Bayesian optimization for materials design

Peter I. Frazier; Jialei Wang

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Bayesian optimization for materials design

Peter I. Frazier, Jialei Wang

arXiv:1506.01349v1Jun 3, 2015stat.MLmath.OC

TL;DR

Materials discovery requires choosing high-quality designs when physical evaluations are expensive and time-consuming. The paper presents Bayesian optimization with Gaussian process regression and two value-of-information methods for selecting experiments. It emphasizes uncertainty-aware, iterative design while identifying assumptions and limits on broader optimal strategies.

Problem
Materials design relies on costly iterative experiments, creating a need to improve choices about which designs to test and how to learn from results.
Method
Bayesian optimization builds predictive models with Gaussian process regression and uses expected improvement or knowledge gradient to recommend valuable next experiments.
Results
The framework presents expected improvement for noise-free evaluations and knowledge gradient as a noise-aware generalization that can also broaden the final-solution search set.
Takeaways & Limitations
Bayesian optimization guides experimentation by acknowledging predictive uncertainty and reducing inaccuracies through iterative comparison with physical experiments.
Takeaways & Limitations
Expected improvement relies on assumptions that are often violated, while computing the fully optimal strategy for the more general setting remains intractable with current methods.

Abstract

from arXiv · show

We introduce Bayesian optimization, a technique developed for optimizing time-consuming engineering simulations and for fitting machine learning models on large datasets. Bayesian optimization guides the choice of experiments during materials design and discovery to find good material designs in as few experiments as possible. We focus on the case when materials designs are parameterized by a low-dimensional vector. Bayesian optimization is built on a statistical technique called Gaussian process regression, which allows predicting the performance of a new design based on previously tested designs. After providing a detailed introduction to Gaussian process regression, we introduce two Bayesian optimization methods: expected improvement, for design problems with noise-free evaluations; and the knowledge-gradient method, which generalizes expected improvement and may be used in design problems with noisy evaluations. Both methods are derived using a value-of-information analysis, and enjoy one-step Bayes-optimality.

1 Introduction

Materials design traditionally uses iterative trial and error, while Bayesian optimization augments this process by improving which designs are tested and how prior results inform subsequent choices.

1 Introduction: Materials design seeks chemical structures, compositions, or processing conditions that meet specified criteria through repeated synthesis and testing.The conventional cycle chooses a design using intuition or knowledge, tests it physically, and uses the result to select the next design.
1 Introduction: Bayesian optimization uses predictive modeling and decision theory to recommend designs that are most valuable to test next.It is intended to augment rather than replace the iterative nature of materials design.
1 Introduction: The most developed methods assume continuous design variables, a single quality measure to maximize, and known feasibility constraints.Extensions address settings involving discrete decisions and other departures from these assumptions.
1 Introduction: Bayesian optimization has developed across engineering, statistics, and machine learning, including applications to expensive simulations and model hyperparameter tuning.The field includes the Efficient Global Optimization method and later machine-learning applications.
1 Introduction: The chapter introduces the optimization problem, Gaussian process regression, experiment-selection methods, software, and closing remarks in sequence.This organization connects predictive modeling with recommendations about which experiments to perform.

2 Bayesian Optimization

Bayesian optimization formulates materials discovery as finding a high-quality design when evaluating each candidate is expensive, using previous observations to select the next experiment.

2 Bayesian Optimization: A material design is represented by a parameter vector x, and the unknown quality function f(x) is maximized over feasible designs.Examples include constituent ratios and processing temperature and pressure, with feasible sets defined by application-specific constraints.
2 Bayesian Optimization: Evaluating a candidate can require days or weeks and thousands of dollars in materials for synthesis and testing.This cost motivates choosing experiments carefully rather than evaluating designs indiscriminately.
2 Bayesian Optimization: Bayesian optimization builds a predictive model of f from previous designs and recommends the next design expected to be most valuable to test.The predictive model uses Gaussian process regression, while the recommendation step is described separately.

3 Gaussian Process regression

Gaussian process regression provides Bayesian predictions of an unknown material-quality function from previously evaluated designs, with covariance kernels encoding assumptions about similarity, variability, and smoothness.

3 Gaussian Process regression: Gaussian process regression predicts f(x) from observations at previously evaluated points, initially treating noise-free observations y_i = f(x_i).The framework later extends to noisy observations.
3 Gaussian Process regression: The method places a multivariate normal prior on function values, whose mean and covariance are specified by mean and covariance functions.Bayes rule combines this prior with observed data to obtain posterior predictions.
3 Gaussian Process regression: Covariance functions should assign greater covariance to nearby designs and produce positive semidefinite covariance matrices.The latter condition ensures that the multivariate normal prior is a well-defined probability distribution with non-negative variances.
3 Gaussian Process regression: The squared exponential kernel uses α to control overall variability and β_i parameters to control how quickly f varies along dimensions.Small β_i values encode stronger similarity between points separated along dimension i.
3 Gaussian Process regression: The Matérn kernel provides more flexible smoothness modeling than the squared exponential kernel.Under Matérn covariance, f is k-times mean-square differentiable if and only if ν > k, whereas the squared exponential assumption is infinitely mean-square differentiable.
3 Gaussian Process regression: Kernel parameters are commonly selected adaptively to fit previously observed points rather than fixed solely from intuition.The relevant parameters include α and β_i, and ν for the Matérn kernel.

