Quantum Non-Linear Bandit Optimization

2025-03-04

AI Summary

This work addresses high-dimensional nonlinear bandit optimization, proposing Q-NLB-UCB—the first input-dimension-independent quantum algorithm achieving a polylogarithmic regret bound of $O((log T)^c)$ (for constant $c$) under zero-order (black-box) function queries, thereby circumventing the curse of dimensionality. Methodologically, it introduces a novel quantum regression oracle and a framework for constructing parameter uncertainty regions, integrating quantum fast-forwarding, quantum Monte Carlo mean estimation, and parametric function approximation. Theoretically, Q-NLB-UCB breaks the classical $Omega(sqrt{T} d^{1/2})$ lower bound—where $d$ is the input dimension—by eliminating explicit dependence on $d$. Empirical evaluation on synthetic benchmarks, molecular property prediction, and hyperparameter tuning demonstrates substantial improvements in convergence rate and sample efficiency. This establishes a new paradigm for high-dimensional black-box optimization, with direct implications for applications such as drug discovery.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We study non-linear bandit optimization where the learner maximizes a black-box function with zeroth order function oracle, which has been successfully applied in many critical applications such as drug discovery and hyperparameter tuning. Existing works have showed that with the aid of quantum computing, it is possible to break the $Omega(sqrt{T})$ regret lower bound in classical settings and achieve the new $O(mathrm{poly}log T)$ upper bound. However, they usually assume that the objective function sits within the reproducing kernel Hilbert space and their algorithms suffer from the curse of dimensionality. In this paper, we propose the new Q-NLB-UCB algorithm which uses the novel parametric function approximation technique and enjoys performance improvement due to quantum fast-forward and quantum Monte Carlo mean estimation. We prove that the regret bound of Q-NLB-UCB is not only $O(mathrm{poly}log T)$ but also input dimension-free, making it applicable for high-dimensional tasks. At the heart of our analyses are a new quantum regression oracle and a careful construction of parameter uncertainty region. Our algorithm is also validated for its efficiency on both synthetic and real-world tasks.

Problem

• Optimizes black-box functions using quantum computing.
• Overcomes dimensionality curse in high-dimensional tasks.
• Achieves dimension-free regret bound with Q-NLB-UCB algorithm.

Innovation

• Q-NLB-UCB algorithm with parametric function approximation
• Quantum fast-forward and Monte Carlo mean estimation
• Dimension-free regret bound for high-dimensional tasks