Chain of Thought via Entropy Injection

A novel approach to improving language model reasoning by dynamically injecting chain-of-thought prompts when the model's uncertainty reaches a critical threshold.

Introduction

Large Language Models (LLMs) have demonstrated impressive capabilities across various tasks, but they still struggle with complex reasoning problems. Chain of Thought (CoT) prompting has emerged as a powerful technique to enhance reasoning by encouraging models to break down problems into intermediate steps. However, traditional CoT approaches apply the same prompting strategy regardless of the model's internal state or the specific challenges of a given problem.

This blog post introduces Entropy-Based Chain of Thought Injection, a dynamic approach that monitors a model's uncertainty during generation and strategically injects CoT prompts precisely when the model needs guidance the most.

Understanding Entropy in Language Models

Before diving into the technique, let's understand what entropy means in the context of language models:

What is Entropy?

In information theory, entropy measures uncertainty or randomness. For language models, entropy reflects the model's confidence in predicting the next token:

Low entropy: The model is confident about what comes next (probability mass concentrated on few tokens)
High entropy: The model is uncertain (probability mass distributed across many tokens)

Mathematically, entropy is calculated as:

def calculate_entropy(logits):
    probs = F.softmax(logits, dim=-1)
    log_probs = F.log_softmax(logits, dim=-1)
    entropy = -torch.sum(probs * log_probs, dim=-1)
    return entropy

This calculation gives us a single value representing the model's uncertainty at each generation step.

The Entropy-Based CoT Injection Technique

The core idea behind this technique is simple yet powerful:

Monitor the model's entropy during text generation
Detect uncertainty spikes that exceed a predefined threshold
Inject a CoT prompt at precisely those moments of high uncertainty
Guide the model through the reasoning process when it needs help most

This approach is adaptive and targeted, providing assistance only when the model struggles, rather than applying a one-size-fits-all prompting strategy.

Implementation Details

The implementation involves several key components:

1. Entropy Calculation

We calculate the entropy of the model's next-token prediction at each step:

entropy = calculate_entropy(next_token_logits)
entropies.append(entropy.item())

2. Threshold Detection

We check if the entropy exceeds our predefined threshold and if we're not in a cooldown period from a previous injection:

if (
    entropy.item() > entropy_threshold
    and cot_injections < max_cot_injections
    and steps_since_cot >= cooldown_steps
):
    # Inject CoT prompt

3. CoT Prompt Injection

When high entropy is detected, we inject a CoT prompt to guide the model:

cot_prompt = " To determine the answer, let's breakdown the problem step by step, then provide a final answer. "

The prompt is injected into the model's context but not included in the final output, serving as temporary guidance.

4. Continued Generation

After injection, the model continues generating with the benefit of the CoT guidance.

Case Study: Numerical Comparison

Let's examine how this technique works on a simple yet illustrative example: determining whether 9.9 is greater than 9.11.

This problem is interesting because it tests the model's understanding of decimal notation. While 9.11 appears larger if we compare the digits naively (9.11 > 9.9), the correct mathematical comparison shows that 9.9 is actually greater (9.9 = 9.90 > 9.11).

Without Entropy-Based Injection

Small language models often struggle with this comparison, frequently answering incorrectly that 9.11 is larger than 9.9.

With Entropy-Based Injection

When we apply entropy-based CoT injection, we observe:

The model begins answering with uncertainty ("Yes, or no.")
It starts structuring its thoughts in steps
When comparing the numbers, entropy spikes at a critical point (reaching 4.98)
The CoT prompt is injected, guiding the model
The model then correctly recognizes that "9.9 is the same as 9.990 and 9.11 is the same as 9.1199"
This leads to the correct conclusion that 9.9 is greater than 9.11

The entropy spike occurred precisely when the model needed to make the crucial realization about decimal representation, and the injection provided the necessary guidance.

Advantages of Entropy-Based CoT Injection

This approach offers several advantages over traditional CoT prompting:

Efficiency: Provides guidance only when needed, reducing unnecessary computation
Adaptivity: Responds to the model's specific struggles with each unique problem
Transparency: Makes the model's uncertainty explicit and observable
Improved Performance: Helps models navigate difficult reasoning steps
Reduced Token Usage: Minimizes prompt overhead by injecting CoT only when necessary

Technical Implementation

The full implementation involves careful token handling and entropy monitoring. Here's a simplified version of the core function:

def entropy_based_cot_injection(prompt, entropy_threshold=4.0, max_length=150):
    input_ids = tokenizer.encode(prompt, return_tensors='pt').to(model.device)
    generated_ids = input_ids.clone()
    
    cot_prompt = " To determine the answer, let's breakdown the problem step by step, then provide a final answer. "
    cot_ids = tokenizer.encode(cot_prompt, return_tensors='pt').to(model.device)
    
    with torch.no_grad():
        for step in range(max_length):
            # Get model predictions
            outputs = model(input_ids=generated_ids)
            next_token_logits = outputs.logits[:, -1, :]
            
            # Calculate entropy
            entropy = calculate_entropy(next_token_logits)
            
            # Check if entropy exceeds threshold
            if entropy.item() > entropy_threshold:
                # Inject CoT prompt
                model_input_ids = torch.cat((input_ids, cot_ids, generated_ids[:, input_ids.size(1):]), dim=1)
                outputs = model(input_ids=model_input_ids)
                next_token_logits = outputs.logits[:, -1, :]
            
            # Generate next token
            next_token_id = sample_next_token(next_token_logits)
            generated_ids = torch.cat((generated_ids, torch.tensor([[next_token_id]], device=model.device)), dim=1)
            
    return tokenizer.decode(generated_ids[0], skip_special_tokens=True)

Practical Applications

This technique can be particularly valuable in several scenarios:

Educational Tools: Helping students work through complex problems
Mathematical Reasoning: Guiding models through multi-step calculations
Logical Deduction: Assisting with complex logical arguments
Decision Making: Supporting structured decision processes
Small Model Enhancement: Improving the reasoning capabilities of smaller, more efficient models

Future Directions

The entropy-based CoT injection approach opens up several exciting research directions:

Dynamic Threshold Adjustment: Adapting the entropy threshold based on the problem type or model behavior
Multiple Injection Types: Using different types of CoT prompts based on the specific reasoning challenge
Hybrid Approaches: Combining entropy-based injection with other techniques like self-consistency or verification
Fine-tuning with Entropy Signals: Training models to be more sensitive to their own uncertainty
Cross-model Guidance: Using a larger model's entropy signals to guide a smaller model

Conclusion

Entropy-Based Chain of Thought Injection represents a significant advancement in how we prompt language models for complex reasoning tasks. By dynamically responding to the model's internal uncertainty, we can provide targeted guidance that improves performance while maintaining efficiency.

This approach demonstrates how monitoring a model's internal state can lead to more intelligent and adaptive prompting strategies, bringing us closer to more reliable AI reasoning systems.

Try It Yourself

You can experiment with this technique using the interactive Colab notebook or explore the GitHub repository for the complete implementation.