[论文解读] Adversarially Robust Optimization with Gaussian Processes

Ilija Bogunovic, Jonathan Scarlett+2 more|arXiv (Cornell University)|Oct 25, 2018

Gaussian Processes and Bayesian Inference被引用 49

一句话总结

This paper introduces StableOpt, a confidence-bound based algorithm for Gaussian process optimization that ensures robustness to adversarial perturbations, with theoretical guarantees and empirical validation.

ABSTRACT

In this paper, we consider the problem of Gaussian process (GP) optimization with an added robustness requirement: The returned point may be perturbed by an adversary, and we require the function value to remain as high as possible even after this perturbation. This problem is motivated by settings in which the underlying functions during optimization and implementation stages are different, or when one is interested in finding an entire region of good inputs rather than only a single point. We show that standard GP optimization algorithms do not exhibit the desired robustness properties, and provide a novel confidence-bound based algorithm StableOpt for this purpose. We rigorously establish the required number of samples for StableOpt to find a near-optimal point, and we complement this guarantee with an algorithm-independent lower bound. We experimentally demonstrate several potential applications of interest using real-world data sets, and we show that StableOpt consistently succeeds in finding a stable maximizer where several baseline methods fail.

研究动机与目标

• Motivate GP optimization under adversarial perturbations and robustness to implementation uncertainty.
• Show that standard GP optimizers lack epsilon-stability and propose a robust alternative.
• Provide theoretical guarantees: finite-sample upper bounds and algorithm-independent lower bounds.
• Demonstrate empirical effectiveness on real datasets and diverse applications.

提出的方法

• Define epsilon-stable inputs and adversarial perturbations using a distance function d and a stability parameter epsilon.
• Use GP posterior mean mu_t and variance sigma_t with confidence bounds (u c b and l c b).
• At each step, select x_t by maximizing the worst-case ucb over perturbations, then choose a perturbation delta_t within Delta_epsilon(x_t) to minimize the lcb before sampling.
• Update GP posterior with the noisy observation of x_t + delta_t and re-compute bounds.
• Provide Theorem 1 giving an upper bound on epsilon-regret with high probability, and Lemma 1 for the confidence parameter beta_t.
• Discuss algorithm-independent lower bound (Theorem 2) and kernel-dependent implications.

实验结果

研究问题

• RQ1 How can GP optimization be made robust to adversarial perturbations within an epsilon-ball?
• RQ2 What are the sample complexity guarantees (upper bounds) for StableOpt to achieve near-optimal epsilon-stability?
• RQ3 How does StableOpt compare to standard GP-UCB and other baselines in achieving robust solutions?
• RQ4 What are the kernel-specific limitations and lower bounds for robust GP optimization?
• RQ5 Can the StableOpt framework be extended to related max-min or robust optimization settings?

主要发现

• Standard GP optimization methods may fail to produce epsilon-stable solutions.
• StableOpt achieves an epsilon-regret of at most eta after T rounds with high probability under suitable conditions (Theorem 1).
• There is an algorithm-independent lower bound showing fundamental limits depending on the kernel (Theorem 2).
• The SE kernel yields near-matching upper and lower bounds up to logarithmic factors for constant noise and budget.
• Empirical evaluations on synthetic, lake data, robot pushing, and group recommendations show StableOpt consistently finds stable maximizers where baselines fail.