The idea that the brain functions in a way that minimizes certain costs pervades theoretical neuroscience. Since a cost function by itself does not predict how the brain finds its minima, additional assumptions about the optimization method are usually made in order to obtain predictions about physiological quantities. In this context, steepest ascent or descent is often suggested as a biologically implemented optimization algorithm. However, the  notion of a steepest direction depends on the choice of a Riemannian metric. Since this choice involves infinite degrees of freedom, the predictive power of models that are based on this principle can be called into question . Despite its great success in machine learning, especially in training deep neural networks, arguments about brain function that rely on gradient-based optimization have to be closely examined .
The idea that the brain functions so as to minimize certain costs pervades theoretical neuroscience. Since 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. Since 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 .
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descent (also called gradient descent)
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a biologically implemented optimization algorithm.
an algorithmic principle of optimization potentially implemented by the brain. In practice, researchers often consider the vector of partial derivatives as the gradient.
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definition of the gradient and the
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this choice involves infinite
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. Despite its great success in machine learning, especially in training deep neural networks, arguments about brain function that rely on gradient-based optimization have to be closely examined
, 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