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What you will learn in Bayesian Optimization in Action

by Quan Nguyen

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Chapter by chapter

Concepts explained in this book

• Bayesian optimizationA Bayesian optimization workflow is an iterative procedure that models an objective, selects candidate points, evaluates them, and updates the observed dataset.
• Gaussian processA Gaussian process is a probabilistic model over functions that defines predictive distributions, including means and uncertainties, at unobserved inputs.
• multifidelity optimizationMultifidelity optimization is optimization that combines evaluations of an objective with evaluations of lower-fidelity approximations having different costs and accuracies.
• objective functionAn objective function is a function whose input is selected to maximize or minimize its output.
• normal distributionA normal distribution is a continuous probability distribution characterized by a mean and variance and having a symmetric bell-shaped density.
• expected improvementExpected improvement is a Bayesian optimization acquisition function that quantifies the expected amount by which evaluating a candidate may improve on the best observed objective value.
• surrogate modelA surrogate model is a predictive model that approximates an objective function from observed evaluations.
• entropyEntropy is an information-theoretic measure of the uncertainty or variability of a random variable, defined for a discrete distribution as the negative sum of each event probability times its base-2 logarithm.
• batch optimizationBatch optimization is an optimization setting in which multiple candidate evaluations are selected or executed together.
• exploration-exploitation trade-offThe exploration-exploitation tradeoff is the tension between investigating uncertain regions and concentrating evaluations in regions already known to perform well.
• mean functionA mean function maps an input to the expected value of a random function before conditioning on observed data.
• probability of improvementProbability of Improvement is a Bayesian optimization acquisition function that assigns each candidate the probability that its objective value will exceed a specified incumbent or improvement threshold.
• pairwise comparisonA pairwise comparison is an observation that indicates which of two candidate inputs has the greater objective or preference value.
• incumbentAn incumbent is the evaluated point with the highest observed objective-function value so far, subject to feasibility when constraints are enforced.
• upper confidence boundAn upper confidence bound is a Bayesian optimization acquisition policy that combines a predictive mean with an uncertainty-based confidence term to select candidate evaluations.
• entropy searchEntropy search is a Bayesian optimization decision procedure that selects evaluations expected to reduce uncertainty about the location of an objective's optimum.
• optimize_acqfoptimize_acqf is a BoTorch function that optimizes an acquisition function to return a high-scoring candidate and its acquisition value.
• acquisition functionAn acquisition score is a numerical measure of the expected usefulness of evaluating a candidate point or batch for identifying an objective-function optimum.
• Thompson samplingThompson sampling is a Bayesian decision policy that samples a plausible model or parameter value from a posterior distribution and selects the action that is optimal under that sample.
• exploitationExploitation is an optimization strategy that evaluates regions believed to have high objective performance based on current information.
• hypervolumeHypervolume is the measure of objective space dominated by a set of multiobjective solutions relative to a reference point.
• black-box solverBlack-box optimization is optimization of an objective whose internal form is unavailable to the optimizer and can only be explored through evaluations.
• variational Gaussian processA variational Gaussian process is a scalable approximation to a Gaussian process whose posterior is represented through variational inference over a finite set of inducing points.
• preference learning problemPreference optimization is optimization of an objective representing a user's preferences when feedback is provided through comparisons rather than numerical evaluations.
• posterior probabilityA posterior distribution is a probability distribution representing updated beliefs about unknown quantities after incorporating observed data.
• Pareto frontierA Pareto frontier is the boundary formed by solutions that are not dominated by any other feasible solution across all objectives.
• max-value entropy searchMax-value Entropy Search is a Bayesian optimization acquisition policy that selects evaluations expected to reduce uncertainty about the global maximum value of an objective.
• covariance functionA covariance function, also called a kernel, specifies the covariance between function values at pairs of inputs in a Gaussian process.
• information theoryInformation theory is the mathematical study of information, uncertainty, and the relationships between probability distributions and observations.
