But how to explain this phenomenon has been largely ignored by existing studies.

Given these considerations, we split our analysis to the case wherebq = s (referred to as the nonbottleneck case) and wherebq = min(M1,,ML 1)(referred to as the bottleneck case).

Consequently,convergenceproofsare more challenging and require aprecise control ofthis perturbation.

The goal of federated learning is to design algorithms in which several agents communicate with a central node, in a privacy-protecting manner, to minimize the average of their loss functions. In this approach, each node not only shares the required computational budget but also has access to a larger data set, which improves the quality of the resulting model. However, this method only develops a common output for all the agents, and therefore, does not adapt the model to each user data. This is an important missing feature especially given the heterogeneity of the underlying data distribution for various agents. In this paper, we study a personalized variant of the federated learning in which our goal is to find a shared initial model in a distributed manner that can be slightly updated by either a current or a new user by performing one or a few steps of gradient descent with respect to its own loss function. This approach keeps all the benefits of the federated learning architecture while leading to a more personalized model for each user. We show this problem can be studied within the Model-Agnostic Meta-Learning (MAML) framework. Inspired by this connection, we propose a personalized variant of the well-known Federated Averaging algorithm and evaluate its performance in terms of gradient norm for non-convex loss functions. Further, we characterize how this performance is affected by the closeness of underlying distributions of user data, measured in terms of distribution distances such as Total Variation and 1-Wasserstein metric.

Over-parameterization is ubiquitous nowadays in training neural networks to benefit both optimization in seeking global optima and generalization in reducing prediction error. However, compressive networks are desired in many real world applications and direct training of small networks may be trapped in local optima. In this paper, instead of pruning or distilling an over-parameterized model to compressive ones, we propose a parsimonious learning approach based on differential inclusions of inverse scale spaces, that generates a family of models from simple to complex ones with a better efficiency and interpretability than stochastic gradient descent in exploring the model space. It enjoys a simple discretization, the Split Linearized Bregman Iterations, with provable global convergence that from any initializations, algorithmic iterations converge to a critical point of empirical risks. One may exploit the proposed method to boost the complexity of neural networks progressively. Numerical experiments with MNIST, Cifar-10/100, and ImageNet are conducted to show the method is promising in training large scale models with a favorite interpretability.