Chapter 3: Acquisition Function Design
This chapter covers Acquisition Function Design. You will learn Expected Improvement, Pareto optimality to multi-objective optimization, and Incorporate constraints into Acquisition Functions.
Expected Improvement・UCB・Multi-Objective Optimization
Learning Objectives
By reading this chapter, you will master the following:
- ✅ Understand the characteristics of four major Acquisition Functions
- ✅ Implement Expected Improvement
- ✅ Apply Pareto optimality to multi-objective optimization
- ✅ Incorporate constraints into Acquisition Functions
- ✅ Explain selection criteria for Acquisition Functions
Reading Time : 25-30 min Code Examples : 7 Exercises : 3
3.1 Fundamentals of Acquisition Functions
What is an Acquisition Function?
Definition : A scoring function that determines which sample should be acquired next
Formula : $$ x^* = \arg\max_{x \in \mathcal{X}} \alpha(x | \mathcal{D}) $$
- $\alpha(x | \mathcal{D})$: Acquisition Function
- $\mathcal{X}$: Search space
- $\mathcal{D}$: Data acquired so far
Four Major Acquisition Functions
1. Expected Improvement (EI)
Principle : Expected value of improvement from the current best value
Formula : $$ \text{EI}(x) = \mathbb{E}[\max(f(x) - f^*, 0)] $$
$$ = \begin{cases} (\mu(x) - f^*)\Phi(Z) + \sigma(x)\phi(Z) & \text{if } \sigma(x) > 0 \ 0 & \text{if } \sigma(x) = 0 \end{cases} $$
where, $$ Z = \frac{\mu(x) - f^*}{\sigma(x)} $$
- $f^*$: Current best value
- $\mu(x)$: Predicted mean
- $\sigma(x)$: Predicted standard deviation
- $\Phi(\cdot)$: Cumulative distribution function of standard normal distribution
- $\phi(\cdot)$: Probability density function of standard normal distribution
Code Example 1: Implementation of Expected Improvement
__PROTECTED_CODE_0__2. Probability of Improvement (PI)
Principle : Probability of improving the current best value
Formula : $$ \text{PI}(x) = P(f(x) \geq f^* + \xi) $$
$$ = \Phi\left(\frac{\mu(x) - f^* - \xi}{\sigma(x)}\right) $$
- $\xi$: Improvement threshold (typically 0.01)
Code Example 2: Implementation of Probability of Improvement
__PROTECTED_CODE_1__3. Upper Confidence Bound (UCB)
Principle : Predicted mean + uncertainty bonus
Formula : $$ \text{UCB}(x) = \mu(x) + \kappa \sigma(x) $$
- $\kappa$: Exploration parameter (typically 1.0-3.0)
Code Example 3: Implementation of UCB
__PROTECTED_CODE_2__4. Thompson Sampling
Principle : Sample from Gaussian Process and select the maximum value
Formula : $$ f(x) \sim \mathcal{GP}(\mu(x), k(x, x’)) $$
$$ x^* = \arg\max_{x \in \mathcal{X}} f(x) $$
Code Example 4: Implementation of Thompson Sampling
__PROTECTED_CODE_3__3.2 Multi-Objective Acquisition Functions
Pareto Optimality
Definition : A solution that does not sacrifice other objectives to improve one objective
Formula : $$ x^* \text{ is Pareto optimal} \iff \nexists x : f_i(x) \geq f_i(x^) \ \forall i \land f_j(x) > f_j(x^) \ \text{for some } j $$
Expected Hypervolume Improvement (EHVI)
Principle : Maximize the expected improvement in hypervolume
Formula : $$ \text{EHVI}(x) = \mathbb{E}[HV(\mathcal{P} \cup {f(x)}) - HV(\mathcal{P})] $$
- $HV(\cdot)$: Hypervolume
- $\mathcal{P}$: Current Pareto set
Code Example 5: Implementation of Multi-Objective Optimization (BoTorch)
__PROTECTED_CODE_4__3.3 Constrained Acquisition Functions
Handling Constraints
Example : Synthesizability constraints, cost constraints
Formula : $$ x^* = \arg\max_{x \in \mathcal{X}} \alpha(x | \mathcal{D}) \cdot P_c(x) $$
- $P_c(x)$: Probability of satisfying constraints
Constrained Expected Improvement : $$ \text{CEI}(x) = \text{EI}(x) \cdot P(c(x) \leq 0) $$
3.4 Case Study: Thermoelectric Materials Exploration
Problem Setting
Objective : Maximize the thermoelectric figure of merit ZT value
ZT value : $$ ZT = \frac{S^2 \sigma T}{\kappa} $$
- $S$: Seebeck coefficient
- $\sigma$: Electrical conductivity
- $T$: Absolute temperature
- $\kappa$: Thermal conductivity
Challenge : Simultaneous optimization of three properties (multi-objective optimization)
Chapter Summary
Comparison Table of Acquisition Functions
| Acquisition Function | Characteristics | Exploration Tendency | Computational Cost | Recommended Use |
|---|---|---|---|---|
| EI | Expected improvement | Balanced | Low | General optimization |
| PI | Improvement probability | Exploitation-focused | Low | Fast exploration |
| UCB | Upper confidence bound | Exploration-focused | Low | Wide-range exploration |
| Thompson | Stochastic | Balanced | Medium | Parallel experiments |
Next Chapter
In Chapter 4, you will learn about Applications and Practice in Materials Exploration : - Active Learning × Bayesian Optimization - Active Learning × High-Throughput Computation - Active Learning × Experimental Robots - Real-World Applications and Career Paths
Chapter 4: Applications and Practice in Materials Exploration →
Exercises
(Omitted: Detailed Implementation of Exercises)
References
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Jones, D. R. et al. (1998). “Efficient Global Optimization of Expensive Black-Box Functions.” Journal of Global Optimization , 13(4), 455-492.
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Daulton, S. et al. (2020). “Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective Bayesian Optimization.” NeurIPS.
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