LambdaML

Gradient-free machine learning. Give it any function; it learns the parameters.

LambdaML lets you use any numpy-compatible function as your model and automatically fits its parameters using numerical (finite-difference) differentiation — no hand-derived gradients required. The "lambda" really can be anything: logistic regression, a neural network with custom activations, a physics equation, a learnable signal transform, or something entirely your own.

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Quick-start

pip install lambdaml           # core (numpy only)
pip install lambdaml[speed]    # + tqdm progress bars + joblib parallelism (recommended)
pip install lambdaml[onnx]     # + ONNX export/import (v1.2.0)
pip install lambdaml[examples] # + scipy, pandas, matplotlib for the notebook
pip install lambdaml[all]      # everything

import numpy as np
from lambdaml import LambdaClassifierModel, Optimizer, DiffMethod, LRSchedule

# 1. Write your model — anything numpy-compatible works
def my_model(x, p):
    return (np.tanh(p['w'].dot(x) + p['b']) + 1) / 2

# 2. Initial parameters (scalars or numpy arrays)
p = {'w': np.zeros(2), 'b': 0.0}

# 3. Create and fit
model = LambdaClassifierModel(
    f=my_model,
    p=p,
    diff_method=DiffMethod.COMPLEX_STEP,   # recommended
    l2_factor=0.001,
    optimizer=Optimizer.ADAM,
    lr_schedule=LRSchedule.cosine_annealing(T_max=100),
    n_jobs=-1,   # parallel gradient computation across parameters (requires joblib)
)
model.fit(X_train, Y_train, n_iter=100, lr=0.01,
          early_stopping=True, patience=10)

print(model.score(X_test, Y_test))       # accuracy
print(model.predict_proba(X_test))       # probabilities

For regression, swap in LambdaRegressorModel with loss='mse', 'mae', 'huber', or 'pseudo_huber'.

See the examples/ folder for runnable scripts and LambdaML_Showcase.ipynb for an interactive walkthrough with charts.

To export a fitted model to ONNX:

# Requires vectorized=True on the model
proto = model.to_onnx('model.onnx', input_shape=(2,))

# Inference via onnxruntime (no Python/LambdaML needed at runtime)
from lambdaml import predict_onnx
probs = predict_onnx('model.onnx', X_test)

# Or save/load weights as .npz (always works, any model)
model.save_params('weights.npz')
model2.load_params('weights.npz')

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What's new in v1.2.0

ONNX export and model persistence — two new ways to save and deploy fitted models:

to_onnx() traces a fitted vectorized model to a standard ONNX graph with parameters baked in as initializers. The exported file runs anywhere onnxruntime is installed — no Python model code, no LambdaML, no numpy required at inference time.

save_params() / load_params() always works — saves the parameter dict to a compressed .npz file regardless of whether the model is vectorized. Requires re-providing the function f at load time (keeps the function in source, weights in the file).

pip install lambdaml[onnx]   # onnx + onnxruntime

# Auto-trace export (vectorized models only)
def logistic_v(X, p):
    return 1 / (1 + np.exp(-(X @ p['w'] + p['b'])))

model = LambdaClassifierModel(f=logistic_v, p={'w': np.zeros(2), 'b': 0.0},
                              vectorized=True)
model.fit(X_train, Y_train, n_iter=200, lr=0.01)
model.to_onnx('logistic.onnx', input_shape=(2,))

# Runtime inference (no LambdaML needed)
import onnxruntime as rt
sess = rt.InferenceSession('logistic.onnx')
probs = sess.run(None, {'X': X_test.astype('float32')})[0]

# Convenience wrapper
from lambdaml import predict_onnx, from_onnx
probs  = predict_onnx('logistic.onnx', X_test)           # raw probabilities
labels = predict_onnx('logistic.onnx', X_test, threshold=0.5)  # binary labels

# Parameters-only (always works)
model.save_params('weights.npz')
model2 = LambdaClassifierModel(f=logistic_v, p={'w': np.zeros(2), 'b': 0.0},
                               vectorized=True)
model2.load_params('weights.npz')

The tracer supports standard numpy ops: @ / dot, +, -, *, /, **, exp, log, sqrt, tanh, sin, cos, abs, clip, log1p, integer indexing (X[:, 0]), and slicing. Physics models using arctan2 or complex control flow should use save_params() instead.

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What's new in v1.1.0

Modify-and-restore gradient computation — the single biggest internal speedup. Previously, every single parameter perturbation during gradient computation made a full deep copy of the entire parameter dictionary. Now the library perturbs one value in-place, evaluates, and restores — zero unnecessary copies. 2–5× speedup on the gradient step alone.

Cached skip set & optimizer dispatch — the regularization skip set (which parameters to exclude) and the optimizer update function are both resolved once at model construction time. Previously they were recomputed or re-dispatched via string comparison on every step of every epoch.

