CCEM: Convex Compositional Reasoning Model—Resolving the Energy Landscape Bottleneck in Combinatorial Reasoning via Convex Optimization

CCEM (Convex Compositional Energy Minimization) is a framework designed to solve the non-convex energy landscape bottleneck in combinatorial reasoning. By using input convex neural networks (ICNNs) to parameterize energy factors and optimizing over convex relaxations of feasible sets, it enables zero-shot generalization—training on small problem instances (e.g., 4×4 sudoku) and applying to large ones (e.g.,9×9,16×16 sudoku) without retraining.

Combinatorial reasoning problems (e.g., sudoku, circuit verification) have exponential solution spaces and complex constraints. Traditional methods often lack generalization or are hard to scale. Energy-based models (EBMs) offer a unified framework (minimizing energy function E(x)=ΣEᵢ(x)), but their non-convex energy landscapes lead to issues like local minima, unstable training, and limited generalization. CCEM addresses this by making the energy landscape convex.

CCEM ensures convex energy landscapes via two key designs:

1. Input Convex Neural Networks (ICNNs): Parameterize each energy factor Eᵢ with non-negative weights and convex activation functions, making Eᵢ convex.
2. Convex Relaxation: Convert discrete constraints (e.g., x∈{0,1}ⁿ) to continuous ones (x∈[0,1]ⁿ) using tight convex relaxation.

Training uses two stages:

• Factor-level Contrastive Learning: Shape local energy basins (positive samples: low energy; negative samples: high energy).
• End-to-End Unrolled Refinement: Unroll the reasoning process (projection gradient descent steps) into the computation graph for end-to-end training.

CCEM’s zero-shot generalization is validated across tasks:

• Sudoku: Trained on 4×4, applied to 9×9/16×16 with higher success than baselines.
• Other tasks: Graph coloring (small→large graphs), circuit verification (small→large circuits), scheduling (small→large problems).

Comparison with baselines:

Method	Generalization	Optimization Efficiency	Training Stability
Standard EBM	Poor	Low	Poor
Graph Neural Networks	Medium	Medium	Medium
Neuro-symbolic Methods	Medium	Medium	Medium
CCEM	Strong	High	Good

Applications:

1. Automatic reasoning systems (general constraint satisfaction, e.g., logic puzzles).
2. Optimization/scheduling (resource allocation, real-time scheduling).
3. Verification/testing (hardware/software validation).
4. Neuro-symbolic AI (combining neural expressiveness with symbolic reliability).

Limitations:

1. Relaxation quality may be loose for some problems.
2. ICNN’s convexity constraints limit expression.
3. Projection introduces discretization errors.
4. Two-stage training is more complex.

CCEM transforms combinatorial reasoning’s non-convex optimization into tractable convex optimization, enabling strong zero-shot generalization. Future directions include adaptive/tighter convex relaxation, hybrid methods, and deeper theoretical analysis of convexity-combinatorial generalization relations. Broader insight: Convexity, often avoided in deep learning, can improve generalization and simplify optimization when combined with problem structure.