Paul Grigas

Wyame Benslimane, Tinghan Ye, Pascal Van Hentenryck et al.

Decision-focused learning (DFL) trains predictive models by optimizing downstream decision quality rather than standalone prediction accuracy. For contextual linear optimization, most existing DFL methods assume offline data and full observations of the objective cost vector. We develop an on-policy learning method for sequential contextual linear optimization under partial feedback, generalizing the standard bandit feedback setting. Our method learns a stochastic predict-then-optimize policy that samples a cost-vector prediction from a conditional distribution and solves the resulting downstream linear optimization problem. To update this distributional model, we introduce a two-component hybrid gradient estimator. The first component is a score function estimator, which provides an unbiased but potentially high-variance policy gradient estimate. The second is a decision-focused plug-in component that uses an auxiliary nuisance estimate of the latent cost vector to exploit the downstream optimization structure, becoming more informative as the estimate improves. We prove an $\mathcal{O}(T^{-1/2})$ bound on the average squared policy-gradient norm, matching the standard non-convex SGD rate. Experiments on top-$k$ selection, shortest path, combinatorial pricing, and a real-data energy-scheduling benchmark show that the hybrid gradient approach achieves lower cumulative regret than contextual-bandit-style baselines across all benchmarks, using both Gaussian and richer conditional generative models. Code is available at https://github.com/Joeyetinghan/on-policy-bandit-dfl.

Liviu Aolaritei, Ricky Huang, Michael I. Jordan et al.

We introduce a diffusion-based uncertainty model for robust optimization on directed graphs, in which perturbations of edge weights propagate along adjacent edges and satisfy conservation constraints at nodes. This topology-aware structure is natural in networked systems where uncertainty is induced by flows and local interactions, including transportation, logistics, communication, and energy networks. We analyze how such diffusive uncertainty reshapes the computational landscape of robust graph optimization. For convex network problems, such as minimum-cost flow and maximum flow, the resulting formulations remain convex and admit polynomial-time solution methods across all diffusion regimes considered. For combinatorial problems, the effect is more delicate. We focus on two canonical combinatorial graph problems, shortest path and the traveling salesman problem (TSP), which provide complementary benchmarks: shortest path is polynomial-time solvable in the nominal setting, whereas TSP is already NP-hard. We show that, for shortest path, propagation depth induces a sharp transition between tractable and intractable robust counterparts. For the traveling salesman problem, robustness often adds no computational complexity beyond ordinary TSP, because the structure of Hamiltonian cycles makes the fixed-tour adversarial problem collapse to explicit formulas. Together, these results show that topology-aware uncertainty can fundamentally change robust combinatorial optimization, with tractability governed by the interaction between propagation, budget geometry, and the structure of feasible solutions.

Heyuan Liu, Paul Grigas

We study an online contextual decision-making problem with resource constraints. At each time period, the decision-maker first predicts a reward vector and resource consumption matrix based on a given context vector and then solves a downstream optimization problem to make a decision. The final goal of the decision-maker is to maximize the summation of the reward and the utility from resource consumption, while satisfying the resource constraints. We propose an algorithm that mixes a prediction step based on the "Smart Predict-then-Optimize (SPO)" method with a dual update step based on mirror descent. We prove regret bounds and demonstrate that the overall convergence rate of our method depends on the $\mathcal{O}(T^{-1/2})$ convergence of online mirror descent as well as risk bounds of the surrogate loss function used to learn the prediction model. Our algorithm and regret bounds apply to a general convex feasible region for the resource constraints, including both hard and soft resource constraint cases, and they apply to a wide class of prediction models in contrast to the traditional settings of linear contextual models or finite policy spaces. We also conduct numerical experiments to empirically demonstrate the strength of our proposed SPO-type methods, as compared to traditional prediction-error-only methods, on multi-dimensional knapsack and longest path instances.

Hyungki Im, Paul Grigas

Learning with label dependent label noise has been extensively explored in both theory and practice; however, dealing with instance (i.e., feature) and label dependent label noise continues to be a challenging task. The difficulty arises from the fact that the noise rate varies for each instance, making it challenging to estimate accurately. The question of whether it is possible to learn a reliable model using only noisy samples remains unresolved. We answer this question with a theoretical analysis that provides matching upper and lower bounds. Surprisingly, our results show that, without any additional assumptions, empirical risk minimization achieves the optimal excess risk bound. Specifically, we derive a novel excess risk bound proportional to the noise level, which holds in very general settings, by comparing the empirical risk minimizers obtained from clean samples and noisy samples. Second, we show that the minimax lower bound for the 0-1 loss is a constant proportional to the average noise rate. Our findings suggest that learning solely with noisy samples is impossible without access to clean samples or strong assumptions on the distribution of the data.

Beichen Wan, Mo Liu, Paul Grigas et al.

