The optimization foundations of deep linear networks have received significant attention lately. However, due to the non-convexity and hierarchical structure, analyzing the regularized loss of deep linear networks remains a challenging task. In this work, we study the local geometric landscape of the regularized squared loss of deep linear networks, providing a deeper understanding of its optimization properties. Specifically, we characterize the critical point set and establish an error-bound property for all critical points under mild conditions. Notably, we identify the sufficient and necessary conditions under which the error bound holds. To support our theoretical findings, we conduct numerical experiments demonstrating that gradient descent exhibits linear convergence when optimizing the regularized loss of deep linear networks.

This paper considers a class of noncooperative games in which the feasible decision sets of all players are coupled together by a coupled inequality constraint. Adopting the variational inequality formulation of the game, we first introduce a new local edge-based equilibrium condition and develop a distributed primal-dual proximal algorithm with full information. Considering challenges when communication delays occur, we devise an asynchronous distributed algorithm to seek a generalized Nash equilibrium. This asynchronous scheme arbitrarily activates one player to start new computations independently at different iteration instants, which means that the picked player can use the involved out-dated information from itself and its neighbors to perform new updates. A distinctive attribute is that the proposed algorithms enable the derivation of new distributed forward-backward-like extensions. In theoretical aspect, we provide explicit conditions on algorithm parameters, for instance, the step-sizes to establish a sublinear convergence rate for the proposed synchronous algorithm. Moreover, the asynchronous algorithm guarantees almost sure convergence in expectation under the same step-size conditions and some standard assumptions. An interesting observation is that our analysis approach improves the convergence rate of prior synchronous distributed forward-backward-based algorithms. Finally, the viability and performance of the proposed algorithms are demonstrated by numerical studies on the networked Cournot competition.

We present a simple proof for bounding the smallest eigenvalue of the empirical covariance in a causal Gaussian process. Along the way, we establish a one-sided tail inequality for Gaussian quadratic forms using a causal decomposition. Our proof only uses elementary facts about the Gaussian distribution and the union bound. We conclude with an example in which we provide a performance guarantee for least squares identification of a vector autoregression.

This tutorial serves as an introduction to recently developed non-asymptotic methods in the theory of -- mainly linear -- system identification. We emphasize tools we deem particularly useful for a range of problems in this domain, such as the covering technique, the Hanson-Wright Inequality and the method of self-normalized martingales. We then employ these tools to give streamlined proofs of the performance of various least-squares based estimators for identifying the parameters in autoregressive models. We conclude by sketching out how the ideas presented herein can be extended to certain nonlinear identification problems.

This paper studies the distributed resilient output containment control of heterogeneous multiagent systems against composite attacks, including denial-of-services (DoS) attacks, false-data injection (FDI) attacks, camouflage attacks, and actuation attacks. Inspired by digital twins, a twin layer (TL) with higher security and privacy is used to decouple the above problem into two tasks: defense protocols against DoS attacks on TL and defense protocols against actuation attacks on cyber-physical layer (CPL). First, considering modeling errors of leader dynamics, we introduce distributed observers to reconstruct the leader dynamics for each follower on TL under DoS attacks. Second, distributed estimators are used to estimate follower states according to the reconstructed leader dynamics on the TL. Third, according to the reconstructed leader dynamics, we design decentralized solvers that calculate the output regulator equations on CPL. Fourth, decentralized adaptive attack-resilient control schemes that resist unbounded actuation attacks are provided on CPL. Furthermore, we apply the above control protocols to prove that the followers can achieve uniformly ultimately bounded (UUB) convergence, and the upper bound of the UUB convergence is determined explicitly. Finally, two simulation examples are provided to show the effectiveness of the proposed control protocols.