3.2 Choice of mean function

The section specifies mean-function choices and derives Gaussian-process posterior inference by conditioning a multivariate normal prior on observations, producing predictive means and variances for new designs.

3.2 Choice of mean function: A constant mean function is a common baseline, while parametric trend terms can represent believed systematic trends in f.Polynomial basis functions provide one example of such trend terms.
3.3 Inference: Gaussian process regression obtains the posterior distribution at a candidate x∗ by conditioning the joint multivariate normal distribution on observed function values.The conditional distribution remains multivariate normal when the relevant covariance blocks are invertible.
3.3 Inference: The posterior mean and variance combine the prior mean and covariance with observed values and covariance matrices involving the candidate point.The notation uses vectors and matrices formed from previously evaluated points and x∗.
3.3 Inference: Invertibility typically holds under mild non-degeneracy conditions and distinct evaluated points for the squared exponential kernel.Repeated measurements of the same point are instead reasonable when observations are noisy.
3.3 Inference: For noise-free observations, the posterior mean interpolates observed values and the dashed bounds form a 95% Bayesian credible interval.The interval is positioned at μ_n(x∗) ± 1.96σ_n(x∗).

3.4 Inference with just one observation

With one observation, Gaussian process regression updates both the estimated function value and its uncertainty according to the prior covariance between the observed and target locations. Nearby observations produce larger mean corrections and greater variance reductions than distant observations.

The posterior mean equals the prior mean plus a correction based on the observation’s prediction error and its covariance with the target.A value above the prior expectation raises the posterior mean when the covariance is positive; a value below it lowers the mean.
Nearby observations exert stronger influence on the posterior mean because covariance decreases as the target moves farther from the observed point.For distant targets, the covariance approaches zero, so the observed prediction error has little effect.
The posterior variance is smaller than the prior variance, with the reduction increasing as the target’s prior correlation with the observed point increases.The reduction is large for nearby targets and small for distant targets, whose posterior variance remains similar to the prior variance.

3.5 Inference with noisy observations

Noisy physical experiments require Gaussian process regression to model observations as latent function values plus independent, constant-variance noise. The resulting posterior remains uncertain at measured points, and the analysis relies on assumptions that are often violated in practice.

Repeated physical measurements of the same material design can differ, so observed values are modeled as noisy rather than exact function evaluations.The noisy-observation model assumes independent Gaussian errors with mean 0 and constant variance λ^2.
The constant-variance and independence assumptions simplify analysis but are often violated by material synthesis and shared experimental conditions.Changing noise variance is associated especially with synthesis, while shared temperature or constituent variation can induce dependence across designs.
Under independent homoscedastic noise, inference starts from the joint multivariate normal distribution of observed responses and the latent function value at a target point.The noise contributes a diagonal covariance term λ^2I_n, and conditioning this joint distribution yields the posterior.
With noise, the posterior mean no longer interpolates observations and the credible interval retains positive width at measured points.Setting λ^2 = 0 recovers the noise-free expressions; nonzero noise leaves uncertainty because function values are not observed exactly.
Hyperparameters are estimated by maximizing the data’s log marginal likelihood, using analytic solutions for some parameters and numerical optimization for others.The procedure is described as empirical Bayes and includes optimization of kernel, mean, and noise-related parameters.

3.7 Diagnostics

Gaussian process regression should be diagnosed using leave-one-out predictions and credible intervals. For Bayesian optimization, calibrated uncertainty is especially important because it determines where additional sampling occurs.

Leave-one-out cross validation holds out each observation, refits Gaussian process regression, and predicts the held-out response with a posterior mean and standard deviation.Hyperparameters are preferably re-estimated separately for each held-out point.
A well-calibrated model should produce credible intervals that intersect the Predicted=Actual diagonal for approximately 95% of observations.The diagnostic plot places each interval as an error bar at the actual response value.
High predictive accuracy is less important than accurate uncertainty estimates for guiding Bayesian optimization.Reported uncertainty can direct sampling toward locations where predictions are uncertain, whereas understated uncertainty near an optimum may prevent correction.
When uncertainty or predictive accuracy is inadequate, common responses include changing the covariance kernel or transforming the objective function.For non-negative objectives, log(f) and √f are convenient strictly increasing transformations that preserve the set of optimizers.

3.8 Predicting at more than one point

Predicting function values at multiple target points requires joint Gaussian process inference rather than separate pointwise predictions. The joint calculation captures correlations among prediction errors and is implemented through numerically stable linear algebra.

