[Paper Review] Bayesian optimisation under uncertain inputs

Rafael Oliveira, Lionel Ott+1 more|arXiv (Cornell University)|Feb 21, 2019

Advanced Bandit Algorithms Research19 citations

TL;DR

This paper proposes uGP-UCB, a Bayesian optimization algorithm that handles uncertain query inputs by modeling input locations as probability distributions using kernel embeddings and Gaussian processes. The method provides theoretical regret bounds under input uncertainty and outperforms conventional approaches in simulated and real-world scenarios with noisy execution and localization errors.

ABSTRACT

Bayesian optimisation (BO) has been a successful approach to optimise functions which are expensive to evaluate and whose observations are noisy. Classical BO algorithms, however, do not account for errors about the location where observations are taken, which is a common issue in problems with physical components. In these cases, the estimation of the actual query location is also subject to uncertainty. In this context, we propose an upper confidence bound (UCB) algorithm for BO problems where both the outcome of a query and the true query location are uncertain. The algorithm employs a Gaussian process model that takes probability distributions as inputs. Theoretical results are provided for both the proposed algorithm and a conventional UCB approach within the uncertain-inputs setting. Finally, we evaluate each method's performance experimentally, comparing them to other input noise aware BO approaches on simulated scenarios involving synthetic and real data.

Motivation & Objective

• To address Bayesian optimization problems where query locations are uncertain due to execution and localization noise.
• To develop a theoretical framework for regret analysis under uncertain inputs in Bayesian optimization.
• To propose a novel algorithm, uGP-UCB, that models uncertain inputs as probability distributions within a Gaussian process framework.
• To empirically evaluate uGP-UCB against existing input-noise-aware BO methods on synthetic and real data.

Proposed method

• Uses kernel embeddings of probability distributions to define a valid kernel for Gaussian processes with uncertain inputs.
• Employs a modified upper confidence bound (UCB) acquisition function that accounts for uncertainty in both input and output observations.
• Applies the uncertain-inputs squared-exponential kernel, derived analytically for Gaussian input distributions, to compute GP covariance.
• Integrates the estimated true query location distribution $ P^L_t $ and execution noise $ P^E_{m{x}} $ into the GP model to predict expected function values.
• Derives theoretical regret bounds for both uGP-UCB and standard IGP-UCB under uncertain inputs using Pinsker’s inequality and reproducing kernel Hilbert space norms.
• Uses Monte Carlo sampling and kernelized expectation approximations to compute the expected function value under input uncertainty.

Experimental results

Research questions

• RQ1 How does input uncertainty in query locations affect the performance of standard Bayesian optimization algorithms?
• RQ2 Can a GP-based acquisition function be adapted to handle input distributions rather than point estimates, improving robustness in noisy execution environments?
• RQ3 What theoretical regret bounds can be established for Bayesian optimization under uncertain inputs?
• RQ4 How does the proposed uGP-UCB algorithm compare empirically to existing input-noise-aware BO methods in terms of convergence and robustness?
• RQ5 What is the impact of modeling input uncertainty via kernel embeddings on optimization efficiency and accuracy?

Key findings

• The uGP-UCB algorithm achieves lower regret than standard IGP-UCB and other input-noise-aware methods in both synthetic and real-world scenarios with uncertain inputs.
• Theoretical regret bounds for uGP-UCB and IGP-UCB under uncertain inputs are derived, showing that regret scales with the Kullback-Leibler divergence between estimated and true input distributions.
• The uncertain-inputs squared-exponential kernel provides an analytical solution for Gaussian input distributions, enabling efficient computation in the GP model.
• Empirical results show that uGP-UCB converges faster and achieves better optimization performance than baseline methods when input uncertainty is high.
• The method outperforms approaches that use unscented transforms or Monte Carlo integration for input marginalization, particularly in high-dimensional or noisy settings.
• Theoretical analysis confirms that the expected function value under input uncertainty is bounded by the GP norm and the KL divergence between input distributions, ensuring stability.