Bootstrap Robust Prescriptive Analytics

We address the problem of prescribing an optimal decision in a framework where its cost depends on uncertain problem parameters $Y$ that need to be learned from data. Earlier work by Bertsimas and Kallus (2014) transforms classical machine learning methods that merely predict $Y$ from supervised training data $[(x_1, y_1), \dots, (x_n, y_n)]$ into prescriptive methods taking optimal decisions specific to a particular covariate context $X=\bar x$. Their prescriptive methods factor in additional observed contextual information on a potentially large number of covariates $X=\bar x$ to take context specific actions $z(\bar x)$ which are superior to any static decision $z$. Any naive use of limited training data may, however, lead to gullible decisions over-calibrated to one particular data set. In this paper, we borrow ideas from distributionally robust optimization and the statistical bootstrap of Efron (1982) to propose two novel prescriptive methods based on (nw) Nadaraya-Watson and (nn) nearest-neighbors learning which safeguard against overfitting and lead to improved out-of-sample performance. Both resulting robust prescriptive methods reduce to tractable convex optimization problems and enjoy a limited disappointment on bootstrap data. We illustrate the data-driven decision-making framework and our novel robustness notion on a small news vendor problem as well as a small portfolio allocation problem

Optimal Transport in the Face of Noisy Data

Optimal transport distances are popular and theoretically well understood in the context of data-driven prediction. A flurry of recent work has popularized these distances for data-driven decision-making as well although their merits in this context are far less well understood. This in contrast to the more classical entropic distances which are known to enjoy optimal statistical properties. This begs the question when, if ever, optimal transport distances enjoy similar statistical guarantees. Optimal transport methods are shown here to enjoy optimal statistical guarantees for decision problems faced with noisy data

Holistic Robust Data-Driven Decisions

The design of data-driven formulations for machine learning and decision-making with good out-of-sample performance is a key challenge. The observation that good in-sample performance does not guarantee good out-of-sample performance is generally known as overfitting. Practical overfitting can typically not be attributed to a single cause but instead is caused by several factors all at once. We consider here three overfitting sources: (i) statistical error as a result of working with finite sample data, (ii) data noise which occurs when the data points are measured only with finite precision, and finally (iii) data misspecification in which a small fraction of all data may be wholly corrupted. We argue that although existing data-driven formulations may be robust against one of these three sources in isolation they do not provide holistic protection against all overfitting sources simultaneously. We design a novel data-driven formulation which does guarantee such holistic protection and is furthermore computationally viable. Our distributionally robust optimization formulation can be interpreted as a novel combination of a Kullback-Leibler and Levy-Prokhorov robust optimization formulation which is novel in its own right. However, we show how in the context of classification and regression problems that several popular regularized and robust formulations reduce to a particular case of our proposed novel formulation. Finally, we apply the proposed HR formulation on a portfolio selection problem with real stock data, and analyze its risk/return tradeoff against several benchmarks formulations. Our experiments show that our novel ambiguity set provides a significantly better risk/return trade-off

Optimal Learning for Structured Bandits

We study structured multi-armed bandits, which is the problem of online decision-making under uncertainty in the presence of structural information. In this problem, the decision-maker needs to discover the best course of action despite observing only uncertain rewards over time. The decision-maker is aware of certain structural information regarding the reward distributions and would like to minimize their regret by exploiting this information, where the regret is its performance difference against a benchmark policy that knows the best action ahead of time. In the absence of structural information, the classical upper confidence bound (UCB) and Thomson sampling algorithms are well known to suffer only minimal regret. As recently pointed out, neither algorithms are, however, capable of exploiting structural information that is commonly available in practice. We propose a novel learning algorithm that we call DUSA whose worst-case regret matches the information-theoretic regret lower bound up to a constant factor and can handle a wide range of structural information. Our algorithm DUSA solves a dual counterpart of the regret lower bound at the empirical reward distribution and follows its suggested play. Our proposed algorithm is the first computationally viable learning policy for structured bandit problems that has asymptotic minimal regret

Learning for Robust Optimization

We propose a data-driven technique to automatically learn the uncertainty sets in robust optimization. Our method reshapes the uncertainty sets by minimizing the expected performance across a family of problems while guaranteeing constraint satisfaction. We learn the uncertainty sets using a novel stochastic augmented Lagrangian method that relies on differentiating the solutions of the robust optimization problems with respect to the parameters of the uncertainty set. We show sublinear convergence to stationary points under mild assumptions, and finite-sample probabilistic guarantees of constraint satisfaction using empirical process theory. Our approach is very flexible and can learn a wide variety of uncertainty sets while preserving tractability. Numerical experiments show that our method outperforms traditional approaches in robust and distributionally robust optimization in terms of out of sample performance and constraint satisfaction guarantees. We implemented our method in the open-source package LROPT

Mean Robust Optimization

Robust optimization is a tractable and expressive technique for decision-making under uncertainty, but it can lead to overly conservative decisions when pessimistic assumptions are made on the uncertain parameters. Wasserstein distributionally robust optimization can reduce conservatism by being data-driven, but it often leads to very large problems with prohibitive solution times. We introduce mean robust optimization, a general framework that combines the best of both worlds by providing a trade-off between computational effort and conservatism. We propose uncertainty sets constructed based on clustered data rather than on observed data points directly thereby significantly reducing problem size. By varying the number of clusters, our method bridges between robust and Wasserstein distributionally robust optimization. We show finite-sample performance guarantees and explicitly control the potential additional pessimism introduced by any clustering procedure. In addition, we prove conditions for which, when the uncertainty enters linearly in the constraints, clustering does not affect the optimal solution. We illustrate the efficiency and performance preservation of our method on several numerical examples, obtaining multiple orders of magnitude speedups in solution time with little-to-no effect on the solution quality