.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples_built/plot_sure_lasso.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_built_plot_sure_lasso.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_built_plot_sure_lasso.py:


SURE-tuned Lasso (no held-out set)
==================================

Stein's Unbiased Risk Estimator (SURE) lets you tune the Lasso regularization
strength when no held-out set exists — denoising, signal recovery, single-fold
settings — by directly estimating the prediction MSE from the *training* data.

This example sets up a sparse-denoising problem with a known noise level σ,
tunes α via :class:`sparho.Sure` + :func:`sparho.hoag_search`, and overlays
the SURE curve against the oracle prediction-MSE curve (which we can only
compute here because we know the noise-free signal).

.. GENERATED FROM PYTHON SOURCE LINES 15-21

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    from sparho import L1, Problem, SquaredLoss, Sure, hoag_search
    from sparho.adapters import SklearnLasso








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Sparse-denoising fixture — 200 obs × 150 features, 12 informative,
i.i.d. Gaussian noise with σ = 0.2.

.. GENERATED FROM PYTHON SOURCE LINES 24-36

.. code-block:: Python

    rng = np.random.default_rng(0)
    n, p, k = 200, 150, 12
    sigma = 0.2
    X = rng.standard_normal((n, p)) / np.sqrt(n)
    beta_star = np.zeros(p)
    support = rng.choice(p, size=k, replace=False)
    beta_star[support] = rng.choice([-1.0, 1.0], size=k) * (1.0 + rng.random(k))
    y_clean = X @ beta_star
    y = y_clean + sigma * rng.standard_normal(n)

    problem = Problem(SquaredLoss(), L1(), X, y)








.. GENERATED FROM PYTHON SOURCE LINES 37-38

Gradient-based search using SURE as the outer criterion.

.. GENERATED FROM PYTHON SOURCE LINES 38-49

.. code-block:: Python

    solver = SklearnLasso(tol=1e-10, max_iter=100_000)
    result = hoag_search(
        problem,
        hp0=1e-2,
        solver=solver,
        criterion=Sure(sigma=sigma, random_state=0),
        n_iter=30,
    )
    alpha_sure = float(result.best_hyperparam)
    print(f"sparho SURE: α* = {alpha_sure:.4g}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    sparho SURE: α* = 0.0009493




.. GENERATED FROM PYTHON SOURCE LINES 50-53

Oracle curve: prediction MSE against the noise-free signal Xβ*. We can only
compute this because the example knows the ground truth; in practice SURE is
what stands in for it.

.. GENERATED FROM PYTHON SOURCE LINES 53-65

.. code-block:: Python

    alphas = np.logspace(-3, 0, 40)
    sure_values, oracle_mse = [], []
    crit = Sure(sigma=sigma, random_state=0)
    for a in alphas:
        sure_values.append(crit.value(problem, float(a), solver))
        coef = solver(problem, float(a)).coef
        err = X @ coef - y_clean
        oracle_mse.append(float(err @ err) / n)

    alpha_oracle = float(alphas[int(np.argmin(oracle_mse))])
    print(f"oracle:      α* = {alpha_oracle:.4g}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    oracle:      α* = 0.001




.. GENERATED FROM PYTHON SOURCE LINES 66-69

Plot both curves on a shared log-α axis. SURE tracks the oracle near the
optimum; the FDMC variance shows up as wiggles at very small α (large
active set ⇒ noisier DOF estimate).

.. GENERATED FROM PYTHON SOURCE LINES 69-83

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(alphas, oracle_mse, "o-", color="C1", label="oracle pred. MSE (uses true signal)")
    ax.plot(alphas, sure_values, "o-", color="C0", label="SURE (data only)")
    ax.axvline(alpha_sure, color="C0", linestyle="--", alpha=0.5, label=f"sparho α* = {alpha_sure:.2g}")
    ax.axvline(
        alpha_oracle, color="C1", linestyle="--", alpha=0.5, label=f"oracle α* = {alpha_oracle:.2g}"
    )
    ax.set_xscale("log")
    ax.set_xlabel(r"$\alpha$")
    ax.set_ylabel("estimated prediction MSE")
    ax.set_title("SURE recovers a near-oracle α without a held-out set")
    ax.legend(fontsize=8)
    fig.tight_layout()
    plt.show()



.. image-sg:: /examples_built/images/sphx_glr_plot_sure_lasso_001.png
   :alt: SURE recovers a near-oracle α without a held-out set
   :srcset: /examples_built/images/sphx_glr_plot_sure_lasso_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.372 seconds)


.. _sphx_glr_download_examples_built_plot_sure_lasso.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_sure_lasso.ipynb <plot_sure_lasso.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_sure_lasso.py <plot_sure_lasso.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_sure_lasso.zip <plot_sure_lasso.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

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