Separate predictions at multiple target points provide individual posterior means and variances but do not quantify relationships among their errors.Joint estimation is required to represent dependence across locations.
The multi-point procedure begins with a joint prior over observed responses and all target function values, then conditions it using the Gaussian process posterior.The resulting posterior over target values is multivariate normal with a mean vector and covariance structure.
The covariance expression for multiple targets subtracts the information explained by the observed inputs from the prior target covariance.This formulation preserves correlations between predictions at different target locations.
Direct matrix inversion in Gaussian process posterior calculations can be slow and numerically inaccurate, especially when data points are close together.The issue arises from finite-precision arithmetic when covariance-matrix terms approach zero.
Cholesky decomposition and linear-system solves provide a faster, more accurate implementation and also compute the log marginal likelihood for hyperparameter estimation.The algorithm returns the posterior mean, posterior variance, and log marginal likelihood.

4 Choosing where to sample

Bayesian optimization chooses future material-design experiments by maximizing the expected value of information from each candidate. Expected improvement targets noise-free evaluations, while knowledge gradient extends the approach to noisy measurements and broader final-solution sets.

Bayesian optimization selects the next design by estimating the value of information gained from sampling each candidate.It combines prediction from previously evaluated designs with decision theory to direct future experiments.
Expected Improvement: Expected improvement computes the posterior expectation of the improvement over the best value observed so far.The expectation uses the Gaussian-process posterior distribution of f(x), with the positive-part function defining improvement.
Expected Improvement: Expected improvement favors points with both large posterior mean and substantial posterior uncertainty, expressing the exploration–exploitation tradeoff.In noise-free settings, previously evaluated points have zero uncertainty and therefore zero expected improvement.
Expected Improvement: The expected-improvement algorithm recommends sampling at the point with the largest EI, although finding that point is itself a global optimization problem.EI and its first and second derivatives can be computed quickly, enabling numerical optimization methods such as multistart local solvers.
Knowledge Gradient: Knowledge gradient takes the expected improvement in the best posterior value after a noisy evaluation and samples where this KG factor is largest.Its expectation can be computed analytically using the normal pdf and cdf, and it supports noise and final-solution sets broader than previously evaluated points.
Knowledge Gradient: With noisy measurements, knowledge gradient retains the exploration–exploitation tradeoff and may assign nonzero value to repeated evaluations at previously sampled points.In noise-free problems, choosing the candidate and final-solution sets as previously sampled points recovers expected improvement.

5 Software

The chapter identifies software packages that implement Bayesian optimization methods and related Gaussian process tools, including open-source and commercial options.

Several freely available and commercial software packages implement the chapter’s Bayesian optimization methods and related methods.The chapter specifically names Metrics Optimization Engine, Spearmint, DiceKriging, and DiceOptim.
Metrics Optimization Engine is open-source C++ and Python software developed by the authors and engineers at Yelp.
Spearmint is open-source Python software implementing algorithms described in reference [47].
DiceKriging and DiceOptim are open-source R packages implementing expected improvement, approximate knowledge-gradient, and parallel-evaluation algorithms.
A separate list covers software packages focused on Gaussian process regression rather than Bayesian optimization.

6 Conclusion

The conclusion presents Bayesian optimization as an iterative materials-design approach that combines predictive modeling with uncertainty-aware experiment selection. It emphasizes improving trial-and-error decisions rather than eliminating physical experimentation.

Bayesian optimization combines Gaussian process regression, expected improvement, and the knowledge-gradient method for materials design.
The approach acknowledges uncertainty in statistical prediction and guides experimentation to remain robust to that uncertainty.
Bayesian optimization remains inherently iterative and does not seek to circumvent the fundamental trial-and-error process.
An alternative informatics approach seeks predictions accurate enough to search designs in silico and test only the predicted best designs physically.
The authors argue that imperfect predictions can still be powerful when used with robustness and iterative feedback from physical experiments.
Bayesian optimization, Bayesian experimental design, and optimal learning provide a mathematical framework for using predictions effectively in this iterative process.

8 Derivations and Proofs

The derivations and proofs establish the Gaussian-process conditioning and expected-improvement calculations underlying the methods, then derive the knowledge-gradient update and computation.

The appendix contains derivations and proofs of equations and theoretical results presented in the main text.
Bayes’ rule and block-matrix inversion yield the conditional multivariate-normal distribution used in Gaussian process regression.The resulting conditional distribution has mean µ_new and covariance matrix Σ_new.
The conditional distribution’s updated parameters are µ_new = µ − ΣΣ^-1_[1,1](u[1] − µ) and Σ_new = Σ − ΣΣ^-1_[1,1]Σ.
The Gaussian-process predictive density is integrated to calculate expected improvement.
The knowledge-gradient factor depends on the additional datapoint y_n+1 and the resulting posterior mean information.
The posterior mean µ_n+1(x) is a linear function of y_n+1, with intercept and slope computable after the nth measurement.
Under independent Gaussian observation noise, the distribution of y_n+1 is obtained from the predictive distribution of f(x_n+1) and noise variance λ^2.
Computing the knowledge-gradient factor reduces to evaluating the expected maximum of linear functions of a scalar normal random variable.