Delaytron: Efficient Learning of Multiclass Classifiers with Delayed Bandit Feedbacks
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In this paper, we present online algorithm called Delaytron for learning multi class classifiers using delayed bandit feedbacks. The sequence of feedback delays {d_t}_t=1^T is unknown to the algorithm. At the t-th round, the algorithm observes an example ๐ฑ_t and predicts a label แปน_t and receives the bandit feedback ๐[แปน_t=y_t] only d_t rounds later. When t+d_t>T, we consider that the feedback for the t-th round is missing. We show that the proposed algorithm achieves regret of ๐ช(โ(2 K/ฮณ[T/2+(2+L^2/R^2โโ_F^2)โ_t=1^Td_t])) when the loss for each missing sample is upper bounded by L. In the case when the loss for missing samples is not upper bounded, the regret achieved by Delaytron is ๐ช(โ(2 K/ฮณ[T/2+2โ_t=1^Td_t+|โณ| T])) where โณ is the set of missing samples in T rounds. These bounds were achieved with a constant step size which requires the knowledge of T and โ_t=1^Td_t. For the case when T and โ_t=1^Td_t are unknown, we use a doubling trick for online learning and proposed Adaptive Delaytron. We show that Adaptive Delaytron achieves a regret bound of ๐ช(โ(T+โ_t=1^Td_t)). We show the effectiveness of our approach by experimenting on various datasets and comparing with state-of-the-art approaches.
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