loss-landscapes

loss-landscapes is a PyTorch library for approximating neural network loss functions, and other related metrics, in low-dimensional subspaces of the model's parameter space. The library makes the production of visualizations such as those seen in Visualizing the Loss Landscape of Neural Nets much easier, aiding the analysis of the geometry of neural network loss landscapes.

This library does not provide plotting facilities, letting the user define how the data should be plotted.

Known limitations: Some normalisation routines produce incorrect results, gradient evaluation has known defects, and the contrib modules cannot be imported. See the strict xfail tests for details before relying on affected functionality.

1. What is a Loss Landscape?

Let L : Parameters -> Real Numbers be a loss function, which maps a point in the model parameter space to a real number. For a neural network with n parameters, the loss function L takes an n-dimensional input. We can define the loss landscape as the set of all n+1-dimensional points (param, L(param)), for all points param in the parameter space. For example, the image below, reproduced from the paper by Li et al (2018), link above, provides a visual representation of what a loss function over a two-dimensional parameter space might look like:

Of course, real machine learning models have a number of parameters much greater than 2, so the parameter space of the model is virtually never two-dimensional. Because we can't print visualizations in more than two dimensions, we cannot hope to visualize the "true" shape of the loss landscape. Instead, a number of techniques exist for reducing the parameter space to one or two dimensions, ranging from dimensionality reduction techniques like PCA, to restricting ourselves to a particular subspace of the overall parameter space. For more details, read Li et al's paper.

2. Base Example: Supervised Loss in Parameter Subspaces

The simplest use case for loss-landscapes is to estimate the value of a supervised loss function in a subspace of a neural network's parameter space. The subspace in question may be a point, a line, or a plane (these subspaces can be meaningfully visualized). Suppose the user has trained a supervised learning model, of type torch.nn.Module, on a dataset consisting of samples X and labels y, by minimizing some loss function. The user now wishes to produce a surface plot alike to the one in section 1.

This is accomplished as follows:

metric = Loss(loss_function, X, y)
landscape = random_plane(model, metric, normalize="filter")

As seen in the example above, the two core concepts in loss-landscapes are metrics and parameter subspaces. The latter define the section of parameter space to be considered, while the former define what quantity is evaluated at each considered point in parameter space, and how it is computed. In the example above, we define a Loss metric over data X and labels y, and instruct loss_landscape to evaluate it in a randomly generated planar subspace.

This would return a 2-dimensional array of loss values, which the user can plot in any desirable way. Example visualizations the user might use this type of data for are shown below.

Check the examples directory for jupyter notebooks with more in-depth examples of what is possible.

3. Metrics and Custom Metrics

The loss-landscapes library can compute any quantity of interest at a collection of points in a parameter subspace, not just loss. This is accomplished using a Metric: a callable object which applies a pre-determined function, such as a cross entropy loss with a specific set of inputs and outputs, to the model. The loss_landscapes.model_metrics package contains a number of metrics that cover common use cases, such as Loss (evaluates a loss function), LossGradient (evaluates the gradient of the loss w.r.t. the model parameters), PrincipalCurvatureEvaluator (evaluates the principal curvatures of the loss function), and more.

Furthermore, the user can add custom metrics by subclassing Metric. As an example, consider the library implementation of Loss, for torch models:

class Metric(abc.ABC):
    """A quantity that can be computed given a model or an agent."""

    def __init__(self):
        super().__init__()

    @abc.abstractmethod
    def __call__(self, model_wrapper: ModelWrapper):
        pass


class Loss(Metric):
    """Computes a specified loss function over specified input-output pairs."""

    def __init__(self, loss_fn, inputs: torch.Tensor, target: torch.Tensor):
        super().__init__()
        self.loss_fn = loss_fn
        self.inputs = inputs
        self.target = target

    def __call__(self, model_wrapper: ModelWrapper) -> float:
        return self.loss_fn(model_wrapper.forward(self.inputs), self.target).item()

The user may create custom Metrics in a similar manner. One complication is that the Metric class' __call__ method is designed to take as input a ModelWrapper rather than a model. This class is internal to the library and exists to facilitate the handling of the myriad of different models a user may pass as inputs to a function such as loss_landscapes.planar_interpolation(). It is sufficient for the user to know that a ModelWrapper is a callable object that can be used to call the model on a given input (see the call_fn argument of the ModelInterface class in the next section). The class also provides a get_model() method that exposes a reference to the underlying model, should the user wish to carry out more complicated operations on it.

In summary, the Metric abstraction adds a great degree of flexibility. An metric defines what quantity dependent on model parameters the user is interested in evaluating, and how to evaluate it. The user could define, for example, a metric that computes an estimate of the expected return of a reinforcement learning agent.

4. More Complex Models

In the general case of a simple supervised learning model, as in the sections above, client code calls functions such as loss_landscapes.linear_interpolation and passes as argument a PyTorch module of type torch.nn.Module.

For more complex cases, such as when the user wants to evaluate the loss landscape as a function of a subset of the model parameters, or the expected return landscape for a RL agent, the user must specify to the loss-landscapes library how to interface with the model (or the agent, on a more general level). This is accomplished using a ModelWrapper object, which hides the implementation details of the model or agent. For general use, the library supplies the GeneralModelWrapper in the loss_landscapes.model_interface.model_wrapper module.

Assume the user wishes to estimate the expected return of some RL agent which provides an agent.act(observation) method for action selection. Then, the example from section 2 becomes as follows:

metric = ExpectedReturnMetric(env, n_samples)
agent_wrapper = GeneralModelWrapper(agent, [agent.q_function, agent.policy], lambda agent, x: agent.act(x))
landscape = random_plane(agent_wrapper, metric, normalize="filter")

5. Installation and Use

The build targets the following combinations:

Compatibility is tested on Linux x86-64 CPU with standard, GIL-enabled CPython, using the latest compatible stable patches in these series. GPU execution, free-threaded builds and alternative interpreters are not covered by these checks.

Install this checkout from the repository root:

python -m pip install .

This installs NumPy and PyTorch automatically. The published PyPI version, 3.0.6, does not declare PyTorch as a dependency; use the checkout installation above for the support matrix listed here.

import loss_landscapes
import loss_landscapes.metrics

See CONTRIBUTING.md for compatibility checks and platform prerequisites.

Development

Development uses uv:

uv sync --all-groups
uv run pre-commit install
uv run pytest

See CONTRIBUTING.md for the full workflow and CHANGELOG.md for release notes.