Differentiable Programming

Solving massive-scale optimization problems requires scalable first-order methods with low per-iteration cost. This tutorial highlights a shift in optimization: using differentiable programming not only to execute algorithms but to learn how to design them. Modern frameworks such as PyTorch, TensorFlow, and JAX enable this paradigm through efficient automatic differentiation. Embedding first-order methods within these systems allows end-to-end training that improves convergence and solution quality. Guided by Fenchel-Rockafellar duality, the tutorial demonstrates how duality-informed iterative schemes such as the alternating direction method of multipliers, and the primal-dual hybrid gradient can be learned and adapted through representative case studies.

State-space models (SSMs) are a widely used tool in time series analysis. In the complex systems that arise from real-world data, it is common to employ particle filtering (PF), an efficient Monte Carlo method for estimating the hidden state corresponding to a sequence of observations. Applying particle filtering requires specifying both the parametric form and the parameters of the system, which are often unknown and must be estimated. Gradient-based optimisation techniques cannot be applied directly to standard particle filters, as the filters themselves are not differentiable. However, several recently proposed methods modify the resampling step to make particle filtering differentiable. In this paper, we present an implementation of several such differentiable particle filters (DPFs) with a unified API built on the popular PyTorch framework. Our implementation makes these algorithms easily accessible to a broader research community and facilitates straightforward comparison between them. We validate our framework by reproducing experiments from several existing studies and demonstrate how DPFs can be applied to address several common challenges with state space modelling.

Combinatory Homomorphic Automatic Differentiation (CHAD) was originally formulated as a semantics-driven source-to-source transformation for reverse-mode automatic differentiation of total functional programs. We extend CHAD to programs with partial operations, data-dependent conditionals, and while-loops, preserving its defining principle of structure-preserving semantics. Our main contribution is the introduction of iteration-extensive indexed categories, which integrate iteration into dependently typed programming languages. Iteration in the base category lifts to parameterized initial algebras in the indexed category, yielding fibred iteration on the op-Grothendieck construction. Its total category is the category of containers associated with the dependently typed target language. This framework characterizes iterative CHAD as the unique iterative Freyd category morphism from the source language's syntactic category to the target language's category of containers that maps each primitive operation to its transposed derivative. Using the universal property of the syntactic model, we prove that the transformed programs compute the reverse-mode derivatives of the original programs. The resulting theory connects fixpoint operators in indexed categories with a structure-preserving construction and correctness proof for iterative CHAD.