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A library for approximating loss landscapes in low-dimensional parameter subspaces

# 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. Other deep learning frameworks are not supported, though a TensorFlow version, loss-landscapes-tf, is planned for a future release.

NOTE: this library is in early development. Bugs are virtually a certainty, and the API is volatile. Do not use this library in production code. For prototyping and research, always use the newest version of the library.

## 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. WIP: Connecting Paths, Saddle Points, and Trajectory Tracking

A number of features are currently under development, but as of yet incomplete.

A number of papers in recent years have shown that loss landscapes of neural networks are dominated by a proliferation of saddle points, that good solutions are better described as large low-loss plateaus than as "well-bottom" points, and that for sufficiently high-dimensional networks, a low-loss path in parameter space can be found between almost any arbitrary pair of minima. In the future, the loss-landscapes library will feature implementations of algorithms for finding such low-loss connecting paths in the loss landscape, as well as tools to facilitate the study of saddle points.

Some sort of trajectory tracking features are also under consideration, though at the time it's unclear what this should actually mean, as the optimization trajectory is implicitly tracked by the user's training loop. Any metric along the optimization trajectory can be tracked with libraries such as ignite for PyTorch.

## 6. Support for Other DL Libraries

The loss-landscapes library was initially designed to be agnostic to the DL framework in use. However, with the increasing number of use cases to cover it became obvious that maintaining the original library-agnostic design was adding too much complexity to the code.

A TensorFlow version, loss-landscapes-tf, is planned for the future.

## 7. Installation and Use

The package is available on PyPI. Install using pip install loss-landscapes. To use the library, import as follows:

import loss_landscapes
import loss_landscapes.metrics