Data-driven optimization aims to translate a machine learning model into decision-making by optimizing decisions on estimated costs. Such a pipeline can be conducted by fitting a distributional model which is then plugged into the target optimization problem. While this fitting can utilize traditional methods such as maximum likelihood, a more recent approach uses estimation-optimization integration that minimizes decision error instead of estimation error. Although intuitive, the statistical benefit of the latter approach is not well understood yet is important to guide the prescriptive usage of machine learning. In this paper, we dissect the performance comparisons between these approaches in terms of the amount of model misspecification. In particular, we show how the integrated approach offers a ``universal double benefit'' on the top two dominating terms of regret when the underlying model is misspecified, while the traditional approach can be advantageous when the model is nearly well-specified. Our comparison is powered by finite-sample tail regret bounds that are derived via new higher-order expansions of regrets and the leveraging of a recent Berry-Esseen theorem.

The increasing complexity of multirotor applications has led to the need of more accurate flight controllers that can reliably predict all forces acting on the robot. Traditional flight controllers model a large part of the forces but do not take so called residual forces into account. A reason for this is that accurately computing the residual forces can be computationally expensive. Incremental Nonlinear Dynamic Inversion (INDI) is a method that computes the difference between different sensor measurements in order to estimate these residual forces. The main issue with INDI is it's reliance on special sensor measurements which can be very noisy. Recent work has also shown that residual forces can be predicted using learning-based methods. In this work, we demonstrate that a learning algorithm can predict a smoother version of INDI outputs without requiring additional sensor measurements. In addition, we introduce a new method that combines learning based predictions with INDI. We also adapt the two approaches to work on quadrotors carrying a slung-type payload. The results show that using a neural network to predict residual forces can outperform INDI while using the combination of neural network and INDI can yield even better results than each method individually.

Machine Learning Force Fields (MLFFs) are a promising alternative to expensive ab initio quantum mechanical molecular simulations. Given the diversity of chemical spaces that are of interest and the cost of generating new data, it is important to understand how MLFFs generalize beyond their training distributions. In order to characterize and better understand distribution shifts in MLFFs, we conduct diagnostic experiments on chemical datasets, revealing common shifts that pose significant challenges, even for large foundation models trained on extensive data. Based on these observations, we hypothesize that current supervised training methods inadequately regularize MLFFs, resulting in overfitting and learning poor representations of out-of-distribution systems. We then propose two new methods as initial steps for mitigating distribution shifts for MLFFs. Our methods focus on test-time refinement strategies that incur minimal computational cost and do not use expensive ab initio reference labels. The first strategy, based on spectral graph theory, modifies the edges of test graphs to align with graph structures seen during training. Our second strategy improves representations for out-of-distribution systems at test-time by taking gradient steps using an auxiliary objective, such as a cheap physical prior. Our test-time refinement strategies significantly reduce errors on out-of-distribution systems, suggesting that MLFFs are capable of and can move towards modeling diverse chemical spaces, but are not being effectively trained to do so. Our experiments establish clear benchmarks for evaluating the generalization capabilities of the next generation of MLFFs. Our code is available at https://tkreiman.github.io/projects/mlff_distribution_shifts/.

The Ride-Pool Matching Problem (RMP) is central to on-demand ride-pooling services, where vehicles must be matched with multiple requests while adhering to service constraints such as pickup delays, detour limits, and vehicle capacity. Most existing RMP solutions assume passengers are picked up and dropped off at their original locations, neglecting the potential for passengers to walk to nearby spots to meet vehicles. This assumption restricts the optimization potential in ride-pooling operations. In this paper, we propose a novel matching method that incorporates extended pickup and drop-off areas for passengers. We first design a tree-based approach to efficiently generate feasible matches between passengers and vehicles. Next, we optimize vehicle routes to cover all designated pickup and drop-off locations while minimizing total travel distance. Finally, we employ dynamic assignment strategies to achieve optimal matching outcomes. Experiments on city-scale taxi datasets demonstrate that our method improves the number of served requests by up to 13\% and average travel distance by up to 21\% compared to leading existing solutions, underscoring the potential of leveraging passenger mobility to significantly enhance ride-pooling service efficiency.

Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers -- Bayesian statistical models mostly based on Gaussian process (GP) priors which seamlessly combine empirical measurements and mechanistic knowledge. As such, they quantify uncertainties arising from e.g. noisy or missing data, unknown PDE parameters or discretization error by design. Prior work has established connections to classical PDE solvers and provided solid theoretical guarantees. However, scaling such methods to large-scale problems remains a fundamental challenge primarily due to dense covariance matrices. Our approach addresses the scalability issues by leveraging the Markov property of many commonly used GP priors. It has been shown that such priors are solutions to stochastic PDEs (SPDEs) which when discretized allow for highly efficient GP regression through sparse linear algebra. In this work, we show how to leverage this prior class to make probabilistic PDE solvers practical, even for large-scale nonlinear PDEs, through greatly accelerated inference mechanisms. Additionally, our approach also allows for flexible and physically meaningful priors beyond what can be modeled with covariance functions. Experiments confirm substantial speedups and accelerated convergence of our physics-informed priors in nonlinear settings.

