Bayesian Optimization under Uncertainty for Training a Scale Parameter in Stochastic Models

Akash Yadav; Ruda Zhang

cs.LGOct 7, 2025

Bayesian Optimization under Uncertainty for Training a Scale Parameter in Stochastic Models

Authors: Akash Yadav, Ruda Zhang

Organizations: University of Houston

Abstract

Hyperparameter tuning is a challenging problem especially when the system itself involves uncertainty. Due to noisy function evaluations, optimization under uncertainty can be computationally expensive. In this paper, we present a novel Bayesian optimization framework tailored for hyperparameter tuning under uncertainty, with a focus on optimizing a scale- or precision-type parameter in stochastic models. The proposed method employs a statistical surrogate for the underlying random variable, enabling analytical evaluation of the expectation operator. Moreover, we derive a closed-form expression for the optimizer of the random acquisition function, which significantly reduces computational cost per iteration. Numerical experiments demonstrate substantial gains in optimization efficiency: about 40-fold reduction in data usage relative to a Monte Carlo-based one-dimensional optimization scheme, and an order of magnitude improvement relative to a Gaussian process-based Bayesian optimization baseline.

Figures & tables

Figure 1: Conditional model of

s|\beta

. The left panel shows the mean of s as a power function of

\beta

for positive and negative exponents, with dashed curves indicating the 2.5th and 97.5th conditional percentiles. The right panel presents the corresponding log–log transformations (

\ln s

vs.

\ln\beta

), where the relationship is linear.

Figure 2: Static problem: raw data and surrogate model fit. (Left) Raw data is highly noisy with a downward linear trend. Plotted points are 1% of the data, uniformly subsampled. (Right) Bayesian GLM constructed during Bayesian optimization captures the GLM regression function accurately with quantified error, using much fewer data points.

Figure 3: Static problem: optimization results under different approaches. (Left) Posterior distribution of

(\beta^{*},f^{*})

well captures the true statistics. (Right) Histogram of

\beta^{*}

posterior samples.

Figure 4: Static problem: progression of Bayesian optimization. (Left) Sample points during BO iterations. (Right) Evolution of

\beta^{*}

posterior distribution and GLM estimate.

Figure 5: Flowchart of the proposed method.

Figure 6: One-dimensional fixed-fixed static system.

Example	Method	Data points	Time (s)	$\mid\beta^{}-\beta^{}_{LR}\mid$	$\sigma(\beta^{*})$
Static	Bayesian GLM	290	0.16	0.18	3.68
	GP-BO (TS)	4,400 $(15.1\times)$	3.4 $(21.2\times)$	5.6	4.5 $(1.2\times)$
	GP-BO (qNEI)	5,100 $(17.6\times)$	3.9 $(24.4\times)$	5.0	3.99 $(1.1\times)$
	1D MC opt.	12,000 $(41.3\times)$	4.0 $(25.0\times)$	3.0	–
Dynamics	Bayesian GLM	280	8.45	0.19	1.6
	GP-BO (TS)	4,400 $(15.7\times)$	48.5 $(5.7\times)$	1.51	1.2 $(0.7\times)$

Table 1: Comparison of data points, Computational time, absolute accuracy, and standard deviation of posterior samples for different methods across three examples. Ratios in parentheses are relative to Bayesian GLM.

Figure 7: Static problem: progression of one-dimensional Monte-Carlo based optimization. (Left)

\beta

sample points during one-dimensional optimization iterations with 1000 MC samples. (Right) Objective function versus

\beta

, with mean trends and local regression estimates.

Figure 8: Static problem: optimization results. (Left)

\beta^{*}

obtained via GP surrogate with heteroscedastic noise and Thompson sampling acquisition function. (Right) Histogram of

\beta^{*}

posterior samples.

Figure 9: Static problem: optimization results. (Left)

\beta^{*}

obtained via GP surrogate with qNEI acquisition function. (Right) Histogram of

\beta^{*}

posterior samples.

Figure 10: Dynamics problem: (a) space structure; (b) loading.

Figure 11: Dynamics problem: raw data and surrogate model fit. (Left) Raw data is highly noisy with a downward linear trend. Plotted points are 1% of the data, uniformly subsampled. (Right) Bayesian GLM captures the GLM regression function accurately with quantified error, using much fewer data points.

Figure 12: Dynamics problem: optimization results under different approaches. (Left) Posterior distribution of

(\beta^{*},f^{*})

well captures the true statistics. (Right) Histogram of

\beta^{*}

posterior samples.

Figure 13: Dynamics problem: progression of Bayesian optimization. (Left) Sample points during BO iterations. (Right) Evolution of

\beta^{*}

posterior distribution and GLM estimate.

Figure 14: Dynamics problem: progression of one-dimensional Monte-Carlo based optimization. (Left)

\beta

sample points during one-dimensional optimization iterations with 1000 MC samples. (Right) Objective function versus

\beta

, with mean trends and local regression estimates.

