Effective Learning Rules as Natural Gradient Descent

We establish that a broad class of effective learning rules—those that improve a scalar performance measure over a given time window—can be expressed as natural gradient descent with respect to an appropriately defined metric. Specifically, parameter updates in this class can always be written as the product of a symmetric positive-definite matrix and the negative gradient of a loss function encoding the task. Given the high level of generality, our findings formally support the idea that the gradient is a fundamental object underlying all learning processes. Our results are valid across a wide range of common settings, including continuous- time, discrete-time, stochastic, and higher-order learning rules, as well as loss functions with explicit time dependence. Beyond providing a unified framework for learning, our results also have practical implications for control as well as experimental neuroscience.