Integer-order calculus often falls short in capturing the long-range dependencies and memory effects found in many real-world processes. Fractional calculus addresses these gaps via fractional-order integrals and derivatives, but fractional-order dynamical systems pose substantial challenges in system identification and optimal control due to the lack of standard control methodologies. In this paper, we theoretically derive the optimal control via linear quadratic regulator (LQR) for fractional-order linear time-invariant (FOLTI) systems and develop an end-to-end deep learning framework based on this theoretical foundation. Our approach establishes a rigorous mathematical model, derives analytical solutions, and incorporates deep learning to achieve data-driven optimal control of FOLTI systems. Our key contributions include: (i) proposing an innovative system identification method control strategy for FOLTI systems, (ii) developing the first end-to-end data-driven learning framework, Fractional-Order Learning for Optimal Control (FOLOC), that learns control policies from observed trajectories, and (iii) deriving a theoretical analysis of sample complexity to quantify the number of samples required for accurate optimal control in complex real-world problems. Experimental results indicate that our method accurately approximates fractional-order system behaviors without relying on Gaussian noise assumptions, pointing to promising avenues for advanced optimal control.

We develop provably safe and convergent reinforcement learning (RL) algorithms for control of nonlinear dynamical systems, bridging the gap between the hard safety guarantees of control theory and the convergence guarantees of RL theory. Recent advances at the intersection of control and RL follow a two-stage, safety filter approach to enforcing hard safety constraints: model-free RL is used to learn a potentially unsafe controller, whose actions are projected onto safe sets prescribed, for example, by a control barrier function. Though safe, such approaches lose any convergence guarantees enjoyed by the underlying RL methods. In this paper, we develop a single-stage, sampling-based approach to hard constraint satisfaction that learns RL controllers enjoying classical convergence guarantees while satisfying hard safety constraints throughout training and deployment. We validate the efficacy of our approach in simulation, including safe control of a quadcopter in a challenging obstacle avoidance problem, and demonstrate that it outperforms existing benchmarks.

Given the success of model-free methods for control design in many problem settings, it is natural to ask how things will change if realistic communication channels are utilized for the transmission of gradients or policies. While the resulting problem has analogies with the formulations studied under the rubric of networked control systems, the rich literature in that area has typically assumed that the model of the system is known. As a step towards bridging the fields of model-free control design and networked control systems, we ask: \textit{Is it possible to solve basic control problems - such as the linear quadratic regulator (LQR) problem - in a model-free manner over a rate-limited channel?} Toward answering this question, we study a setting where a worker agent transmits quantized policy gradients (of the LQR cost) to a server over a noiseless channel with a finite bit-rate. We propose a new algorithm titled Adaptively Quantized Gradient Descent (\texttt{AQGD}), and prove that above a certain finite threshold bit-rate, \texttt{AQGD} guarantees exponentially fast convergence to the globally optimal policy, with \textit{no deterioration of the exponent relative to the unquantized setting}. More generally, our approach reveals the benefits of adaptive quantization in preserving fast linear convergence rates, and, as such, may be of independent interest to the literature on compressed optimization.

We study the problem of learning decentralized linear quadratic regulator when the system model is unknown a priori. We propose an online learning algorithm that adaptively designs a control policy as new data samples from a single system trajectory become available. Our algorithm design uses a disturbance-feedback representation of state-feedback controllers coupled with online convex optimization with memory and delayed feedback. We show that our controller enjoys an expected regret that scales as $\sqrt{T}$ with the time horizon $T$ for the case of partially nested information pattern. For more general information patterns, the optimal controller is unknown even if the system model is known. In this case, the regret of our controller is shown with respect to a linear sub-optimal controller. We validate our theoretical findings using numerical experiments.

