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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⋆. 

We focus on the task of learning a single index model with respect to the isotropic Gaussian distribution in d dimensions. Prior work has shown that the sample complexity of learning the hidden direction is governed by the information exponent k of the link function. Ben Arous et al. showed that d^k samples suffice for learning and that this is tight for online SGD. However, the CSQ lowerbound for gradient based methods only shows that d^{k/2} samples are necessary. In this work, we close the gap between the upper and lower bounds by showing that online SGD on a smoothed loss learns the hidden direction with the correct number of samples. We also draw connections to statistical analyses of tensor PCA and to the implicit regularization effects of minibatch SGD on empirical losses. 

Traditional analyses of gradient descent show that when the largest eigenvalue of the Hessian, also known as the sharpness S(θ), is bounded by 2/η, training is "stable" and the training loss decreases monotonically. Recent works, however, have observed that this assumption does not hold when training modern neural networks with full batch or large batch gradient descent. Most recently, Cohen et al. (2021) observed two important phenomena. The first, dubbed progressive sharpening, is that the sharpness steadily increases throughout training until it reaches the instability cutoff 2/η. The second, dubbed edge of stability, is that the sharpness hovers at 2/η for the remainder of training while the loss continues decreasing, albeit non-monotonically. We demonstrate that, far from being chaotic, the dynamics of gradient descent at the edge of stability can be captured by a cubic Taylor expansion: as the iterates diverge in direction of the top eigenvector of the Hessian due to instability, the cubic term in the local Taylor expansion of the loss function causes the curvature to decrease until stability is restored. This property, which we call self-stabilization, is a general property of gradient descent and explains its behavior at the edge of stability. A key consequence of self-stabilization is that gradient descent at the edge of stability implicitly follows projected gradient descent (PGD) under the constraint S(θ)≤2/η. Our analysis provides precise predictions for the loss, sharpness, and deviation from the PGD trajectory throughout training, which we verify both empirically in a number of standard settings and theoretically under mild conditions. Our analysis uncovers the mechanism for gradient descent's implicit bias towards stability. 

Significant theoretical work has established that in specific regimes, neural networks trained by gradient descent behave like kernel methods. However, in practice, it is known that neural networks strongly outperform their associated kernels. In this work, we explain this gap by demonstrating that there is a large class of functions which cannot be efficiently learned by kernel methods but can be easily learned with gradient descent on a two layer neural network outside the kernel regime by learning representations that are relevant to the target task. We also demonstrate that these representations allow for efficient transfer learning, which is impossible in the kernel regime. Specifically, we consider the problem of learning polynomials which depend on only a few relevant directions, i.e. of the form $f(x)=g(Ux)$ where $U: \R^d \to \R^r$ with $d≫r$. When the degree of f⋆ is p, it is known that n≍dp samples are necessary to learn f⋆ in the kernel regime. Our primary result is that gradient descent learns a representation of the data which depends only on the directions relevant to f. This results in an improved sample complexity of n≍d2r+drp. Furthermore, in a transfer learning setup where the data distributions in the source and target domain share the same representation U but have different polynomial heads we show that a popular heuristic for transfer learning has a target sample complexity independent of d.