Per-sample-gradients

What is it?

Per-sample-gradient computation is computing the gradient for each and every sample in a batch of data. It is a useful quantity in differential privacy, meta-learning, and optimization research.

Note

This tutorial requires PyTorch 2.0.0 or later.

import torch
import torch.nn as nn
import torch.nn.functional as F
torch.manual_seed(0)

# Here's a simple CNN and loss function:

class SimpleCNN(nn.Module):
    def __init__(self):
        super(SimpleCNN, self).__init__()
        self.conv1 = nn.Conv2d(1, 32, 3, 1)
        self.conv2 = nn.Conv2d(32, 64, 3, 1)
        self.fc1 = nn.Linear(9216, 128)
        self.fc2 = nn.Linear(128, 10)

    def forward(self, x):
        x = self.conv1(x)
        x = F.relu(x)
        x = self.conv2(x)
        x = F.relu(x)
        x = F.max_pool2d(x, 2)
        x = torch.flatten(x, 1)
        x = self.fc1(x)
        x = F.relu(x)
        x = self.fc2(x)
        output = F.log_softmax(x, dim=1)
        output = x
        return output

def loss_fn(predictions, targets):
    return F.nll_loss(predictions, targets)

Let’s generate a batch of dummy data and pretend that we’re working with an MNIST dataset. The dummy images are 28 by 28 and we use a minibatch of size 64.

device = 'cuda'

num_models = 10
batch_size = 64
data = torch.randn(batch_size, 1, 28, 28, device=device)

targets = torch.randint(10, (64,), device=device)

In regular model training, one would forward the minibatch through the model, and then call .backward() to compute gradients. This would generate an ‘average’ gradient of the entire mini-batch:

model = SimpleCNN().to(device=device)
predictions = model(data)  # move the entire mini-batch through the model

loss = loss_fn(predictions, targets)
loss.backward()  # back propagate the 'average' gradient of this mini-batch

In contrast to the above approach, per-sample-gradient computation is equivalent to:

• for each individual sample of the data, perform a forward and a backward pass to get an individual (per-sample) gradient.

def compute_grad(sample, target):
    sample = sample.unsqueeze(0)  # prepend batch dimension for processing
    target = target.unsqueeze(0)

    prediction = model(sample)
    loss = loss_fn(prediction, target)

    return torch.autograd.grad(loss, list(model.parameters()))


def compute_sample_grads(data, targets):
    """ manually process each sample with per sample gradient """
    sample_grads = [compute_grad(data[i], targets[i]) for i in range(batch_size)]
    sample_grads = zip(*sample_grads)
    sample_grads = [torch.stack(shards) for shards in sample_grads]
    return sample_grads

per_sample_grads = compute_sample_grads(data, targets)

sample_grads[0] is the per-sample-grad for model.conv1.weight. model.conv1.weight.shape is [32, 1, 3, 3]; notice how there is one gradient, per sample, in the batch for a total of 64.

print(per_sample_grads[0].shape)

torch.Size([64, 32, 1, 3, 3])

Per-sample-grads, the efficient way, using function transforms

We can compute per-sample-gradients efficiently by using function transforms.

The torch.func function transform API transforms over functions. Our strategy is to define a function that computes the loss and then apply transforms to construct a function that computes per-sample-gradients.

We’ll use the torch.func.functional_call function to treat an nn.Module like a function.

First, let’s extract the state from model into two dictionaries, parameters and buffers. We’ll be detaching them because we won’t use regular PyTorch autograd (e.g. Tensor.backward(), torch.autograd.grad).

from torch.func import functional_call, vmap, grad

params = {k: v.detach() for k, v in model.named_parameters()}
buffers = {k: v.detach() for k, v in model.named_buffers()}

Next, let’s define a function to compute the loss of the model given a single input rather than a batch of inputs. It is important that this function accepts the parameters, the input, and the target, because we will be transforming over them.