We consider the sequential experimental design problem in the predict-then-optimize paradigm. In this paradigm, the outputs of the prediction model are used as coefficient vectors in a downstream linear optimization problem. Traditional sequential experimental design aims to control the input variables (features) so that the improvement in prediction accuracy from each experimental outcome (label) is maximized. However, in the predict-then-optimize setting, performance is ultimately evaluated based on the decision loss induced by the downstream optimization, rather than by prediction error. This mismatch between prediction accuracy and decision loss renders traditional decision-blind designs inefficient. To address this issue, we propose a directional-based metric to quantify predictive uncertainty. This metric does not require solving an optimization oracle and is therefore computationally tractable. We show that the resulting sequential design criterion enjoys strong consistency and convergence guarantees. Under a broad class of distributions, we demonstrate that our directional uncertainty-based design attains an earlier stopping time than decision-blind designs. This advantage is further supported by real-world experiments on an LLM job allocation problem.

Hyungki Im, Wyame Benslimane, Paul Grigas

We study an extension of contextual stochastic linear optimization (CSLO) that, in contrast to most of the existing literature, involves inequality constraints that depend on uncertain parameters predicted by a machine learning model. To handle the constraint uncertainty, we use contextual uncertainty sets constructed via methods like conformal prediction. Given a contextual uncertainty set method, we introduce the "Smart Predict-then-Optimize with Robust Constraints" (SPO-RC) loss, a feasibility-sensitive adaptation of the SPO loss that measures decision error of predicted objective parameters. We also introduce a convex surrogate, SPO-RC+, and prove Fisher consistency with SPO-RC. To enhance performance, we train on truncated datasets where true constraint parameters lie within the uncertainty sets, and we correct the induced sample selection bias using importance reweighting techniques. Through experiments on fractional knapsack and alloy production problem instances, we demonstrate that SPO-RC+ effectively handles uncertainty in constraints and that combining truncation with importance reweighting can further improve performance.

Wyame Benslimane, Paul Grigas

We propose a new methodology for parameterized constrained robust optimization, an important class of optimization problems under uncertainty, based on learning with a self-supervised penalty-based loss function. Whereas supervised learning requires pre-solved instances for training, our approach leverages a custom loss function derived from the exact penalty method in optimization to learn an approximation, typically defined by a neural network model, of the parameterized optimal solution mapping. Additionally, we adapt our approach to robust constrained combinatorial optimization problems by incorporating a surrogate linear cost over mixed integer domains, and a smooth approximations thereof, into the final layer of the network architecture. We perform computational experiments to test our approach on three different applications: multidimensional knapsack with continuous variables, combinatorial multidimensional knapsack with discrete variables, and an inventory management problem. Our results demonstrate that our self-supervised approach is able to effectively learn neural network approximations whose inference time is significantly smaller than the computation time of traditional solvers for this class of robust optimization problems. Furthermore, our results demonstrate that by varying the penalty parameter we are able to effectively balance the trade-off between sub-optimality and robust feasibility of the obtained solutions.

Mo Liu, Paul Grigas, Heyuan Liu et al.

We develop the first active learning method for contextual linear optimization. Specifically, we introduce a label acquisition algorithm that sequentially decides whether to request the ``labels'' of feature samples from an unlabeled data stream, where the labels correspond to the coefficients of the objective in the linear optimization. Our method is the first to be directly informed by the decision loss induced by the predicted coefficients, referred to as the Smart Predict-then-Optimize (SPO) loss. Motivated by the structure of the SPO loss, our algorithm adopts a margin-based criterion utilizing the concept of distance to degeneracy. In particular, we design an efficient active learning algorithm with theoretical excess risk (i.e., generalization) guarantees. We derive upper bounds on the label complexity, defined as the number of samples whose labels are acquired to achieve a desired small level of SPO risk. These bounds show that our algorithm has a much smaller label complexity than the naive supervised learning approach that labels all samples, particularly when the SPO loss is minimized directly on the collected data. To address the discontinuity and nonconvexity of the SPO loss, we derive label complexity bounds under tractable surrogate loss functions. Under natural margin conditions, these bounds also outperform naive supervised learning. Using the SPO+ loss, a specialized surrogate of the SPO loss, we establish even tighter bounds under separability conditions. Finally, we present numerical evidence showing the practical value of our algorithms in settings such as personalized pricing and the shortest path problem.

