Bilevel optimization is characterized by a two-level optimization structure, where the upper-level problem is constrained by optimal lower-level solutions, and such structures are prevalent in real-world problems. The constraint by optimal lower-level solutions poses significant challenges, especially in noisy, constrained, and derivative-free settings, as repeating lower-level optimizations is sample inefficient and predicted lower-level solutions may be suboptimal. We present BILevel Bayesian Optimization (BILBO), a novel Bayesian optimization algorithm for general bilevel problems with blackbox functions, which optimizes both upper- and lower-level problems simultaneously, without the repeated lower-level optimization required by existing methods. BILBO samples from confidence-bounds based trusted sets, which bounds the suboptimality on the lower level. Moreover, BILBO selects only one function query per iteration, where the function query selection strategy incorporates the uncertainty of estimated lower-level solutions and includes a conditional reassignment of the query to encourage exploration of the lower-level objective. The performance of BILBO is theoretically guaranteed with a sublinear regret bound for commonly used kernels and is empirically evaluated on several synthetic and real-world problems.

Though some research efforts have been dedicated to constrained Bayesian optimization (BO), there remains a notable absence of a principled approach with a theoretical performance guarantee in the decoupled setting. Such a setting involves independent evaluations of the objective function and constraints at different inputs, and is hence a relaxation of the commonly-studied coupled setting where functions must be evaluated together. As a result, the decoupled setting requires an adaptive selection between evaluating either the objective function or a constraint, in addition to selecting an input (in the coupled setting). This paper presents a novel constrained BO algorithm with a provable performance guarantee that can address the above relaxed setting. Specifically, it considers the fundamental trade-off between exploration and exploitation in constrained BO, and, interestingly, affords a noteworthy connection to active learning. The performance of our proposed algorithms is also empirically evaluated using several synthetic and real-world optimization problems.

Research on optimizing the risk measure of a blackbox function using Gaussian processes, especially Bayesian optimization (BO) of risk measures, has become increasingly important due to the inevitable presence of uncontrollable variables in real-world applications. Nevertheless, existing works on BO of risk measures start the optimization from scratch for every new task without considering the results of prior tasks. In contrast, its vanilla BO counterpart has received a thorough investigation on utilizing prior tasks to speed up the current task through the body of works on meta-BO which, however, have not considered risk measures. To bridge this gap, this paper presents the first algorithm for meta-BO of risk measures (i.e., value-at-risk (VaR) and the conditional VaR), namely meta-VBO, by introducing a novel adjustment to the upper confidence bound acquisition function. Our proposed algorithm exhibits two desirable properties: (i) invariance to scaling and vertical shifting of the blackbox function and (ii) robustness to prior harmful tasks. We provide a theoretical performance guarantee for our algorithm and empirically demonstrate its performance using several synthetic function benchmarks and real-world objective functions.

In this paper, we formalize the active set ordering problem, which involves actively discovering a set of inputs based on their orderings determined by expensive evaluations of a blackbox function. We then propose the mean prediction (MP) algorithm and theoretically analyze it in terms of the regret of predicted pairwise orderings between inputs. Notably, as a special case of this framework, we can cast Bayesian optimization as an active set ordering problem by recognizing that maximizers can be identified solely by comparison rather than by precisely estimating the function evaluations. As a result, we are able to construct the popular Gaussian process upper confidence bound (GP-UCB) algorithm through the lens of ordering with several nuanced insights. We empirically validate the performance of our proposed solution using various synthetic functions and real-world datasets.

The Nash equilibrium (NE) is a classic solution concept for normal-form games that is stable under potential unilateral deviations by self-interested agents. Bayesian optimization (BO) has been used to find NE in continuous general-sum games with unknown costly-to-sample utility functions in a sample-efficient manner. This paper presents the first no-regret BO algorithm that is sample-efficient in finding pure NE by leveraging theory on high probability confidence bounds with Gaussian processes and the maximum information gain of kernel functions. Unlike previous works, our algorithm is theoretically guaranteed to converge to the optimal solution (i.e., NE). We also introduce the novel setting of applying BO to finding mixed NE in unknown discrete general-sum games and show that our theoretical framework is general enough to be extended naturally to this setting by developing a no-regret BO algorithm that is sample-efficient in finding mixed NE. We empirically show that our algorithms are competitive w.r.t. suitable baselines in finding NE.

Many real-world experimental design problems (a) evaluate multiple experimental conditions in parallel and (b) replicate each condition multiple times due to large and heteroscedastic observation noise. Given a fixed total budget, this naturally induces a trade-off between evaluating more unique conditions while replicating each of them fewer times vs. evaluating fewer unique conditions and replicating each more times. Moreover, in these problems, practitioners may be risk-averse and hence prefer an input with both good average performance and small variability. To tackle both challenges, we propose the Batch Thompson Sampling for Replicable Experimental Design (BTS-RED) framework, which encompasses three algorithms. Our BTS-RED-Known and BTS-RED-Unknown algorithms, for, respectively, known and unknown noise variance, choose the number of replications adaptively rather than deterministically such that an input with a larger noise variance is replicated more times. As a result, despite the noise heteroscedasticity, both algorithms enjoy a theoretical guarantee and are asymptotically no-regret. Our Mean-Var-BTS-RED algorithm aims at risk-averse optimization and is also asymptotically no-regret. We also show the effectiveness of our algorithms in two practical real-world applications: precision agriculture and AutoML.

