When Is a Draft Accepted? A Theory of Acceptance in Speculative Decoding

Speculative decoding accelerates large language model (LLM) inference by using a lightweight draft model to generate multiple candidate tokens that are verified by the target model in a single forward pass. Its speedup is largely determined by the acceptance length, yet existing draft-model training methods mainly optimize cross-entropy or Kullback-Leibler (KL) divergence as proxies. These objectives encourage distribution matching but do not directly optimize acceptance length, and the acceptance mechanism also differs between greedy and sampling-based decoding. In this work, we propose acceptance-length-aware training losses that directly optimize the expected number of accepted tokens within a speculative window. For greedy verification, we derive an expected accepted length (EAL) loss that explicitly maximizes expected acceptance length. For sampling-based decoding, we introduce a window total variation (WTV) loss that optimizes the overlap between temperature-scaled draft and target distributions while accounting for sequential acceptance dependencies. Both objectives can be further combined with a group-relative reinforcement learning stage (GRPO) using simulated acceptance length as the reward. Experiments across different target and draft models, tasks, and decoding settings show that our losses consistently improve acceptance length over KL-based training. WTV provides particularly strong gains under sampling-based decoding, while EAL better matches greedy verification. These results show that directly optimizing the acceptance objective, with losses tailored to the decoding mode, is more effective than conventional distribution-matching objectives.

Speculative decoding accelerates large language model (LLM) inference. It uses a small draft model to propose a tree of future tokens. A larger target model then verifies these tokens in a single batched forward pass. Despite the growing body of work on speculative methods, the degree to which the cognitive characteristics of a task affect acceptance probability remains largely unexplored. We present an empirical study of tree-based speculative decoding acceptance dynamics. Our study spans four well-established NLP benchmark domains: code generation, mathematical reasoning, logical reasoning, and open-ended chat. For this, we use TinyLlama-1.1B as the draft model against Llama-2-7B-Chat-GPTQ as the target. Over 99,768 speculative nodes collected from 200 prompts, we derive per-domain acceptance rates, expected accepted lengths, depth-acceptance profiles, and entropy-acceptance correlations. We find that task type is a stronger predictor of acceptance than tree depth. Furthermore, only the chat domain consistently yields an expected accepted length exceeding 1.0 token per step. We also show that the entropy-acceptance correlation is consistently negative but weak across all domains (rho in [-0.20, -0.15]). Counterintuitively, chat produces the highest entropy yet the highest acceptance rate. We attribute this divergence to the lexical predictability of RLHF-aligned register. These findings have direct implications for domain-aware speculation budgets and draft-model selection strategies. Index Terms--speculative decoding, large language model inference, tree attention, draft model, acceptance probability, LLM efficiency

Single-stream autoregressive decoding of large language models is bound by memory bandwidth: each generated token requires one full forward pass through the target model, and successive passes cannot be parallelized. Speculative decoding restructures this computation: a small draft model proposes $K$ tokens autoregressively, the target model scores all of them in one batched pass, and a rejection-sampling rule provably preserves the target model's output distribution. We present a from-scratch, device-agnostic (CUDA/MPS/CPU) implementation and an empirical study across five draft/target backend configurations on a consumer Apple-silicon laptop. Distribution equivalence is verified at three levels, culminating in a two-sample test over roughly 9,200 real-model tokens per method ($χ^2 = 162.5$, dof $= 200$, $p = 0.976$) and exact greedy-sequence agreement. The best configuration reaches a measured $1.61\times$ wall-clock speedup at $K=6$, on an acceptance profile declining from 69.7% at $K=1$ to 37.8% at the optimum, while three of five configurations decelerate, either because the draft fails to out-speed a small target or because the quantized Metal backend executes "parallel" verification serially, an effect we isolate and quantify. The failures are as instructive as the successes: speculative decoding pays off only when verification is genuinely batch-parallel and the draft/target latency gap is real.