  Unifying seemingly disparate algorithmic ideas to produce better performing
algorithms has been a longstanding goal in reinforcement learning. As a primary
example, TD($\lambda$) elegantly unifies one-step TD prediction with Monte
Carlo methods through the use of eligibility traces and the trace-decay
parameter $\lambda$. Currently, there are a multitude of algorithms that can be
used to perform TD control, including Sarsa, $Q$-learning, and Expected Sarsa.
These methods are often studied in the one-step case, but they can be extended
across multiple time steps to achieve better performance. Each of these
algorithms is seemingly distinct, and no one dominates the others for all
problems. In this paper, we study a new multi-step action-value algorithm
called $Q(\sigma)$ which unifies and generalizes these existing algorithms,
while subsuming them as special cases. A new parameter, $\sigma$, is introduced
to allow the degree of sampling performed by the algorithm at each step during
its backup to be continuously varied, with Sarsa existing at one extreme (full
sampling), and Expected Sarsa existing at the other (pure expectation).
$Q(\sigma)$ is generally applicable to both on- and off-policy learning, but in
this work we focus on experiments in the on-policy case. Our results show that
an intermediate value of $\sigma$, which results in a mixture of the existing
algorithms, performs better than either extreme. The mixture can also be varied
dynamically which can result in even greater performance.
