docs/source_en/Instruction/GRPO/AdvancedResearch/RLOO.md

REINFORCE Leave-One-Out (RLOO)

REINFORCE Leave-One-Out (RLOO) is a reinforcement learning algorithm based on the classic REINFORCE policy-gradient method. It constructs an unbiased advantage baseline via the Leave-One-Out (LOO) technique.

Algorithm Overview

For clarity, we explain RLOO by contrasting it with GRPO (Group Relative Policy Optimization).

Both GRPO and RLOO estimate advantages via intra-group comparisons to avoid the high variance of a global baseline. Their core differences are mainly in the following aspects:

Difference 1: How the Advantage Baseline Is Constructed

1. GRPO (Group Relative Policy Optimization)

For each prompt, GRPO generates $G$ response samples and normalizes rewards using the group mean and standard deviation:

$${\widehat{A}}_i=\frac{R_i-\text{mean}(\{R_j{\}}_{j=1}^G)}{\text{std}(\{R_j{\}}_{j=1}^G)}\backslash hat\{A\}\_\{i\}\; =\; \backslash frac\{R\_i\; -\; \backslash text\{mean\}(\backslash \{R\_j\backslash \}\_\{j=1\}^G)\}\{\backslash text\{std\}(\backslash \{R\_j\backslash \}\_\{j=1\}^G)\}$$

Where:

• $R_i$ is the reward of the $i$-th sample
• $\text{mean}({R_j}{j=1}^G) = \frac{1}{G}\sum{j=1}^G R_j$ is the group mean
• $\text{std}({R_j}_{j=1}^G)$ is the group standard deviation

2. RLOO (REINFORCE Leave-One-Out)

For each prompt, RLOO generates $K$ response samples and constructs the baseline via Leave-One-Out, i.e., for the $i$-th sample, the baseline is the mean of the other $K-1$ samples:

$${\widehat{A}}_i=R_i-\frac{1}{K-1}\sum _{j\mathrm{\ne }i}{R}_j\backslash hat\{A\}\_\{i\}\; =\; R\_i\; -\; \backslash frac\{1\}\{K-1\}\backslash sum\_\{j\; \backslash neq\; i\}\; R\_j$$

This can be equivalently rewritten as:

$${\widehat{A}}_i=\frac{K}{K-1}(R_i-\bar{R})\backslash hat\{A\}\_\{i\}\; =\; \backslash frac\{K\}\{K-1\}\; \backslash left(R\_i\; -\; \backslash bar\{R\}\backslash right)$$

where $\bar{R} = \frac{1}{K}\sum_{j=1}^K R_j$ is the group mean reward.

Note: We use $K$ here to match the notation in the paper. It has the same meaning as $G$ in GRPO and corresponds to the configuration parameter num_generations.

Why Leave-One-Out?

The key advantage is unbiasedness. For the $i$-th sample, its reward $R_i$ is independent of the baseline $\frac{1}{K-1}\sum_{j \neq i} R_j$, hence the advantage estimate is unbiased. In contrast, using the mean including itself as the baseline introduces bias.

Difference 2: How KL Regularization Is Applied

To prevent the policy from drifting too far from the reference policy, both algorithms introduce KL divergence regularization, but in different ways:

GRPO: Adds KL divergence as an independent regularization term to the loss:

$$\mathcal{L}(\theta )=-\mathbb{E}[{\widehat{A}}_i\log {\pi }_{\theta }(a_i\mathrm{\mid }{s}_i)]+\beta \cdot \text{KL}({\pi }_{\theta }\mathrm{\parallel }{\pi }_{\text{ref}})\backslash mathcal\{L\}(\backslash theta)\; =\; -\backslash mathbb\{E\}\backslash left[\backslash hat\{A\}\_i\; \backslash log\; \backslash pi\_\backslash theta(a\_i|s\_i)\backslash right]\; +\; \backslash beta\; \backslash cdot\; \backslash text\{KL\}(\backslash pi\_\backslash theta\; \backslash Vert\; \backslash pi\_\{\backslash text\{ref\}\})$$

RLOO: Integrates KL divergence directly into the reward, constructing a modified reward:

$$R_i^{\mathrm{\prime }}=R_i-\beta \cdot \text{KL}({\pi }_{\theta }\mathrm{\parallel }{\pi }_{\text{ref}})R\text{'}\_i\; =\; R\_i\; -\; \backslash beta\; \backslash cdot\; \backslash text\{KL\}(\backslash pi\_\backslash theta\; \backslash Vert\; \backslash pi\_\{\backslash text\{ref\}\})$$

where $\beta$ is the KL coefficient (parameter beta), and $\pi_{\text{ref}}$ is the reference policy (typically an SFT model or the initial policy).

Parameter Configuration

RLOO training can be enabled based on GRPOTrainer by setting the following parameters:

# Basic RLOO configuration
--advantage_estimator rloo  # Use RLOO's leave-one-out advantage estimator
--kl_in_reward true         # Integrate KL divergence into the reward (default for RLOO)

You can refer to this script for training.

Important Parameters

• --advantage_estimator: Choose the advantage estimator

  • grpo (default): standardize using group mean and standard deviation
  • rloo: construct the baseline via Leave-One-Out

• --kl_in_reward: Controls where the KL term is applied

  • false: KL as a separate regularization term in the loss (GRPO style)
  • true: subtract KL directly from the reward to form a modified reward (RLOO style)

• --num_generations: Number of samples per prompt, i.e., $K$

• --beta: KL regularization coefficient $\beta$

  • Controls how conservatively the policy updates

Other parameters are consistent with the GRPO arguments.