Multi-objective Markov decision processes are a special kind of multi-objective optimization problem that involves sequential decision making while satisfying the Markov property of stochastic processes. Multi-objective reinforcement learning methods address this problem by fusing the reinforcement learning paradigm with multi-objective optimization techniques. One major drawback of these methods is the lack of adaptability to non-stationary dynamics in the environment. This is because they adopt optimization procedures that assume stationarity to evolve a coverage set of policies that can solve the problem. This paper introduces a developmental optimization approach that can evolve the policy coverage set while exploring the preference space over the defined objectives in an online manner. We propose a novel multi-objective reinforcement learning algorithm that can robustly evolve a convex coverage set of policies in an online manner in non-stationary environments. We compare the proposed algorithm with two state-of-the-art multi-objective reinforcement learning algorithms in stationary and non-stationary environments. Results showed that the proposed algorithm significantly outperforms the existing algorithms in non-stationary environments while achieving comparable results in stationary environments.

Federated learning (FL) has emerged as an appealing machine learning approach to deal with massive raw data generated at multiple mobile devices, {which needs to aggregate the training model parameter of every mobile device at one base station (BS) iteratively}. For parameter aggregating in FL, over-the-air computation is a spectrum-efficient solution, which allows all mobile devices to transmit their parameter-mapped signals concurrently to a BS. Due to heterogeneous channel fading and noise, there exists difference between the BS's received signal and its desired signal, measured as the mean-squared error (MSE). To minimize the MSE, we propose to jointly optimize the signal amplification factors at the BS and the mobile devices as well as the data size (the number of data samples involved in local training) at every mobile device. The formulated problem is challenging to solve due to its non-convexity. To find the optimal solution, with some simplification on cost function and variable replacement, which still preserves equivalence, we transform the changed problem to be a bi-level problem equivalently. For the lower-level problem, optimal solution is found by enumerating every candidate solution from the Karush-Kuhn-Tucker (KKT) condition. For the upper-level problem, the optimal solution is found by exploring its piecewise convexity. Numerical results show that our proposed method can greatly reduce the MSE and can help to improve the training performance of FL compared with benchmark methods.

Graphical models have exhibited their performance in numerous tasks ranging from biological analysis to recommender systems. However, graphical models with hub nodes are computationally difficult to fit, particularly when the dimension of the data is large. To efficiently estimate the hub graphical models, we introduce a two-phase algorithm. The proposed algorithm first generates a good initial point via a dual alternating direction method of multipliers (ADMM), and then warm starts a semismooth Newton (SSN) based augmented Lagrangian method (ALM) to compute a solution that is accurate enough for practical tasks. The sparsity structure of the generalized Jacobian ensures that the algorithm can obtain a nice solution very efficiently. Comprehensive experiments on both synthetic data and real data show that it obviously outperforms the existing state-of-the-art algorithms. In particular, in some high dimensional tasks, it can save more than 70\% of the execution time, meanwhile still achieves a high-quality estimation.

Efficient multi-robot task allocation (MRTA) is fundamental to various time-sensitive applications such as disaster response, warehouse operations, and construction. This paper tackles a particular class of these problems that we call MRTA-collective transport or MRTA-CT -- here tasks present varying workloads and deadlines, and robots are subject to flight range, communication range, and payload constraints. For large instances of these problems involving 100s-1000's of tasks and 10s-100s of robots, traditional non-learning solvers are often time-inefficient, and emerging learning-based policies do not scale well to larger-sized problems without costly retraining. To address this gap, we use a recently proposed encoder-decoder graph neural network involving Capsule networks and multi-head attention mechanism, and innovatively add topological descriptors (TD) as new features to improve transferability to unseen problems of similar and larger size. Persistent homology is used to derive the TD, and proximal policy optimization is used to train our TD-augmented graph neural network. The resulting policy model compares favorably to state-of-the-art non-learning baselines while being much faster. The benefit of using TD is readily evident when scaling to test problems of size larger than those used in training.

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 instead is caused by several factors all at once. 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. We argue that 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 which does guarantee such holistic protection and is furthermore computationally viable. Our distributionally robust optimization formulation can be interpreted as a novel combination of a Kullback-Leibler and Levy-Prokhorov robust optimization formulation which is novel in its own right. However, we show how in the context of classification and regression problems that several popular regularized and robust formulations reduce to a particular case of our proposed novel formulation. Finally, we apply the proposed HR formulation on a portfolio selection problem with real stock data, and analyze its risk/return tradeoff against several benchmarks formulations. Our experiments show that our novel ambiguity set provides a significantly better risk/return trade-off.

