arXiv subjects

Kamélia Daudel

Publications and source records attributed to Kamélia Daudel.

7 recordsLinked to original sources

arXiv2026

Control Variate Design for Score-Function Gradient Estimators in Importance-Weighted Variational Inference

Importance-weighted variational inference approximates intractable densities by optimizing multi-sample Monte Carlo objectives whose tightness increases with the number of importance samples $N$. When reparameterization is unavailable, optimization relies on score-function (REINFORCE) gradient estimators. In this paper, we provide a comprehensive analysis of score-function gradient estimators for importance-weighted objectives by characterizing their asymptotic expectation and variance. We establish that standard variational inference for Monte Carlo objectives (VIMCO) gradient estimators suffer from a vanishing signal-to-noise ratio (SNR) as $N$ increases, which arises from a suboptimal control variate choice and can impair optimization. To address this issue, we derive a general mechanism for constructing VIMCO gradient estimators that avoid SNR collapse. We identify the High-Effective Sample Size optimal variance importance sampling estimator as belonging to this category of estimators and establish its $\sqrt{N}$ SNR scaling. This estimator does not, however, have zero variance at optimality and achieving its $\sqrt{N}$ SNR rate may require prohibitively large $N$ when the importance weights are degenerate. We overcome these limitations by introducing VIMCO gradient estimators that attain zero variance at optimality and support importance weight tempering. Experiments with discrete latent-variable and state-space models validate the theoretical predictions and illustrate the benefits of our estimators.

stat.ML↗

arXiv2026

Learning with Importance Weighted Variational Inference

Several variational bounds involving importance weighting ideas generalize the Evidence Lower BOund (ELBO) for marginal likelihood optimization, such as the Importance-weighted Auto-Encoder (IWAE), Variational Rényi (VR) and VR-IWAE bounds. Yet, it remains unclear how the joint choice of bound and gradient estimator impacts the behavior of the resulting variational inference (VI) algorithms. This paper provides a unified theoretical comparison of reparameterized (REP) and doubly-reparameterized (DREP) gradient estimators tied to the IWAE, VR and VR-IWAE bounds. Through asymptotic analyses of the Signal-to-Noise Ratio as the number of Monter Carlo samples $N$ goes to infinity, we identify a bias-variance tradeoff in these gradient estimators and we formally justify the superiority of DREP over REP in importance-weighted VI. An additional asymptotic analysis for challenging regimes, where both $N$ and the Kullback-Leibler divergence between the variational and posterior densities go to infinity, indicates that importance-weighted VI gradient estimators point in a well-founded direction even when the variational approximation deteriorates. Together, these complementary results characterize the optimization trajectory in importance-weighted VI from poor initialization to final convergence. Importantly, our proof techniques establish general theoretical tools for the study of sample means ratios whose scope extend beyond VI and constitute an independent contribution to the field of Monte Carlo methods.

stat.ML↗

arXiv2023

Alpha-divergence Variational Inference Meets Importance Weighted Auto-Encoders: Methodology and Asymptotics

Several algorithms involving the Variational Rényi (VR) bound have been proposed to minimize an alpha-divergence between a target posterior distribution and a variational distribution. Despite promising empirical results, those algorithms resort to biased stochastic gradient descent procedures and thus lack theoretical guarantees. In this paper, we formalize and study the VR-IWAE bound, a generalization of the Importance Weighted Auto-Encoder (IWAE) bound. We show that the VR-IWAE bound enjoys several desirable properties and notably leads to the same stochastic gradient descent procedure as the VR bound in the reparameterized case, but this time by relying on unbiased gradient estimators. We then provide two complementary theoretical analyses of the VR-IWAE bound and thus of the standard IWAE bound. Those analyses shed light on the benefits or lack thereof of these bounds. Lastly, we illustrate our theoretical claims over toy and real-data examples.

stat.ML↗

arXiv2023

Monotonic Alpha-divergence Minimisation for Variational Inference

In this paper, we introduce a novel family of iterative algorithms which carry out $α$-divergence minimisation in a Variational Inference context. They do so by ensuring a systematic decrease at each step in the $α$-divergence between the variational and the posterior distributions. In its most general form, the variational distribution is a mixture model and our framework allows us to simultaneously optimise the weights and components parameters of this mixture model. Our approach permits us to build on various methods previously proposed for $α$-divergence minimisation such as Gradient or Power Descent schemes and we also shed a new light on an integrated Expectation Maximization algorithm. Lastly, we provide empirical evidence that our methodology yields improved results on several multimodal target distributions and on a real data example.

stat.CO↗

arXiv2021

Mixture weights optimisation for Alpha-Divergence Variational Inference

This paper focuses on $α$-divergence minimisation methods for Variational Inference. More precisely, we are interested in algorithms optimising the mixture weights of any given mixture model, without any information on the underlying distribution of its mixture components parameters. The Power Descent, defined for all $α\neq 1$, is one such algorithm and we establish in our work the full proof of its convergence towards the optimal mixture weights when $α<1$. Since the $α$-divergence recovers the widely-used forward Kullback-Leibler when $α\to 1$, we then extend the Power Descent to the case $α= 1$ and show that we obtain an Entropic Mirror Descent. This leads us to investigate the link between Power Descent and Entropic Mirror Descent: first-order approximations allow us to introduce the Renyi Descent, a novel algorithm for which we prove an $O(1/N)$ convergence rate. Lastly, we compare numerically the behavior of the unbiased Power Descent and of the biased Renyi Descent and we discuss the potential advantages of one algorithm over the other.

math.ST↗

arXiv2021

The $f$-Divergence Expectation Iteration Scheme

This paper introduces the $f$-EI$(ϕ)$ algorithm, a novel iterative algorithm which operates on measures and performs $f$-divergence minimisation in a Bayesian framework. We prove that for a rich family of values of $(f,ϕ)$ this algorithm leads at each step to a systematic decrease in the $f$-divergence and show that we achieve an optimum. In the particular case where we consider a weighted sum of Dirac measures and the $α$-divergence, we obtain that the calculations involved in the $f$-EI$(ϕ)$ algorithm simplify to gradient-based computations. Empirical results support the claim that the $f$-EI$(ϕ)$ algorithm serves as a powerful tool to assist Variational methods.

math.ST↗

arXiv2020

Infinite-dimensional gradient-based descent for alpha-divergence minimisation

This paper introduces the $(α, Γ)$-descent, an iterative algorithm which operates on measures and performs $α$-divergence minimisation in a Bayesian framework. This gradient-based procedure extends the commonly-used variational approximation by adding a prior on the variational parameters in the form of a measure. We prove that for a rich family of functions $Γ$, this algorithm leads at each step to a systematic decrease in the $α$-divergence and derive convergence results. Our framework recovers the Entropic Mirror Descent algorithm and provides an alternative algorithm that we call the Power Descent. Moreover, in its stochastic formulation, the $(α, Γ)$-descent allows to optimise the mixture weights of any given mixture model without any information on the underlying distribution of the variational parameters. This renders our method compatible with many choices of parameters updates and applicable to a wide range of Machine Learning tasks. We demonstrate empirically on both toy and real-world examples the benefit of using the Power descent and going beyond the Entropic Mirror Descent framework, which fails as the dimension grows.

math.ST↗