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Single-Index Models are high-dimensional regression problems with planted structure, whereby labels depend on an unknown one-dimensional projection of the input via a generic, non-linear, and potentially non-deterministic transformation. As such, they encompass a broad class of statistical inference tasks, and provide a rich template to study statistical and computational trade-offs in the high-dimensional regime. While the information-theoretic sample complexity to recover the hidden direction is lin- ear in the dimension d, we show that computationally efficient algorithms, both within the Statistical Query (SQ) and the Low-Degree Polynomial (LDP) framework, necessarily require Ω(dk⋆/2) samples, where k⋆ is a “generative” exponent associated with the model that we explicitly characterize. Moreover, we show that this sample complexity is also sufficient, by establishing matching upper bounds using a partial-trace algorithm. Therefore, our results pro- vide evidence of a sharp computational-to-statistical gap (under both the SQ and LDP class) whenever k⋆ > 2. To complete the study, we construct smooth and Lipschitz deterministic target functions with arbitrarily large generative exponents k⋆.
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Lattice-Based Methods Surpass Sum-of-Squares in Clustering
Clustering is a fundamental primitive in unsupervised learning which gives rise to a rich class of computationally-challenging inference tasks. In this work, we focus on the canonical task of clustering d-dimensional Gaussian mixtures with unknown (and possibly degenerate) covariance. Recent works (Ghosh et al. ’20; Mao, Wein ’21; Davis, Diaz, Wang ’21) have established lower bounds against the class of low-degree polynomial methods and the sum-of-squares (SoS) hierarchy for recovering certain hidden structures planted in Gaussian clustering instances. Prior work on many similar inference tasks portends that such lower bounds strongly suggest the presence of an inherent statistical-to-computational gap for clustering, that is, a parameter regime where the clustering task is statistically possible but no polynomial-time algorithm succeeds. One special case of the clustering task we consider is equivalent to the problem of finding a planted hypercube vector in an otherwise random subspace. We show that, perhaps surprisingly, this particular clustering model does not exhibit a statistical-to-computational gap, despite the aforementioned low-degree and SoS lower bounds. To achieve this, we give an algorithm based on Lenstra–Lenstra–Lovász lattice basis reduction which achieves the statistically-optimal sample complexity of d + 1 samples. This result extends the class of problems whose conjectured statistical-to-computational gaps can be “closed” by “brittle” polynomial-time algorithms, highlighting the crucial but subtle role of noise in the onset of statistical-to-computational gaps.
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- PAR ID:
- 10329471
- Date Published:
- Journal Name:
- conference on learning theory
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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