  Restless bandits are an important class of problems with applications in
recommender systems, active learning, revenue management and other areas. We
consider infinite-horizon discounted restless bandits with many arms where a
fixed proportion of arms may be pulled in each period and where arms share a
finite state space. Although an average-case-optimal policy can be computed via
stochastic dynamic programming, the computation required grows exponentially
with the number of arms $N$. Thus, it is important to find scalable policies
that can be computed efficiently for large $N$ and that are near optimal in
this regime, in the sense that the optimality gap (i.e. the loss of expected
performance against an optimal policy) per arm vanishes for large $N$. However,
the most popular approach, the Whittle index, requires a hard-to-verify
indexability condition to be well-defined and another hard-to-verify condition
to guarantee a $o(N)$ optimality gap. We present a method resolving these
difficulties. By replacing a global Lagrange multiplier used by the Whittle
index with a sequence of Lagrangian multipliers, one per time period up to a
finite truncation point, we derive a class of policies, called fluid-balance
policies, that have a $O(\sqrt{N})$ optimality gap. Unlike the Whittle index,
fluid-balance policies do not require indexability to be well-defined and their
$O(\sqrt{N})$ optimality gap bound holds universally without sufficient
conditions. We also demonstrate empirically that fluid-balance policies provide
state-of-the-art performance on specific problems.
