Equation 23 · The Main Technical Approaches to AI Alignment, Compared
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol hat A_i,t
hat ,t is part of the quantity the equation computes from the expression on the right.
Symbol tilde r_i
tilde is one of the signed contributions combined to compute the quantity on the left.
Symbol r_i
occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol r_1
is one of the signed contributions combined to compute the quantity on the left.
Symbol r_G
is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subtraction
Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
Numerator: r_i - mean(r_1,ldots,r_G)
The complete quantity above the fraction bar.
Denominator: std(r_1,ldots,r_G)
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
RLVR is the newest of the six and the only one that does not fit a learned model of anyone’s judgment at all. Where every technique above trains a proxy — a reward model, an AI judge, a decomposition scheme, a weak label — RLVR restricts itself to tasks where the reward can be computed directly and automatically: a unit test passes, a final numeric answer matches, a proof checker accepts. Shao and colleagues’ DeepSeekMath paper introduced Group Relative Policy Optimization, the reinforcement learning algorithm used throughout most subsequent RLVR work, replacing the learned value function used in standard policy-gradient methods with a group-relative advantage estimated directly from a batch…
Read the full surrounding passage
RLVR is the newest of the six and the only one that does not fit a learned model of anyone’s judgment at all. Where every technique above trains a proxy — a reward model, an AI judge, a decomposition scheme, a weak label — RLVR restricts itself to tasks where the reward can be computed directly and automatically: a unit test passes, a final numeric answer matches, a proof checker accepts. Shao and colleagues’ DeepSeekMath paper introduced Group Relative Policy Optimization, the reinforcement learning algorithm used throughout most subsequent RLVR work, replacing the learned value function used in standard policy-gradient methods with a group-relative advantage estimated directly from a batch of sampled outputs and their rule-based rewards, . normalizing each sampled response’s reward against the mean and standard deviation of a group of G responses to the same prompt rather than against a separately trained critic network [ 13 ] . Lambert and colleagues, building the fully open Tulu 3 post-training recipe, named the general approach explicitly, describing it as “a novel method we call Reinforcement Learning with Verifiable Rewards,” and used it alongside supervised fine-tuning and preference optimization rather than as a wholesale replacement for either [ 15 ] . The clearest large-scale demonstration is DeepSeek-R1: Guo and colleagues report training a model with reinforcement learning alone against a purely rule-based reward — combining an answer-correctness check with a format check, and deliberately avoiding a learned reward model because, in their account, a neural reward model “may suffer from reward hacking in large-scale reinforcement learning” — and observed pass@1 accuracy on the AIME 2024 competition-mathematics benchmark rise from 15.6 percent to 71.0 percent over training, reaching 86.7 percent with majority voting across 64 samples [ 14 ] . Formally, the reward itself is as simple as the reward model above was elaborate:
Sources cited in the surrounding passage
- [13] DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models ↗
- [15] Tulu 3: Pushing Frontiers in Open Language Model Post-Training ↗
- [14] DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning ↗
These citations give research context. Read each source to check which claims it supports.
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