parax 0.4.13

Declarative, parametric modelling in JAX

Parax is a parametric modelling library built on top of JAX and Equinox.

At its core, the library provides a Parameter class which inherits from eqx.Module and wraps a JAX array. A parameter can be marked as fixed for training, as well as assigned arbitrary metadata.

However, the library also provides additional helpers, including parax.partition, parax.Module and parax.Operator. partition caters for easy partitioning of module's that contain Parameter objects. Parameter and Module together provide an experience similar to PyTorch with its torch.nn.Parameter, but in a more Equinox/JAX-friendly style. Operator caters for composable operations over arbitrary arguments (such as Module's), allowing for easy PyTree feature extraction.

Parax
Author	Gary Allen
Homepage	github.com/parax/parax
Docs	gvcallen.github.io/parax

Features

• Easy parameter fixing: Parameters can be marked as fixed and the resultant modules partitioned using parax.partition.
• Encapsulated constraints and scaling: Optional scaling and transformations are abstracted away by applying them when the parameter object is cast to a JAX array. This can be used, for example, to enforce positivity or arbitrary constraints during optimization.
• Parameter transforms: Arbitrary transforms can be applied to parameters using myparam.transformed(bij). This applies a transform to both the parameter and its underlying distribution (if any).
• Arbitrary metadata support: While Parax natively caters for common metadata such as distributions, bijectors, scaling, bounds and a name, arbitrary metadata can also be attached for more complex modelling purposes (for example, in the scientific domain it is common to want to attach units to a parameter).
• Extended Equinox module: Parax provides parax.Module, which extends eqx.Module to allow for easy inspection, updating, fixing, freeing, or mapping of parameters and their metadata deep within complex models using simple string paths and bulk with_* methods. For example, parax.Module.named_params() returns a dictionary of parameters with names based on string paths.
• (experimental) Composable PyTree operations: Parax provides parax.Operator, which caters for composable, parametric operations over arbitrary arguments. This can be very useful in manipulating domain-specific parax.Module objects in a parameter-aware manner.
• (experimental) Model saving and loading. By employing methods to serialize distreqx distributions and transformation/bijections, Parax provides (experimental) support to directly save (pickle) models using parax.load and parax.save, as long as they align to certain rules.

Installation

Parax can be installed using pip directly:

pip install parax

Overview

The Parameter class is designed to be used as if it was a JAX array. The raw value inside a parameter is therefore stored in "latent" space i.e. untransformed and unscaled. However, parameters eagerly cast to JAX arrays, at which point any transformation/bijection and scaling is applied. This completely abstracts the underlying latent value (to be used in optimization) from the user, bypassing the need to explicitly apply the transform.

To make optimization easy, Parax also comes with a built-in parax.partition function, which partitions a model into trainable parameters. If a model is built purely using Parameter's, this removes the need for any conditional logical that would usually be done manually during eqx.partition.

Further, Parax also provides an extended version of Equinox's Module in parax.Module. This allows for parameter-aware module inspection and manipulation. For example, parameters can easily be flattened, updated using a single string assigned using the hierarchy, and mapped in batches.

The library is mainly intended for use in domain-specific scientific modeling, but can easily be applied to broader applications.

Example 1: Enforcing bounds

The following example creates a parax.Parameter that is strictly bounded between 0.0 and 1.0, and whose physical value follows a normal distribution.

import jax.numpy as jnp
import parax as prx
from distreqx.bijectors import Sigmoid

normal_param = prx.Normal(0.5, 0.1, transform=Sigmoid())
assert normal_param.latent_value == 0.0
assert normal_param.value == 0.5
assert jnp.array(normal_param) == 0.5

Example 2: Manipulating parameters

parax.Module is designed as a lightweight add-on to equinox.Module with added parameter manipulation, mapping and inspection routines.

The following example defines a nested module, and then fixes one of the parameters, and adds a probability distribution to another.

