A number of machine learning tasks entail a high degree of invariance: the data distribution does not change if we act on the data with a certain group of transformations. For instance, labels of images are invariant under translations of the images. Certain neural network architectures -- for instance, convolutional networks -- are believed to owe their success to the fact that they exploit such invariance properties. With the objective of quantifying the gain achieved by invariant architectures, we introduce two classes of models: invariant random features and invariant kernel methods. The latter includes, as a special case, the neural tangent kernel for convolutional networks with global average pooling. We consider uniform covariates distributions on the sphere and hypercube and a general invariant target function. We characterize the test error of invariant methods in a high-dimensional regime in which the sample size and number of hidden units scale as polynomials in the dimension, for a class of groups that we call `degeneracy $\alpha$', with $\alpha \leq 1$. We show that exploiting invariance in the architecture saves a $d^\alpha$ factor ($d$ stands for the dimension) in sample size and number of hidden units to achieve the same test error as for unstructured architectures. Finally, we show that output symmetrization of an unstructured kernel estimator does not give a significant statistical improvement; on the other hand, data augmentation with an unstructured kernel estimator is equivalent to an invariant kernel estimator and enjoys the same improvement in statistical efficiency.

Modern machine learning models such as deep neural networks with high accuracy tend to be miscalibrated: The predicted top probability (confidence) does not reflect the actual accuracy of the model, and tends to be over-confident. For example, a WideResNet 32 on CIFAR100 has on average a predicted top probability of 87%, while the actual test accuracy is only 72% (Guo et al., 2017). As the confidence is often comprehended as an estimate of the true accuracy, such over-confidence could be dangerous, especially in risk-sensitive domains such as medical AI (Begoli et al., 2019), self-driving cars (Michelmore et al., 2018), and so on. To address this issue, there is a growing line of research on improving the calibration of models, by either performing recalibration of well-trained models to adjust the confidence scores (Platt et al., 1999; Zadrozny and Elkan; Naeini et al., 2015; Guo et al., 2017), or by averaging the predictions over multiple models to make the confidence scores more accurate (Lakshminarayanan et al., 2016; Gal and Ghahramani, 2016). These methods in general can reduce the over-confidence and improve the calibration of the model, while preserving (or even improving) the model's accuracy (Ovadia et al., 2019). Despite these progresses, the more fundamental question of why such over-confidence happens for vanillaly trained models remains not satisfactorily understood.

Consider the classical supervised learning problem: we are given data $(y_i,{\boldsymbol x}_i)$, $i\le n$, with $y_i$ a response and ${\boldsymbol x}_i\in {\mathcal X}$ a covariates vector, and try to learn a model $f:{\mathcal X}\to{\mathbb R}$ to predict future responses. Random features methods map the covariates vector ${\boldsymbol x}_i$ to a point ${\boldsymbol \phi}({\boldsymbol x}_i)$ in a higher dimensional space ${\mathbb R}^N$, via a random featurization map ${\boldsymbol \phi}$. We study the use of random features methods in conjunction with ridge regression in the feature space ${\mathbb R}^N$. This can be viewed as a finite-dimensional approximation of kernel ridge regression (KRR), or as a stylized model for neural networks in the so called lazy training regime. We define a class of problems satisfying certain spectral conditions on the underlying kernels, and a hypercontractivity assumption on the associated eigenfunctions. These conditions are verified by classical high-dimensional examples. Under these conditions, we prove a sharp characterization of the error of random features ridge regression. In particular, we address two fundamental questions: $(1)$~What is the generalization error of KRR? $(2)$~How big $N$ should be for the random features approximation to achieve the same error as KRR? In this setting, we prove that KRR is well approximated by a projection onto the top $\ell$ eigenfunctions of the kernel, where $\ell$ depends on the sample size $n$. We show that the test error of random features ridge regression is dominated by its approximation error and is larger than the error of KRR as long as $N\le n^{1-\delta}$ for some $\delta>0$. We characterize this gap. For $N\ge n^{1+\delta}$, random features achieve the same error as the corresponding KRR, and further increasing $N$ does not lead to a significant change in test error.

For a certain scaling of the initialization of stochastic gradient descent (SGD), wide neural networks (NN) have been shown to be well approximated by reproducing kernel Hilbert space (RKHS) methods. Recent empirical work showed that, for some classification tasks, RKHS methods can replace NNs without a large loss in performance. On the other hand, two-layers NNs are known to encode richer smoothness classes than RKHS and we know of special examples for which SGD-trained NN provably outperform RKHS. This is true even in the wide network limit, for a different scaling of the initialization. How can we reconcile the above claims? For which tasks do NNs outperform RKHS? If feature vectors are nearly isotropic, RKHS methods suffer from the curse of dimensionality, while NNs can overcome it by learning the best low-dimensional representation. Here we show that this curse of dimensionality becomes milder if the feature vectors display the same low-dimensional structure as the target function, and we precisely characterize this tradeoff. Building on these results, we present a model that can capture in a unified framework both behaviors observed in earlier work. We hypothesize that such a latent low-dimensional structure is present in image classification. We test numerically this hypothesis by showing that specific perturbations of the training distribution degrade the performances of RKHS methods much more significantly than NNs.

