Compiling to recurrent neurons

Discrete structures are currently second-class in differentiable programming. Since functions over discrete structures lack overt derivatives, differentiable programs do not differentiate through them and limit where they can be used. For example, when programming a neural network, conditionals and iteration cannot be used everywhere; they can break the derivatives necessary for gradient-based learning to work. This limits the class of differentiable algorithms we can directly express, imposing restraints on how we build neural networks and differentiable programs more generally. However, these restraints are not fundamental. Recent work shows conditionals can be first-class, by compiling them into differentiable form as linear neurons. Similarly, this work shows iteration can be first-class -- by compiling to linear recurrent neurons. We present a minimal typed, higher-order and linear programming language with iteration called $\textsf{Cajal}\scriptstyle(\mathbb{\multimap}, \mathbb{2}, \mathbb{N})$. We prove its programs compile correctly to recurrent neurons, allowing discrete algorithms to be expressed in a differentiable form compatible with gradient-based learning. With our implementation, we conduct two experiments where we link these recurrent neurons against a neural network solving an iterative image transformation task. This determines part of its function prior to learning. As a result, the network learns faster and with greater data-efficiency relative to a neural network programmed without first-class iteration. A key lesson is that recurrent neurons enable a rich interplay between learning and the discrete structures of ordinary programming.

Compiling to linear neurons

We don't program neural networks directly. Instead, we rely on an indirect style where learning algorithms, like gradient descent, determine a neural network's function by learning from data. This indirect style is often a virtue; it empowers us to solve problems that were previously impossible. But it lacks discrete structure. We can't compile most algorithms into a neural network -- even if these algorithms could help the network learn. This limitation occurs because discrete algorithms are not obviously differentiable, making them incompatible with the gradient-based learning algorithms that determine a neural network's function. To address this, we introduce $\textsf{Cajal}$: a typed, higher-order and linear programming language intended to be a minimal vehicle for exploring a direct style of programming neural networks. We prove $\textsf{Cajal}$ programs compile to linear neurons, allowing discrete algorithms to be expressed in a differentiable form compatible with gradient-based learning. With our implementation of $\textsf{Cajal}$, we conduct several experiments where we link these linear neurons against other neural networks to determine part of their function prior to learning. Linking with these neurons allows networks to learn faster, with greater data-efficiency, and in a way that's easier to debug. A key lesson is that linear programming languages provide a path towards directly programming neural networks, enabling a rich interplay between learning and the discrete structures of ordinary programming.

Towards Neural Functional Program Evaluation

This paper explores the capabilities of current transformer-based language models for program evaluation of simple functional programming languages. We introduce a new program generation mechanism that allows control over syntactic sugar for semantically equivalent programs. T5 experiments reveal that neural functional program evaluation performs surprisingly well, achieving high 90% exact program match scores for most in-distribution and out-of-distribution tests. Using pretrained T5 weights has significant advantages over random initialization. We present and evaluate on three datasets to study generalization abilities that are specific to functional programs based on: type, function composition, and reduction steps. Code and data are publicly available at https://github.com/ElementAI/neural-interpreters.