[Paper Review] Graph-Dependent Implicit Regularisation for Distributed Stochastic Subgradient Descent

Dominic Richards, Patrick Rebeschini|arXiv (Cornell University)|Sep 18, 2018

Stochastic Gradient Optimization Techniques34 references4 citations

TL;DR

This paper proposes graph-dependent implicit regularization for distributed stochastic subgradient descent (Distributed SGD) in multi-agent convex optimization, achieving central-like generalization rates without explicit regularization. By tuning step sizes and using early stopping based on graph topology, the method maintains statistical guarantees up to logarithmic factors, avoiding projections or dual methods.

ABSTRACT

We propose graph-dependent implicit regularisation strategies for distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity, and smoothness, we establish statistical learning rates that retain, up to logarithmic terms, centralised statistical guarantees through implicit regularisation (step size tuning and early stopping) with appropriate dependence on the graph topology. Our approach avoids the need for explicit regularisation in decentralised learning problems, such as adding constraints to the empirical risk minimisation rule. Particularly for distributed methods, the use of implicit regularisation allows the algorithm to remain simple, without projections or dual methods. To prove our results, we establish graph-independent generalisation bounds for Distributed SGD that match the centralised setting (using algorithmic stability), and we establish graph-dependent optimisation bounds that are of independent interest. We present numerical experiments to show that the qualitative nature of the upper bounds we derive can be representative of real behaviours.

Motivation & Objective

• To close the gap in statistical generalization guarantees for decentralized, unregularized distributed learning methods.
• To establish that implicit regularization—via step size tuning and early stopping—can achieve central-like learning rates in distributed settings without explicit constraints or penalty terms.
• To derive graph-dependent optimization bounds and graph-independent generalization bounds for Distributed SGD using algorithmic stability.
• To demonstrate that the theoretical upper bounds align with empirical behavior through numerical experiments.

Proposed method

• Proposes a distributed stochastic subgradient descent algorithm where agents update locally and exchange model parameters with neighbors over a communication graph.
• Introduces graph-dependent implicit regularization by tuning the step size as $ \eta = \frac{1}{\beta + 1/\rho} $, where $ \rho $ depends on the graph's spectral gap via $ \sigma_2(P) $.
• Employs algorithmic stability to derive graph-independent generalization bounds matching central settings, even in decentralized setups.
• Uses telescoping sum techniques and subgradient boundedness to derive optimization error bounds dependent on the graph's spectral properties.
• Establishes that the choice of $ \eta $ cancels the $ \|\overline{X}^{s+1} - \overline{X}^s\| $ term, enabling tight convergence control.
• Analyzes the expected suboptimality gap $ \mathbb{E}[\overline{F}(\frac{1}{t}\sum X_v^{s+1}) - \overline{F}(X^\star)] $ under smoothness and bounded gradient noise assumptions.

Experimental results

Research questions

• RQ1 Can implicit regularization via step size tuning and early stopping achieve central-like generalization in decentralized, unregularized distributed learning?
• RQ2 How does the graph topology influence the optimization and generalization performance of Distributed SGD?
• RQ3 Can generalization bounds for Distributed SGD be derived independently of the graph structure, using algorithmic stability?
• RQ4 What is the interplay between gradient noise, step size, and convergence rate in decentralized settings?
• RQ5 Can theoretical upper bounds on error be shown to reflect real-world algorithmic behavior?

Key findings

• The method achieves a generalization error bound of the form $ \frac{\rho}{2}\sigma^2 + \frac{(\beta + 1/\rho)G^2}{2t} + \mathcal{O}\left(\frac{\log((t+1)\sqrt{n})}{1 - \sigma_2(P)}\right) $, matching central settings up to logarithmic factors.
• The choice $ \eta = \frac{1}{\beta + 1/\rho} $ cancels the $ \|\overline{X}^{s+1} - \overline{X}^s\| $ term in the optimization error, enabling tighter convergence.
• Graph-independent generalization bounds are established via algorithmic stability, showing that Distributed SGD can achieve central-like generalization without explicit regularization.
• Graph-dependent optimization bounds are derived, showing convergence rates that depend on the spectral gap $ \sigma_2(P) $ of the communication matrix $ P $.
• Numerical experiments confirm that the qualitative behavior of the theoretical upper bounds aligns with observed algorithmic performance.
• The approach avoids projections and dual methods, maintaining simplicity while achieving strong statistical guarantees in decentralized settings.