• multi-parameter optimizationMultiobjective optimization simultaneously optimizes multiple objectives, generally seeking solutions that represent trade-offs among them.
• costQuerying cost is the computational, monetary, or resource expense incurred by evaluating an objective at a specified fidelity.
• multifidelity max-value entropy searchMultifidelity Max-value Entropy Search is a Bayesian optimization policy that selects both input locations and fidelity levels to reduce uncertainty about the objective's maximum while accounting for evaluation cost.
• structured dataStructured data is data organized according to meaningful internal arrangement, relationships, or spatial structure rather than as an unstructured collection of independent values.
• constrained expected improvementConstrainedExpectedImprovement is a BoTorch acquisition-function class that evaluates expected improvement while accounting for constraint feasibility probabilities.
• probability of feasibilityThe probability of feasibility is the probability, under a predictive distribution for a constraint function, that a candidate satisfies the specified constraint.
• nondominated solutionA nondominated point is a multiobjective solution for which no other considered solution is at least as good in every objective and better in one.
• preferenceA user preference is a person's relative desirability judgment over products, experiences, or other alternatives.
• stochastic gradient descentMini-batch gradient descent is an optimization technique that updates model parameters using loss and gradient estimates computed from small subsets of the training data.
• inducing pointAn inducing point is a selected input location whose associated function values summarize information from a larger dataset for a sparse Gaussian process approximation.
• constraintA constraint is an equality or inequality condition that candidate solutions must satisfy in an optimization problem.
• evidence lower boundThe evidence lower bound is a tractable variational objective that lower-bounds the marginal log likelihood and is optimized in approximate Bayesian inference.
• Sobol sequenceA Sobol sequence is a low-discrepancy numerical sequence that distributes points evenly across a multidimensional domain.
• baselineA reference point is a fixed objective-space point relative to which dominated hypervolume and hypervolume improvements are computed.
• hyperparameter tuningHyperparameter tuning is the process of selecting machine-learning configuration values that optimize a specified performance objective.
• prior probabilityA prior distribution is a probability distribution representing beliefs about an unknown quantity before incorporating observations.
• optimizationA method for selecting inputs that maximize or minimize an objective function.
• multi-armed banditA multi-armed bandit is a sequential decision problem in which an agent repeatedly selects among actions with uncertain reward distributions to balance information gathering and reward seeking.
• meanAn expected value is the probability-weighted average value of a random variable.
• length scaleA length scale is a kernel parameter that controls how rapidly function correlations change as inputs become separated.
• DominatedPartitioningDominatedPartitioning is a BoTorch class that represents the partition of objective space dominated by a set of points.
• ExpectedHypervolumeImprovementAn expected hypervolume improvement policy selects evaluations according to their expected increase in the dominated hypervolume of a multiobjective solution set.
• feature extractor modelA feature extractor is a model component that transforms structured inputs into informative numerical features for a downstream model.
• likelihoodA likelihood function quantifies how compatible observed data are with different values of a model's parameters or latent quantities.
• matérn 5/2 kernelA Matérn 5/2 kernel is a covariance kernel with Matérn smoothness parameter equal to five halves, producing twice-differentiable sample-function behavior under common parameterizations.
• output scaleAn output scale is a covariance amplitude parameter that controls the range of a Gaussian process's function values and predictions.
• radial basis function kernelAn RBF kernel is a radial basis function covariance or similarity function whose value decreases with distance between inputs according to a length-scale parameter.
• mixed search spaceA mixed search space contains both continuous input dimensions and discrete choices among alternatives.
• optimize_acqf_mixedoptimize_acqf_mixed is a BoTorch function that optimizes an acquisition function over search spaces combining continuous variables with fixed discrete alternatives.
• constant functionA constant mean function is a Gaussian-process mean function that assigns the same expected value to every input.
• marginal likelihoodA marginal likelihood is the probability or density of observed data after integrating over latent variables or model parameters.
• multifidelity matérn kernelA multifidelity Matérn kernel is a Matérn covariance function extended to model observations from multiple fidelity levels and their correlations.
