  We consider the problem of designing an allocation rule or an "online
learning algorithm" for a class of bandit problems in which the set of control
actions available at each time $s$ is a convex, compact subset of
$\mathbb{R}^d$. Upon choosing an action $x$ at time $s$, the algorithm obtains
a noisy value of the unknown and time-varying function $f_s$ evaluated at $x$.
The "regret" of an algorithm is the gap between its expected reward, and the
reward earned by a strategy which has the knowledge of the function $f_s$ at
each time $s$ and hence chooses the action $x_s$ that maximizes $f_s$.
  For this non-stationary bandit problem set-up, we consider two variants of
the Kiefer Wolfowitz (KW) algorithm i) KW with fixed step-size $\beta$, and ii)
KW with sliding window of length $L$. We show that if the number of times that
the function $f_s$ varies during time $T$ is $o(T)$, and if the learning rates
of the proposed algorithms are chosen "optimally", then the regret of the
proposed algorithms is $o(T)$, and hence the algorithms are asymptotically
efficient.
