最佳化半監督式歸一化指數函式抽樣

Abstract

在最佳次抽樣方法中，標籤資訊的缺失對抽樣機率的估計構成顯著挑戰，特別是在進行分類任務時，傳統方法往往依賴完整的回應資料。為解決此問題，本文提出一套應用於 softmax 回歸模型的半監督 A-/L-最適次抽樣框架。該方法於基準約束條件下推導出理論上的最適抽樣機率，並進一步探討其在平衡類別回應分布方面的統計意涵。同時，我們亦考量不受拘束條件影響的抽樣方法，提出以最小化漸近預測均方誤差（Asymptotic Mean Squared Prediction Error, MSPE）為目標的抽樣策略，使所構建之抽樣機率對模型約束具備更高的穩健性。理論結果經由模擬數據與實證資料驗證，皆顯示本方法能有效提升預測準確率與計算效率。

Missing label information presents a significant challenge for optimal subsampling methods, which typically rely on complete response data to compute sampling probabilities. In this study, we propose a semi-supervised A-/L-optimal subsampling framework for softmax regression that effectively addresses this issue. We derive the optimal subsampling probabilities under the baseline constraint and highlight their role in balancing categorical responses. In addition, we explore constraint-invariant subsampling by minimizing the asymptotic mean squared prediction error (MSPE), enabling the construction of subsampling probabilities for each observation, which is robust to model constraint choices. Our theoretical findings are supported by simulations and real-data applications, demonstrating improvements in both prediction accuracy and computational efficiency.