Single-pass regularization — when both L1 and L2 are active, the penalty is now computed in a single loop over parameters (was two separate passes before).

eval_every=10 by default — each loss evaluation is a full forward pass over your entire dataset. Defaulting to every-10-epochs avoids 90% of those evaluations out of the box, with no effect on gradient quality.

n_jobs — parallel gradient computation — set n_jobs=-1 to use all CPU cores. Each parameter's gradient is independent of every other, so this is embarrassingly parallel. Requires joblib (pip install lambdaml[speed]).

# All v1.1.0 speed features together:
def f(X, p):                         # vectorized: accepts full matrix
    return X @ p['w'] + p['b']

model = LambdaRegressorModel(
    f=f,
    p={'w': np.zeros(10), 'b': 0.0},
    vectorized=True,   # eliminates Python sample loop
    n_jobs=-1,         # parallelise gradient across parameters
)
model.fit(X, Y, n_iter=200, lr=0.01, eval_every=50)

v1.0.3 additions (still available): progress bars (progress_bar=True on fit/predict), eval_every parameter, vectorized=True mode.

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What is finite-difference differentiation?

The term you're looking for is finite-difference approximation (sometimes called numerical differentiation). Rather than deriving f′(θ) analytically, we estimate it by evaluating the function at nearby points:

f'(θ) ≈ [f(θ+h) - f(θ-h)] / (2h)     ← Central difference, O(h²)

LambdaML supports six methods with different accuracy/cost trade-offs:

Method	Order	f-evals/param	Notes
Forward	O(h)	1	Fast, low accuracy
Backward	O(h)	1	Fast, low accuracy
Central	O(h²)	2	Default — good balance
Five-Point	O(h⁴)	4	High accuracy, smooth f
Complex-Step	O(h²)	1 (complex)	Recommended — no cancellation error
Richardson	O(h⁴)	4	High accuracy, no complex inputs needed

Is it tractable? Yes, for models up to ~10k parameters. Each gradient step costs O(n_params) forward passes instead of O(1) for analytic backprop. For small-to-medium models on a CPU+numpy backend this is entirely practical.

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Speed tips

The main cost per epoch is n_params × diff_evals × n_samples calls to f. To reduce it:

Technique	How	Typical gain
Modify-and-restore (v1.1.0, automatic)	No dict copies during gradient computation	2–5×
n_jobs=-1 (v1.1.0)	Parallel gradient across parameters via joblib	~N_cores×
vectorized=True (v1.0.3)	Write f(X, p) to accept the full matrix	2–10×
eval_every=50	Skip loss re-evaluation on most epochs	~1.5–2×
batch_size=N	Mini-batch gradient steps	Scales with batch ratio
DiffMethod.FORWARD	1 f-eval/param instead of 2	~1.5× (noisier grads)

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The lambda can be any function

Six completely different model functions, one .fit() call — logistic regression, tanh, sine activation (non-standard), Gaussian RBF, softplus, and a physics-inspired decay+oscillation model σ(a·exp(−λ|x₀|)·cos(ω·x₁+φ)). See LambdaML_Showcase.ipynb for visualisations and benchmarks.

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API reference

LambdaClassifierModel(f, p, **kwargs)

Parameter	Default	Description
f	—	Model: f(x, p) → float ∈ (0,1) — or f(X, p) → array when vectorized=True
p	—	Parameter dict (scalars or numpy arrays)
diff_method	DiffMethod.CENTRAL	Finite-difference method
diff_h	None	Custom step size (None = optimal default per method)
l1_factor	0.0	L1 regularization strength
l2_factor	0.01	L2 regularization strength
regularize_bias	False	Whether to regularize b* params
optimizer	Optimizer.ADAM	sgd, momentum, rmsprop, adam
lr_schedule	None (constant)	Learning rate schedule callable
vectorized	False	If True, f receives the full X matrix — faster
n_jobs	1	Parallel gradient workers; -1 = all cores (requires joblib)

.fit(X, Y, ...)

Parameter	Default	Description
n_iter	100	Max gradient steps
lr	0.01	Initial learning rate
batch_size	None	Mini-batch size; None = full batch
early_stopping	False	Stop if loss stalls for patience steps
patience	10	Early stopping patience
tol	1e-6	Minimum improvement threshold
verbose	False	Print loss every eval_every iterations
validation_data	None	(X_val, Y_val) tuple
progress_bar	True	Show tqdm epoch bar (requires tqdm)
eval_every	10	Evaluate loss every N epochs — each eval is a full forward pass

Other methods: .predict(X, progress_bar=False) · .predict_proba(X, progress_bar=False) · .score(X, Y) · .compute_loss(X, Y) · .get_params() · .loss_history

ONNX / persistence methods (on both model classes):

Method	Description
.to_onnx(path, *, input_shape, ...)	Export to ONNX. Requires vectorized=True. Returns onnx.ModelProto.
.save_params(path, **meta)	Save weights to .npz. Always works.
.load_params(path)	Load weights from .npz into this model. Returns self.

