标题： 单指标模型中使用聚合的最优速率和自适应性 显示英文标题

标题： Optimal rates and adaptation in the single-index model using aggregation

摘要： 我们希望在单指标模型中恢复回归函数。通过使用带局部多项式估计量的聚合算法，我们特别回答了 Stone (1982) 提出的问题 2 的第二部分关于最优收敛速度的问题。这里构造的过程具有很强的适应性：它既能适应链接函数的光滑性，又能适应未知指标。此外，该过程在设计分布上局部自适应。我们提出了局部多项式估计量的新上界（这些结果本身具有独立兴趣），允许设计相当一般化。通过数值模拟研究了该算法的行为。特别是，我们通过经验表明，它比经验风险最小化有显著改进。 显示英文摘要

摘要： We want to recover the regression function in the single-index model. Using an aggregation algorithm with local polynomial estimators, we answer in particular to the second part of Question~2 from Stone (1982) on the optimal convergence rate. The procedure constructed here has strong adaptation properties: it adapts both to the smoothness of the link function and to the unknown index. Moreover, the procedure locally adapts to the distribution of the design. We propose new upper bounds for the local polynomial estimator (which are results of independent interest) that allows a fairly general design. The behavior of this algorithm is studied through numerical simulations. In particular, we show empirically that it improves strongly over empirical risk minimization.