Let a smooth function $f: T^d \times T^d \to \mathbb{R}$ over the $d$-torus and $\beta>0$. Consider the min-max objective functional $F_\beta(\mu, \nu) = \iint f d\mu d\nu + \beta^{-1} H(\mu) - \beta^{-1} H(\nu)$ over $\mathcal{P}(T^d) \times \mathcal{P}(T^d)$, where $H$ denotes the negative differential entropy. Its unique saddle point defines the entropy-regularized mixed Nash equilibrium of a two-player zero-sum game, and its Wasserstein gradient descent-ascent flow $(\mu_t, \nu_t)$ corresponds to the mean-field limit of a Langevin descent-ascent dynamics. Do $\mu_t$ and $\nu_t$ converge (weakly, say) as $t \to \infty$, for any $f$ and $\beta$? This rather natural qualitative question is still open, and it is not clear whether it can be addressed using the tools currently available for the analysis of dynamics in Wasserstein space. Even though the simple trick of using a different timescale for the ascent versus the descent is known to guarantee convergence, we propose this question as a toy setting to further our understanding of the Wasserstein geometry for optimization.

Deep learning succeeds by doing hierarchical feature learning, yet tuning hyper-parameters (HP) such as initialization scales, learning rates etc., only give indirect control over this behavior. In this paper, we introduce a key notion to predict and control feature learning: the angle $\theta_\ell$ between the feature updates and the backward pass (at layer index $\ell$). We show that the magnitude of feature updates after one GD step, at any training time, can be expressed via a simple and general *feature speed formula* in terms of this angle $\theta_\ell$, the loss decay, and the magnitude of the backward pass. This angle $\theta_\ell$ is controlled by the conditioning of the layer-to-layer Jacobians and at random initialization, it is determined by the spectrum of a certain kernel, which coincides with the Neural Tangent Kernel when $\ell=\text{depth}$. Given $\theta_\ell$, the feature speed formula provides us with rules to adjust HPs (scales and learning rates) so as to satisfy certain dynamical properties, such as feature learning and loss decay. We investigate the implications of our approach for ReLU MLPs and ResNets in the large width-then-depth limit. Relying on prior work, we show that in ReLU MLPs with iid initialization, the angle degenerates with depth as $\cos(\theta_\ell)=\Theta(1/\sqrt{\ell})$. In contrast, ResNets with branch scale $O(1/\sqrt{\text{depth}})$ maintain a non-degenerate angle $\cos(\theta_\ell)=\Theta(1)$. We use these insights to recover key properties of known HP scalings (such as $\mu$P), and also introduce a new HP scaling for large depth ReLU MLPs with favorable theoretical properties.

The largest eigenvalue of the Hessian, or sharpness, of neural networks is a key quantity to understand their optimization dynamics. In this paper, we study the sharpness of deep linear networks for univariate regression. Minimizers can have arbitrarily large sharpness, but not an arbitrarily small one. Indeed, we show a lower bound on the sharpness of minimizers, which grows linearly with depth. We then study the properties of the minimizer found by gradient flow, which is the limit of gradient descent with vanishing learning rate. We show an implicit regularization towards flat minima: the sharpness of the minimizer is no more than a constant times the lower bound. The constant depends on the condition number of the data covariance matrix, but not on width or depth. This result is proven both for a small-scale initialization and a residual initialization. Results of independent interest are shown in both cases. For small-scale initialization, we show that the learned weight matrices are approximately rank-one and that their singular vectors align. For residual initialization, convergence of the gradient flow for a Gaussian initialization of the residual network is proven. Numerical experiments illustrate our results and connect them to gradient descent with non-vanishing learning rate.

Mean-field Langevin dynamics (MLFD) is a class of interacting particle methods that tackle convex optimization over probability measures on a manifold, which are scalable, versatile, and enjoy computational guarantees. However, some important problems -- such as risk minimization for infinite width two-layer neural networks, or sparse deconvolution -- are originally defined over the set of signed, rather than probability, measures. In this paper, we investigate how to extend the MFLD framework to convex optimization problems over signed measures. Among two known reductions from signed to probability measures -- the lifting and the bilevel approaches -- we show that the bilevel reduction leads to stronger guarantees and faster rates (at the price of a higher per-iteration complexity). In particular, we investigate the convergence rate of MFLD applied to the bilevel reduction in the low-noise regime and obtain two results. First, this dynamics is amenable to an annealing schedule, adapted from [Suzuki et al., 2023], that results in polynomial convergence rates to a fixed multiplicative accuracy. Second, we investigate the problem of learning a single neuron with the bilevel approach and obtain local exponential convergence rates that depend polynomially on the dimension and noise level (to compare with the exponential dependence that would result from prior analyses).