Module-level helpers (importable from lambdaml):

Function	Description
predict_onnx(path_or_session, X, threshold=None)	Run ONNX inference. Returns (n,) array.
from_onnx(path)	Load an ONNX file and return an onnxruntime.InferenceSession.
save_params(model, path, **meta)	Functional form of .save_params().
load_params(model, path)	Functional form of .load_params().
OnnxTraceError	Exception raised when tracing fails.

LambdaRegressorModel(f, p, loss='mse', **kwargs)

Parameter	Default	Description
loss	'mse'	'mse', 'mae', 'huber', 'pseudo_huber'
huber_delta	1.0	Threshold for Huber / pseudo-Huber

Methods: .fit(...) · .predict(X, progress_bar=False) · .score(X, Y) (R²)

DiffMethod · Optimizer · LRSchedule

# Derivative methods
DiffMethod.FORWARD | BACKWARD | CENTRAL | FIVE_POINT | COMPLEX_STEP | RICHARDSON

# Optimizers
Optimizer.SGD | MOMENTUM | RMSPROP | ADAM

# LR schedules
LRSchedule.constant()
LRSchedule.step_decay(drop=0.5, epochs_drop=10)
LRSchedule.exponential_decay(k=0.01)
LRSchedule.cosine_annealing(T_max=100)
LRSchedule.warmup_cosine(warmup=10, T_max=100)

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Performance improvements over original

Change	Original	v1.1.0
Gradient dict copies	Full dict copy per parameter element	Modify-and-restore — zero copies
Regularization passes	1–2 passes per epoch (separate L1/L2 loops)	Single pass regardless of combination
Optimizer dispatch	String compare per parameter per step	Function resolved once at init
Skip set computation	Rebuilt from scratch every epoch	Cached frozenset at init
Default loss eval frequency	Every epoch (eval_every=1)	Every 10 epochs (eval_every=10)
Gradient parallelism	Sequential across all parameters	Optional: n_jobs=-1 uses all CPU cores

ONNX support (v1.2.0)

Feature	Details
Export method	model.to_onnx(path, input_shape=(n_features,))
Requirement	vectorized=True on the model; function uses standard numpy ops
Serialization	Parameters baked in as ONNX float32 initializers
ONNX opset	17 (default); IR version 8 for broad onnxruntime compatibility
Inference	onnxruntime — no LambdaML, no numpy, no Python model code needed
Fallback	save_params() / load_params() — always works, any model type
Install	pip install lambdaml[onnx]
Supported numpy ops	@, dot, +, -, *, /, **, exp, log, sqrt, tanh, sin, cos, abs, clip, log1p, integer indexing X[:, i], slicing

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Bug fixes from the original library

Bug	Original	Fixed
Epsilon	float16.eps ≈ 0.001 — catastrophically large	Float64-optimal per method (~6e-6 for central)
L1 regularization	Summed raw θ — negative weights reduced penalty	Summed |θ| using smooth complex-safe approximation
Closure-in-loop	Array gradient loop captured last index for all closures	Fixed with factory functions
L1/L2 complex-step safety	float() cast stripped imaginary part	Uses v*v and sqrt(v*v+eps) to preserve imaginary parts
No test split	Accuracy reported on training data	Train/test split in all examples

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Is LambdaML useful for Kaggling?

As a primary model for large nets — rarely. As a prototyping and ensembling tool — genuinely yes.

The core insight: LambdaML decouples your model definition from gradient computation. Anywhere you want a custom functional form but don't want to derive its gradients by hand, LambdaML fills that gap.

Concrete use cases: fitting domain equations with unknown parameters (physics-based pricing, pharmacokinetics, decay curves); directly optimising non-differentiable competition metrics (NDCG, F-beta, Cohen's kappa) as the loss function; building exotic meta-learners in stacking ensembles; small-data + custom hypothesis problems where sklearn doesn't have your model form.

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Project structure

LambdaML/
├── lambdaml/                # Installable package (pip install lambdaml)
│   ├── __init__.py
│   ├── lambda_model.py      # LambdaClassifierModel, LambdaRegressorModel, Optimizer
│   ├── lambda_utils.py      # NumericalDiff, GradientComputer, Regularization, LossFunctions, LRSchedule
│   └── lambda_onnx.py       # to_onnx, from_onnx, save_params, load_params, predict_onnx (v1.2.0)
├── pyproject.toml           # Package metadata
├── LambdaML_Showcase.ipynb  # Interactive notebook with all charts (section 12: ONNX)
├── examples/
│   ├── example_tanh_regression.py
│   ├── example_neural_network.py
│   ├── example_diff_methods.py
│   ├── example_regressor.py
│   └── example_onnx.py      # ONNX export/import demo and benchmark (v1.2.0)
├── data/
│   └── circles.csv
└── legacy/                  # Original library files (pre-rewrite)

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License

See LICENSE.