Deep learning methods operate in regimes that defy the traditional statistical mindset. The neural network architectures often contain more parameters than training samples, and are so rich that they can interpolate the observed labels, even if the latter are replaced by pure noise. Despite their huge complexity, the same architectures achieve small generalization error on real data. This phenomenon has been rationalized in terms of a so-called `double descent' curve. As the model complexity increases, the generalization error follows the usual U-shaped curve at the beginning, first decreasing and then peaking around the interpolation threshold (when the model achieves vanishing training error). However, it descends again as model complexity exceeds this threshold. The global minimum of the generalization error is found in this overparametrized regime, often when the number of parameters is much larger than the number of samples. Far from being a peculiar property of deep neural networks, elements of this behavior have been demonstrated in much simpler settings, including linear regression with random covariates. In this paper we consider the problem of learning an unknown function over the $d$-dimensional sphere $\mathbb S^{d-1}$, from $n$ i.i.d. samples $(\boldsymbol x_i, y_i) \in \mathbb S^{d-1} \times \mathbb R$, $i \le n$. We perform ridge regression on $N$ random features of the form $\sigma(\boldsymbol w_a^{\mathsf T}\boldsymbol x)$, $a \le N$. This can be equivalently described as a two-layers neural network with random first-layer weights. We compute the precise asymptotics of the generalization error, in the limit $N, n, d \to \infty$ with $N/d$ and $n/d$ fixed. This provides the first analytically tractable model that captures all the features of the double descent phenomenon.

We study the supervised learning problem under either of the following two models: (1) Feature vectors ${\boldsymbol x}_i$ are $d$-dimensional Gaussians and responses are $y_i = f_*({\boldsymbol x}_i)$ for $f_*$ an unknown quadratic function; (2) Feature vectors ${\boldsymbol x}_i$ are distributed as a mixture of two $d$-dimensional centered Gaussians, and $y_i$'s are the corresponding class labels. We use two-layers neural networks with quadratic activations, and compare three different learning regimes: the random features (RF) regime in which we only train the second-layer weights; the neural tangent (NT) regime in which we train a linearization of the neural network around its initialization; the fully trained neural network (NN) regime in which we train all the weights in the network. We prove that, even for the simple quadratic model of point (1), there is a potentially unbounded gap between the prediction risk achieved in these three training regimes, when the number of neurons is smaller than the ambient dimension. When the number of neurons is larger than the number of dimensions, the problem is significantly easier and both NT and NN learning achieve zero risk.

We consider learning two layer neural networks using stochastic gradient descent. The mean-field description of this learning dynamics approximates the evolution of the network weights by an evolution in the space of probability distributions in $R^D$ (where $D$ is the number of parameters associated to each neuron). This evolution can be defined through a partial differential equation or, equivalently, as the gradient flow in the Wasserstein space of probability distributions. Earlier work shows that (under some regularity assumptions), the mean field description is accurate as soon as the number of hidden units is much larger than the dimension $D$. In this paper we establish stronger and more general approximation guarantees. First of all, we show that the number of hidden units only needs to be larger than a quantity dependent on the regularity properties of the data, and independent of the dimensions. Next, we generalize this analysis to the case of unbounded activation functions, which was not covered by earlier bounds. We extend our results to noisy stochastic gradient descent. Finally, we show that kernel ridge regression can be recovered as a special limit of the mean field analysis.

Multi-layer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a non-convex high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties? In this paper we consider a simple case, namely two-layers neural networks, and prove that -in a suitable scaling limit- SGD dynamics is captured by a certain non-linear partial differential equation (PDE) that we call distributional dynamics (DD). We then consider several specific examples, and show how DD can be used to prove convergence of SGD to networks with nearlyideal generalization error. This description allows to 'average-out' some of the complexities of the landscape of neural networks, and can be used to prove a general convergence result for noisy SGD.

We consider the problem of estimating a large rank-one tensor ${\boldsymbol u}^{\otimes k}\in({\mathbb R}^{n})^{\otimes k}$, $k\ge 3$ in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio $\lambda_{Bayes}= O(1)$ above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably no polynomial-time algorithm is known that achieved this goal unless $\lambda\ge C n^{(k-2)/4}$ and even powerful semidefinite programming relaxations appear to fail for $1\ll \lambda\ll n^{(k-2)/4}$. In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-$k$ homogeneous polynomial over the unit sphere in $n$ dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions $n$, and give exact formulas for the exponential growth rate. We show that (for $\lambda$ larger than a constant) critical points are either very close to the unknown vector ${\boldsymbol u}$, or are confined in a band of width $\Theta(\lambda^{-1/(k-1)})$ around the maximum circle that is orthogonal to ${\boldsymbol u}$. For local maxima, this band shrinks to be of size $\Theta(\lambda^{-1/(k-2)})$. These `uninformative' local maxima are likely to cause the failure of optimization algorithms.

A number of statistical estimation problems can be addressed by semidefinite programs (SDP). While SDPs are solvable in polynomial time using interior point methods, in practice generic SDP solvers do not scale well to high-dimensional problems. In order to cope with this problem, Burer and Monteiro proposed a non-convex rank-constrained formulation, which has good performance in practice but is still poorly understood theoretically. In this paper we study the rank-constrained version of SDPs arising in MaxCut and in synchronization problems. We establish a Grothendieck-type inequality that proves that all the local maxima and dangerous saddle points are within a small multiplicative gap from the global maximum. We use this structural information to prove that SDPs can be solved within a known accuracy, by applying the Riemannian trust-region method to this non-convex problem, while constraining the rank to be of order one. For the MaxCut problem, our inequality implies that any local maximizer of the rank-constrained SDP provides a $ (1 - 1/(k-1)) \times 0.878$ approximation of the MaxCut, when the rank is fixed to $k$. We then apply our results to data matrices generated according to the Gaussian ${\mathbb Z}_2$ synchronization problem, and the two-groups stochastic block model with large bounded degree. We prove that the error achieved by local maximizers undergoes a phase transition at the same threshold as for information-theoretically optimal methods.