Luc Brogat-Motte

Luc Brogat-Motte, Alessandro Rudi, Céline Brouard et al.

We propose and analyse a reduced-rank method for solving least-squares regression problems with infinite dimensional output. We derive learning bounds for our method, and study under which setting statistical performance is improved in comparison to full-rank method. Our analysis extends the interest of reduced-rank regression beyond the standard low-rank setting to more general output regularity assumptions. We illustrate our theoretical insights on synthetic least-squares problems. Then, we propose a surrogate structured prediction method derived from this reduced-rank method. We assess its benefits on three different problems: image reconstruction, multi-label classification, and metabolite identification.

Tamim El Ahmad, Luc Brogat-Motte, Pierre Laforgue et al.

Leveraging the kernel trick in both the input and output spaces, surrogate kernel methods are a flexible and theoretically grounded solution to structured output prediction. If they provide state-of-the-art performance on complex data sets of moderate size (e.g., in chemoinformatics), these approaches however fail to scale. We propose to equip surrogate kernel methods with sketching-based approximations, applied to both the input and output feature maps. We prove excess risk bounds on the original structured prediction problem, showing how to attain close-to-optimal rates with a reduced sketch size that depends on the eigendecay of the input/output covariance operators. From a computational perspective, we show that the two approximations have distinct but complementary impacts: sketching the input kernel mostly reduces training time, while sketching the output kernel decreases the inference time. Empirically, our approach is shown to scale, achieving state-of-the-art performance on benchmark data sets where non-sketched methods are intractable.

Luc Brogat-Motte, Riccardo Bonalli, Alessandro Rudi

Identification of nonlinear dynamical systems is crucial across various fields, facilitating tasks such as control, prediction, optimization, and fault detection. Many applications require methods capable of handling complex systems while providing strong learning guarantees for safe and reliable performance. However, existing approaches often focus on simplified scenarios, such as deterministic models, known diffusion, discrete systems, one-dimensional dynamics, or systems constrained by strong structural assumptions such as linearity. This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled stochastic differential equations with non-uniform diffusion. We assume regularity of the coefficients within a Sobolev space, allowing for broad applicability to various dynamical systems in robotics, finance, climate modeling, and biology. Leveraging the Fokker-Planck equation, we split the estimation into two tasks: (a) estimating system dynamics for a finite set of controls, and (b) estimating coefficients that govern those dynamics. We provide strong theoretical guarantees, including finite-sample bounds for \(L^2\), \(L^\infty\), and risk metrics, with learning rates adaptive to coefficients' regularity, similar to those in nonparametric least-squares regression literature. The practical effectiveness of our approach is demonstrated through extensive numerical experiments. Our method is available as an open-source Python library.

Luc Brogat-Motte, Alessandro Rudi, Riccardo Bonalli

We address the problem of safely learning controlled stochastic dynamics from discrete-time trajectory observations, ensuring system trajectories remain within predefined safe regions during both training and deployment. Safety-critical constraints of this kind are crucial in applications such as autonomous robotics, finance, and biomedicine. We introduce a method that ensures safe exploration and efficient estimation of system dynamics by iteratively expanding an initial known safe control set using kernel-based confidence bounds. After training, the learned model enables predictions of the system's dynamics and permits safety verification of any given control. Our approach requires only mild smoothness assumptions and access to an initial safe control set, enabling broad applicability to complex real-world systems. We provide theoretical guarantees for safety and derive adaptive learning rates that improve with increasing Sobolev regularity of the true dynamics. Experimental evaluations demonstrate the practical effectiveness of our method in terms of safety, estimation accuracy, and computational efficiency.

Luc Brogat-Motte, Rémi Flamary, Céline Brouard et al.

This paper introduces a novel and generic framework to solve the flagship task of supervised labeled graph prediction by leveraging Optimal Transport tools. We formulate the problem as regression with the Fused Gromov-Wasserstein (FGW) loss and propose a predictive model relying on a FGW barycenter whose weights depend on inputs. First we introduce a non-parametric estimator based on kernel ridge regression for which theoretical results such as consistency and excess risk bound are proved. Next we propose an interpretable parametric model where the barycenter weights are modeled with a neural network and the graphs on which the FGW barycenter is calculated are additionally learned. Numerical experiments show the strength of the method and its ability to interpolate in the labeled graph space on simulated data and on a difficult metabolic identification problem where it can reach very good performance with very little engineering.

Luc Brogat-Motte, Alessandro Rudi, Céline Brouard et al.

A powerful and flexible approach to structured prediction consists in embedding the structured objects to be predicted into a feature space of possibly infinite dimension by means of output kernels, and then, solving a regression problem in this output space. A prediction in the original space is computed by solving a pre-image problem. In such an approach, the embedding, linked to the target loss, is defined prior to the learning phase. In this work, we propose to jointly learn a finite approximation of the output embedding and the regression function into the new feature space. For that purpose, we leverage a priori information on the outputs and also unexploited unsupervised output data, which are both often available in structured prediction problems. We prove that the resulting structured predictor is a consistent estimator, and derive an excess risk bound. Moreover, the novel structured prediction tool enjoys a significantly smaller computational complexity than former output kernel methods. The approach empirically tested on various structured prediction problems reveals to be versatile and able to handle large datasets.

Pierre Laforgue, Alex Lambert, Luc Brogat-Motte et al.

Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with infinite dimensional output spaces unlocks many more applications, such as functional regression, structured output prediction, and structured data representation. However, these sophisticated schemes crucially rely on the kernel trick in the output space, so that most of previous works have focused on the square norm loss function, completely neglecting robustness issues that may arise in such surrogate problems. To overcome this limitation, this paper develops a duality approach that allows to solve OVK machines for a wide range of loss functions. The infinite dimensional Lagrange multipliers are handled through a Double Representer Theorem, and algorithms for $ε$-insensitive losses and the Huber loss are thoroughly detailed. Robustness benefits are emphasized by a theoretical stability analysis, as well as empirical improvements on structured data applications.