Theoretical limits on the speed of learning inverse models explain the rate of adaptation in arm reaching tasks
Authors: Barradas Victor R. , Koike Yasuharu , Schweighofer Nicolas 
Published: September 2022

An essential aspect of human motor learning is the formation of inverse models, which map desired actions to motor commands. Inverse models can be learned by adjusting parameters in neural circuits to minimize errors in the performance of motor tasks through gradient descent. However, the theory of gradient descent establishes an upper limit on the learning speed, above which learning becomes unstable. Specifically, the eigenvalues of the Hessian of the error surface around a minimum determine the maximum speed of learning in a given task. Here, we use this theoretical framework to analyze the speed of learning in different inverse model learning architectures in a set of isometric arm-reaching tasks. We show theoretically that, for these isometric tasks, the error surface and, thus the speed of learning, are determined by the shapes of 1) the force manipulability ellipsoid of the arm and 2) the distribution of targets in the task. In particular, rounder manipulability ellipsoids generate a rounder error surface, allowing for faster learning of the inverse model. Rounder target distributions have a similar effect. We tested these predictions experimentally in a virtual quasi-isometric reaching task with a visuomotor transformation. The experimental results were consistent with our theoretical predictions. Furthermore, our analysis accounts for the speed of learning in previous experiments with incompatible and compatible virtual surgery tasks, and in visuomotor rotation tasks with different numbers of targets. By identifying aspects of a task that influence the speed of learning, our results provide theoretical principles for the design of motor tasks that allow for faster learning.

Author Summary

When facing a new or changing environment, humans need to learn a new internal inverse model to generate fast and accurate movements. Different computational architectures based on supervised learning via gradient descent have been proposed to explain the acquisition of these inverse models. Although the theory of gradient descent and its results regarding the speed of learning are well developed in Machine Learning, this framework has seldom been applied to computational models of human motor learning. In this study, we found that applying this theoretical framework to a set of isometric reaching tasks clearly reveals aspects of the motor task that can speed up or slow down learning. In particular, we found that, in isometric tasks, the force manipulability ellipsoid of the arm and the distribution of force targets determine the speed of learning. These theoretical results allowed us to generate testable hypotheses about the speed of learning in different motor task conditions, which we successfully confirmed experimentally. We believe that our methods and results could open new lines of research to systematically identify aspects of motor learning tasks that can be exploited to enhance the speed of learning and to design new tasks that are easy to learn.