nlls-gram 0.4.1

Gram/dual-form Levenberg-Marquardt nonlinear least-squares solvers for JAX

nlls_gram

Levenberg-Marquardt nonlinear least-squares solvers for JAX, with dense Gram/dual solves and optional matrix-free iterative solves.

GramLevenbergMarquardt minimizes ||r(params)||^2 for a user-supplied residual_fn(params, batch), where params is any JAX pytree (a flat array, a dict, nnx.state(model, nnx.Param), ...). It follows an init/update protocol: update(params, state, batch) returns the new params pytree (same structure), the next state, and an LMInfo. The default dense solver factors the small n x n Gram (dual) system, which is useful when there are many more parameters than residual rows. Optional iterative solvers provide matrix-free CG in Gram or normal form and LSMR on the damped least-squares formulation.

The solver interface is general JAX — it knows nothing about flax/nnx/optax — and the package depends on jax plus lineax for LSMR. Dtypes flow from your params/residual, and the damping state follows the residual dtype; JAX decides float32 vs float64 via jax_enable_x64. init_damping must be positive; use a small positive value for near Gauss-Newton behavior.

Install

uv add nlls-gram

For local development on an NVIDIA CUDA 13 machine, use the optional gpu dependency group:

uv sync --group gpu

That group is for this repository's development and GPU tests; it is not a published nlls-gram[gpu] extra. Users who want to run the optimizer on a GPU should install the JAX accelerator build that matches their hardware alongside nlls-gram, for example:

uv add nlls-gram "jax[cuda13]"

See the JAX installation guide for the current CUDA, ROCm, TPU, and CPU installation choices.

Minimal example

Fit y = a * exp(b * x) to noise-free data generated from (a, b) = (2, -1), using a plain dict pytree of parameters. With JAX's default configuration, this runs in float32:

import jax
import jax.numpy as jnp

from nlls_gram import GramLevenbergMarquardt


# residual_fn(params, batch) -> 1-D residual array; the solver minimizes its SSQ.
def residual_fn(params, batch):
    x, y = batch
    return params["a"] * jnp.exp(params["b"] * x) - y


x = jnp.linspace(0.0, 2.0, 20)
y = 2.0 * jnp.exp(-1.0 * x)

params = {"a": 1.0, "b": 0.0}
solver = GramLevenbergMarquardt(residual_fn, init_damping=1e-2)
lm_state = solver.init()


# The solver does not jit internally; wrap the train step yourself.
@jax.jit
def train_step(params, lm_state, batch):
    return solver.update(params, lm_state, batch)


for _ in range(50):
    params, lm_state, info = train_step(params, lm_state, (x, y))

print(params["a"], params["b"])  # ~2.0, ~-1.0
print(params["a"].dtype, info.loss.dtype)  # float32 float32

Float64 example

Enable x64 before creating arrays, then initialize the data and parameters as float64:

import jax

jax.config.update("jax_enable_x64", True)

import jax.numpy as jnp

from nlls_gram import GramLevenbergMarquardt

dtype = jnp.float64


def residual_fn(params, batch):
    x, y = batch
    return params["a"] * jnp.exp(params["b"] * x) - y


x = jnp.linspace(0.0, 2.0, 20, dtype=dtype)
y = 2.0 * jnp.exp(-1.0 * x)

params = {
    "a": jnp.asarray(1.0, dtype=dtype),
    "b": jnp.asarray(0.0, dtype=dtype),
}

solver = GramLevenbergMarquardt(residual_fn, init_damping=1e-2)
lm_state = solver.init()

for _ in range(50):
    params, lm_state, info = solver.update(params, lm_state, (x, y))

print(params["a"], params["b"])  # ~2.0, ~-1.0
print(params["a"].dtype, info.loss.dtype, info.damping.dtype)

Fletcher regularization

The default regularization="identity" uses the classic LM damping matrix lambda * I. If parameters are badly scaled, regularization="fletcher" can help by damping each parameter direction in proportion to diag(J.T @ J). The diagonal is clipped before use, with defaults fletcher_min_diagonal=1e-6 and fletcher_max_diagonal=1e6, so nearly unused or extremely sensitive parameter directions do not dominate the Gram solve.

import jax.numpy as jnp

from nlls_gram import GramLevenbergMarquardt

x = jnp.linspace(0.0, 2.0, 50)
y = 2.0 * jnp.exp(-1.0 * x)
parameter_scale = 1e-3


def residual_fn(params, batch):
    x, y = batch
    b = parameter_scale * params["b_scaled"]
    return params["a"] * jnp.exp(b * x) - y


def iterations_to_threshold(regularization):
    params = {"a": 1.0, "b_scaled": 0.0}
    solver = GramLevenbergMarquardt(
        residual_fn,
        init_damping=1e-2,
        regularization=regularization,
        fletcher_min_diagonal=1e-6,
        fletcher_max_diagonal=1e6,
    )
    lm_state = solver.init()
    for iteration in range(1, 51):
        params, lm_state, info = solver.update(params, lm_state, (x, y))
        if float(info.loss) < 1e-8:
            return iteration
    return None


print(iterations_to_threshold("identity"))  # ~16
print(iterations_to_threshold("fletcher"))  # ~4

Iterative solves

The default linear_solver="cholesky" materializes the dense Gram matrix and uses a Cholesky factorization. For larger identity-regularized problems, use an iterative solver with JAX JVP/VJP linearization instead of materializing J:

Iterative solvers default to a small fixed iteration budget: iterative_tol=0.0, iterative_atol=0.0, and iterative_maxiter=8. This avoids extra tolerance-driven convergence work and is intended for low-rank local linear solves. Set a positive iterative_tol or iterative_atol if you want early convergence checks instead.

