aiqbench analysis
This paper is theoretical machine learning work on agnostic and positive-unlabeled learning of Boolean multi-index models, such as intersections of halfspaces, under Gaussian-smooth input distributions. Its main contribution is a partitioning primitive that improves dimension dependence for several learning tasks, most notably reducing the exponential dependence on k for functions of k halfspaces. For practitioners, the results are complexity bounds and Statistical Query lower bounds rather than deployable methods, so their value lies in clarifying what is achievable in principle.
Gaussian Partitioning for Reliable Learning of Multi-Index Models
arXiv:2611.05264v1 Announce Type: cross Abstract: We introduce Gaussian partitioning, an algorithmic primitive for distributions smooth with respect to a Gaussian. It partitions space into polyhedral regions so that almost all mass lies in thin strips restricting a few orthogonal directions. Within each strip, the conditional distribution approximately matches low-degree Gaussian moments in the unrestricted directions. Our construction builds on logconcave partitioning of Diakonikolas, Kane, and Ma (FOCS 2026). Under input marginals smooth with respect to the standard Gaussian, we obtain agnostic learners with error $(1+c)\mathrm{opt}+\epsilon$ for broad classes of Boolean multi-index models, including functions of halfspaces, low-dimensional convex concepts, and low-dimensional polynomial threshold functions. For functions of $k$ halfspaces, both agnostic and testable agnostic learners achieve this error with dimension dependence $d^{\mathrm{poly}(k/c)}$, improving the previous exponential dependence on $k/c$ in the agnostic exponent. For learning from positive examples with imperfect unlabeled data (PUI), we give algorithms and Statistical Query (SQ) lower bounds that qualitatively characterize complexity through Gaussian moment matching. For fixed smoothness and positive-mass parameters, we learn intersections of $k$ halfspaces with dimension dependence $d^{O(\log(k/\epsilon))}$ and matching SQ lower bounds, improving the previous $d^{\mathrm{poly}(1/\epsilon)}$ upper bound even for one halfspace. A general reduction from agnostic reliable learning to PUI yields negative reliable learners for these intersections with the same dimension dependence, for fixed smoothness. Finally, for PQ learning and testable learning with distribution shift, we prove qualitatively optimal SQ lower bounds of $d^{\Omega(1/\epsilon^2)}$, even for intersections of two halfspaces with Gaussian training data and arbitrary test distributions.
Abstract:We introduce Gaussian partitioning, an algorithmic primitive for distributions smooth with respect to a Gaussian. It partitions space into polyhedral regions so that almost all mass lies in thin strips restricting a few orthogonal directions. Within each strip, the conditional distribution approximately matches low-degree Gaussian moments in the unrestricted directions. Our construction builds on logconcave partitioning of Diakonikolas, Kane, and Ma (FOCS 2026). Under input marginals smooth with respect to the standard Gaussian, we obtain agnostic learners with error $(1+c)\mathrm{opt}+\epsilon$ for broad classes of Boolean multi-index models, including functions of halfspaces, low-dimensional convex concepts, and low-dimensional polynomial threshold functions. For functions of $k$ halfspaces, both agnostic and testable agnostic learners achieve this error with dimension dependence $d^{\mathrm{poly}(k/c)}$, improving the previous exponential dependence on $k/c$ in the agnostic exponent. For learning from positive examples with imperfect unlabeled data (PUI), we give algorithms and Statistical Query (SQ) lower bounds that qualitatively characterize complexity through Gaussian moment matching. For fixed smoothness and positive-mass parameters, we learn intersections of $k$ halfspaces with dimension dependence $d^{O(\log(k/\epsilon))}$ and matching SQ lower bounds, improving the previous $d^{\mathrm{poly}(1/\epsilon)}$ upper bound even for one halfspace. A general reduction from agnostic reliable learning to PUI yields negative reliable learners for these intersections with the same dimension dependence, for fixed smoothness. Finally, for PQ learning and testable learning with distribution shift, we prove qualitatively optimal SQ lower bounds of $d^{\Omega(1/\epsilon^2)}$, even for intersections of two halfspaces with Gaussian training data and arbitrary test distributions.
Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG)
arXiv:2611.05264 [cs.DS]
(or arXiv:2611.05264v1 [cs.DS] for this version)
https://doi.org/10.48550/arXiv.2611.05264 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 342 words