Test-Time Compute and GRPO in Practice: From PPO to Critic-Free Reinforcement Learning

Introduction: The Pre-training Wall and the Dawn of Test-Time Scaling

For the past several years, the foundational law of frontier LLM development was Chinchilla's Pre-training Scaling Laws: stack deeper transformer layers, ingest multi-trillion token corpora, and burn increasingly massive GPU clusters.

However, entering 2026, this brute-force approach has encountered formidable physical and thermodynamic bottlenecks:

1. The Human Knowledge Depletion Wall: High signal-to-noise public internet text has been virtually exhausted. Naive ingestion of low-quality synthetic web dumps risks "model collapse" and entropy degeneration during unsupervised pre-training.
2. Diminishing Returns on Model Scale: Pushing model parameters from 70B to 700B incurs exponential spikes in capital expenditures, cluster interconnect overhead, and power consumption, yet returns only marginal improvements on everyday reasoning.

As pre-training scaling slows, frontier reasoning engines (such as OpenAI o1/o3 and DeepSeek-R1) have ignited a secondary growth curve: Test-Time Compute Scaling Laws.

graph LR
    subgraph Traditional Paradigm: One-Shot Pre-training Inference
        A1["Complex Math/Coding Prompt"] --> A2["70B~400B Dense Base LLM"] --> A3["Greedy Decoding (Prone to Hallucinations)"]
    end
    subgraph Reasoning Paradigm: Test-Time Compute Scaling
        B1["Complex Math/Coding Prompt"] --> B2["Compact Base Model"] --> B3["Extended Chain-of-Thought (CoT)"] --> B4["Self-Verification & Backtracking"] --> B5["Deterministic Accurate Solution"]
    end

Rather than spending millions of dollars during pre-training to memorize answers to every conceivable question, test-time scaling trains models to allocate dynamic computation at inference time—thinking, calculating, and self-correcting before providing a response.

I. The Three Regimes of Test-Time Compute Scaling

In modern literature, extending test-time compute falls into three primary architectural regimes:

The breakthrough of DeepSeek-R1 lies in fusing sequential chain-of-thought expansion with critic-free reinforcement learning, proving that pure rule-based RL can induce deep reasoning behaviors without manually engineered step-by-step PRMs.

II. From PPO to GRPO: Why Traditional RLHF Breaks on Long Reasoning Traces

For years, PPO (Proximal Policy Optimization) was the standard algorithm for post-training alignment. However, when applied to reasoning models with 10k+ token outputs, PPO collapses under severe infrastructure and mathematical constraints.

1. The Quad-Model VRAM Explosion

A classic PPO training setup requires hosting and synchronizing four distinct neural networks:

• Actor Model ($\pi_\theta$): The trainable policy generating tokens.
• Critic Model ($V_\phi$): Typically matching the Actor's size, tasked with estimating scalar state values $V(s)$.
• Reference Model ($\pi_{ref}$): A frozen copy computing per-token KL divergence to prevent policy drift.
• Reward Model ($R_\psi$): A frozen network scoring terminal outputs.

graph TD
    subgraph Traditional PPO Architecture
        P1["Actor Model (Trainable)"]
        P2["Critic Model (Trainable - Massive VRAM)"]
        P3["Reference Model (Frozen)"]
        P4["Reward Model (Frozen)"]
    end
    subgraph GRPO Architecture
        G1["Actor Model (Trainable)"]
        G2["Ref Weights / Analytical KL Calculation"]
        G3["Deterministic Environment (Python Sandbox / Unit Tests / Matcher)"]
    end

In a 70B parameter setup, loading the Actor and Critic alongside their respective AdamW optimizer states easily demands over 600GB of VRAM. This forces teams to deploy complex tensor and pipeline parallelism merely to fit the training loop. For low-level driver and memory bus topology guidelines, consult our NVIDIA GPU Package Architecture Deep Dive.

2. Value Function Drift Across Long Horizons

When a model reasons through intricate mathematical proofs, trajectories stretch across 8,000 to 16,000 tokens. Training a Critic to accurately predict the expected discounted return at every intermediate token is mathematically fragile. Critic errors amplify gradient variance, causing loss values to explode into NaNs.

III. Mathematical Derivation of GRPO: The Critic-Free Revolution

GRPO (Group Relative Policy Optimization) was pioneered by DeepSeek in the DeepSeekMath paper and scaled in DeepSeek-R1.

Its core thesis is remarkably elegant: Eliminate the Critic network entirely, sample a group of completions for each prompt, and use the group's empirical distribution as the baseline.

