  We consider a stochastic lost-sales inventory control system with a lead time
$L$ over a planning horizon $T$. Supply is uncertain, and is a function of the
order quantity (due to random yield/capacity, etc). We aim to minimize the
$T$-period cost, a problem that is known to be computationally intractable even
under known distributions of demand and supply. In this paper, we assume that
both the demand and supply distributions are unknown and develop a
computationally efficient online learning algorithm. We show that our algorithm
achieves a regret (i.e. the performance gap between the cost of our algorithm
and that of an optimal policy over $T$ periods) of $O(L+\sqrt{T})$ when
$L\geq\log(T)$. We do so by 1) showing our algorithm cost is higher by at most
$O(L+\sqrt{T})$ for any $L\geq 0$ compared to an optimal constant-order policy
under complete information (a well-known and widely-used algorithm) and 2)
leveraging its known performance guarantee from the existing literature. To the
best of our knowledge, a finite-sample $O(\sqrt{T})$ (and polynomial in $L$)
regret bound when benchmarked against an optimal policy is not known before in
the online inventory control literature. A key challenge in this learning
problem is that both demand and supply data can be censored; hence only
truncated values are observable. We circumvent this challenge by showing that
the data generated under an order quantity $q^2$ allows us to simulate the
performance of not only $q^2$ but also $q^1$ for all $q^1<q^2$, a key
observation to obtain sufficient information even under data censoring. By
establishing a high probability coupling argument, we are able to evaluate and
compare the performance of different order policies at their steady state
within a finite time horizon. Since the problem lacks convexity, we develop an
active elimination method that adaptively rules out suboptimal solutions.
