Re-implement ROC-AUC. (#6747)

* Re-implement ROC-AUC.

* Binary
* MultiClass
* LTR
* Add documents.

This PR resolves a few issues:
  - Define a value when the dataset is invalid, which can happen if there's an
  empty dataset, or when the dataset contains only positive or negative values.
  - Define ROC-AUC for multi-class classification.
  - Define weighted average value for distributed setting.
  - A correct implementation for learning to rank task.  Previous
  implementation is just binary classification with averaging across groups,
  which doesn't measure ordered learning to rank.

tests/python/test_eval_metrics.py

@@ -123,3 +123,90 @@ class TestEvalMetrics:
         gamma_dev = float(booster.eval(xgb.DMatrix(X, y)).split(":")[1].split(":")[0])
         skl_gamma_dev = mean_gamma_deviance(y, score)
         np.testing.assert_allclose(gamma_dev, skl_gamma_dev, rtol=1e-6)
+
+    def run_roc_auc_binary(self, tree_method, n_samples):
+        import numpy as np
+        from sklearn.datasets import make_classification
+        from sklearn.metrics import roc_auc_score
+
+        rng = np.random.RandomState(1994)
+        n_samples = n_samples
+        n_features = 10
+
+        X, y = make_classification(
+            n_samples,
+            n_features,
+            n_informative=n_features,
+            n_redundant=0,
+            random_state=rng
+        )
+        Xy = xgb.DMatrix(X, y)
+        booster = xgb.train(
+            {
+                "tree_method": tree_method,
+                "eval_metric": "auc",
+                "objective": "binary:logistic",
+            },
+            Xy,
+            num_boost_round=8,
+        )
+        score = booster.predict(Xy)
+        skl_auc = roc_auc_score(y, score)
+        auc = float(booster.eval(Xy).split(":")[1])
+        np.testing.assert_allclose(skl_auc, auc, rtol=1e-6)
+
+        X = rng.randn(*X.shape)
+        score = booster.predict(xgb.DMatrix(X))
+        skl_auc = roc_auc_score(y, score)
+        auc = float(booster.eval(xgb.DMatrix(X, y)).split(":")[1])
+        np.testing.assert_allclose(skl_auc, auc, rtol=1e-6)
+
+    @pytest.mark.skipif(**tm.no_sklearn())
+    @pytest.mark.parametrize("n_samples", [4, 100, 1000])
+    def test_roc_auc(self, n_samples):
+        self.run_roc_auc_binary("hist", n_samples)
+
+    def run_roc_auc_multi(self, tree_method, n_samples):
+        import numpy as np
+        from sklearn.datasets import make_classification
+        from sklearn.metrics import roc_auc_score
+
+        rng = np.random.RandomState(1994)
+        n_samples = n_samples
+        n_features = 10
+        n_classes = 4
+
+        X, y = make_classification(
+            n_samples,
+            n_features,
+            n_informative=n_features,
+            n_redundant=0,
+            n_classes=n_classes,
+            random_state=rng
+        )
+
+        Xy = xgb.DMatrix(X, y)
+        booster = xgb.train(
+            {
+                "tree_method": tree_method,
+                "eval_metric": "auc",
+                "objective": "multi:softprob",
+                "num_class": n_classes,
+            },
+            Xy,
+            num_boost_round=8,
+        )
+        score = booster.predict(Xy)
+        skl_auc = roc_auc_score(y, score, average="weighted", multi_class="ovr")
+        auc = float(booster.eval(Xy).split(":")[1])
+        np.testing.assert_allclose(skl_auc, auc, rtol=1e-6)
+
+        X = rng.randn(*X.shape)
+        score = booster.predict(xgb.DMatrix(X))
+        skl_auc = roc_auc_score(y, score, average="weighted", multi_class="ovr")
+        auc = float(booster.eval(xgb.DMatrix(X, y)).split(":")[1])
+        np.testing.assert_allclose(skl_auc, auc, rtol=1e-6)
+
+    @pytest.mark.parametrize("n_samples", [4, 100, 1000])
+    def test_roc_auc_multi(self, n_samples):
+        self.run_roc_auc_multi("hist", n_samples)