scikit-learn
diff --git a/‎sklearn/linear_model/_cd_fast.pyx
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+14-4Lines changed: 14 additions & 4 deletions b/‎sklearn/linear_model/_cd_fast.pyx
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+14-4Lines changed: 14 additions & 4 deletions
diff --git a/‎sklearn/linear_model/tests/test_coordinate_descent.py
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+50Lines changed: 50 additions & 0 deletions b/‎sklearn/linear_model/tests/test_coordinate_descent.py
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+50Lines changed: 50 additions & 0 deletions
@@ -244,10 +244,20 @@ def enet_coordinate_descent(floating[::1] w,
         else:
             # for/else, runs if for doesn't end with a `break`
             with gil:
-                warnings.warn("Objective did not converge. You might want to "
-                              "increase the number of iterations. Duality "
-                              "gap: {}, tolerance: {}".format(gap, tol),
-                              ConvergenceWarning)
+                message = (
+                    "Objective did not converge. You might want to increase "
+                    "the number of iterations, check the scale of the "
+                    "features or consider increasing regularisation. "
+                    f"Duality gap: {gap:.3e}, tolerance: {tol:.3e}"
+                )
+                if alpha < np.finfo(np.float64).eps:
+                    message += (
+                        " Linear regression models with null weight for the "
+                        "l1 regularization term are more efficiently fitted "
+                        "using one of the solvers implemented in "
+                        "sklearn.linear_model.Ridge/RidgeCV instead."
+                    )
+                warnings.warn(message, ConvergenceWarning)
 
     return w, gap, tol, n_iter + 1
 
 
@@ -1419,3 +1419,53 @@ def test_enet_sample_weight_does_not_overwrite_sample_weight(check_input):
     reg.fit(X, y, sample_weight=sample_weight, check_input=check_input)
 
     assert_array_equal(sample_weight, sample_weight_1_25)
+
+
+@pytest.mark.parametrize("ridge_alpha", [1e-1, 1., 1e6])
+@pytest.mark.parametrize("normalize", [True, False])
+def test_enet_ridge_consistency(normalize, ridge_alpha):
+    # Check that ElasticNet(l1_ratio=0) converges to the same solution as Ridge
+    # provided that the value of alpha is adapted.
+    #
+    # XXX: this test does not pass for weaker regularization (lower values of
+    # ridge_alpha): it could be either a problem of ElasticNet or Ridge (less
+    # likely) and depends on the dataset statistics: lower values for
+    # effective_rank are more problematic in particular.
+
+    rng = np.random.RandomState(42)
+    X, y = make_regression(
+        n_samples=100,
+        n_features=300,
+        effective_rank=100,
+        n_informative=50,
+        random_state=rng,
+    )
+    sw = rng.uniform(low=0.01, high=2, size=X.shape[0])
+
+    ridge = Ridge(
+        alpha=ridge_alpha,
+        normalize=normalize,
+    ).fit(X, y, sample_weight=sw)
+
+    enet = ElasticNet(
+        alpha=ridge_alpha / sw.sum(),
+        normalize=normalize,
+        l1_ratio=0.,
+        max_iter=1000,
+    )
+    # Even when the ElasticNet model has actually converged, the duality gap
+    # convergence criterion is never met when l1_ratio is 0 and for any value
+    # of the `tol` parameter. The convergence message should point the user to
+    # Ridge instead:
+    expected_msg = (
+        r"Objective did not converge\. .* "
+        r"Linear regression models with null weight for the "
+        r"l1 regularization term are more efficiently fitted "
+        r"using one of the solvers implemented in "
+        r"sklearn\.linear_model\.Ridge/RidgeCV instead\."
+    )
+    with pytest.warns(ConvergenceWarning, match=expected_msg):
+        enet.fit(X, y, sample_weight=sw)
+
+    assert_allclose(ridge.coef_, enet.coef_)
+    assert_allclose(ridge.intercept_, enet.intercept_)