In contrast with classical dynamic programming-based methods, our method can search over non-stationary policy functions, and can directly compute optimal infinite horizon action sequences from a given state. The central idea in our method is the construction of a mapping between infinite horizon action sequences and real numbers in a bounded interval. This construction enables us to formulate an optimization problem for directly computing optimal infinite horizon action sequences, without requiring a policy function. We demonstrate the effectiveness of our approach by applying it to nonlinear optimal control problems. Overall, our contributions provide a novel theoretical framework for formulating and solving reinforcement learning problems.

The increasing significance of digital twin technology across engineering and industrial domains, such as aerospace, infrastructure, and automotive, is undeniable. However, the lack of detailed application-specific information poses challenges to its seamless implementation in practical systems. Data-driven models play a crucial role in digital twins, enabling real-time updates and predictions by leveraging data and computational models. Nonetheless, the fidelity of available data and the scarcity of accurate sensor data often hinder the efficient learning of surrogate models, which serve as the connection between physical systems and digital twin models. To address this challenge, we propose a novel framework that begins by developing a robust multi-fidelity surrogate model, subsequently applied for tracking digital twin systems. Our framework integrates polynomial correlated function expansion (PCFE) with the Gaussian process (GP) to create an effective surrogate model called H-PCFE. Going a step further, we introduce deep-HPCFE, a cascading arrangement of models with different fidelities, utilizing nonlinear auto-regression schemes. These auto-regressive schemes effectively address the issue of erroneous predictions from low-fidelity models by incorporating space-dependent cross-correlations among the models. To validate the efficacy of the multi-fidelity framework, we first assess its performance in uncertainty quantification using benchmark numerical examples. Subsequently, we demonstrate its applicability in the context of digital twin systems.

We show that running gradient descent with variable learning rate guarantees loss $f(x) \leq 1.1 \cdot f(x^*) + \epsilon$ for the logistic regression objective, where the error $\epsilon$ decays exponentially with the number of iterations and polynomially with the magnitude of the entries of an arbitrary fixed solution $x^*$. This is in contrast to the common intuition that the absence of strong convexity precludes linear convergence of first-order methods, and highlights the importance of variable learning rates for gradient descent. We also apply our ideas to sparse logistic regression, where they lead to an exponential improvement of the sparsity-error tradeoff.

We study federated learning (FL) -- especially cross-silo FL -- with non-convex loss functions and data from people who do not trust the server or other silos. In this setting, each silo (e.g. hospital) must protect the privacy of each person's data (e.g. patient's medical record), even if the server or other silos act as adversarial eavesdroppers. To that end, we consider inter-silo record-level (ISRL) differential privacy (DP), which requires silo~$i$'s communications to satisfy record/item-level DP. We propose novel ISRL-DP algorithms for FL with heterogeneous (non-i.i.d.) silo data and two classes of Lipschitz continuous loss functions: First, we consider losses satisfying the Proximal Polyak-Lojasiewicz (PL) inequality, which is an extension of the classical PL condition to the constrained setting. In contrast to our result, prior works only considered unconstrained private optimization with Lipschitz PL loss, which rules out most interesting PL losses such as strongly convex problems and linear/logistic regression. Our algorithms nearly attain the optimal strongly convex, homogeneous (i.i.d.) rate for ISRL-DP FL without assuming convexity or i.i.d. data. Second, we give the first private algorithms for non-convex non-smooth loss functions. Our utility bounds even improve on the state-of-the-art bounds for smooth losses. We complement our upper bounds with lower bounds. Additionally, we provide shuffle DP (SDP) algorithms that improve over the state-of-the-art central DP algorithms under more practical trust assumptions. Numerical experiments show that our algorithm has better accuracy than baselines for most privacy levels. All the codes are publicly available at: https://github.com/ghafeleb/Private-NonConvex-Federated-Learning-Without-a-Trusted-Server.

In this paper, we consider the problem of learning a linear regression model on a data domain of interest (target) given few samples. To aid learning, we are provided with a set of pre-trained regression models that are trained on potentially different data domains (sources). Assuming a representation structure for the data generating linear models at the sources and the target domains, we propose a representation transfer based learning method for constructing the target model. The proposed scheme is comprised of two phases: (i) utilizing the different source representations to construct a representation that is adapted to the target data, and (ii) using the obtained model as an initialization to a fine-tuning procedure that re-trains the entire (over-parameterized) regression model on the target data. For each phase of the training method, we provide excess risk bounds for the learned model compared to the true data generating target model. The derived bounds show a gain in sample complexity for our proposed method compared to the baseline method of not leveraging source representations when achieving the same excess risk, therefore, theoretically demonstrating the effectiveness of transfer learning for linear regression.

