On the choice of metric in gradient-based theories of brain function.

Wulfram Gerstner,Simone Carlo Surace,Jean-Pascal Pfister,Johanni Brea

doi:10.1371/journal.pcbi.1007640

Wulfram Gerstner, Simone Carlo Surace + Show 2 more

Open Access

https://doi.org/10.1371/journal.pcbi.1007640

Copy DOI

Abstract

This is a PLOS Computational Biology Education paper.The idea that the brain functions so as to minimize certain costs pervades theoretical neuroscience. Because a cost function by itself does not predict how the brain finds its minima, additional assumptions about the optimization method need to be made to predict the dynamics of physiological quantities. In this context, steepest descent (also called gradient descent) is often suggested as an algorithmic principle of optimization potentially implemented by the brain. In practice, researchers often consider the vector of partial derivatives as the gradient. However, the definition of the gradient and the notion of a steepest direction depend on the choice of a metric. Because the choice of the metric involves a large number of degrees of freedom, the predictive power of models that are based on gradient descent must be called into question, unless there are strong constraints on the choice of the metric. Here, we provide a didactic review of the mathematics of gradient descent, illustrate common pitfalls of using gradient descent as a principle of brain function with examples from the literature, and propose ways forward to constrain the metric.

Highlights

The minimization of costs is a widespread approach in theoretical neuroscience [1,2,3,4,5]
The idea that biological systems are operating at some kind of optimality is old and is often mathematically formalized as the minimization of a cost function
The present paper tries to explain that even the simplest dynamical system, steepest descent, depends on the choice of a Riemannian metric

Summary

Author summary

A good skier may choose to follow the steepest direction to move as quickly as possible from the mountain peak to the base. Steepest descent in an abstract sense is an appealing idea to describe adaptation and learning in the brain. A scientist may hypothesize that synaptic or neuronal variables change in the direction of steepest descent in an abstract error landscape during learning of a new task or memorization of a new concept. Many scientists are taught that the steepest direction can be found by computing the vector of partial derivatives. The vector of partial derivatives is equal to the steepest direction only if the angles in the abstract space are measured in a particular way. Didactic review of the mathematics of finding steepest directions in abstract spaces, illustrate the pitfalls with examples from the neuroscience literature, and propose guidelines to constrain the way angles are measured in these spaces

Introduction

Conclusions