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Exploiting Gradients in Bayesian Inference of Expensive Simulators

Šimon Soldát, Václav Šmídl

arXiv:2610.12076v1cs.LGstat.ML

TL;DR

Expensive simulator evaluations make full posterior inference difficult under limited simulation budgets, while available simulator gradients are rarely used for inference. The paper incorporates gradients into Gaussian-process surrogates for Bayesian optimization-based inference and finds that cost-adjusted benefits depend on differentiation mode and problem dimensions.

  • Problem

    Expensive simulator evaluations make posterior characterization challenging under limited simulation budgets, and increasingly available simulator gradients remain largely untapped in simulation-based inference.

    Source: §I · pp. 1–2

  • Method

    The paper incorporates gradient-enhanced Gaussian-process surrogates into BOSIP for Bayesian simulation-based inference.

    Source: §I · p. 2, B. Surrogate-Based Posterior… · p. 3

  • Results

    Gradient-enhanced surrogates improve posterior approximation across benchmarks, but only reverse-mode differentiation retains performance improvements after accounting for gradient costs.

    Source: §IV · p. 7, §V · p. 8

  • Takeaways & Limitations

    Gradient-enhanced surrogates are most beneficial when parameter dimensionality exceeds proxy dimensionality, where reverse-mode differentiation is computationally efficient.

    Source: §V · p. 8

    After accounting for computational expense, forward-mode speed-ups do not justify their added cost, unlike reverse-mode gains.

    Source: §V · p. 8

Abstract

from arXiv · show

Simulators based on differential equations are ubiquitous in science and engineering. They are often used in simulation-based inference to evaluate the posterior distribution of the input parameters based on real-world observations of the simulator outputs. However, inference becomes challenging when individual simulator evaluations are computationally expensive. In such cases, a Bayesian optimization-based active learning approach with Gaussian process surrogate models has been used to maximize the information obtained from a limited simulation budget. Recently, gradients of simulator outputs with respect to input parameters have become increasingly available, yet they are rarely exploited for inference. Even though we only need to learn the simulator input-output relationship, gradient information can provide an additional valuable signal to guide the active learning procedure. This is of particular interest in the case of expensive simulators, when sample efficiency is crucial. In this paper, we demonstrate how incorporating gradient information into the Gaussian process surrogate accelerates Bayesian optimization-based inference under a limited simulation budget. Our results show significant improvement in convergence speed from using gradient information. For reverse-mode differentiation, the inference efficiency gains are maintained when accounting for the additional computational cost. In contrast, for forward-mode differentiation, the inference speed-up does not outweigh the computational costs. These results indicate that gradient-enhanced surrogates are beneficial primarily in problems where the number of parameters exceeds the output dimensionality, where reverse-mode differentiation is efficient.

I. INTRODUCTION

The paper extends BOSIP, a Bayesian optimization framework for learning full simulator-input posteriors, with gradient-enhanced Gaussian processes. It reports that gradients improve posterior convergence, with cost-adjusted benefits depending on differentiation mode and problem dimensions.

I. INTRODUCTION

  • Expensive simulator evaluations make exhaustive posterior exploration infeasible under limited simulation budgets, requiring careful selection of parameter configurations.A single forward simulation may require hours of compute in practical settings.Source: §I · p. 1
  • BOSIP targets the full parameter posterior rather than only its mode, propagating surrogate uncertainty through the likelihood to guide information-focused acquisition.Maximum Variance and Expected Integrated Variance are examples of its specialized acquisition functions.Source: §I · p. 1
  • In the toy comparison, classic BO concentrates evaluations near the posterior mode, whereas BOSIP spreads them across high-probability regions to learn posterior shape.The example has a uniform prior, linear simulator, Gaussian likelihood, and observation zo = 0.Source: Figure 1 · p. 2
  • Reverse-mode gradient computation scales with output dimensionality, while forward-mode cost scales with input dimensionality, making their relative efficiency depend on problem dimensions.Gradients are increasingly available through automatic differentiation, but remain largely untapped in simulation-based inference.Source: §I · p. 2
  • The paper incorporates derivative information into Gaussian processes, allowing gradients to inform predictions of simulator function values.Existing techniques for incorporating derivative information into GPs support this gradient-enhanced surrogate approach,, [24].Source: §I · p. 2
  • Gradient-enhanced BOSIP consistently improves posterior approximation over gradient-free baselines across seven benchmarks, with reverse-mode gains retained after accounting for computational cost.Forward-mode speed-ups do not outweigh their computational costs; reverse-mode is most efficient when parameter dimensionality exceeds output dimensionality.Source: §I · p. 2

II. SIMULATOR INFERENCE VIA BAYESIAN OPTIMIZATION

This section recaps the theoretical and methodological framework of BOSIP, previously proposed by the authors.

