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 \emph{function} that belongs to a reproducing kernel Hilbert space (RKHS). To incentivize coordination while respecting network heterogeneity, we impose nonlinear proximity constraints. To solve the constrained stochastic program, we propose applying a functional variant of stochastic primal-dual (Arrow-Hurwicz) method which yields a decentralized algorithm. To handle the fact that agents' functions have complexity proportional to time (owing to the RKHS parameterization), 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), when used with constant step-sizes, yields $\mathcal{O}(\sqrt{T})$ attenuation in sub-optimality and exactly satisfies the constraints in the long run, which improves upon the state of the art rates for vector-valued problems.
翻译:我们考虑在分散的多样化网络中学习:代理商试图将集合整个网络的数据的螺旋函数最小化,而只是能够访问本地的数据流。我们关注的是代理商试图估计属于复制核心Hilbert空间(RKHS)的回归 emph{forpy} 的情况。为了激励协调,同时尊重网络的异质性,我们施加了非线性近距离限制。为了解决限制的随机程序,我们提议应用一个功能变异的随机原始(Arrow-Hurwicz)方法,该方法产生一种分散的算法。为了处理代理商的功能与时间成比例的复杂性(根据RKHS参数化),我们根据对代理商本地观测的内核评估,将原始的螺旋体放大到细小空间上。由此产生的计划,在使用恒定的步尺寸时,将产生 $\mathcal{O}sqrt{t} 。要处理一个事实,即代理商的功能与时间成比例的复杂性,我们预测将原始温度稳定在水平上,从而改善矢量的矢量控制。