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

# loss-landscapes

`loss-landscapes` is a library for approximating neural network loss functions, and other related metrics, along low-dimensional subspaces of the model 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.

Currently, `loss-landscapes` only supports PyTorch models, but support for other DL libraries (TensorFlow in particular) is planned for future releases.

## 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. Loss in Parameter Subspaces

This library facilitates the computation of a neural network model's loss landscape in low-dimensional subspaces of the parameter space. It does not provide plotting facilities, letting the user define how the data should be plotted, and is designed to support any deep learning library (in principle - currently only PyTorch is supported). As an example, it allows a PyTorch user to produce data for a plot such as the one seen above by simply calling

```evaluator = LossEvaluator(loss_function, X, y)
landscape = random_plane(model, evaluator, normalize='filter')
```

This would return a 2-dimensional array of loss values, which the user can plot in any desirable way. Below are a contour plot and a surface plot made in `matplotlib`, drawn over the same loss landscape data, that demonstrate what a loss landscape along a planar parameter subspace could look like. Check the `examples` directory for `jupyter` notebooks with more in-depth examples of what is possible.

## 3. Evaluators and Custom Evaluators

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 an `Evaluator`: a callable object which applies a pre-determined function, such as a cross entropy loss with a specific set of inputs and outputs, at every point. The `loss_landscapes.evaluators` package contains a number of evaluators that cover common use cases, such as `LossEvaluator` (evaluates a loss function), `GradientEvaluator` (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 evaluators by subclassing `Evaluator`. As an example, consider the library implementation of `LossEvaluator`, for `torch` models:

```class LossEvaluator(SupervisedTorchEvaluator):
""" Computes a specified loss function over specified input-output pairs. """
def __init__(self, supervised_loss_fn, inputs, target):
super().__init__(supervised_loss_fn, inputs, target)

def __call__(self, model) -> np.ndarray:
return self.loss_fn(model(self.inputs), self.target).clone().detach().numpy()
```

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

## 4. Model Wrappers

In the general case, client code calls functions such as `loss_landscapes.linear_interpolation` and passes as argument a reference to a deep learning model. The library implementation wraps the model using a `ModelWrapper` - an abstraction layer that hides the model's implementation details, which are specific to the DL library being used in the client code. This process is not visible to the user and in most cases can safely be ignored.

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, the user can import a model wrapper from the `loss_landscapes.model_interface` package and explicitly wrap it before passing it to a function such as `linear_interpolation`. Some of the wrappers allow for highly granular control over which model parameters are ignored.

## 5. Trajectory Tracking

The library also enables trajectory tracking in a simple, model-agnostic fashion. A `TrajectoryTracker` object stores a model parameter history, which the user can update by passing the model to the tracker, which extracts the model's current state and appends it to the trajectory history.

More importantly, a trajectory tracker can compute low-dimensional approximations of the trajectory, using random directions, PCA directions (see Li et al., 2018), and so on. A good use case for trajectory tracking is to overlay the optimization trajectory on a landscape contour plot, which can easily be accomplished using `matplotlib`.

## 6. Work in Progress: Connecting Paths and Saddle Points

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.

## 7. Support for Other DL Libraries

Once the currently envisioned features are complete, the first priority will be adding support for TensorFlow.

## 8. 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.evaluators  # for the base Evaluator class
import loss_landscapes.evaluators.torch  # for the pre-defined PyTorch evaluators
```