Bart P. G. Van Parys

Zhenyuan Liu, Bart P. G. Van Parys, Henry Lam

In data-driven optimization, sample average approximation (SAA) is known to suffer from the so-called optimizer's curse that causes an over-optimistic evaluation of the solution performance. We argue that a special type of distributionallly robust optimization (DRO) formulation offers theoretical advantages in correcting for this optimizer's curse compared to simple ``margin'' adjustments to SAA and other DRO approaches: It attains a statistical bound on the out-of-sample performance, for a wide class of objective functions and distributions, that is nearly tightest in terms of exponential decay rate. This DRO uses an ambiguity set based on a Kullback Leibler (KL) divergence smoothed by the Wasserstein or Lévy-Prokhorov (LP) distance via a suitable distance optimization. Computationally, we also show that such a DRO, and its generalized versions using smoothed $f$-divergence, are not harder than DRO problems based on $f$-divergence or Wasserstein distances, rendering our DRO formulations both statistically optimal and computationally viable.

Jannis Kurtz, Bart P. G. van Parys

Optimization problems routinely depend on uncertain parameters that must be predicted before a decision is made. Classical robust and regret formulations are designed to handle erroneous predictions and can provide statistical error bounds in simple settings. However, when predictions lack rigorous error bounds (as is typical of modern machine learning methods) classical robust models often yield vacuous guarantees, while regret formulations can paradoxically produce decisions that are more optimistic than even a nominal solution. We introduce Globalized Adversarial Regret Optimization (GARO), a decision framework that controls adversarial regret, defined as the gap between the worst-case cost and the oracle robust cost, uniformly across all possible uncertainty set sizes. By design, GARO delivers absolute or relative performance guarantees against an oracle with full knowledge of the prediction error, without requiring any probabilistic calibration of the uncertainty set. We show that GARO equipped with a relative rate function generalizes the classical adaptation method of Lepski to downstream decision problems. We derive exact tractable reformulations for problems with affine worst-case cost functions and polyhedral norm uncertainty sets, and provide a discretization and a constraint-generation algorithm with convergence guarantees for general settings. Finally, experiments demonstrate that GARO yields solutions with a more favorable trade-off between worst-case and mean out-of-sample performance, as well as stronger global performance guarantees.

Liviu Aolaritei, Bart P. G. Van Parys, Henry Lam et al.

Importance Sampling (IS) is a widely used variance reduction technique for enhancing the efficiency of Monte Carlo methods, particularly in rare-event simulation and related applications. Despite its power, the performance of IS is often highly sensitive to the choice of the proposal distribution and frequently requires stochastic calibration techniques. While the design and analysis of IS have been extensively studied in estimation settings, applying IS within stochastic optimization introduces a unique challenge: the decision and the IS distribution are mutually dependent, creating a circular optimization structure. This interdependence complicates both the analysis of convergence for decision iterates and the efficiency of the IS scheme. In this paper, we propose an iterative gradient-based algorithm that jointly updates the decision variable and the IS distribution without requiring time-scale separation between the two. Our method achieves the lowest possible asymptotic variance and guarantees global convergence under convexity of the objective and mild assumptions on the IS distribution family. Furthermore, we show that these properties are preserved under linear constraints by incorporating a recent variant of Nesterov's dual averaging method.

Amine Bennouna, Bart P. G. Van Parys

We study the problem of designing optimal learning and decision-making formulations when only historical data is available. Prior work typically commits to a particular class of data-driven formulation and subsequently tries to establish out-of-sample performance guarantees. We take here the opposite approach. We define first a sensible yard stick with which to measure the quality of any data-driven formulation and subsequently seek to find an optimal such formulation. Informally, any data-driven formulation can be seen to balance a measure of proximity of the estimated cost to the actual cost while guaranteeing a level of out-of-sample performance. Given an acceptable level of out-of-sample performance, we construct explicitly a data-driven formulation that is uniformly closer to the true cost than any other formulation enjoying the same out-of-sample performance. We show the existence of three distinct out-of-sample performance regimes (a superexponential regime, an exponential regime and a subexponential regime) between which the nature of the optimal data-driven formulation experiences a phase transition. The optimal data-driven formulations can be interpreted as a classically robust formulation in the superexponential regime, an entropic distributionally robust formulation in the exponential regime and finally a variance penalized formulation in the subexponential regime. This final observation unveils a surprising connection between these three, at first glance seemingly unrelated, data-driven formulations which until now remained hidden.

Bart P. G. Van Parys

Classical Kullback-Leibler or entropic distances are known to enjoy certain desirable statistical properties in the context of decision-making with noiseless data. However, in most practical situations the data available to a decision maker is subject to a certain amount of measurement noise. We hence study here data-driven prescription problems in which the data is corrupted by a known noise source. We derive efficient data-driven formulations in this noisy regime and indicate that they enjoy an entropic optimal transport interpretation. Finally, we show that these efficient robust formulations are tractable in several interesting settings by exploiting a classical representation result by Strassen.

Bart P. G. Van Parys, Negin Golrezaei

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 convex structural information regarding the reward distributions; that is, the decision-maker knows the reward distributions of the arms belong to a convex compact set. In the presence such structural information, they then 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 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 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. We show that this idea leads to the first computationally viable learning policy with asymptotic minimal regret for various structural information, including well-known structured bandits such as linear, Lipschitz, and convex bandits, and novel structured bandits that have not been studied in the literature due to the lack of a unified and flexible framework.