# FwFMRegressor

Field-weighted Factorization Machine for regression.

The model equation is defined as:

\\\[\\hat{y}(x) = w\_{0} + \\sum\_{j=1}^{p} w\_{j} x\_{j} + \\sum\_{j=1}^{p} \\sum\_{j'=j+1}^{p} r\_{f\_j, f\_{j'}} \\langle \\mathbf{v}\_j, \\mathbf{v}\_{j'} \\rangle x\_{j} x\_{j'}\\\]

Where \\(f\_j\\) and \\(f\_{j'}\\) are \\(j\\) and \\(j'\\) fields, respectively, and \\(\\mathbf{v}\_j\\) and \\(\\mathbf{v}\_{j'}\\) are \\(j\\) and \\(j'\\) latent vectors, respectively.

For more efficiency, this model automatically one-hot encodes strings features considering them as categorical variables. Field names are inferred from feature names by taking everything before the first underscore: `feature_name.split('_')\[0\]`.

## Parameters

- **n\_factors**

  *Default* → `10`

  Dimensionality of the factorization or number of latent factors.
- **weight\_optimizer**

  *Type* → `optim.base.Optimizer | None`

  *Default* → `None`

  The sequential optimizer used for updating the feature weights. Note that the intercept is handled separately.
- **latent\_optimizer**

  *Type* → `optim.base.Optimizer | None`

  *Default* → `None`

  The sequential optimizer used for updating the latent factors.
- **int\_weight\_optimizer**

  *Type* → `optim.base.Optimizer | None`

  *Default* → `None`

  The sequential optimizer used for updating the field pairs interaction weights.
- **loss**

  *Type* → `optim.losses.RegressionLoss | None`

  *Default* → `None`

  The loss function to optimize for.
- **sample\_normalization**

  *Default* → `False`

  Whether to divide each element of `x` by `x`'s L2-norm.
- **l1\_weight**

  *Default* → `0.0`

  Amount of L1 regularization used to push weights towards 0.
- **l2\_weight**

  *Default* → `0.0`

  Amount of L2 regularization used to push weights towards 0.
- **l1\_latent**

  *Default* → `0.0`

  Amount of L1 regularization used to push latent weights towards 0.
- **l2\_latent**

  *Default* → `0.0`

  Amount of L2 regularization used to push latent weights towards 0.
- **intercept**

  *Default* → `0.0`

  Initial intercept value.
- **intercept\_lr**

  *Type* → `optim.base.Scheduler | float`

  *Default* → `0.01`

  Learning rate scheduler used for updating the intercept. An instance of `optim.schedulers.Constant` is used if a `float` is passed. No intercept will be used if this is set to 0.
- **weight\_initializer**

  *Type* → `optim.initializers.Initializer | None`

  *Default* → `None`

  Weights initialization scheme. Defaults to `optim.initializers.Zeros()`.
- **latent\_initializer**

  *Type* → `optim.initializers.Initializer | None`

  *Default* → `None`

  Latent factors initialization scheme. Defaults to `optim.initializers.Normal(mu=.0, sigma=.1, random_state=self.random_state)`.
- **clip\_gradient**

  *Default* → `1000000000000.0`

  Clips the absolute value of each gradient value.
- **seed**

  *Type* → `int | None`

  *Default* → `None`

  Randomization seed used for reproducibility.

## Attributes

- **weights**

  The current weights assigned to the features.
- **latents**

  The current latent weights assigned to the features.
- **interaction\_weights**

  The current interaction strengths of field pairs.

## Examples

```python
from river import facto

dataset = (
    ({'user': 'Alice', 'item': 'Superman'}, 8),
    ({'user': 'Alice', 'item': 'Terminator'}, 9),
    ({'user': 'Alice', 'item': 'Star Wars'}, 8),
    ({'user': 'Alice', 'item': 'Notting Hill'}, 2),
    ({'user': 'Alice', 'item': 'Harry Potter '}, 5),
    ({'user': 'Bob', 'item': 'Superman'}, 8),
    ({'user': 'Bob', 'item': 'Terminator'}, 9),
    ({'user': 'Bob', 'item': 'Star Wars'}, 8),
    ({'user': 'Bob', 'item': 'Notting Hill'}, 2)
)

model = facto.FwFMRegressor(
    n_factors=10,
    intercept=5,
    seed=42,
)

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

model.predict_one({'Bob': 1, 'Harry Potter': 1})
```

```text
5.236501
```

```python
report = model.debug_one({'Bob': 1, 'Harry Potter': 1})

print(report)
```

```text
Name                                    Value      Weight     Contribution
                            Intercept    1.00000    5.23426        5.23426
Bob(Harry Potter) - Harry Potter(Bob)    1.00000    0.00224        0.00224
                         Harry Potter    1.00000    0.00000        0.00000
                                  Bob    1.00000    0.00000        0.00000
```

## Methods

> [!NOTE]
>
> **debug\_one**
>
> Debugs the output of the FM regressor.
>
> **Parameters**
>
> - **x** — `dict`
> - **decimals** — `int` — defaults to `5`
>
> **Returns**
>
> `str`: A table which explains the output.

> [!NOTE]
>
> **learn\_one**
>
> Fits to a set of features `x` and a real-valued target `y`.
>
> **Parameters**
>
> - **x** — `dict[base.typing.FeatureName, Any]`
> - **y** — `base.typing.RegTarget`
> - **w** — defaults to `1.0`

> [!NOTE]
>
> **predict\_one**
>
> Predict the output of features `x`.
>
> **Parameters**
>
> - **x**
>
> **Returns**
>
> The prediction.

## References

- [Junwei Pan, Jian Xu, Alfonso Lobos Ruiz, Wenliang Zhao, Shengjun Pan, Yu Sun, and Quan Lu, 2018, April. Field-weighted Factorization Machines for Click-Through Rate Prediction in Display Advertising. In Proceedings of the 2018 World Wide Web Conference on World Wide Web. International World Wide Web Conferences Steering Committee, (pp. 1349–1357).](<https://doi.org/10.1145/3178876.3186040>)