arXiv subjects

Nicolas Chopin

Publications and source records attributed to Nicolas Chopin.

At least 19 recordsLinked to original sources

arXiv2026

Self-Consistency via Marginal Sharpening

Inference-time sampling can elicit strong reasoning abilities from language models without additional training. Existing power-sampling methods do so by sharpening the distribution over full generated outputs, favoring completions that are individually likely under the model. We argue that this is the wrong object to target for reasoning: a completion entangles a reasoning trace with a final answer, whereas what matters is whether an answer is supported by many plausible reasoning paths. We therefore shift the target from the full-output distribution to the sharpened answer marginal, making self-consistency an inference-time objective rather than a post-hoc voting criterion. Surprisingly, this marginal target admits an efficient approximation: we propose a simple, purely autoregressive parallel sampling algorithm that approximately samples from the sharpened answer marginal, eliciting stronger performance than standard power sampling on mathematics and coding benchmarks while being orders of magnitude faster.

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arXiv2026

On the complexity of standard and waste-free SMC samplers

We establish finite sample bounds for the error of standard and waste-free SMC samplers. Our results cover estimates of both expectations and normalising constants of the target distributions. We consider first an arbitrary sequence of distributions, and then specialise our results to tempering sequences. We use our results to derive the complexity of SMC samplers with respect to the parameters of the problem, such as $T$, the number of target distributions, in the general case, or $d$, the dimension of the ambient space, in the tempering case. We use these bounds to derive practical recommendations for the implementation of SMC samplers for end users.

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arXiv2026

A forward-only scheme for online learning of proposal distributions in particle filters

We introduce a new online approach for constructing proposal distributions in particle filters using a forward scheme. Our method progressively incorporates future observations to refine proposals. This is in contrast to backward-scheme algorithms that require access to the entire dataset, such as the iterated auxiliary particle filters (Guarniero et al., 2017, arXiv:1511.06286) and controlled sequential Monte Carlo (Heng et al., 2020, arXiv:1708.08396) which leverage all future observations through backward recursion. In comparison, our forward scheme achieves a gradual improvement of proposals that converges toward the proposal targeted by these backward methods. We show that backward approaches can be numerically unstable even in simple settings. Our forward method, however, offers significantly greater robustness with only a minor trade-off in performance, measured by the variance of the marginal likelihood estimator. Numerical experiments on both simulated and real data illustrate the enhanced stability of our forward approach.

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arXiv2025

Convergence of a class of gradient-free optimisation schemes when the objective function is noisy, irregular, or both

We investigate the convergence properties of a class of iterative algorithms designed to minimize a potentially non-smooth and noisy objective function, which may be algebraically intractable and whose values may be obtained as the output of a black box. The algorithms considered can be cast under the umbrella of a generalised gradient descent recursion, where the gradient is that of a smooth approximation of the objective function. The framework we develop includes as special cases model-based and mollification methods, two classical approaches to zero-th order optimisation. The convergence results are obtained under very weak assumptions on the regularity of the objective function and involve a trade-off between the degree of smoothing and size of the steps taken in the parameter updates. As expected, additional assumptions are required in the stochastic case. We illustrate the relevance of these algorithms and our convergence results through a challenging classification example from machine learning.

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arXiv2025

Extrapolation of Tempered Posteriors

Tempering is a popular tool in Bayesian computation, being used to transform a posterior distribution $p_1$ into a reference distribution $p_0$ that is more easily approximated. Several algorithms exist that start by approximating $p_0$ and proceed through a sequence of intermediate distributions $p_t$ until an approximation to $p_1$ is obtained. Our contribution reveals that high-quality approximation of terms up to $p_1$ is not essential, as knowledge of the intermediate distributions enables posterior quantities of interest to be extrapolated. Specifically, we establish conditions under which posterior expectations are determined by their associated tempered expectations on any non-empty $t$ interval. Harnessing this result, we propose novel methodology for approximating posterior expectations based on extrapolation and smoothing of tempered expectations, which we implement as a post-processing variance-reduction tool for sequential Monte Carlo.

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arXiv2025

Least squares variational inference

Variational inference consists in finding the best approximation of a target distribution within a certain family, where `best' means (typically) smallest Kullback-Leiber divergence. We show that, when the approximation family is exponential, the best approximation is the solution of a fixed-point equation. We introduce LSVI (Least-Squares Variational Inference), a Monte Carlo variant of the corresponding fixed-point recursion, where each iteration boils down to ordinary least squares regression and does not require computing gradients. We show that LSVI is equivalent to stochastic mirror descent; we use this insight to derive convergence guarantees. We introduce various ideas to improve LSVI further when the approximation family is Gaussian, leading to a $O(d^3)$ complexity in the dimension $d$ of the target in the full-covariance case, and a $O(d)$ complexity in the mean-field case. We show that LSVI outperforms state-of-the-art methods in a range of examples, while remaining gradient-free, that is, it does not require computing gradients.

