynomial threshold functions up to error $OPT_{{PTF}_\ell} + \epsilon$ takes time at least $d^{\tilde{\Omega}(\ell^{2-\beta}/\epsilon^{2-\beta})}$. $OPT_{LTF}$ and $OPT_{{PTF}_\ell}$ denote the best error achievable by any halfspace or polynomial threshold function, respectively. Our lower bounds qualitively match algorithmic guarantees and (nearly) recover known lower bounds based on non-worst-case assumptions. Previously, such hardness results [Daniely16, DKPZ21] were based on average-case complexity assumptions or restricted to the statistical query model. Our work gives the first hardness results basing these fundamental learning problems on worst-case complexity assumptions. It is inspired by a sequence of recent works showing hardness of learning well-separated Gaussian mixtures based on worst-case lattice problems.