Learning Costs for Structured Monge Displacements

Optimal transport theory has provided machine learning with several tools to infer a push-forward map between densities from samples. While this theory has recently seen tremendous methodological developments in machine learning, its practical implementation remains notoriously difficult, because it is plagued by both computational and statistical challenges. Because of such difficulties, existing approaches rarely depart from the default choice of estimating such maps with the simple squared-Euclidean distance as the ground cost, $c(x,y)=\|x-y\|^2_2$. We follow a different path in this work, with the motivation of \emph{learning} a suitable cost structure to encourage maps to transport points along engineered features. We extend the recently proposed Monge-Bregman-Occam pipeline~\citep{cuturi2023monge}, that rests on an alternative cost formulation that is also cost-invariant $c(x,y)=h(x-y)$, but which adopts a more general form as $h=\tfrac12 \ell_2^2+\tau$, where $\tau$ is an appropriately chosen regularizer. We first propose a method that builds upon proximal gradient descent to generate ground truth transports for such structured costs, using the notion of $h$-transforms and $h$-concave potentials. We show more generally that such a method can be extended to compute $h$-transforms for entropic potentials. We study a regularizer that promotes transport displacements in low-dimensional spaces, and propose to learn such a basis change using Riemannian gradient descent on the Stiefel manifold. We show that these changes lead to estimators that are more robust and easier to interpret.

The Adversarial Consistency of Surrogate Risks for Binary Classification

We study the consistency of surrogate risks for robust binary classification. It is common to learn robust classifiers by adversarial training, which seeks to minimize the expected $0$-$1$ loss when each example can be maliciously corrupted within a small ball. We give a simple and complete characterization of the set of surrogate loss functions that are \emph{consistent}, i.e., that can replace the $0$-$1$ loss without affecting the minimizing sequences of the original adversarial risk, for any data distribution. We also prove a quantitative version of adversarial consistency for the $\rho$-margin loss. Our results reveal that the class of adversarially consistent surrogates is substantially smaller than in the standard setting, where many common surrogates are known to be consistent.

Minimax estimation of discontinuous optimal transport maps: The semi-discrete case

We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $\mathbb R^d$, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution $Q$ is a discrete measure supported on a finite number of points in $\mathbb R^d$. We study a computationally efficient estimator initially proposed by Pooladian and Niles-Weed (2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate $n^{-1/2}$, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.

Optimal transport map estimation in general function spaces

We consider the problem of estimating the optimal transport map between a (fixed) source distribution $P$ and an unknown target distribution $Q$, based on samples from $Q$. The estimation of such optimal transport maps has become increasingly relevant in modern statistical applications, such as generative modeling. At present, estimation rates are only known in a few settings (e.g. when $P$ and $Q$ have densities bounded above and below and when the transport map lies in a H\"older class), which are often not reflected in practice. We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure $P$ satisfies a Poincar\'e inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for bounded densities and H\"older transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when $P$ is the normal distribution and the transport map is given by an infinite-width shallow neural network.

Perturbation Analysis of Neural Collapse

Training deep neural networks for classification often includes minimizing the training loss beyond the zero training error point. In this phase of training, a "neural collapse" behavior has been observed: the variability of features (outputs of the penultimate layer) of within-class samples decreases and the mean features of different classes approach a certain tight frame structure. Recent works analyze this behavior via idealized unconstrained features models where all the minimizers exhibit exact collapse. However, with practical networks and datasets, the features typically do not reach exact collapse, e.g., because deep layers cannot arbitrarily modify intermediate features that are far from being collapsed. In this paper, we propose a richer model that can capture this phenomenon by forcing the features to stay in the vicinity of a predefined features matrix (e.g., intermediate features). We explore the model in the small vicinity case via perturbation analysis and establish results that cannot be obtained by the previously studied models. For example, we prove reduction in the within-class variability of the optimized features compared to the predefined input features (via analyzing gradient flow on the "central-path" with minimal assumptions), analyze the minimizers in the near-collapse regime, and provide insights on the effect of regularization hyperparameters on the closeness to collapse. We support our theory with experiments in practical deep learning settings.

