This study investigates the misclassification excess risk bound in the context of 1-bit matrix completion, a significant problem in machine learning involving the recovery of an unknown matrix from a limited subset of its entries. Matrix completion has garnered considerable attention in the last two decades due to its diverse applications across various fields. Unlike conventional approaches that deal with real-valued samples, 1-bit matrix completion is concerned with binary observations. While prior research has predominantly focused on the estimation error of proposed estimators, our study shifts attention to the prediction error. This paper offers theoretical analysis regarding the prediction errors of two previous works utilizing the logistic regression model: one employing a max-norm constrained minimization and the other employing nuclear-norm penalization. Significantly, our findings demonstrate that the latter achieves the minimax-optimal rate without the need for an additional logarithmic term. These novel results contribute to a deeper understanding of 1-bit matrix completion by shedding light on the predictive performance of specific methodologies.

这项研究探讨了在1位矩阵补全中的错分过量风险界，1位矩阵补全是机器学习中一个重要的问题，涉及从一个有限子集的条目中恢复未知矩阵。与处理实值样本的传统方法不同，1位矩阵补全关注的是二进制观测。通过理论分析两个先前采用逻辑回归模型的方法的预测误差，本文证明了后者在不需要额外的对数项的情况下实现了极小化最优速率，这些新的结果有助于更深入地理解1位矩阵补全，并揭示了特定方法的预测性能。

1比特矩阵补全的误分类风险上界