Approximation Algorithms for Distributionally Robust Stochastic Optimization with Black-Box Distributions

Two-stage stochastic optimization is a framework for modeling uncertainty, where we have a probability distribution over possible realizations of the data, called scenarios, and decisions are taken in two stages: we make first-stage decisions knowing only the underlying distribution and before a scenario is realized, and may take additional second-stage recourse actions after a scenario is realized. The goal is typically to minimize the total expected cost. A criticism of this model is that the underlying probability distribution is itself often imprecise! To address this, a versatile approach that has been proposed is the {\em distributionally robust 2-stage model}: given a collection of probability distributions, our goal now is to minimize the maximum expected total cost with respect to a distribution in this collection. We provide a framework for designing approximation algorithms in such settings when the collection is a ball around a central distribution and the central distribution is accessed {\em only via a sampling black box}. We first show that one can utilize the {\em sample average approximation} (SAA) method to reduce the problem to the case where the central distribution has {\em polynomial-size} support. We then show how to approximately solve a fractional relaxation of the SAA (i.e., polynomial-scenario central-distribution) problem. By complementing this via LP-rounding algorithms that provide {\em local} (i.e., per-scenario) approximation guarantees, we obtain the {\em first} approximation algorithms for the distributionally robust versions of a variety of discrete-optimization problems including set cover, vertex cover, edge cover, facility location, and Steiner tree, with guarantees that are, except for set cover, within $O(1)$-factors of the guarantees known for the deterministic version of the problem

Prepare for the Expected Worst: Algorithms for Reconfigurable Resources Under Uncertainty

In this paper we study how to optimally balance cheap inflexible resources with more expensive, reconfigurable resources despite uncertainty in the input problem. Specifically, we introduce the MinEMax model to study "build versus rent" problems. In our model different scenarios appear independently. Before knowing which scenarios appear, we may build rigid resources that cannot be changed for different scenarios. Once we know which scenarios appear, we are allowed to rent reconfigurable but expensive resources to use across scenarios. Although computing the objective in our model might seem to require enumerating exponentially-many possibilities, we show it is well estimated by a surrogate objective which is representable by a polynomial-size LP. In this surrogate objective we pay for each scenario only to the extent that it exceeds a certain threshold. Using this objective we design algorithms that approximately-optimally balance inflexible and reconfigurable resources for several NP-hard covering problems. For example, we study variants of minimum spanning and Steiner trees, minimum cuts, and facility location. Up to constants, our approximation guarantees match those of previously-studied algorithms for demand-robust and stochastic two-stage models. Lastly, we demonstrate that our problem is sufficiently general to smoothly interpolate between previous demand-robust and stochastic two-stage problems

Approximation Algorithms for Distributionally Robust Stochastic Optimization

Two-stage stochastic optimization is a widely used framework for modeling uncertainty, where we have a probability distribution over possible realizations of the data, called scenarios, and decisions are taken in two stages: we take first-stage actions knowing only the underlying distribution and before a scenario is realized, and may take additional second-stage recourse actions after a scenario is realized. The goal is typically to minimize the total expected cost. A common criticism levied at this model is that the underlying probability distribution is itself often imprecise. To address this, an approach that is quite versatile and has gained popularity in the stochastic-optimization literature is the two-stage distributionally robust stochastic model: given a collection D of probability distributions, our goal now is to minimize the maximum expected total cost with respect to a distribution in D. There has been almost no prior work however on developing approximation algorithms for distributionally robust problems where the underlying scenario collection is discrete, as is the case with discrete-optimization problems. We provide frameworks for designing approximation algorithms in such settings when the collection D is a ball around a central distribution, defined relative to two notions of distance between probability distributions: Wasserstein metrics (which include the L_1 metric) and the L_infinity metric. Our frameworks yield efficient algorithms even in settings with an exponential number of scenarios, where the central distribution may only be accessed via a sampling oracle. For distributionally robust optimization under a Wasserstein ball, we first show that one can utilize the sample average approximation (SAA) method (solve the distributionally robust problem with an empirical estimate of the central distribution) to reduce the problem to the case where the central distribution has a polynomial-size support, and is represented explicitly. This follows because we argue that a distributionally robust problem can be reduced in a novel way to a standard two-stage stochastic problem with bounded inflation factor, which enables one to use the SAA machinery developed for two-stage stochastic problems. Complementing this, we show how to approximately solve a fractional relaxation of the SAA problem (i.e., the distributionally robust problem obtained by replacing the original central distribution with its empirical estimate). Unlike in two-stage {stochastic, robust} optimization with polynomially many scenarios, this turns out to be quite challenging. We utilize a variant of the ellipsoid method for convex optimization in conjunction with several new ideas to show that the SAA problem can be approximately solved provided that we have an (approximation) algorithm for a certain max-min problem that is akin to, and generalizes, the k-max-min problem (find the worst-case scenario consisting of at most k elements) encountered in two-stage robust optimization. We obtain such an algorithm for various discrete-optimization problems; by complementing this via rounding algorithms that provide local (i.e., per-scenario) approximation guarantees, we obtain the first approximation algorithms for the distributionally robust versions of a variety of discrete-optimization problems including set cover, vertex cover, edge cover, facility location, and Steiner tree, with guarantees that are, except for set cover, within O(1)-factors of the guarantees known for the deterministic version of the problem. For distributionally robust optimization under an L_infinity ball, we consider a fractional relaxation of the problem, and replace its objective function with a proxy function that is pointwise close to the true objective function (within a factor of 2). We then show that we can efficiently compute approximate subgradients of the proxy function, provided that we have an algorithm for the problem of computing the t worst scenarios under a given first-stage decision, given an integer t. We can then approximately minimize the proxy function via a variant of the ellipsoid method, and thus obtain an approximate solution for the fractional relaxation of the distributionally robust problem. Complementing this via rounding algorithms with local guarantees, we obtain approximation algorithms for distributionally robust versions of various covering problems, including set cover, vertex cover, edge cover, and facility location, with guarantees that are within O(1)-factors of the guarantees known for their deterministic versions

