Speculative Decoding: Lossless Multi-Token Generation

Draft several tokens with a fast model, score draft prefixes with the target model in parallel, then use modified rejection/residual sampling so the sampled distribution matches target-model decoding.

published · difficulty 4/5 · 18 min read

Intuition

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Autoregressive decoding is painfully sequential: one forward pass per token.

Speculative decoding gets around this by splitting work into two roles:

a fast draft model proposes a chunk of $k$ next tokens,
the expensive target model scores the prompt plus each draft prefix in parallel.

In the full algorithm, the target produces $k+1$ next-token distributions: one for the current prefix, one after each draft prefix, and an extra distribution that can supply a new target token if every draft token is accepted. If draft tokens pass the target/draft probability-ratio checks, you accept a prefix of the draft and "skip ahead" in the sequence. If one fails, you reject at the first failed token and repair that position with a residual sample so the target distribution is preserved. The key promise is that this is lossless: the final distribution of generated text is exactly what you would have sampled from the target model alone.

So the win is systems-level: fewer expensive sequential target steps, same output distribution.

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Math

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Equation 1Inspect equation object\alpha_i = \min (1, (p_i[x_i])/(q_i[x_i])).Equation 2Inspect equation objectx_i \sim Normalize (\max(0, p_i - q_i)).

Let $q_i$ be the draft distribution at position $i$ and $p_i$ be the target distribution. This page follows Leviathan et al.'s notation where $p$ is the target and $q$ is the draft; Chen et al. use the opposite symbols. If the draft proposes token $x_i$ , the acceptance probability is:

\alpha_i = \min\!\left(1, \frac{p_i[x_i]}{q_i[x_i]}\right).

If a proposed token is rejected, you sample from a residual distribution that corrects for what the draft already "used up":

x_i \sim \mathrm{Normalize}\!\left(\max(0, p_i - q_i)\right).

This rejection/residual mechanism is what makes the method lossless: accepted tokens come from $q$ but are filtered to match $p$ , and rejected positions are repaired to restore the exact $p$ distribution.

A rough intuition for speedup: if the draft is often right, you accept a long prefix of the $k$ proposed tokens, and the target model advances several tokens per verification step.

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Code

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TraceMatch variables to symbols before reading the implementation.Leave with a runnable witness for the math.

Code witness 1import numpy as np target_p = np.array([0.45, 0.30, 0.15, 0.10]) draft_q = np.array([0.30, 0....python

First check the one-token correction mechanism. Even though some samples come from the draft model, the accepted-or-repaired output should match the target distribution. Then simulate the accepted-prefix length that creates the latency upside.

import numpy as np

target_p = np.array([0.45, 0.30, 0.15, 0.10])
draft_q = np.array([0.30, 0.45, 0.20, 0.05])
rng = np.random.default_rng(0)

def residual_distribution(p, q):
    repair_mass = np.maximum(0.0, p - q)
    return repair_mass / repair_mass.sum()

def speculative_one_token(p, q):
    draft = rng.choice(len(q), p=q)
    accept_prob = min(1.0, p[draft] / q[draft])
    if rng.random() < accept_prob:
        return draft, True
    repair = rng.choice(len(p), p=residual_distribution(p, q))
    return repair, False

samples = np.zeros_like(target_p)
accepted = 0
for _ in range(200_000):
    token, was_accepted = speculative_one_token(target_p, draft_q)
    samples[token] += 1
    accepted += int(was_accepted)

print("target distribution:   ", np.round(target_p, 3))
print("speculative samples:   ", np.round(samples / samples.sum(), 3))
print("draft-token acceptance:", round(accepted / samples.sum(), 3))

def expected_accepted(alpha, k):
    # Expected length of consecutive acceptances (prefix), capped at k.
    if alpha == 1.0:
        return float(k)
    return alpha * (1.0 - alpha**k) / (1.0 - alpha)

def simulate(alpha, k, trials=50000, seed=0):
    rng = np.random.RandomState(seed)
    total = 0
    for _ in range(trials):
        a = 0
        for _ in range(k):
            if rng.rand() < alpha:
                a += 1
            else:
                break
        total += a
    return total / trials

for alpha in [0.3, 0.6, 0.85]:
    for k in [2, 4, 8]:
        theo = expected_accepted(alpha, k)
        mc = simulate(alpha, k)
        print(f"alpha={alpha:.2f} k={k:>2}  E[accepted] theo={theo:.3f}  mc={mc:.3f}")

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Interactive Demo

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Live Concept Demo

Explore Speculative Decoding: Lossless Multi-Token Generation

The stage is code-native and interactive. Use it to test the explanation against the mechanism.

difficulty 4/5undergraduatecode-aligned

Demo inquiry checkpoint

Manipulate one control and predict the visible change.