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arXiv2025

Adaptive stratified Monte Carlo using decision trees

It has been known for a long time that stratification is one possible strategy to obtain higher convergence rates for the Monte Carlo estimation of integrals over the hyper-cube $[0, 1]^s$ of dimension $s$. However, stratified estimators such as Haber's are not practical as $s$ grows, as they require $\mathcal{O}(k^s)$ evaluations for some $k\geq 2$. We propose an adaptive stratification strategy, where the strata are derived from a a decision tree applied to a preliminary sample. We show that this strategy leads to higher convergence rates, that is, the corresponding estimators converge at rate $\mathcal{O}(N^{-1/2-r})$ for some $r>0$ for certain classes of functions. Empirically, we show through numerical experiments that the method may improve on standard Monte Carlo even when $s$ is large.

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arXiv2024

Saddlepoint Monte Carlo and its Application to Exact Ecological Inference

Assuming X is a random vector and A a non-invertible matrix, one sometimes need to perform inference while only having access to samples of Y = AX. The corresponding likelihood is typically intractable. One may still be able to perform exact Bayesian inference using a pseudo-marginal sampler, but this requires an unbiased estimator of the intractable likelihood. We propose saddlepoint Monte Carlo, a method for obtaining an unbiased estimate of the density of Y with very low variance, for any model belonging to an exponential family. Our method relies on importance sampling of the characteristic function, with insights brought by the standard saddlepoint approximation scheme with exponential tilting. We show that saddlepoint Monte Carlo makes it possible to perform exact inference on particularly challenging problems and datasets. We focus on the ecological inference problem, where one observes only aggregates at a fine level. We present in particular a study of the carryover of votes between the two rounds of various French elections, using the finest available data (number of votes for each candidate in about 60,000 polling stations over most of the French territory). We show that existing, popular approximate methods for ecological inference can lead to substantial bias, which saddlepoint Monte Carlo is immune from. We also present original results for the 2024 legislative elections on political centre-to-left and left-to-centre conversion rates when the far-right is present in the second round. Finally, we discuss other exciting applications for saddlepoint Monte Carlo, such as dealing with aggregate data in privacy or inverse problems.

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arXiv2024

Gradient-free optimization via integration

We develop and analyse an approach to optimize functions $l\colon \mathbb{R}^d \rightarrow \mathbb{R}$ not assumed to be convex, differentiable or even continuous. The algorithm belongs to the class of model-based search methods. The idea is to fit recursively $l$ to a parametric family of distributions, using a Bayesian update followed by a reprojection back onto the chosen family. Remarkably, reprojection in our scenario boils down to computing expectations, which can be simply approximated through Monte Carlo. We show that when the family of distributions is appropriately chosen this approach can be interpreted as an implicit time-inhomogeneous gradient descent algorithm on a sequence of smoothed approximations of $l$, providing a route to establishing convergence. We establish new results for generic inhomogeneous gradient descent algorithms, which we specialise to the model-based search algorithm in the Gaussian scenario. We illustrate the performance of the algorithm on a challenging classification task in machine learning.

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arXiv2024

Logarithmic Smoothing for Pessimistic Off-Policy Evaluation, Selection and Learning

This work investigates the offline formulation of the contextual bandit problem, where the goal is to leverage past interactions collected under a behavior policy to evaluate, select, and learn new, potentially better-performing, policies. Motivated by critical applications, we move beyond point estimators. Instead, we adopt the principle of pessimism where we construct upper bounds that assess a policy's worst-case performance, enabling us to confidently select and learn improved policies. Precisely, we introduce novel, fully empirical concentration bounds for a broad class of importance weighting risk estimators. These bounds are general enough to cover most existing estimators and pave the way for the development of new ones. In particular, our pursuit of the tightest bound within this class motivates a novel estimator (LS), that logarithmically smooths large importance weights. The bound for LS is provably tighter than its competitors, and naturally results in improved policy selection and learning strategies. Extensive policy evaluation, selection, and learning experiments highlight the versatility and favorable performance of LS.

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arXiv2024

Towards a turnkey approach to unbiased Monte Carlo estimation of smooth functions of expectations

Given a smooth function $f$, we develop a general approach to turn Monte Carlo samples with expectation $m$ into an unbiased estimate of $f(m)$. Specifically, we develop estimators that are based on randomly truncating the Taylor series expansion of $f$ and estimating the coefficients of the truncated series. We derive their properties and propose a strategy to set their tuning parameters -- which depend on $m$ -- automatically, with a view to make the whole approach simple to use. We develop our methods for the specific functions $f(x)=\log x$ and $f(x)=1/x$, as they arise in several statistical applications such as maximum likelihood estimation of latent variable models and Bayesian inference for un-normalised models. Detailed numerical studies are performed for a range of applications to determine how competitive and reliable the proposed approach is.

