ProxSkip: Yes! Local Gradient Steps Provably Lead to Communication Acceleration! Finally! (2202.09357v2)

Abstract: We introduce ProxSkip -- a surprisingly simple and provably efficient method for minimizing the sum of a smooth ($f$) and an expensive nonsmooth proximable ($\psi$) function. The canonical approach to solving such problems is via the proximal gradient descent (ProxGD) algorithm, which is based on the evaluation of the gradient of $f$ and the prox operator of $\psi$ in each iteration. In this work we are specifically interested in the regime in which the evaluation of prox is costly relative to the evaluation of the gradient, which is the case in many applications. ProxSkip allows for the expensive prox operator to be skipped in most iterations: while its iteration complexity is $\mathcal{O}\left(\kappa \log \frac{1}{\varepsilon}\right)$, where $\kappa$ is the condition number of $f$, the number of prox evaluations is $\mathcal{O}\left(\sqrt{\kappa} \log \frac{1}{\varepsilon}\right)$ only. Our main motivation comes from federated learning, where evaluation of the gradient operator corresponds to taking a local GD step independently on all devices, and evaluation of prox corresponds to (expensive) communication in the form of gradient averaging. In this context, ProxSkip offers an effective acceleration of communication complexity. Unlike other local gradient-type methods, such as FedAvg, SCAFFOLD, S-Local-GD and FedLin, whose theoretical communication complexity is worse than, or at best matching, that of vanilla GD in the heterogeneous data regime, we obtain a provable and large improvement without any heterogeneity-bounding assumptions.

• The paper introduces ProxSkip, a technique that leverages proximal methods to reduce communication rounds in federated learning.
• It establishes theoretical bounds on expected error and convergence rates through recursive inequalities under strong convexity.
• The approach significantly improves convergence in heterogeneous data settings, optimizing client-server communication efficiency.

ProxSkip: An Effective Communication-Acceleration Technique for Federated Learning

The paper presents "ProxSkip," a communication-acceleration technique specifically designed for federated learning. Federated learning presents distinct challenges due to the frequent need for communication between clients and a central server, which can introduce inefficiencies. ProxSkip aims to mitigate this bottleneck by incorporating proximal point methods, ensuring more efficient and faster convergence while reducing communication overhead.

Methodology and Key Findings

ProxSkip introduces a communication-efficient algorithm that amalgamates ideas from proximal methods and adaptive updates. The core principle revolves around leveraging local updates more effectively and skipping certain immediate communication rounds when the local models are progressing satisfactorily.

The authors provide an in-depth theoretical analysis of ProxSkip's performance. They include detailed derivations of certain lemmas and propositions that underpin the method's efficacy. A key aspect of the analysis is the introduction of mathematical expressions that provide bounds on the expected error, which demonstrates ProxSkip's capability to reduce this error rate efficiently compared to traditional methods.

Analytical Insights

The paper derives multiple recursive inequalities that elucidate the behavior of the expected error and consensus deviation over iterations. The authors analyze the convergence properties using stochastic analysis and iterative updates grounded in strong convexity and smoothness conditions. The derived recurrence relations reveal that ProxSkip achieves improved convergence rates, specifically in settings where the client data is highly heterogeneous.

Moreover, the authors develop a probabilistic framework that allows local updates to either follow a typical stochastic gradient descent-like path or skip particular updates, based on a probability parameter $p$. This adaptability enables the algorithm to control the balance between computation and communication.

Practical Implications

The communication reduction realized by ProxSkip has profound implications for federated learning environments, particularly those constrained by limited communication bandwidth. In collaborative scenarios, such as those found in mobile networks or distributed sensor networks, efficient communication is paramount. ProxSkip's ability to minimize communication without compromising convergence can lead to enhanced deployment in such contexts.

Theoretical Implications and Future Directions

The theoretical contributions of this paper extend to the broader understanding of combining proximal algorithms with skipping strategies in distributed contexts. The adaptive nature of ProxSkip suggests that similar techniques could be further explored and potentially integrated with other optimization frameworks beyond federated learning.

Future research could focus on refining the probabilistic model governing the "skip" decisions, potentially incorporating dynamic learning algorithms that adjust the probability parameter $p$ in response to the observed data distribution and model behavior. Additionally, exploring the application of ProxSkip in asynchronous settings where communication delays are non-uniform could further enhance its practical utility.

In conclusion, this paper offers a novel approach to tackle communication inefficiencies in federated learning through ProxSkip. Its blend of theoretical rigor and practical considerations sets a precedent for future exploration and elaboration in distributed optimization algorithms. The insights gained from this work may pave the way for more resilient and efficient federated learning protocols in diverse application domains.