01Choose lensTrace a quantity

02ObserveDemo state pending

03GroundName the equation, invariant, or control that explains it.

04CarryNext: Long Context Engineering: RoPE Scaling, KV Compression & Memory Optimization

Choose what to inspect in Speculative Decoding: Lossless Multi-Token Generation. This shared fallback is an observation guide, not evidence of learning.

Loading interactive demo...

Use the demo to see how acceptance probability, draft length, and distribution mismatch affect expected speedup, and why verification is "free" only if your target model can score the chunk efficiently.

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Object

4/4 sections ready

Concept: Speculative Decoding: Lossless Multi-Token Generation

What is the smallest example that makes Speculative Decoding: Lossless Multi-Token Generation click without losing the math?

BeforeMaximum LikelihoodNow4/4 sections readyTryManipulate one control and predict the visible change.NextLong Context Engineering: RoPE Scaling, KV Compression & Memory Optimization

Object contextLLM Systems

StateLLM Systems

2 sources attachedChecking local snapshot

concept:llm-systems/speculative-decoding

Math2 equation objects\alpha_i = \min (1, (p_i[x_i])/(q_i[x_i])).CodeSpeculative Decoding: Lossless Multi-Token Generation code witness 1target_p = np.array([0.45, 0.30, 0.15, 0.10])DemoSpeculative Decoding: Lossless Multi-Token Generation predictionManipulate one control and predict the visible change.

ConceptLearner lens

Speculative Decoding: Lossless Multi-Token Generation

What is the smallest example that makes Speculative Decoding: Lossless Multi-Token Generation click without losing the math?

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4/4 sections ready

Carry inMaximum Likelihood

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Work hereSpeculative Decoding: Lossless Multi-Token Generation

Carry outLong Context Engineering: RoPE Scaling, KV Compression & Memory Optimization

The next edge should feel earned: use the demo prediction here before following Long Context Engineering: RoPE Scaling, KV Compression & Memory Optimization.

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ConceptSpeculative Decoding: Lossless Multi-Token GenerationLLM Systems

01MechanismInspect motionSee how the idea changes form before adding more explanation.02VisualChoose a cueBind one visible mark to one mathematical role.03SourcesCheck boundary2 source links attached to the object.04ClaimAudit evidenceTreat the page claim as something to verify.05PracticeAttempt transferWrite before feedback so the model has something to repair.06RoomAsk groundedAttach the question to the same exact object.

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Demo notes open01 / Intuition

Editorial systems illustration of draft tokens passing through verifier gates with accepted and rejected branches.

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4/4 stages readyDemo notes connected

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Inspection depth2/4

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Object - ConceptSpeculative Decoding: Lossless Multi-Token GenerationQuestion

What is the smallest example that makes Speculative Decoding: Lossless Multi-Token Generation click without losing the math?

concept:llm-systems/speculative-decoding

Boundary

sources: leviathan-2022-speculative-decoding, chen-2023-speculative-sampling

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2 selected-object sources shown first; 2 references total.

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selected object source · paper · 2022Fast Inference from Transformers via Speculative DecodingLeviathan, Kalman, and Matias

Located CF editorial boundary

Introduces speculative decoding as parallel draft-prefix scoring with modified rejection/residual sampling that preserves the target distribution.

Used here as

Leviathan defines gamma draft tokens from a faster approximation model, parallel target scoring of draft prefixes, accept/reject filtering, residual sampling from norm(max(0, p - q)), and...

Caveat

Checks lossless sampling under standardized sampling distributions; Chen qualifies the guarantee as within hardware numerics. Not a universal latency guarantee. Code is a toy finite-distr...

Open source

selected object source · paper · 2023Accelerating Large Language Model Decoding with Speculative SamplingChen et al.

Located CF editorial boundary

Independently grounds K-token drafts, parallel target scoring, modified rejection sampling, residual repair, and measured large-model decoding speedups.

Used here as

Leviathan defines gamma draft tokens from a faster approximation model, parallel target scoring of draft prefixes, accept/reject filtering, residual sampling from norm(max(0, p - q)), and...