• binary searchBinary search is a divide-and-conquer search algorithm that repeatedly compares a target with the middle element of a sorted sequence and discards one half of the remaining search interval.
• natural gradient descentNatural gradient descent is an optimization method that updates parameters using gradients adjusted to the geometry of the model's probability distribution.
• utility functionUtility is the value assigned to an outcome according to how desirable or useful it is for an optimization goal.
• inequality constraintAn inequality constraint is a constraint expressed as an inequality that limits the admissible values of one or more variables.
• correlationCorrelation is a numerical measure of the strength and direction of dependence between variables.
• neural network-Gaussian process modelA neural network-Gaussian process model combines a neural network transformation of inputs with Gaussian process prediction on the transformed representation.
• uniform distributionA uniform distribution assigns equal probability density across a specified interval or finite set of outcomes.
• confidence intervalA confidence interval is an interval constructed to provide a specified long-run coverage level for an unknown parameter.
• multifidelity bayesopt loopA multifidelity BayesOpt policy is a Bayesian optimization decision rule that selects query locations and fidelity levels by balancing information about the objective against evaluation cost.
• covariance matrixA covariance matrix is a matrix whose diagonal contains variances and whose off-diagonal entries contain covariances between components of a multivariate random variable.
• bayesian statisticsA Bayesian update is the revision of a prior belief into a posterior belief using observed data and a likelihood.
• batch acquisition scoreA batch acquisition score is a numerical measure of the expected usefulness of evaluating a group of candidate points together.
• cyclic optimizationCyclic optimization is an iterative optimization method that successively refines individual components or members of a candidate solution until convergence.
• predictive standard deviationA posterior standard deviation is the spread of a posterior predictive distribution around its posterior mean.
• uncertainty quantificationUncertainty quantification is the assignment or estimation of numerical uncertainty associated with predictions or inferred quantities.
• automatic relevance determinationAutomatic relevance determination is a kernel modeling technique that assigns separate length scales to input dimensions to represent dimension-specific variability.
• dominated regionDominated space is the region of objective space whose values are dominated by one or more candidate solutions.
• uncertaintyUncertainty is the lack of certainty about the value or behavior of an unknown quantity or function.
• linear regressionLinear regression is a model that represents a response as a linear combination of input features and fitted coefficients.
• credible intervalA credible interval is a Bayesian interval containing a specified posterior probability for an uncertain quantity.
• noisy function evaluationA noisy observation is an observed label or function value corrupted by stochastic or measurement noise.
• smoothnessSmoothness is the property that nearby or similar inputs produce correlated and therefore similar function values.
• mean vectorA mean vector is a vector containing the expected value of each component of a multivariate probability distribution.
• variational parameterA variational parameter is a parameter that specifies a variational distribution or another component of a variational approximation.
• return on investmentA return on investment measure is a ratio of profit or benefit gained from an investment to the cost of making that investment.
• batch expected improvementBatch expected improvement is a batch Bayesian optimization acquisition function that scores multiple candidate evaluations by their joint expected improvement over the incumbent.
• batch upper confidence boundBatch upper confidence bound is a batch Bayesian optimization acquisition policy that scores multiple candidates using predictive means and uncertainty.
• fidelity correlationA fidelity correlation is a parameter describing how informative observations at one fidelity are about function values at another fidelity.
• fit_gpytorch_mllfit_gpytorch_mll is a BoTorch helper function that fits a Gaussian-process model by optimizing a marginal log-likelihood object.
• multifidelity gaussian processA multifidelity Gaussian process is a Gaussian-process model that represents correlations between observations obtained from different fidelity levels of an objective function.
• SingleTaskMultiFidelityGPSingleTaskMultiFidelityGP is a BoTorch Gaussian-process model class for learning a single objective from observations annotated with fidelity information.
• varianceVariance is the expected squared deviation of a random variable from its mean and measures the spread of its distribution.
