We consider a learning system based on the conventional multiplicative weight (MW) rule that combines experts' advice to predict a sequence of true outcomes. It is assumed that one of the experts is malicious and aims to impose the maximum loss on the system. The loss of the system is naturally defined to be the aggregate absolute difference between the sequence of predicted outcomes and the true outcomes. We consider this problem under both offline and online settings. In the offline setting where the malicious expert must choose its entire sequence of decisions a priori, we show somewhat surprisingly that a simple greedy policy of always reporting false prediction is asymptotically optimal with an approximation ratio of $1+O(\sqrt{\frac{\ln N}{N}})$, where $N$ is the total number of prediction stages. In particular, we describe a policy that closely resembles the structure of the optimal offline policy. For the online setting where the malicious expert can adaptively make its decisions, we show that the optimal online policy can be efficiently computed by solving a dynamic program in $O(N^2)$. Our results provide a new direction for vulnerability assessment of commonly used learning algorithms to adversarial attacks where the threat is an integral part of the system.

In this paper, we propose a cost-aware cascading bandits model, a new variant of multi-armed ban- dits with cascading feedback, by considering the random cost of pulling arms. In each step, the learning agent chooses an ordered list of items and examines them sequentially, until certain stopping condition is satisfied. Our objective is then to max- imize the expected net reward in each step, i.e., the reward obtained in each step minus the total cost in- curred in examining the items, by deciding the or- dered list of items, as well as when to stop examina- tion. We study both the offline and online settings, depending on whether the state and cost statistics of the items are known beforehand. For the of- fline setting, we show that the Unit Cost Ranking with Threshold 1 (UCR-T1) policy is optimal. For the online setting, we propose a Cost-aware Cas- cading Upper Confidence Bound (CC-UCB) algo- rithm, and show that the cumulative regret scales in O(log T ). We also provide a lower bound for all {\alpha}-consistent policies, which scales in {\Omega}(log T ) and matches our upper bound. The performance of the CC-UCB algorithm is evaluated with both synthetic and real-world data.

In this paper, we investigate cost-aware joint learning and optimization for multi-channel opportunistic spectrum access in a cognitive radio system. We investigate a discrete time model where the time axis is partitioned into frames. Each frame consists of a sensing phase, followed by a transmission phase. During the sensing phase, the user is able to sense a subset of channels sequentially before it decides to use one of them in the following transmission phase. We assume the channel states alternate between busy and idle according to independent Bernoulli random processes from frame to frame. To capture the inherent uncertainty in channel sensing, we assume the reward of each transmission when the channel is idle is a random variable. We also associate random costs with sensing and transmission actions. Our objective is to understand how the costs and reward of the actions would affect the optimal behavior of the user in both offline and online settings, and design the corresponding opportunistic spectrum access strategies to maximize the expected cumulative net reward (i.e., reward-minus-cost). We start with an offline setting where the statistics of the channel status, costs and reward are known beforehand. We show that the the optimal policy exhibits a recursive double threshold structure, and the user needs to compare the channel statistics with those thresholds sequentially in order to decide its actions. With such insights, we then study the online setting, where the statistical information of the channels, costs and reward are unknown a priori. We judiciously balance exploration and exploitation, and show that the cumulative regret scales in O(log T). We also establish a matched lower bound, which implies that our online algorithm is order-optimal. Simulation results corroborate our theoretical analysis.