arXiv subjects

Bart Van Parys

Publications and source records attributed to Bart Van Parys.

12 recordsLinked to original sources

arXiv2026

Learning Decision-Focused Uncertainty Sets in Robust Optimization

We propose a data-driven technique to automatically learn contextual uncertainty sets in robust optimization, resulting in excellent worst-case and average-case performance while also guaranteeing constraint satisfaction. Our method reshapes the uncertainty sets by minimizing the expected performance across a contextual family of problems, subject to conditional-value-at-risk constraints. Our approach is very flexible, and can learn a wide variety of uncertainty sets while preserving tractability. We solve the constrained learning problem using a stochastic augmented Lagrangian method that relies on differentiating the solutions of the robust optimization problems with respect to the parameters of the uncertainty set. Due to the nonsmooth and nonconvex nature of the augmented Lagrangian function, we apply the nonsmooth conservative implicit function theorem to establish convergence to a critical point, which is a feasible solution of the constrained problem under mild assumptions. Using empirical process theory, we show finite-sample probabilistic guarantees of constraint satisfaction for the resulting solutions. 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.

math.OC↗

arXiv2026

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.

math.OC↗

arXiv2025

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 is caused by several factors simultaneously. 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. 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 that guarantees such holistic protection and is computationally viable. Our distributionally robust optimization formulation can be interpreted as a novel combination of a Kullback-Leibler and Lévy-Prokhorov robust optimization formulation. In the context of classification and regression problems, we show that several popular regularized and robust formulations naturally reduce to a particular case of our proposed novel formulation. Finally, we apply the proposed HR formulation to two real-life applications and study it alongside several benchmarks: (1) training neural networks on healthcare data, where we analyze various robustness and generalization properties in the presence of noise, labeling errors, and scarce data, (2) a portfolio selection problem with real stock data, and analyze the risk/return tradeoff under the natural severe distribution shift of the application.

stat.ML↗

arXiv2024

From Distributional Robustness to Robust Statistics: A Confidence Sets Perspective

We establish a connection between distributionally robust optimization (DRO) and classical robust statistics. We demonstrate that this connection arises naturally in the context of estimation under data corruption, where the goal is to construct ``minimal'' confidence sets for the unknown data-generating distribution. Specifically, we show that a DRO ambiguity set, based on the Kullback-Leibler divergence and total variation distance, is uniformly minimal, meaning it represents the smallest confidence set that contains the unknown distribution with at a given confidence power. Moreover, we prove that when parametric assumptions are imposed on the unknown distribution, the ambiguity set is never larger than a confidence set based on the optimal estimator proposed by Huber. This insight reveals that the commonly observed conservatism of DRO formulations is not intrinsic to these formulations themselves but rather stems from the non-parametric framework in which these formulations are employed.

math.OC↗

arXiv2023

Certified Robust Neural Networks: Generalization and Corruption Resistance

Recent work have demonstrated that robustness (to "corruption") can be at odds with generalization. Adversarial training, for instance, aims to reduce the problematic susceptibility of modern neural networks to small data perturbations. Surprisingly, overfitting is a major concern in adversarial training despite being mostly absent in standard training. We provide here theoretical evidence for this peculiar "robust overfitting" phenomenon. Subsequently, we advance a novel distributionally robust loss function bridging robustness and generalization. We demonstrate both theoretically as well as empirically the loss to enjoy a certified level of robustness against two common types of corruption--data evasion and poisoning attacks--while ensuring guaranteed generalization. We show through careful numerical experiments that our resulting holistic robust (HR) training procedure yields SOTA performance. Finally, we indicate that HR training can be interpreted as a direct extension of adversarial training and comes with a negligible additional computational burden. A ready-to-use python library implementing our algorithm is available at https://github.com/RyanLucas3/HR_Neural_Networks.

stat.ML↗

arXiv2021

Bootstrap Robust Prescriptive Analytics

We address the problem of prescribing an optimal decision in a framework where the cost function depends on uncertain problem parameters that need to be learned from data. Earlier work proposed prescriptive formulations based on supervised machine learning methods. These prescriptive methods can factor in contextual information on a potentially large number of covariates to take context specific actions which are superior to any static decision. When working with noisy or corrupt data, however, such nominal prescriptive methods can be prone to adverse overfitting phenomena and fail to generalize on out-of-sample data. In this paper we combine ideas from robust optimization and the statistical bootstrap to propose novel prescriptive methods which safeguard against overfitting. We show indeed that a particular entropic robust counterpart to such nominal formulations guarantees good performance on synthetic bootstrap data. As bootstrap data is often a sensible proxy to actual out-of-sample data, our robust counterpart can be interpreted to directly encourage good out-of-sample performance. The associated robust prescriptive methods furthermore reduce to convenient tractable convex optimization problems in the context of local learning methods such as nearest neighbors and Nadaraya-Watson learning. We illustrate our data-driven decision-making framework and our novel robustness notion on a small newsvendor problem.

math.OC↗

arXiv2020

Sparse Classification: a scalable discrete optimization perspective

We formulate the sparse classification problem of $n$ samples with $p$ features as a binary convex optimization problem and propose a cutting-plane algorithm to solve it exactly. For sparse logistic regression and sparse SVM, our algorithm finds optimal solutions for $n$ and $p$ in the $10,000$s within minutes. On synthetic data our algorithm achieves perfect support recovery in the large sample regime. Namely, there exists a $n_0$ such that the algorithm takes a long time to find the optimal solution and does not recover the correct support for $n 0$.

