How Sequential Algorithm Portfolios can benefit Black Box Optimization

TL;DR

This work addresses the inefficiency of committing to a single optimizer in black-box optimization by introducing sequential budget-heterogeneous portfolios that allocate a fixed budget across multiple algorithms. A data-driven greedy approach, augmented with a budget-penalty, constructs portfolios that exploit algorithm complementarity and variance reduction, achieving consistent improvements over single-algorithm baselines on the COCO/BBOB benchmark. Analyses across diverse settings reveal a backbone generalist (e.g., $\\ellq$-CMA-ES) complemented by specialists, with restart dynamics emerging as a key driver of performance. The findings suggest practical benefits for restart strategies and warm-started execution, and point to integration opportunities with AutoML pipelines for robust, adaptive optimization.

Abstract

In typical black-box optimization applications, the available computational budget is often allocated to a single algorithm, typically chosen based on user preference with limited knowledge about the problem at hand or according to some expert knowledge. However, we show that splitting the budget across several algorithms yield significantly better results. This approach benefits from both algorithm complementarity across diverse problems and variance reduction within individual functions, and shows that algorithm portfolios do NOT require parallel evaluation capabilities. To demonstrate the advantage of sequential algorithm portfolios, we apply it to the COCO data archive, using over 200 algorithms evaluated on the BBOB test suite. The proposed sequential portfolios consistently outperform single-algorithm baselines, achieving relative performance gains of over 14%, and offering new insights into restart mechanisms and potential for warm-started execution strategies.

Figures (14)

• Figure 1: Empirical attainment plots for BBOB functions using CMA-ES from nevergrad. The horizontal axis shows function evaluations ($\mathcal{B}\xspace$), and the vertical axis represents normalized precisions ($\mathcal{E}\xspace$). The red curve represent the performance of a portfolio consisting only of one pair algorithm-budget.
• Figure 2: Performance comparison between the portfolio and baseline algorithm over increasing total budgets, using a subset of COCO’s algorithm set. Background elements represent the composition of the resulting portfolio. We observe that performance gains become more prominent as the total budget increases, and the selected portfolios diversify accordingly.
• Figure 3: The plot shows: (blue) the best-performing algorithm for each function; (ref) the single best algorithm across all functions; and (green) the performance of the constructed portfolio for a total budget of 10,000. Although the portfolio may not outperform the per-function best algorithm on every instance, it achieves a higher overall performance by leveraging complementary strengths across the function set.
• Figure 4: Normalized Shapley values indicating the contribution of each algorithm to each individual function in the portfolio. Higher values suggest greater importance of the corresponding algorithm in improving performance for a given function.
• Figure 5: Effect of the penalty function $\pi(b)$ on greedy portfolio construction across different weights $w$ and powers $p$.