In this paper we consider a nonconvex unconstrained optimization problem minimizing a twice differentiable objective function with H\"older continuous Hessian. Specifically, we first propose a Newton-conjugate gradient (Newton-CG) method for finding an approximate first-order stationary point (FOSP) of this problem, assuming the associated the H\"older parameters are explicitly known. Then we develop a parameter-free Newton-CG method without requiring any prior knowledge of these parameters. To the best of our knowledge, this method is the first parameter-free second-order method achieving the best-known iteration and operation complexity for finding an approximate FOSP of this problem. Furthermore, we propose a Newton-CG method for finding an approximate second-order stationary point (SOSP) of the considered problem with high probability and establish its iteration and operation complexity. Finally, we present preliminary numerical results to demonstrate the superior practical performance of our parameter-free Newton-CG method over a well-known regularized Newton method.

Many real-world decision processes are modeled by optimization problems whose defining parameters are unknown and must be inferred from observable data. The Predict-Then-Optimize framework uses machine learning models to predict unknown parameters of an optimization problem from features before solving. Recent works show that decision quality can be improved in this setting by solving and differentiating the optimization problem in the training loop, enabling end-to-end training with loss functions defined directly on the resulting decisions. However, this approach can be inefficient and requires handcrafted, problem-specific rules for backpropagation through the optimization step. This paper proposes an alternative method, in which optimal solutions are learned directly from the observable features by predictive models. The approach is generic, and based on an adaptation of the Learning-to-Optimize paradigm, from which a rich variety of existing techniques can be employed. Experimental evaluations show the ability of several Learning-to-Optimize methods to provide efficient, accurate, and flexible solutions to an array of challenging Predict-Then-Optimize problems.

Prediction of mortality in intensive care unit (ICU) patients is an important task in critical care medicine. Prior work in creating mortality risk models falls into two major categories: domain-expert-created scoring systems, and black box machine learning (ML) models. Both of these have disadvantages: black box models are unacceptable for use in hospitals, whereas manual creation of models (including hand-tuning of logistic regression parameters) relies on humans to perform high-dimensional constrained optimization, which leads to a loss in performance. In this work, we bridge the gap between accurate black box models and hand-tuned interpretable models. We build on modern interpretable ML techniques to design accurate and interpretable mortality risk scores. We leverage the largest existing public ICU monitoring datasets, namely the MIMIC III and eICU datasets. By evaluating risk across medical centers, we are able to study generalization across domains. In order to customize our risk score models, we develop a new algorithm, GroupFasterRisk, which has several important benefits: (1) it uses hard sparsity constraint, allowing users to directly control the number of features; (2) it incorporates group sparsity to allow more cohesive models; (3) it allows for monotonicity correction on models for including domain knowledge; (4) it produces many equally-good models at once, which allows domain experts to choose among them. GroupFasterRisk creates its risk scores within hours, even on the large datasets we study here. GroupFasterRisk's risk scores perform better than risk scores currently used in hospitals, and have similar prediction performance to black box ML models (despite being much sparser). Because GroupFasterRisk produces a variety of risk scores and handles constraints, it allows design flexibility, which is the key enabler of practical and trustworthy model creation.

Same-Day Delivery (SDD) services aim to maximize the fulfillment of online orders while minimizing delivery delays but are beset by operational uncertainties such as those in order volumes and courier planning. Our work aims to enhance the operational efficiency of SDD by focusing on the ultra-fast Order Dispatching Problem (ODP), which involves matching and dispatching orders to couriers within a centralized warehouse setting, and completing the delivery within a strict timeline (e.g., within minutes). We introduce important extensions to ultra-fast ODP such as order batching and explicit courier assignments to provide a more realistic representation of dispatching operations and improve delivery efficiency. As a solution method, we primarily focus on NeurADP, a methodology that combines Approximate Dynamic Programming (ADP) and Deep Reinforcement Learning (DRL), and our work constitutes the first application of NeurADP outside of the ride-pool matching problem. NeurADP is particularly suitable for ultra-fast ODP as it addresses complex one-to-many matching and routing intricacies through a neural network-based VFA that captures high-dimensional problem dynamics without requiring manual feature engineering as in generic ADP methods. We test our proposed approach using four distinct realistic datasets tailored for ODP and compare the performance of NeurADP against myopic and DRL baselines by also making use of non-trivial bounds to assess the quality of the policies. Our numerical results indicate that the inclusion of order batching and courier queues enhances the efficiency of delivery operations and that NeurADP significantly outperforms other methods. Detailed sensitivity analysis with important parameters confirms the robustness of NeurADP under different scenarios, including variations in courier numbers, spatial setup, vehicle capacity, and permitted delay time.

