Learning with rejection has been a prototypical model for studying the human-AI interaction on prediction tasks. Upon the arrival of a sample instance, the model first uses a rejector to decide whether to accept and use the AI predictor to make a prediction or reject and defer the sample to humans. Learning such a model changes the structure of the original loss function and often results in undesirable non-convexity and inconsistency issues. For the classification with rejection problem, several works develop consistent surrogate losses for the joint learning of the predictor and the rejector, while there have been fewer works for the regression counterpart. This paper studies the regression with rejection (RwR) problem and investigates a no-rejection learning strategy that uses all the data to learn the predictor. We first establish the consistency for such a strategy under the weak realizability condition. Then for the case without the weak realizability, we show that the excessive risk can also be upper bounded with the sum of two parts: prediction error and calibration error. Lastly, we demonstrate the advantage of such a proposed learning strategy with empirical evidence.

As artificial intelligence (AI) systems play an increasingly prominent role in human decision-making, challenges surface in the realm of human-AI interactions. One challenge arises from the suboptimal AI policies due to the inadequate consideration of humans disregarding AI recommendations, as well as the need for AI to provide advice selectively when it is most pertinent. This paper presents a sequential decision-making model that (i) takes into account the human's adherence level (the probability that the human follows/rejects machine advice) and (ii) incorporates a defer option so that the machine can temporarily refrain from making advice. We provide learning algorithms that learn the optimal advice policy and make advice only at critical time stamps. Compared to problem-agnostic reinforcement learning algorithms, our specialized learning algorithms not only enjoy better theoretical convergence properties but also show strong empirical performance.

This paper presents fast first-order methods for solving linear programs (LPs) approximately. We adapt online linear programming algorithms to offline LPs and obtain algorithms that avoid any matrix multiplication. We also introduce a variable-duplication technique that copies each variable $K$ times and reduces the optimality gap and constraint violation by a factor of $\sqrt{K}$. Furthermore, we show how online algorithms can be effectively integrated into sifting, a column generation scheme for large-scale LPs. Numerical experiments demonstrate that our methods can serve as either an approximate direct solver, or an initialization subroutine for exact LP solving.

In this paper, we study the predict-then-optimize problem where the output of a machine learning prediction task is used as the input of some downstream optimization problem, say, the objective coefficient vector of a linear program. The problem is also known as predictive analytics or contextual linear programming. The existing approaches largely suffer from either (i) optimization intractability (a non-convex objective function)/statistical inefficiency (a suboptimal generalization bound) or (ii) requiring strong condition(s) such as no constraint or loss calibration. We develop a new approach to the problem called maximum optimality margin which designs the machine learning loss function by the optimality condition of the downstream optimization. The max-margin formulation enjoys both computational efficiency and good theoretical properties for the learning procedure. More importantly, our new approach only needs the observations of the optimal solution in the training data rather than the objective function, which makes it a new and natural approach to the inverse linear programming problem under both contextual and context-free settings; we also analyze the proposed method under both offline and online settings, and demonstrate its performance using numerical experiments.

Contextual optimization, also known as predict-then-optimize or prescriptive analytics, considers an optimization problem with the presence of covariates (context or side information). The goal is to learn a prediction model (from the training data) that predicts the objective function from the covariates, and then in the test phase, solve the optimization problem with the covariates but without the observation of the objective function. In this paper, we consider a risk-sensitive version of the problem and propose a generic algorithm design paradigm called predict-then-calibrate. The idea is to first develop a prediction model without concern for the downstream risk profile or robustness guarantee, and then utilize calibration (or recalibration) methods to quantify the uncertainty of the prediction. While the existing methods suffer from either a restricted choice of the prediction model or strong assumptions on the underlying data, we show the disentangling of the prediction model and the calibration/uncertainty quantification has several advantages. First, it imposes no restriction on the prediction model and thus fully unleashes the potential of off-the-shelf machine learning methods. Second, the derivation of the risk and robustness guarantee can be made independent of the choice of the prediction model through a data-splitting idea. Third, our paradigm of predict-then-calibrate applies to both (risk-sensitive) robust and (risk-neutral) distributionally robust optimization (DRO) formulations. Theoretically, it gives new generalization bounds for the contextual LP problem and sheds light on the existing results of DRO for contextual LP. Numerical experiments further reinforce the advantage of the predict-then-calibrate paradigm in that an improvement on either the prediction model or the calibration model will lead to a better final performance.

We study the learning problem of revealed preference in a stochastic setting: a learner observes the utility-maximizing actions of a set of agents whose utility follows some unknown distribution, and the learner aims to infer the distribution through the observations of actions. The problem can be viewed as a single-constraint special case of the inverse linear optimization problem. Existing works all assume that all the agents share one common utility which can easily be violated under practical contexts. In this paper, we consider two settings for the underlying utility distribution: a Gaussian setting where the customer utility follows the von Mises-Fisher distribution, and a $\delta$-corruption setting where the customer utility distribution concentrates on one fixed vector with high probability and is arbitrarily corrupted otherwise. We devise Bayesian approaches for parameter estimation and develop theoretical guarantees for the recovery of the true parameter. We illustrate the algorithm performance through numerical experiments.

In this paper, we study the bandits with knapsacks (BwK) problem and develop a primal-dual based algorithm that achieves a problem-dependent logarithmic regret bound. The BwK problem extends the multi-arm bandit (MAB) problem to model the resource consumption, and the existing BwK literature has been mainly focused on deriving asymptotically optimal distribution-free regret bounds. We first study the primal and dual linear programs underlying the BwK problem. From this primal-dual perspective, we discover symmetry between arms and knapsacks, and then propose a new notion of suboptimality measure for the BwK problem. The suboptimality measure highlights the important role of knapsacks in determining algorithm regret and inspires the design of our two-phase algorithm. In the first phase, the algorithm identifies the optimal arms and the binding knapsacks, and in the second phase, it exhausts the binding knapsacks via playing the optimal arms through an adaptive procedure. Our regret upper bound involves the proposed suboptimality measure and it has a logarithmic dependence on length of horizon $T$ and a polynomial dependence on $m$ (the numbers of arms) and $d$ (the number of knapsacks). To the best of our knowledge, this is the first problem-dependent logarithmic regret bound for solving the general BwK problem.

In this paper, we develop a simple and fast online algorithm for solving a class of binary integer linear programs (LPs) arisen in the general resource allocation problem. The algorithm requires only one single pass through the input data and is free of doing any matrix inversion. It can be viewed as both an approximate algorithm for solving binary integer LPs and a fast algorithm for solving online LP problems. The algorithm is inspired by an equivalent form of the dual problem of the relaxed LP and it essentially performs (one-pass) projected stochastic subgradient descent in the dual space. We analyze the algorithm under two different models, stochastic input and random permutation, with minimal technical assumptions on the input data. The algorithm achieves $O\left(m \sqrt{n}\right)$ expected regret under the stochastic input model and $O\left((m+\log n)\sqrt{n}\right)$ expected regret under the random permutation model, and it achieves $O(m \sqrt{n})$ expected constraint violation under both models, where $n$ is the number of decision variables and $m$ is the number of constraints. In addition, we employ the notion of permutational Rademacher complexity and derive regret bounds for two earlier online LP algorithms for comparison. Both algorithms improve the regret bound with a factor of $\sqrt{m}$ by paying more computational cost. Furthermore, we demonstrate how to convert the possibly infeasible solution to a feasible one through a randomized procedure. Numerical experiments illustrate the general applicability and effectiveness of the algorithms.