Created lasso_regression.py

machine_learning/lasso_regression.py

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+"""
+Lasso regression is a type of linear regression that adds a regularization term to the
+ordinary least squares (OLS) objective function. This regularization term is the
+L1 norm of the coefficients, which encourages sparsity in the model parameters. The
+objective function for Lasso regression is given by:
+
+minimize ||y - Xβ||² + λ||β||₁
+
+where:
+- y is the response vector,
+- X is the design matrix,
+- β is the vector of coefficients,
+- λ (lambda) is the regularization parameter controlling the strength of the penalty.
+
+Lasso regression can be solved using coordinate descent or other optimization techniques.
+
+References:
+    - https://en.wikipedia.org/wiki/Lasso_(statistics)
+    - https://en.wikipedia.org/wiki/Regularization_(mathematics)
+"""
+
+import numpy as np
+
+
+class LassoRegression:
+    __slots__ = "alpha", "params", "tol", "max_iter"
+
+    def __init__(
+        self, alpha: float = 1.0, tol: float = 1e-4, max_iter: int = 1000
+    ) -> None:
+        """
+        Initializes the Lasso regression model.
+
+        @param alpha:    regularization strength; must be a positive float
+        @param tol:      tolerance for stopping criteria
+        @param max_iter: maximum number of iterations
+        @raises ValueError: if alpha is not positive
+        """
+        if alpha <= 0:
+            raise ValueError("Regularization strength must be positive")
+
+        self.alpha = alpha
+        self.tol = tol
+        self.max_iter = max_iter
+        self.params = None
+
+    @staticmethod
+    def _soft_thresholding(rho: float, alpha: float) -> float:
+        """
+        Applies the soft thresholding operator.
+
+        @param rho:    the value to be thresholded
+        @param alpha:  the regularization parameter
+        @returns:      the thresholded value
+        """
+        return np.sign(rho) * max(0, abs(rho) - alpha)
+
+    def fit(self, x: np.ndarray, y: np.ndarray) -> None:
+        """
+        Fits the Lasso regression model to the data.
+
+        @param x: the design matrix (features)
+        @param y: the response vector (target)
+        @raises ArithmeticError: if x isn't full rank, can't compute coefficients
+        """
+        n_samples, n_features = x.shape
+        self.params = np.zeros(n_features)
+
+        for _ in range(self.max_iter):
+            params_old = self.params.copy()
+            for j in range(n_features):
+                # Compute the residual
+                residual = y - x @ self.params + x[:, j] * self.params[j]
+                # Update the j-th coefficient using soft thresholding
+                self.params[j] = self._soft_thresholding(
+                    x[:, j].T @ residual / n_samples, self.alpha / n_samples
+                )
+
+            # Check for convergence
+            if np.linalg.norm(self.params - params_old, ord=1) < self.tol:
+                break
+
+    def predict(self, x: np.ndarray) -> np.ndarray:
+        """
+        Predicts the response values for the given input data.
+
+        @param X: the design matrix (features) for prediction
+        @returns: the predicted response values
+        @raises ArithmeticError: if this function is called before the model parameters are fit
+        """
+        if self.params is None:
+            raise ArithmeticError("Predictor hasn't been fit yet")
+
+        return x @ self.params
+
+
+def main() -> None:
+    """
+    Fit a Lasso regression model to predict a target variable using synthetic data.
+    """
+    import matplotlib.pyplot as plt
+    from sklearn.datasets import make_regression
+
+    # Create synthetic data
+    X, y = make_regression(n_samples=100, n_features=10, noise=0.1)
+
+    lasso_reg = LassoRegression(alpha=0.1)
+    lasso_reg.fit(X, y)
+
+    predictions = lasso_reg.predict(X)
+
+    plt.scatter(y, predictions, alpha=0.5)
+    plt.xlabel("True Values")
+    plt.ylabel("Predicted Values")
+    plt.title("Lasso Regression Predictions")
+    plt.plot([y.min(), y.max()], [y.min(), y.max()], color="red", linewidth=2)
+    plt.show()
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    main()