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We study active learning methods for single index models of the form $$F({\bm x}) = f(\langle {\bm w}, {\bm x}\rangle)$$, where $$f:\mathbb{R} \to \mathbb{R}$$ and $${\bx,\bm w} \in \mathbb{R}^d$$. In addition to their theoretical interest as simple examples of non-linear neural networks, single index models have received significant recent attention due to applications in scientific machine learning like surrogate modeling for partial differential equations (PDEs). Such applications require sample-efficient active learning methods that are robust to adversarial noise. I.e., that work even in the challenging agnostic learning setting. We provide two main results on agnostic active learning of single index models. First, when $$f$$ is known and Lipschitz, we show that $$\tilde{O}(d)$$ samples collected via {statistical leverage score sampling} are sufficient to learn a near-optimal single index model. Leverage score sampling is simple to implement, efficient, and already widely used for actively learning linear models. Our result requires no assumptions on the data distribution, is optimal up to log factors, and improves quadratically on a recent $${O}(d^{2})$$ bound of \cite{gajjar2023active}. Second, we show that $$\tilde{O}(d)$$ samples suffice even in the more difficult setting when $$f$$ is \emph{unknown}. Our results leverage tools from high dimensional probability, including Dudley's inequality and dual Sudakov minoration, as well as a novel, distribution-aware discretization of the class of Lipschitz functions.
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This content will become publicly available on January 22, 2026
Omnipredicting Single-Index Models with Multi-Index Models
Recent work on supervised learning [GKR+22] defined the notion of omnipredictors, i.e., predictor functions p over features that are simultaneously competitive for minimizing a family of loss functions against a comparator class . Omniprediction requires approximating the Bayes-optimal predictor beyond the loss minimization paradigm, and has generated significant interest in the learning theory community. However, even for basic settings such as agnostically learning single-index models (SIMs), existing omnipredictor constructions require impractically-large sample complexities and runtimes, and output complex, highly-improper hypotheses. Our main contribution is a new, simple construction of omnipredictors for SIMs. We give a learner outputting an omnipredictor that is ε-competitive on any matching loss induced by a monotone, Lipschitz link function, when the comparator class is bounded linear predictors. Our algorithm requires ≈ε−4 samples and runs in nearly-linear time, and its sample complexity improves to ≈ε−2 if link functions are bi-Lipschitz. This significantly improves upon the only prior known construction, due to [HJKRR18, GHK+23], which used ≳ε−10 samples. We achieve our construction via a new, sharp analysis of the classical Isotron algorithm [KS09, KKKS11] in the challenging agnostic learning setting, of potential independent interest. Previously, Isotron was known to properly learn SIMs in the realizable setting, as well as constant-factor competitive hypotheses under the squared loss [ZWDD24]. As they are based on Isotron, our omnipredictors are multi-index models with ≈ε−2 prediction heads, bringing us closer to the tantalizing goal of proper omniprediction for general loss families and comparators.
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- Award ID(s):
- 2505865
- PAR ID:
- 10631523
- Publisher / Repository:
- https://doi.org/10.48550/arXiv.2411.13083
- Date Published:
- ISSN:
- 2411.13083
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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