First Analysis of Local GD on Heterogeneous Data (1909.04715v2)

Abstract: We provide the first convergence analysis of local gradient descent for minimizing the average of smooth and convex but otherwise arbitrary functions. Problems of this form and local gradient descent as a solution method are of importance in federated learning, where each function is based on private data stored by a user on a mobile device, and the data of different users can be arbitrarily heterogeneous. We show that in a low accuracy regime, the method has the same communication complexity as gradient descent.

An Analysis of Local Gradient Descent in Federated Learning with Heterogeneous Data

The paper "First Analysis of Local GD on Heterogeneous Data" conducts an exhaustive investigation into the convergence properties of local gradient descent (GD) when applied to federated learning scenarios with heterogeneous data distributions. It addresses a crucial problem in contemporary machine learning: optimizing models where data is decentralized across multiple devices, each holding potentially distinct and private data distributions, and where communication between these devices is limited due to privacy and logistical concerns.

Convergence Analysis

The principal contribution of this paper is its theoretical analysis of local GD's convergence properties in such federated settings. The authors provide the first convergence results for local GD under these circumstances, offering insights into the communication complexity that arises. It is demonstrated that in low accuracy regimes, local GD possesses the same communication complexity as regular gradient descent.

Key Assumptions and Findings

The research assumes that each function $f_m$ is $L$-smooth and convex, providing bounds on the consistency of gradient directions across multiple data holders. The paper argues against the bounded gradient assumption common in previous studies, claiming it fails to yield meaningful convergence bounds, particularly in non-i.i.d data contexts prevalent in federated learning.

Two crucial metrics are introduced: iterate variance $(V_t)$ and a measure of dissimilarity at the optimum $(\sigma^2)$, to evaluate the convergence based on variance in local gradients. The analysis demonstrates that local GD can achieve convergence rates comparable to minibatch SGD when the data is identically distributed, with distinctions drawn between communication complexities based on the heterogeneity level of data involved.

Local GD vs Standard Methods

The paper outlines conditions under which local GD matches the communication complexity of gradient descent in achieving approximate solutions. Specifically, it finds that for accuracies above a certain threshold $(\epsilon \geq \frac{3\sigma^2}{L})$, local GD can efficiently exploit the model averaging framework to deliver analogous performance to standard GD in terms of communications, which is advantageous when rapid model convergence is required.

Furthermore, under certain circumstances, local GD demonstrates efficient handling of communication trade-offs compared to minibatch SGD, with the potential for communication savings in discrete federated learning settings through controlled synchronization intervals.

Experimental Validation

Empirical validation is conducted via logistic regression experiments over datasets demonstrating non-i.i.d partitioning, reflecting real-world federated learning challenges. The results affirm the theoretical claims, with local GD exhibiting efficient convergence in terms of communication rounds, especially when computational costs are prioritized over communication.

Implications and Future Directions

The paper provides a foundational analysis essential for advancing federated learning frameworks where local computation is preferred over costly communications. By highlighting the conditions under which local GD can be efficaciously applied, it opens avenues for further refinement of federated algorithms that reduce client-server interactions, thereby improving privacy and reducing overhead.

The paper also sets the stage for future research into optimizing communication strategies within distributed machine learning, potentially incorporating strategic client selection, diverse learning rates, and asynchronous updating to extend the utility of local GD. This exploration can guide the development of systems that address the federated learning needs in more complex settings, such as varying device capabilities and data distributions, capturing the dynamics of real-world applications more accurately.

This paper stands as an important step toward enabling practical federated learning applications where concerns of privacy, data distribution heterogeneity, and communication efficiency converge, necessitating advancements in local optimization methods.