Adaptive optimizers such as Adam (Kingma & Ba, 2015) have been central to the success of large language models. However, they often require maintaining optimizer states throughout training, which can result in memory requirements several times greater than the model footprint. This overhead imposes constraints on scalability and computational efficiency. Stochastic Gradient Descent (SGD), in contrast, is a stateless optimizer, as it does not track state variables during training. Consequently, it achieves optimal memory efficiency. However, its capability in LLM training is limited (Zhao et al., 2024b) In this work, we show that pre-processing SGD using normalization and whitening in a stateless manner can achieve similar performance as Adam for LLM training, while maintaining the same memory footprint of SGD. Specifically, we show that normalization stabilizes gradient distributions, and whitening counteracts the local curvature of the loss landscape. This results in SWAN (SGD with Whitening And Normalization), a stochastic optimizer that eliminates the need to store any optimizer states. Empirically, SWAN achieves 50% reduction on total end-to-end memory compared to Adam. Under the memory-efficienct LLaMA training benchmark of (Zhao et al., 2024a), SWAN reaches the same evaluation perplexity using half as many tokens for 350M and 1.3B model.

Training large language models (LLMs) commonly relies on adaptive optimizers such as Adam (Kingma & Ba 2015), which accelerate convergence through moment estimates but incur substantial memory overhead. Recent stateless approaches such as SWAN (Ma et al., 2024) have shown that appropriate preprocessing of instantaneous gradient matrices can match the performance of adaptive methods without storing optimizer states. Building on this insight, we introduce \emph{gradient multi-normalization}, a principled framework for designing stateless optimizers that normalize gradients with respect to multiple norms simultaneously. Whereas standard first-order methods can be viewed as gradient normalization under a single norm (Bernstein & Newhouse, 2024), our formulation generalizes this perspective to a multi-norm setting. We derive an efficient alternating scheme that enforces these normalization constraints and show that our procedure can produce, up to an arbitrary precision, a fixed-point of the problem. This unifies and extends prior stateless optimizers, showing that SWAN arises as a specific instance with particular norm choices. Leveraging this principle, we develop SinkGD, a lightweight matrix optimizer that retains the memory footprint of SGD while substantially reducing computation relative to whitening-based methods. On the memory-efficient LLaMA training benchmark (Zhao et al., 2024), SinkGD achieves state-of-the-art performance, reaching the same evaluation perplexity as Adam using only 40\% of the training tokens.

In this paper, we investigate the impact of test-time adversarial attacks on linear regression models and determine the optimal level of robustness that any model can reach while maintaining a given level of standard predictive performance (accuracy). Through quantitative estimates, we uncover fundamental tradeoffs between adversarial robustness and accuracy in different regimes. We obtain a precise characterization which distinguishes between regimes where robustness is achievable without hurting standard accuracy and regimes where a tradeoff might be unavoidable. Our findings are empirically confirmed with simple experiments that represent a variety of settings. This work covers feature covariance matrices and attack norms of any nature, extending previous works in this area.

We propose a novel formalism for describing Structural Causal Models (SCMs) as fixed-point problems on causally ordered variables, eliminating the need for Directed Acyclic Graphs (DAGs), and establish the weakest known conditions for their unique recovery given the topological ordering (TO). Based on this, we design a two-stage causal generative model that first infers in a zero-shot manner a valid TO from observations, and then learns the generative SCM on the ordered variables. To infer TOs, we propose to amortize the learning of TOs on synthetically generated datasets by sequentially predicting the leaves of graphs seen during training. To learn SCMs, we design a transformer-based architecture that exploits a new attention mechanism enabling the modeling of causal structures, and show that this parameterization is consistent with our formalism. Finally, we conduct an extensive evaluation of each method individually, and show that when combined, our model outperforms various baselines on generated out-of-distribution problems.

In this work we study the robustness to adversarial attacks, of early-stopping strategies on gradient-descent (GD) methods for linear regression. More precisely, we show that early-stopped GD is optimally robust (up to an absolute constant) against Euclidean-norm adversarial attacks. However, we show that this strategy can be arbitrarily sub-optimal in the case of general Mahalanobis attacks. This observation is compatible with recent findings in the case of classification Vardi et al. (2022) that show that GD provably converges to non-robust models. To alleviate this issue, we propose to apply instead a GD scheme on a transformation of the data adapted to the attack. This data transformation amounts to apply feature-depending learning rates and we show that this modified GD is able to handle any Mahalanobis attack, as well as more general attacks under some conditions. Unfortunately, choosing such adapted transformations can be hard for general attacks. To the rescue, we design a simple and tractable estimator whose adversarial risk is optimal up to within a multiplicative constant of 1.1124 in the population regime, and works for any norm.

