Goose: Anisotropic Speculation Trees for Training-Free Speculative Decoding

Speculative decoding accelerates large language model inference by drafting multiple candidate tokens and verifying them in a single forward pass. Candidates are organized as a tree: deeper trees accept more tokens per step, but adding depth requires sacrificing breadth (fallback options) under a fixed verification budget. Existing training-free methods draft from a single token source and shape their trees without distinguishing candidate quality across origins. We observe that two common training-free token sources -- n-gram matches copied from the input context, and statistical predictions from prior forward passes -- differ sharply in acceptance rate (~6x median gap, range 2-18x across five models and five benchmarks). We prove that when such a quality gap exists, the optimal tree is anisotropic (asymmetric): reliable tokens should form a deep chain while unreliable tokens spread as wide branches, raising the depth ceiling of balanced trees. We realize this structure in GOOSE, a training-free framework that builds an adaptive spine tree: a deep chain of high-acceptance context-matched tokens with wide branches of low-acceptance alternatives at each node. The resulting tree provably accepts at least as many tokens per step as either source alone. On five LLMs (7B-33B) and five benchmarks, GOOSE achieves 1.9-4.3x lossless speedup, outperforming balanced-tree baselines by 12-33% under the same budget.