For fixed-budget CG in residual space, use:

solver = GramLevenbergMarquardt(
    residual_fn,
    init_damping=1e-2,
    linear_solver="cg",
    formulation="gram",
    iterative_tol=0.0,
    iterative_atol=0.0,
    iterative_maxiter=8,
)

For fixed-budget CG in parameter space, use:

solver = GramLevenbergMarquardt(
    residual_fn,
    init_damping=1e-2,
    linear_solver="cg",
    formulation="normal",
    iterative_tol=0.0,
    iterative_atol=0.0,
    iterative_maxiter=8,
)

CG currently supports only regularization="identity". formulation="gram" solves in residual space, so the Krylov vectors have length equal to the number of residuals. formulation="normal" solves in parameter space, so the Krylov vectors have length equal to the number of parameters. Both use matrix-free JVPs for J @ v and VJPs/linear transposes for J.T @ u; choose the formulation based on which vector space is smaller and better conditioned.

linear_solver="lsmr" uses Lineax LSMR on the damped least-squares problem directly:

min_s ||J s + r||^2 + lambda ||s||^2

For fixed-budget LSMR, use:

solver = GramLevenbergMarquardt(
    residual_fn,
    init_damping=1e-2,
    linear_solver="lsmr",
    iterative_tol=0.0,
    iterative_atol=0.0,
    iterative_maxiter=8,
    lsmr_conlim=float("inf"),
)

It uses the augmented operator [J; sqrt(lambda) I], so matrix-vector products call JAX JVPs for J @ s and transposed products call VJPs/linear transposes for J.T @ u. LSMR does not use the Gram or normal formulation. Its default lsmr_conlim=float("inf") prevents condition-limit early termination; Lineax still computes LSMR's internal norm estimates each iteration. Iterative solvers can reduce memory and factorization cost on larger dense GPU problems, but each iteration performs matrix-free Jacobian-vector and transpose-vector products, so the Cholesky default remains better for small residual dimensions.

Geodesic acceleration

Geodesic acceleration is off by default. When enabled, the solver uses analytic JAX forward-mode JVPs to build an accelerated candidate; it does not use finite differences.

solver = GramLevenbergMarquardt(
    residual_fn,
    init_damping=1e-2,
    geodesic_acceleration=True,
)

The accelerated candidate is used only when its acceleration ratio, 2 * ||a|| / ||v||, is at or below a positive geodesic_acceptance_ratio and its loss is no worse than the plain LM velocity candidate. Otherwise the update automatically falls back to the velocity step. Use LMInfo.used_geodesic, LMInfo.acceleration_ratio, LMInfo.loss_old, LMInfo.loss_candidate, and LMInfo.damping_factor to tune damping and geodesic behavior.

params can be any pytree. With Flax NNX, pass nnx.state(model, nnx.Param) as params and write residual_fn(state, batch) using nnx.merge; the solver itself stays NNX-agnostic.

Filtering / freezing parameters

update optimizes exactly the params pytree you pass. For Flax NNX transfer learning, construct or load the full module first, choose the trainable leaves with an NNX filter, and pass only that trainable state to the solver. This mirrors the wrt argument used by nnx.Optimizer: wrt means "differentiate and update these leaves", while ... captures the already-initialized frozen remainder. Install Flax in your project to run this example.

import jax
import jax.numpy as jnp
from flax import nnx

from nlls_gram import GramLevenbergMarquardt


class ExpModel(nnx.Module):
    def __init__(self):
        self.a = nnx.Param(jnp.asarray(1.0))
        self.b = nnx.Param(jnp.asarray(-1.0))

    def __call__(self, x):
        return self.a[...] * jnp.exp(self.b[...] * x)


x = jnp.linspace(0.0, 2.0, 20)
y = 2.0 * jnp.exp(-1.0 * x)

model = ExpModel()
wrt = nnx.PathContains("a")  # train "a"; keep all other initialized state fixed
graphdef, trainable, frozen = nnx.split(model, wrt, ...)


def residual_fn(trainable, batch):
    x, y = batch
    model = nnx.merge(graphdef, trainable, frozen)
    return model(x) - y


solver = GramLevenbergMarquardt(residual_fn, init_damping=1e-2)
lm_state = solver.init()
for _ in range(50):
    trainable, lm_state, info = solver.update(trainable, lm_state, (x, y))

model = nnx.merge(graphdef, trainable, frozen)
print(model.a[...], model.b[...])  # ~2.0, -1.0

For built-in NNX layers, set both computation and parameter initialization dtypes when you want an all-float64 model:

layer = nnx.Linear(
    1,
    1,
    dtype=jnp.float64,
    param_dtype=jnp.float64,
    rngs=nnx.Rngs(0),
)

Benchmarks

Optional pytest-benchmark checks live outside the normal test suite and do not run in CI by default:

uv run --group benchmark pytest benchmarks --benchmark-only

For a larger RBF-style interpolation profile with CPU/GPU, Cholesky/CG/LSMR, Gram/normal CG, and geodesic on/off variants:

uv run --group benchmark --group gpu pytest \
  benchmarks/test_large_interpolation_benchmark.py --benchmark-only

For a small classic geodesic-acceleration convergence benchmark based on the GSL modified Rosenbrock example:

uv run --group benchmark pytest \
  benchmarks/test_classic_geodesic_benchmark.py --benchmark-only

On machines with a CUDA-enabled JAX install, the optional GPU test checks that a jitted geodesic update runs on a GPU device:

uv run --group gpu pytest tests/test_gpu.py

Documentation

Full docs: https://highdimensionaleconlab.github.io/nlls_gram/

License

MIT