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Toward generalizable learning of all (linear) first-order methods via memory augmented Transformers

Sanchayan Dutta, Suvrit Sra

TL;DR

This work demonstrates that memory-augmented Transformers (Memformers) can implement and learn the full class of linear first-order methods (LFOMs), including gradient descent, momentum methods, and conjugate gradient descent, by maintaining and updating memory across layers. The authors provide a theoretical framework showing how Memformers can realize LFOM iterations and CGD-like updates, and they validate these claims with empirical experiments on random linear regression tasks, where Memformers achieve competitive or superior performance under various preconditioning and architectural setups. They further enhance robustness and adaptation through multi-head attention and mixture-of-experts, enabling test-time adaptation to distribution shifts (OOD) and demonstrating LFOMs as learnable meta-algorithms with generalization guarantees. Overall, the paper positions Memformers as versatile, learnable optimizers capable of capturing a wide range of first-order methods and adapting to changing data distributions, with potential implications for accelerating optimization in large-scale models and in-context learning scenarios.

Abstract

We show that memory-augmented Transformers can implement the entire class of linear first-order methods (LFOMs), a class that contains gradient descent (GD) and more advanced methods such as conjugate gradient descent (CGD), momentum methods and all other variants that linearly combine past gradients. Building on prior work that studies how Transformers simulate GD, we provide theoretical and empirical evidence that memory-augmented Transformers can learn more advanced algorithms. We then take a first step toward turning the learned algorithms into actually usable methods by developing a mixture-of-experts (MoE) approach for test-time adaptation to out-of-distribution (OOD) samples. Lastly, we show that LFOMs can themselves be treated as learnable algorithms, whose parameters can be learned from data to attain strong performance.

Toward generalizable learning of all (linear) first-order methods via memory augmented Transformers

TL;DR

This work demonstrates that memory-augmented Transformers (Memformers) can implement and learn the full class of linear first-order methods (LFOMs), including gradient descent, momentum methods, and conjugate gradient descent, by maintaining and updating memory across layers. The authors provide a theoretical framework showing how Memformers can realize LFOM iterations and CGD-like updates, and they validate these claims with empirical experiments on random linear regression tasks, where Memformers achieve competitive or superior performance under various preconditioning and architectural setups. They further enhance robustness and adaptation through multi-head attention and mixture-of-experts, enabling test-time adaptation to distribution shifts (OOD) and demonstrating LFOMs as learnable meta-algorithms with generalization guarantees. Overall, the paper positions Memformers as versatile, learnable optimizers capable of capturing a wide range of first-order methods and adapting to changing data distributions, with potential implications for accelerating optimization in large-scale models and in-context learning scenarios.

Abstract

We show that memory-augmented Transformers can implement the entire class of linear first-order methods (LFOMs), a class that contains gradient descent (GD) and more advanced methods such as conjugate gradient descent (CGD), momentum methods and all other variants that linearly combine past gradients. Building on prior work that studies how Transformers simulate GD, we provide theoretical and empirical evidence that memory-augmented Transformers can learn more advanced algorithms. We then take a first step toward turning the learned algorithms into actually usable methods by developing a mixture-of-experts (MoE) approach for test-time adaptation to out-of-distribution (OOD) samples. Lastly, we show that LFOMs can themselves be treated as learnable algorithms, whose parameters can be learned from data to attain strong performance.

Paper Structure

This paper contains 29 sections, 9 theorems, 70 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Work
  4. Background and Problem Setup
  5. Linear Transformers on Random Linear Regression
  6. Linear First-Order Methods (LFOMs)
  7. Memformers Can Implement LFOMs
  8. Single Dynamic Memory for CGD-like Algorithms
  9. Implementing k Steps of LFOM with Separate Memory Registers at Each Layer $\ell$
  10. Experimental Results
  11. Influence of Batch Size on Performance
  12. Multi-Headed Attention and Out-of-Distribution (OOD) Performance
  13. OOD Adaptation using Multi-Headed Attention
  14. Test-Time Adaptation and Gated Memory
  15. LFOMs as Learnable Algorithms
  16. ...and 14 more sections

Key Result

Lemma 2.1

Let an $L$-layer linear transformer be parameterized by $\mathbf{A}_0, \dots, \mathbf{A}_{L-1}$, as in params_Thm3. Let $y_\ell^{(n+1)} = [\mathbf{Z}_\ell]_{(d+1),(n+1)}$ for $\ell = 1, \dots, L$; then, where the sequence $\{\mathbf{w}_{\ell}^{\mathrm{gd}}\}$ is defined as $\mathbf{w}_{0}^{\mathrm{gd}} = 0$ and for $\ell = 1, \dots, L-1$: with the empirical least-squares loss (with $\mathbf{X} :

Figures (18)

  • Figure 1: CGD-like Memformer (\ref{['eq:dynamic-mem']}) without preconditioning ($\mathbf{A}_\ell = \mathbf{I}$) vs. actual CGD running separately on each test sample. Test data is drawn from the same distribution as the training data.
  • Figure 2: CGD-like Memformer (\ref{['eq:dynamic-mem']}) with preconditioning ($\mathbf{A}_\ell\neq \mathbf{I}$). This yields a more general LFOM-like scheme, often outperforming CGD, Nesterov AGM and momentum GD. Test data is independently drawn from the same distribution as training data.
  • Figure 3: LFOM Memformer (\ref{['LFOM_memory']}) vs. CGD, Nesterov AGM and momentum GD (Pre = non-trivial preconditioners).
  • Figure 4: LFOM Memformer with GD++ (\ref{['LFOM_memory']}) vs. CGD, where the $\mathbf{B}_\ell$ blocks \ref{['params_Thm3']} approximate the Hessian inverse (quasi-Newton).
  • Figure 5: LFOM Memformer \ref{['LFOM_memory']} with scalar preconditioners $\Gamma_\ell$ vs. CGD performance on small batch size ($B = 1$). The Memformer demonstrates superior performance.
  • ...and 13 more figures

Theorems & Definitions (12)

  • Lemma 2.1: Lemma 1, ahn2024transformers
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 4.1: Multi-Head Memformer with Soft Gating
  • Theorem 5.1: Statistical Learnability of LFOMs in the In-Context Setting
  • Theorem
  • Theorem
  • Theorem
  • proof
  • ...and 2 more