Atoms#

Atoms are pre-built mathematical functions that can be used in optimization objectives and constraints. They extend the base Atom class and provide efficient implementations of common optimization functions.

What are Atoms?#

Atoms represent mathematical functions with specific properties:

Smoothness: Whether the function is differentiable everywhere
Proxability: Whether the function has an efficient proximal operator

These properties determine which optimization algorithms can be used with the atom.

Available Atoms#

L1 Norm#

The L1Norm atom represents the L1 (Manhattan) norm: \(\|x\|_1 = \sum_i |x_i|\).

from rlaopt.atoms import L1Norm
from rlaopt.expression import Variable

x = Variable((10,), name='x')
l1 = L1Norm(x, scaling=0.1)  # 0.1 * ||x||_1

L2 Norm#

The L2Norm atom represents the L2 (Euclidean) norm: \(\|x\|_2 = \sqrt{\sum_i x_i^2}\).

from rlaopt.atoms import L2Norm
from rlaopt.expression import Variable

x = Variable((10,), name='x')
l2 = L2Norm(x, scaling=0.1)  # 0.1 * ||x||_2

Quadratic Form#

The QuadForm atom represents a quadratic form defined by a matrix \(Q\): \(x^\top Q x\).

import torch

from rlaopt.atoms import QuadForm
from rlaopt.expression import Variable

x = Variable((10,), name='x')
Q = torch.randn(10, 10)
Q = Q.T @ Q
qf = QuadForm(x, Q)  # x^T Q x

The quadratic form is smooth, but it is not proxable, since its proximal operator would require solving a linear system.

Elastic Net#

The ElasticNet atom combines L1 and L2 regularization: \(\lambda_1 \|x\|_1 + \lambda_2 \|x\|_2^2\).

from rlaopt.atoms import ElasticNet
from rlaopt.expression import Variable

x = Variable((10,), name='x')
elastic = ElasticNet(x, l1_scaling=0.1, l2_scaling=0.01)

Other Atoms#

rlaopt provides many other atoms:

SumSquares: Sum of squares (squared L2 norm)
Box: Box constraints
Halfspace: Halfspace constraints
Polyhedron: Polyhedral constraints
L1NormBall: L1-norm ball constraint
L2NormBall: L2-norm ball constraint
LinearRegression: Linear (least-squares) regression loss
LogisticRegression: Logistic regression loss
PoissonRegression: Poisson regression loss
HuberRegression: Huber regression loss
MultinomialRegression: Multinomial regression loss

See the Atoms Module section for a complete list.

Using Atoms in Objectives#

Atoms can be combined with expressions to build optimization objectives:

from rlaopt.expression import Variable
from rlaopt.atoms import L1Norm, SumSquares

x = Variable((10,), name='x')
A = Variable((5, 10), name='A')
b = Variable((5,), name='b')

# Build objective: ||Ax - b||^2 + lambda * ||x||_1
residual = A @ x - b
objective = SumSquares(residual) + L1Norm(x, scaling=0.1)

Atom Properties#

Each atom provides information about its properties:

atom = L1Norm(x)

# Check if smooth
print(atom.is_smooth())  # False

# Check if proxable
print(atom.is_proxable())  # True

These properties help determine which solvers can be used with the atom.

Atoms

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