Practical Bayesian Algorithm Execution via Posterior Sampling

TL;DR

Observing that, in many tasks, the property of interest corresponds to a target set of points defined by the function, PS-BAX is introduced, a simple, effective, and scalable BAX method based on posterior sampling, offering new insights into posterior sampling as an algorithm design paradigm.

Abstract

We consider Bayesian algorithm execution (BAX), a framework for efficiently selecting evaluation points of an expensive function to infer a property of interest encoded as the output of a base algorithm. Since the base algorithm typically requires more evaluations than are feasible, it cannot be directly applied. Instead, BAX methods sequentially select evaluation points using a probabilistic numerical approach. Current BAX methods use expected information gain to guide this selection. However, this approach is computationally intensive. Observing that, in many tasks, the property of interest corresponds to a target set of points defined by the function, we introduce PS-BAX, a simple, effective, and scalable BAX method based on posterior sampling. PS-BAX is applicable to a wide range of problems, including many optimization variants and level set estimation. Experiments across diverse tasks demonstrate that PS-BAX performs competitively with existing baselines while being significantly faster, simpler to implement, and easily parallelizable, setting a strong baseline for future research. Additionally, we establish conditions under which PS-BAX is asymptotically convergent, offering new insights into posterior sampling as an algorithm design paradigm.

Figures (7)

• Figure 1: Depiction of PS-BAX (Algorithm \ref{['alg:ps']}) for the level-set estimation problem. We plot the objective function $f$ (black line), the current available data $\mathcal{D}_{n-1}$ (black points), the threshold (grey dashed line), the posterior distribution $p(f \mid \mathcal{D}_{n-1})$ (blue line and light blue region), a sample from the posterior $\tilde{f}_n \sim p(f \mid \mathcal{D}_{n-1})$ (green line), the corresponding sampled target set $X_n=\mathcal{O}_{\mathcal{A}}(\tilde{f}_n)$ (green region) (this is the set of inputs where the green line is above the threshold), the variance of $p(f \mid \mathcal{D}_{n-1})$ (green line, bottom row), and the next point to evaluate selected by PS-BAX $x_{n} \in X_n$ (input marked by the vertical red line). The key step is computing the target set $X_n$ using the sampled function $\tilde{f}_n$, which generalizes posterior sampling for standard BO.
• Figure 2: Results for Local Optimization, showing the log10 inference regret achieved by the compared algorithms (lower values indicate better performance). The left and right panels present results for the Hartmann-6D and Ackley-10D functions, respectively. On Hartmann-6D, PS-BAX and EI perform comparably, both outperforming INFO-BAX. On Ackley-10D, PS-BAX achieves significantly better results than the rest of the algorithms.
• Figure 3: Results for Level Set Estimation, showing the F1 score (where higher is better). The left and right panels present results for the Himmelblau test function and the topographic mapping problem, respectively. In the former problem, all algorithms perform similarly, while in the latter, PS-BAX outperforms all baselines.
• Figure 4: Depiction of the INFO-BAX (left) and PS-BAX (right) algorithms on the topographic level set estimation problem described in Section \ref{['exp:level_set']}. Each figure shows the ground truth super-level set (small black dots), the points evaluated after 100 iterations (green and blue dots for INFO-BAX and PS-BAX, respectively), and the estimated level set from the final posterior mean (red dots). PS-BAX provides an accurate estimate of the level set, whereas INFO-BAX misses a significant portion.
• Figure 5: Results for Top-$k$ Estimation, showing the Jaccard distance (where lower is better). The left panel presents results for the 3-dimensional Rosenbrock test function with $k=4$, while the right panel shows results for the real-world protein design GB1 dataset with $k=10$. In both problems, PS-BAX performs similarly to INFO-BAX.

Theorems & Definitions (10)

• Definition 1
• Theorem 1
• Corollary 1
• Theorem 2
• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof