Welcome to Randal Douc's wiki

• Abstract: Several variational bounds involving importance weighting ideas have been proposed to generalize and improve on the Evidence Lower BOund (ELBO) in the context of maximum likelihood optimization, such as the Importance Weighted Auto-Encoder (IWAE), Variational Rényi (VR) and VR-IWAE bounds. Learning the parameters of interest using these bounds typically involves stochastic gradient-based variational inference procedures. Yet, it remains unclear how the joint choice of bound and gradient estimator impacts the behavior of the resulting algorithms.
• In this talk, we study reparameterized and doubly-reparameterized gradient estimators tied to the IWAE, VR and VR-IWAE bounds. Our asymptotic analyses provide a unified comparison of these estimators under mild assumptions, allowing us to identify their respective strengths. Additional asymptotic analyses reveal a new perspective on challenging regimes where the variational approximation deteriorates: even in such settings, importance-weighted gradient estimators can still be used to learn the parameters of interest. Consequently, our work motivates further exploration of importance weighting as a principle for designing and analyzing variational inference algorithms. In addition, our proof techniques establish general theoretical tools that apply more broadly within importance weighting and are of independent interest. We complement our theoretical contributions with experiments illustrating our findings.

• Abstract: General state-space models (SSMs) are widely used in statistical machine learning and are among the most classical generative models for sequential time-series data. SSMs, comprising latent Markovian states, can be subjected to variational inference (VI), but standard VI methods like the importance-weighted autoencoder (IWAE) lack functionality for streaming data. To enable online VI in SSMs when the observations are received in real time, we propose maximising an IWAE-type variational lower bound on the asymptotic contrast function, rather than the standard IWAE ELBO, using stochastic approximation. Unlike the recursive maximum likelihood method, which directly maximises the asymptotic contrast, our approach, called online sequential IWAE (OSI-WAE), allows for online learning of both model parameters and a Markovian recognition model for inferring latent states. By approximating filter state posteriors and their derivatives using sequential Monte Carlo (SMC) methods, we create a particle-based framework for online VI in SSMs. This approach is more theoretically well-founded than recently proposed online variational SMC methods. We provide rigorous theoretical results on the learning objective and a numerical study demonstrating the method’s efficiency in learning model parameters and particle proposal kernels.

• Abstract : Importance sampling is a well-known Monte Carlo method used to estimate expectations under a target distribution by drawing weighted samples from a proposal distribution. However, for intricate target distributions, such as multi-modal distributions in high-dimensional spaces, the method becomes inefficient unless the proposal distribution is carefully designed.
• In this talk, we introduce an adaptive framework for constructing efficient proposal distributions related to the recent framework of mirror descent. Our algorithm enhances exploration of the target distribution by combining global sampling strategies with a delayed weighting mechanism. This delayed weighting is essential, as it enables immediate resampling in regions where the proposal distribution is poorly suited. We establish that the proposed scheme exhibits geometric convergence under mild assumptions.

• Abstract: This paper addresses the problem of approximating an unknown probability distribution with density $f$ - which can only be evaluated up to an unknown scaling factor - with the help of a sequential algorithm that produces at each iteration $n\geq 1$ an estimated density $q_n$. The proposed method optimizes the Kullback-Leibler divergence using a mirror descent (MD) algorithm directly on the space of density functions, while a stochastic approximation technique helps to manage between algorithm complexity and variability. One of the key innovations of this work is the theoretical guarantee that is provided for an algorithm with a fixed MD learning rate \(\eta \in (0,1 )\). The main result is that the sequence \(q_n\) converges almost surely to the target density \(f\) uniformly on compact sets. Through numerical experiments, we show that fixing the learning rate \(\eta \in (0,1 )\) significantly improves the algorithm's performance, particularly in the context of multi-modal target distributions where a small value of $\eta$ allows to increase the chance of finding all modes. Additionally, we propose a particle subsampling method to enhance computational efficiency and compare our method against other approaches through numerical experiments.

