The idea/description of this method is fully taken from John Wentworth's Applied Linear Algebra lecture series, specifically Lecture 2

Training deep neural networks involves navigating high-dimensional loss landscapes. Understanding the curvature of these landscapes via the Hessian of the loss function can provide insights into the optimization dynamics. However, computing the full Hessian can be prohibitively expensive. In this post, I describe a method (described by John Wentworth in his lecture series) for efficiently computing the top eigenvalues and eigenvectors of the loss Hessian using PyTorch's autograd and SciPy's sparse linear algebra utilities.

Hessian-vector product

The core idea hinges upon the Hessian-vector product (HVP). Given a vector , the HVP is defined as  , where  is the Hessian matrix. This product can be computed efficiently using automatic differentiation without forming the full Hessian. The process can be outlined as:

  1. Compute the gradient of the loss with respect to model parameters: 
  2. Compute the dot product of  and 
  3. Compute the gradient of  with respect to the model parameters, which gives the HVP

Lanczos Iteration and eigsh

eigsh from scipy.sparse.linalg implements the Lanczos iteration, which finds the top eigenvalues and eigenvectors of a symmetric matrix. It requires matrix-vector multiplication as the main computation, making it ideal for large matrices where full matrix factorizations are infeasible.

Using LinearOperator

To interface with eigsh, we need a mechanism to represent our Hessian as a linear operator that supports matrix-vector multiplication. SciPy's LinearOperator serves this purpose, allowing us to define a matrix implicitly by its action on vectors without forming the matrix explicitly.


Given a PyTorch model, loss function, and training data, the approach is to:

  1. Accumulate a subset of the training data (as many batches as specified)
  2. Define the HVP using PyTorch's autograd
  3. Construct a LinearOperator using the HVP
  4. Call eigsh with this linear operator to compute the top eigenvalues and eigenvectors

Appendix: Python code

You can find this code as a GitHub gist here also.

import torch as t
from torch.autograd import grad
from scipy.sparse.linalg import LinearOperator, eigsh
import numpy as np
def get_hessian_eigenvectors(model, loss_fn, train_data_loader, num_batches, device, n_top_vectors, param_extract_fn):
   model: a pytorch model
   loss_fn: a pytorch loss function
   train_data_loader: a pytorch data loader
   num_batches: number of batches to use for the hessian calculation
   device: the device to use for the hessian calculation
   n_top_vectors: number of top eigenvalues / eigenvectors to return
   param_extract_fn: a function that takes a model and returns a list of parameters to compute the hessian with respect to (pass None to use all parameters)
   returns: a tuple of (eigenvalues, eigenvectors)
   eigenvalues: a numpy array of the top eigenvalues, arranged in increasing order
   eigenvectors: a numpy array of the top eigenvectors, arranged in increasing order, shape (n_top_vectors, num_params)
   param_extract_fn = param_extract_fn or (lambda x: x.parameters())
   num_params = sum(p.numel() for p in param_extract_fn(model))
   subset_images, subset_labels = [], []
   for batch_idx, (images, labels) in enumerate(train_data_loader):
       if batch_idx >= num_batches:
   subset_images =
   subset_labels =
   def compute_loss():
       output = model(subset_images)
       return loss_fn(output, subset_labels)
   def hessian_vector_product(vector):
       grad_params = grad(compute_loss(), param_extract_fn(model), create_graph=True)
       flat_grad =[g.view(-1) for g in grad_params])
       grad_vector_product = t.sum(flat_grad * vector)
       hvp = grad(grad_vector_product, param_extract_fn(model), retain_graph=True)
       return[g.contiguous().view(-1) for g in hvp])
   def matvec(v):
       v_tensor = t.tensor(v, dtype=t.float32, device=device)
       return hessian_vector_product(v_tensor).cpu().detach().numpy()
   linear_operator = LinearOperator((num_params, num_params), matvec=matvec)
   eigenvalues, eigenvectors = eigsh(linear_operator, k=n_top_vectors, tol=0.001, which='LM', return_eigenvectors=True)
   eigenvectors = np.transpose(eigenvectors)
   return eigenvalues, eigenvectors

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5 comments, sorted by Click to highlight new comments since: Today at 1:19 PM

If you're interested in approximating Hessian-vector products efficiently for frontier-size models, this recent Anthropic paper describes a mechanism for doing so.

Ah nice, thanks! This looks really interesting and useful

You should train both a feedforward network and a CNN on image classification on imagenet, to see if we see that the hessian of the CNN is more similar to the identity after training rather than the feedforward because of the image understanding priors.

Given that this method returns a numeric matrix, then it must be an Hessian evaluated at a point or the average Hessian of many points. Is the result the Hessian averaged over all training data? Is this average useful rather than just cancelling out high and low Hessian values

The method described does not explicitly compute the full Hessian matrix. Instead, it derives the top eigenvalues and eigenvectors of the Hessian. The implementation accumulates a large batch from a dataloader by concatenating n_batches of the typical batch size. This is an approximation to estimate the genuine loss/gradient on the complete dataset more closely. If you have a large and high-variance dataset, averaging gradients over multiple batches might be better. This is because the loss calculated from a single, accumulated batch may not be adequately representative of the entire dataset's true loss.