  Prices generated by automated price experimentation algorithms often display
wild fluctuations, leading to unfavorable customer perceptions and violations
of individual fairness: e.g., the price seen by a customer can be significantly
higher than what was seen by her predecessors, only to fall once again later.
To address this concern, we propose demand learning under a monotonicity
constraint on the sequence of prices, within the framework of stochastic convex
optimization with bandit feedback.
  Our main contribution is the design of the first sublinear-regret algorithms
for monotonic price experimentation for smooth and strongly concave revenue
functions under noisy as well as noiseless bandit feedback. The monotonicity
constraint presents a unique challenge: since any increase (or decrease) in the
decision-levels is final, an algorithm needs to be cautious in its exploration
to avoid over-shooting the optimum. At the same time, minimizing regret
requires that progress be made towards the optimum at a sufficient pace.
Balancing these two goals is particularly challenging under noisy feedback,
where obtaining sufficiently accurate gradient estimates is expensive. Our key
innovation is to utilize conservative gradient estimates to adaptively tailor
the degree of caution to local gradient information, being aggressive far from
the optimum and being increasingly cautious as the prices approach the optimum.
Importantly, we show that our algorithms guarantee the same regret rates (up to
logarithmic factors) as the best achievable rates of regret without the
monotonicity requirement.
