Abstract-- This paper presents a novel framework for analyzing Incremental-Input-to-State Stability (δISS) based on the idea of using rewards as "test functions." Whereas control theory traditionally deals with Lyapunov functions that satisfy a time-decrease condition, reinforcement learning (RL) value functions are constructed by exponentially decaying a Lipschitz reward function that may be non-smooth and unbounded on both sides. Thus, these RL-style value functions cannot be directly understood as Lyapunov certificates. We develop a new equivalence between a variant of incremental input-to-state stability of a closed-loop system under given a policy, and the regularity of RL-style value functions under adversarial selection of a H older-continuous reward function. This result highlights that the regularity of value functions, and their connection to incremental stability, can be understood in a way that is distinct from the traditional Lyapunov-based approach to certifying stability in control theory.

Contextual bandit is a class of online learning problems that can be viewed as a simple reinforcement learning problem without transition. For a completely understanding of contextual bandit problems, we refer the readers to the Chapter 4 of [Bubeck et al., 2012]. Here we include the main idea for completeness. In contextual bandit problems, the agent needs to find out the best action given some observed context (a.k.a the optimal policy in reinforcement learning). Formally, we define S as the context set and K as the number of action.

In the face of evolving cyber threats such as malware, ransomware and phishing, autonomous cybersecurity defense (ACD) systems have become essential for real-time threat detection and response with optional human intervention. However, existing ACD systems rely on limiting assumptions, particularly the stationarity of the underlying network dynamics. In real-world scenarios, network topologies can change due to actions taken by attackers or defenders, system failures, or time evolution of networks, leading to failures in the adaptive capabilities of current defense agents. Moreover, many agents are trained on static environments, resulting in overfitting to specific topologies, which hampers their ability to generalize to out-of-distribution network topologies. This work addresses these challenges by exploring methods for developing agents to learn generalizable policies across dynamic network environments -- general ACD (GACD).

Stochastic approximation (SA) is an iterative algorithm to find the fixed point of an operator given noisy samples of this operator. SA appears in many areas such as optimization and Reinforcement Learning (RL). When implemented in practice, the noise that appears in the update of RL algorithms is naturally Markovian. Furthermore, in some settings, such as gradient TD, SA is employed in a two-time-scale manner. The mix of Markovian noise along with the two-time-scale structure results in an algorithm which is complex to analyze theoretically. In this paper, we characterize a tight convergence bound for the iterations of linear two-time-scale SA with Markovian noise. Our results show the convergence behavior of this algorithm given various choices of step sizes. Applying our result to the well-known TDC algorithm, we show the first $O(1/\epsilon)$ sample complexity for the convergence of this algorithm, outperforming all the previous work. Similarly, our results can be applied to establish the convergence behavior of a variety of RL algorithms, such as TD-learning with Polyak averaging, GTD, and GTD2.

This research focuses on the estimation of a non-parametric regression function designed for data with simultaneous time and space dependencies. In such a context, we study the Trend Filtering, a nonparametric estimator introduced by \cite{mammen1997locally} and \cite{rudin1992nonlinear}. For univariate settings, the signals we consider are assumed to have a kth weak derivative with bounded total variation, allowing for a general degree of smoothness. In the multivariate scenario, we study a $K$-Nearest Neighbor fused lasso estimator as in \cite{padilla2018adaptive}, employing an ADMM algorithm, suitable for signals with bounded variation that adhere to a piecewise Lipschitz continuity criterion. By aligning with lower bounds, the minimax optimality of our estimators is validated. A unique phase transition phenomenon, previously uncharted in Trend Filtering studies, emerges through our analysis. Both Simulation studies and real data applications underscore the superior performance of our method when compared with established techniques in the existing literature.

Energy cost is increasingly crucial in the modern computing industry with the wide deployment of large-scale machine learning models and language models. For the firms that provide computing services, low energy consumption is important both from the perspective of their own market growth and the government's regulations. In this paper, we study the energy benefits of quantum computing vis-a-vis classical computing. Deviating from the conventional notion of quantum advantage based solely on computational complexity, we redefine advantage in an energy efficiency context. Through a Cournot competition model constrained by energy usage, we demonstrate quantum computing firms can outperform classical counterparts in both profitability and energy efficiency at Nash equilibrium. Therefore quantum computing may represent a more sustainable pathway for the computing industry. Moreover, we discover that the energy benefits of quantum computing economies are contingent on large-scale computation. Based on real physical parameters, we further illustrate the scale of operation necessary for realizing this energy efficiency advantage.

Perfect synchronization in distributed machine learning problems is inefficient and even impossible due to the existence of latency, package losses and stragglers. We propose a Robust Fully-Asynchronous Stochastic Gradient Tracking method (R-FAST), where each device performs local computation and communication at its own pace without any form of synchronization. Different from existing asynchronous distributed algorithms, R-FAST can eliminate the impact of data heterogeneity across devices and allow for packet losses by employing a robust gradient tracking strategy that relies on properly designed auxiliary variables for tracking and buffering the overall gradient vector. More importantly, the proposed method utilizes two spanning-tree graphs for communication so long as both share at least one common root, enabling flexible designs in communication architectures. We show that R-FAST converges in expectation to a neighborhood of the optimum with a geometric rate for smooth and strongly convex objectives; and to a stationary point with a sublinear rate for general non-convex settings. Extensive experiments demonstrate that R-FAST runs 1.5-2 times faster than synchronous benchmark algorithms, such as Ring-AllReduce and D-PSGD, while still achieving comparable accuracy, and outperforms existing asynchronous SOTA algorithms, such as AD-PSGD and OSGP, especially in the presence of stragglers.

A minimum dominating set in a graph is a minimum set of vertices such that every vertex of the graph either belongs to it, or is adjacent to one vertex of this set. This mathematical object is of high relevance in a number of applications related to social networks analysis, design of wireless networks, coding theory, and data mining, among many others. When vertex weights are given, minimizing the total weight of the dominating set gives rise to a problem variant known as the minimum weight dominating set problem. To solve this problem, we introduce a hybrid matheuristic combining a tabu search with an integer programming solver. The latter is used to solve subproblems in which only a fraction of the decision variables, selected relatively to the search history, are left free while the others are fixed. Moreover, we introduce an adaptive penalty to promote the exploration of intermediate infeasible solutions during the search, enhance the algorithm with perturbations and node elimination procedures, and exploit richer neighborhood classes. Extensive experimental analyses on a variety of instance classes demonstrate the good performance of the algorithm, and the contribution of each component in the success of the search is analyzed.

The Minimum Weight Dominating Set (MWDS) problem is an important generalization of the Minimum Dominating Set (MDS) problem with extensive applications. This paper proposes a new local search algorithm for the MWDS problem, which is based on two new ideas. The first idea is a heuristic called two-level configuration checking (CC2), which is a new variant of a recent powerful configuration checking strategy (CC) for effectively avoiding the recent search paths. The second idea is a novel scoring function based on the frequency of being uncovered of vertices. Our algorithm is called CC2FS, according to the names of the two ideas. The experimental results show that, CC2FS performs much better than some state-of-the-art algorithms in terms of solution quality on a broad range of MWDS benchmarks.