Efficient Multinomial Logistic Bandit Algorithm Using Frequent Directions Matrix Sketching
Researchers propose EOFD-MLogB, a new algorithm that improves computational efficiency for multinomial logistic bandits by integrating frequent directions matrix sketching. The method reduces per-round time complexity from O(K³d³) to O(Kd(m+K)²) while maintaining near-optimal regret bounds. This advancement makes high-dimensional bandit problems computationally tractable for practical applications.
The paper addresses a key computational bottleneck in multinomial logistic bandits (MLogB), where existing UCB-type algorithms like OFUL-MLogB achieve theoretically sound regret bounds but require prohibitive computational resources per round. The proposed EOFD-MLogB algorithm integrates frequent directions matrix sketching to maintain a low-rank SVD sketch of the accumulated Hessian, transforming expensive operations into simpler one-dimensional root-finding tasks and smaller eigenvalue computations. The algorithm achieves dominant per-round time complexity of O(Kd(m+K)²) and space complexity of O(Kd(m+K)), where m is a sketch size much smaller than the dimension d. Theoretical analysis shows the regret bound is Õ(ΔT(Kd ln ΔT + m)√T), where the sketching error factor ΔT depends on the Hessian's spectral properties. When the Hessian is approximately low-rank, the regret approaches that of the original OFUL-MLogB algorithm, and experiments confirm both computational efficiency and competitive empirical performance.
What's missing
The paper does not discuss practical applications or domains where multinomial logistic bandits are commonly deployed (e.g., recommendation systems, online advertising, clinical trials). Additionally, the conditions under which the Hessian exhibits approximate low-rank structure in real-world problems are not elaborated, which would help practitioners assess when this method provides the most benefit.
What different sources said
- arXiv cs.LGCenter
Efficient Multinomial Logistic Bandit via Frequent Directions
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