Franck Gabriel

NeurIPS Conference 2022 Conference Paper

Feature Learning in $L_2$-regularized DNNs: Attraction/Repulsion and Sparsity

• Arthur Jacot
• Eugene Golikov
• Clement Hongler
• Franck Gabriel

We study the loss surface of DNNs with $L_{2}$ regularization. Weshow that the loss in terms of the parameters can be reformulatedinto a loss in terms of the layerwise activations $Z_{\ell}$ of thetraining set. This reformulation reveals the dynamics behind featurelearning: each hidden representations $Z_{\ell}$ are optimal w. r. t. to an attraction/repulsion problem and interpolate between the inputand output representations, keeping as little information from theinput as necessary to construct the activation of the next layer. For positively homogeneous non-linearities, the loss can be furtherreformulated in terms of the covariances of the hidden representations, which takes the form of a partially convex optimization over a convexcone. This second reformulation allows us to prove a sparsity result forhomogeneous DNNs: any local minimum of the $L_{2}$-regularized losscan be achieved with at most $N(N+1)$ neurons in each hidden layer(where $N$ is the size of the training set). We show that this boundis tight by giving an example of a local minimum that requires $N^{2}/4$hidden neurons. But we also observe numerically that in more traditionalsettings much less than $N^{2}$ neurons are required to reach theminima.

STOC Conference 2021 Invited Paper

Neural tangent kernel: convergence and generalization in neural networks (invited paper)

• Arthur Jacot
• Franck Gabriel
• Clément Hongler

ICML Conference 2020 Conference Paper

Implicit Regularization of Random Feature Models

• Arthur Jacot
• Berfin Simsek
• Francesco Spadaro
• Clément Hongler
• Franck Gabriel

Random Features (RF) models are used as efficient parametric approximations of kernel methods. We investigate, by means of random matrix theory, the connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a Gaussian RF model with $P$ features, $N$ data points, and a ridge $\lambda$, we show that the average (i. e. expected) RF predictor is close to a KRR predictor with an \emph{effective ridge} $\tilde{\lambda}$. We show that $\tilde{\lambda} > \lambda$ and $\tilde{\lambda} \searrow \lambda$ monotonically as $P$ grows, thus revealing the \emph{implicit regularization effect} of finite RF sampling. We then compare the risk (i. e. test error) of the $\tilde{\lambda}$-KRR predictor with the average risk of the $\lambda$-RF predictor and obtain a precise and explicit bound on their difference. Finally, we empirically find an extremely good agreement between the test errors of the average $\lambda$-RF predictor and $\tilde{\lambda}$-KRR predictor.

NeurIPS Conference 2020 Conference Paper

Kernel Alignment Risk Estimator: Risk Prediction from Training Data

• Arthur Jacot
• Berfin Simsek
• Francesco Spadaro
• Clement Hongler
• Franck Gabriel

We study the risk (i. e. generalization error) of Kernel Ridge Regression (KRR) for a kernel $K$ with ridge $\lambda>0$ and i. i. d. observations. For this, we introduce two objects: the Signal Capture Threshold (SCT) and the Kernel Alignment Risk Estimator (KARE). The SCT $\vartheta_{K, \lambda}$ is a function of the data distribution: it can be used to identify the components of the data that the KRR predictor captures, and to approximate the (expected) KRR risk. This then leads to a KRR risk approximation by the KARE $\rho_{K, \lambda}$, an explicit function of the training data, agnostic of the true data distribution. We phrase the regression problem in a functional setting. The key results then follow from a finite-size adaptation of the resolvent method for general Wishart random matrices. Under a natural universality assumption (that the KRR moments depend asymptotically on the first two moments of the observations) we capture the mean and variance of the KRR predictor. We numerically investigate our findings on the Higgs and MNIST datasets for various classical kernels: the KARE gives an excellent approximation of the risk. This supports our universality hypothesis. Using the KARE, one can compare choices of Kernels and hyperparameters directly from the training set. The KARE thus provides a promising data-dependent procedure to select Kernels that generalize well.

ICLR Conference 2020 Conference Paper

The asymptotic spectrum of the Hessian of DNN throughout training

• Arthur Jacot
• Franck Gabriel
• Clément Hongler

The dynamics of DNNs during gradient descent is described by the so-called Neural Tangent Kernel (NTK). In this article, we show that the NTK allows one to gain precise insight into the Hessian of the cost of DNNs: we obtain a full characterization of the asymptotics of the spectrum of the Hessian, at initialization and during training.

NeurIPS Conference 2018 Conference Paper

Neural Tangent Kernel: Convergence and Generalization in Neural Networks

• Arthur Jacot
• Franck Gabriel
• Clement Hongler

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function (which maps input vectors to output vectors) follows the so-called kernel gradient associated with a new object, which we call the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.