Federated Learning (FL), a distributed learning paradigm that scales on-device learning collaboratively, has emerged as a promising approach for decentralized AI applications. Local optimization methods such as Federated Averaging (FedAvg) are the most prominent methods for FL applications. Despite their simplicity and popularity, the theoretical understanding of local optimization methods is far from clear. This dissertation aims to advance the theoretical foundation of local methods in the following three directions. First, we establish sharp bounds for FedAvg, the most popular algorithm in Federated Learning. We demonstrate how FedAvg may suffer from a notion we call iterate bias, and how an additional third-order smoothness assumption may mitigate this effect and lead to better convergence rates. We explain this phenomenon from a Stochastic Differential Equation (SDE) perspective. Second, we propose Federated Accelerated Stochastic Gradient Descent (FedAc), the first principled acceleration of FedAvg, which provably improves the convergence rate and communication efficiency. Our technique uses on a potential-based perturbed iterate analysis, a novel stability analysis of generalized accelerated SGD, and a strategic tradeoff between acceleration and stability. Third, we study the Federated Composite Optimization problem, which extends the classic smooth setting by incorporating a shared non-smooth regularizer. We show that direct extensions of FedAvg may suffer from the "curse of primal averaging," resulting in slow convergence. As a solution, we propose a new primal-dual algorithm, Federated Dual Averaging, which overcomes the curse of primal averaging by employing a novel inter-client dual averaging procedure.

Federated Learning (FL) is a distributed learning paradigm which scales on-device learning collaboratively and privately. Standard FL algorithms such as Federated Averaging (FedAvg) are primarily geared towards smooth unconstrained settings. In this paper, we study the Federated Composite Optimization (FCO) problem, where the objective function in FL includes an additive (possibly) non-smooth component. Such optimization problems are fundamental to machine learning and arise naturally in the context of regularization (e.g., sparsity, low-rank, monotonicity, and constraint). To tackle this problem, we propose different primal/dual averaging approaches and study their communication and computation complexities. Of particular interest is Federated Dual Averaging (FedDualAvg), a federated variant of the dual averaging algorithm. FedDualAvg uses a novel double averaging procedure, which involves gradient averaging step in standard dual averaging and an average of client updates akin to standard federated averaging. Our theoretical analysis and empirical experiments demonstrate that FedDualAvg outperforms baselines for FCO.