• unit cubeA unit cube, or unit hypercube, is the multidimensional hypercube whose coordinate values range from zero to one in every dimension.
• MNIST datasetThe MNIST dataset is a labeled collection of handwritten digit images commonly used for machine-learning training and evaluation.
• ard_num_dimsard_num_dims is a GPyTorch covariance-function parameter specifying the number of input dimensions for which separate kernel length scales are maintained.
• gradient descentGradient descent is an iterative optimization technique that updates parameters in the direction opposite the objective gradient.
• pairwise comparison datasetA pairwise comparison dataset is a collection of outcomes indicating which member of each pair of alternatives was preferred.
• cumulative distribution functionA cumulative distribution function method returns the probability that a random variable is less than or equal to a specified value.
• function evaluationA function evaluation, or query, is the computation of an objective function's output at a specified input.
• ProbabilityOfImprovementProbabilityOfImprovement is a BoTorch acquisition-module class that computes probability-of-improvement scores from a Gaussian process and an incumbent value.
• grid searchGrid search is an optimization technique that evaluates points arranged on a regular grid over the search space.
• random searchAn optimization strategy that evaluates randomly selected candidate locations without adaptively modeling the objective.
• samplingUniform sampling draws points so that equal-size regions of the sampling domain have equal probability.
• adamAdam is an adaptive stochastic-gradient optimization algorithm that maintains moving estimates of first and second moments of gradients.
• multifidelity datasetA multifidelity training dataset is a collection of objective observations accompanied by metadata identifying the fidelity at which each observation was obtained.
• qNoisyExpectedImprovementNoisy expected improvement is an expected-improvement acquisition variant designed to account for substantial uncertainty or observation noise in the surrogate model.
• curse of dimensionalityThe curse of dimensionality is the phenomenon in which computational or statistical costs grow exponentially as the number of dimensions increases.
• approximate distributionA variational distribution is an optimized probability distribution that approximates an intractable posterior distribution in variational inference.
• variational strategyVariationalStrategy is a GPyTorch strategy class that constructs approximate Gaussian process predictions from inducing points and variational parameters.
• digital imageImage data is numerical or symbolic data representing visual content, typically as arrays of pixel values.
• MaxPosteriorSamplingMaxPosteriorSampling is a BoTorch sampler that draws posterior samples at candidate points and selects candidates maximizing those samples.
• SobolEngineSobolEngine is a PyTorch quasi-random sampling class that generates Sobol-sequence points in a unit cube.
• returnCumulative reward is the total reward accumulated across a sequence of actions or time steps.
• InverseCostWeightedUtilityInverseCostWeightedUtility is a BoTorch utility implementation that weights information or value by the inverse of evaluation cost.
• UCB parameterA UCB parameter is a positive scalar that weights predictive uncertainty in an upper confidence bound score and controls its exploration tendency.
• parallelismParallelism is the execution of independent processes simultaneously to reduce total completion time.
• neural networkA neural network is a machine-learning model composed of interconnected computational layers that transform inputs into outputs.
• expected posterior entropyExpected posterior entropy is the expected uncertainty remaining in a posterior distribution after incorporating a prospective observation.
• numerical instabilityNumerical instability is computational behavior in which small numerical errors cause unreliable or invalid results in a calculation.
• GaussianLikelihoodGaussianLikelihood is a GPyTorch likelihood module that models observations with Gaussian noise.
• feasible solutionA feasible point or solution is a candidate assignment that satisfies all constraints in a constrained optimization problem.
• NaturalVariationalDistributionNaturalVariationalDistribution is a GPyTorch variational-distribution implementation whose parameters are represented for natural-gradient optimization.
• PairwiseGPPairwiseGP is a BoTorch model class for Gaussian-process inference from pairwise comparison data.
• PairwiseLaplaceMarginalLogLikelihoodPairwiseLaplaceMarginalLogLikelihood is a BoTorch marginal-log-likelihood implementation for Gaussian-process models trained on pairwise comparisons.
