Kernel Density Estimators in Large Dimensions

This paper studies Kernel density estimation for a high-dimensional distribution $\rho(x)$. Traditional approaches have focused on the limit of large number of data points $n$ and fixed dimension $d$. We analyze instead the regime where both the number $n$ of data points $y_i$ and their dimensionality $d$ grow with a fixed ratio $\alpha=(\log n)/d$. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density $\hat \rho_h^{\mathcal {D}}(x)=\frac{1}{n h^d}\sum_{i=1}^n K\left(\frac{x-y_i}{h}\right)$, depending on the bandwidth $h$: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, $h_{CLT}(\alpha)$, we find that the CLT breaks down. The statistics of $\hat \rho_h^{\mathcal {D}}(x)$ for a fixed $x$ drawn from $\rho(x)$ is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value $h_G(\alpha)$, we find that $\hat \rho_h^{\mathcal {D}}(x)$ is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. Our findings reveal limitations of classical approaches, show the relevance of these new statistical regimes, and offer new insights for Kernel density estimation in high-dimensional settings.

Cascade of phase transitions in the training of Energy-based models

In this paper, we investigate the feature encoding process in a prototypical energy-based generative model, the Restricted Boltzmann Machine (RBM). We start with an analytical investigation using simplified architectures and data structures, and end with numerical analysis of real trainings on real datasets. Our study tracks the evolution of the model's weight matrix through its singular value decomposition, revealing a series of phase transitions associated to a progressive learning of the principal modes of the empirical probability distribution. The model first learns the center of mass of the modes and then progressively resolve all modes through a cascade of phase transitions. We first describe this process analytically in a controlled setup that allows us to study analytically the training dynamics. We then validate our theoretical results by training the Bernoulli-Bernoulli RBM on real data sets. By using data sets of increasing dimension, we show that learning indeed leads to sharp phase transitions in the high-dimensional limit. Moreover, we propose and test a mean-field finite-size scaling hypothesis. This shows that the first phase transition is in the same universality class of the one we studied analytically, and which is reminiscent of the mean-field paramagnetic-to-ferromagnetic phase transition.

From Zero to Hero: How local curvature at artless initial conditions leads away from bad minima

We investigate the optimization dynamics of gradient descent in a non-convex and high-dimensional setting, with a focus on the phase retrieval problem as a case study for complex loss landscapes. We first study the high-dimensional limit where both the number $M$ and the dimension $N$ of the data are going to infinity at fixed signal-to-noise ratio $\alpha = M/N$. By analyzing how the local curvature changes during optimization, we uncover that for intermediate $\alpha$, the Hessian displays a downward direction pointing towards good minima in the first regime of the descent, before being trapped in bad minima at the end. Hence, the local landscape is benign and informative at first, before gradient descent brings the system into a uninformative maze. The transition between the two regimes is associated to a BBP-type threshold in the time-dependent Hessian. Through both theoretical analysis and numerical experiments, we show that in practical cases, i.e. for finite but even very large $N$, successful optimization via gradient descent in phase retrieval is achieved by falling towards the good minima before reaching the bad ones. This mechanism explains why successful recovery is obtained well before the algorithmic transition corresponding to the high-dimensional limit. Technically, this is associated to strong logarithmic corrections of the algorithmic transition at large $N$ with respect to the one expected in the $N\to\infty$ limit. Our analysis sheds light on such a new mechanism that facilitate gradient descent dynamics in finite large dimensions, also highlighting the importance of good initialization of spectral properties for optimization in complex high-dimensional landscapes.

Dynamical Regimes of Diffusion Models

Using statistical physics methods, we study generative diffusion models in the regime where the dimension of space and the number of data are large, and the score function has been trained optimally. Our analysis reveals three distinct dynamical regimes during the backward generative diffusion process. The generative dynamics, starting from pure noise, encounters first a 'speciation' transition where the gross structure of data is unraveled, through a mechanism similar to symmetry breaking in phase transitions. It is followed at later time by a 'collapse' transition where the trajectories of the dynamics become attracted to one of the memorized data points, through a mechanism which is similar to the condensation in a glass phase. For any dataset, the speciation time can be found from a spectral analysis of the correlation matrix, and the collapse time can be found from the estimation of an 'excess entropy' in the data. The dependence of the collapse time on the dimension and number of data provides a thorough characterization of the curse of dimensionality for diffusion models. Analytical solutions for simple models like high-dimensional Gaussian mixtures substantiate these findings and provide a theoretical framework, while extensions to more complex scenarios and numerical validations with real datasets confirm the theoretical predictions.

On the Impact of Overparameterization on the Training of a Shallow Neural Network in High Dimensions

We study the training dynamics of a shallow neural network with quadratic activation functions and quadratic cost in a teacher-student setup. In line with previous works on the same neural architecture, the optimization is performed following the gradient flow on the population risk, where the average over data points is replaced by the expectation over their distribution, assumed to be Gaussian.We first derive convergence properties for the gradient flow and quantify the overparameterization that is necessary to achieve a strong signal recovery. Then, assuming that the teachers and the students at initialization form independent orthonormal families, we derive a high-dimensional limit for the flow and show that the minimal overparameterization is sufficient for strong recovery. We verify by numerical experiments that these results hold for more general initializations.