math.OC↗

arXiv2020

Sparse Regression: Scalable algorithms and empirical performance

In this paper, we review state-of-the-art methods for feature selection in statistics with an application-oriented eye. Indeed, sparsity is a valuable property and the profusion of research on the topic might have provided little guidance to practitioners. We demonstrate empirically how noise and correlation impact both the accuracy - the number of correct features selected - and the false detection - the number of incorrect features selected - for five methods: the cardinality-constrained formulation, its Boolean relaxation, $\ell_1$ regularization and two methods with non-convex penalties. A cogent feature selection method is expected to exhibit a two-fold convergence, namely the accuracy and false detection rate should converge to $1$ and $0$ respectively, as the sample size increases. As a result, proper method should recover all and nothing but true features. Empirically, the integer optimization formulation and its Boolean relaxation are the closest to exhibit this two properties consistently in various regimes of noise and correlation. In addition, apart from the discrete optimization approach which requires a substantial, yet often affordable, computational time, all methods terminate in times comparable with the \verb|glmnet| package for Lasso. We released code for methods that were not publicly implemented. Jointly considered, accuracy, false detection and computational time provide a comprehensive assessment of each feature selection method and shed light on alternatives to the Lasso-regularization which are not as popular in practice yet.

stat.ME↗

arXiv2017

Sparse High-Dimensional Regression: Exact Scalable Algorithms and Phase Transitions

We present a novel binary convex reformulation of the sparse regression problem that constitutes a new duality perspective. We devise a new cutting plane method and provide evidence that it can solve to provable optimality the sparse regression problem for sample sizes n and number of regressors p in the 100,000s, that is two orders of magnitude better than the current state of the art, in seconds. The ability to solve the problem for very high dimensions allows us to observe new phase transition phenomena. Contrary to traditional complexity theory which suggests that the difficulty of a problem increases with problem size, the sparse regression problem has the property that as the number of samples $n$ increases the problem becomes easier in that the solution recovers 100% of the true signal, and our approach solves the problem extremely fast (in fact faster than Lasso), while for small number of samples n, our approach takes a larger amount of time to solve the problem, but importantly the optimal solution provides a statistically more relevant regressor. We argue that our exact sparse regression approach presents a superior alternative over heuristic methods available at present.

math.OC↗

arXiv2017

Sparse Hierarchical Regression with Polynomials

We present a novel method for exact hierarchical sparse polynomial regression. Our regressor is that degree $r$ polynomial which depends on at most $k$ inputs, counting at most $\ell$ monomial terms, which minimizes the sum of the squares of its prediction errors. The previous hierarchical sparse specification aligns well with modern big data settings where many inputs are not relevant for prediction purposes and the functional complexity of the regressor needs to be controlled as to avoid overfitting. We present a two-step approach to this hierarchical sparse regression problem. First, we discard irrelevant inputs using an extremely fast input ranking heuristic. Secondly, we take advantage of modern cutting plane methods for integer optimization to solve our resulting reduced hierarchical $(k, \ell)$-sparse problem exactly. The ability of our method to identify all $k$ relevant inputs and all $\ell$ monomial terms is shown empirically to experience a phase transition. Crucially, the same transition also presents itself in our ability to reject all irrelevant features and monomials as well. In the regime where our method is statistically powerful, its computational complexity is interestingly on par with Lasso based heuristics. The presented work fills a void in terms of a lack of powerful disciplined nonlinear sparse regression methods in high-dimensional settings. Our method is shown empirically to scale to regression problems with $n\approx 10,000$ observations for input dimension $p\approx 1,000$.

math.OC↗

arXiv2016

Multivariate Chebyshev Inequality with Estimated Mean and Variance

A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this paper we present a generalization of this result to multiple dimensions where the only requirement is that the samples are independent and identically distributed. Furthermore, we show that as the number of samples tends to infinity our inequality converges to the theoretical multi-dimensional Chebyshev bound.

stat.ME↗

arXiv2015

Security Constrained Optimal Power Flow with Distributionally Robust Chance Constraints

The growing amount of fluctuating renewable infeeds and market liberalization increases uncertainty in power system operation. To capture the influence of fluctuations in operational planning, we model the forecast errors of the uncertain in-feeds as random variables and formulate a security constrained optimal power flow using chance constraints. The chance constraints limit the probability of violations of technical constraints, such as generation and transmission limits, but require a tractable reformulation. In this paper, we discuss different analytical reformulations of the chance constraints, based on a given set of assumptions concerning the forecast error distributions. In particular, we discuss reformulations that do not assume a normal distribution, and admit an analytical reformulation given only a mean vector and covariance matrix. We illustrate our method with a case study of the IEEE 118 bus system, based on real data from the European system. The different reformulations are compared in terms of both achieved empirical violation probability and operational cost, which allows us to provide a suggestion for the most appropriate reformulation in an optimal power flow setting. For a large number of uncertainty sources, it is observed that the distributions of the line flows and generator outputs can be close to normal, even though the power injections are not normally distributed.

math.OC↗