The Sketched Wasserstein Distance for mixture distributions

The Sketched Wasserstein Distance ($W^S$) is a new probability distance specifically tailored to finite mixture distributions. Given any metric $d$ defined on a set $\mathcal{A}$ of probability distributions, $W^S$ is defined to be the most discriminative convex extension of this metric to the space $\mathcal{S} = \textrm{conv}(\mathcal{A})$ of mixtures of elements of $\mathcal{A}$. Our representation theorem shows that the space $(\mathcal{S}, W^S)$ constructed in this way is isomorphic to a Wasserstein space over $\mathcal{X} = (\mathcal{A}, d)$. This result establishes a universality property for the Wasserstein distances, revealing them to be uniquely characterized by their discriminative power for finite mixtures. We exploit this representation theorem to propose an estimation methodology based on Kantorovich--Rubenstein duality, and prove a general theorem that shows that its estimation error can be bounded by the sum of the errors of estimating the mixture weights and the mixture components, for any estimators of these quantities. We derive sharp statistical properties for the estimated $W^S$ in the case of $p$-dimensional discrete $K$-mixtures, which we show can be estimated at a rate proportional to $\sqrt{K/N}$, up to logarithmic factors. We complement these bounds with a minimax lower bound on the risk of estimating the Wasserstein distance between distributions on a $K$-point metric space, which matches our upper bound up to logarithmic factors. This result is the first nearly tight minimax lower bound for estimating the Wasserstein distance between discrete distributions. Furthermore, we construct $\sqrt{N}$ asymptotically normal estimators of the mixture weights, and derive a $\sqrt{N}$ distributional limit of our estimator of $W^S$ as a consequence. Simulation studies and a data analysis provide strong support on the applicability of the new Sketched Wasserstein Distance.

The Consistency of Adversarial Training for Binary Classification

Robustness to adversarial perturbations is of paramount concern in modern machine learning. One of the state-of-the-art methods for training robust classifiers is adversarial training, which involves minimizing a supremum-based surrogate risk. The statistical consistency of surrogate risks is well understood in the context of standard machine learning, but not in the adversarial setting. In this paper, we characterize which supremum-based surrogates are consistent for distributions absolutely continuous with respect to Lebesgue measure in binary classification. Furthermore, we obtain quantitative bounds relating adversarial surrogate risks to the adversarial classification risk. Lastly, we discuss implications for the $\cH$-consistency of adversarial training.

An improved central limit theorem and fast convergence rates for entropic transportation costs

We prove a central limit theorem for the entropic transportation cost between subgaussian probability measures, centered at the population cost. This is the first result which allows for asymptotically valid inference for entropic optimal transport between measures which are not necessarily discrete. In the compactly supported case, we complement these results with new, faster, convergence rates for the expected entropic transportation cost between empirical measures. Our proof is based on strengthening convergence results for dual solutions to the entropic optimal transport problem.

Entropic estimation of optimal transport maps

We develop a computationally tractable method for estimating the optimal map between two distributions over $\mathbb{R}^d$ with rigorous finite-sample guarantees. Leveraging an entropic version of Brenier's theorem, we show that our estimator -- the barycentric projection of the optimal entropic plan -- is easy to compute using Sinkhorn's algorithm. As a result, unlike current approaches for map estimation, which are slow to evaluate when the dimension or number of samples is large, our approach is parallelizable and extremely efficient even for massive data sets. Under smoothness assumptions on the optimal map, we show that our estimator enjoys comparable statistical performance to other estimators in the literature, but with much lower computational cost. We showcase the efficacy of our proposed estimator through numerical examples. Our proofs are based on a modified duality principle for entropic optimal transport and on a method for approximating optimal entropic plans due to Pal (2019).

Plugin Estimation of Smooth Optimal Transport Maps

We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived from our observations, appropriately extended so that they define functions on $\mathbb{R}^d$. When the underlying map is assumed to be Lipschitz, we show that computing the optimal coupling between the empirical measures, and extending it using linear smoothers, already gives a minimax optimal estimator. When the underlying map enjoys higher regularity, we show that the optimal coupling between appropriate nonparametric density estimates yields faster rates. Our work also provides new bounds on the risk of corresponding plugin estimators for the quadratic Wasserstein distance, and we show how this problem relates to that of estimating optimal transport maps using stability arguments for smooth and strongly convex Brenier potentials. As an application of our results, we derive a central limit theorem for a density plugin estimator of the squared Wasserstein distance, which is centered at its population counterpart when the underlying distributions have sufficiently smooth densities. In contrast to known central limit theorems for empirical estimators, this result easily lends itself to statistical inference for Wasserstein distances.