As the use of machine learning (ML) models is becoming increasingly popular in many real-world applications, there are practical challenges that need to be addressed for model maintenance. One such challenge is to "undo" the effect of a specific subset of dataset used for training a model. This specific subset may contain malicious or adversarial data injected by an attacker, which affects the model performance. Another reason may be the need for a service provider to remove data pertaining to a specific user to respect the user's privacy. In both cases, the problem is to "unlearn" a specific subset of the training data from a trained model without incurring the costly procedure of retraining the whole model from scratch. Towards this goal, this paper presents a Markov chain Monte Carlo-based machine unlearning (MCU) algorithm. MCU helps to effectively and efficiently unlearn a trained model from subsets of training dataset. Furthermore, we show that with MCU, we are able to explain the effect of a subset of a training dataset on the model prediction. Thus, MCU is useful for examining subsets of data to identify the adversarial data to be removed. Similarly, MCU can be used to erase the lineage of a user's personal data from trained ML models, thus upholding a user's "right to be forgotten". We empirically evaluate the performance of our proposed MCU algorithm on real-world phishing and diabetes datasets. Results show that MCU can achieve a desirable performance by efficiently removing the effect of a subset of training dataset and outperform an existing algorithm that utilizes the remaining dataset.

This paper investigates the problem of fairly trading off between payoff and model rewards in collaborative machine learning (ML) where parties aggregate their datasets together to obtain improved ML models over that of each party. Supposing parties can afford the optimal model trained on the aggregated dataset, we propose an allocation scheme that distributes the payoff fairly. Notably, the same scheme can be derived from two different approaches based on (a) desirable properties of the parties' payoffs or (b) that of the underlying payoff flows from one party to another. While the former is conceptually simpler, the latter can be used to handle the practical constraint on the budgets of parties. In particular, we propose desirable properties for achieving a fair adjustment of the payoff flows that can trade off between the model reward's performance and the payoff reward. We empirically demonstrate that our proposed scheme is a sensible solution in several scenarios of collaborative ML with different budget constraints.

This paper presents an information-theoretic framework for unifying active learning problems: level set estimation (LSE), Bayesian optimization (BO), and their generalized variant. We first introduce a novel active learning criterion that subsumes an existing LSE algorithm and achieves state-of-the-art performance in LSE problems with a continuous input domain. Then, by exploiting the relationship between LSE and BO, we design a competitive information-theoretic acquisition function for BO that has interesting connections to upper confidence bound and max-value entropy search (MES). The latter connection reveals a drawback of MES which has important implications on not only MES but also on other MES-based acquisition functions. Finally, our unifying information-theoretic framework can be applied to solve a generalized problem of LSE and BO involving multiple level sets in a data-efficient manner. We empirically evaluate the performance of our proposed algorithms using synthetic benchmark functions, a real-world dataset, and in hyperparameter tuning of machine learning models.

This paper presents a novel approach to top-k ranking Bayesian optimization (top-k ranking BO) which is a practical and significant generalization of preferential BO to handle top-k ranking and tie/indifference observations. We first design a surrogate model that is not only capable of catering to the above observations, but is also supported by a classic random utility model. Another equally important contribution is the introduction of the first information-theoretic acquisition function in BO with preferential observation called multinomial predictive entropy search (MPES) which is flexible in handling these observations and optimized for all inputs of a query jointly. MPES possesses superior performance compared with existing acquisition functions that select the inputs of a query one at a time greedily. We empirically evaluate the performance of MPES using several synthetic benchmark functions, CIFAR-10 dataset, and SUSHI preference dataset.

Value-at-risk (VaR) is an established measure to assess risks in critical real-world applications with random environmental factors. This paper presents a novel VaR upper confidence bound (V-UCB) algorithm for maximizing the VaR of a black-box objective function with the first no-regret guarantee. To realize this, we first derive a confidence bound of VaR and then prove the existence of values of the environmental random variable (to be selected to achieve no regret) such that the confidence bound of VaR lies within that of the objective function evaluated at such values. Our V-UCB algorithm empirically demonstrates state-of-the-art performance in optimizing synthetic benchmark functions, a portfolio optimization problem, and a simulated robot task.

