Procrastinated Tree Search: Black-box Optimization with Delayed, Noisy, and Multi-Fidelity Feedback

TL;DR

This paper introduces PCTS, a flexible tree search method for black-box optimization that effectively handles delayed, noisy, and multi-fidelity feedback, with theoretical regret bounds and empirical validation.

Contribution

The paper proposes PCTS, an extension of HOO, incorporating delay and noise tolerance, along with a generic regret analysis and practical algorithms for complex feedback scenarios.

Findings

PCTS achieves regret bounds similar to non-delayed HOO for small delays.

PCTS outperforms existing methods in synthetic and hyperparameter tuning tasks.

Theoretical analysis quantifies regret under various delay and fidelity conditions.

Abstract

In black-box optimization problems, we aim to maximize an unknown objective function, where the function is only accessible through feedbacks of an evaluation or simulation oracle. In real-life, the feedbacks of such oracles are often noisy and available after some unknown delay that may depend on the computation time of the oracle. Additionally, if the exact evaluations are expensive but coarse approximations are available at a lower cost, the feedbacks can have multi-fidelity. In order to address this problem, we propose a generic extension of hierarchical optimistic tree search (HOO), called ProCrastinated Tree Search (PCTS), that flexibly accommodates a delay and noise-tolerant bandit algorithm. We provide a generic proof technique to quantify regret of PCTS under delayed, noisy, and multi-fidelity feedbacks. Specifically, we derive regret bounds of PCTS enabled with delayed-UCB1 (DUCB1) and delayed-UCB-V (DUCBV) algorithms. Given a horizon $T$, PCTS retains the regret bound of non-delayed HOO for expected delay of $O(\log T)$ and worsens by $O(T^{\frac{1-\alpha}{d+2}})$ for expected delays of $O(T^{1-\alpha})$ for $\alpha \in (0,1]$. We experimentally validate on multiple synthetic functions and hyperparameter tuning problems that PCTS outperforms the state-of-the-art black-box optimization methods for feedbacks with different noise levels, delays, and fidelity.