1. Group Sampling and Normalized Advantage

For any input query $q$, the policy $\pi_{\theta_{old}}$ generates a group of $G$ distinct candidate completions:

$${o_1, o_2, \dots, o_G} \sim \pi_{\theta_{old}}(q)$$

The verification environment (e.g., a regex answer parser or a compiler test runner) assigns scalar rewards to each completion:

$${r_1, r_2, \dots, r_G}$$

Rather than evaluating an absolute value network $V(s)$, GRPO computes the relative advantage $A_i$ of completion $o_i$ normalized against its peers:

$$A_i = \frac{r_i - \text{mean}({r_1, \dots, r_G})}{\text{std}({r_1, \dots, r_G}) + \epsilon}$$

• If $o_i$ outperforms the group average, $A_i > 0$, reinforcing the token trajectory.
• If $o_i$ underperforms, $A_i < 0$, penalizing the trajectory.
• Normalizing by the standard deviation dynamically stabilizes variance across batches.

2. The GRPO Objective Function

Retaining the clipped surrogate mechanism from PPO, GRPO optimizes the following objective:

$$\mathcal{J}{GRPO}(\theta) = \mathbb{E}{q \sim P(Q), {o_i}{i=1}^G \sim \pi{\theta_{old}}(q)} \left[ \frac{1}{G} \sum_{i=1}^{G} \frac{1}{|o_i|} \sum_{t=1}^{|o_i|} \left( \min \left( \frac{\pi_\theta(o_{i,t} \mid q, o_{i,<t})}{\pi_{\theta_{old}}(o_{i,t} \mid q, o_{i,<t})} A_{i,t}, \; \text{clip}\left(\frac{\pi_\theta(o_{i,t} \mid q, o_{i,<t})}{\pi_{\theta_{old}}(o_{i,t} \mid q, o_{i,<t})}, 1-\epsilon, 1+\epsilon\right) A_{i,t} \right) - \beta D_{KL}(\pi_\theta \parallel \pi_{ref}) \right) \right]$$

where the per-token KL divergence approximation is computed directly:

$$D_{KL} = \frac{\pi_{ref}(o_{i,t} \mid \cdot)}{\pi_\theta(o_{i,t} \mid \cdot)} - \log \frac{\pi_{ref}(o_{i,t} \mid \cdot)}{\pi_\theta(o_{i,t} \mid \cdot)} - 1$$

This architectural shift achieves two immediate advantages:

1. Over 50% Reduction in Training Memory: Eliminating the Critic model discards its weights, gradients, and optimizer states.
2. Zero Value-Model Misalignment: The baseline is derived directly from empirical sampling statistics, providing stable convergence.

IV. The "Aha Moment" and the Emergence of Self-Reflection

In the DeepSeek-R1-Zero experiment, researchers demonstrated that without a single human SFT demonstration, training a pure base model with GRPO induced complex reasoning behaviors:

1. Deterministic Rule-Based Rewards

Instead of relying on a learned neural reward model (which often suffers from reward gaming), the environment enforces strict rule-based verifiers:

1. Accuracy Reward: Verifies if the mathematical result matches ground truth or if the unit tests pass in a sandboxed interpreter (1.0 for success, 0.0 for failure).
2. Format Reward: Enforces that reasoning is encapsulated strictly between <think> and </think> tags, with the final answer inside <answer>.

2. Spontaneous Emergence of Self-Correction

After several thousand iterations, as trajectory length expanded, logs revealed models discovering self-correction unprompted:

User: Is 9.11 larger than 9.9?
Assistant:
<think>
First, compare 9.11 and 9.9.
The integer part is both 9.
Now compare the tenths place: 9.11 has 1 in the tenths place, while 9.9 has 9 in the tenths place.
Wait! Let me rethink this... Am I confusing decimal places with software version numbers?
In software engineering, v9.11 is newer than v9.9, but mathematically, 9.9 = 9.90.
Since 9.90 > 9.11, 9.9 is strictly larger.
Let me double check the calculation: 9.9 - 9.11 = 0.79 > 0. Yes, absolutely correct.
</think>
<answer>
No, 9.9 is larger than 9.11.
</answer>

From a reinforcement learning perspective, exploratory paths that verified intermediate results achieved a higher pass rate on difficult tasks than one-shot guesses. The group-relative advantage mechanism amplified these self-questioning trajectories.

