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From Noise to Diversity: Random Embedding Injection in LLM Reasoning

This paper demonstrates that injecting training-free random embedding vectors into Large Language Model inputs effectively enhances reasoning performance by increasing early-stage token diversity and widening Pass@N, a mechanism that improves both inference accuracy and training outcomes without requiring learned soft prompts.

Original authors: Heejun Kim, Seungpil Lee, Jewon Yeom, Jaewon Sok, Seonghyeon Park, Jeongjae Park, Taesup Kim, Sundong Kim

Published 2026-05-13

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CC BY 4.0

Original authors: Heejun Kim, Seungpil Lee, Jewon Yeom, Jaewon Sok, Seonghyeon Park, Jeongjae Park, Taesup Kim, Sundong Kim

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a brilliant but very rigid librarian (the Large Language Model) who is great at solving math problems, but sometimes gets stuck in a rut. They always follow the same path to the answer, and if that path leads to a dead end, they give up.

For a long time, researchers tried to fix this by "training" the librarian with special, custom-written notes (called Soft Prompts) to guide them toward better solutions. They assumed the magic was in the content of those notes.

This paper asks a different question: What if the magic isn't the notes themselves, but the simple act of handing the librarian a new, random piece of paper before they start?

Here is the breakdown of their discovery, Random Soft Prompts (RSP), using everyday analogies:

1. The "Random Noise" Experiment

Instead of training a special note, the researchers just grabbed a handful of completely random, meaningless scribbles (random numbers) and attached them to the front or back of the math problem.

The Surprise: Even though these scribbles contained zero useful information, the librarian suddenly started solving problems better. In some cases, they performed just as well as if they had been given a carefully trained, perfect note.
The Lesson: The improvement didn't come from learning anything new. It came from the act of injecting something new into the mix.

2. How It Works: The "Branching Path" Analogy

Think of the librarian's thought process as walking down a hallway with many doors.

Without the Random Paper: The librarian walks straight down the main hall, opens the first door they see, and keeps walking that path. If that path is a dead end, they fail.
With the Random Paper: The random scribbles act like a sudden, confusing gust of wind at the very start of the hallway.
- Early Stage (The Branch): Because of this "wind," the librarian pauses and looks at different doors than they usually would. They might open a door they never considered before. This creates a "branch" in their thinking, leading to a completely different route to the solution.
- Later Stage (The Stabilization): As the librarian walks further down this new path, the "wind" (the random paper) fades away because the hallway gets longer and the librarian's memory of the start gets diluted. They settle into a confident stride on this new path.

3. The "Rolling Dice" Effect (Pass@N)

The paper found that if you ask the librarian to try the same problem 10 times, but give them a different random scribble each time, you get 10 different paths.

The Result: Even if 9 of those paths are wrong, the fact that you forced the librarian to try 10 different starting points means you are much more likely to find the one correct path among them.
The Catch: If you use the same random scribble for all 10 tries, they all take the same wrong path. The magic only works if the "wind" changes every time.

4. Why This Matters

It's Free: You don't need to spend weeks training the model or teaching it new facts. You just add random noise.
It's Structural: The paper proves that the "magic" of these advanced AI tricks often comes from the structure of adding extra tokens, not the specific content of those tokens. It's like how shaking a box of puzzle pieces might help you find the right corner piece faster, even if the shaking itself doesn't solve the puzzle.
It Helps Training: They also showed that using this random "wind" during the model's training (teaching phase) helps the model learn faster and better, similar to how a coach might vary drills to keep an athlete from getting bored or stuck in a routine.

Summary

The paper claims that randomness is a powerful tool. By injecting random, meaningless vectors into the AI's input, you force it to explore different "roads" to the answer early on. Once it picks a road, it stays on it, but because you started with a different road, you might end up at the destination more often than if you just stuck to the usual route. It turns out that sometimes, shaking things up is just as effective as teaching something new.

Technical Summary: From Noise to Diversity: Random Embedding Injection in LLM Reasoning

Problem Statement

Recent research in Large Language Model (LLM) reasoning has focused on Soft Prompts, which insert trainable continuous embedding vectors into model inputs to steer behavior. While these methods (e.g., Prefix-Tuning, SoftCoT, LTPO) have demonstrated performance gains, a critical ambiguity remains: is the improvement driven by the learned content of the optimized vectors, or by the structural act of injection itself? Prior work attributes gains to learned representations, yet theoretical constraints suggest soft prompts only bias attention in fixed directions without altering relative attention patterns among content tokens. Furthermore, existing methods require domain-specific training, are sensitive to hyperparameters, and lack a unified theoretical explanation for their efficacy.

