Sketch-Guided Equality Saturation: Scaling Equality Saturation to Complex Optimizations of Functional Programs

TL;DR

This paper introduces sketch-guided equality saturation, a technique that enables scalable and efficient optimization of complex functional programs by incorporating programmer-provided sketches to guide rewrite sequences.

Contribution

It presents novel encoding methods for polymorphically typed lambda calculi and a semi-automatic sketch-guided approach to scale equality saturation for complex optimizations.

Findings

Efficient lambda calculus encoding reduces runtime and memory by orders of magnitude.

Unguided equality saturation struggles with complex optimizations, only finding simple ones.

Sketch-guided approach finds all seven complex matrix multiplication optimizations quickly and with minimal resources.

Abstract

Generating high-performance code for diverse hardware and application domains is challenging. Functional array programming languages with patterns like map and reduce have been successfully combined with term rewriting to define and explore optimization spaces. However, deciding what sequence of rewrites to apply is hard and has a huge impact on the performance of the rewritten program. Equality saturation avoids the issue by exploring many possible ways to apply rewrites, efficiently representing many equivalent programs in an e-graph data structure. Equality saturation has some limitations when rewriting functional language terms, as currently naive encodings of the lambda calculus are used. We present new techniques for encoding polymorphically typed lambda calculi, and show that the efficient encoding reduces the runtime and memory consumption of equality saturation by orders of magnitude. Moreover, equality saturation does not yet scale to complex compiler optimizations. These emerge from long rewrite sequences of thousands of rewrite steps, and may use pathological combinations of rewrite rules that cause the e-graph to quickly grow too large. This paper introduces \emph{sketch-guided equality saturation}, a semi-automatic technique that allows programmers to provide program sketches to guide rewriting. Sketch-guided equality saturation is evaluated for seven complex matrix multiplication optimizations, including loop blocking, vectorization, and multi-threading. Even with efficient lambda calculus encoding, unguided equality saturation can locate only the two simplest of these optimizations, the remaining five are undiscovered even with an hour of compilation time and 60GB of RAM. By specifying three or fewer sketch guides all seven optimizations are found in seconds of compilation time, using under 1GB of RAM, and generating high performance code.