Adaptive Batch Sizes for Active Learning A Probabilistic Numerics Approach

TL;DR

This work proposes a novel Probabilistic Numerics framework that adaptively changes batch sizes, framing batch selection as a quadrature task and demonstrates that this approach significantly enhances learning efficiency and flexibility in diverse Bayesian batch active learning and Bayesian optimization applications.

Abstract

Active learning parallelization is widely used, but typically relies on fixing the batch size throughout experimentation. This fixed approach is inefficient because of a dynamic trade-off between cost and speed -- larger batches are more costly, smaller batches lead to slower wall-clock run-times -- and the trade-off may change over the run (larger batches are often preferable earlier). To address this trade-off, we propose a novel Probabilistic Numerics framework that adaptively changes batch sizes. By framing batch selection as a quadrature task, our integration-error-aware algorithm facilitates the automatic tuning of batch sizes to meet predefined quadrature precision objectives, akin to how typical optimizers terminate based on convergence thresholds. This approach obviates the necessity for exhaustive searches across all potential batch sizes. We also extend this to scenarios with constrained active learning and constrained optimization, interpreting constraint violations as reductions in the precision requirement, to subsequently adapt batch construction. Through extensive experiments, we demonstrate that our approach significantly enhances learning efficiency and flexibility in diverse Bayesian batch active learning and Bayesian optimization applications.

Figures (6)

• Figure 1: We fix the quadrature precision instead of batch size. The batch size changes adaptively to meet the predefined precision requirement. Our method, AdaBatAL, efficiently determines the optimal number of batch sizes and their querying positions without requiring a brute-force search of all possible batch sizes. AdaBatAL also offers adaptive batch sizes for constrained active learning and constrained Bayesian optimization.
• Figure 2: Constrained batch active learning. As the increased violation risk $\epsilon_\text{vio}$ propagates to the tolerance $\epsilon_\text{LP}$, reward maximization is subsequently prioritized over quadrature, resulting in safe batch samples.
• Figure 3: Batch Bayesian optimization results on Hartmann ($d=6$): (a) convergence plot with ($n \leq 5$). (b) batch size variability ($n \leq 100$). The tolerance is set ($\epsilon = 10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}$). (c) Total queries vs. simple regret at the last iteration results of (a)(b). For fixed batch size methods, the mean batch size of AdaBatAL is used ($n = 5, 30, 50, 73, 90$). The plot shows mean $\pm$ standard error of the mean.
• Figure 4: Tolerance effect on constrained batch BO on Branin ($d=2$): the balance between (a) violation rate and expected reward, and (b) worst-case error and log determinant. (c) Tolerance adaptively controls violation rate, and (d) outperforms the fixed cases. (a)(b)(c) are the two Y-axis plots where the color and arrow indicate which Y axis to see.
• Figure 5: Convergence plot of both constrained batch active learning and Bayesian optimization results across 5 synthetic functions and 7 real-world tasks . $d$ is the dimension, $c$ is the number of unknown constraints. Negative log marginal likelihood (NLML) for active learning tasks, log regret or log best observations for optimization task. Lines and shaded area denote mean ± 1 standard error.

Theorems & Definitions (2)

• Proposition 1
• proof : Proof of Proposition \ref{['prop:lp']}