Franck Gabriel

Franck Gabriel, François Ged, Maria Han Veiga et al.

Diffusion models now set the benchmark in high-fidelity generative sampling, yet they can, in principle, be prone to memorization. In this case, their learned score overfits the finite dataset so that the reverse-time SDE samples are mostly training points. In this paper, we interpret the empirical score as a noisy version of the true score and show that its covariance matrix is asymptotically a re-weighted data PCA. In large dimension, the small time limit makes the noise variance blow up while simultaneously reducing spatial correlation. To reduce this variance, we introduce a kernel-smoothed empirical score and analyze its bias-variance trade-off. We derive asymptotic bounds on the Kullback-Leibler divergence between the true distribution and the one generated by the modified reverse SDE. Regularization on the score has the same effect as increasing the size of the training dataset, and thus helps prevent memorization. A spectral decomposition of the forward diffusion suggests better variance control under some regularity conditions of the true data distribution. Reverse diffusion with kernel-smoothed empirical score can be reformulated as a gradient descent drifted toward a Log-Exponential Double-Kernel Density Estimator (LED-KDE). This perspective highlights two regularization mechanisms taking place in denoising diffusions: an initial Gaussian kernel first diffuses mass isotropically in the ambient space, while a second kernel applied in score space concentrates and spreads that mass along the data manifold. Hence, even a straightforward regularization-without any learning-already mitigates memorization and enhances generalization. Numerically, we illustrate our results with several experiments on synthetic and MNIST datasets.

Mario Geiger, Arthur Jacot, Stefano Spigler et al.

Supervised deep learning involves the training of neural networks with a large number $N$ of parameters. For large enough $N$, in the so-called over-parametrized regime, one can essentially fit the training data points. Sparsity-based arguments would suggest that the generalization error increases as $N$ grows past a certain threshold $N^{*}$. Instead, empirical studies have shown that in the over-parametrized regime, generalization error keeps decreasing with $N$. We resolve this paradox through a new framework. We rely on the so-called Neural Tangent Kernel, which connects large neural nets to kernel methods, to show that the initialization causes finite-size random fluctuations $\|f_{N}-\bar{f}_{N}\|\sim N^{-1/4}$ of the neural net output function $f_{N}$ around its expectation $\bar{f}_{N}$. These affect the generalization error $ε_{N}$ for classification: under natural assumptions, it decays to a plateau value $ε_{\infty}$ in a power-law fashion $\sim N^{-1/2}$. This description breaks down at a so-called jamming transition $N=N^{*}$. At this threshold, we argue that $\|f_{N}\|$ diverges. This result leads to a plausible explanation for the cusp in test error known to occur at $N^{*}$. Our results are confirmed by extensive empirical observations on the MNIST and CIFAR image datasets. Our analysis finally suggests that, given a computational envelope, the smallest generalization error is obtained using several networks of intermediate sizes, just beyond $N^{*}$, and averaging their outputs.

Arthur Jacot, Franck Gabriel, Clément 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 $f_θ$ (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: 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 it 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 prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function $f_θ$ 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.