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Many decision-making problems feature multiple objectives. In such problems, it is not always possible to know the preferences of a decision-maker for different objectives. However, it is often possible to observe the behavior of decision-makers. In multi-objective decision-making, preference inference is the process of inferring the preferences of a decision-maker for different objectives. This research proposes a Dynamic Weight-based Preference Inference (DWPI) algorithm that can infer the preferences of agents acting in multi-objective decision-making problems, based on observed behavior trajectories in the environment. The proposed method is evaluated on three multi-objective Markov decision processes: Deep Sea Treasure, Traffic, and Item Gathering. The performance of the proposed DWPI approach is compared to two existing preference inference methods from the literature, and empirical results demonstrate significant improvements compared to the baseline algorithms, in terms of both time requirements and accuracy of the inferred preferences. The Dynamic Weight-based Preference Inference algorithm also maintains its performance when inferring preferences for sub-optimal behavior demonstrations. In addition to its impressive performance, the Dynamic Weight-based Preference Inference algorithm does not require any interactions during training with the agent whose preferences are inferred, all that is required is a trajectory of observed behavior.

Conditional Average Treatment Effects (CATE) estimation is one of the main challenges in causal inference with observational data. In addition to Machine Learning based-models, nonparametric estimators called meta-learners have been developed to estimate the CATE with the main advantage of not restraining the estimation to a specific supervised learning method. This task becomes, however, more complicated when the treatment is not binary as some limitations of the naive extensions emerge. This paper looks into meta-learners for estimating the heterogeneous effects of multi-valued treatments. We consider different meta-learners, and we carry out a theoretical analysis of their error upper bounds as functions of important parameters such as the number of treatment levels, showing that the naive extensions do not always provide satisfactory results. We introduce and discuss meta-learners that perform well as the number of treatments increases. We empirically confirm the strengths and weaknesses of those methods with synthetic and semi-synthetic datasets.

In this paper, we propose, analyze and implement efficient time parallel methods for the Cahn-Hilliard (CH) equation. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of the CH equation has. The CH equation generally needs to be simulated for a very long time to get the solution of phase coarsening stage. Therefore it is desirable to accelerate the computation using parallel method in time. We present linear and nonlinear Parareal methods for the CH equation depending on the choice of fine approximation. We illustrate our results by numerical experiments.

Quadrotors are agile flying robots that are challenging to control. Considering the full dynamics of quadrotors during motion planning is crucial to achieving good solution quality and small tracking errors during flight. Optimization-based methods scale well with high-dimensional state spaces and can handle dynamic constraints directly, therefore they are often used in these scenarios. The resulting optimization problem is notoriously difficult to solve due to its nonconvex constraints. In this work, we present an analysis of four solvers for nonlinear trajectory optimization (KOMO, direct collocation with SCvx, direct collocation with CasADi, Crocoddyl) and evaluate their performance in scenarios where the solvers are tasked to find minimum-effort solutions to geometrically complex problems and problems requiring highly dynamic solutions. Benchmarking these methods helps to determine the best algorithm structures for these kinds of problems.

Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the nascent theory of parameterized complexity has allowed us to push beyond the P-NP ``dichotomy'' in classical computational complexity and identify the exact boundaries of tractability for numerous problems, there is no analogue in the domain of sample complexity that could push beyond efficient PAC learnability. As our core contribution, we fill this gap by developing a theory of parameterized PAC learning which allows us to shed new light on several recent PAC learning results that incorporated elements of parameterized complexity. Within the theory, we identify not one but two notions of fixed-parameter learnability that both form distinct counterparts to the class FPT -- the core concept at the center of the parameterized complexity paradigm -- and develop the machinery required to exclude fixed-parameter learnability. We then showcase the applications of this theory to identify refined boundaries of tractability for CNF and DNF learning as well as for a range of learning problems on graphs.

Depth completion from RGB images and sparse Time-of-Flight (ToF) measurements is an important problem in computer vision and robotics. While traditional methods for depth completion have relied on stereo vision or structured light techniques, recent advances in deep learning have enabled more accurate and efficient completion of depth maps from RGB images and sparse ToF measurements. To evaluate the performance of different depth completion methods, we organized an RGB+sparse ToF depth completion competition. The competition aimed to encourage research in this area by providing a standardized dataset and evaluation metrics to compare the accuracy of different approaches. In this report, we present the results of the competition and analyze the strengths and weaknesses of the top-performing methods. We also discuss the implications of our findings for future research in RGB+sparse ToF depth completion. We hope that this competition and report will help to advance the state-of-the-art in this important area of research. More details of this challenge and the link to the dataset can be found at //mipi-challenge.org/MIPI2023.

We consider the computations of the action ground state for a rotating nonlinear Schr\"odinger equation. It reads as a minimization of the action functional under the Nehari constraint. In the focusing case, we identify an equivalent formulation of the problem which simplifies the constraint. Based on it, we propose a normalized gradient flow method with asymptotic Lagrange multiplier and establish the energy-decaying property. Popular optimization methods are also applied to gain more efficiency. In the defocusing case, we prove that the ground state can be obtained by the unconstrained minimization. Then the direct gradient flow method and unconstrained optimization methods are applied. Numerical experiments show the convergence and accuracy of the proposed methods in both cases, and comparisons on the efficiency are discussed. Finally, the relation between the action and the energy ground states are numerically investigated.

We propose a numerical method to spline-interpolate discrete signals and then apply the integral transforms to the corresponding analytical spline functions. This represents a robust and computationally efficient technique for estimating the Laplace transform for noisy data. We revisited a Meijer-G symbolic approach to compute the Laplace transform and alternative approaches to extend canonical observed time-series. A discrete quantization scheme provides the foundation for rapid and reliable estimation of the inverse Laplace transform. We derive theoretic estimates for the inverse Laplace transform of analytic functions and demonstrate empirical results validating the algorithmic performance using observed and simulated data. We also introduce a generalization of the Laplace transform in higher dimensional space-time. We tested the discrete LT algorithm on data sampled from analytic functions with known exact Laplace transforms. The validation of the discrete ILT involves using complex functions with known analytic ILTs.

Precision medicine seeks to discover an optimal personalized treatment plan and thereby provide informed and principled decision support, based on the characteristics of individual patients. With recent advancements in medical imaging, it is crucial to incorporate patient-specific imaging features in the study of individualized treatment regimes. We propose a novel, data-driven method to construct interpretable image features which can be incorporated, along with other features, to guide optimal treatment regimes. The proposed method treats imaging information as a realization of a stochastic process, and employs smoothing techniques in estimation. We show that the proposed estimators are consistent under mild conditions. The proposed method is applied to a dataset provided by the Alzheimer's Disease Neuroimaging Initiative.

We consider evolutionary systems, i.e. systems of linear partial differential equations arising from the mathematical physics. For these systems there exists a general solution theory in exponentially weighted spaces which can be exploited in the analysis of numerical methods. The numerical method considered in this paper is a discontinuous Galerkin method in time combined with a conforming Galerkin method in space. Building on our recent paper, we improve some of the results, study the dependence of the numerical solution on the weight-parameter, consider a reformulation and post-processing of its numerical solution. As a by-product we provide error estimates for the dG-C0 method. Numerical simulations support the theoretical findings.