Decision-focused learning (DFL) is an emerging paradigm in machine learning which trains a model to optimize decisions, integrating prediction and optimization in an end-to-end system. This paradigm holds the promise to revolutionize decision-making in many real-world applications which operate under uncertainty, where the estimation of unknown parameters within these decision models often becomes a substantial roadblock. This paper presents a comprehensive review of DFL. It provides an in-depth analysis of the various techniques devised to integrate machine learning and optimization models, introduces a taxonomy of DFL methods distinguished by their unique characteristics, and conducts an extensive empirical evaluation of these methods proposing suitable benchmark dataset and tasks for DFL. Finally, the study provides valuable insights into current and potential future avenues in DFL research.

Deterministic model predictive control (MPC), while powerful, is often insufficient for effectively controlling autonomous systems in the real-world. Factors such as environmental noise and model error can cause deviations from the expected nominal performance. Robust MPC algorithms aim to bridge this gap between deterministic and uncertain control. However, these methods are often excessively difficult to tune for robustness due to the nonlinear and non-intuitive effects that controller parameters have on performance. To address this challenge, a unifying perspective on differentiable optimization for control is presented, which enables derivation of a general, differentiable tube-based MPC algorithm. The proposed approach facilitates the automatic and real-time tuning of robust controllers in the presence of large uncertainties and disturbances.

We present SCQPTH: a differentiable first-order splitting method for convex quadratic programs. The SCQPTH framework is based on the alternating direction method of multipliers (ADMM) and the software implementation is motivated by the state-of-the art solver OSQP: an operating splitting solver for convex quadratic programs (QPs). The SCQPTH software is made available as an open-source python package and contains many similar features including efficient reuse of matrix factorizations, infeasibility detection, automatic scaling and parameter selection. The forward pass algorithm performs operator splitting in the dimension of the original problem space and is therefore suitable for large scale QPs with $100-1000$ decision variables and thousands of constraints. Backpropagation is performed by implicit differentiation of the ADMM fixed-point mapping. Experiments demonstrate that for large scale QPs, SCQPTH can provide a $1\times - 10\times$ improvement in computational efficiency in comparison to existing differentiable QP solvers.

Learning directed acyclic graphs (DAGs) to identify causal relations underlying observational data is crucial but also poses significant challenges. Recently, topology-based methods have emerged as a two-step approach to discovering DAGs by first learning the topological ordering of variables and then eliminating redundant edges, while ensuring that the graph remains acyclic. However, one limitation is that these methods would generate numerous spurious edges that require subsequent pruning. To overcome this limitation, in this paper, we propose an improvement to topology-based methods by introducing limited time series data, consisting of only two cross-sectional records that need not be adjacent in time and are subject to flexible timing. By incorporating conditional instrumental variables as exogenous interventions, we aim to identify descendant nodes for each variable. Following this line, we propose a hierarchical topological ordering algorithm with conditional independence test (HT-CIT), which enables the efficient learning of sparse DAGs with a smaller search space compared to other popular approaches. The HT-CIT algorithm greatly reduces the number of edges that need to be pruned. Empirical results from synthetic and real-world datasets demonstrate the superiority of the proposed HT-CIT algorithm.

Researchers constantly strive to explore larger and more complex search spaces in various scientific studies and physical experiments. However, such investigations often involve sophisticated simulators or time-consuming experiments that make exploring and observing new design samples challenging. Previous works that target such applications are typically sample-inefficient and restricted to vector search spaces. To address these limitations, this work proposes a constrained multi-objective optimization (MOO) framework, called BREATHE, that searches not only traditional vector-based design spaces but also graph-based design spaces to obtain best-performing graphs. It leverages second-order gradients and actively trains a heteroscedastic surrogate model for sample-efficient optimization. In a single-objective vector optimization application, it leads to 64.1% higher performance than the next-best baseline, random forest regression. In graph-based search, BREATHE outperforms the next-best baseline, i.e., a graphical version of Gaussian-process-based Bayesian optimization, with up to 64.9% higher performance. In a MOO task, it achieves up to 21.9$\times$ higher hypervolume than the state-of-the-art method, multi-objective Bayesian optimization (MOBOpt). BREATHE also outperforms the baseline methods on most standard MOO benchmark applications.