∇Continuous Function
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Foundation Lab

Test-Time Compute & Inference Scaling

The paradigm behind o1: spend more compute at inference for harder problems

Concept 74 of 100Scaling & AlignmentPhase 11
Open New NotebookAll Foundations
#74Test-TimeScaling & Alignment
key equation\text{Quality} \sim \log(\text{inference compute})

Selected Foundation Object

Keep the equation fixed; move through the evidence.

Concept 74 of 100Test-TimeScaling & Alignment / Phase 11: Frontier research & scaling
Current question

Train-time and test-time compute are substitutes: you can trade one for the other

\text{Quality} \sim \log(\text{inference compute})
PredictionCommit before the demo.

Ask what should change when the equation is manipulated, then let the visualization test that expectation.

EvidenceCompare local witness and source.

Use the runnable panel, the key equation, and canonical papers as separate forms of evidence for the same object.

InvariantName what survives notation changes.

The useful learning product is the reusable mechanism you can carry into another model, paper, or engineering tradeoff.

Next moveContinue in the newer notebook.

This atlas page keeps the working demo; the domain notebook carries the fuller Intuition -> Math -> Code -> Demo sequence.

Open demoInspect equationOpen newer notebookNext foundation

Why It Matters for Modern Models

  • The paradigm behind o1: spend more compute at inference for harder problems
  • Enables adaptive compute: easy questions are fast, hard ones "think longer"
  • May be more efficient than pure pretraining scaling for reasoning tasks

What Tutorials Skip

What is still poorly explained in textbooks and papers:

  • Train-time and test-time compute are substitutes: you can trade one for the other
  • Verifiers (reward models) let you search through many candidate solutions
  • Tree search over reasoning steps explores the space of possible derivations

Interactive Visualization

Core Math (Optional Deep Dive)

If you want intuition first, start with the key equation and the visualization. Come back here for the full walkthrough.

Key Equation
Quality∼log⁡(inference compute)\text{Quality} \sim \log(\text{inference compute})

Test-time scaling trades compute for quality at inference:

Quality∼log⁡(inference compute)\text{Quality} \sim \log(\text{inference compute})

Best-of-N sampling: Generate N responses, select best via verifier:

y∗=arg⁡max⁡y∈{y1,...,yN}V(y)y^* = \arg\max_{y \in \{y_1, ..., y_N\}} V(y)

Process Reward Models score intermediate steps:

RPRM(s1,...,sk)=∏i=1kP(correct∣s1,...,si)R_{PRM}(s_1, ..., s_k) = \prod_{i=1}^k P(\text{correct} | s_1, ..., s_i)

Monte Carlo Tree Search for reasoning:

UCB(s)=V(s)+clog⁡NparentNsUCB(s) = V(s) + c\sqrt{\frac{\log N_{parent}}{N_s}}

Canonical Papers

Scaling LLM Test-Time Compute Optimally

Snell et al.2024arXiv
Read paper →

Let's Verify Step by Step

Lightman et al.2023arXiv
Read paper →

Connections

Prerequisites

↗Scaling⚖RLHF

Enables

💭CoT📋PRMs🌲MCTS-LLM

Next Moves

Choose the next question to carry this object forward.