Recently, compositional optimization (CO) has gained popularity because of its applications in distributionally robust optimization (DRO) and many other machine learning problems. Large-scale and distributed availability of data demands the development of efficient federated learning (FL) algorithms for solving CO problems. Developing FL algorithms for CO is particularly challenging because of the compositional nature of the objective. Moreover, current state-of-the-art methods to solve such problems rely on large batch gradients (depending on the solution accuracy) not feasible for most practical settings. To address these challenges, in this work, we propose efficient FedAvg-type algorithms for solving non-convex CO in the FL setting. We first establish that vanilla FedAvg is not suitable to solve distributed CO problems because of the data heterogeneity in the compositional objective at each client which leads to the amplification of bias in the local compositional gradient estimates. To this end, we propose a novel FL framework FedDRO that utilizes the DRO problem structure to design a communication strategy that allows FedAvg to control the bias in the estimation of the compositional gradient. A key novelty of our work is to develop solution accuracy-independent algorithms that do not require large batch gradients (and function evaluations) for solving federated CO problems. We establish $\mathcal{O}(\epsilon^{-2})$ sample and $\mathcal{O}(\epsilon^{-3/2})$ communication complexity in the FL setting while achieving linear speedup with the number of clients. We corroborate our theoretical findings with empirical studies on large-scale DRO problems.

Neglecting the effect that decisions have on individuals (and thus, on the underlying data distribution) when designing algorithmic decision-making policies may increase inequalities and unfairness in the long term--even if fairness considerations were taken in the policy design process. In this paper, we propose a novel framework for achieving long-term group fairness in dynamical systems, in which current decisions may affect an individual's features in the next step, and thus, future decisions. Specifically, our framework allows us to identify a time-independent policy that converges, if deployed, to the targeted fair stationary state of the system in the long-term, independently of the initial data distribution. We model the system dynamics with a time-homogeneous Markov chain and optimize the policy leveraging the Markov chain convergence theorem to ensure unique convergence. We provide examples of different targeted fair states of the system, encompassing a range of long-term goals for society and policy makers. Furthermore, we show how our approach facilitates the evaluation of different long-term targets by examining their impact on the group-conditional population distribution in the long term and how it evolves until convergence.

We present the concept of a Generalized Feedback Nash Equilibrium (GFNE) in dynamic games, extending the Feedback Nash Equilibrium concept to games in which players are subject to state and input constraints. We formalize necessary and sufficient conditions for (local) GFNE solutions at the trajectory level, which enable the development of efficient numerical methods for their computation. Specifically, we propose a Newton-style method for finding game trajectories which satisfy necessary conditions for an equilibrium, which can then be checked against sufficiency conditions. We show that the evaluation of the necessary conditions in general requires computing a series of nested, implicitly-defined derivatives, which quickly becomes intractable. To this end, we introduce an approximation to the necessary conditions which is amenable to efficient evaluation, and in turn, computation of solutions. We term the solutions to the approximate necessary conditions Generalized Feedback Quasi-Nash Equilibria (GFQNE), and we introduce numerical methods for their computation. In particular, we develop a Sequential Linear-Quadratic Game approach, in which a LQ local approximation of the game is solved at each iteration. The development of this method relies on the ability to compute a GFNE to inequality- and equality-constrained LQ games, and therefore specific methods for the solution of these special cases are developed in detail. We demonstrate the effectiveness of the proposed solution approach on a dynamic game arising in an autonomous driving application.

This paper concerns a class of DC composite optimization problems which, as an extension of convex composite optimization problems and DC programs with nonsmooth components, often arises in robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) by computing in each step an inexact minimizer of a strongly convex majorization constructed with a partial linearization of their objective functions at the current iterate, and establish the convergence of the generated iterate sequence under the Kurdyka-\L\"ojasiewicz (KL) property of a potential function. In particular, by leveraging the composite structure, we provide a verifiable condition for the potential function to have the KL property of exponent $1/2$ at the limit point, so for the iterate sequence to have a local R-linear convergence rate. Finally, we apply the proposed iLPA to a robust factorization model for matrix completions with outliers and non-uniform sampling, and numerical comparison with the Polyak subgradient method confirms its superiority in terms of computing time and quality of solutions.

We consider the problem of nonlinear stochastic optimal control. This problem is thought to be fundamentally intractable owing to Bellman's infamous "curse of dimensionality". We present a result that shows that repeatedly solving an open-loop deterministic problem from the current state, similar to Model Predictive Control (MPC), results in a feedback policy that is $O(\epsilon^4)$ near to the true global stochastic optimal policy. Furthermore, empirical results show that solving the Stochastic Dynamic Programming (DP) problem is highly susceptible to noise, even when tractable, and in practice, the MPC-type feedback law offers superior performance even for stochastic systems.

The availability of representative datasets is an essential prerequisite for many successful artificial intelligence and machine learning models. However, in real life applications these models often encounter scenarios that are inadequately represented in the data used for training. There are various reasons for the absence of sufficient data, ranging from time and cost constraints to ethical considerations. As a consequence, the reliable usage of these models, especially in safety-critical applications, is still a tremendous challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches. Knowledge augmented machine learning approaches offer the possibility of compensating for deficiencies, errors, or ambiguities in the data, thus increasing the generalization capability of the applied models. Even more, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-driven models with existing knowledge. The identified approaches are structured according to the categories knowledge integration, extraction and conformity. In particular, we address the application of the presented methods in the field of autonomous driving.