To theoretically understand the behavior of trained deep neural networks, it is necessary to study the dynamics induced by gradient methods from a random initialization. However, the nonlinear and compositional structure of these models make these dynamics difficult to analyze. To overcome these challenges, large-width asymptotics have recently emerged as a fruitful viewpoint and led to practical insights on real-world deep networks. For two-layer neural networks, it has been understood via these asymptotics that the nature of the trained model radically changes depending on the scale of the initial random weights, ranging from a kernel regime (for large initial variance) to a feature learning regime (for small initial variance). For deeper networks more regimes are possible, and in this paper we study in detail a specific choice of "small" initialization corresponding to "mean-field" limits of neural networks, which we call integrable parameterizations (IPs). First, we show that under standard i.i.d. zero-mean initialization, integrable parameterizations of neural networks with more than four layers start at a stationary point in the infinite-width limit and no learning occurs. We then propose various methods to avoid this trivial behavior and analyze in detail the resulting dynamics. In particular, one of these methods consists in using large initial learning rates, and we show that it is equivalent to a modification of the recently proposed maximal update parameterization µP. We confirm our results with numerical experiments on image classification tasks, which additionally show a strong difference in behavior between various choices of activation functions that is not yet captured by theory.

In supervised learning, the regularization path is sometimes used as a convenient theoretical proxy for the optimization path of gradient descent initialized from zero. In this paper, we study a modification of the regularization path for infinite-width 2-layer ReLU neural networks with nonzero initial distribution of the weights at different scales. By exploiting a link with unbalanced optimal-transport theory, we show that, despite the non-convexity of the 2-layer network training, this problem admits an infinite-dimensional convex counterpart. We formulate the corresponding functional-optimization problem and investigate its main properties. In particular, we show that, as the scale of the initialization ranges between $0$ and $+\infty$, the associated path interpolates continuously between the so-called kernel and rich regimes. Numerical experiments confirm that, in our setting, the scaling path and the final states of the optimization path behave similarly, even beyond these extreme points.

We study the computation of doubly regularized Wasserstein barycenters, a recently introduced family of entropic barycenters governed by inner and outer regularization strengths. Previous research has demonstrated that various regularization parameter choices unify several notions of entropy-penalized barycenters while also revealing new ones, including a special case of debiased barycenters. In this paper, we propose and analyze an algorithm for computing doubly regularized Wasserstein barycenters. Our procedure builds on damped Sinkhorn iterations followed by exact maximization/minimization steps and guarantees convergence for any choice of regularization parameters. An inexact variant of our algorithm, implementable using approximate Monte Carlo sampling, offers the first non-asymptotic convergence guarantees for approximating Wasserstein barycenters between discrete point clouds in the free-support/grid-free setting.

We study the convergence to local Nash equilibria of gradient methods for two-player zero-sum differentiable games. It is well-known that, in the continuous-time setting, such dynamics converge locally when $S \succ 0$ and may diverge when $S=0$, where $S\succeq 0$ is the symmetric part of the Jacobian at equilibrium that accounts for the "potential" component of the game. We show that these dynamics also converge as soon as $S$ is nonzero (*partial curvature*) and the eigenvectors of the antisymmetric part $A$ are in general position with respect to the kernel of $S$. We then study the convergence rate when $S \ll A$ and prove that it typically depends on the *average* of the eigenvalues of $S$, instead of the minimum as an analogy with minimization problems would suggest. To illustrate our results, we consider the problem of computing mixed Nash equilibria of continuous games. We show that, thanks to partial curvature, conic particle methods -- which optimize over both weights and supports of the mixed strategies -- generically converge faster than fixed-support methods. For min-max games, it is thus beneficial to add degrees of freedom "with curvature": this can be interpreted as yet another benefit of over-parameterization.

Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals. To solve this task, a min-entropy estimator relative to the Wiener measure in path space was introduced in [Lavenant et al., 2021], and shown to consistently recover the dynamics of a large class of drift-diffusion processes from the solution of an infinite dimensional convex optimization problem. In this paper, we introduce a grid-free algorithm to compute this estimator. Our method consists in a family of point clouds (one per snapshot) coupled via Schrödinger bridges which evolve with noisy gradient descent. We study the mean-field limit of the dynamics and prove its global convergence to the desired estimator. Overall, this leads to an inference method with end-to-end theoretical guarantees that solves an interpretable model for trajectory inference. We also present how to adapt the method to deal with mass variations, a useful extension when dealing with single cell RNA-sequencing data where cells can branch and die.