Meng Qi, Paul Grigas, Zuo-Jun Max Shen

Many real-world optimization problems involve uncertain parameters with probability distributions that can be estimated using contextual feature information. In contrast to the standard approach of first estimating the distribution of uncertain parameters and then optimizing the objective based on the estimation, we propose an integrated conditional estimation-optimization (ICEO) framework that estimates the underlying conditional distribution of the random parameter while considering the structure of the optimization problem. We directly model the relationship between the conditional distribution of the random parameter and the contextual features, and then estimate the probabilistic model with an objective that aligns with the downstream optimization problem. We show that our ICEO approach is asymptotically consistent under moderate regularity conditions and further provide finite performance guarantees in the form of generalization bounds. Computationally, performing estimation with the ICEO approach is a non-convex and often non-differentiable optimization problem. We propose a general methodology for approximating the potentially non-differentiable mapping from estimated conditional distribution to the optimal decision by a differentiable function, which greatly improves the performance of gradient-based algorithms applied to the non-convex problem. We also provide a polynomial optimization solution approach in the semi-algebraic case. Numerical experiments are also conducted to show the empirical success of our approach in different situations including with limited data samples and model mismatches.

Heyuan Liu, Paul Grigas

The predict-then-optimize framework is fundamental in practical stochastic decision-making problems: first predict unknown parameters of an optimization model, then solve the problem using the predicted values. A natural loss function in this setting is defined by measuring the decision error induced by the predicted parameters, which was named the Smart Predict-then-Optimize (SPO) loss by Elmachtoub and Grigas [arXiv:1710.08005]. Since the SPO loss is typically nonconvex and possibly discontinuous, Elmachtoub and Grigas [arXiv:1710.08005] introduced a convex surrogate, called the SPO+ loss, that importantly accounts for the underlying structure of the optimization model. In this paper, we greatly expand upon the consistency results for the SPO+ loss provided by Elmachtoub and Grigas [arXiv:1710.08005]. We develop risk bounds and uniform calibration results for the SPO+ loss relative to the SPO loss, which provide a quantitative way to transfer the excess surrogate risk to excess true risk. By combining our risk bounds with generalization bounds, we show that the empirical minimizer of the SPO+ loss achieves low excess true risk with high probability. We first demonstrate these results in the case when the feasible region of the underlying optimization problem is a polyhedron, and then we show that the results can be strengthened substantially when the feasible region is a level set of a strongly convex function. We perform experiments to empirically demonstrate the strength of the SPO+ surrogate, as compared to standard $\ell_1$ and squared $\ell_2$ prediction error losses, on portfolio allocation and cost-sensitive multi-class classification problems.

Alfonso Lobos, Paul Grigas, Zheng Wen

We consider an online revenue maximization problem over a finite time horizon subject to lower and upper bounds on cost. At each period, an agent receives a context vector sampled i.i.d. from an unknown distribution and needs to make a decision adaptively. The revenue and cost functions depend on the context vector as well as some fixed but possibly unknown parameter vector to be learned. We propose a novel offline benchmark and a new algorithm that mixes an online dual mirror descent scheme with a generic parameter learning process. When the parameter vector is known, we demonstrate an $O(\sqrt{T})$ regret result as well an $O(\sqrt{T})$ bound on the possible constraint violations. When the parameter is not known and must be learned, we demonstrate that the regret and constraint violations are the sums of the previous $O(\sqrt{T})$ terms plus terms that directly depend on the convergence of the learning process.

Paul Grigas, Alfonso Lobos, Nathan Vermeersch

The Frank-Wolfe method and its extensions are well-suited for delivering solutions with desirable structural properties, such as sparsity or low-rank structure. We introduce a new variant of the Frank-Wolfe method that combines Frank-Wolfe steps and steepest descent steps, as well as a novel modification of the "Frank-Wolfe gap" to measure convergence in the non-convex case. We further extend this method to incorporate in-face directions for preserving structured solutions as well as block coordinate steps, and we demonstrate computational guarantees in terms of the modified Frank-Wolfe gap for all of these variants. We are particularly motivated by the application of this methodology to the training of neural networks with sparse properties, and we apply our block coordinate method to the problem of $\ell_1$ regularized neural network training. We present the results of several numerical experiments on both artificial and real datasets demonstrating significant improvements of our method in training sparse neural networks.

Othman El Balghiti, Adam N. Elmachtoub, Paul Grigas et al.

The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem, and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters, in contrast to the prediction error of the parameters. This loss function was recently introduced in Elmachtoub and Grigas (2022) and referred to as the Smart Predict-then-Optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out-of-sample, in the context of the SPO loss. Since the SPO loss is non-convex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points, but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions, and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation.