A common approach is to apply a spectral algorithm to initialize the parameters of interest, and then run a locally convergent nonconvex optimization algorithm; alternatively, one could even run the nonconvex algorithm from a random initialization. These nonconvex optimization algorithms have been extensively analyzed under Gaussian assumptions on the covariates (see, e.g. Balakrishnan et al. (2017); Klusowski et al. (2019); Chen et al. (2019); Kwon and Caramanis (2020)), and it is known that they can exhibit favorable behavior globally in some settings. In many tasks such as ranking (Bradley and Terry, 1952), crowd-labeling (Dawid and Skene, 1979) and web search (Chen et al., 2013), however, measurements are taken as pairwise comparisons between entities, which renders the covariates inherently discrete. These designs or covariates are far from Gaussian, and it is natural to ask how standard nonconvex optimization algorithms now perform. Motivated by this question, we study how the celebrated expectation-maximization (EM) algorithm behaves when estimating a mixture of symmetric linear regressions under the pairwise comparison design.

In this study, we have developed an incremental machine learning (ML) method that efficiently obtains the optimal model when a small number of instances or features are added or removed. This problem holds practical importance in model selection, such as cross-validation (CV) and feature selection. Among the class of ML methods known as linear estimators, there exists an efficient model update framework called the low-rank update that can effectively handle changes in a small number of rows and columns within the data matrix. However, for ML methods beyond linear estimators, there is currently no comprehensive framework available to obtain knowledge about the updated solution within a specific computational complexity. In light of this, our study introduces a method called the Generalized Low-Rank Update (GLRU) which extends the low-rank update framework of linear estimators to ML methods formulated as a certain class of regularized empirical risk minimization, including commonly used methods such as SVM and logistic regression. The proposed GLRU method not only expands the range of its applicability but also provides information about the updated solutions with a computational complexity proportional to the amount of dataset changes. To demonstrate the effectiveness of the GLRU method, we conduct experiments showcasing its efficiency in performing cross-validation and feature selection compared to other baseline methods.

The availability of accurate localization is critical for multi-robot exploration strategies; noisy or inconsistent localization causes failure in meeting exploration objectives. We aim to achieve high localization accuracy with contemporary exploration map belief and vice versa without needing global localization information. This paper proposes a novel simultaneous exploration and localization (SEAL) approach, which uses Gaussian Processes (GP)-based information fusion for maximum exploration while performing communication graph optimization for relative localization. Both these cross-dependent objectives were integrated through the Rao-Blackwellization technique. Distributed linearized convex hull optimization is used to select the next-best unexplored region for distributed exploration. SEAL outperformed cutting-edge methods on exploration and localization performance in extensive ROS-Gazebo simulations, illustrating the practicality of the approach in real-world applications.

This study investigates the effects of anthropometric attributes, biological sex, and posture on translational body kinematic responses in translational vibrations. In total, 35 participants were recruited. Perturbations were applied on a standard car seat using a motion-based platform with 0.1 to 12.0 Hz random noise signals, with 0.3 m/s2 rms acceleration, for 60 seconds. Multiple linear regression models (three basic models and one advanced model, including interactions between predictors) were created to determine the most influential predictors of peak translational gains in the frequency domain per body segment (pelvis, trunk, and head). The models introduced experimentally manipulated factors (motion direction, posture, measured anthropometric attributes, and biological sex) as predictors. Effects of included predictors on the model fit were estimated. Basic linear regression models could explain over 70% of peak body segments' kinematic body response (where the R2 adjusted was 0.728). The inclusion of additional predictors (posture, body height and weight, and biological sex) did enhance the model fit, but not significantly (R2 adjusted was 0.730). The multiple stepwise linear regression, including interactions between predictors, accounted for the data well with an adjusted R2 of 0.907. The present study shows that perturbation direction and body segment kinematics are crucial factors influencing peak translational gains. Besides the body segments' response, perturbation direction was the strongest predictor. Adopted postures and biological sex do not significantly affect kinematic responses.

Most implementations of Bayesian additive regression trees (BART) one-hot encode categorical predictors, replacing each one with several binary indicators, one for every level or category. Regression trees built with these indicators partition the discrete set of categorical levels by repeatedly removing one level at a time. Unfortunately, the vast majority of partitions cannot be built with this strategy, severely limiting BART's ability to partially pool data across groups of levels. Motivated by analyses of baseball data and neighborhood-level crime dynamics, we overcame this limitation by re-implementing BART with regression trees that can assign multiple levels to both branches of a decision tree node. To model spatial data aggregated into small regions, we further proposed a new decision rule prior that creates spatially contiguous regions by deleting a random edge from a random spanning tree of a suitably defined network. Our re-implementation, which is available in the flexBART package, often yields improved out-of-sample predictive performance and scales better to larger datasets than existing implementations of BART.