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arXiv2023

Fast Slate Policy Optimization: Going Beyond Plackett-Luce

An increasingly important building block of large scale machine learning systems is based on returning slates; an ordered lists of items given a query. Applications of this technology include: search, information retrieval and recommender systems. When the action space is large, decision systems are restricted to a particular structure to complete online queries quickly. This paper addresses the optimization of these large scale decision systems given an arbitrary reward function. We cast this learning problem in a policy optimization framework and propose a new class of policies, born from a novel relaxation of decision functions. This results in a simple, yet efficient learning algorithm that scales to massive action spaces. We compare our method to the commonly adopted Plackett-Luce policy class and demonstrate the effectiveness of our approach on problems with action space sizes in the order of millions.

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arXiv2023

A connection between Tempering and Entropic Mirror Descent

This paper explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.

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arXiv2023

Debiasing Piecewise Deterministic Markov Process samplers using couplings

Monte Carlo methods -- such as Markov chain Monte Carlo (MCMC) and piecewise deterministic Markov process (PDMP) samplers -- provide asymptotically exact estimators of expectations under a target distribution. There is growing interest in alternatives to this asymptotic regime, in particular in constructing estimators that are exact in the limit of an infinite amount of computing processors, rather than in the limit of an infinite number of Markov iterations. In particular, Jacob et al. (2020) introduced coupled MCMC estimators to remove the non-asymptotic bias, resulting in MCMC estimators that can be embarrassingly parallelised. In this work, we extend the estimators of Jacob et al. (2020) to the continuous-time context and derive couplings for the bouncy, the boomerang and the coordinate samplers. Some preliminary empirical results are included that demonstrate the reasonable scaling of our method with the dimension of the target.

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arXiv2023

Higher-order stochastic integration through cubic stratification

We propose two novel unbiased estimators of the integral $\int_{[0,1]^{s}}f(u) du$ for a function $f$, which depend on a smoothness parameter $r\in\mathbb{N}$. The first estimator integrates exactly the polynomials of degrees $p<r$ and achieves the optimal error $n^{-1/2-r/s}$ (where $n$ is the number of evaluations of $f$) when $f$ is $r$ times continuously differentiable. The second estimator is computationally cheaper but it is restricted to functions that vanish on the boundary of $[0,1]^s$. The construction of the two estimators relies on a combination of cubic stratification and control ariates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of $n$.

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arXiv2023

Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusions

This paper is concerned with online filtering of discretely observed nonlinear diffusion processes. Our approach is based on the fully adapted auxiliary particle filter, which involves Doob's $h$-transforms that are typically intractable. We propose a computational framework to approximate these $h$-transforms by solving the underlying backward Kolmogorov equations using nonlinear Feynman-Kac formulas and neural networks. The methodology allows one to train a locally optimal particle filter prior to the data-assimilation procedure. Numerical experiments illustrate that the proposed approach can be orders of magnitude more efficient than state-of-the-art particle filters in the regime of highly informative observations, when the observations are extreme under the model, or if the state dimension is large.

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arXiv2023

PAC-Bayesian Offline Contextual Bandits With Guarantees

This paper introduces a new principled approach for off-policy learning in contextual bandits. Unlike previous work, our approach does not derive learning principles from intractable or loose bounds. We analyse the problem through the PAC-Bayesian lens, interpreting policies as mixtures of decision rules. This allows us to propose novel generalization bounds and provide tractable algorithms to optimize them. We prove that the derived bounds are tighter than their competitors, and can be optimized directly to confidently improve upon the logging policy offline. Our approach learns policies with guarantees, uses all available data and does not require tuning additional hyperparameters on held-out sets. We demonstrate through extensive experiments the effectiveness of our approach in providing performance guarantees in practical scenarios.

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arXiv2023

Scalable Bayesian bi-level variable selection in generalized linear models

Motivated by a real-world application in cardiology, we develop an algorithm to perform Bayesian bi-level variable selection in a generalized linear model, for datasets that may be large both in terms of the number of individuals and the number of predictors. Our algorithm relies on the waste-free SMC Sequential Monte Carlo methodology of Dau and Chopin (2022), a new proposal mechanism to deal with the constraints specific to bi-level selection (which forbid to select an individual predictor if its group is not selected), and the ALA (approximate Laplace approximation) approach of Rossell et al. (2021). We show in our numerical study that the algorithm may offer reliable performance on large datasets within a few minutes, on both simulated data and real data related to the aforementioned cardiology application.

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