Franck Gabriel

Arthur Jacot, Eugene Golikov, Clément Hongler et al.

We study the loss surface of DNNs with $L_{2}$ regularization. We show that the loss in terms of the parameters can be reformulated into a loss in terms of the layerwise activations $Z_{\ell}$ of the training set. This reformulation reveals the dynamics behind feature learning: each hidden representations $Z_{\ell}$ are optimal w.r.t. to an attraction/repulsion problem and interpolate between the input and output representations, keeping as little information from the input as necessary to construct the activation of the next layer. For positively homogeneous non-linearities, the loss can be further reformulated in terms of the covariances of the hidden representations, which takes the form of a partially convex optimization over a convex cone. This second reformulation allows us to prove a sparsity result for homogeneous DNNs: any local minimum of the $L_{2}$-regularized loss can 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 bound is tight by giving an example of a local minimum that requires $N^{2}/4$ hidden neurons. But we also observe numerically that in more traditional settings much less than $N^{2}$ neurons are required to reach the minima.

Arthur Jacot, François Ged, Berfin Şimşek et al.

The dynamics of Deep Linear Networks (DLNs) is dramatically affected by the variance $σ^2$ of the parameters at initialization $θ_0$. For DLNs of width $w$, we show a phase transition w.r.t. the scaling $γ$ of the variance $σ^2=w^{-γ}$ as $w\to\infty$: for large variance ($γ<1$), $θ_0$ is very close to a global minimum but far from any saddle point, and for small variance ($γ>1$), $θ_0$ is close to a saddle point and far from any global minimum. While the first case corresponds to the well-studied NTK regime, the second case is less understood. This motivates the study of the case $γ\to +\infty$, where we conjecture a Saddle-to-Saddle dynamics: throughout training, gradient descent visits the neighborhoods of a sequence of saddles, each corresponding to linear maps of increasing rank, until reaching a sparse global minimum. We support this conjecture with a theorem for the dynamics between the first two saddles, as well as some numerical experiments.

Sylvain Carré, Franck Gabriel, Clément Hongler et al.

Modern mathematics is built on the idea that proofs should be translatable into formal proofs, whose validity is an objective question, decidable by a computer. Yet, in practice, proofs are informal and may omit many details. An agent considers a proof valid if they trust that it could be expanded into a machine-verifiable proof. A proof's validity can thus become a subjective matter and lead to a debate, which may be difficult to settle. Hence, while the concept of valid proof is well-defined, the process to establish validity is itself a complex multi-agent problem. We introduce the SPRIG protocol. SPRIG allows agents to propose and verify succinct and informative proofs in a decentralized fashion; the trust is established by agents being able to request more details in the proof steps; debates, if they arise, must isolate details of proofs and, if they persist, go down to machine-level details, where they are automatically settled. A structure of bounties and stakes is set to incentivize agents to act in good faith. We propose a game-theoretic discussion of SPRIG, showing how agents with various types of information interact, leading to a proof tree with an appropriate level of detail and to the invalidation of wrong proofs, and we discuss resilience against various attacks. We then analyze a simplified model, characterize its equilibria and compute the agents' level of trust. SPRIG is designed to run as a smart contract on a blockchain platform. This allows anonymous agents to participate in the verification debate, and to contribute with their information. The smart contract mediates the interactions, settles debates, and guarantees that bounties and stakes are paid as specified. SPRIG enables new applications, such as the issuance of bounties for open problems, and the creation of derivatives markets, allowing agents to inject more information pertaining to proofs.

Arthur Jacot, Berfin Şimşek, Francesco Spadaro et al.

We study the risk (i.e. generalization error) of Kernel Ridge Regression (KRR) for a kernel $K$ with ridge $λ>0$ 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 $\vartheta_{K,λ}$ 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 $ρ_{K, λ}$, 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 analysis of the Stieltjes transform of 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, thus supporting our universality assumption. 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.

Arthur Jacot, Berfin Şimşek, Francesco Spadaro et al.

Random Feature (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 $λ$, we show that the average (i.e. expected) RF predictor is close to a KRR predictor with an effective ridge $\tildeλ$. We show that $\tildeλ > λ$ and $\tildeλ \searrow λ$ monotonically as $P$ grows, thus revealing the implicit regularization effect of finite RF sampling. We then compare the risk (i.e. test error) of the $\tildeλ$-KRR predictor with the average risk of the $λ$-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 $λ$-RF predictor and $\tildeλ$-KRR predictor.

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. When the NTK is fixed during training, we obtain a full characterization of the asymptotics of the spectrum of the Hessian, at initialization and during training. In the so-called mean-field limit, where the NTK is not fixed during training, we describe the first two moments of the Hessian at initialization.

Arthur Jacot, Franck Gabriel, François Ged et al.

We analyze architectural features of Deep Neural Networks (DNNs) using the so-called Neural Tangent Kernel (NTK), which describes the training and generalization of DNNs in the infinite-width setting. In this setting, we show that for fully-connected DNNs, as the depth grows, two regimes appear: "order", where the (scaled) NTK converges to a constant, and "chaos", where it converges to a Kronecker delta. Extreme order slows down training while extreme chaos hinders generalization. Using the scaled ReLU as a nonlinearity, we end up in the ordered regime. In contrast, Layer Normalization brings the network into the chaotic regime. We observe a similar effect for Batch Normalization (BN) applied after the last nonlinearity. We uncover the same order and chaos modes in Deep Deconvolutional Networks (DC-NNs). Our analysis explains the appearance of so-called checkerboard patterns and border artifacts. Moving the network into the chaotic regime prevents checkerboard patterns; we propose a graph-based parametrization which eliminates border artifacts; finally, we introduce a new layer-dependent learning rate to improve the convergence of DC-NNs. We illustrate our findings on DCGANs: the ordered regime leads to a collapse of the generator to a checkerboard mode, which can be avoided by tuning the nonlinearity to reach the chaotic regime. As a result, we are able to obtain good quality samples for DCGANs without BN.