  Off-policy reinforcement learning with eligibility traces is challenging
because of the discrepancy between target policy and behavior policy. One
common approach is to measure the difference between two policies in a
probabilistic way, such as importance sampling and tree-backup. However,
existing off-policy learning methods based on probabilistic policy measurement
are inefficient when utilizing traces under a greedy target policy, which is
ineffective for control problems. The traces are cut immediately when a
non-greedy action is taken, which may lose the advantage of eligibility traces
and slow down the learning process. Alternatively, some non-probabilistic
measurement methods such as General Q($\lambda$) and Naive Q($\lambda$) never
cut traces, but face convergence problems in practice. To address the above
issues, this paper introduces a new method named TBQ($\sigma$), which
effectively unifies the tree-backup algorithm and Naive Q($\lambda$). By
introducing a new parameter $\sigma$ to illustrate the \emph{degree} of
utilizing traces, TBQ($\sigma$) creates an effective integration of
TB($\lambda$) and Naive Q($\lambda$) and continuous role shift between them.
The contraction property of TB($\sigma$) is theoretically analyzed for both
policy evaluation and control settings. We also derive the online version of
TBQ($\sigma$) and give the convergence proof. We empirically show that, for
$\epsilon\in(0,1]$ in $\epsilon$-greedy policies, there exists some degree of
utilizing traces for $\lambda\in[0,1]$, which can improve the efficiency in
trace utilization for off-policy reinforcement learning, to both accelerate the
learning process and improve the performance.
