Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

ParallelFlow: Parallelizing Linear Transformers via Flow Discretization

Nicola Muca Cirone, Cristopher Salvi

TL;DR

This paper targets scalable long-context sequence modeling by addressing the quadratic-time bottleneck of attention through a principled reframing of linear transformers as matrix-valued state-space models (SSMs). It introduces Parallel Flows, a framework that decouples temporal dynamics from implementation constraints and connects discrete chunked computations to flows governed by controlled differential equations, with links to rough path theory. The authors present a generalized low-rank Delta Rule for rank-$R$ updates and a signature-kernel inspired algorithm that achieves favorable parallel scaling, alongside an alternative flow representation as a product of exponentials. While practical hardware limitations (notably with Triton) temper empirical gains, the work lays a solid theoretical and architectural groundwork for hardware-efficient, parallel-in-time pipelines and outlines concrete future directions, including adaptive step sizes and higher-order solvers, to push scalable sequence modeling forward.

Abstract

We present a theoretical framework for analyzing linear attention models through matrix-valued state space models (SSMs). Our approach, Parallel Flows, provides a perspective that systematically decouples temporal dynamics from implementation constraints, enabling independent analysis of critical algorithmic components: chunking, parallelization, and information aggregation. Central to this framework is the reinterpretation of chunking procedures as computations of the flows governing system dynamics. This connection establishes a bridge to mathematical tools from rough path theory, opening the door to new insights into sequence modeling architectures. As a concrete application, we analyze DeltaNet in a generalized low-rank setting motivated by recent theoretical advances. Our methods allow us to design simple, streamlined generalizations of hardware-efficient algorithms present in the literature, and to provide completely different ones, inspired by rough paths techniques, with provably lower complexity. This dual contribution demonstrates how principled theoretical analysis can both explain existing practical methods and inspire fundamentally new computational approaches.

ParallelFlow: Parallelizing Linear Transformers via Flow Discretization

TL;DR

This paper targets scalable long-context sequence modeling by addressing the quadratic-time bottleneck of attention through a principled reframing of linear transformers as matrix-valued state-space models (SSMs). It introduces Parallel Flows, a framework that decouples temporal dynamics from implementation constraints and connects discrete chunked computations to flows governed by controlled differential equations, with links to rough path theory. The authors present a generalized low-rank Delta Rule for rank- updates and a signature-kernel inspired algorithm that achieves favorable parallel scaling, alongside an alternative flow representation as a product of exponentials. While practical hardware limitations (notably with Triton) temper empirical gains, the work lays a solid theoretical and architectural groundwork for hardware-efficient, parallel-in-time pipelines and outlines concrete future directions, including adaptive step sizes and higher-order solvers, to push scalable sequence modeling forward.

Abstract

We present a theoretical framework for analyzing linear attention models through matrix-valued state space models (SSMs). Our approach, Parallel Flows, provides a perspective that systematically decouples temporal dynamics from implementation constraints, enabling independent analysis of critical algorithmic components: chunking, parallelization, and information aggregation. Central to this framework is the reinterpretation of chunking procedures as computations of the flows governing system dynamics. This connection establishes a bridge to mathematical tools from rough path theory, opening the door to new insights into sequence modeling architectures. As a concrete application, we analyze DeltaNet in a generalized low-rank setting motivated by recent theoretical advances. Our methods allow us to design simple, streamlined generalizations of hardware-efficient algorithms present in the literature, and to provide completely different ones, inspired by rough paths techniques, with provably lower complexity. This dual contribution demonstrates how principled theoretical analysis can both explain existing practical methods and inspire fundamentally new computational approaches.

Paper Structure

This paper contains 21 sections, 10 theorems, 88 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Linear attention transformers as associative linear RNNs
  3. Sequential vs parallel form tradeoffs
  4. Gated linear transformers and SSMs
  5. Contributions
  6. CDEs: a unifying framework for autoregressive sequence models.
  7. Flows as series of iterated integrals
  8. Parallel Flows
  9. Low-Rank Delta Rule
  10. Discretization and Intra-Chunk computation
  11. A signature kernel inspired algorithm
  12. An alternative representation of the flow as product of exponentials
  13. Numerics
  14. Conclusions and Future Directions
  15. Theoretical Proofs
  16. ...and 6 more sections

Key Result

Proposition 2.1

Given matrix-valued paths $\boldsymbol{\omega}, \boldsymbol{\xi}: [0, 1] \to \mathbb{R}^{d \times d}$, the CDE defined by can be solved on any interval $[s, t] \subseteq [0,1]$ as

Figures (2)

  • Figure 1: Pytorch code showing a simple implementation of the algorithm.
  • Figure 2: Pytorch code showing a simple implementation of the sigDelta algorithm.

Theorems & Definitions (17)

  • Proposition 2.1: Flow
  • Remark 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Proposition 3.4: Triangular Tensor Inversion
  • Theorem 3.5
  • Proposition A.1
  • proof
  • Proposition A.2
  • proof
  • ...and 7 more