  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.
