We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve longrun-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the "price of non-stationarity," which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one.

We consider a single-product revenue management problem where, given an initial inventory, the objective is to dynamically adjust prices over a finite sales horizon to maximize expected revenues. Realized demand is observed over time, but the underlying functional relationship between price and mean demand rate that governs these observations (otherwise known as the demand function or demand curve) is not known. We consider two instances of this problem: (i) a setting where the demand function is assumed to belong to a known parametric family with unknown parameter values; and (ii) a setting where the demand function is assumed to belong to a broad class of functions that need not admit any parametric representation. In each case we develop policies that learn the demand function "on the fly," and optimize prices based on that. The performance of these algorithms is measured in terms of the regret: the revenue loss relative to the maximal revenues that can be extracted when the demand function is known prior to the start of the selling season. We derive lower bounds on the regret that hold for any admissible pricing policy, and then show that our proposed algorithms achieve a regret that is "close" to this lower bound. The magnitude of the regret can be interpreted as the economic value of prior knowledge on the demand function, manifested as the revenue loss due to model uncertainty. AbstractWe consider a single product revenue management problem where, given an initial inventory, the objective is to dynamically adjust prices over a finite sales horizon to maximize expected revenues. Realized demand is observed over time, but the underlying functional relationship between price and mean demand rate that governs these observations (otherwise known as the demand function or demand curve), is not known. We consider two instances of this problem:i.) a setting where the demand function is assumed to belong to a known parametric family with unknown parameter values; and ii.) a setting where the demand function is assumed to belong to a broad class of functions that need not admit any parametric representation. In each case we develop policies that learn the demand function "on the fly," and optimize prices based on that. The performance of these algorithms is measured in terms of the regret: the revenue loss relative to the maximal revenues that can be extracted when the demand function is known prior to the start of the selling season. We derive lower bounds on the regret that hold for any admissible pricing policy, and then show that our proposed algorithms achieve a regret that is "close" to this lower bound. The magnitude of the regret can be interpreted as the economic value of prior knowledge on the demand function; manifested as the revenue loss due to model uncertainty.

In a multi-armed bandit problem, a gambler needs to choose at each round one of K arms, each characterized by an unknown reward distribution. The objective is to maximize cumulative expected earnings over a planning horizon of length T, and performance is measured in terms of regret relative to a (static) oracle that knows the identity of the best arm a priori. This problem has been studied extensively when the reward distributions do not change over time, and uncertainty essentially amounts to identifying the optimal arm. We complement this literature by developing a flexible non-parametric model for temporal uncertainty in the rewards. The extent of temporal uncertainty is measured via the cumulative mean change in the rewards over the horizon, a metric we refer to as temporal variation, and regret is measured relative to a (dynamic) oracle that plays the pointwise optimal action at each period. Assuming that nature can choose any sequence of mean rewards such that their temporal variation does not exceed V (a temporal uncertainty budget), we characterize the complexity of this problem via the minimax regret, which depends on V (the hardness of the problem), the horizon length T, and the number of arms K. History: Former designation of this paper was SSy-2018-015.R1.

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavytraffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called "interchange-of-limits" for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.

Security of the Ekert protocol is proven against individual attacks where an eavesdropper is allowed to share any density matrix with the two communicating parties. The density matrix spans all of the photon number states of both receivers, as well as a probe state of arbitrary dimensionality belonging to the eavesdropper. Using this general eavesdropping strategy, we show that the Shannon information on the final key, after error correction and privacy amplification, can be made exponentially small. This is done by finding a bound on the eavesdropper's average collision probability. We find that the average collision probability for the Ekert protocol is the same as that of the BB84 protocol for single photons, indicating that there is no analog in the Ekert protocol to photon splitting attacks. We then compare the communication rate of both protocols as a function of distance, and show that the Ekert protocol has potential for much longer communication distances, up to 170km, in the presence of realistic detector dark counts and channel loss. Finally, we propose a slightly more complicated scheme based on entanglement swapping that can lead to even longer distances of communication. The limiting factor in this new scheme is the fiber loss, which imposes very slow communication rates at longer distances.