  This paper studies bandit convex optimization with constraints, where the
learner aims to generate a sequence of decisions under partial information of
loss functions such that the cumulative loss is reduced as well as the
cumulative constraint violation is simultaneously reduced. We adopt the
cumulative \textit{hard} constraint violation as the metric of constraint
violation, which is defined by $\sum_{t=1}^{T} \max\{g_t(\boldsymbol{x}_t),
0\}$. Owing to the maximum operator, a strictly feasible solution cannot cancel
out the effects of violated constraints compared to the conventional metric
known as \textit{long-term} constraints violation. We present a penalty-based
proximal gradient descent method that attains a sub-linear growth of both
regret and cumulative hard constraint violation, in which the gradient is
estimated with a two-point function evaluation. Precisely, our algorithm
attains $O(d^2T^{\max\{c,1-c\}})$ regret bounds and $O(d^2T^{1-\frac{c}{2}})$
cumulative hard constraint violation bounds for convex loss functions and
time-varying constraints, where $d$ is the dimensionality of the feasible
region and $c\in[\frac{1}{2}, 1)$ is a user-determined parameter. We also
extend the result for the case where the loss functions are strongly convex and
show that both regret and constraint violation bounds can be further reduced.