Robert M. Freund, Paul Grigas, Rahul Mazumder

Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists when the data is non-separable, but fails to exist when the data is separable. First-order methods are the algorithms of choice for solving large-scale instances of the logistic regression problem. In this paper, we introduce a pair of condition numbers that measure the degree of non-separability or separability of a given dataset in the setting of binary classification, and we study how these condition numbers relate to and inform the properties and the convergence guarantees of first-order methods. When the training data is non-separable, we show that the degree of non-separability naturally enters the analysis and informs the properties and convergence guarantees of two standard first-order methods: steepest descent (for any given norm) and stochastic gradient descent. Expanding on the work of Bach, we also show how the degree of non-separability enters into the analysis of linear convergence of steepest descent (without needing strong convexity), as well as the adaptive convergence of stochastic gradient descent. When the training data is separable, first-order methods rather curiously have good empirical success, which is not well understood in theory. In the case of separable data, we demonstrate how the degree of separability enters into the analysis of $\ell_2$ steepest descent and stochastic gradient descent for delivering approximate-maximum-margin solutions with associated computational guarantees as well. This suggests that first-order methods can lead to statistically meaningful solutions in the separable case, even though the ML solution does not exist.

Adam N. Elmachtoub, Paul Grigas

Many real-world analytics problems involve two significant challenges: prediction and optimization. Due to the typically complex nature of each challenge, the standard paradigm is predict-then-optimize. By and large, machine learning tools are intended to minimize prediction error and do not account for how the predictions will be used in the downstream optimization problem. In contrast, we propose a new and very general framework, called Smart "Predict, then Optimize" (SPO), which directly leverages the optimization problem structure, i.e., its objective and constraints, for designing better prediction models. A key component of our framework is the SPO loss function which measures the decision error induced by a prediction. Training a prediction model with respect to the SPO loss is computationally challenging, and thus we derive, using duality theory, a convex surrogate loss function which we call the SPO+ loss. Most importantly, we prove that the SPO+ loss is statistically consistent with respect to the SPO loss under mild conditions. Our SPO+ loss function can tractably handle any polyhedral, convex, or even mixed-integer optimization problem with a linear objective. Numerical experiments on shortest path and portfolio optimization problems show that the SPO framework can lead to significant improvement under the predict-then-optimize paradigm, in particular when the prediction model being trained is misspecified. We find that linear models trained using SPO+ loss tend to dominate random forest algorithms, even when the ground truth is highly nonlinear.

Robert M. Freund, Paul Grigas, Rahul Mazumder

Motivated principally by the low-rank matrix completion problem, we present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. This is accomplished by a new approach to generating ``in-face" directions at each iteration, as well as through new choice rules for selecting between in-face and ``regular" Frank-Wolfe steps. Our framework for generating in-face directions generalizes the notion of away-steps introduced by Wolfe. In particular, the in-face directions always keep the next iterate within the minimal face containing the current iterate. We present computational guarantees for the new method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply the new method to the matrix completion problem, where low-dimensional faces correspond to low-rank matrices. We present computational results that demonstrate the effectiveness of our methodological approach at producing nearly-optimal solutions of very low rank. On both artificial and real datasets, we demonstrate significant speed-ups in computing very low-rank nearly-optimal solutions as compared to either the Frank-Wolfe method or its traditional away-step variant.

Robert M. Freund, Paul Grigas, Rahul Mazumder

In this paper we analyze boosting algorithms in linear regression from a new perspective: that of modern first-order methods in convex optimization. We show that classic boosting algorithms in linear regression, namely the incremental forward stagewise algorithm (FS$_\varepsilon$) and least squares boosting (LS-Boost($\varepsilon$)), can be viewed as subgradient descent to minimize the loss function defined as the maximum absolute correlation between the features and residuals. We also propose a modification of FS$_\varepsilon$ that yields an algorithm for the Lasso, and that may be easily extended to an algorithm that computes the Lasso path for different values of the regularization parameter. Furthermore, we show that these new algorithms for the Lasso may also be interpreted as the same master algorithm (subgradient descent), applied to a regularized version of the maximum absolute correlation loss function. We derive novel, comprehensive computational guarantees for several boosting algorithms in linear regression (including LS-Boost($\varepsilon$) and FS$_\varepsilon$) by using techniques of modern first-order methods in convex optimization. Our computational guarantees inform us about the statistical properties of boosting algorithms. In particular they provide, for the first time, a precise theoretical description of the amount of data-fidelity and regularization imparted by running a boosting algorithm with a prespecified learning rate for a fixed but arbitrary number of iterations, for any dataset.

Robert M. Freund, Paul Grigas, Rahul Mazumder

Boosting methods are highly popular and effective supervised learning methods which combine weak learners into a single accurate model with good statistical performance. In this paper, we analyze two well-known boosting methods, AdaBoost and Incremental Forward Stagewise Regression (FS$_\varepsilon$), by establishing their precise connections to the Mirror Descent algorithm, which is a first-order method in convex optimization. As a consequence of these connections we obtain novel computational guarantees for these boosting methods. In particular, we characterize convergence bounds of AdaBoost, related to both the margin and log-exponential loss function, for any step-size sequence. Furthermore, this paper presents, for the first time, precise computational complexity results for FS$_\varepsilon$.