Metric learning seeks a transformation of the feature space that enhances prediction quality for a given task. In this work we provide PAC-style sample complexity rates for supervised metric learning. We give matching lower- and upper-bounds showing that sample complexity scales with the representation dimension when no assumptions are made about the underlying data distribution. In addition, by leveraging the structure of the data distribution, we provide rates fine-tuned to a specific notion of the intrinsic complexity of a given dataset, allowing us to relax the dependence on representation dimension. We show both theoretically and empirically that augmenting the metric learning optimization criterion with a simple norm-based regularization is important and can help adapt to a dataset’s intrinsic complexity yielding better generalization, thus partly explaining the empirical success of similar regularizations reported in previous works.

Mammalian cerebral cortex is accepted as being critical for voluntary motor control, but what functions depend on cortex is still unclear. Here we used rapid, reversible optogenetic inhibition to test the role of cortex during a head-fixed task in which mice reach, grab, and eat a food pellet. Sudden cortical inhibition blocked initiation or froze execution of this skilled prehension behavior, but left untrained forelimb movements unaffected. Unexpectedly, kinematically normal prehension occurred immediately after cortical inhibition even during rest periods lacking cue and pellet. This 'rebound' prehension was only evoked in trained and food-deprived animals, suggesting that a motivation-gated motor engram sufficient to evoke prehension is activated at inhibition's end. These results demonstrate the necessity and sufficiency of cortical activity for enacting a learned skill.