• Abstract: Denoising diffusion models have driven significant progress in the field of Bayesian inverse problems. Recent approaches use pre-trained diffusion models as priors to solve a wide range of such problems, only leveraging inference-time compute and thereby eliminating the need to retrain task-specific models on the same dataset. To approximate the posterior of a Bayesian inverse problem, a diffusion model samples from a sequence of intermediate posterior distributions, each with an intractable likelihood function. This work proposes a novel mixture approximation of these intermediate distributions. Since direct gradient-based sampling of these mixtures is infeasible due to intractable terms, we propose a practical method based on Gibbs sampling. We validate our approach through extensive experiments on image inverse problems, utilizing both pixel- and latent-space diffusion priors, as well as on source separation with an audio diffusion model.

• Abstract: Minimizing the alpha-divergence is a compelling way to approximate an unnormalized density with an exponential family distribution. To establish convergence properties for monotonic alpha-divergence minimization algorithms, it is helpful to understand how the objective behaves. In particular, we would like to know if its level sets are compact. This presentation investigates the behavior of the alpha-divergence as the parameter approaches the boundary of the parameter space, and as its norm goes to infinity. We connect this limit behavior to a key property of the approximating exponential family, which we call “strong minimality”. This property is sufficient to guarantee the compactness of the level sets.

• Abstract: Sequential Monte Carlo (SMC) methods are general iterative stochastic algorithms to generate

samples from a sequence of probability distributions. They proceed by recursively moving samples through the bridge via three crucial steps: importance sampling, resampling, and rejuvenating. The last step usually consists in running independent Markov chains, and only keeping the final states for the next iterations.

Waste-free SMC [Dau and Chopin, 2022] avoids discarding the intermediary samples.

We establish a finite-sample complexity bound for waste-free SMC, our proof generalises the approach of Marion et al [2023]. This bound prescribes sufficient conditions on the size of the particle system for the sampler to return estimated expectations $\hat{\pi}_T(f)$ such that $|\hat{\pi}_T − \pi_T(f)| ≤ \varepsilon$ with probability at least $1 − \eta$, for any bounded test function $f$.

We demonstrate that waste-free SMC enjoys lower complexity than SMC differing from a logarithmic factor $O(\log(T\eta^{-1} \varepsilon^{-2}))$.

We show that our analysis allows practitioners to tune the particle budget according to the specific statistical objective in hand. In particular, when the goal is to control errors for expec- tation estimates under $\hat{\pi}_T$ of bounded functions, the sampler can be run with an intentionally reduced number of particles in earlier iterations while still preserving finite-sample guarantees on the estimates. In doing so, we highlight an important distinction between those two objectives: estimating normal- isation constants is more challenging than estimating expectations of bounded functions.

* Abstract: Sequential Monte Carlo (SMC) methods are general iterative stochastic algorithms to generate samples from a sequence of probability distributions. They proceed by recursively moving samples through the bridge via three crucial steps: importance sampling, resampling, and rejuvenating. The last step usually consists in running independent Markov chains, and only keeping the final states for the next iterations. Waste-free SMC [Dau and Chopin, 2022] avoids discarding the intermediary samples. We establish a finite-sample complexity bound for waste-free SMC, our proof generalises the approach of Marion et al [2023]. This bound prescribes sufficient conditions on the size of the particle system for the sampler to return estimated expectations $\hat{\pi}_T(f)$ such that $|\hat{\pi}_T − \pi_T(f)| ≤ \varepsilon$ with probability at least $1 − \eta$, for any bounded test function $f$. We demonstrate that waste-free SMC enjoys lower complexity than SMC differing from a logarithmic factor $O(\log(T\eta^{-1} \varepsilon^{-2}))$. We show that our analysis allows practitioners to tune the particle budget according to the specific statistical objective in hand. In particular, when the goal is to control errors for expectation estimates under $\hat{\pi}_T$ of bounded functions, the sampler can be run with an intentionally reduced number of particles in earlier iterations while still preserving finite-sample guarantees on the estimates. In doing so, we highlight an important distinction between those two objectives: estimating normalisation constants is more challenging than estimating expectations of bounded functions.