A gradient processing and optimisation library in JAX.

## Project description

# Optax

## Introduction

Optax is a gradient processing and optimization library for JAX.

Optax is designed to facilitate research by providing building blocks that can be easily recombined in custom ways.

Our goals are to

- Provide simple, well-tested, efficient implementations of core components.
- Improve research productivity by enabling to easily combine low level ingredients into custom optimisers (or other gradient processing components).
- Accelerate adoption of new ideas by making it easy for anyone to contribute.

We favour focusing on small composable building blocks that can be effectively combined into custom solutions. Others may build upon these basic components more complicated abstractions. Whenever reasonable, implementations prioritise readability and structuring code to match standard equations, over code reuse.

An initial prototype of this library was made available in JAX's experimental
folder as `jax.experimental.optix`

. Given the wide adoption across DeepMind
of `optix`

, and after a few iterations on the API, `optix`

was eventually moved
out of `experimental`

as a standalone open-source library, renamed `optax`

.

Documentation on Optax can be found at optax.readthedocs.io.

## Installation

You can install the latest released version of Optax from PyPI via:

```
pip install optax
```

or you can install the latest development version from GitHub:

```
pip install git+https://github.com/deepmind/optax.git
```

## Quickstart

Optax contains implementations of many popular optimizers and
loss functions.
For example the following code snippet uses the Adam optimizer from `optax.adam`

and the mean squared error from `optax.l2_loss`

. We initialize the optimizer
state using the `init`

function and `params`

of the model.

```
optimizer = optax.adam(learning_rate)
# Obtain the `opt_state` that contains statistics for the optimizer.
params = {'w': jnp.ones((num_weights,))}
opt_state = optimizer.init(params)
```

To write the update loop we need a loss function that can be differentiated by
Jax (with `jax.grad`

in this
example) to obtain the gradients.

```
compute_loss = lambda params, x, y: optax.l2_loss(params['w'].dot(x), y)
grads = jax.grad(compute_loss)(params, xs, ys)
```

The gradients are then converted via `optimizer.update`

to obtain the updates
that should be applied to the current params to obtain the new ones.
`optax.apply_updates`

is a convinience utility to do this.

```
updates, opt_state = optimizer.update(grads, opt_state)
params = optax.apply_updates(params, updates)
```

You can continue the quick start in the Optax quickstart notebook.

## Components

We refer to the docs for a detailed list of available Optax components. Here, we highlight the main categories of buiilding blocks provided by Optax.

### Gradient Transformations (transform.py)

One of the key building blocks of `optax`

is a `GradientTransformation`

.

Each transformation is defined two functions:

`state = init(params)`

`grads, state = update(grads, state, params=None)`

The `init`

function initializes a (possibly empty) set of statistics (aka state)
and the `update`

function transforms a candidate gradient given some statistics,
and (optionally) the current value of the parameters.

For example:

```
tx = scale_by_rms()
state = tx.init(params) # init stats
grads, state = tx.update(grads, state, params) # transform & update stats.
```

### Composing Gradient Transformations (combine.py)

The fact that transformations take candidate gradients as input and return processed gradients as output (in contrast to returning the updated parameters) is critical to allow to combine arbitrary transformations into a custom optimiser / gradient processor, and also allows to combine transformations for different gradients that operate on a shared set of variables.

For instance, `chain`

combines them sequentially, and returns a
new `GradientTransformation`

that applies several transformations in sequence.

For example:

```
my_optimiser = chain(
clip_by_global_norm(max_norm),
scale_by_adam(eps=1e-4),
scale(-learning_rate))
```

### Wrapping Gradient Transformations (wrappers.py)

Optax also provides several wrappers that take a `GradientTransformation`

as
input and return a new `GradientTransformation`

that modifies the behaviour
of the inner transformation in a specific way.

For instance the `flatten`

wrapper flattens gradients into a single large vector
before applying the inner GradientTransformation. The transformed updated are
then unflattened before being returned to the user. This can be used to reduce
the overhead of performing many calculations on lots of small variables,
at the cost of increasing memory usage.

For example:

```
my_optimiser = flatten(adam(learning_rate))
```

Other examples of wrappers include accumulating gradients over multiple steps, or applying the inner transformation only to specific parameters or at specific steps.

### Schedules (schedule.py)

Many popular transformations use time dependent components, e.g. to anneal
some hyper-parameter (e.g. the learning rate). Optax provides for this purpose
`schedules`

that can be used to decay scalars as a function of a `step`

count.

