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We consider the problem of learning a realization of a partially observed bilinear dynamical system (BLDS) from noisy input-output data. Given a single trajectory of input-output samples, we provide an algorithm and a finite time analysis for learning the system’s Markov-like parameters, from which a balanced realization of the bilinear system can be obtained. The stability of BLDS depends on the sequence of inputs used to excite the system. Moreover, our identification algorithm regresses the outputs to highly correlated, nonlinear, and heavy-tailed covariates. These properties, unique to partially observed bilinear dynamical systems, pose significant challenges to the analysis of our algorithm for learning the unknown dynamics. We address these challenges and provide high probability error bounds on our identification algorithm under a uniform stability assumption. Our analysis provides insights into system theoretic quantities that affect learning accuracy and sample complexity. Lastly, we perform numerical experiments with synthetic data to reinforce these insights. 

This paper investigates ML systems serving a group of users, with multiple models/services, each aimed at specializing to a sub-group of users. We consider settings where upon deploying a set of services, users choose the one minimizing their personal losses and the learner iteratively learns by interacting with diverse users. Prior research shows that the outcomes of learning dynamics, which comprise both the services' adjustments and users' service selections, hinge significantly on the initial conditions. However, finding good initial conditions faces two main challenges:(i)\emph {Bandit feedback:} Typically, data on user preferences are not available before deploying services and observing user behavior;(ii)\emph {Suboptimal local solutions:} The total loss landscape (ie, the sum of loss functions across all users and services) is not convex and gradient-based algorithms can get stuck in poor local minima. We address these challenges with a randomized algorithm to adaptively select a minimal set of users for data collection in order to initialize a set of services. Under mild assumptions on the loss functions, we prove that our initialization leads to a total loss within a factor of the\textit {globally optimal total loss, with complete user preference data}, and this factor scales logarithmically in the number of services. This result is a generalization of the well-known k-means++ guarantee to a broad problem class which is also of independent interest. The theory is complemented by experiments on real as well as semi-synthetic datasets.