Figure 15: Dynamic problem: optimization results. (Left)

\beta^{*}

obtained via GP surrogate with heteroscedastic noise and Thompson sampling acquisition function. (Right) Histogram of

\beta^{*}

posterior samples.

Figure 16: Dynamic problem: optimization results. (Left)

\beta^{*}

obtained via GP surrogate with qNEI acquisition function. (Right) Histogram of

\beta^{*}

posterior samples.

Figure 17: Diagnosis of residual distributions. (Left) Statistics of the residual distribution conditional on

\beta

. (Right) Histogram of the residual distribution for

\beta=150

, and the best-fit PDFs using log-normal and Gamma distributions.

Figure 18: Non-power-law problem: raw data and surrogate model fit. (Left) Raw data is highly noisy with a downward linear trend. Plotted points are 1% of the data, uniformly subsampled. (Right) Bayesian GLM captures the local regression function accurately near the optima using much fewer data points.

Figure 19: Non-power-law problem: optimization results under different approaches. (Left) Posterior distribution of

(\beta^{*},f^{*})

well captures the true statistics. (Right) Histogram of posterior samples of

\beta^{*}

, indicating residual bias due to posterior mass falling outside the optimal region.

Figure 20: Non-power-law problem: progression of Bayesian optimization. (Left) Sample points during BO iterations. (Right) Evolution of

\beta^{*}

posterior distribution and GLM estimate.

Figure 21: Non-power-law problem: progression of one-dimensional Monte-Carlo based optimization. (Left)

\beta

sample points during one-dimensional optimization iterations with 1000 MC samples. (Right) Objective function versus

\beta

, with mean trends and local regression estimates.

Figure 22: Non-power-law problem: optimization results. (Left)

\beta^{*}

obtained via GP surrogate with heteroscedastic noise and Thompson sampling acquisition function. (Right) Histogram of

\beta^{*}

posterior samples.

Figure 23: Non-power-law problem: optimization results. (Left)

\beta^{*}

obtained via GP surrogate with qNEI acquisition function. (Right) Histogram of

\beta^{*}

posterior samples.

Example	Metric	Run 1	Run 2	Run 3
Static	$\hat{\beta}$	98	98	97
Static	# evaluations	280	290	290
Dynamics	$\hat{\beta}$	40	43	43
Dynamics	# evaluations	280	290	290
Non power-law	$\hat{\beta}$	31	30	30
Non power-law	# evaluations	300	300	290

Table 2: Initialization sensitivity study across three examples. Three runs correspond to different initial datasets (all other settings fixed).

Example	Method	Mean $\hat{\beta}^{*}\pm$ std	Mean $\sigma(\beta^{*})\pm$ std	Mean Data Points
Static ( $\beta^{*}_{LR}=103$ )	Bayesian GLM	$97.9\pm 1.63$	$3.82\pm 0.27$	$309$
	GP-BO (TS)	$101.3\pm 6.26$	$6.44\pm 3.39$	$5100$
	GP-BO (qNEI)	$97.2\pm 5.37$	$6.94\pm 4.04$	$5100$
Dynamics ( $\beta^{*}_{LR}=40$ )	Bayesian GLM	$41.8\pm 1.34$	$1.78\pm 0.20$	$340$
	GP-BO (TS)	$37.2\pm 1.66$	$1.32\pm 0.46$	$4727$
	GP-BO (qNEI)	$38.5\pm 2.80$	$1.45\pm 0.68$	$4407$

Table 3: Multi-seed statistical evaluation over

N_{\mathrm{seed}}=15

independent runs. For each method and example, we report the mean and standard deviation of the estimated optimal hyperparameter, the standard deviation of the posterior samples,

\sigma(\beta^{*})

, and the mean number of data points.

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Expected Hypervolume Maximization for Multiobjective Optimization under Uncertainties

Victor Trappler

Mines Saint-Étienne MSE, LIMOS, FAYOL-ENSMSE, FAYOL-ENSMSE

Surrogate-Assisted Optimization Bayesian Optimization Active Learning Multi-Objective Optimization

Open paper AskPDF arXiv

The problem of multiobjective optimization under uncertainties is often approached by taking the expectation of each objective. In this work, we propose instead to formulate this as a Bayesian decision problem and to rely on the expected value of the hypervolume, which is to be maximized with respect to a finite set of input points. We show that this can be performed using methods based on gradients in a stochastic optimization framework, provided that care is taken with respect to dominated points. Moreover, in the absence of readily available differentiable code, we propose to use Gaussian Processes as differentiable surrogate models, in order to perform the optimization. An additional contribution in this work are some active learning strategies, through acquisition functions which helps construct a surrogate model well-designed for the multiobjective optimization problem at stake. These strategies are compared on simple analytical problems to assess their performances.

Abstract

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