Kilian Fatras

Alexander Tong, Nikolay Malkin, Kilian Fatras et al. · mila, utoronto

We present simulation-free score and flow matching ([SF]$^2$M), a simulation-free objective for inferring stochastic dynamics given unpaired samples drawn from arbitrary source and target distributions. Our method generalizes both the score-matching loss used in the training of diffusion models and the recently proposed flow matching loss used in the training of continuous normalizing flows. [SF]$^2$M interprets continuous-time stochastic generative modeling as a Schrödinger bridge problem. It relies on static entropy-regularized optimal transport, or a minibatch approximation, to efficiently learn the SB without simulating the learned stochastic process. We find that [SF]$^2$M is more efficient and gives more accurate solutions to the SB problem than simulation-based methods from prior work. Finally, we apply [SF]$^2$M to the problem of learning cell dynamics from snapshot data. Notably, [SF]$^2$M is the first method to accurately model cell dynamics in high dimensions and can recover known gene regulatory networks from simulated data. Our code is available in the TorchCFM package at https://github.com/atong01/conditional-flow-matching.

Alexander Tong, Kilian Fatras, Nikolay Malkin et al. · mila

Continuous normalizing flows (CNFs) are an attractive generative modeling technique, but they have been held back by limitations in their simulation-based maximum likelihood training. We introduce the generalized conditional flow matching (CFM) technique, a family of simulation-free training objectives for CNFs. CFM features a stable regression objective like that used to train the stochastic flow in diffusion models but enjoys the efficient inference of deterministic flow models. In contrast to both diffusion models and prior CNF training algorithms, CFM does not require the source distribution to be Gaussian or require evaluation of its density. A variant of our objective is optimal transport CFM (OT-CFM), which creates simpler flows that are more stable to train and lead to faster inference, as evaluated in our experiments. Furthermore, we show that when the true OT plan is available, our OT-CFM method approximates dynamic OT. Training CNFs with CFM improves results on a variety of conditional and unconditional generation tasks, such as inferring single cell dynamics, unsupervised image translation, and Schrödinger bridge inference.

Alexia Jolicoeur-Martineau, Kilian Fatras, Tal Kachman

Tabular data is hard to acquire and is subject to missing values. This paper introduces a novel approach for generating and imputing mixed-type (continuous and categorical) tabular data utilizing score-based diffusion and conditional flow matching. In contrast to prior methods that rely on neural networks to learn the score function or the vector field, we adopt XGBoost, a widely used Gradient-Boosted Tree (GBT) technique. To test our method, we build one of the most extensive benchmarks for tabular data generation and imputation, containing 27 diverse datasets and 9 metrics. Through empirical evaluation across the benchmark, we demonstrate that our approach outperforms deep-learning generation methods in data generation tasks and remains competitive in data imputation. Notably, it can be trained in parallel using CPUs without requiring a GPU. Our Python and R code is available at https://github.com/SamsungSAILMontreal/ForestDiffusion.

Alexia Jolicoeur-Martineau, Emy Gervais, Kilian Fatras et al.

Ensemble methods combine the predictions of multiple models to improve performance, but they require significantly higher computation costs at inference time. To avoid these costs, multiple neural networks can be combined into one by averaging their weights. However, this usually performs significantly worse than ensembling. Weight averaging is only beneficial when different enough to benefit from combining them, but similar enough to average well. Based on this idea, we propose PopulAtion Parameter Averaging (PAPA): a method that combines the generality of ensembling with the efficiency of weight averaging. PAPA leverages a population of diverse models (trained on different data orders, augmentations, and regularizations) while slowly pushing the weights of the networks toward the population average of the weights. We also propose PAPA variants (PAPA-all, and PAPA-2) that average weights rarely rather than continuously; all methods increase generalization, but PAPA tends to perform best. PAPA reduces the performance gap between averaging and ensembling, increasing the average accuracy of a population of models by up to 0.8% on CIFAR-10, 1.9% on CIFAR-100, and 1.6% on ImageNet when compared to training independent (non-averaged) models.

Avishek Joey Bose, Tara Akhound-Sadegh, Guillaume Huguet et al.

The computational design of novel protein structures has the potential to impact numerous scientific disciplines greatly. Toward this goal, we introduce FoldFlow, a series of novel generative models of increasing modeling power based on the flow-matching paradigm over $3\mathrm{D}$ rigid motions -- i.e. the group $\text{SE}(3)$ -- enabling accurate modeling of protein backbones. We first introduce FoldFlow-Base, a simulation-free approach to learning deterministic continuous-time dynamics and matching invariant target distributions on $\text{SE}(3)$. We next accelerate training by incorporating Riemannian optimal transport to create FoldFlow-OT, leading to the construction of both more simple and stable flows. Finally, we design FoldFlow-SFM, coupling both Riemannian OT and simulation-free training to learn stochastic continuous-time dynamics over $\text{SE}(3)$. Our family of FoldFlow, generative models offers several key advantages over previous approaches to the generative modeling of proteins: they are more stable and faster to train than diffusion-based approaches, and our models enjoy the ability to map any invariant source distribution to any invariant target distribution over $\text{SE}(3)$. Empirically, we validate FoldFlow, on protein backbone generation of up to $300$ amino acids leading to high-quality designable, diverse, and novel samples.