The relevance of optimal transport methods to machine learning has long been hindered by two salient limitations. First, the $O(n^3)$ computational cost of standard sample-based solvers (when used on batches of $n$ samples) is prohibitive. Second, the mass conservation constraint makes OT solvers too rigid in practice: because they must match \textit{all} points from both measures, their output can be heavily influenced by outliers. A flurry of recent works in OT has addressed these computational and modelling limitations, but has resulted in two separate strains of methods: While the computational outlook was much improved by entropic regularization, more recent $O(n)$ linear-time \textit{low-rank} solvers hold the promise to scale up OT further. On the other hand, modelling rigidities have been eased owing to unbalanced variants of OT, that rely on penalization terms to promote, rather than impose, mass conservation. The goal of this paper is to merge these two strains, to achieve the promise of \textit{both} versatile/scalable unbalanced/low-rank OT solvers. We propose custom algorithms to implement these extensions for the linear OT problem and its Fused-Gromov-Wasserstein generalization, and demonstrate their practical relevance to challenging spatial transcriptomics matching problems.

Triangular flows, also known as Knöthe-Rosenblatt measure couplings, comprise an important building block of normalizing flow models for generative modeling and density estimation, including popular autoregressive flows such as real-valued non-volume preserving transformation models (Real NVP). We present statistical guarantees and sample complexity bounds for triangular flow statistical models. In particular, we establish the statistical consistency and the finite sample convergence rates of the minimum Kullback-Leibler divergence statistical estimator of the Knöthe-Rosenblatt measure coupling using tools from empirical process theory. Our results highlight the anisotropic geometry of function classes at play in triangular flows, shed light on optimal coordinate ordering, and lead to statistical guarantees for Jacobian flows. We conduct numerical experiments to illustrate the practical implications of our theoretical findings.

We propose a new conditional dependence measure and a statistical test for conditional independence. The measure is based on the difference between analytic kernel embeddings of two well-suited distributions evaluated at a finite set of locations. We obtain its asymptotic distribution under the null hypothesis of conditional independence and design a consistent statistical test from it. We conduct a series of experiments showing that our new test outperforms state-of-the-art methods both in terms of type-I and type-II errors even in the high dimensional setting.

The ability to align points across two related yet incomparable point clouds (e.g. living in different spaces) plays an important role in machine learning. The Gromov-Wasserstein (GW) framework provides an increasingly popular answer to such problems, by seeking a low-distortion, geometry-preserving assignment between these points. As a non-convex, quadratic generalization of optimal transport (OT), GW is NP-hard. While practitioners often resort to solving GW approximately as a nested sequence of entropy-regularized OT problems, the cubic complexity (in the number $n$ of samples) of that approach is a roadblock. We show in this work how a recent variant of the OT problem that restricts the set of admissible couplings to those having a low-rank factorization is remarkably well suited to the resolution of GW: when applied to GW, we show that this approach is not only able to compute a stationary point of the GW problem in time $O(n^2)$, but also uniquely positioned to benefit from the knowledge that the initial cost matrices are low-rank, to yield a linear time $O(n)$ GW approximation. Our approach yields similar results, yet orders of magnitude faster computation than the SoTA entropic GW approaches, on both simulated and real data.

The matching principles behind optimal transport (OT) play an increasingly important role in machine learning, a trend which can be observed when OT is used to disambiguate datasets in applications (e.g. single-cell genomics) or used to improve more complex methods (e.g. balanced attention in transformers or self-supervised learning). To scale to more challenging problems, there is a growing consensus that OT requires solvers that can operate on millions, not thousands, of points. The low-rank optimal transport (LOT) approach advocated in \cite{scetbon2021lowrank} holds several promises in that regard, and was shown to complement more established entropic regularization approaches, being able to insert itself in more complex pipelines, such as quadratic OT. LOT restricts the search for low-cost couplings to those that have a low-nonnegative rank, yielding linear time algorithms in cases of interest. However, these promises can only be fulfilled if the LOT approach is seen as a legitimate contender to entropic regularization when compared on properties of interest, where the scorecard typically includes theoretical properties (statistical complexity and relation to other methods) or practical aspects (debiasing, hyperparameter tuning, initialization). We target each of these areas in this paper in order to cement the impact of low-rank approaches in computational OT.

We present eigenvalue decay estimates of integral operators associated with compositional dot-product kernels. The estimates improve on previous ones established for power series kernels on spheres. This allows us to obtain the volumes of balls in the corresponding reproducing kernel Hilbert spaces. We discuss the consequences on statistical estimation with compositional dot product kernels and highlight interesting trade-offs between the approximation error and the statistical error depending on the number of compositions and the smoothness of the kernels.