This paper presents two Bayesian optimization (BO) algorithms with theoretical performance guarantee to maximize the conditional value-at-risk (CVaR) of a black-box function: CV-UCB and CV-TS which are based on the well-established principle of optimism in the face of uncertainty and Thompson sampling, respectively. To achieve this, we develop an upper confidence bound of CVaR and prove the no-regret guarantee of CV-UCB by utilizing an interesting connection between CVaR and value-at-risk (VaR). For CV-TS, though it is straightforwardly performed with Thompson sampling, bounding its Bayesian regret is non-trivial because it requires a tail expectation bound for the distribution of CVaR of a black-box function, which has not been shown in the literature. The performances of both CV-UCB and CV-TS are empirically evaluated in optimizing CVaR of synthetic benchmark functions and simulated real-world optimization problems.

This paper presents Gaussian process meta-learning (GPML) for few-shot regression, which explicitly exploits the distance between regression problems/tasks using a novel task kernel. It contrasts sharply with the popular metric-based meta-learning approach which is based on the distance between data inputs or their embeddings in the few-shot learning literature. Apart from the superior predictive performance by capturing the diversity of different tasks, GPML offers a set of representative tasks that are useful for understanding the task distribution. We empirically demonstrate the performance and interpretability of GPML in several few-shot regression problems involving a multimodal task distribution and real-world datasets.

Information-based Bayesian optimization (BO) algorithms have achieved state-of-the-art performance in optimizing a black-box objective function. However, they usually require several approximations or simplifying assumptions (without clearly understanding their effects on the BO performance) and/or their generalization to batch BO is computationally unwieldy, especially with an increasing batch size. To alleviate these issues, this paper presents a novel trusted-maximizers entropy search (TES) acquisition function: It measures how much an input query contributes to the information gain on the maximizer over a finite set of trusted maximizers, i.e., inputs optimizing functions that are sampled from the Gaussian process posterior belief of the objective function. Evaluating TES requires either only a stochastic approximation with sampling or a deterministic approximation with expectation propagation, both of which are investigated and empirically evaluated using synthetic benchmark objective functions and real-world optimization problems, e.g., hyperparameter tuning of a convolutional neural network and synthesizing physically realizable faces to fool a black-box face recognition system. Though TES can naturally be generalized to a batch variant with either approximation, the latter is amenable to be scaled to a much larger batch size in our experiments.

The problem of inverse reinforcement learning (IRL) is relevant to a variety of tasks including value alignment and robot learning from demonstration. Despite significant algorithmic contributions in recent years, IRL remains an ill-posed problem at its core; multiple reward functions coincide with the observed behavior and the actual reward function is not identifiable without prior knowledge or supplementary information. This paper presents an IRL framework called Bayesian optimization-IRL (BO-IRL) which identifies multiple solutions that are consistent with the expert demonstrations by efficiently exploring the reward function space. BO-IRL achieves this by utilizing Bayesian Optimization along with our newly proposed kernel that (a) projects the parameters of policy invariant reward functions to a single point in a latent space and (b) ensures nearby points in the latent space correspond to reward functions yielding similar likelihoods. This projection allows the use of standard stationary kernels in the latent space to capture the correlations present across the reward function space. Empirical results on synthetic and real-world environments (model-free and model-based) show that BO-IRL discovers multiple reward functions while minimizing the number of expensive exact policy optimizations.

This paper studies the problem of approximately unlearning a Bayesian model from a small subset of the training data to be erased. We frame this problem as one of minimizing the Kullback-Leibler divergence between the approximate posterior belief of model parameters after directly unlearning from erased data vs. the exact posterior belief from retraining with remaining data. Using the variational inference (VI) framework, we show that it is equivalent to minimizing an evidence upper bound which trades off between fully unlearning from erased data vs. not entirely forgetting the posterior belief given the full data (i.e., including the remaining data); the latter prevents catastrophic unlearning that can render the model useless. In model training with VI, only an approximate (instead of exact) posterior belief given the full data can be obtained, which makes unlearning even more challenging. We propose two novel tricks to tackle this challenge. We empirically demonstrate our unlearning methods on Bayesian models such as sparse Gaussian process and logistic regression using synthetic and real-world datasets.

Existing inverse reinforcement learning (IRL) algorithms have assumed each expert’s demonstrated trajectory to be produced by only a single reward function. This paper presents a novel generalization of the IRL problem that allows each trajectory to be generated by multiple locally consistent reward functions, hence catering to more realistic and complex experts’ behaviors. Solving our generalized IRL problem thus involves not only learning these reward functions but also the stochastic transitions between them at any state (including unvisited states). By representing our IRL problem with a probabilistic graphical model, an expectation-maximization (EM) algorithm can be devised to iteratively learn the different reward functions and the stochastic transitions between them in order to jointly improve the likelihood of the expert’s demonstrated trajectories. As a result, the most likely partition of a trajectory into segments that are generated from different locally consistent reward functions selected by EM can be derived. Empirical evaluation on synthetic and real-world datasets shows that our IRL algorithm outperforms the state-of-the-art EM clustering with maximum likelihood IRL, which is, interestingly, a reduced variant of our approach.