  We consider infinite-horizon $\gamma$-discounted Markov Decision Processes,
for which it is known that there exists a stationary optimal policy. We
consider the algorithm Value Iteration and the sequence of policies
$\pi_1,...,\pi_k$ it implicitely generates until some iteration $k$. We provide
performance bounds for non-stationary policies involving the last $m$ generated
policies that reduce the state-of-the-art bound for the last stationary policy
$\pi_k$ by a factor $\frac{1-\gamma}{1-\gamma^m}$. In particular, the use of
non-stationary policies allows to reduce the usual asymptotic performance
bounds of Value Iteration with errors bounded by $\epsilon$ at each iteration
from $\frac{\gamma}{(1-\gamma)^2}\epsilon$ to
$\frac{\gamma}{1-\gamma}\epsilon$, which is significant in the usual situation
when $\gamma$ is close to 1. Given Bellman operators that can only be computed
with some error $\epsilon$, a surprising consequence of this result is that the
problem of "computing an approximately optimal non-stationary policy" is much
simpler than that of "computing an approximately optimal stationary policy",
and even slightly simpler than that of "approximately computing the value of
some fixed policy", since this last problem only has a guarantee of
$\frac{1}{1-\gamma}\epsilon$.