V. Hands-on Implementation: A Minimal GRPO Training Pipeline

Using Hugging Face's TRL (Transformer Reinforcement Learning) library, here is an end-to-end runnable script training a lightweight base model (such as Qwen/Qwen2.5-1.5B-Instruct) with GRPO:

1. Environment Setup

pip install torch transformers trl peft datasets accelerate

2. Complete Training Code

import re
import torch
from datasets import Dataset
from transformers import AutoTokenizer, AutoModelForCausalLM
from trl import GRPOTrainer, GRPOConfig

# 1. Prepare deterministic verification dataset
train_data = [
    {
        "prompt": "Solve this equation: 3 * x + 7 = 22. What is x? Present your reasoning inside <think> and final value in <answer>.",
        "target": "5"
    },
    {
        "prompt": "A train travels 180 km in 3 hours. What is its speed in km/h? Think first in <think>, give value in <answer>.",
        "target": "60"
    },
    {
        "prompt": "If a square has an area of 64 cm^2, what is its perimeter in cm? Reason in <think>, answer in <answer>.",
        "target": "32"
    }
] * 100  # Expand dataset scale

dataset = Dataset.from_list(train_data)

# 2. Define deterministic rule-based reward functions
def correctness_reward_func(prompts, completions, target, **kwargs):
    """Verify if the content in <answer> strictly matches ground truth."""
    rewards = []
    for completion, true_target in zip(completions, target):
        match = re.search(r"<answer>(.*?)</answer>", completion, re.DOTALL)
        if match:
            pred = match.group(1).strip()
            rewards.append(2.0 if pred == true_target.strip() else 0.0)
        else:
            rewards.append(0.0)
    return rewards

def format_reward_func(completions, **kwargs):
    """Reward proper reasoning tag encapsulation."""
    rewards = []
    pattern = r"^<think>.*?</think>\s*<answer>.*?</answer>$"
    for completion in completions:
        if re.search(pattern, completion.strip(), re.DOTALL):
            rewards.append(0.5)
        else:
            rewards.append(0.0)
    return rewards

# 3. Load model and tokenizer
model_id = "Qwen/Qwen2.5-1.5B-Instruct"
tokenizer = AutoTokenizer.from_pretrained(model_id)
if tokenizer.pad_token is None:
    tokenizer.pad_token = tokenizer.eos_token

# 4. Configure GRPO Hyperparameters
training_args = GRPOConfig(
    output_dir="./grpo_output_qwen",
    learning_rate=2e-5,
    per_device_train_batch_size=2,
    gradient_accumulation_steps=4,
    num_generations=4,          # Group size G=4
    max_prompt_length=256,
    max_completion_length=1024,  # Ample space for CoT exploration
    temperature=0.7,
    warmup_ratio=0.1,
    logging_steps=10,
    max_steps=100,
    save_strategy="steps",
    save_steps=50,
    bf16=True,
    report_to="none"
)

# 5. Launch the Critic-Free Trainer
trainer = GRPOTrainer(
    model=model_id,
    reward_funcs=[correctness_reward_func, format_reward_func],
    args=training_args,
    train_dataset=dataset,
)

print("🚀 Launching critic-free GRPO reinforcement learning pipeline...")
trainer.train()

For foundational training workflows and adapter memory tuning, review our Comprehensive LLM Fine-Tuning Guide.

VI. Production Pitfalls: Overthinking and Dynamic Compute Governance

Deploying reasoning models in production requires addressing these operational considerations:

1. The Overthinking Trap: When asked trivial factual queries ("What is the capital of France?"), models may output 800 tokens of self-questioning, adding seconds of unnecessary Time-To-First-Token (TTFT) latency.
2. Two-Stage Routing Architecture:
   • Direct standard conversational requests and retrieval tasks to lightweight models or RAG retrieval pipelines.
   • Reserve thinking endpoints for complex logical synthesis, math, and code generation with bounded token limits (max_thinking_tokens). For high-throughput infrastructure setup, refer to our vLLM Production Serving Guide.

Frequently Asked Questions (FAQ)

Q1: How does GRPO converge reliably without an explicit Critic network?

GRPO replaces the parametric state-value estimation of the Bellman equation with empirical Monte Carlo group sampling. By generating a group of $G$ responses for the same prompt, the group mean serves as a dynamic, unbiased baseline. As long as the group size is sufficient ($G \ge 4 \sim 8$), the normalized advantage $\frac{r_i - \mu}{\sigma}$ accurately signals relative trajectory quality.

Q2: Does GRPO eliminate the need for Supervised Fine-Tuning (SFT) data entirely?

While DeepSeek-R1-Zero proved that cold-start reasoning can emerge from pure RL, practical production workflows benefit significantly from a lightweight initial SFT phase. Pure RL on raw base models frequently generates multilingual gibberish, formatting anomalies, and infinite repetition early in training. Starting with a few thousand curated chain-of-thought demonstrations accelerates convergence by over 5x while preserving readability.

Q3: What is the fundamental difference between GRPO and DPO?

DPO (Direct Preference Optimization) is an offline supervised preference algorithm operating on static pairs of chosen and rejected responses $(y_w, y_l)$. It cannot discover novel reasoning pathways absent from the static dataset. GRPO is an active, online reinforcement learning algorithm where the model generates real-time samples evaluated dynamically by verifiable environment rewards, enabling open-ended exploration and spontaneous self-correction.

Originally published at Nobita Talks AI on blog.llmgo.top.