Learning Neural Network Subspaces

Welcome to the codebase for Learning Neural Network Subspaces by Mitchell Wortsman, Maxwell Horton, Carlos Guestrin, Ali Farhadi, Mohammad Rastegari.

Abstract

Recent observations have advanced our understanding of the neural network optimization landscape, revealing the existence of (1) paths of high accuracy containing diverse solutions and (2) wider minima offering improved performance. Previous methods observing diverse paths require multiple training runs. In contrast we aim to leverage both property (1) and (2) with a single method and in a single training run. With a similar computational cost as training one model, we learn lines, curves, and simplexes of high-accuracy neural networks. These neural network subspaces contain diverse solutions that can be ensembled, approaching the ensemble performance of independently trained networks without the training cost. Moreover, using the subspace midpoint boosts accuracy, calibration, and robustness to label noise, outperforming Stochastic Weight Averaging.

Code Overview

In this repository we walk through learning neural network subspaces with PyTorch. We will ground the discussion with learning a line of neural networks. In our code, a line is defined by endpoints weight and weight1 and a point on the line is given by w = (1 - alpha) * weight + alpha * weight1 for some alpha in [0,1].

Algorithm 1 (see paper) works as follows:

1. weight and weight1 are initialized independently.
2. For each batch data, targets, alpha is chosen uniformly from [0,1] and the weights w = (1 - alpha) * weight + alpha * weight1 are used in the forward pass.
3. The regularization term is computed (see Eq. 3).
4. With loss.backward() and optimizer.step() the endpoints weight and weight1 are updated.

Instead of using a regular nn.Conv2d we instead use a SubspaceConv (found in modes/modules.py).

class SubspaceConv(nn.Conv2d):
    def forward(self, x):
        w = self.get_weight()
        x = F.conv2d(
            x,
            w,
            self.bias,
            self.stride,
            self.padding,
            self.dilation,
            self.groups,
        )
        return x

For each subspace type (lines, curves, and simplexes) the function get_weight must be implemented. For lines we use:

class TwoParamConv(SubspaceConv):
    def __init__(self, *args, **kwargs):
        super().__init__(*args, **kwargs)
        self.weight1 = nn.Parameter(torch.zeros_like(self.weight))

    def initialize(self, initialize_fn):
        initialize_fn(self.weight1)

class LinesConv(TwoParamConv):
    def get_weight(self):
        w = (1 - self.alpha) * self.weight + self.alpha * self.weight1
        return w

Note that the other endpoint weight is instantiated and initialized by nn.Conv2d. Also note that there is an equivalent implementation for batch norm layers also found in modes/modules.py.

Now we turn to the training logic which appears in trainers/train_one_dim_subspaces.py. In the snippet below we assume we are not training with the layerwise variant (args.layerwise = False) and we are drawing only one sample from the subspace (args.num_samples = 1).

for batch_idx, (data, target) in enumerate(train_loader):
    data, target = data.to(args.device), target.to(args.device)

    alpha = np.random.uniform(0, 1)
    for m in model.modules():
        if isinstance(m, nn.Conv2d) or isinstance(m, nn.BatchNorm2d):
            setattr(m, f"alpha", alpha)

    optimizer.zero_grad()
    output = model(data)
    loss = criterion(output, target)

All that's left is to compute the regularization term and call backward. For lines, this is given by the snippet below.

    num = 0.0
    norm = 0.0
    norm1 = 0.0
    for m in model.modules():
        if isinstance(m, nn.Conv2d):
            num += (self.weight * self.weight1).sum()
            norm += self.weight.pow(2).sum()
            norm1 += self.weight1.pow(2).sum()
    loss += args.beta * (num.pow(2) / (norm * norm1))

    loss.backward()

    optimizer.step()

Training Lines, Curves, and Simplexes

We now walkthrough generating the plots in Figures 4 and 5 of the paper. Before running code please install PyTorch and Tensorboard (for making plots you will also need tex on your computer). Note that this repository differs from that used to generate the figures in the paper, as the latter leveraged Apple's internal tools. Accordingly there may be some bugs and we encourage you to submit an issue or send an email if you run into any problems.

In this example walkthrough we consider TinyImageNet, which we download to ~/data using a script such as this. To run standard training and ensemble the trained models, use the following command:

python experiment_configs/tinyimagenet/ensembles/train_ensemble_members.py
python experiment_configs/tinyimagenet/ensembles/eval_ensembles.py

Note that if your data is not in ~/data please change the paths in these experiment configs. Logs and checkpoints be saved in learning-subspaces-results, although this path can also be changed.

For one dimensional subspaces, use the following command to train:

python experiment_configs/tinyimagenet/one_dimensional_subspaces/train_lines.py
python experiment_configs/tinyimagenet/one_dimensional_subspaces/train_lines_layerwise.py
python experiment_configs/tinyimagenet/one_dimensional_subspaces/train_curves.py

To evaluate (i.e. generate the data for Figure 4) use:

python experiment_configs/tinyimagenet/one_dimensional_subspaces/eval_lines.py
python experiment_configs/tinyimagenet/one_dimensional_subspaces/eval_lines_layerwise.py
python experiment_configs/tinyimagenet/one_dimensional_subspaces/eval_curves.py

We recommend looking at the experiment config files before running, which can be modified to change the type of model, number of random seeds. The default in these configs is 2 random seeds.

Analogously, to train simplexes use:

python experiment_configs/tinyimagenet/simplexes/train_simplexes.py
python experiment_configs/tinyimagenet/simplexes/train_simplexes_layerwise.py

For generating plots like those in Figure 4 and 5 use:

python analyze_results/tinyimagenet/one_dimensional_subspaces.py
python analyze_results/tinyimagenet/simplexes.py

Equivalent configs exist for other datasets, and the configs can be modified to add label noise, experiment with other models, and more. Also, if there is any functionality missing from this repository that you would like please also submit an issue.

Bibtex

@article{wortsman2021learning,
  title={Learning Neural Network Subspaces},
  author={Wortsman, Mitchell and Horton, Maxwell and Guestrin, Carlos and Farhadi, Ali and Rastegari, Mohammad},
  journal={arXiv preprint arXiv:2102.10472},
  year={2021}
}