Methodology: Random Soft Prompts (RSP)

To disentangle the contribution of learned content from the act of injection, the authors propose Random Soft Prompts (RSPs). RSPs are a training-free intervention where a sequence of random embedding vectors is appended to the input.

Construction: RSP vectors are sampled independently from an isotropic Gaussian distribution fitted to the entrywise mean ( $\mu_E$ ) and variance ( $\sigma_E$ ) of the pretrained token embedding table.
$h_j \sim \mathcal{N}(\mu_E \mathbf{1}_d, \sigma_E^2 \mathbf{I}_d)$
Mechanism: The method relies on two theoretical phases:
1. Exploration (Early Decoding): When the Key-Value (KV) cache is short, the attention mass ( $w$ ) allocated to the random tokens is significant. Because the RSP is isotropic, it perturbs the hidden state in all directions, potentially shifting the top- $K$ token predictions and opening diverse reasoning trajectories that differ from the baseline.
2. Annealing (Later Decoding): As autoregressive generation proceeds, the KV cache grows. Theoretical analysis (Theorem 1) demonstrates that the upper bound on the attention mass allocated to the fixed-length RSP sequence decays monotonically as the number of real tokens ( $n$ ) increases. Consequently, the influence of the RSP naturally dilutes, allowing the model to commit to a single completion path.
Implementation: RSPs are injected as a suffix (default), prefix, or infix. Crucially, they are freshly resampled for every inference rollout or training trajectory, ensuring independence.

Key Contributions

Isolation of Injection Effects: RSP provides a unified lens to study the structural effect of soft prompt injection. By removing the training step entirely, the authors demonstrate that the act of injecting random vectors alone can induce accuracy comparable to optimized soft prompts.
Theoretical and Empirical Validation: The paper establishes a theoretical framework showing how attention mass decays with KV cache growth (Theorem 1) and how isotropic coverage ensures reachability of diverse branching states. Empirically, RSPs replicate the early-decoding entropy signatures of optimized methods (LTPO, SoftCoT, TTSV) without any learned parameters.
Extension to Training: The authors extend RSP from inference to DAPO (a reinforcement learning training algorithm), demonstrating that independent RSP resampling during rollouts composes with loss-level diversity preservation to yield further performance gains.

Experimental Results

The authors evaluated RSP on mathematical reasoning benchmarks (MATH-500, GSM8K, AIME24) using models like Qwen2.5-Math and LLaMA-3.1.

Accuracy Gains: RSP achieves accuracy comparable to or exceeding optimized soft prompts (TTSV, SoftCoT, LTPO) without training. Notably, on format-mismatched scenarios (applying ChatML templates to base models not trained for them), RSP recovered significant accuracy (e.g., +29.0 percentage points on Qwen2.5-Math-1.5B for MATH-500).
Diversity and Pass@N:
- Token-Level: RSP induces a distinct shift in the first-token distribution (Spearman $\rho = 0.709$ vs. 1.0 for temperature scaling), expanding support beyond the baseline's top-10 tokens.
- Trajectory-Level: RSP generates semantically diverse reasoning paths (higher inter-cluster distance) while maintaining within-group coherence.
- Outcome-Level: Pass@N (probability of at least one correct answer in $N$ attempts) scales significantly with independent resampling. Sharing a single RSP across samples yields no gain, confirming that diversity arises from the stochastic injection, not a fixed perturbation.
Training Integration: In DAPO training, adding RSP to rollouts resulted in a higher peak average accuracy (54.36% vs. 53.20% for DAPO alone) and delayed performance degradation, suggesting improved exploration-exploitation dynamics.

Significance and Claims

The paper claims that a substantial portion of the success attributed to "learned" soft prompts is actually a structural by-product of the injection mechanism itself. The act of introducing random, isotropic perturbations at the input layer:

Lifts early-stage token diversity, enabling the model to explore reasoning trajectories that might be inaccessible via standard temperature scaling (which preserves rank order).
Automatically attenuates as generation proceeds, preventing the random noise from destabilizing the final answer.
Serves as a control that reveals the limitations of prior work's assumption that learned content is the primary driver of reasoning improvements.

The authors conclude that RSP offers a simple, training-free alternative for enhancing reasoning diversity and accuracy, particularly in scenarios where format alignment is poor or where independent trajectory exploration is critical for RL-based training. They note limitations regarding the theoretical explanation of layer-wise decay and the applicability to non-mathematical tasks, framing the work as a foundational analysis of injection dynamics rather than a deployed solution.