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We study a model-free federated linear quadratic regulator (LQR) problem where M agents with unknown, distinct yet similar dynamics collaboratively learn an optimal policy to minimize an average quadratic cost while keeping their data private. To exploit the similarity of the agents' dynamics, we propose to use federated learning (FL) to allow the agents to periodically communicate with a central server to train policies by leveraging a larger dataset from all the agents. With this setup, we seek to understand the following questions: (i) Is the learned common policy stabilizing for all agents? (ii) How close is the learned common policy to each agent's own optimal policy? (iii) Can each agent learn its own optimal policy faster by leveraging data from all agents? To answer these questions, we propose a federated and model-free algorithm named FedLQR. Our analysis overcomes numerous technical challenges, such as heterogeneity in the agents' dynamics, multiple local updates, and stability concerns. We show that FedLQR produces a common policy that, at each iteration, is stabilizing for all agents. We provide bounds on the distance between the common policy and each agent's local optimal policy. Furthermore, we prove that when learning each agent's optimal policy, FedLQR achieves a sample complexity reduction proportional to the number of agents M in a low-heterogeneity regime, compared to the single-agent setting. 

Context.Light bridges are bright, long, and narrow features that are typically connected to the formation or decay processes of sunspots and pores.

Aims.The interaction of magnetic fields and plasma flows is investigated in the trailing part of an active region, where pores and magnetic knots evolve into a complex sunspot. The goal is to identify the photospheric and chromospheric processes, which transform the mainly vertical magnetic fields of pores into a sunspot with multiple umbral cores, light bridges, and rudimentary penumbrae.

Methods.Conducting observations with a broad variety of telescopes and instruments provides access to different atmospheric layers and the changing morphology of features connected to strong magnetic fields. While the Helioseismic and Magnetic Imager (HMI) of the Solar Dynamics Observatory (SDO) provides full-disk continuum images and line-of-sight magnetograms, the fine structure and flows around a pore can be deduced from high-resolution observations in various wavelengths as provided by theGoodeSolar Telescope (GST) at the Big Bear Solar Observatory (BBSO). Horizontal proper motions are evaluated applying local correlation tracking (LCT) to the available time series, whereas the connectivity of sunspot features can be established using the background-subtracted activity maps (BaSAMs).

Results.Photospheric flow maps indicate radial outflows, where the light bridge connects to the surrounding granulation, whereas inflows are present at the border of the pores. In contrast, the chromospheric flow maps show strong radial outflows at superpenumbral scales, even in the absence of a penumbra in the photosphere. The region in between the two polarities is characterized by expanding granules creating strong divergence centers. Variations in BaSAMs follow locations of significant and persistent changes in and around pores. The resulting maps indicate low variations along the light bridge, as well as thin hairlines connecting the light bridge to the pores and strong variations at the border of pores. Various BaSAMs demonstrate the interaction of pores with the surrounding supergranular cell. The Hαline-of-sight velocity maps provide further insights into the flow structure, with twisted motions along some of the radial filaments around the pore with the light bridge. Furthermore, flows along filaments connecting the two polarities of the active region are pronounced in the line-of-sight velocity maps.

Conclusions.The present observations reveal that even small-scale changes of plasma motions in and around pores are conducive to transform pores into sunspots. In addition, chromospheric counterparts of penumbral filaments appear much earlier than the penumbral filaments in the photosphere. Penumbra formation is aided by a stable magnetic feature that anchors the advection of magnetic flux and provides a connection to the surrounding supergranular cell, whereas continuously emerging flux and strong light bridges are counteragents that affect the appearance and complexity of sunspots and their penumbrae.

 

We study the problem of learning a linear system model from the observations of M clients. The catch: Each client is observing data from a different dynamical system. This work addresses the question of how multiple clients collaboratively learn dynamical models in the presence of heterogeneity. We pose this problem as a federated learning problem and characterize the tension between achievable performance and system heterogeneity. Furthermore, our federated sample complexity result provides a constant factor improvement over the single agent setting. Finally, we describe a meta federated learning algorithm, FedSysID, that leverages existing federated algorithms at the client level. 

We address the problem of learning linear system models from observing multiple trajectories from different system dynamics. This framework encompasses a collaborative scenario where several systems seeking to estimate their dynamics are partitioned into clusters according to their system similarity. Thus, the systems within the same cluster can benefit from the observations made by the others. Considering this framework, we present an algorithm where each system alternately estimates its cluster identity and performs an estimation of its dynamics. This is then aggregated to update the model of each cluster. We show that under mild assumptions, our algorithm correctly estimates the cluster identities and achieves an approximate sample complexity that scales inversely with the number of systems in the cluster, thus facilitating a more efficient and personalized system identification process.