  Recent theoretical studies have shown that heavy-tails can emerge in
stochastic optimization due to `multiplicative noise', even under surprisingly
simple settings, such as linear regression with Gaussian data. While these
studies have uncovered several interesting phenomena, they consider
conventional stochastic optimization problems, which exclude decentralized
settings that naturally arise in modern machine learning applications. In this
paper, we study the emergence of heavy-tails in decentralized stochastic
gradient descent (DE-SGD), and investigate the effect of decentralization on
the tail behavior. We first show that, when the loss function at each
computational node is twice continuously differentiable and strongly convex
outside a compact region, the law of the DE-SGD iterates converges to a
distribution with polynomially decaying (heavy) tails. To have a more explicit
control on the tail exponent, we then consider the case where the loss at each
node is a quadratic, and show that the tail-index can be estimated as a
function of the step-size, batch-size, and the topological properties of the
network of the computational nodes. Then, we provide theoretical and empirical
results showing that DE-SGD has heavier tails than centralized SGD. We also
compare DE-SGD to disconnected SGD where nodes distribute the data but do not
communicate. Our theory uncovers an interesting interplay between the tails and
the network structure: we identify two regimes of parameters (stepsize and
network size), where DE-SGD can have lighter or heavier tails than disconnected
SGD depending on the regime. Finally, to support our theoretical results, we
provide numerical experiments conducted on both synthetic data and neural
networks.