Caveat

Checks lossless sampling under standardized sampling distributions; Chen qualifies the guarantee as within hardware numerics. Not a universal latency guarantee. Code is a toy finite-distr...

Open source

Claim Review

Draft several tokens with a fast model, score draft prefixes with the target model in parallel, then use modified rejection/residual sampling so the sampled distribution matches target-model decoding.

Object - ConceptSpeculative Decoding: Lossless Multi-Token GenerationQuestion

What is the smallest example that makes Speculative Decoding: Lossless Multi-Token Generation click without losing the math?

concept:llm-systems/speculative-decoding

Boundary

sources: leviathan-2022-speculative-decoding, chen-2023-speculative-sampling

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1 CF editorial source-scope review recorded

Publisher-side editorial review is not independent replication. Claims without it still need exact source-support review. 2 references and 3 local witnesses are available for inspection.

Speculative decoding uses a fast draft model to propose multiple tokens and a target model to verify them in parallel, using rejection/residual sampling so the final output distribution matches target-model decoding.

Registered CF editorial boundary

Fast Inference from Transformers via Speculative DecodingIntroduces speculative decoding as parallel draft-prefix scoring with modified rejection/residual sampling that preserves the target distribution.Accelerating Large Language Model Decoding with Speculative SamplingIndependently grounds K-token drafts, parallel target scoring, modified rejection sampling, residual repair, and measured large-model decoding spee...

Used here as

Leviathan defines gamma draft tokens from a faster approximation model, parallel target scoring of draft prefixes, accept/reject filtering, residual sampling from norm(max(0, p - q)), and proves target-distr...

Local witness

source-span-leviathan-2022-speculative-decoding source-span-chen-2023-speculative-sampling math-object-1 math-object-2 code-witness-1 interactive-demo

Equation 1

\alpha_i = \min\!\left(1, \frac{p_i[x_i]}{q_i[x_i]}\right).

Equation 2

x_i \sim \mathrm{Normalize}\!\left(\max(0, p_i - q_i)\right).

Code witness 1import numpy as np target_p = np.array([0.45, 0.30, 0.15, 0.10]) draft_q = np.array([0.30, 0....

Caveat

Checks lossless sampling under standardized sampling distributions; Chen qualifies the guarantee as within hardware numerics. Not a universal latency guarantee. Code is a toy finite-distribution/residual che...

Review stateCF editorial source-scope reviewClaim metadata: source checkedPublisher-side editorial review only; not independent replication. Check caveats and exact source scope.

Leviathan 2022 supports gamma-token drafting, parallel target scoring of draft prefixes, accept/reject filtering, residual sampling from norm(max(0, p - q)), and a proof of target-distribution sampling. Chen 2023 independently supports K-token drafts, parallel scoring, modified rejection/residual sampling, and hardware-numerics-qualified preservation, with reversed p/q notation. Page math follows Leviathan; code is a toy residual/distribution plus prefix check, and demo is toy speedup only.

Reviewer: codex+oracle; reviewed 2026-05-07

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Object - ConceptSpeculative Decoding: Lossless Multi-Token GenerationQuestion

What is the smallest example that makes Speculative Decoding: Lossless Multi-Token Generation click without losing the math?

concept:llm-systems/speculative-decoding

Boundary

sources: leviathan-2022-speculative-decoding, chen-2023-speculative-sampling

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Selected object routeAsk from this object; carry one invariant back.sources: leviathan-2022-speculative-decoding, chen-2023-speculative-sampling

ObjectConceptSpeculative Decoding: Lossless Multi-Token Generation
PredictBefore revealSpeculative Decoding: Lossless Multi-Token Generation prediction
WitnessCompare codeSpeculative Decoding: Lossless Multi-Token Generation code witness 1
RoomAsk groundedChecking local snapshot

ConceptSpeculative Decoding: Lossless Multi-Token GenerationLLM Systems

Code witness comparisonSpeculative Decoding: Lossless Multi-Token Generation code witness 1target_p = np.array([0.45, 0.30, 0.15, 0.10])Prediction before revealSpeculative Decoding: Lossless Multi-Token Generation predictionManipulate one control and predict the visible change.

Grounded room questionWhat is the smallest example that makes Speculative Decoding: Lossless Multi-Token Generation click without losing the math?Checking local snapshot

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conceptLLM Systems

Speculative Decoding: Lossless Multi-Token Generation

Anchored question

What is the smallest example that makes Speculative Decoding: Lossless Multi-Token Generation click without losing the math?

