Simon Barthelmé

NeurIPS Conference 2025 Conference Paper

Least squares variational inference

• Yvann Le Fay
• Nicolas Chopin
• Simon Barthelmé

Variational inference seeks the best approximation of a target distribution within a chosen family, where "best" means minimizing Kullback-Leibler divergence. When the approximation family is exponential, the optimal approximation satisfies a fixed-point equation. We introduce LSVI (Least Squares Variational Inference), a gradient-free, Monte Carlo-based scheme for the fixed-point recursion, where each iteration boils down to performing ordinary least squares regression on tempered log-target evaluations under the variational approximation. We show that LSVI is equivalent to biased stochastic natural gradient descent and use this to derive convergence rates with respect to the numbers of samples and iterations. When the approximation family is Gaussian, LSVI involves inverting the Fisher information matrix, whose size grows quadratically with dimension $d$. We exploit the regression formulation to eliminate the need for this inversion, yielding $O(d^3)$ complexity in the full-covariance case and $O(d)$ in the mean-field case. Finally, we numerically demonstrate LSVI’s performance on various tasks, including logistic regression, discrete variable selection, and Bayesian synthetic likelihood, showing competitive results with state-of-the-art methods, even when gradients are unavailable.

JMLR Journal 2019 Journal Article

Determinantal Point Processes for Coresets

• Nicolas Tremblay
• Simon Barthelmé
• Pierre-Olivier Amblard

When faced with a data set too large to be processed all at once, an obvious solution is to retain only part of it. In practice this takes a wide variety of different forms, and among them “coresets” are especially appealing. A coreset is a (small) weighted sample of the original data that comes with the following guarantee: a cost function can be evaluated on the smaller set instead of the larger one, with low relative error. For some classes of problems, and via a careful choice of sampling distribution (based on the so-called “sensitivity” metric), iid random sampling has turned to be one of the most successful methods for building coresets efficiently. However, independent samples are sometimes overly redundant, and one could hope that enforcing diversity would lead to better performance. The difficulty lies in proving coreset properties in non-iid samples. We show that the coreset property holds for samples formed with determinantal point processes (DPP). DPPs are interesting because they are a rare example of repulsive point processes with tractable theoretical properties, enabling us to prove general coreset theorems. We apply our results to both the $k$-means and the linear regression problems, and give extensive empirical evidence that the small additional computational cost of DPP sampling comes with superior performance over its iid counterpart. Of independent interest, we also provide analytical formulas for the sensitivity in the linear regression and $1$-means cases. [abs] [ pdf ][ bib ] &copy JMLR 2019. ( edit, beta )

NeurIPS Conference 2015 Conference Paper

Bounding errors of Expectation-Propagation

• Guillaume Dehaene
• Simon Barthelmé

Expectation Propagation is a very popular algorithm for variational inference, but comes with few theoretical guarantees. In this article, we prove that the approximation errors made by EP can be bounded. Our bounds have an asymptotic interpretation in the number n of datapoints, which allows us to study EP's convergence with respect to the true posterior. In particular, we show that EP converges at a rate of $O(n^{-2})$ for the mean, up to an order of magnitude faster than the traditional Gaussian approximation at the mode. We also give similar asymptotic expansions for moments of order 2 to 4, as well as excess Kullback-Leibler cost (defined as the additional KL cost incurred by using EP rather than the ideal Gaussian approximation). All these expansions highlight the superior convergence properties of EP. Our approach for deriving those results is likely applicable to many similar approximate inference methods. In addition, we introduce bounds on the moments of log-concave distributions that may be of independent interest.

ICML Conference 2011 Conference Paper

ABC-EP: Expectation Propagation for Likelihoodfree Bayesian Computation

• Simon Barthelmé
• Nicolas Chopin