  We study a stochastic bandit problem with a general unknown reward function
and a general unknown constraint function. Both functions can be non-linear
(even non-convex) and are assumed to lie in a reproducing kernel Hilbert space
(RKHS) with a bounded norm. This kernelized bandit setup strictly generalizes
standard multi-armed bandits and linear bandits. In contrast to safety-type
hard constraints studied in prior works, we consider soft constraints that may
be violated in any round as long as the cumulative violations are small, which
is motivated by various practical applications. Our ultimate goal is to study
how to utilize the nature of soft constraints to attain a finer
complexity-regret-constraint trade-off in the kernelized bandit setting. To
this end, leveraging primal-dual optimization, we propose a general framework
for both algorithm design and performance analysis. This framework builds upon
a novel sufficient condition, which not only is satisfied under general
exploration strategies, including \emph{upper confidence bound} (UCB),
\emph{Thompson sampling} (TS), and new ones based on \emph{random exploration},
but also enables a unified analysis for showing both sublinear regret and
sublinear or even zero constraint violation. We demonstrate the superior
performance of our proposed algorithms via numerical experiments based on both
synthetic and real-world datasets. Along the way, we also make the first
detailed comparison between two popular methods for analyzing constrained
bandits and Markov decision processes (MDPs) by discussing the key difference
and some subtleties in the analysis, which could be of independent interest to
the communities.
