On the Complexity of Learning Sparse Functions with Statistical and Gradient Queries

The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$--potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.

Dimension-free deterministic equivalents for random feature regression

In this work we investigate the generalization performance of random feature ridge regression (RFRR). Our main contribution is a general deterministic equivalent for the test error of RFRR. Specifically, under a certain concentration property, we show that the test error is well approximated by a closed-form expression that only depends on the feature map eigenvalues. Notably, our approximation guarantee is non-asymptotic, multiplicative, and independent of the feature map dimension -- allowing for infinite-dimensional features. We expect this deterministic equivalent to hold broadly beyond our theoretical analysis, and we empirically validate its predictions on various real and synthetic datasets. As an application, we derive sharp excess error rates under standard power-law assumptions of the spectrum and target decay. In particular, we provide a tight result for the smallest number of features achieving optimal minimax error rate.

Asymptotics of Random Feature Regression Beyond the Linear Scaling Regime

Recent advances in machine learning have been achieved by using overparametrized models trained until near interpolation of the training data. It was shown, e.g., through the double descent phenomenon, that the number of parameters is a poor proxy for the model complexity and generalization capabilities. This leaves open the question of understanding the impact of parametrization on the performance of these models. How does model complexity and generalization depend on the number of parameters $p$? How should we choose $p$ relative to the sample size $n$ to achieve optimal test error? In this paper, we investigate the example of random feature ridge regression (RFRR). This model can be seen either as a finite-rank approximation to kernel ridge regression (KRR), or as a simplified model for neural networks trained in the so-called lazy regime. We consider covariates uniformly distributed on the $d$-dimensional sphere and compute sharp asymptotics for the RFRR test error in the high-dimensional polynomial scaling, where $p,n,d \to \infty$ while $p/ d^{\kappa_1}$ and $n / d^{\kappa_2}$ stay constant, for all $\kappa_1 , \kappa_2 \in \mathbb{R}_{>0}$. These asymptotics precisely characterize the impact of the number of random features and regularization parameter on the test performance. In particular, RFRR exhibits an intuitive trade-off between approximation and generalization power. For $n = o(p)$, the sample size $n$ is the bottleneck and RFRR achieves the same performance as KRR (which is equivalent to taking $p = \infty$). On the other hand, if $p = o(n)$, the number of random features $p$ is the limiting factor and RFRR test error matches the approximation error of the random feature model class (akin to taking $n = \infty$). Finally, a double descent appears at $n= p$, a phenomenon that was previously only characterized in the linear scaling $\kappa_1 = \kappa_2 = 1$.

A non-asymptotic theory of Kernel Ridge Regression: deterministic equivalents, test error, and GCV estimator

We consider learning an unknown target function $f_*$ using kernel ridge regression (KRR) given i.i.d. data $(u_i,y_i)$, $i\leq n$, where $u_i \in U$ is a covariate vector and $y_i = f_* (u_i) +\varepsilon_i \in \mathbb{R}$. A recent string of work has empirically shown that the test error of KRR can be well approximated by a closed-form estimate derived from an `equivalent' sequence model that only depends on the spectrum of the kernel operator. However, a theoretical justification for this equivalence has so far relied either on restrictive assumptions -- such as subgaussian independent eigenfunctions -- , or asymptotic derivations for specific kernels in high dimensions. In this paper, we prove that this equivalence holds for a general class of problems satisfying some spectral and concentration properties on the kernel eigendecomposition. Specifically, we establish in this setting a non-asymptotic deterministic approximation for the test error of KRR -- with explicit non-asymptotic bounds -- that only depends on the eigenvalues and the target function alignment to the eigenvectors of the kernel. Our proofs rely on a careful derivation of deterministic equivalents for random matrix functionals in the dimension free regime pioneered by Cheng and Montanari (2022). We apply this setting to several classical examples and show an excellent agreement between theoretical predictions and numerical simulations. These results rely on having access to the eigendecomposition of the kernel operator. Alternatively, we prove that, under this same setting, the generalized cross-validation (GCV) estimator concentrates on the test error uniformly over a range of ridge regularization parameter that includes zero (the interpolating solution). As a consequence, the GCV estimator can be used to estimate from data the test error and optimal regularization parameter for KRR.

Six Lectures on Linearized Neural Networks

In these six lectures, we examine what can be learnt about the behavior of multi-layer neural networks from the analysis of linear models. We first recall the correspondence between neural networks and linear models via the so-called lazy regime. We then review four models for linearized neural networks: linear regression with concentrated features, kernel ridge regression, random feature model and neural tangent model. Finally, we highlight the limitations of the linear theory and discuss how other approaches can overcome them.

SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics

We investigate the time complexity of SGD learning on fully-connected neural networks with isotropic data. We put forward a complexity measure -- the leap -- which measures how "hierarchical" target functions are. For $d$-dimensional uniform Boolean or isotropic Gaussian data, our main conjecture states that the time complexity to learn a function $f$ with low-dimensional support is $\tilde\Theta (d^{\max(\mathrm{Leap}(f),2)})$. We prove a version of this conjecture for a class of functions on Gaussian isotropic data and 2-layer neural networks, under additional technical assumptions on how SGD is run. We show that the training sequentially learns the function support with a saddle-to-saddle dynamic. Our result departs from [Abbe et al. 2022] by going beyond leap 1 (merged-staircase functions), and by going beyond the mean-field and gradient flow approximations that prohibit the full complexity control obtained here. Finally, we note that this gives an SGD complexity for the full training trajectory that matches that of Correlational Statistical Query (CSQ) lower-bounds.

