  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.
