Outcome Weighted Learning on Data with Low Intrinsic Dimension

Precision medicine aims to personalize treatment by identifying the optimal individualized treatment rule (ITR) that maximizes expected clinical outcomes. Outcome weighted learning (OWL) is an approach that has caught attention for estimating optimal ITR by leveraging each patient’s unique characteristics within a weighted classification framework. The curse of dimensionality in statistical learning significantly impacts the convergence rates of OWL algorithms, especially when analyzing high-dimensional biomedical data. We establish improved learning rates for OWL algorithms with varying Gaussian kernels, assuming the data lie on a low-dimensional intrinsic structure characterized by the box-counting dimension. Under noise or Besov smoothness assumptions, we derive a novel oracle inequality for OWL and obtain fast learning rates for the hinge loss function and q-norm hinge loss function (1 < q ≤ 2), where the rates explicitly depend on the box-counting dimension of the prognostic variable domain. Additionally, we select the hyperparameters of OWL adaptively based on the data through a training-validation procedure. The same learning rates are obtained without assuming any prior knowledge of the data distribution.