Tiago Salvador, Kilian Fatras, Ioannis Mitliagkas et al.

Unsupervised Domain Adaptation (UDA) aims at classifying unlabeled target images leveraging source labeled ones. In this work, we consider the Partial Domain Adaptation (PDA) variant, where we have extra source classes not present in the target domain. Most successful algorithms use model selection strategies that rely on target labels to find the best hyper-parameters and/or models along training. However, these strategies violate the main assumption in PDA: only unlabeled target domain samples are available. Moreover, there are also inconsistencies in the experimental settings - architecture, hyper-parameter tuning, number of runs - yielding unfair comparisons. The main goal of this work is to provide a realistic evaluation of PDA methods with the different model selection strategies under a consistent evaluation protocol. We evaluate 7 representative PDA algorithms on 2 different real-world datasets using 7 different model selection strategies. Our two main findings are: (i) without target labels for model selection, the accuracy of the methods decreases up to 30 percentage points; (ii) only one method and model selection pair performs well on both datasets. Experiments were performed with our PyTorch framework, BenchmarkPDA, which we open source.

Clément Bonet, Kimia Nadjahi, Thibault Séjourné et al.

Optimal transport (OT) is a powerful framework to compare probability measures, a fundamental task in many statistical and machine learning problems. Substantial advances have been made in designing OT variants which are either computationally and statistically more efficient or robust. Among them, sliced OT distances have been extensively used to mitigate optimal transport's cubic algorithmic complexity and curse of dimensionality. In parallel, unbalanced OT was designed to allow comparisons of more general positive measures, while being more robust to outliers. In this paper, we bridge the gap between those two concepts and develop a general framework for efficiently comparing positive measures. We notably formulate two different versions of sliced unbalanced OT, and study the associated topology and statistical properties. We then develop a GPU-friendly Frank-Wolfe like algorithm to compute the corresponding loss functions, and show that the resulting methodology is modular as it encompasses and extends prior related work. We finally conduct an empirical analysis of our loss functions and methodology on both synthetic and real datasets, to illustrate their computational efficiency, relevance and applicability to real-world scenarios including geophysical data.

Kilian Fatras, Hiroki Naganuma, Ioannis Mitliagkas

It is common in computer vision to be confronted with domain shift: images which have the same class but different acquisition conditions. In domain adaptation (DA), one wants to classify unlabeled target images using source labeled images. Unfortunately, deep neural networks trained on a source training set perform poorly on target images which do not belong to the training domain. One strategy to improve these performances is to align the source and target image distributions in an embedded space using optimal transport (OT). However OT can cause negative transfer, i.e. aligning samples with different labels, which leads to overfitting especially in the presence of label shift between domains. In this work, we mitigate negative alignment by explaining it as a noisy label assignment to target images. We then mitigate its effect by appropriate regularization. We propose to couple the MixUp regularization \citep{zhang2018mixup} with a loss that is robust to noisy labels in order to improve domain adaptation performance. We show in an extensive ablation study that a combination of the two techniques is critical to achieve improved performance. Finally, we evaluate our method, called \textsc{mixunbot}, on several benchmarks and real-world DA problems.

Charles Guille-Escuret, Hiroki Naganuma, Kilian Fatras et al.

Understanding the optimization dynamics of neural networks is necessary for closing the gap between theory and practice. Stochastic first-order optimization algorithms are known to efficiently locate favorable minima in deep neural networks. This efficiency, however, contrasts with the non-convex and seemingly complex structure of neural loss landscapes. In this study, we delve into the fundamental geometric properties of sampled gradients along optimization paths. We focus on two key quantities, which appear in the restricted secant inequality and error bound. Both hold high significance for first-order optimization. Our analysis reveals that these quantities exhibit predictable, consistent behavior throughout training, despite the stochasticity induced by sampling minibatches. Our findings suggest that not only do optimization trajectories never encounter significant obstacles, but they also maintain stable dynamics during the majority of training. These observed properties are sufficiently expressive to theoretically guarantee linear convergence and prescribe learning rate schedules mirroring empirical practices. We conduct our experiments on image classification, semantic segmentation and language modeling across different batch sizes, network architectures, datasets, optimizers, and initialization seeds. We discuss the impact of each factor. Our work provides novel insights into the properties of neural network loss functions, and opens the door to theoretical frameworks more relevant to prevalent practice.

Alexia Jolicoeur-Martineau, Kilian Fatras, Ke Li et al.

