  In Multi-Agent Reinforcement Learning (MARL), multiple agents interact with a
common environment, as also with each other, for solving a shared problem in
sequential decision-making. It has wide-ranging applications in gaming,
robotics, finance, etc. In this work, we derive a novel law of iterated
logarithm for a family of distributed nonlinear stochastic approximation
schemes that is useful in MARL. In particular, our result describes the
convergence rate on almost every sample path where the algorithm converges.
This result is the first of its kind in the distributed setup and provides
deeper insights than the existing ones, which only discuss convergence rates in
the expected or the CLT sense. Importantly, our result holds under
significantly weaker assumptions: neither the gossip matrix needs to be doubly
stochastic nor the stepsizes square summable. As an application, we show that,
for the stepsize $n^{-\gamma}$ with $\gamma \in (0, 1),$ the distributed TD(0)
algorithm with linear function approximation has a convergence rate of
$O(\sqrt{n^{-\gamma} \ln n })$ a.s.; for the $1/n$ type stepsize, the same is
$O(\sqrt{n^{-1} \ln \ln n})$ a.s. These decay rates do not depend on the graph
depicting the interactions among the different agents.