The kidney paired donation (KPD) program provides an innovative solution to overcome incompatibility challenges in kidney transplants by matching incompatible donor-patient pairs and facilitating kidney exchanges. To address unequal access to transplant opportunities, there are two widely used fairness criteria: group fairness and individual fairness. However, these criteria do not consider protected patient features, which refer to characteristics legally or ethically recognized as needing protection from discrimination, such as race and gender. Motivated by the calibration principle in machine learning, we introduce a new fairness criterion: the matching outcome should be conditionally independent of the protected feature, given the sensitization level. We integrate this fairness criterion as a constraint within the KPD optimization framework and propose a computationally efficient solution. Theoretically, we analyze the associated price of fairness using random graph models. Empirically, we compare our fairness criterion with group fairness and individual fairness through both simulations and a real-data example.

We consider the problem of preprocessing an $n\times n$ matrix M, and supporting queries that, for any vector v, returns the matrix-vector product Mv. This problem has been extensively studied in both theory and practice: on one side, practitioners have developed algorithms that are highly efficient in practice, whereas theoreticians have proven that the problem cannot be solved faster than naive multiplication in the worst-case. This lower bound holds even in the average-case, implying that existing average-case analyses cannot explain this gap between theory and practice. Therefore, we study the problem for structured matrices. We show that for $n\times n$ matrices of VC-dimension d, the matrix-vector multiplication problem can be solved with $\tilde{O}(n^2)$ preprocessing and $\tilde O(n^{2-1/d})$ query time. Given the low constant VC-dimensions observed in most real-world data, our results posit an explanation for why the problem can be solved so much faster in practice. Moreover, our bounds hold even if the matrix does not have a low VC-dimension, but is obtained by (possibly adversarially) corrupting at most a subquadratic number of entries of any unknown low VC-dimension matrix. Our results yield the first non-trivial upper bounds for many applications. In previous works, the online matrix-vector hypothesis (conjecturing that quadratic time is needed per query) was used to prove many conditional lower bounds, showing that it is impossible to compute and maintain high-accuracy estimates for shortest paths, Laplacian solvers, effective resistance, and triangle detection in graphs subject to node insertions and deletions in subquadratic time. Yet, via a reduction to our matrix-vector-multiplication result, we show we can maintain the aforementioned problems efficiently if the input is structured, providing the first subquadratic upper bounds in the high-accuracy regime.

Autonomous motion planning under unknown nonlinear dynamics presents significant challenges. An agent needs to continuously explore the system dynamics to acquire its properties, such as reachability, in order to guide system navigation adaptively. In this paper, we propose a hybrid planning-control framework designed to compute a feasible trajectory toward a target. Our approach involves partitioning the state space and approximating the system by a piecewise affine (PWA) system with constrained control inputs. By abstracting the PWA system into a directed weighted graph, we incrementally update the existence of its edges via affine system identification and reach control theory, introducing a predictive reachability condition by exploiting prior information of the unknown dynamics. Heuristic weights are assigned to edges based on whether their existence is certain or remains indeterminate. Consequently, we propose a framework that adaptively collects and analyzes data during mission execution, continually updates the predictive graph, and synthesizes a controller online based on the graph search outcomes. We demonstrate the efficacy of our approach through simulation scenarios involving a mobile robot operating in unknown terrains, with its unknown dynamics abstracted as a single integrator model.

The Bellman equation and its continuous-time counterpart, the Hamilton-Jacobi-Bellman (HJB) equation, serve as necessary conditions for optimality in reinforcement learning and optimal control. While the value function is known to be the unique solution to the Bellman equation in tabular settings, we demonstrate that this uniqueness fails to hold in continuous state spaces. Specifically, for linear dynamical systems, we prove the Bellman equation admits at least $\binom{2n}{n}$ solutions, where $n$ is the state dimension. Crucially, only one of these solutions yields both an optimal policy and a stable closed-loop system. We then demonstrate a common failure mode in value-based methods: convergence to unstable solutions due to the exponential imbalance between admissible and inadmissible solutions. Finally, we introduce a positive-definite neural architecture that guarantees convergence to the stable solution by construction to address this issue.

We study the problem of robustly estimating the edge density of Erd\H{o}s-R\'enyi random graphs $G(n, d^\circ/n)$ when an adversary can arbitrarily add or remove edges incident to an $\eta$-fraction of the nodes. We develop the first polynomial-time algorithm for this problem that estimates $d^\circ$ up to an additive error $O([\sqrt{\log(n) / n} + \eta\sqrt{\log(1/\eta)} ] \cdot \sqrt{d^\circ} + \eta \log(1/\eta))$. Our error guarantee matches information-theoretic lower bounds up to factors of $\log(1/\eta)$. Moreover, our estimator works for all $d^\circ \geq \Omega(1)$ and achieves optimal breakdown point $\eta = 1/2$. Previous algorithms [AJK+22, CDHS24], including inefficient ones, incur significantly suboptimal errors. Furthermore, even admitting suboptimal error guarantees, only inefficient algorithms achieve optimal breakdown point. Our algorithm is based on the sum-of-squares (SoS) hierarchy. A key ingredient is to construct constant-degree SoS certificates for concentration of the number of edges incident to small sets in $G(n, d^\circ/n)$. Crucially, we show that these certificates also exist in the sparse regime, when $d^\circ = o(\log n)$, a regime in which the performance of previous algorithms was significantly suboptimal.