Adaptive Kernel Learning in Heterogeneous Networks

Abstract

We consider learning in decentralized heterogeneous networks: agents seek to minimize a convex functional that aggregates data across the network, while only having access to their local data streams. We focus on the case where agents seek to estimate a regression function that belongs to a reproducing kernel Hilbert space (RKHS). To incentivize coordination while respecting network heterogeneity, we impose nonlinear proximity constraints. The resulting constrained stochastic optimization problem is solved using the functional variant of stochastic primal-dual (Arrow-Hurwicz) method which yields a decentralized algorithm. In order to avoid the model complexity from growing linearly over time, we project the primal iterates onto subspaces greedily constructed from kernel evaluations of agents' local observations. The resulting scheme, dubbed Heterogeneous Adaptive Learning with Kernels (HALK), allows us, for the first time, to characterize the precise trade-off between the optimality gap, constraint violation, and the model complexity. In particular, the proposed algorithm can be tuned to achieve zero constraint violation, an optimality gap of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (T <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> +α) after T iterations, where the number of elements retained in the dictionary is determined by 1/α. Simulations on a correlated spatio-temporal field estimation validate our theoretical results, which are corroborated in practice for networked oceanic sensing buoys estimating temperature and salinity from depth measurements.