Notice how parameters can be initialized with floats or default parameters - parax.Module automatically applies the as_param converter, and also deep-copies any mutatable parameter objects to avoid the Python "mutable default" trap.

import jax.numpy as jnp
import parax as prx

class Quadratic(prx.Module):
    a: prx.Parameter = 0.0
    b: prx.Parameter = 0.0
    c: prx.Parameter = prx.Fixed(0.0)

    def __call__(self, x: jnp.ndarray) -> jnp.ndarray:
        return self.a * (x ** 2) + self.b * x + self.c

class NthRootOfModule(prx.Module):
    # Create a module that takes the n'th root of another module
    nth_root: prx.Parameter

    # Define the module we are wrapping and mark its parameters
    # as "transparent" in the hierachy (so we dont have an "of_" prefix)
    of: prx.Module = prx.field(transparent=True)

    def __call__(self, x: jnp.ndarray) -> jnp.ndarray:
        return jnp.power(self.of(x), 1.0/self.n)

# Create a module and print its parameter names
cube_root_quadratic = NthRootOfModule(3, Quadratic(2.0, -3.0))
print(cube_root_quadratic.param_names()) # prints ['nth_root', 'a', 'b']

# Updated 'a' to have a normal distribution and 'nth_root' to be fixed to 2.5.
# prx.Normal is a factory that creates a `distreqx.distribution.Normal`
statistical_2p5_root_quadratic = cube_root_quadratic.with_params(
    nth_root=prx.Fixed(2.5),
    a=prx.Normal(3.0, 1.0),
)

Example 3: Optimizing a model

In this example, we define a simple quadratic model ($y = ax^2 + bx + c$) and derive directly from eqx.Module. We fix the y-intercept, leave the other coefficients free, partition using prx.partition, and use optimistix to fit the model to some noisy data.

import jax
import jax.numpy as jnp
import equinox as eqx
import optimistix as optx

import parax as prx
from parax.parameters import Free, Fixed

# 1. Define the Parametric Model
class Quadratic(eqx.Module):
    """A generic quadratic curve: y = a*x^2 + b*x + c"""
    
    a: prx.Parameter
    b: prx.Parameter
    c: prx.Parameter

    def __call__(self, x: jnp.ndarray) -> jnp.ndarray:
        return self.a * (x ** 2) + self.b * x + self.c
    
# We pass in free/fixed parameters without metadata using factories.
# Note that `parax.Module` would allow us to simply pass `a=1.5` for free parameters.
model = Quadratic(a=Free(1.5), b=Free(0.5), c=Fixed(10.0))

# 2. Generate some dummy "ground truth" data with noise
x_true = jnp.linspace(-5.0, 5.0, 100)
y_true = 3.0 * (x_true ** 2) - 2.0 * x_true + 10.0 # True a=3.0, b=-2.0
y_true = y_true + jax.random.normal(jax.random.key(0), x_true.shape)

# 3. Partition the model into free and fixed parameters
params, static = prx.partition(model)

# 4. Define the loss Function
def loss_fn(params, args=None):
    model = eqx.combine(params, static)
    y_pred = model(x_true)
    return jnp.mean((y_pred - y_true)**2)

# 5. Run the BFGS optimizer
solver = optx.LBFGS(rtol=1e-6, atol=1e-6)
solution = optx.minimise(
    fn=loss_fn,
    y0=params,
    solver=solver,
    args=(x_true, y_true, static),
)

# 6. Recombine to get the final fitted model
fitted_model = eqx.combine(solution.value, static)

print(f"Fitted 'a': {jnp.array(fitted_model.a):.8f} (Expected ~3.0)")
print(f"Fitted 'b': {jnp.array(fitted_model.b):.8f} (Expected ~-2.0)")
print(f"Fixed 'c':  {jnp.array(fitted_model.c):.8f} (Remained 10.0)")
print(f'Final loss: {loss_fn(fitted_model)}')
print(solution.result)