> For the complete documentation index, see [llms.txt](https://docs.zama.org/concrete-ml/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.zama.org/concrete-ml/0.6-1/built-in-models/linear.md).

# Linear Models

Concrete-ML provides several of the most popular linear models for `regression` and `classification` that can be found in [Scikit-learn](https://scikit-learn.org/stable/):

|                                                        Concrete-ML                                                        |                                                                         scikit-learn                                                                         |
| :-----------------------------------------------------------------------------------------------------------------------: | :----------------------------------------------------------------------------------------------------------------------------------------------------------: |
|   [LinearRegression](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.linear_model.md#class-linearregression)   |    [LinearRegression](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression)    |
| [LogisticRegression](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.linear_model.md#class-logisticregression) | [LogisticRegression](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression) |
|               [LinearSVC](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.svm.md#class-linearsvc)              |                        [LinearSVC](https://scikit-learn.org/stable/modules/generated/sklearn.svm.LinearSVC.html#sklearn.svm.LinearSVC)                       |
|               [LinearSVR](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.svm.md#class-linearsvr)              |                        [LinearSVR](https://scikit-learn.org/stable/modules/generated/sklearn.svm.LinearSVR.html#sklearn.svm.LinearSVR)                       |
|        [PoissonRegressor](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.glm.md#class-poissonregressor)       |    [PoissonRegressor](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.PoissonRegressor.html#sklearn.linear_model.PoissonRegressor)    |
|        [TweedieRegressor](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.glm.md#class-tweedieregressor)       |    [TweedieRegressor](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.TweedieRegressor.html#sklearn.linear_model.TweedieRegressor)    |
|          [GammaRegressor](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.glm.md#class-gammaregressor)         |       [GammaRegressor](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.GammaRegressor.html#sklearn.linear_model.GammaRegressor)       |
|              [Lasso](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.linear_model.md#class-lasso)              |                     [Lasso](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.Lasso.html#sklearn.linear_model.Lasso)                    |
|              [Ridge](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.linear_model.md#class-ridge)              |                     [Ridge](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge)                    |
|         [ElasticNet](/concrete-ml/0.6-1/developer-guide/api/concrete.ml.sklearn.linear_model.md#class-elasticnet)         |             [ElasticNet](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.ElasticNet.html#sklearn.linear_model.ElasticNet)             |

Using these models in FHE is extremely similar to what can be done with scikit-learn's [API](https://scikit-learn.org/stable/modules/classes.html#module-sklearn.linear_model), making it easy for data scientists who are used to this framework to get started with Concrete-ML.

Models are also compatible with some of scikit-learn's main workflows, such as `Pipeline()` and `GridSearch()`.

## Quantization parameters

The `n_bits` parameter controls the bit-width of the inputs and weights of the linear models. When non-linear mapping is applied by the model, such as *exp* or *sigmoid*, currently Concrete-ML applies it on the client-side, on clear-text values that are the decrypted output of the linear part of the model. Thus, Linear Models do not use table lookups, and can, therefore, use high precision integers for weight and inputs. The `n_bits` parameter can be set to `8` or more bits for models with up to 300 input dimensions. When the input has more dimensions, `n_bits` must be reduced to `6-7`. Accuracy and R2 scores are preserved down to `n_bits=6`, compared to the non-quantized float models from scikit-learn.

## Example

Here is an example below of how to use a LogisticRegression model in FHE on a simple data set for classification. A more complete example can be found in the [LogisticRegression notebook](/concrete-ml/0.6-1/built-in-models/ml_examples.md).

```python
import numpy
from tqdm import tqdm
from sklearn.datasets import make_classification
from sklearn.model_selection import train_test_split

from concrete.ml.sklearn import LogisticRegression

# Create the data for classification:
X, y = make_classification(
    n_features=30,
    n_redundant=0,
    n_informative=2,
    random_state=2,
    n_clusters_per_class=1,
    n_samples=250,
)

# Retrieve train and test sets:
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.4, random_state=42)

# Instantiate the model:
model = LogisticRegression(n_bits=8)

# Fit the model:
model.fit(X_train, y_train)

# Evaluate the model on the test set in clear:
y_pred_clear = model.predict(X_test)

# Compile the model:
model.compile(X_train)

# Perform the inference in FHE:
y_pred_fhe = model.predict(X_test, execute_in_fhe=True)

# Assert that FHE predictions are the same as the clear predictions:
print(
    f"{(y_pred_fhe == y_pred_clear).sum()} examples over {len(y_pred_fhe)} "
    "have a FHE inference equal to the clear inference."
)

# Output:
#  100 examples over 100 have a FHE inference equal to the clear inference
```

We can then plot the decision boundary of the classifier and compare those results with a scikit-learn model executed in clear. The complete code can be found in the [LogisticRegression notebook](/concrete-ml/0.6-1/built-in-models/ml_examples.md).

![Sklearn model decision boundaries](https://3176746334-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F75MldzZY6YYabEQfXDIR%2Fuploads%2Fgit-blob-ab8b41a760c7f9fd84dceb810e83700bfa8d8d4f%2Flogistic_regression_clear.png?alt=media) ![FHE model decision boundarires](https://3176746334-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F75MldzZY6YYabEQfXDIR%2Fuploads%2Fgit-blob-9846bd126365d512379fb7bf39eacccfb52d84c2%2Flogistic_regression_fhe.png?alt=media)

The overall accuracy scores are identical (93%) between the scikit-learn model (executed in the clear) and the Concrete-ML one (executed in FHE). In fact, quantization has little impact on the decision boundaries, as linear models are able to consider large precision numbers when quantizing inputs and weights in Concrete-ML. Additionally, as the linear models do not use PBS, the FHE computations are always exact, meaning the FHE predictions are always identical to the quantized clear ones.


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