Questions of budget forcing

Let me address each one in detail:

For Q1, Compute Allowance During Inference:

From the paper's experimental results, they tested different compute budgets:

# Example compute budgets from their experiments
compute_settings = {
    'MATH500': [512, 2048],  # tokens
    'AIME24': [512, 2048, 8192],  # tokens
    'GPQA': [1024, 4096]  # tokens
}

Their findings for optimal compute windows:

• Base performance: ~2000 tokens
• Good performance: 4000-6000 tokens
• Diminishing returns: >8000 tokens

They found that the optimal compute time varies by task:

For Q2, Scaling Behavior:

The paper shows that performance does improve with compute, but not linearly. They observed:

For example on AIME24:

For Q3, Control and Optimization:

They evaluated control through three metrics:

def evaluate_control(model, settings):
    metrics = {
        'Control': measure_adherence_to_budget(),
        'Scaling': measure_performance_improvement(),
        'Performance': measure_max_performance()
    }
    
    # Control metric (how well budget is maintained)
    Control = sum(within_budget) / total_samples
    
    # Scaling metric (improvement per compute)
    Scaling = (performance_delta) / (compute_delta)
    
    # Performance metric (absolute performance)
    Performance = max(accuracies)

For finding optimal settings:

1. Grid search over token budgets
2. Evaluated on development set
3. Found sweet spots where:
   • Control > 90% (budget adherence)
   • Scaling is positive
   • Performance improves meaningfully

Their optimization strategy:

def optimize_budget_forcing(model, task):
    results = []
    for max_tokens in [1000, 2000, 4000, 8000]:
        for min_tokens in [max_tokens * 0.5, max_tokens * 0.75]:
            performance = evaluate_with_budget(
                model, 
                task,
                min_tokens=min_tokens,
                max_tokens=max_tokens
            )
            results.append({
                'settings': (min_tokens, max_tokens),
                'performance': performance
            })
    
    return find_optimal_settings(results)

Key Findings for Optimization:

1. Task-specific tuning is important
2. Start with conservative budgets
3. Gradually increase until diminishing returns
4. Monitor for:
   • Solution quality
   • Computation cost
   • Consistency of results

Best practices they discovered:

class OptimalBudgetForcing:
    def __init__(self, task_type):
        self.base_tokens = self.get_base_tokens(task_type)
        self.max_multiplier = 4  # Maximum 4x base tokens
        self.extension_step = "Wait"  # Token to extend thinking
        
    def get_base_tokens(self, task_type):
        return {
            'simple_math': 2000,
            'competition_math': 4000,
            'science': 3000
        }[task_type]

The paper demonstrates that finding the right balance requires:

1. Understanding the task complexity
2. Monitoring performance curves
3. Setting appropriate budgets
4. Regular evaluation of the trade-offs