Allow max samples to be higher

Signed-off-by: Adam Li <adam2392@gmail.com>

sklearn/ensemble/_forest.py

@@ -104,14 +104,18 @@ def _get_n_samples_bootstrap(n_samples, max_samples):
     """
     Get the number of samples in a bootstrap sample.
 
+    The expected total number of unique samples in a bootstrap sample is
+    required to be at most ``n_samples - 1``.
+    This is equivalent to the expected number of out-of-bag samples being at
+    least 1.
+
     Parameters
     ----------
     n_samples : int
         Number of samples in the dataset.
     max_samples : int or float
         The maximum number of samples to draw from the total available:
-            - if float, this indicates a fraction of the total and should be
-              the interval `(0.0, 1.0]`;
+            - if float, this indicates a fraction of the total;
             - if int, this indicates the exact number of samples;
             - if None, this indicates the total number of samples.
 
@@ -124,12 +128,21 @@ def _get_n_samples_bootstrap(n_samples, max_samples):
         return n_samples
 
     if isinstance(max_samples, Integral):
-        if max_samples > n_samples:
-            msg = "`max_samples` must be <= n_samples={} but got value {}"
-            raise ValueError(msg.format(n_samples, max_samples))
+        expected_oob_samples = (1 - np.exp(-max_samples / n_samples)) * n_samples
+        if expected_oob_samples >= n_samples - 1:
+            raise ValueError(
+                "The expected number of unique samples in the bootstrap sample"
+                f" must be at most {n_samples - 1}. It is: {expected_oob_samples}"
+            )
         return max_samples
 
     if isinstance(max_samples, Real):
+        expected_oob_samples = (1 - np.exp(-max_samples)) * n_samples
+        if expected_oob_samples >= n_samples - 1:
+            raise ValueError(
+                "The expected number of unique samples in the bootstrap sample"
+                f" must be at most {n_samples - 1}. It is: {expected_oob_samples}"
+            )
         return max(round(n_samples * max_samples), 1)
 
 

sklearn/ensemble/tests/test_forest.py

@@ -1660,7 +1660,10 @@ def test_max_samples_bootstrap(name):
 def test_large_max_samples_exception(name):
     # Check invalid `max_samples`
     est = FOREST_CLASSIFIERS_REGRESSORS[name](bootstrap=True, max_samples=int(1e9))
-    match = "`max_samples` must be <= n_samples=6 but got value 1000000000"
+    # TODO: remove the following line when the issue is fixed
+    # https://github.com/scikit-learn/scikit-learn/issues/28507
+    # match = "`max_samples` must be <= n_samples=6 but got value 1000000000"
+    match = "The expected number of unique samples"
     with pytest.raises(ValueError, match=match):
         est.fit(X, y)