Quentin Duchemin

Quentin Duchemin, Guillaume Obozinski

The perspective of developing trustworthy AI for critical applications in science and engineering requires machine learning techniques that are capable of estimating their own uncertainty. In the context of regression, instead of estimating a conditional mean, this can be achieved by producing a predictive interval for the output, or to even learn a model of the conditional probability $p(y|x)$ of an output $y$ given input features $x$. While this can be done under parametric assumptions with, e.g. generalized linear model, these are typically too strong, and non-parametric models offer flexible alternatives. In particular, for scalar outputs, learning directly a model of the conditional cumulative distribution function of $y$ given $x$ can lead to more precise probabilistic estimates, and the use of proper scoring rules such as the weighted interval score (WIS) and the continuous ranked probability score (CRPS) lead to better coverage and calibration properties. This paper introduces novel algorithms for learning probabilistic regression trees for the WIS or CRPS loss functions. These algorithms are made computationally efficient thanks to an appropriate use of known data structures - namely min-max heaps, weight-balanced binary trees and Fenwick trees. Through numerical experiments, we demonstrate that the performance of our methods is competitive with alternative approaches. Additionally, our methods benefit from the inherent interpretability and explainability of trees. As a by-product, we show how our trees can be used in the context of conformal prediction and explain why they are particularly well-suited for achieving group-conditional coverage guarantees.

Xiaoxia Zhang, Quentin Duchemin, Kangning Liu et al.

Neural networks are increasingly used to estimate parameters in quantitative MRI, in particular in magnetic resonance fingerprinting. Their advantages over the gold standard non-linear least square fitting are their superior speed and their immunity to the non-convexity of many fitting problems. We find, however, that in heterogeneous parameter spaces, i.e. in spaces in which the variance of the estimated parameters varies considerably, good performance is hard to achieve and requires arduous tweaking of the loss function, hyper parameters, and the distribution of the training data in parameter space. Here, we address these issues with a theoretically well-founded loss function: the Cramér-Rao bound (CRB) provides a theoretical lower bound for the variance of an unbiased estimator and we propose to normalize the squared error with respective CRB. With this normalization, we balance the contributions of hard-to-estimate and not-so-hard-to-estimate parameters and areas in parameter space, and avoid a dominance of the former in the overall training loss. Further, the CRB-based loss function equals one for a maximally-efficient unbiased estimator, which we consider the ideal estimator. Hence, the proposed CRB-based loss function provides an absolute evaluation metric. We compare a network trained with the CRB-based loss with a network trained with the commonly used means squared error loss and demonstrate the advantages of the former in numerical, phantom, and in vivo experiments.

Quentin Duchemin, Yohann De Castro, Claire Lacour

Despite the ubiquity of U-statistics in modern Probability and Statistics, their non-asymptotic analysis in a dependent framework may have been overlooked. In a recent work, a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains has been proved. In this paper, we put this theoretical breakthrough into action by pushing further the current state of knowledge in three different active fields of research. First, we establish a new exponential inequality for the estimation of spectra of trace class integral operators with MCMC methods. The novelty is that this result holds for kernels with positive and negative eigenvalues, which is new as far as we know. In addition, we investigate generalization performance of online algorithms working with pairwise loss functions and Markov chain samples. We provide an online-to-batch conversion result by showing how we can extract a low risk hypothesis from the sequence of hypotheses generated by any online learner. We finally give a non-asymptotic analysis of a goodness-of-fit test on the density of the invariant measure of a Markov chain. We identify some classes of alternatives over which our test based on the $L_2$ distance has a prescribed power.

Quentin Duchemin, Yohann de Castro, Claire Lacour

We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $π$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{é} who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein's type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.

Quentin Duchemin, Yohann de Castro

We introduce Markov Random Geometric Graphs (MRGGs), a growth model for temporal dynamic networks. It is based on a Markovian latent space dynamic: consecutive latent points are sampled on the Euclidean Sphere using an unknown Markov kernel; and two nodes are connected with a probability depending on a unknown function of their latent geodesic distance. More precisely, at each stamp-time $k$ we add a latent point $X_k$ sampled by jumping from the previous one $X_{k-1}$ in a direction chosen uniformly $Y_k$ and with a length $r_k$ drawn from an unknown distribution called the latitude function. The connection probabilities between each pair of nodes are equal to the envelope function of the distance between these two latent points. We provide theoretical guarantees for the non-parametric estimation of the latitude and the envelope functions.We propose an efficient algorithm that achieves those non-parametric estimation tasks based on an ad-hoc Hierarchical Agglomerative Clustering approach. As a by product, we show how MRGGs can be used to detect dependence structure in growing graphs and to solve link prediction problems.

Quentin Duchemin

We introduce the Markov Stochastic Block Model (MSBM): a growth model for community based networks where node attributes are assigned through a Markovian dynamic. We rely on HMMs' literature to design prediction methods that are robust to local clustering errors. We focus specifically on the link prediction and collaborative filtering problems and we introduce a new model selection procedure to infer the number of hidden clusters in the network. Our approaches for reliable prediction in MSBMs are not algorithm-dependent in the sense that they can be applied using your favourite clustering tool. In this paper, we use a recent SDP method to infer the hidden communities and we provide theoretical guarantees. In particular, we identify the relevant signal-to-noise ratio (SNR) in our framework and we prove that the misclassification error decays exponentially fast with respect to this SNR.