Graph-Dependent Implicit Regularisation for Distributed Stochastic Subgradient Descent
This work addresses decentralised learning problems for multi-agent systems by simplifying algorithms through implicit regularisation, though it is incremental as it builds on existing distributed SGD methods.
The paper tackles the problem of achieving centralised statistical guarantees in distributed stochastic subgradient descent for convex multi-agent learning by proposing graph-dependent implicit regularisation strategies, resulting in learning rates that retain centralised guarantees up to logarithmic terms without explicit regularisation.
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.