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Inevitable Trade-off between Watermark Strength and Speculative Sampling Efficiency for Language Models
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Proposes a two-reweight framework that combines unbiased watermarking with speculative sampling, proves a no-go theorem that watermark strength and sampling efficiency cannot both be maximized for vocabulary size above two, and gives two algorithms (MWS, MSE) trading one off against the other.
Problem
Unbiased watermarking of LLM outputs slows generation, while speculative sampling accelerates generation; it is unclear whether both can be achieved together without losing watermark strength or sampling efficiency.
Approach
Reweight both target and draft distributions with the same watermark code so that the watermarked draft and target overlap more, then apply speculative sampling. A no-go theorem shows that keeping both the full watermark strength and the full speculative efficiency is impossible for non-trivial reweighting when the vocabulary exceeds two; two algorithms each keep one property.
Key results
- MSE matches VSpS sampling efficiency (AATPS about 1.551 at K=1 vs 1.5508 for VSpS) but yields much lower watermark strength (ANLPPT maximin-LLR 0.640 vs 1.71 for VUW with DeltaGumbel at K=1).
- MWS matches VUW watermark strength (ANLPPT maximin-LLR 1.71 for both at K=1 DeltaGumbel) while achieving AATPS 1.51 vs 1.0 for VUW and lower PTT (23.24 vs 30.40 ms).
- At K=4 with DeltaGumbel, MWS gives AATPS 1.987 vs 2.132 for VSpS, showing only a marginal efficiency gap relative to VSpS while keeping watermark strength.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MSE instead of VSpS with DeltaGumbel watermarking | speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | AATPS: 1.5508\pm0.0017 -> 1.5494\pm0.0017 | No clear effect |
| Use MSE instead of VSpS with Gamma watermarking | speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | AATPS: 1.5508\pm0.0017 -> 1.5504\pm0.0017 | No clear effect |
| Use MWS rather than VUW with DeltaGumbel watermarking | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | ANLPPT (U Score): 0.376\pm0.009 -> 0.374\pm0.009 | No clear effect |
| Use MWS rather than VUW with Gamma watermarking | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | ANLPPT (U Score): 0.097\pm0.002 -> 0.098\pm0.002 | No clear effect |
| Use MWS rather than VSpS | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | AATPS: 1.5508\pm0.0017 -> 1.5105\pm0.0017 | Hurt |
| Use MWS rather than VSpS | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | AATPS: 1.5508\pm0.0017 -> 1.5312\pm0.0017 | Hurt |
| Use MSE instead of VUW with DeltaGumbel watermarking | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | ANLPPT (U Score): 0.376\pm0.009 -> 0.153\pm0.004 | Hurt |
| Use MSE instead of VUW with Gamma watermarking | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | ANLPPT (U Score): 0.097\pm0.002 -> 0.0433\pm0.0012 | Hurt |
| Use MSE instead of VUW with DeltaGumbel watermarking | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | ANLPPT (maximin-LLR): 1.71\pm0.03 -> 0.640\pm0.014 | Hurt |
| Use MSE instead of VUW with Gamma watermarking | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | ANLPPT (maximin-LLR): 0.272\pm0.005 -> 0.0605\pm0.0019 | Hurt |
| Use MWS instead of VUW with DeltaGumbel watermarking | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization, K=1 | PTT | Helped |
| Use MSE instead of VUW | watermarked speculative sampling with LLaMa-7b · CNN/DailyMail summarization | LOGPPL | No clear effect |
Surprising findings
- Naively combining speculative sampling with unbiased watermarking is not sufficient: maintaining both watermark strength and efficiency is provably impossible for vocabulary size above two under the framework.
- The MSE method sharply reduces watermark strength (e.g., ANLPPT maximin-LLR 0.640 vs 1.71 for DeltaGumbel at K=1), showing the trade-off is substantial in practice.
Limitations
- Theory is restricted to the two reweight framework; other frameworks might yield different results.
- Experiments use small models (Llama-7b/13b with Llama-68m draft) and basic speculative sampling, not state-of-the-art acceleration.
- MWS reduces speculative efficiency, and its slower speed may create a side channel revealing the watermarking algorithm.
Experimental setup
Datasets: CNN/DailyMail
Benchmarks: CNN/DailyMail text summarization open-ended text generation
Metrics: AATPS ANLPPT (U score, maximin-LLR) PTT (ms per token) LOGPPL
Baselines: Basic Sampling Vanilla Unbiased Watermark (VUW) Vanilla Speculative Sampling (VSpS)
Builds on: speculative sampling unbiased watermarking (Kuditipudi-style reweighting with context code history) DeltaGumbel reweight Gamma reweight
Closest papers by meaning
- Break the Trade-off Between Watermark Strength and Speculative Sampling Efficiency for Language Models (ICLR 2026 · similarity 0.95)
- A Resilient and Accessible Distribution-Preserving Watermark for Large Language Models (ICML 2024 · similarity 0.91)
- Watermarks in the Sand: Impossibility of Strong Watermarking for Language Models (ICML 2024 · similarity 0.91)
- Theoretically Grounded Framework for LLM Watermarking: A Distribution-Adaptive Approach (NeurIPS 2025 · similarity 0.90)
- On the Learnability of Watermarks for Language Models (ICLR 2024 · similarity 0.90)
- Unbiased Watermark for Large Language Models (ICLR 2024 · similarity 0.90)
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