This work addresses the problem of high-dimensional classification by exploring the generalized Bayesian logistic regression method under a sparsity-inducing prior distribution. The method involves utilizing a fractional power of the likelihood resulting the fractional posterior. Our study yields concentration results for the fractional posterior, not only on the joint distribution of the predictor and response variable but also for the regression coefficients. Significantly, we derive novel findings concerning misclassification excess risk bounds using sparse generalized Bayesian logistic regression. These results parallel recent findings for penalized methods in the frequentist literature. Furthermore, we extend our results to the scenario of model misspecification, which is of critical importance.

翻译：本文通过探索在稀疏诱导先验分布下的广义贝叶斯逻辑回归方法，解决了高维分类问题。该方法利用似然函数的分数幂，从而得到分数后验分布。我们的研究不仅得出了关于预测变量和响应变量联合分布的分数后验的集中性结果，还得到了回归系数的集中性结果。重要的是，我们利用稀疏广义贝叶斯逻辑回归推导出了关于误分类超额风险界的新颖结论。这些结果与频率学派文献中针对惩罚化方法的最新成果具有相似性。此外，我们将研究结果扩展到了模型误设的情境，这一情境具有至关重要的意义。