The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design choices which are hard to justify in principle and whose effectiveness often goes unexplained. Research debt is increasing and many papers are found not to be reproducible. This thesis is a survey that covers some recent work attempting to study machine learning categorically. Category theory is a branch of abstract mathematics that has found successful applications in many fields, both inside and outside mathematics. Acting as a lingua franca of mathematics and science, category theory might be able to give a unifying structure to the field of machine learning. This could solve some of the aforementioned problems. In this work, we mainly focus on the application of category theory to deep learning. Namely, we discuss the use of categorical optics to model gradient-based learning, the use of categorical algebras and integral transforms to link classical computer science to neural networks, the use of functors to link different layers of abstraction and preserve structure, and, finally, the use of string diagrams to provide detailed representations of neural network architectures.

We propose a categorical semantics for 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, and different architectures, shedding new light on their similarities and differences. Furthermore, our approach to learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realised in the discrete setting of Boolean and polynomial circuits. We demonstrate the practical significance of our framework with an implementation in Python.

We improve the framework of open games with agency by showing how the players' counterfactual analysis giving rise to Nash equilibria can be described in the dynamics of the game itself (hence diegetically), getting rid of devices such as equilibrium predicates. This new approach overlaps almost completely with the way gradient-based learners are specified and trained. Indeed, we show feedback propagation in games can be seen as a form of backpropagation, with a crucial difference explaining the distinctive character of the phenomenology of non-cooperative games. We outline a functorial construction of arena of games, show players form a subsystem over it, and prove that their 'fixpoint behaviours' are Nash equilibria.