Source boundaryInspect source ids: leviathan-2022-speculative-decoding, chen-2023-speculative-samplingStable content-object key attached

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Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo.

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The learner can state the mechanism in their own words

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01PredictionChecking browser-local route memory

02EvidenceChecking for a carried observation

03BoundaryInspect source ids: leviathan-2022-speculative-decoding, chen-2023-speculative-sampling

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What would resolve this

The learner can state the mechanism in their own words
The learner can name the prerequisite that would repair confusion
The learner can predict how the mechanism changes under one perturbation

Object-attached AI handoff

I am working in Continuous Function's research reading room. Object: concept - Speculative Decoding: Lossless Multi-Token Generation Object key: concept:llm-systems/speculative-decoding Context: LLM Systems Anchor id: concept/concept-notebook/llm-systems/speculative-decoding Open question: What is the smallest example that makes Speculative Decoding: Lossless Multi-Token Generation click without losing the math? Evidence to inspect: - Source ids to inspect: leviathan-2022-speculative-decoding, chen-2023-speculative-sampling - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan Deterministic role lenses for this object: - Boundary: fixed perspectives, not people, community contributions, or independent review - Source-checking summary: Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Teach/transfer move: Turn the mechanism into one sentence that predicts a neighboring concept. - Assumptions: - Source ids leviathan-2022-speculative-decoding, chen-2023-speculative-sampling must support the exact object, not just the surrounding topic. - The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. - The concept explanation is local atlas prose until checked against its math, code, and source support. - Prerequisite gaps should become a repair route, not a reason to leave the object vague. - Role-lens requests: - Learner: ask for "Ask what would make "Speculative Decoding: Lossless Multi-Token Generation" feel predictable rather than familiar." | assumption: Source ids leviathan-2022-speculative-decoding, chen-2023-speculative-sampling must support the exact object, not just the surrounding topic. | next action: The learner can state the mechanism in their own words - Researcher: ask for "Source ids to inspect: leviathan-2022-speculative-decoding, chen-2023-speculative-sampling" | assumption: The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. | next action: The learner can name the prerequisite that would repair confusion - Experimenter: ask for "Choose one variable or condition to perturb before asking for an explanation." | assumption: The concept explanation is local atlas prose until checked against its math, code, and source support. | next action: The learner can predict how the mechanism changes under one perturbation - Professor: ask for "Find the smallest transferable rule a learner could reuse without the AI." | assumption: Prerequisite gaps should become a repair route, not a reason to leave the object vague. | next action: Teach or transfer: Turn the mechanism into one sentence that predicts a neighboring concept. What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action. Current deterministic role lens for this object: - Role lens: Learner - Evidence request: Ask what would make "Speculative Decoding: Lossless Multi-Token Generation" feel predictable rather than familiar. - Assumption to keep visible: Source ids leviathan-2022-speculative-decoding, chen-2023-speculative-sampling must support the exact object, not just the surrounding topic. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Next action: The learner can state the mechanism in their own words

concept/concept-notebook/llm-systems/speculative-decoding
concept:llm-systems/speculative-decoding

Learning Map

Before / Now / Try / Next

BeforeMaximum Likelihood

NowIntuition → Math → Code → Demo

TryManipulate one control and predict the visible change.

NextLong Context Engineering: RoPE Scaling, KV Compression & Memory Optimization

Learnerpicture first Researchersource check Experimentertest the demo Professorname invariant

01Predict 02Ground 03Code 04Carry

01Intuitionready
02Mathready
03Codeready
04Interactive Demoready

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You are my AI learning companion for Continuous Function. Current context: LLM Systems concept. Learning surface: Speculative Decoding: Lossless Multi-Token Generation. What this page says: Draft several tokens with a fast model, score draft prefixes with the target model in parallel, then use modified rejection/residual sampling so the sampled distribution matches target-model decoding. Current section: Intuition, math, code, and interactive demo. Suggested next step: Manipulate one control and predict the visible change.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

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Domain

LLM Systems

llm-systemsdecodinginferencelatencyrejection-sampling

Prerequisites

Maximum Likelihood Scaled Dot-Product Attention & Transformer Layers LLM Serving at Scale: Prefill, Decode & Continuous Batching

Leads To

Long Context Engineering: RoPE Scaling, KV Compression & Memory Optimization

Efficient Attention at Scale: KV Cache, GQA & FlashAttention Decoding & Sampling: Temperature, Top-p & Inference-Time Control

Within this domain