Categorical Foundations of Gradient-Based Learning

Abstract

We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy, shedding new light on their similarities and differences. Our approach to gradient-based learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realized in the discrete setting of boolean circuits. Finally, we demonstrate the practical significance of our framework with an implementation in Python.

Monoidal Reverse Differential Categories

“…Then in particular, by Theorem 4.17, the coKleisli category associated to these models is a CRDC. By the results of [11], this means that one could apply supervised learning techniques to these examples. This possibility of combining quantum computation with supervised learning is an exciting direction we hope will be pursued in the future.…”

“…Specifically, a CRDC is equivalent to giving a CDC in which the subcategory of linear maps in each simple slice has a transpose operator, which categorically speaking is a special type of of dagger structure. The explicit connection with supervised learning was then made in [11], which showed how to describe several supervised-learning techniques in the abstract setting of a CRDC.…”

Categorical composable cryptography

“…In the extended version, we model categorically situations where there is a notion of distance between resources, and instead of exact resource conversions one either studies approximate transformations or sequences of transformations that succeed in the limit. In the extended version, we discuss a variant of a construction on monoidal categories, used in special cases in [31] and discussed in more detail in [23,33], that allows one to declare some resources free and thus enlarge the set of possible resource conversions.…”

Categories of Differentiable Polynomial Circuits for Machine Learning

“…Reverse Derivative Categories [10] have recently been introduced as a formalism to study abstractly the concept of differentiable functions. As explored in [11], it turns out that this framework is suitable to give a categorical semantics for gradient-based learning. In this approach, models-as for instance neural networks-correspond to morphisms in some RDC.…”