Sparse multiresolution representations with adaptive kernels

Reproducing kernel Hilbert spaces (RKHSs) are key elements of many non-parametric tools successfully used in signal processing, statistics, and machine learning. In this work, we aim to address three issues of the classical RKHS based techniques. First, they require the RKHS to be known a priori, which is unrealistic in many applications. Furthermore, the choice of RKHS affects the shape and smoothness of the solution, thus impacting its performance. Second, RKHSs are ill-equipped to deal with heterogeneous degrees of smoothness, i.e., with functions that are smooth in some parts of their domain but vary rapidly in others. Finally, the computational complexity of evaluating the solution of these methods grows with the number of data points, rendering these techniques infeasible for many applications. Though kernel learning, local kernel adaptation, and sparsity have been used to address these issues, many of these approaches are computationally intensive or forgo optimality guarantees. We tackle these problems by leveraging a novel integral representation of functions in RKHSs that allows for arbitrary centers and different kernels at each center. To address the complexity issues, we then write the function estimation problem as a sparse functional program that explicitly minimizes the support of the representation leading to low complexity solutions. Despite their non-convexity and infinite dimensionality, we show these problems can be solved exactly and efficiently by leveraging duality, and we illustrate this new approach in simulated and real data

Optimally Compressed Nonparametric Online Learning

Batch training of machine learning models based on neural networks is now well established, whereas to date streaming methods are largely based on linear models. To go beyond linear in the online setting, nonparametric methods are of interest due to their universality and ability to stably incorporate new information via convexity or Bayes' Rule. Unfortunately, when used online, nonparametric methods suffer a "curse of dimensionality" which precludes their use: their complexity scales at least with the time index. We survey online compression tools which bring their memory under control and attain approximate convergence. The asymptotic bias depends on a compression parameter that trades off memory and accuracy. Further, the applications to robotics, communications, economics, and power are discussed, as well as extensions to multi-agent systems

Federated Classification using Parsimonious Functions in Reproducing Kernel Hilbert Spaces

Federated learning forms a global model using data collected from a federation agent. This type of learning has two main challenges: the agents generally don't collect data over the same distribution, and the agents have limited capabilities of storing and transmitting data. Therefore, it is impractical for each agent to send the entire data over the network. Instead, each agent must form a local model and decide what information is fundamental to the learning problem, which will be sent to a central unit. The central unit can then form the global model using only the information received from the agents. We propose a method that tackles these challenges. First each agent forms a local model using a low complexity reproducing kernel Hilbert space representation. From the model the agents identify the fundamental samples which are sent to the central unit. The fundamental samples are obtained by solving the dual problem. The central unit then forms the global model. We show that the solution of the federated learner converges to that of the centralized learner asymptotically as the sample size increases. The performance of the proposed algorithm is evaluated using experiments with both simulated data and real data sets from an activity recognition task, for which the data is collected from a wearable device. The experimentation results show that the accuracy of our method converges to that of a centralized learner with increasing sample size