• preference-based dataPreference-based data consists of observations that order or compare alternatives without providing their numerical objective values.
• infinite-dimensional spaceAn infinite-dimensional space is a mathematical space requiring infinitely many coordinates or basis elements to represent its elements.
• GPUA graphics processing unit is a hardware processor optimized for highly parallel numerical operations such as matrix multiplication.
• budgetAn evaluation budget is an upper limit on the cumulative resources or cost that may be spent on objective-function evaluations during an optimization run.
• posterior mean maximizerA posterior mean maximizer is the input that maximizes the predictive posterior mean of an objective function.
• regularizationA loss function quantifies the discrepancy between a model's predictions and desired outcomes or data, with lower values indicating better fit.
• covarianceCovariance is a measure of how two random variables vary together.
• random variableA random variable is a variable whose value is determined by an uncertain outcome or probabilistic process.
• random vectorA random vector is an ordered collection of random variables represented as a vector.
• linear cost modelA linear cost model represents evaluation cost as a fixed component plus a weighted term that varies linearly with fidelity.
• quadratic mean functionA quadratic mean function maps an input to an expected value represented by a second-degree polynomial with learned coefficients.
• FixedFeatureAcquisitionFunctionFixedFeatureAcquisitionFunction is a BoTorch wrapper that evaluates or optimizes an acquisition function while holding selected input features at fixed values.
• optimal valueAn optimal value is the objective-function value attained at an optimizer.
• optimizerAn optimizer is an input location at which an objective function attains its maximum or minimum over the domain under consideration.
• Bayesian modelA Bayesian model represents uncertain quantities with probability distributions and updates beliefs using observed data and Bayes' theorem.
• preference modelingPreference learning is the process of learning a user's preferences from judgments or comparisons rather than necessarily from numerical evaluations.
• prior knowledgePrior knowledge is information or assumptions about an objective function that are available before observing all relevant data.
• feasible regionA feasible region is the subset of the search space containing inputs that satisfy all specified optimization constraints.
• policyA policy is a decision rule that determines an action from the information available to an optimization procedure.
• reward probabilityA reward probability is the probability that an action produces a reward in a stochastic reward process.
• optimize_acqf_cyclicoptimize_acqf_cyclic is a BoTorch function that optimizes an acquisition function through cyclic updates of batch candidates subject to constraints such as bounds.
• probabilistic predictionA probabilistic prediction is a prediction represented by a probability distribution rather than a single point value.
• ExactMarginalLogLikelihoodExactMarginalLogLikelihood is a GPyTorch module that computes the exact marginal log likelihood of training data under a Gaussian process model.
• Kullback–Leibler divergenceKullback–Leibler divergence is a nonnegative statistical divergence that quantifies how one probability distribution differs from another.
• pixelA pixel is an individual spatial element representing visual information at a location in an image.
• infeasible pointAn infeasible point is a candidate input that violates at least one constraint and therefore cannot be accepted as a solution.
• similarity metricA similarity metric is a numerical rule for quantifying how similar two data points are.
• conditioningConditioning is the mathematical operation of updating a probability distribution after fixing or observing the value of one or more variables.
• matérn kernelA Matérn kernel is a covariance function whose smoothness is controlled by a parameter that determines the finite differentiability of functions drawn from the associated Gaussian process.
• aerostructural optimizationAerostructural optimization is the computational optimization of an aircraft's aerodynamic and structural design variables to improve performance.
• search spaceA search space is the set of input locations or controllable settings over which an objective function is optimized.
• preference elicitationPreference elicitation is the process of obtaining information about a person's preferences through explicit responses or choices.
• noiseNoise is unwanted variation or corruption in observed feedback that obscures the underlying signal of interest.
• constraint-aware bayesopt policyConstrained Bayesian optimization is Bayesian optimization that models and accounts for constraints when selecting candidate evaluations.
• constraint functionA constraint function is a function whose value determines whether a candidate satisfies a specified optimization constraint.

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