Sinkhorn Distributionally Robust Optimization

We study distributionally robust optimization (DRO) with Sinkhorn distance -- a variant of Wasserstein distance based on entropic regularization. We derive convex programming dual reformulation for a general nominal distribution. Compared with Wasserstein DRO, it is computationally tractable for a larger class of loss functions, and its worst-case distribution is more reasonable for practical applications. To solve the dual reformulation, we develop a stochastic mirror descent algorithm using biased gradient oracles and analyze its convergence rate. Finally, we provide numerical examples using synthetic and real data to demonstrate its superior performance.Comment: 56 pages, 8 figure

Distributionally Robust Joint Chance-Constrained Optimization for Networked Microgrids Considering Contingencies and Renewable Uncertainty

In light of a reliable and resilient power system under extreme weather and natural disasters, networked microgrids integrating local renewable resources have been adopted extensively to supply demands when the main utility experiences blackouts. However, the stochastic nature of renewables and unpredictable contingencies are difficult to address with the deterministic energy management framework. The paper proposes a comprehensive distributionally robust joint chance-constrained (DR-JCC) framework that incorporates microgrid island, power flow, distributed batteries and voltage control constraints. All chance constraints are solved jointly and each one is assigned to an optimized violation rate. To highlight, the JCC problem with the optimized violation rates has been recognized to be NP-hard and challenging to be solved. This paper proposes a novel evolutionary algorithm that successfully tackles the problem and reduces the solution conservativeness (i.e. operation cost) by around 50% comparing with the baseline Bonferroni Approximation. Considering the imperfect solar power forecast, we construct three data-driven ambiguity sets to model uncertain forecast error distributions. The solution is thus robust for any distribution in sets with the shared moment and shape assumptions. The proposed method is validated by robustness tests based on those sets and firmly secures the solution robustness.Comment: Accepted by IEEE Transactions on Smart Gri

A distributionally robust perspective on uncertainty quantification and chance constrained programming

The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones

Safe Zero-Shot Model-Based Learning and Control: A Wasserstein Distributionally Robust Approach

This paper explores distributionally robust zero-shot model-based learning and control using Wasserstein ambiguity sets. Conventional model-based reinforcement learning algorithms struggle to guarantee feasibility throughout the online learning process. We address this open challenge with the following approach. Using a stochastic model-predictive control (MPC) strategy, we augment safety constraints with affine random variables corresponding to the instantaneous empirical distributions of modeling error. We obtain these distributions by evaluating model residuals in real time throughout the online learning process. By optimizing over the worst case modeling error distribution defined within a Wasserstein ambiguity set centered about our empirical distributions, we can approach the nominal constraint boundary in a provably safe way. We validate the performance of our approach using a case study of lithium-ion battery fast charging, a relevant and safety-critical energy systems control application. Our results demonstrate marked improvements in safety compared to a basic learning model-predictive controller, with constraints satisfied at every instance during online learning and control.Comment: In review for CDC2

A Survey of Contextual Optimization Methods for Decision Making under Uncertainty

Recently there has been a surge of interest in operations research (OR) and the machine learning (ML) community in combining prediction algorithms and optimization techniques to solve decision-making problems in the face of uncertainty. This gave rise to the field of contextual optimization, under which data-driven procedures are developed to prescribe actions to the decision-maker that make the best use of the most recently updated information. A large variety of models and methods have been presented in both OR and ML literature under a variety of names, including data-driven optimization, prescriptive optimization, predictive stochastic programming, policy optimization, (smart) predict/estimate-then-optimize, decision-focused learning, (task-based) end-to-end learning/forecasting/optimization, etc. Focusing on single and two-stage stochastic programming problems, this review article identifies three main frameworks for learning policies from data and discusses their strengths and limitations. We present the existing models and methods under a uniform notation and terminology and classify them according to the three main frameworks identified. Our objective with this survey is to both strengthen the general understanding of this active field of research and stimulate further theoretical and algorithmic advancements in integrating ML and stochastic programming

Zeroth-Order Methods for Convex-Concave Minmax Problems: Applications to Decision-Dependent Risk Minimization

Min-max optimization is emerging as a key framework for analyzing problems of robustness to strategically and adversarially generated data. We propose a random reshuffling-based gradient free Optimistic Gradient Descent-Ascent algorithm for solving convex-concave min-max problems with finite sum structure. We prove that the algorithm enjoys the same convergence rate as that of zeroth-order algorithms for convex minimization problems. We further specialize the algorithm to solve distributionally robust, decision-dependent learning problems, where gradient information is not readily available. Through illustrative simulations, we observe that our proposed approach learns models that are simultaneously robust against adversarial distribution shifts and strategic decisions from the data sources, and outperforms existing methods from the strategic classification literature.Comment: 32 pages, 5 figure