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 𝐱_t and predicts a label ỹ_t and receives the bandit feedback 𝕀[ỹ_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 𝒪(√(2 K/γ[T/2+(2+L^2/R^2‖‖_F^2)∑_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 𝒪(√(2 K/γ[T/2+2∑_t=1^Td_t+|ℳ| T])) where ℳ 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 ∑_t=1^Td_t. For the case when T and ∑_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 𝒪(√(T+∑_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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