Personalized Federated Learning via Variance-Aware Nonparametric Empirical Bayes

We develop a new approach to Personalized Federated Learning across heterogeneous clients using Nonparametric Empirical Bayes (NPEB). Leveraging the asymptotic normality of local parameter estimates obtained from Empirical Risk Minimization or M-estimation, our method formulates these estimates as noisy observations to estimate an unknown shared prior via Nonparametric Maximum Likelihood. A key challenge in applying NPEB in this setting is that existing approaches assume known fixed variances, which is not true in practice. To address this, we introduce a Variance-Aware Nonparametric Empirical Bayes (VANEB) framework that leverages the parameter-dependent asymptotic variance of local M-estimators. A key technical contribution is a generalized Tweedie's formula for this heteroskedastic setting. We then establish non-asymptotic error rates for density estimation in the average squared Hellinger distance and derive an oracle denoising inequality that provides error bounds for our estimator. While our theoretical guarantees are rooted in the asymptotic regime of M-estimators, we empirically explore heuristic extensions of VANEB to modern federated learning settings involving Deep Neural Networks (DNNs). For DNNs, we propose VANEB-head and VANEB-FT, which personalize the last fully connected layer via an NPEB step using an approximate diagonal variance estimator. We show that our method has strong performance on popular vision datasets MNIST and CIFAR-10, using a convolutional neural network architecture.

Heterogeneous transfer learning for high dimensional regression with feature mismatch

We consider the problem of transferring knowledge from a source, or proxy, domain to a new target domain for learning a high-dimensional regression model with possibly different features. Recently, the statistical properties of homogeneous transfer learning have been investigated. However, most homogeneous transfer and multi-task learning methods assume that the target and proxy domains have the same feature space, limiting their practical applicability. In applications, target and proxy feature spaces are frequently inherently different, for example, due to the inability to measure some variables in the target data-poor environments. Conversely, existing heterogeneous transfer learning methods do not provide statistical error guarantees, limiting their utility for scientific discovery. We propose a two-stage method that involves learning the relationship between the missing and observed features through a projection step in the proxy data and then solving a joint penalized regression optimization problem in the target data. We develop an upper bound on the method's parameter estimation risk and prediction risk, assuming that the proxy and the target domain parameters are sparsely different. Our results elucidate how estimation and prediction error depend on the complexity of the model, sample size, the extent of overlap, and correlation between matched and mismatched features.