II. SIMULATOR INFERENCE VIA BAYESIAN OPTIMIZATION

  • The section recaps the BOSIP theoretical and methodological framework introduced by the authors.It refers readers to that work for a comprehensive description of the method.Source: §II · p. 3
  • The prior BOSIP work discusses the method's design choices in detail.Source: §II · p. 3
  • The cited work presents a unifying perspective on related work.Source: §II · p. 3

A. Problem Formulation

The problem is to infer simulator input parameters consistent with noisy observations, expressed as a posterior distribution. BOSIP addresses the expense of characterizing that posterior by using a surrogate and information-guided sequential queries.

A. Problem Formulation

  • The inference target is the posterior distribution of simulator inputs given observed data from a noisy measurement of the real system.The observation model describes measurement uncertainty; the formulation omits direct dependence of the observation on x for simplicity.Source: A. Problem Formulation · p. 3
  • The posterior is expressed using the likelihood, parameter prior, and evidence, with the evidence constant with respect to x.The unnormalized posterior is the likelihood multiplied by the prior; its normalization constant can be recovered post hoc if needed.Source: A. Problem Formulation · p. 3
  • The likelihood can be evaluated for parameters x using the simulator and a closed-form observation model, but full posterior characterization requires many evaluations.Source: A. Problem Formulation · p. 3
  • BOSIP uses a cheap probabilistic surrogate for the simulator and Bayesian optimization to select parameter queries that maximize information about the posterior.Source: A. Problem Formulation · p. 3

B. Surrogate-Based Posterior Approximation

BOSIP approximates a proxy of the simulator outputs with a surrogate, then uses its predictions and uncertainty to estimate the unnormalized posterior and guide acquisition.

B. Surrogate-Based Posterior Approximation

  • The inference goal is to learn the probability density function of the parameter posterior, represented through an unnormalized posterior target.Source: B. Surrogate-Based Posterior… · p. 3
  • BOSIP's surrogate approximates a proxy variable derived from simulator outputs, with a proxy mapping followed by a complementary likelihood mapping.The proxy mapping allows a flexible choice of surrogate target.Source: B. Surrogate-Based Posterior… · p. 3
  • The surrogate predictive distribution estimates the proxy variable for given parameters and supports an unnormalized posterior estimate.Source: B. Surrogate-Based Posterior… · p. 3
  • Marginalizing over predictive uncertainty yields a likelihood estimate without further simulator evaluations, while propagated uncertainty provides uncertainty in the posterior estimate for active acquisition.Source: B. Surrogate-Based Posterior… · p. 3

C. Gaussian Process Surrogate

The Gaussian process surrogate models each proxy dimension independently and provides a closed-form predictive distribution. Its mean and covariance are determined by the prior mean and kernel, with hyperparameters selected by MAP estimation.

C. Gaussian Process Surrogate

  • Each proxy dimension is modeled by an independent Gaussian process with a closed-form normal predictive distribution.The prior mean and kernel define the GP; the kernel encodes assumptions about response-surface smoothness and structure.Source: C. Gaussian Process Surrogate · p. 4
  • The predictive mean uses the prior mean, training-target residuals, and covariances between training inputs and the test input.K is the training-input covariance matrix, k(x) is the covariance vector, δj contains training targets, and m contains prior means at training inputs.Source: C. Gaussian Process Surrogate · p. 4
  • Hyperparameters are selected by MAP estimation, with each proxy dimension receiving its own set of parameters.These include simulation-noise deviation, kernel lengthscales, and kernel amplitude; the prior is user-defined.Source: C. Gaussian Process Surrogate · p. 4
  • For a Gaussian likelihood with proxy δ = y, the posterior mean and variance can be computed in closed form.The GP prior mean is typically zero unless prior knowledge about the simulator is available.Source: C. Gaussian Process Surrogate · p. 4

D. Active Data Acquisition

Active data acquisition selects simulator inputs sequentially to improve posterior characterization under a limited simulation budget. BOSIP’s MaxVar acquisition uses posterior variance to balance exploration and exploitation.

D. Active Data Acquisition

  • The BO loop fits the surrogate, maximizes the acquisition function to choose the next input, and evaluates the simulator to augment the dataset.The cycle continues until the simulation budget is exhausted.Source: D. Active Data Acquisition · p. 4
  • The acquisition function balances exploration of high predictive-uncertainty regions with exploitation of regions of large estimated posterior probability.It can be evaluated and maximized using the surrogate without running the simulator.Source: D. Active Data Acquisition · p. 4
  • MaxVar selects the point that maximizes the posterior variance.In BOSIP, this variance arises after the predictive distribution is transformed through the likelihood and multiplied by the prior.Source: D. Active Data Acquisition · p. 4

III. INCORPORATING GRADIENT INFORMATION

The gradient-enhanced surrogate conditions jointly on simulator proxy values and their parameter derivatives. The procedure adapts the acquisition and posterior calculations to use the augmented model, while the study benchmarks MaxVar alone.