We introduce an extension of the Optimal Transport problem when multiple costs are involved. Considering each cost as an agent, we aim to share equally between agents the work of transporting one distribution to another. To do so, we minimize the transportation cost of the agent who works the most. Another point of view is when the goal is to partition equitably goods between agents according to their heterogeneous preferences. Here we aim to maximize the utility of the least advantaged agent. This is a fair division problem. Like Optimal Transport, the problem can be cast as a linear optimization problem. When there is only one agent, we recover the Optimal Transport problem. When two agents are considered, we are able to recover Integral Probability Metrics defined by $\alpha$-Hölder functions, which include the widely-known Dudley metric. To the best of our knowledge, this is the first time a link is given between the Dudley metric and Optimal Transport. We provide an entropic regularization of that problem which leads to an alternative algorithm faster than the standard linear program.

This paper tackles the problem of adversarial examples from a game theoretic point of view. We study the open question of the existence of mixed Nash equilibria in the zero-sum game formed by the attacker and the classifier. While previous works usually allow only one player to use randomized strategies, we show the necessity of considering randomization for both the classifier and the attacker. We demonstrate that this game has no duality gap, meaning that it always admits approximate Nash equilibria. We also provide the first optimization algorithms to learn a mixture of classifiers that approximately realizes the value of this game, \emph{i.e.} procedures to build an optimally robust randomized classifier.

Several recent applications of optimal transport (OT) theory to machine learning have relied on regularization, notably entropy and the Sinkhorn algorithm. Because matrix-vector products are pervasive in the Sinkhorn algorithm, several works have proposed to \textit{approximate} kernel matrices appearing in its iterations using low-rank factors. Another route lies instead in imposing low-nonnegative rank constraints on the feasible set of couplings considered in OT problems, with no approximations on cost nor kernel matrices. This route was first explored by \citet{forrow2018statistical}, who proposed an algorithm tailored for the squared Euclidean ground cost, using a proxy objective that can be solved through the machinery of regularized 2-Wasserstein barycenters. Building on this, we introduce in this work a generic approach that aims at solving, in full generality, the OT problem under low-nonnegative rank constraints with arbitrary costs. Our algorithm relies on an explicit factorization of low-rank couplings as a product of \textit{sub-coupling} factors linked by a common marginal; similar to an NMF approach, we alternatively updates these factors. We prove the non-asymptotic stationary convergence of this algorithm and illustrate its efficiency on benchmark experiments.

This work considers noise removal from images, focusing on the well-known K-SVD denoising algorithm. This sparsity-based method was proposed in 2006, and for a short while it was considered as state-of-the-art. However, over the years it has been surpassed by other methods, including the recent deep-learning-based newcomers. The question we address in this paper is whether K-SVD was brought to its peak in its original conception, or whether it can be made competitive again. The approach we take in answering this question is to redesign the algorithm to operate in a supervised manner. More specifically, we propose an end-to-end deep architecture with the exact K-SVD computational path, and train it for optimized denoising. Our work shows how to overcome difficulties arising in turning the K-SVD scheme into a differentiable, and thus learnable, machine. With a small number of parameters to learn and while preserving the original K-SVD essence, the proposed architecture is shown to outperform the classical K-SVD algorithm substantially, and getting closer to recent state-of-the-art learning-based denoising methods. Adopting a broader context, this work touches on themes around the design of deep-learning solutions for image processing tasks, while paving a bridge between classic methods and novel deep-learning-based ones.

We present a description of the function space and the smoothness class associated with a convolutional network using the machinery of reproducing kernel Hilbert spaces. We show that the mapping associated with a convolutional network expands into a sum involving elementary functions akin to spherical harmonics. This functional decomposition can be related to the functional ANOVA decomposition in nonparametric statistics. Building off our functional characterization of convolutional networks, we obtain statistical bounds highlighting an interesting trade-off between the approximation error and the estimation error.

Although Sinkhorn divergences are now routinely used in data sciences to compare probability distributions, the computational effort required to compute them remains expensive, growing in general quadratically in the size $n$ of the support of these distributions. Indeed, solving optimal transport (OT) with an entropic regularization requires computing a $n\times n$ kernel matrix (the neg-exponential of a $n\times n$ pairwise ground cost matrix) that is repeatedly applied to a vector. We propose to use instead ground costs of the form $c(x,y)=-\log\dotp{\varphi(x)}{\varphi(y)}$ where $\varphi$ is a map from the ground space onto the positive orthant $\RR^r_+$, with $r\ll n$. This choice yields, equivalently, a kernel $k(x,y)=\dotp{\varphi(x)}{\varphi(y)}$, and ensures that the cost of Sinkhorn iterations scales as $O(nr)$. We show that usual cost functions can be approximated using this form. Additionaly, we take advantage of the fact that our approach yields approximation that remain fully differentiable with respect to input distributions, as opposed to previously proposed adaptive low-rank approximations of the kernel matrix, to train a faster variant of OT-GAN~\cite{salimans2018improving}.