For example you may use a polynomial schedule (with `power=1`

) to decay
a hyper-parameter linearly over a number of steps:

```
schedule_fn = polynomial_schedule(
init_value=1., end_value=0., power=1, transition_steps=5)
for step_count in range(6):
print(schedule_fn(step_count)) # [1., 0.8, 0.6, 0.4, 0.2, 0.]
```

Schedules are used by certain gradient transformation, for instance:

```
schedule_fn = polynomial_schedule(
init_value=-learning_rate, end_value=0., power=1, transition_steps=5)
optimiser = chain(
clip_by_global_norm(max_norm),
scale_by_adam(eps=1e-4),
scale_by_schedule(schedule_fn))
```

### Popular optimisers (alias.py)

In addition to the low level building blocks we also provide aliases for popular
optimisers built using these components (e.g. RMSProp, Adam, AdamW, etc, ...).
These are all still instances of a `GradientTransformation`

, and can therefore
be further combined with any of the individual building blocks.

For example:

```
def adamw(learning_rate, b1, b2, eps, weight_decay):
return chain(
scale_by_adam(b1=b1, b2=b2, eps=eps),
scale_and_decay(-learning_rate, weight_decay=weight_decay))
```

### Applying updates (update.py)

After transforming an update using a `GradientTransformation`

or any custom
manipulation of the update, you will typically apply the update to a set
of parameters. This can be done trivially using `tree_map`

.

For convenience, we expose an `apply_updates`

function to apply updates to
parameters. The function just adds the updates and the parameters together,
i.e. `tree_map(lambda p, u: p + u, params, updates)`

.

```
updates, state = tx.update(grads, state, params) # transform & update stats.
new_params = optax.apply_updates(params, updates) # update the parameters.
```

Note that separating gradient transformations from the parameter update is
critical to support composing sequence of transformations (e.g. `chain`

), as
well as combine multiple updates to the same parameters (e.g. in multi-task
settings where different tasks need different sets of gradient transformations).

### Losses (loss.py)

Optax provides a number of standard losses used in deep learning, such as
`l2_loss`

, `softmax_cross_entropy`

, `cosine_distance`

, etc.

```
loss = huber_loss(predictions, targets)
```

The losses accept batches as inputs, however they perform no reduction across the batch dimension(s). This is trivial to do in JAX, for example:

```
avg_loss = jnp.mean(huber_loss(predictions, targets))
sum_loss = jnp.sum(huber_loss(predictions, targets))
```

### Second Order (second_order.py)

Computing the Hessian or Fisher information matrices for neural networks is typically intractable due to the quadratic memory requirements. Solving for the diagonals of these matrices is often a better solution. The library offers functions for computing these diagonals with sub-quadratic memory requirements.

### Stochastic gradient estimators (stochastic_gradient_estimators.py)

Stochastic gradient estimators compute Monte Carlo estimates of gradients of the expectation of a function under a distribution with respect to the distribution's parameters.

Unbiased estimators, such as the score function estimator (REINFORCE),
pathwise estimator (reparameterization trick) or measure valued estimator,
are implemented: `score_function_jacobians`

, `pathwise_jacobians`

and ` measure_valued_jacobians`

. Their applicability (both in terms of functions and
distributions) is discussed in their respective documentation.

Stochastic gradient estimators can be combined with common control variates for
variance reduction via `control_variates_jacobians`

. For provided control
variates see `delta`

and `moving_avg_baseline`

.

The result of a gradient estimator or `control_variates_jacobians`

contains the
Jacobians of the function with respect to the samples from the input
distribution. These can then be used to update distributional parameters, or
to assess gradient variance.

Example of how to use the `pathwise_jacobians`

estimator:

```
dist_params = [mean, log_scale]
function = lambda x: jnp.sum(x * weights)
jacobians = pathwise_jacobians(
function, dist_params,
utils.multi_normal, rng, num_samples)
mean_grads = jnp.mean(jacobians[0], axis=0)
log_scale_grads = jnp.mean(jacobians[1], axis=0)
grads = [mean_grads, log_scale_grads]
optim_update, optim_state = optim.update(grads, optim_state)
updated_dist_params = optax.apply_updates(dist_params, optim_update)
```

where `optim`

is an optax optimizer.

## Citing Optax

Optax is part of the DeepMind JAX Ecosystem, to cite Optax please use the DeepMind JAX Ecosystem citation.

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