III. INCORPORATING GRADIENT INFORMATION

  • The gradient-enhanced GP jointly models each proxy dimension’s function values and parameter gradients, propagating derivative information into proxy-value predictions.The augmented kernel incorporates covariances among function values, gradients, and their cross-covariances.Source: §III · pp. 4–5
  • The augmented training targets include function values and gradients, producing an n(1 + d) × n(1 + d) kernel matrix.The augmented kernel is determined by the original kernel and its derivatives, without adding kernel hyperparameters.Source: §III · p. 5
  • Proxy gradients are computed by the chain rule, combining the proxy-mapping Jacobian with the simulator Jacobian obtained through forward- or reverse-mode differentiation.The proxy-mapping Jacobian is computed analytically from the proxy definition.Source: §III · p. 5
  • Gradient observations add d gradient-noise deviations per proxy dimension to account for uncertainty in each partial derivative.These deviations are added to the corresponding diagonal entries of the kernel matrix in place of simulation noise for gradient observations.Source: §III · p. 5
  • The adapted procedure updates the augmented model and acquisition after each simulator evaluation, then estimates the posterior from the marginal proxy predictive distribution.The gradient information improves the posterior estimate by propagating through the augmented covariance structure into proxy-value predictions.Source: §III · p. 5
  • For MaxVar, the acquisition uses the marginal predictive distribution of the proxy; adapting more complex acquisitions requires marginalizing over gradients and is left for future work.The study benchmarks only MaxVar.Source: §III · p. 5

IV. EXPERIMENTS

Across seven benchmarks, gradient-enhanced BOSIP improves posterior approximation, with cost-adjusted gains for reverse-mode but not forward-mode differentiation. Benefits are larger in higher dimensions, while SIR results show that proxy design can affect numerical stability.

IV. EXPERIMENTS

  • The seven benchmarks comprise four analytical toy problems and three PDE-governed simulators, including 2D and 3D problems; experiments use 20 runs with three initial points each.Budgets are 100 simulations for analytical problems and 200 for PDE simulators; 6D problems use 500.Source: §IV · pp. 5–6
  • In higher-dimensional benchmarks, reverse-mode gradient benefits are larger, although the 6D experiments do not fully converge within their 500-simulation budget.The benchmark study extends analytical problems to 6D; those experiments omit kernel hyperparameter optimization.Source: Figure 3 · p. 6, §IV · pp. 6–7
  • Posterior approximation quality is measured by the total variation distance between the BOSIP approximation and the benchmark’s true posterior.The distance is approximated by Monte Carlo integration.Source: §IV · pp. 6–7
  • On SIR, smoothing the proxy response surface greatly improves gradient-enhanced performance and mitigates numerical issues caused by sharp ridges and huge gradients.Gradient-enhanced runs are truncated at 50 simulations because the numerical issues impede further convergence.Source: Figure 4 · p. 7, §IV · p. 7
  • Reverse-mode gradient-enhanced BOSIP retains better posterior-approximation performance than the gradient-free baseline after computational-cost adjustment, while forward-mode cost-adjusted performance is worse.The cost adjustment assumes a gradient pass costs roughly one simulation; actual gradient costs vary.Source: Figure 2 · p. 6, §IV · p. 7
  • Gradient-enhanced runs are less efficient per total information collected, but reverse-mode’s lower added cost yields improved cost-adjusted performance over the gradient-free approach.The data-adjusted score accounts for simulator output values and gradient components.Source: §IV · p. 7

V. DISCUSSION

Gradient-enhanced surrogates improve posterior convergence, but their practical value depends on gradient-computation costs and is strongest when parameter dimensionality exceeds proxy dimensionality. Gaussian-process scalability remains a challenge as datasets grow.

V. DISCUSSION

  • Gradient-enhanced surrogates accelerate posterior convergence and improve approximation quality, but only reverse-mode retains gains after gradient costs; forward-mode overhead favors additional gradient-free simulations.The reported improvements span analytical and realistic benchmarks.Source: §V · p. 8
  • Gradient information is primarily worthwhile when parameter dimensionality exceeds proxy-variable dimensionality, with reverse-mode providing gradients across all input dimensions at the same cost.Source: §V · p. 8
  • Gaussian-process surrogates become computationally demanding as the number of data points increases, motivating work on scalability or alternative gradient-enabled surrogates.This challenge matters because high-dimensional problems that benefit most from gradients typically require larger simulation budgets to converge.Source: §V · p. 8
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