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.

翻译：本研究探讨了1-bit矩阵补全背景下的误分类过剩风险界，这是一个涉及从有限观测条目中恢复未知矩阵的机器学习重要问题。在过去二十年中，矩阵补全因其在多个领域的广泛应用而备受关注。与处理实值样本的传统方法不同，1-bit矩阵补全关注的是二元观测数据。先前研究主要关注所提出估计量的估计误差，而本研究则将重点转向预测误差。本文针对两项基于逻辑回归模型的先前工作提供了预测误差的理论分析：一项采用最大范数约束最小化方法，另一项采用核范数罚函数方法。值得关注的是，我们的研究结果表明，后者在无需额外对数项的情况下达到了极小化最优速率。这些新颖的结论通过揭示特定方法的预测性能，为深入理解1-bit矩阵补全提供了新的见解。