Spectrum of inner-product kernel matrices in the polynomial regime and multiple descent phenomenon in kernel ridge regression

We study the spectrum of inner-product kernel matrices, i.e., $n \times n$ matrices with entries $h (\langle \textbf{x}_i ,\textbf{x}_j \rangle/d)$ where the $( \textbf{x}_i)_{i \leq n}$ are i.i.d.~random covariates in $\mathbb{R}^d$. In the linear high-dimensional regime $n \asymp d$, it was shown that these matrices are well approximated by their linearization, which simplifies into the sum of a rescaled Wishart matrix and identity matrix. In this paper, we generalize this decomposition to the polynomial high-dimensional regime $n \asymp d^\ell,\ell \in \mathbb{N}$, for data uniformly distributed on the sphere and hypercube. In this regime, the kernel matrix is well approximated by its degree-$\ell$ polynomial approximation and can be decomposed into a low-rank spike matrix, identity and a `Gegenbauer matrix' with entries $Q_\ell (\langle \textbf{x}_i , \textbf{x}_j \rangle)$, where $Q_\ell$ is the degree-$\ell$ Gegenbauer polynomial. We show that the spectrum of the Gegenbauer matrix converges in distribution to a Marchenko-Pastur law. This problem is motivated by the study of the prediction error of kernel ridge regression (KRR) in the polynomial regime $n \asymp d^\kappa, \kappa >0$. Previous work showed that for $\kappa \not\in \mathbb{N}$, KRR fits exactly a degree-$\lfloor \kappa \rfloor$ polynomial approximation to the target function. In this paper, we use our characterization of the kernel matrix to complete this picture and compute the precise asymptotics of the test error in the limit $n/d^\kappa \to \psi$ with $\kappa \in \mathbb{N}$. In this case, the test error can present a double descent behavior, depending on the effective regularization and signal-to-noise ratio at level $\kappa$. Because this double descent can occur each time $\kappa$ crosses an integer, this explains the multiple descent phenomenon in the KRR risk curve observed in several previous works.

The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks

It is currently known how to characterize functions that neural networks can learn with SGD for two extremal parameterizations: neural networks in the linear regime, and neural networks with no structural constraints. However, for the main parametrization of interest (non-linear but regular networks) no tight characterization has yet been achieved, despite significant developments. We take a step in this direction by considering depth-2 neural networks trained by SGD in the mean-field regime. We consider functions on binary inputs that depend on a latent low-dimensional subspace (i.e., small number of coordinates). This regime is of interest since it is poorly understood how neural networks routinely tackle high-dimensional datasets and adapt to latent low-dimensional structure without suffering from the curse of dimensionality. Accordingly, we study SGD-learnability with $O(d)$ sample complexity in a large ambient dimension $d$. Our main results characterize a hierarchical property, the "merged-staircase property", that is both necessary and nearly sufficient for learning in this setting. We further show that non-linear training is necessary: for this class of functions, linear methods on any feature map (e.g., the NTK) are not capable of learning efficiently. The key tools are a new "dimension-free" dynamics approximation result that applies to functions defined on a latent space of low-dimension, a proof of global convergence based on polynomial identity testing, and an improvement of lower bounds against linear methods for non-almost orthogonal functions.

Learning with convolution and pooling operations in kernel methods

Recent empirical work has shown that hierarchical convolutional kernels inspired by convolutional neural networks (CNNs) significantly improve the performance of kernel methods in image classification tasks. A widely accepted explanation for the success of these architectures is that they encode hypothesis classes that are suitable for natural images. However, understanding the precise interplay between approximation and generalization in convolutional architectures remains a challenge. In this paper, we consider the stylized setting of covariates (image pixels) uniformly distributed on the hypercube, and fully characterize the RKHS of kernels composed of single layers of convolution, pooling, and downsampling operations. We then study the gain in sample efficiency of kernel methods using these kernels over standard inner-product kernels. In particular, we show that 1) the convolution layer breaks the curse of dimensionality by restricting the RKHS to `local' functions; 2) local pooling biases learning towards low-frequency functions, which are stable by small translations; 3) downsampling may modify the high-frequency eigenspaces but leaves the low-frequency part approximately unchanged. Notably, our results quantify how choosing an architecture adapted to the target function leads to a large improvement in the sample complexity.

Minimum complexity interpolation in random features models

Despite their many appealing properties, kernel methods are heavily affected by the curse of dimensionality. For instance, in the case of inner product kernels in $\mathbb{R}^d$, the Reproducing Kernel Hilbert Space (RKHS) norm is often very large for functions that depend strongly on a small subset of directions (ridge functions). Correspondingly, such functions are difficult to learn using kernel methods. This observation has motivated the study of generalizations of kernel methods, whereby the RKHS norm -- which is equivalent to a weighted $\ell_2$ norm -- is replaced by a weighted functional $\ell_p$ norm, which we refer to as $\mathcal{F}_p$ norm. Unfortunately, tractability of these approaches is unclear. The kernel trick is not available and minimizing these norms requires to solve an infinite-dimensional convex problem. We study random features approximations to these norms and show that, for $p>1$, the number of random features required to approximate the original learning problem is upper bounded by a polynomial in the sample size. Hence, learning with $\mathcal{F}_p$ norms is tractable in these cases. We introduce a proof technique based on uniform concentration in the dual, which can be of broader interest in the study of overparametrized models.