Neural networks trained to minimize the logistic (a.k.a. cross-entropy) loss with gradient-based methods are observed to perform well in many supervised classification tasks. Towards understanding this phenomenon, we analyze the training and generalization behavior of infinitely wide two-layer neural networks with homogeneous activations. We show that the limits of the gradient flow on exponentially tailed losses can be fully characterized as a max-margin classifier in a certain non-Hilbertian space of functions. In presence of hidden low-dimensional structures, the resulting margin is independent of the ambiant dimension, which leads to strong generalization bounds. In contrast, training only the output layer implicitly solves a kernel support vector machine, which a priori does not enjoy such an adaptivity. Our analysis of training is non-quantitative in terms of running time but we prove computational guarantees in simplified settings by showing equivalences with online mirror descent. Finally, numerical experiments suggest that our analysis describes well the practical behavior of two-layer neural networks with ReLU activation and confirm the statistical benefits of this implicit bias.

The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which can be solved to $\epsilon$-accuracy by adding an entropic regularization of order $\epsilon$ and using for instance Sinkhorn's algorithm. In this work, we propose instead to estimate it with the Sinkhorn divergence, which is also built on entropic regularization but includes debiasing terms. We show that, for smooth densities, this estimator has a comparable sample complexity but allows higher regularization levels, of order $\epsilon^{1/2}$, which leads to improved computational complexity bounds and a strong speedup in practice. Our theoretical analysis covers the case of both randomly sampled densities and deterministic discretizations on uniform grids. We also propose and analyze an estimator based on Richardson extrapolation of the Sinkhorn divergence which enjoys improved statistical and computational efficiency guarantees, under a condition on the regularity of the approximation error, which is in particular satisfied for Gaussian densities. We finally demonstrate the efficiency of the proposed estimators with numerical experiments.

The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between \emph{one-dimensional random projections} of the two measures. However, the topological, statistical, and computational consequences of this technique have not yet been well-established. In this paper, we aim at bridging this gap and derive various theoretical properties of sliced probability divergences. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of a sliced divergence does not depend on the problem dimension. We finally apply our general results to several base divergences, and illustrate our theory on both synthetic and real data experiments.

Optimal transport (OT) and maximum mean discrepancies (MMD) are now routinely used in machine learning to compare probability measures. We focus in this paper on Sinkhorn divergences (SDs), a regularized variant of OT distances which can interpolate, depending on the regularization strength $\varepsilon$, between OT ($\varepsilon=0$) and MMD ($\varepsilon=\infty$). Although the tradeoff induced by that regularization is now well understood computationally (OT, SDs and MMD require respectively $O(n^3\log n)$, $O(n^2)$ and $n^2$ operations given a sample size $n$), much less is known in terms of their sample complexity, namely the gap between these quantities, when evaluated using finite samples vs. their respective densities. Indeed, while the sample complexity of OT and MMD stand at two extremes, $1/n^{1/d}$ for OT in dimension $d$ and $1/\sqrt{n}$ for MMD, that for SDs has only been studied empirically. In this paper, we (i) derive a bound on the approximation error made with SDs when approximating OT as a function of the regularizer $\varepsilon$, (ii) prove that the optimizers of regularized OT are bounded in a Sobolev (RKHS) ball independent of the two measures and (iii) provide the first sample complexity bound for SDs, obtained,by reformulating SDs as a maximization problem in a RKHS. We thus obtain a scaling in $1/\sqrt{n}$ (as in MMD), with a constant that depends however on $\varepsilon$, making the bridge between OT and MMD complete.

In a series of recent theoretical works, it was shown that strongly over-parameterized neural networks trained with gradient-based methods could converge exponentially fast to zero training loss, with their parameters hardly varying. In this work, we show that this lazy training'' phenomenon is not specific to over-parameterized neural networks, and is due to a choice of scaling, often implicit, that makes the model behave as its linearization around the initialization, thus yielding a model equivalent to learning with positive-definite kernels. Through a theoretical analysis, we exhibit various situations where this phenomenon arises in non-convex optimization and we provide bounds on the distance between the lazy and linearized optimization paths. Our numerical experiments bring a critical note, as we observe that the performance of commonly used non-linear deep convolutional neural networks in computer vision degrades when trained in the lazy regime. This makes it unlikely thatlazy training'' is behind the many successes of neural networks in difficult high dimensional tasks.

Many tasks in machine learning and signal processing can be solved by minimizing a convex function of a measure. This includes sparse spikes deconvolution or training a neural network with a single hidden layer. For these problems, we study a simple minimization method: the unknown measure is discretized into a mixture of particles and a continuous-time gradient descent is performed on their weights and positions. This is an idealization of the usual way to train neural networks with a large hidden layer. We show that, when initialized correctly and in the many-particle limit, this gradient flow, although non-convex, converges to global minimizers. The proof involves Wasserstein gradient flows, a by-product of optimal transport theory. Numerical experiments show that this asymptotic behavior is already at play for a reasonable number of particles, even in high dimension.