We develop an asymptotic framework to compare the test performance of (personalized) federated learning algorithms whose purpose is to move beyond algorithmic convergence arguments. To that end, we study a high-dimensional linear regression model to elucidate the statistical properties (per client test error) of loss minimizers. Our techniques and model allow precise predictions about the benefits of personalization and information sharing in federated scenarios, including that Federated Averaging with simple client fine-tuning achieves identical asymptotic risk to more intricate meta-learning approaches and outperforms naive Federated Averaging. We evaluate and corroborate these theoretical predictions on federated versions of the EMNIST, CIFAR-100, Shakespeare, and Stack Overflow datasets.

In non-private stochastic convex optimization, stochastic gradient methods converge much faster on interpolation problems—namely, problems where there exists a solution that simultaneously minimizes all of the sample losses—than on non-interpolating ones; similar improvements are not known in the private setting. In this paper, we investigate differentially private stochastic optimization in the interpolation regime. First, we show that without additional assumptions, interpolation problems do not exhibit an improved convergence rates with differential privacy. However, when the functions exhibit quadratic growth around the optimum, we show (near) exponential improvements in the private sample complexity. In particular, we propose an adaptive algorithm that improves the sample complexity to achieve expected error $\alpha$ from $\frac{d}{\diffp \sqrt{\alpha}}$ to $\frac{1}{\alpha^\rho} + \frac{d}{\diffp} \log\paren{\frac{1}{\alpha}}$ for any fixed $\rho >0$, while retaining the standard minimax-optimal sample complexity for non-interpolation problems. We prove a lower bound that shows the dimension-dependent term in the expression above is tight. Furthermore, we provide a superefficiency result which demonstrates the necessity of the polynomial term for adaptive algorithms: any algorithm that has a polylogarithmic sample complexity for interpolation problems cannot achieve the minimax-optimal rates for the family of non-interpolation problems.

The Approximate-Proximal Point (APROX) family of model-based stochastic optimization algorithms improve over standard stochastic gradient methods, as they are robust to step size choices, adaptive to problem difficulty, converge on a broader range of problems than stochastic gradient methods, and converge very fast on interpolation problems, all while retaining nice minibatching properties \cite{AsiDu19siopt,AsiChChDu20}. In this paper, we propose an acceleration scheme for the APROX family and provide non-asymptotic convergence guarantees, which are order-optimal in all problem-dependent constants and provide even larger minibatching speedups. For interpolation problems where the objective satisfies additional growth conditions, we show that our algorithm achieves linear convergence rates for a wide range of stepsizes. In this setting, we also prove matching lower bounds, identifying new fundamental constants and showing the optimality of the APROX family. We corroborate our theoretical results with empirical testing to demonstrate the gains accurate modeling, acceleration, and minibatching provide.

We extend the Approximate-Proximal Point (aProx) family of model-based methods for solving stochastic convex optimization problems, including stochastic subgradient, proximal point, and bundle methods, to the minibatch setting. To do this, we propose two minibatched algorithms for which we prove a non-asymptotic upper bound on the rate of convergence, revealing a linear speedup in minibatch size. In contrast to standard stochastic gradient methods, these methods may have linear speedup in the minibatch setting even for non-smooth functions. Our algorithms maintain the desirable traits characteristic of the aProx family, such as robustness to initial step size choice. Additionally, we show improved convergence rates for "interpolation" problems, which (for example) gives a new parallelization strategy for alternating projections. We corroborate our theoretical results with extensive empirical testing, which demonstrates the gains provided by accurate modeling and minibatching.