  Online Learning to Rank (OLTR) methods optimize ranking models by directly
interacting with users, which allows them to be very efficient and responsive.
All OLTR methods introduced during the past decade have extended on the
original OLTR method: Dueling Bandit Gradient Descent (DBGD). Recently, a
fundamentally different approach was introduced with the Pairwise
Differentiable Gradient Descent (PDGD) algorithm. To date the only comparisons
of the two approaches are limited to simulations with cascading click models
and low levels of noise. The main outcome so far is that PDGD converges at
higher levels of performance and learns considerably faster than DBGD-based
methods. However, the PDGD algorithm assumes cascading user behavior,
potentially giving it an unfair advantage. Furthermore, the robustness of both
methods to high levels of noise has not been investigated. Therefore, it is
unclear whether the reported advantages of PDGD over DBGD generalize to
different experimental conditions. In this paper, we investigate whether the
previous conclusions about the PDGD and DBGD comparison generalize from ideal
to worst-case circumstances. We do so in two ways. First, we compare the
theoretical properties of PDGD and DBGD, by taking a critical look at
previously proven properties in the context of ranking. Second, we estimate an
upper and lower bound on the performance of methods by simulating both ideal
user behavior and extremely difficult behavior, i.e., almost-random
non-cascading user models. Our findings show that the theoretical bounds of
DBGD do not apply to any common ranking model and, furthermore, that the
performance of DBGD is substantially worse than PDGD in both ideal and
worst-case circumstances. These results reproduce previously published findings
about the relative performance of PDGD vs. DBGD and generalize them to
extremely noisy and non-cascading circumstances.
