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

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.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 also an appealing idea to describe adaptation and learning in the brain. For example, 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. There is, however, a pitfall in this reasoning: a multitude of steepest directions exists for any abstract error landscape because the steepest direction depends on how angles are measured, and it may be unclear how angles should be measured. Many scientists are taught that the steepest direction can be found by computing the vector of partial derivatives. But 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. In this article, we provide a 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.