  Despite the popularity of policy gradient methods, they are known to suffer
from large variance and high sample complexity. To mitigate this, we introduce
SoftTreeMax -- a generalization of softmax that takes planning into account. In
SoftTreeMax, we extend the traditional logits with the multi-step discounted
cumulative reward, topped with the logits of future states. We consider two
variants of SoftTreeMax, one for cumulative reward and one for exponentiated
reward. For both, we analyze the gradient variance and reveal for the first
time the role of a tree expansion policy in mitigating this variance. We prove
that the resulting variance decays exponentially with the planning horizon as a
function of the expansion policy. Specifically, we show that the closer the
resulting state transitions are to uniform, the faster the decay. In a
practical implementation, we utilize a parallelized GPU-based simulator for
fast and efficient tree search. Our differentiable tree-based policy leverages
all gradients at the tree leaves in each environment step instead of the
traditional single-sample-based gradient. We then show in simulation how the
variance of the gradient is reduced by three orders of magnitude, leading to
better sample complexity compared to the standard policy gradient. On Atari,
SoftTreeMax demonstrates up to 5x better performance in a faster run time
compared to distributed PPO. Lastly, we demonstrate that high reward correlates
with lower variance.
