Getting to the Point. Index Sets and Parallelism-Preserving Autodiff for Pointful Array Programming (2104.05372v1)

Abstract: We present a novel programming language design that attempts to combine the clarity and safety of high-level functional languages with the efficiency and parallelism of low-level numerical languages. We treat arrays as eagerly-memoized functions on typed index sets, allowing abstract function manipulations, such as currying, to work on arrays. In contrast to composing primitive bulk-array operations, we argue for an explicit nested indexing style that mirrors application of functions to arguments. We also introduce a fine-grained typed effects system which affords concise and automatically-parallelized in-place updates. Specifically, an associative accumulation effect allows reverse-mode automatic differentiation of in-place updates in a way that preserves parallelism. Empirically, we benchmark against the Futhark array programming language, and demonstrate that aggressive inlining and type-driven compilation allows array programs to be written in an expressive, "pointful" style with little performance penalty.

• The paper introduces Dex, which fuses typed indexing with an effect system to enable parallel and efficient automatic differentiation.
• It employs a novel strategy by treating arrays as eagerly memoized functions, supporting advanced operations like currying.
• Dex demonstrates competitive performance against Futhark, bridging high-level abstraction and low-level numerical efficiency.

An Overview of "Getting to the Point."

The paper "Getting to the Point." introduces Dex, a novel programming language synthesizing high-level functional language clarity and safety with the efficiency and parallelism typical of low-level numerical languages. Dex stands out by treating arrays as eagerly-memoized functions, facilitating abstract function manipulations—such as currying—directly on arrays. Unlike traditional array programming approaches that rely on composing primitive operations, Dex promotes an explicit nested indexing style reflecting function-to-argument application paradigms.

Key Contributions

The authors make several significant contributions:

1. Typed Indexing: Dex introduces typed indexing to array manipulation, allowing for concise yet flexible expressions. Typed index sets effectively draw parallels between arrays and functions, merging array and functional programming paradigms.
2. Effect Systems for Parallelism: Dex incorporates a novel effect system, specializing in an associative accumulation effect. This system supports clear and automatically parallelized in-place updates, capturing parallelism even in reverse-mode automatic differentiation (AD), which is crucial for high-performance numerical computing.
3. Function and Array Duality: By typing array index sets, Dex aligns the semantics of arrays with those of functions, allowing functions to be used in array-like constructs while retaining parallelization opportunities and type safety.
4. Defunctionalization: Dex employs a simplification pass which removes higher-order functions from the intermediate representation, facilitating efficient compilation and parallel execution. This process, which reduces expressions to a first-order representation, is essential for code generation on modern computing architectures.
5. Efficient Automatic Differentiation: Dex is designed around efficient automatic differentiation, offering both forward-mode and reverse-mode AD through linearization and transposition. The differentiation process maintains structural linearity, optimizing both computational cost and parallel execution capabilities.
6. Future Direction in Automatic Differentiation: By allowing state mutation and leveraging effects for AD, Dex solves typical AD challenges without sacrificing performance or parallelism. The design of its effect system ensures that all constructs within Dex are amenable to effective differentiation and efficient execution.

Empirical Performance

The paper benchmarks Dex against Futhark, demonstrating that Dex's approach allows expressive array programming without significant performance penalties. This demonstrates that Dex's type-driven compilation approach, using aggressive inlining and type-directed optimization, can achieve competitive performance, maintaining expressiveness in user programs.

Practical and Theoretical Implications

The theoretical implications of Dex are manifold, contributing to both functional programming languages and the automatic differentiation community. Dex bridges the gap between high-level abstraction and low-level performance, suggesting future directions in language design focusing on typed effects and novel indexing approaches.

On a practical level, Dex opens new avenues for high-performance computing, especially in areas requiring robust parallel computation, such as machine learning and scientific simulations. Its ability to preserve expressiveness while enabling automatic differentiation natively is particularly valuable in these domains.

Future Developments

The potential future developments in Dex focus on enhancing its optimization strategies and extending its type system for more expressive power while preserving computational efficiency. Continuous improvements in automatic parallelization and differentiation are also anticipated, alongside broader adoption in performance-critical application domains.

In summary, "Getting to the Point." presents a coherent and technically sound advancement in programming language design, with significant implications for both theoretical exploration and practical application in computationally intensive fields. Dex offers a promising blueprint for future programming languages aiming to reconcile expressiveness, safety, and performance.