The last few months I’ve been doing a deep-dive into information geometry and I’ve really, thoroughly enjoyed it. Understanding models in higher-dimensions is nearly impossible (for me at least) without breaking them down this way. I used a Fisher information matrix approximation to “watch” a model train and then compared it to other models by measuring “alignment” via top-k FIM eigenvalues from the final, trained manifolds.
What resulted was, essentially, that task manifolds develop shared features in parameter space. I started using composites of the FIM top-k eigenvalues from separate models as initialization points for training (with noise perturbations to give GD room to work), and it positively impacted the models themselves to train faster, with better accuracy, and fewer a…
The last few months I’ve been doing a deep-dive into information geometry and I’ve really, thoroughly enjoyed it. Understanding models in higher-dimensions is nearly impossible (for me at least) without breaking them down this way. I used a Fisher information matrix approximation to “watch” a model train and then compared it to other models by measuring “alignment” via top-k FIM eigenvalues from the final, trained manifolds.
What resulted was, essentially, that task manifolds develop shared features in parameter space. I started using composites of the FIM top-k eigenvalues from separate models as initialization points for training (with noise perturbations to give GD room to work), and it positively impacted the models themselves to train faster, with better accuracy, and fewer active dimensions when compared to random initialization.
Some of that is obvious- of course if you initialize with some representation of a model’s features you’re going to train faster and better. But in some cases, it wasn’t. Some FIM top-k eigenvalues were strictly orthogonal between two tasks- and including both of them in a composite initialization only resulted in interference and noise. Only tasks that genuinely shared features could be used in composites.
Furthermore, I started dialing up and down the representation of the FIM data in the composite initialization and found that, in some cases, reducing the representation of some manifold’s FIM top-k eigenspace matrix in the composite actually resulted in better performance by the under-represented model. Faster training, fewer active dimensions, and better accuracy.
This is enormously computationally expensive in order to get those modest gains- but the direction of my research has never been about making bigger, better models but rather understanding how models form through gradient descent and how shared features develop in similar tasks.
This has led to some very fun experiments and I’m continuing forward- but it has me wondering, has anyone else been down this road? Is anyone else engaging with the geometry of their models? If so, what have you learned from it?
Edit: Adding visualization shared in the comments: https://imgur.com/a/sR6yHM1