Haowei Wang

Shouri Hu, Haowei Wang, Zhongxiang Dai et al.

The expected improvement (EI) is one of the most popular acquisition functions for Bayesian optimization (BO) and has demonstrated good empirical performances in many applications for the minimization of simple regret. However, under the evaluation metric of cumulative regret, the performance of EI may not be competitive, and its existing theoretical regret upper bound still has room for improvement. To adapt the EI for better performance under cumulative regret, we introduce a novel quantity called the evaluation cost which is compared against the acquisition function, and with this, develop the expected improvement-cost (EIC) algorithm. In each iteration of EIC, a new point with the largest acquisition function value is sampled, only if that value exceeds its evaluation cost. If none meets this criteria, the current best point is resampled. This evaluation cost quantifies the potential downside of sampling a point, which is important under the cumulative regret metric as the objective function value in every iteration affects the performance measure. We establish in theory a high-probability regret upper bound of EIC based on the maximum information gain, which is tighter than the bound of existing EI-based algorithms. It is also comparable to the regret bound of other popular BO algorithms such as Thompson sampling (GP-TS) and upper confidence bound (GP-UCB). We further perform experiments to illustrate the improvement of EIC over several popular BO algorithms.

Shixiong Wang, Haowei Wang, Xinke Li et al.

Trustworthy machine learning aims at combating distributional uncertainties in training data distributions compared to population distributions. Typical treatment frameworks include the Bayesian approach, (min-max) distributionally robust optimization (DRO), and regularization. However, three issues have to be raised: 1) the prior distribution in the Bayesian method and the regularizer in the regularization method are difficult to specify; 2) the DRO method tends to be overly conservative; 3) all the three methods are biased estimators of the true optimal cost. This paper studies a new framework that unifies the three approaches and addresses the three challenges above. The asymptotic properties (e.g., consistencies and asymptotic normalities), non-asymptotic properties (e.g., generalization bounds and unbiasedness), and solution methods of the proposed model are studied. The new model reveals the trade-off between the robustness to the unseen data and the specificity to the training data. Experiments on various real-world tasks validate the superiority of the proposed learning framework.

Shixiong Wang, Haowei Wang

Bayesian methods, distributionally robust optimization methods, and regularization methods are three pillars of trustworthy machine learning combating distributional uncertainty, e.g., the uncertainty of an empirical distribution compared to the true underlying distribution. This paper investigates the connections among the three frameworks and, in particular, explores why these frameworks tend to have smaller generalization errors. Specifically, first, we suggest a quantitative definition for "distributional robustness", propose the concept of "robustness measure", and formalize several philosophical concepts in distributionally robust optimization. Second, we show that Bayesian methods are distributionally robust in the probably approximately correct (PAC) sense; in addition, by constructing a Dirichlet-process-like prior in Bayesian nonparametrics, it can be proven that any regularized empirical risk minimization method is equivalent to a Bayesian method. Third, we show that generalization errors of machine learning models can be characterized using the distributional uncertainty of the nominal distribution and the robustness measures of these machine learning models, which is a new perspective to bound generalization errors, and therefore, explain the reason why distributionally robust machine learning models, Bayesian models, and regularization models tend to have smaller generalization errors in a unified manner.