Bagging is an ensemble technique commonly used in statistics and machine learning to improve the performance of prediction procedures. In this paper, we examine the predictive risk of variants of bagged predictors in the proportional asymptotic regime, where the ratio of features to observations converges to a constant. Specifically, using classical results of simple random sampling, we propose a general strategy for analyzing prediction risk under squared error loss for bagged predictors. We specialize our strategy to obtain accurate Derive asymptotic risk. In addition, it provides a general cross-validation procedure for choosing the optimal subsample size for bagging, and is useful for mitigating non-monotonic behavior of sample size limit risk (i.e., double or multiple descent). I will explain gender. In demonstrating proposed procedures for bagged ridge and ridgeless predictors, we exhaustively explore the oracle properties of optimal subsample sizes and provide detailed comparisons between different bagging variants.

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