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Abstract:Mixed multinomial logits are discrete mixtures introduced several decades ago to model the probability of choosing an attribute from $p$ possible candidates, in heterogeneous populations. The model has recently attracted attention in the AI literature, under the name softmax mixtures, where it is routinely used in the final layer of a neural network to map a large number $p$ of vectors in $\mathbb{R}^L$ to a probability vector. Despite its wide applicability and empirical success, statistically optimal estimators of the mixture parameters, obtained via algorithms whose running time scales polynomially in $L$, are not known. This paper provides a solution to this problem for contemporary applications, such as large language models, in which the mixture has a large number $p$ of support points, and the size $N$ of the sample observed from the mixture is also large. Our proposed estimator combines two classical estimators, obtained respectively via a method of moments (MoM) and the expectation-minimization (EM) algorithm. Although both estimator types have been studied, from a theoretical perspective, for Gaussian mixtures, no similar results exist for softmax mixtures for either procedure. We develop a new MoM parameter estimator based on latent moment estimation that is tailored to our model, and provide the first theoretical analysis for a MoM-based procedure in softmax mixtures. Although consistent, MoM for softmax mixtures can exhibit poor numerical performance, as observed other mixture models. Nevertheless, as MoM is provably in a neighborhood of the target, it can be used as warm start for any iterative algorithm. We study in detail the EM algorithm, and provide its first theoretical analysis for softmax mixtures. Our final proposal for parameter estimation is the EM algorithm with a MoM warm start.

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Abstract:We propose a procedure for estimating the Schr\"odinger bridge between two probability distributions. Unlike existing approaches, our method does not require iteratively simulating forward and backward diffusions or training neural networks to fit unknown drifts. Instead, we show that the potentials obtained from solving the static entropic optimal transport problem between the source and target samples can be modified to yield a natural plug-in estimator of the time-dependent drift that defines the bridge between two measures. Under minimal assumptions, we show that our proposal, which we call the \emph{Sinkhorn bridge}, provably estimates the Schr\"odinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure. Our approach combines results from the areas of sampling, and theoretical and statistical entropic optimal transport.

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Abstract:The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension $d$ of the problem increases, the number of iterations required to ensure convergence within a desired error in the $W_2$ metric scales in proportion to $d$ or $\sqrt{d}$. In this paper, we argue that, despite this poor scaling of the $W_2$ error for the full set of variables, the behavior for a small number of variables can be significantly better: a number of iterations proportional to $K$, up to logarithmic terms in $d$, often suffices for the algorithm to converge to within a desired $W_2$ error for all $K$-marginals. We refer to this effect as delocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. Our analysis relies on a novel $W_{2,\ell^\infty}$ metric to measure convergence. A key technical challenge we address is the lack of a one-step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.

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Abstract:We present an introduction to the field of statistical optimal transport, based on lectures given at \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XLIX.

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Authors:Parnian Kassraie, Aram-Alexandre Pooladian, Michal Klein, James Thornton, Jonathan Niles-Weed, Marco Cuturi

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Abstract:Optimal transport (OT) has profoundly impacted machine learning by providing theoretical and computational tools to realign datasets. In this context, given two large point clouds of sizes $n$ and $m$ in $\mathbb{R}^d$, entropic OT (EOT) solvers have emerged as the most reliable tool to either solve the Kantorovich problem and output a $n\times m$ coupling matrix, or to solve the Monge problem and learn a vector-valued push-forward map. While the robustness of EOT couplings/maps makes them a go-to choice in practical applications, EOT solvers remain difficult to tune because of a small but influential set of hyperparameters, notably the omnipresent entropic regularization strength $\varepsilon$. Setting $\varepsilon$ can be difficult, as it simultaneously impacts various performance metrics, such as compute speed, statistical performance, generalization, and bias. In this work, we propose a new class of EOT solvers (ProgOT), that can estimate both plans and transport maps. We take advantage of several opportunities to optimize the computation of EOT solutions by dividing mass displacement using a time discretization, borrowing inspiration from dynamic OT formulations, and conquering each of these steps using EOT with properly scheduled parameters. We provide experimental evidence demonstrating that ProgOT is a faster and more robust alternative to standard solvers when computing couplings at large scales, even outperforming neural network-based approaches. We also prove statistical consistency of our approach for estimating optimal transport maps.

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Authors:Michal Klein, Aram-Alexandre Pooladian, Pierre Ablin, Eugène Ndiaye, Jonathan Niles-Weed, Marco Cuturi

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Abstract: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.

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Abstract: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.

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Abstract: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.

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Abstract: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.

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Abstract: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.

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