Robustness to distributional shift is one of the key challenges of contemporary machine learning. Attaining such robustness is the goal of distributionally robust optimization, which seeks a solution to an optimization problem that is worst-case robust under a specified distributional shift of an uncontrolled covariate. In this paper, we study such a problem when the distributional shift is measured via the maximum mean discrepancy (MMD). For the setting of zeroth-order, noisy optimization, we present a novel distributionally robust Bayesian optimization algorithm (DRBO). Our algorithm provably obtains sub-linear robust regret in various settings that differ in how the uncertain covariate is observed. We demonstrate the robust performance of our method on both synthetic and real-world benchmarks.
Gaussian processes are an important regression tool with excellent analytic properties which allow for direct integration of derivative observations. However, vanilla GP methods scale cubically in the amount of observations. In this work, we propose a novel approach for scaling GP regression with derivatives based on quadrature Fourier features. We then prove deterministic, non-asymptotic and exponentially fast decaying error bounds which apply for both the approximated kernel as well as the approximated posterior. To furthermore illustrate the practical applicability of our method, we then apply it to ODIN, a recently developed algorithm for ODE parameter inference. In an extensive experiments section, all results are empirically validated, demonstrating the speed, accuracy, and practical applicability of this approach.
We consider the problem of optimizing an unknown (typically non-convex) function with a bounded norm in some Reproducing Kernel Hilbert Space (RKHS), based on noisy bandit feedback. We consider a novel variant of this problem in which the point evaluations are not only corrupted by random noise, but also adversarial corruptions. We introduce an algorithm Fast-Slow GP-UCB based on Gaussian process methods, randomized selection between two instances labeled "fast" (but non-robust) and "slow" (but robust), enlarged confidence bounds, and the principle of optimism under uncertainty. We present a novel theoretical analysis upper bounding the cumulative regret in terms of the corruption level, the time horizon, and the underlying kernel, and we argue that certain dependencies cannot be improved. We observe that distinct algorithmic ideas are required depending on whether one is required to perform well in both the corrupted and non-corrupted settings, and whether the corruption level is known or not.
We consider robust optimization problems, where the goal is to optimize an unknown objective function against the worst-case realization of an uncertain parameter. For this setting, we design a novel sample-efficient algorithm GP-MRO, which sequentially learns about the unknown objective from noisy point evaluations. GP-MRO seeks to discover a robust and randomized mixed strategy, that maximizes the worst-case expected objective value. To achieve this, it combines techniques from online learning with nonparametric confidence bounds from Gaussian processes. Our theoretical results characterize the number of samples required by GP-MRO to discover a robust near-optimal mixed strategy for different GP kernels of interest. We experimentally demonstrate the performance of our algorithm on synthetic datasets and on human-assisted trajectory planning tasks for autonomous vehicles. In our simulations, we show that robust deterministic strategies can be overly conservative, while the mixed strategies found by GP-MRO significantly improve the overall performance.
Partial monitoring is a rich framework for sequential decision making under uncertainty that generalizes many well known bandit models, including linear, combinatorial and dueling bandits. We introduce information directed sampling (IDS) for stochastic partial monitoring with a linear reward and observation structure. IDS achieves adaptive worst-case regret rates that depend on precise observability conditions of the game. Moreover, we prove lower bounds that classify the minimax regret of all finite games into four possible regimes. IDS achieves the optimal rate in all cases up to logarithmic factors, without tuning any hyper-parameters. We further extend our results to the contextual and the kernelized setting, which significantly increases the range of possible applications.
Meta-learning can successfully acquire useful inductive biases from data, especially when a large number of meta-tasks are available. Yet, its generalization properties to unseen tasks are poorly understood. Particularly if the number of meta-tasks is small, this raises concerns for potential overfitting. We provide a theoretical analysis using the PAC-Bayesian framework and derive novel generalization bounds for meta-learning with unbounded loss functions and Bayesian base learners. Using these bounds, we develop a class of PAC-optimal meta-learning algorithms with performance guarantees and a principled meta-regularization. When instantiating our PAC-optimal hyper-posterior (PACOH) with Gaussian processes as base learners, the resulting approach consistently outperforms several popular meta-learning methods, both in terms of predictive accuracy and the quality of its uncertainty estimates.
Despite the recent surge of interest in designing and guaranteeing mathematical formulations of fairness, virtually all existing notions of algorithmic fairness fail to be adaptable to the intricacies and nuances of the decision-making context at hand. We argue that capturing such factors is an inherently human task, as it requires knowledge of the social background in which machine learning tools impact real people's outcomes and a deep understanding of the ramifications of automated decisions for decision subjects and society. In this work, we present a framework to construct a context-dependent mathematical formulation of fairness utilizing people's judgment of fairness. We utilize the theoretical model of Heidari et al. (2019)---which shows that most existing formulations of algorithmic fairness are special cases of economic models of Equality of Opportunity (EOP)---and present a practical human-in-the-loop approach to pinpoint the fairness notion in the EOP family that best captures people's perception of fairness in the given context. To illustrate our framework, we run human-subject experiments designed to learn the parameters of Heidari et al.'s EOP model (including circumstance, desert, and utility) in a hypothetical recidivism decision-making scenario. Our work takes an initial step toward democratizing the formulation of fairness and utilizing human-judgment to tackle a fundamental shortcoming of automated decision-making systems: that the machine on its own is incapable of understanding and processing the human aspects and social context of its decisions.