Differentiable Entropy Regularization: A Complexity-Aware Approach for Neural Optimization

We introduce the first differentiable approximation of range-partition entropy, a complexity measure from computational geometry that directly bounds algorithmic runtime. Unlike architectural modifications, our method is a complementary regularizer that provides orthogonal efficiency gains when combined with existing optimizations. We establish theoretical guarantees in computational geometry, achieving 4--5× provable speedups on convex hull and triangulation with <0.2% error. On ImageNet-1K with ViT-Base, entropy regularization achieves 80.1% top-1 accuracy at 80% sparsity (1.60× standalone speedup), and when combined with FlashAttention yields 2.07× speedup versus 1.63× for FlashAttention alone. On large language models (LLaMA-2 7B, Mistral-7B, Phi-2), we achieve 1.48--1.60× inference speedups at 70--75% sparsity with minimal quality degradation (ROUGE-L drops of 0.3--0.4 points, perplexity increase of 0.9). Unlike prior regularization methods that target output distributions, we directly minimize representation complexity, yielding both efficiency gains and improved robustness through semantically structured sparsity patterns (IoU 0.73 vs 0.41 for magnitude pruning, CIFAR-100-C mCE 48.7 vs 55.4). Benefits are strongest for geometry and vision transformers, with more modest but measurable gains on LLMs, demonstrating that complexity regularization offers a principled pathway to joint efficiency-robustness optimization.