Wavelet Conditional Renormalization Group

We develop a multiscale approach to estimate high-dimensional probability distributions from a dataset of physical fields or configurations observed in experiments or simulations. In this way we can estimate energy functions (or Hamiltonians) and efficiently generate new samples of many-body systems in various domains, from statistical physics to cosmology. Our method -- the Wavelet Conditional Renormalization Group (WC-RG) -- proceeds scale by scale, estimating models for the conditional probabilities of "fast degrees of freedom" conditioned by coarse-grained fields. These probability distributions are modeled by energy functions associated with scale interactions, and are represented in an orthogonal wavelet basis. WC-RG decomposes the microscopic energy function as a sum of interaction energies at all scales and can efficiently generate new samples by going from coarse to fine scales. Near phase transitions, it avoids the "critical slowing down" of direct estimation and sampling algorithms. This is explained theoretically by combining results from RG and wavelet theories, and verified numerically for the Gaussian and $\varphi^4$ field theories. We show that multiscale WC-RG energy-based models are more general than local potential models and can capture the physics of complex many-body interacting systems at all length scales. This is demonstrated for weak-gravitational-lensing fields reflecting dark matter distributions in cosmology, which include long-range interactions with long-tail probability distributions. WC-RG has a large number of potential applications in non-equilibrium systems, where the underlying distribution is not known {\it a priori}. Finally, we discuss the connection between WC-RG and deep network architectures.

Optimal learning rate schedules in high-dimensional non-convex optimization problems

Learning rate schedules are ubiquitously used to speed up and improve optimisation. Many different policies have been introduced on an empirical basis, and theoretical analyses have been developed for convex settings. However, in many realistic problems the loss-landscape is high-dimensional and non convex -- a case for which results are scarce. In this paper we present a first analytical study of the role of learning rate scheduling in this setting, focusing on Langevin optimization with a learning rate decaying as $\eta(t)=t^{-\beta}$. We begin by considering models where the loss is a Gaussian random function on the $N$-dimensional sphere ($N\rightarrow \infty$), featuring an extensive number of critical points. We find that to speed up optimization without getting stuck in saddles, one must choose a decay rate $\beta<1$, contrary to convex setups where $\beta=1$ is generally optimal. We then add to the problem a signal to be recovered. In this setting, the dynamics decompose into two phases: an \emph{exploration} phase where the dynamics navigates through rough parts of the landscape, followed by a \emph{convergence} phase where the signal is detected and the dynamics enter a convex basin. In this case, it is optimal to keep a large learning rate during the exploration phase to escape the non-convex region as quickly as possible, then use the convex criterion $\beta=1$ to converge rapidly to the solution. Finally, we demonstrate that our conclusions hold in a common regression task involving neural networks.

Transformed CNNs: recasting pre-trained convolutional layers with self-attention

Vision Transformers (ViT) have recently emerged as a powerful alternative to convolutional networks (CNNs). Although hybrid models attempt to bridge the gap between these two architectures, the self-attention layers they rely on induce a strong computational bottleneck, especially at large spatial resolutions. In this work, we explore the idea of reducing the time spent training these layers by initializing them as convolutional layers. This enables us to transition smoothly from any pre-trained CNN to its functionally identical hybrid model, called Transformed CNN (T-CNN). With only 50 epochs of fine-tuning, the resulting T-CNNs demonstrate significant performance gains over the CNN (+2.2% top-1 on ImageNet-1k for a ResNet50-RS) as well as substantially improved robustness (+11% top-1 on ImageNet-C). We analyze the representations learnt by the T-CNN, providing deeper insights into the fruitful interplay between convolutions and self-attention. Finally, we experiment initializing the T-CNN from a partially trained CNN, and find that it reaches better performance than the corresponding hybrid model trained from scratch, while reducing training time.

Sifting out the features by pruning: Are convolutional networks the winning lottery ticket of fully connected ones?

Pruning methods can considerably reduce the size of artificial neural networks without harming their performance. In some cases, they can even uncover sub-networks that, when trained in isolation, match or surpass the test accuracy of their dense counterparts. Here we study the inductive bias that pruning imprints in such "winning lottery tickets". Focusing on visual tasks, we analyze the architecture resulting from iterative magnitude pruning of a simple fully connected network (FCN). We show that the surviving node connectivity is local in input space, and organized in patterns reminiscent of the ones found in convolutional networks (CNN). We investigate the role played by data and tasks in shaping the architecture of pruned sub-networks. Our results show that the winning lottery tickets of FCNs display the key features of CNNs. The ability of such automatic network-simplifying procedure to recover the key features "hand-crafted" in the design of CNNs suggests interesting applications to other datasets and tasks, in order to discover new and efficient architectural inductive biases.

ConViT: Improving Vision Transformers with Soft Convolutional Inductive Biases

Convolutional architectures have proven extremely successful for vision tasks. Their hard inductive biases enable sample-efficient learning, but come at the cost of a potentially lower performance ceiling. Vision Transformers (ViTs) rely on more flexible self-attention layers, and have recently outperformed CNNs for image classification. However, they require costly pre-training on large external datasets or distillation from pre-trained convolutional networks. In this paper, we ask the following question: is it possible to combine the strengths of these two architectures while avoiding their respective limitations? To this end, we introduce gated positional self-attention (GPSA), a form of positional self-attention which can be equipped with a "soft" convolutional inductive bias. We initialize the GPSA layers to mimic the locality of convolutional layers, then give each attention head the freedom to escape locality by adjusting a gating parameter regulating the attention paid to position versus content information. The resulting convolutional-like ViT architecture, ConViT, outperforms the DeiT on ImageNet, while offering a much improved sample efficiency. We further investigate the role of locality in learning by first quantifying how it is encouraged in vanilla self-attention layers, then analyzing how it is escaped in GPSA layers. We conclude by presenting various ablations to better understand the success of the ConViT. Our code and models are released publicly.