Diffusion Models (DMs) are powerful generative models that add Gaussian noise to the data and learn to remove it. We wanted to determine which noise distribution (Gaussian or non-Gaussian) led to better generated data in DMs. Since DMs do not work by design with non-Gaussian noise, we built a framework that allows reversing a diffusion process with non-Gaussian location-scale noise. We use that framework to show that the Gaussian distribution performs the best over a wide range of other distributions (Laplace, Uniform, t, Generalized-Gaussian).

Kilian Fatras

Optimal transport (OT) is known to be sensitive against outliers because of its marginal constraints. Outlier robust OT variants have been proposed based on the definition that outliers are samples which are expensive to move. In this paper, we show that this definition is restricted by considering the case where outliers are closer to the target measure than clean samples. We show that outlier robust OT fully transports these outliers leading to poor performances in practice. To tackle these outliers, we propose to detect them by relying on a classifier trained with adversarial training to classify source and target samples. A sample is then considered as an outlier if the prediction from the classifier is different from its assigned label. To decrease the influence of these outliers in the transport problem, we propose to either remove them from the problem or to increase the cost of moving them by using the classifier prediction. We show that we successfully detect these outliers and that they do not influence the transport problem on several experiments such as gradient flows, generative models and label propagation.

Kilian Fatras, Thibault Séjourné, Nicolas Courty et al.

Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same minibatch strategy coupled with unbalanced optimal transport can yield more robust behavior. We discuss the associated theoretical properties, such as unbiased estimators, existence of gradients and concentration bounds. Our experimental study shows that in challenging problems associated to domain adaptation, the use of unbalanced optimal transport leads to significantly better results, competing with or surpassing recent baselines.

Kilian Fatras, Younes Zine, Szymon Majewski et al.

Optimal transport distances have become a classic tool to compare probability distributions and have found many applications in machine learning. Yet, despite recent algorithmic developments, their complexity prevents their direct use on large scale datasets. To overcome this challenge, a common workaround is to compute these distances on minibatches i.e. to average the outcome of several smaller optimal transport problems. We propose in this paper an extended analysis of this practice, which effects were previously studied in restricted cases. We first consider a large variety of Optimal Transport kernels. We notably argue that the minibatch strategy comes with appealing properties such as unbiased estimators, gradients and a concentration bound around the expectation, but also with limits: the minibatch OT is not a distance. To recover some of the lost distance axioms, we introduce a debiased minibatch OT function and study its statistical and optimisation properties. Along with this theoretical analysis, we also conduct empirical experiments on gradient flows, generative adversarial networks (GANs) or color transfer that highlight the practical interest of this strategy.

Jean-Christophe Burnel, Kilian Fatras, Nicolas Courty

Adversarial examples are a hot topic due to their abilities to fool a classifier's prediction. There are two strategies to create such examples, one uses the attacked classifier's gradients, while the other only requires access to the clas-sifier's prediction. This is particularly appealing when the classifier is not full known (black box model). In this paper, we present a new method which is able to generate natural adversarial examples from the true data following the second paradigm. Based on Generative Adversarial Networks (GANs) [5], it reweights the true data empirical distribution to encourage the classifier to generate ad-versarial examples. We provide a proof of concept of our method by generating adversarial hyperspectral signatures on a remote sensing dataset.

Kilian Fatras, Younes Zine, Rémi Flamary et al.

Optimal transport distances are powerful tools to compare probability distributions and have found many applications in machine learning. Yet their algorithmic complexity prevents their direct use on large scale datasets. To overcome this challenge, practitioners compute these distances on minibatches {\em i.e.} they average the outcome of several smaller optimal transport problems. We propose in this paper an analysis of this practice, which effects are not well understood so far. We notably argue that it is equivalent to an implicit regularization of the original problem, with appealing properties such as unbiased estimators, gradients and a concentration bound around the expectation, but also with defects such as loss of distance property. Along with this theoretical analysis, we also conduct empirical experiments on gradient flows, GANs or color transfer that highlight the practical interest of this strategy.

Kilian Fatras, Bharath Bhushan Damodaran, Sylvain Lobry et al.

Noisy labels often occur in vision datasets, especially when they are obtained from crowdsourcing or Web scraping. We propose a new regularization method, which enables learning robust classifiers in presence of noisy data. To achieve this goal, we propose a new adversarial regularization scheme based on the Wasserstein distance. Using this distance allows taking into account specific relations between classes by leveraging the geometric properties of the labels space. Our Wasserstein Adversarial Regularization (WAR) encodes a selective regularization, which promotes smoothness of the classifier between some classes, while preserving sufficient complexity of the decision boundary between others. We first discuss how and why adversarial regularization can be used in the context of label noise and then show the effectiveness of our method on five datasets corrupted with noisy labels: in both benchmarks and real datasets, WAR outperforms the state-of-the-art competitors.