# Shallow Queries, Mature Values: Depth-Asynchronous Self-Speculation for Looped Transformers

arXiv: https://arxiv.org/abs/2609.34538 · HTML: https://mlsanity.com/paper/2609.34538/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 23 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [Fast Inference from Transformers via Speculative Decoding](https://mlsanity.com/paper/2211.17192/) (2022)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Accelerating Large Language Model Decoding with Speculative Sampling](https://mlsanity.com/paper/2302.01318/) (2023)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [LayerSkip: Enabling Early Exit Inference and Self-Speculative Decoding](https://mlsanity.com/paper/2404.16710/) (2024)
- [Break the Sequential Dependency of LLM Inference Using Lookahead Decoding](https://mlsanity.com/paper/2402.02057/) (2024)
- [Looped Transformers as Programmable Computers](https://mlsanity.com/paper/2301.13196/) (2023)
- [Relaxed Recursive Transformers: Effective Parameter Sharing with Layer-wise LoRA](https://mlsanity.com/paper/2410.20672/) (2024)
- [SPEED: Speculative Pipelined Execution for Efficient Decoding](https://mlsanity.com/paper/2310.12072/) (2023)
- [Turning Trash into Treasure: Accelerating Inference of Large Language Models with Token Recycling](https://mlsanity.com/paper/2408.08696/) (2024)
- [Teaching Pretrained Language Models to Think Deeper with Retrofitted Recurrence](https://mlsanity.com/paper/2511.07384/) (2025)
- [SAM Decoding: Speculative Decoding via Suffix Automaton](https://mlsanity.com/paper/2411.10666/) (2024)
- [SCOUT: Teaching Pre-trained Language Models to Enhance Reasoning via Flow Chain-of-Thought](https://mlsanity.com/paper/2505.24181/) (2025)
- [Efficient Parallel Samplers for Recurrent-Depth Models and Their Connection to Diffusion Language Models](https://mlsanity.com/paper/2510.14961/) (2025)
- [Depth-adaptive Inference of Looped Language Models via Continuous Depth Batching](https://mlsanity.com/paper/2608.09444/) (2026)
- [LoopSpec: Pipelined Self-Speculative Decoding for Looped Transformers](https://mlsanity.com/paper/2609.17184/) (2026)

## Abstract (arXiv)

Looped Transformers reuse a shared block across recurrent depths, making autoregressive decoding expensive because every generated token requires many sequential recurrent passes. Self-speculative decoders reduce this cost by drafting at an early depth and verifying at full depth, but typically bind draft computation to prefix representations from the same recurrent depth. We find that queries and keys approach their final-depth representations earlier than values, and controlled prefix-channel interventions show that mature values substantially improve shallow draft predictions. Motivated by this asymmetry, we introduce Depth-Asynchronous Self-Speculation (DAS), which decouples the depth of draft computation from the depth of verified-prefix representations it reads. Its Mature-V primitive lets shallow queries retrieve full-depth prefix values without additional recurrent computation. We further develop DAS-Wave, which combines depth-asynchronous prefix reads with carried parallel refinement, progressive block growth, and an independent full-depth verifier. Across four recurrent-model checkpoints and mathematics and code workloads, DAS-Wave achieves 4.00--6.96× mean throughput speedup over paired full-depth autoregressive decoding in the same inference stack. These results identify prefix-information depth as an effective design axis for recurrent self-speculation.