This paper considers a novel multi-agent linear stochastic approximation algorithm driven by Markovian noise and general consensus-type interaction, in which each agent evolves according to its local stochastic approximation process which depends on the information from its neighbors. The interconnection structure among the agents is described by a time-varying directed graph. While the convergence of consensus-based stochastic approximation algorithms when the interconnection among the agents is described by doubly stochastic matrices (at least in expectation) has been studied, less is known about the case when the interconnection matrix is simply stochastic. For any uniformly strongly connected graph sequences whose associated interaction matrices are stochastic, the paper derives finite-time bounds on the mean-square error, defined as the deviation of the output of the algorithm from the unique equilibrium point of the associated ordinary differential equation. For the case of interconnection matrices being stochastic, the equilibrium point can be any unspecified convex combination of the local equilibria of all the agents in the absence of communication. Both the cases with constant and time-varying step-sizes are considered. In the case when the convex combination is required to be a straight average and interaction between any pair of neighboring agents may be uni-directional, so that doubly stochastic matrices cannot be implemented in a distributed manner, the paper proposes a push-sum-type distributed stochastic approximation algorithm and provides its finite-time bound for the time-varying step-size case by leveraging the analysis for the consensus-type algorithm with stochastic matrices and developing novel properties of the push-sum algorithm. Distributed temporal difference learning is discussed as an illustrative application.

We propose extrinsic and intrinsic deep neural network architectures as general frameworks for deep learning on manifolds. Specifically, extrinsic deep neural networks (eDNNs) preserve geometric features on manifolds by utilizing an equivariant embedding from the manifold to its image in the Euclidean space. Moreover, intrinsic deep neural networks (iDNNs) incorporate the underlying intrinsic geometry of manifolds via exponential and log maps with respect to a Riemannian structure. Consequently, we prove that the empirical risk of the empirical risk minimizers (ERM) of eDNNs and iDNNs converge in optimal rates. Overall, The eDNNs framework is simple and easy to compute, while the iDNNs framework is accurate and fast converging. To demonstrate the utilities of our framework, various simulation studies, and real data analyses are presented with eDNNs and iDNNs.

Adversarial examples can easily degrade the classification performance in neural networks. Empirical methods for promoting robustness to such examples have been proposed, but often lack both analytical insights and formal guarantees. Recently, some robustness certificates have appeared in the literature based on system theoretic notions. This work proposes an incremental dissipativity-based robustness certificate for neural networks in the form of a linear matrix inequality for each layer. We also propose an equivalent spectral norm bound for this certificate which is scalable to neural networks with multiple layers. We demonstrate the improved performance against adversarial attacks on a feed-forward neural network trained on MNIST and an Alexnet trained using CIFAR-10.

Recently, many cooperative distributed multi-agent reinforcement learning (MARL) algorithms have been proposed in the literature. In this work, we study the effect of adversarial attacks on a network that employs a consensus-based MARL algorithm. We show that an adversarial agent can persuade all the other agents in the network to implement policies that optimize an objective that it desires. In this sense, the standard consensus-based MARL algorithms are fragile to attacks.

The main goal of this study is to extract a set of brain networks in multiple time-resolutions to analyze the connectivity patterns among the anatomic regions for a given cognitive task. We suggest a deep architecture which learns the natural groupings of the connectivity patterns of human brain in multiple time-resolutions. The suggested architecture is tested on task data set of Human Connectome Project (HCP) where we extract multi-resolution networks, each of which corresponds to a cognitive task. At the first level of this architecture, we decompose the fMRI signal into multiple sub-bands using wavelet decompositions. At the second level, for each sub-band, we estimate a brain network extracted from short time windows of the fMRI signal. At the third level, we feed the adjacency matrices of each mesh network at each time-resolution into an unsupervised deep learning algorithm, namely, a Stacked De- noising Auto-Encoder (SDAE). The outputs of the SDAE provide a compact connectivity representation for each time window at each sub-band of the fMRI signal. We concatenate the learned representations of all sub-bands at each window and cluster them by a hierarchical algorithm to find the natural groupings among the windows. We observe that each cluster represents a cognitive task with a performance of 93% Rand Index and 71% Adjusted Rand Index. We visualize the mean values and the precisions of the networks at each component of the cluster mixture. The mean brain networks at cluster centers show the variations among cognitive tasks and the precision of each cluster shows the within cluster variability of networks, across the subjects.