Note - because the model was originally written to handle batches, we’ll use torch.unsqueeze to add a batch dimension.

def compute_loss(params, buffers, sample, target):
    batch = sample.unsqueeze(0)
    targets = target.unsqueeze(0)

    predictions = functional_call(model, (params, buffers), (batch,))
    loss = loss_fn(predictions, targets)
    return loss

Now, let’s use the grad transform to create a new function that computes the gradient with respect to the first argument of compute_loss (i.e. the params).

ft_compute_grad = grad(compute_loss)

The ft_compute_grad function computes the gradient for a single (sample, target) pair. We can use vmap to get it to compute the gradient over an entire batch of samples and targets. Note that in_dims=(None, None, 0, 0) because we wish to map ft_compute_grad over the 0th dimension of the data and targets, and use the same params and buffers for each.

ft_compute_sample_grad = vmap(ft_compute_grad, in_dims=(None, None, 0, 0))

Finally, let’s used our transformed function to compute per-sample-gradients:

ft_per_sample_grads = ft_compute_sample_grad(params, buffers, data, targets)

we can double check that the results using grad and vmap match the results of hand processing each one individually:

for per_sample_grad, ft_per_sample_grad in zip(per_sample_grads, ft_per_sample_grads.values()):
    assert torch.allclose(per_sample_grad, ft_per_sample_grad, atol=3e-3, rtol=1e-5)

A quick note: there are limitations around what types of functions can be transformed by vmap. The best functions to transform are ones that are pure functions: a function where the outputs are only determined by the inputs, and that have no side effects (e.g. mutation). vmap is unable to handle mutation of arbitrary Python data structures, but it is able to handle many in-place PyTorch operations.

Performance comparison

Curious about how the performance of vmap compares?

Currently the best results are obtained on newer GPU’s such as the A100 (Ampere) where we’ve seen up to 25x speedups on this example, but here are some results on our build machines:

def get_perf(first, first_descriptor, second, second_descriptor):
    """takes torch.benchmark objects and compares delta of second vs first."""
    second_res = second.times[0]
    first_res = first.times[0]

    gain = (first_res-second_res)/first_res
    if gain < 0: gain *=-1
    final_gain = gain*100

    print(f"Performance delta: {final_gain:.4f} percent improvement with {first_descriptor} ")

from torch.utils.benchmark import Timer

without_vmap = Timer(stmt="compute_sample_grads(data, targets)", globals=globals())
with_vmap = Timer(stmt="ft_compute_sample_grad(params, buffers, data, targets)",globals=globals())
no_vmap_timing = without_vmap.timeit(100)
with_vmap_timing = with_vmap.timeit(100)

print(f'Per-sample-grads without vmap {no_vmap_timing}')
print(f'Per-sample-grads with vmap {with_vmap_timing}')

get_perf(with_vmap_timing, "vmap", no_vmap_timing, "no vmap")

Per-sample-grads without vmap <torch.utils.benchmark.utils.common.Measurement object at 0x7fe5d02aef20>
compute_sample_grads(data, targets)
  87.80 ms
  1 measurement, 100 runs , 1 thread
Per-sample-grads with vmap <torch.utils.benchmark.utils.common.Measurement object at 0x7fe5d00d3d90>
ft_compute_sample_grad(params, buffers, data, targets)
  8.69 ms
  1 measurement, 100 runs , 1 thread
Performance delta: 910.0004 percent improvement with vmap

There are other optimized solutions (like in https://github.com/pytorch/opacus) to computing per-sample-gradients in PyTorch that also perform better than the naive method. But it’s cool that composing vmap and grad give us a nice speedup.

In general, vectorization with vmap should be faster than running a function in a for-loop and competitive with manual batching. There are some exceptions though, like if we haven’t implemented the vmap rule for a particular operation or if the underlying kernels weren’t optimized for older hardware (GPUs). If you see any of these cases, please let us know by opening an issue at on GitHub.