Loucas PillaudVivien
Research interests
My main research interests are convex optimization, statistics and PDEs. More precisely, here is are a selection of research topics I am interested in:
Stochastic approximations in Hilbert spaces
Kernel methods
Stochastic Differential Equations (and PDEs) and how they can model machine learning problems
Interacting particle systems
Publications
A. Varre, L. PillaudVivien, N. Flammarion. Last iterate convergence of SGD for LeastSquares in the Interpolation regime. [arxiv:2102.03183, pdf], Preprint, 2020. [Show Abstract]
Abstract: Motivated by the recent successes of neural networks that have the ability to fit the data perfectly and generalize well, we study the noiseless model in the fundamental leastsquares setup. We assume that an optimum predictor fits perfectly inputs and outputs ⟨θ∗,ϕ(X)⟩=Y, where ϕ(X) stands for a possibly infinite dimensional nonlinear feature map. To solve this problem, we consider the estimator given by the last iterate of stochastic gradient descent (SGD) with constant stepsize. In this context, our contribution is two fold: (i) from a (stochastic) optimization perspective, we exhibit an archetypal problem where we can show explicitly the convergence of SGD final iterate for a nonstrongly convex problem with constant stepsize whereas usual results use some form of average and (ii) from a statistical perspective, we give explicit nonasymptotic convergence rates in the overparameterized setting and leverage a finegrained parameterization of the problem to exhibit polynomial rates that can be faster than O(1/T). The link with reproducing kernel Hilbert spaces is established.
L. PillaudVivien, F. Bach, T. Lelievre, A. Rudi, G. Stoltz. Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions. [arxiv:1910.14564, pdf], accepted in Proceedings of the International Conference on Artificial Intelligence and Statistics (AISTATS), 2019. [Show Abstract]
Abstract: Poincaré inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincaré constant, for which the inequality is tight, is related to the typical convergence rate of diffusions to their equilibrium measure. In this paper, we show both theoretically and experimentally that, given sufficiently many samples of a measure, we can estimate its Poincaré constant. As a byproduct of the estimation of the Poincaré constant, we derive an algorithm that captures a low dimensional representation of the data by finding directions which are difficult to sample. These directions are of crucial importance for sampling or in fields like molecular dynamics, where they are called reaction coordinates. Their knowledge can leverage, with a simple conditioning step, computational bottlenecks by using importance sampling techniques.
T. Lelievre, L. PillaudVivien, J. Reygner. Central Limit Theorem for stationary FlemingViot particle systems in finite spaces. [arXiv:1806.04490, pdf], ALEA Latin American Journal of Probability and Mathematical Statistics, 2018. [Show Abstract]
Abstract: We consider the FlemingViot particle system associated with a continuoustime Markov chain in a finite space. Assuming irreducibility, it is known that the particle system possesses a unique stationary distribution, under which its empirical measure converges to the quasistationary distribution of the Markov chain. We complement this Law of Large Numbers with a Central Limit Theorem. Our proof essentially relies on elementary computations on the infinitesimal generator of the FlemingViot particle system, and involves the socalled πreturn process in the expression of the asymptotic variance. Our work can be seen as an infinitetime version, in the setting of finite space Markov chains, of recent results by Cérou, Delyon, Guyader and Rousset [ arXiv:1611.00515, arXiv:1709.06771].
L. PillaudVivien, A. Rudi, F. Bach. Statistical Optimality of Stochastic Gradient Descent on Hard Learning Problems through Multiple Passes. [arXiv:1805.10074, pdf, poster], Advances in Neural Information Processing Systems (NeurIPS), 2018. [Show Abstract]
Abstract: We consider stochastic gradient descent (SGD) for leastsquares regression with potentially several passes over the data. While several passes have been widely reported to perform practically better in terms of predictive performance on unseen data, the existing theoretical analysis of SGD suggests that a single pass is statistically optimal. While this is true for lowdimensional easy problems, we show that for hard problems, multiple passes lead to statistically optimal predictions while single pass does not; we also show that in these hard models, the optimal number of passes over the data increases with sample size. In order to define the notion of hardness and show that our predictive performances are optimal, we consider potentially infinitedimensional models and notions typically associated to kernel methods, namely, the decay of eigenvalues of the covariance matrix of the features and the complexity of the optimal predictor as measured through the covariance matrix. We illustrate our results on synthetic experiments with nonlinear kernel methods and on a classical benchmark with a linear model.
L. PillaudVivien, A. Rudi, F. Bach. Exponential convergence of testing error for stochastic gradient methods. [arXiv:1712.04755, pdf, video, poster], Proceedings of the International Conference on Learning Theory (COLT), 2018. [Show Abstract]
Abstract: We consider binary classification problems with positive definite kernels and square loss, and study the convergence rates of stochastic gradient methods. We show that while the excess testing loss (squared loss) converges slowly to zero as the number of observations (and thus iterations) goes to infinity, the testing error (classification error) converges exponentially fast if lownoise conditions are assumed.
PhD Thesis
I will defend my thesis on Friday, October 30, at 3 pm, at INRIA.
You can download the final version of the manuscript via this link [Thesis].
You can also have a look at the slides. (Available soon !)
Some Presentations
Review
Reviewer for Journals:
Reviewer for Conferences:
