Multifidelity max-value entropy search

As explained in Bayesian Optimization in Action

Multifidelity 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.

Multifidelity Max-value Entropy Search (multifidelity MES) is a cost-aware Bayesian optimization policy for expensive objectives with cheaper, imperfect approximations. At each step, it chooses both an input location and a fidelity level, aiming to learn about the true objective’s maximum without spending the full budget on every evaluation. section 9.3.2

The policy treats a query’s information about the highest true objective value as its benefit and its evaluation price as its cost. Its acquisition score is return on investment: information gain about the optimum per unit cost. Maximizing this score can favor low-fidelity evaluations when they are cheap and informative, while favoring high fidelity when low-fidelity evidence is insufficient or nearly as costly. section 9.3.2

Figure 9.10 The formula of the ROI acquisition score for multifidelity optimization. This score quantifies the amount of information a query provides about the objective’s optimum for each unit of cost.

Use the ROI score to compare every candidate input-fidelity pair on a common scale: expected information about the optimum divided by query cost. section 9.3.2

A practical optimization loop is:

• Model the cost of querying each candidate at each available fidelity. section 9.3.2
• Generate candidate inputs at the target fidelity for the MES information calculation, then construct a cost-aware multifidelity MES acquisition function. section 9.3.2
• Optimize over the input domain while restricting the fidelity coordinate to the available levels, and evaluate the selected pair. section 9.3.2
• Track budget spent and recommend the input that maximizes the posterior mean at target fidelity rather than relying on the best observed high-fidelity value. section 9.4

Multiple approximations of Branin

With two low-fidelity Branin approximations and the ground truth, multifidelity MES applies the same information-per-cost rule across all three choices. The approximations have fidelity values 0.1 and 0.3, with costs 0.3 and 0.5, while the true function has fidelity 1 and cost 1.2; higher fidelity more closely resembles the ground truth. section 9.6

Assess progress by the true objective value at the posterior-mean maximizer as budget is spent. A high-fidelity incumbent is misleading: low-fidelity observations may locate the promising region even before any high-fidelity observation exists. Repeated runs can then be averaged to compare multifidelity MES with single-fidelity MES. section 9.4 section 9.5

How Bayesian Optimization in Action covers it

How deep it goes

Develops ROI scoring, fidelity-selection cases, BoTorch construction, and budget-based evaluation without deriving MES internals.

Examples

Uses low- versus high-fidelity toy query choices and AugmentedBranin with two low-fidelity approximations.

What you can do after

Apply it via BoTorch to optimize a cost-aware multifidelity acquisition function and evaluate recommendations under a budget.

How it connects

Connects Bayesian optimization with Max-value Entropy Search, acquisition functions, Gaussian-process beliefs, posterior means, and cost models.

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Related concepts

• max-value entropy searchcontrasts with · Max-value Entropy Search is a Bayesian optimization acquisition policy that selects evaluations expected to reduce uncertainty about the global maximum value of an objective.
• multifidelity optimizationkind of · Multifidelity optimization combines evaluations of an objective at different levels of accuracy and cost to reduce optimization expense.
• multifidelity bayesopt loopkind of · A multifidelity BayesOpt loop is an iterative Bayesian optimization procedure that fits a model to observations from multiple fidelities and selects both an input and fidelity for the next evaluation.

This explainer was generated by Manning's knowledge graph from the text of Bayesian Optimization in Action. The section links point to the parts of the book it draws on.