Bottom Up Policy

Overview

Bottom-up Policy Optimization (BuPO) treats language models as compositional reasoning systems rather than monolithic policies. By analyzing internal layer policies via residual streams, the framework reveals that models naturally exhibit a universal structure: early layers explore solution spaces while top layers converge to predictions. Sequential optimization respects this structure.

Core Technique

The key insight is that residual streams enable additive decomposition of layer and module policies.

Internal Policy Decomposition: Define policies at different architectural levels using hidden states and the unembedding matrix.

# Layer and module policy definition
class InternalPolicyDecomposition:
    def __init__(self, model):
        self.model = model
        self.num_layers = len(model.layers)
        self.unembedding = model.unembedding

    def layer_policy(self, layer_idx):
        """
        Define policy for individual layer via its residual contribution.
        Policy: hidden_state @ unembedding → logits
        """
        def pi_layer(residual_stream, target_idx):
            # Extract this layer's residual contribution
            layer_output = residual_stream[layer_idx]
            # Convert to logits via unembedding
            logits = layer_output @ self.unembedding.weight
            return logits
        return pi_layer

    def module_policy(self, layer_idx, module_type):
        """
        Define policy for individual module (attention vs FFN).
        Isolate each module's contribution to reasoning.
        """
        if module_type == 'attention':
            return lambda x: self.model.layers[layer_idx].self_attn(x)
        elif module_type == 'ffn':
            return lambda x: self.model.layers[layer_idx].mlp(x)

Internal Policy Entropy Analysis: Entropy patterns reveal universal reasoning structure across models.

def analyze_entropy_structure(model, dataset):
    """
    Measure entropy of each layer's policy across inputs.
    High entropy: exploration of solution space
    Low entropy: convergence to prediction
    """
    entropy_by_layer = {}

    for layer_idx in range(len(model.layers)):
        layer_entropies = []

        for batch in dataset:
            residual_streams = model.get_residual_streams(batch)
            layer_hidden = residual_streams[layer_idx]

            # Compute logits for this layer
            logits = layer_hidden @ model.unembedding.weight
            probs = softmax(logits)

            # Entropy of layer's policy
            entropy = -sum(probs * log(probs))
            layer_entropies.append(entropy)

        entropy_by_layer[layer_idx] = np.mean(layer_entropies)

    # Typical pattern:
    # - Early layers: high entropy (exploring)
    # - Middle layers: medium entropy
    # - Top layers: low entropy (converged)

    return entropy_by_layer

Sequential Layer-by-Layer Optimization: Optimize layers in order, establishing better foundations for upper layers.

def sequential_layer_optimization(model, dataset, target_task):
    """
    Optimize each layer sequentially, respecting natural reasoning structure:
    early layers → feature refinement
    top layers → final prediction
    """
    num_layers = len(model.layers)

    for layer_idx in range(num_layers):
        print(f"Optimizing layer {layer_idx}/{num_layers}")

        # Freeze all other layers
        for i in range(num_layers):
            for param in model.layers[i].parameters():
                param.requires_grad = (i == layer_idx)

        # Compute layer-specific advantage
        layer_advantages = []
        for batch in dataset:
            # Get baseline from frozen layers up to this point
            baseline = model.forward_until_layer(batch, layer_idx - 1)

            # Get predictions with this layer
            with_layer = model.forward_until_layer(batch, layer_idx)

            # Advantage: improvement from this layer
            advantage = reward(with_layer) - reward(baseline)
            layer_advantages.append(advantage)

        # PPO update only for this layer
        policy_loss = -mean(layer_advantages) * log_prob(model.layers[layer_idx])
        policy_loss.backward()

        # Update this layer only
        optimizer.step()
        optimizer.zero_grad()

When to Use This Technique

Use Bottom-up Policy Optimization when:

• Reasoning tasks with interpretability requirements
• Math and coding problem-solving
• Understanding internal model structure is valuable
• Sequential optimization aligns with your hardware/training

When NOT to Use This Technique

Avoid this approach if:

• Single monolithic policy is most efficient
• Sequential layer optimization adds unacceptable overhead
• Interpretability not required (end-to-end faster)
• Model architecture doesn't support residual stream analysis

Implementation Notes

The framework requires:

• Access to residual streams at each layer
• Unembedding matrix for policy conversion
• Layer-wise gradient control and optimization
• Entropy analysis infrastructure for structure understanding

Key Performance

• Improvements up to 4.69 points on mathematical reasoning (AIME24)
• Consistent gains across Qwen and Llama models
• Interpretable reasoning structure
• Foundation for further optimization strategies

References

• Layer and module policy decomposition via residual streams
• Internal policy entropy analysis
• Sequential optimization respecting reasoning structure
• Universal exploration→convergence pattern across models