Scaling Laws Code

Overview

This research establishes the first comprehensive scaling law framework for multilingual code pre-training. Unlike language-agnostic approaches, the work reveals that programming languages have fundamentally different scaling characteristics and transfer behaviors. These insights enable principled token allocation strategies.

Core Technique

The key insight is that different programming languages exhibit distinct scaling relationships and transfer properties.

Language-Specific Scaling Laws: Each language exhibits power-law relationships with distinct exponents.

# Scaling law analysis per programming language
import numpy as np

class LanguageScalingLaws:
    def __init__(self):
        self.languages = ['python', 'java', 'javascript', 'c', 'c++', 'rust', 'go']
        self.scaling_laws = {}

    def fit_scaling_law(self, language, model_sizes, perplexities):
        """
        Fit power law: L = a * (N + D)^(-b)
        where N = model size, D = data size, L = loss
        """
        # Log-linear regression
        log_sizes = np.log(model_sizes)
        log_losses = np.log(perplexities)

        coefficients = np.polyfit(log_sizes, log_losses, deg=1)
        # coefficients: [exponent_b, intercept_a]

        self.scaling_laws[language] = {
            'exponent': abs(coefficients[0]),  # b term
            'category': self.categorize_language(language)
        }

        return self.scaling_laws[language]

    def categorize_language(self, language):
        """
        Interpreted vs compiled affects scaling behavior.
        """
        if language in ['python', 'javascript']:
            return 'interpreted'
        else:
            return 'compiled'

    def analyze_scaling_differences(self):
        """
        Interpreted languages scale better with model/data increase
        than compiled languages.
        """
        interpreted = [
            law for lang, law in self.scaling_laws.items()
            if law['category'] == 'interpreted'
        ]
        compiled = [
            law for lang, law in self.scaling_laws.items()
            if law['category'] == 'compiled'
        ]

        print(f"Interpreted avg exponent: {np.mean([l['exponent'] for l in interpreted]):.3f}")
        print(f"Compiled avg exponent: {np.mean([l['exponent'] for l in compiled]):.3f}")
        # Typical result: Interpreted benefit more from scale

Synergy Gain Matrix: Cross-lingual transfer depends on language similarity.

def compute_synergy_gains(language_pairs, training_results):
    """
    Measure cross-lingual transfer effects.
    Syntactically similar languages show positive transfer.
    """
    synergy_matrix = {}

    for lang1, lang2 in language_pairs:
        # Joint training performance
        joint_performance = training_results[(lang1, lang2)]

        # Individual training performance
        individual_performance = (
            training_results[lang1] + training_results[lang2]
        ) / 2

        # Synergy = improvement from joint training
        synergy = joint_performance - individual_performance

        # Syntactic similarity metric
        similarity = compute_syntactic_similarity(lang1, lang2)

        synergy_matrix[(lang1, lang2)] = {
            'synergy_gain': synergy,
            'syntax_similarity': similarity
        }

    return synergy_matrix

def analyze_transfer_patterns(synergy_matrix):
    """
    Similar languages (Java-C#) show positive transfer.
    Dissimilar languages show minimal or negative transfer.
    """
    positive_synergies = [
        (pair, data['synergy_gain']) for pair, data in synergy_matrix.items()
        if data['synergy_gain'] > 0
    ]
    print("Positive transfer pairs (high syntax similarity):")
    for pair, gain in positive_synergies:
        print(f"  {pair}: +{gain:.3f}")

Parallel Pairing Strategy: Concatenating code with translations improves multilingual translation.

def parallel_pairing_training(code_datasets):
    """
    For each code snippet, include translation to related language.
    Improves multilingual translation with favorable scaling.
    """
    paired_data = []

    for code_sample in code_datasets['python']:
        # Original code
        original = code_sample

        # Translation to Java (similar-enough for transfer)
        translated = translate_code(code_sample, 'python', 'java')

        # Pair them in training data
        paired_data.append({
            'source': original,
            'translation': translated,
            'source_lang': 'python',
            'target_lang': 'java'
        })

    return paired_data

Proportion-Dependent Allocation Strategy: Optimal token allocation balances utility and saturation properties.

def optimal_token_allocation(languages, total_tokens, scaling_laws):
    """
    Allocate tokens based on:
    1. Scaling utility (languages benefiting more get more)
    2. Saturation rate (fast-saturating languages get less)
    3. Cross-lingual synergies
    """
    allocations = {}

    # Compute utility weight per language
    utility_weights = {}
    for lang in languages:
        law = scaling_laws[lang]

        # High exponent = benefits more from scale
        scaling_benefit = law['exponent']

        # Saturation rate (compiled langs saturate faster)
        if law['category'] == 'compiled':
            saturation_penalty = 0.7  # Reduce allocation
        else:
            saturation_penalty = 1.0  # Full allocation

        utility_weights[lang] = scaling_benefit * saturation_penalty

    # Normalize weights to sum to total_tokens
    total_weight = sum(utility_weights.values())
    for lang in languages:
        allocations[lang] = int(
            (utility_weights[lang] / total_weight) * total_tokens
        )

    return allocations

When to Use This Technique

Use these scaling laws when:

• Pre-training on multilingual codebases
• Deciding language-specific model sizes
• Allocating computational budget across languages
• Predicting multilingual code model performance

When NOT to Use This Technique

Avoid this approach if:

• Single-language code models (simpler approach)
• Scaling laws unavailable for your languages
• Transfer between your specific languages unpredictable
• Computational budget not constraining

Implementation Notes

The framework requires:

• 1000+ experiments across language-size combinations
• Scaling law fitting infrastructure
• Cross-lingual transfer measurement
• Parallel pairing data generation
• Token allocation optimization

Key Performance

• Language-specific exponents range 0.07-0.15
• Positive transfer for syntactically similar languages
• Parallel pairing improves translation with favorable scaling
• Optimized allocation outperforms uniform distribution

References

• Language-specific power-law scaling relationships
• Interpreted vs compiled language scaling differences
• Synergy gain matrix from cross-lingual training
• Parallel pairing for translation improvement
• Proportion-dependent token allocation strategy