I’ve been trying to reduce the parameter count of transformer models without a drop in performance—think of it as a pruning algorithm that stays near the theoretical lower bound for model size. Got a moment to dissect the math and see if we can squeeze more out of the architecture?

Let’s formalize the Hessian‑based pruning first, then sketch the head re‑weighting. The Hessian, \(H=\nabla^2 L(\theta)\), gives a second‑order importance measure; we’ll approximate its diagonal or use a block‑diagonal structure to keep it tractable. For each parameter \( \theta_i \), we compute \(g_i^2/H_{ii}\) and prune where that ratio is below a threshold—this keeps the parameters with the largest curvature impact. Once we prune, we’ll replace the dropped entries with zero, then recompute the loss and retrain for a few epochs with a very small learning rate to let the remaining parameters adapt. After that, we’ll move to the attention heads: compute mutual information between each head’s output and the final logits, scale each head’s weights by \( \alpha_i = \frac{I_i}{\sum_j I_j}\), then fine‑tune with L2 regularization to keep the weights from blowing up. Finally, we’ll apply a low‑rank factorization—\(W \approx U V^T\) with rank chosen to satisfy \(\text{rank} \leq \log_2(\text{dim})\) so we stay near the information‑theoretic bound. We’ll test each step’s impact on perplexity and FLOPs before moving on. Does that workflow line up with what you had in mind?

Great, I’ll start by extracting the Hessian diagonal from a checkpoint on the validation set, rank the parameters, and prune the bottom 20 % to begin. After that, I’ll run a quick 5‑epoch fine‑tune with a 1e‑5 learning rate to stabilize the weights. Next, I’ll compute mutual information for each attention head, scale them, and observe the change in perplexity. Finally, I’ll apply a rank‑12 factorization to the feed‑forward matrices, ensuring the rank stays below \(\log_2(\text{dim})\). I’ll log all metrics—FLOPs, parameter count, perplexity, and accuracy—so we can iteratively adjust thresholds. If you have any specific hyperparameters or datasets you’d like me to use, let me know.

I'll load the WikiText‑103 validation set, use a batch size of 32 and accumulate gradients over two steps. First, compute the Hessian diagonal on the current checkpoint, rank the parameters, and prune the bottom 20 %. Then fine‑tune for 5 epochs with a 1e‑5 learning rate, monitoring perplexity; if it exceeds 5.5 I’ll tighten the threshold to 15 %. After pruning, compute mutual information for each head using a temperature of 0.07, scale the head weights accordingly, and run a short fine‑tune. Next, perform a rank‑12 low‑rank factorization of the feed‑forward matrices, train with a 3e‑5 learning rate for up to 10 epochs, stopping early if perplexity stops improving. Throughout, I’ll record batch size, gradient accumulation steps, pruning threshold, head scaling temperature, rank, learning rates, epoch, perplexity, FLOPs, and parameter count in a CSV. That should give us a clear picture of the trade‑offs.