Implementation:Online ml River Facto HOFM

Knowledge Sources	Online_ml_River
Domains	Online_Learning, Factorization_Machines, Polynomial_Regression
Last Updated	2026-02-08 16:00 GMT

Overview

Higher-Order Factorization Machines (HOFM) generalize standard FM to model interactions of arbitrary degree, not just pairwise interactions.

Description

HOFM extends Factorization Machines to capture higher-order feature interactions beyond pairs. The degree parameter controls the maximum order of interactions to model. For degree 3, the model captures linear terms, pairwise interactions, and three-way interactions. Each feature maintains separate latent vectors for each interaction order (2, 3, ..., degree). The model sums all interaction terms up to the specified degree. This allows HOFM to learn complex non-linear relationships while maintaining computational efficiency through factorization. Like standard FM, it automatically one-hot encodes string features and uses gradient descent for optimization.

Usage

Use HOFM when relationships between features involve complex multi-way interactions that cannot be captured by pairwise terms alone. This is useful in domains like computational biology, social network analysis, or recommendation systems where three-way or higher-order interactions are meaningful. Be aware that higher degrees increase model complexity and computational cost, so start with degree 3 and increase only if necessary.

Code Reference

Source Location

Repository: Online_ml_River
File: river/facto/hofm.py

Signature

class HOFMRegressor(
    degree=3,
    n_factors=10,
    weight_optimizer: optim.base.Optimizer | None = None,
    latent_optimizer: optim.base.Optimizer | None = None,
    loss: optim.losses.RegressionLoss | None = None,
    sample_normalization=False,
    l1_weight=0.0,
    l2_weight=0.0,
    l1_latent=0.0,
    l2_latent=0.0,
    intercept=0.0,
    intercept_lr: optim.base.Scheduler | float = 0.01,
    weight_initializer: optim.initializers.Initializer | None = None,
    latent_initializer: optim.initializers.Initializer | None = None,
    clip_gradient=1e12,
    seed: int | None = None,
)

class HOFMClassifier(
    degree=3,
    n_factors=10,
    weight_optimizer: optim.base.Optimizer | None = None,
    latent_optimizer: optim.base.Optimizer | None = None,
    loss: optim.losses.BinaryLoss | None = None,
    sample_normalization=False,
    l1_weight=0.0,
    l2_weight=0.0,
    l1_latent=0.0,
    l2_latent=0.0,
    intercept=0.0,
    intercept_lr: optim.base.Scheduler | float = 0.01,
    weight_initializer: optim.initializers.Initializer | None = None,
    latent_initializer: optim.initializers.Initializer | None = None,
    clip_gradient=1e12,
    seed: int | None = None,
)

Import

from river import facto

I/O Contract

Input
Parameter	Type	Description
x	dict	Feature dictionary with feature names and values
y	float/bool	Target value for regression or classification

Output
Method	Return Type	Description
predict_one(x)	float/bool	Predicted value or class label
predict_proba_one(x)	dict	Class probabilities (classifier only)
learn_one(x, y)	None	Incrementally updates the model

Usage Examples

from river import facto

dataset = (
    ({'user': 'Alice', 'item': 'Superman', 'time': .12}, 8),
    ({'user': 'Alice', 'item': 'Terminator', 'time': .13}, 9),
    ({'user': 'Alice', 'item': 'Star Wars', 'time': .14}, 8),
    ({'user': 'Alice', 'item': 'Notting Hill', 'time': .15}, 2),
    ({'user': 'Bob', 'item': 'Superman', 'time': .13}, 8),
)

model = facto.HOFMRegressor(
    degree=3,
    n_factors=10,
    intercept=5,
    seed=42,
)

for x, y in dataset:
    model.learn_one(x, y)

prediction = model.predict_one({'user': 'Bob', 'item': 'Harry Potter', 'time': .14})
print(f"Prediction: {prediction}")  # 5.311745

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