Define best_iteration only if early stopping is used. (#9403)

* Define `best_iteration` only if early stopping is used.

This is the behavior specified by the document but not honored in the actual code.

- Don't set the attributes if there's no early stopping.
- Clean up the code for callbacks, and replace assertions with proper exceptions.
- Assign the attributes when early stopping `save_best` is used.
- Turn the attributes into Python properties.

---------

Co-authored-by: Philip Hyunsu Cho <chohyu01@cs.washington.edu>

tests/python/test_training_continuation.py

@@ -100,8 +100,8 @@ class TestTrainingContinuation:
         res2 = mean_squared_error(
             y_2class,
             gbdt_04.predict(
-                dtrain_2class, iteration_range=(0, gbdt_04.best_iteration + 1)
-            )
+                dtrain_2class, iteration_range=(0, gbdt_04.num_boosted_rounds())
+            ),
         )
         assert res1 == res2
 
@@ -112,7 +112,7 @@ class TestTrainingContinuation:
         res2 = mean_squared_error(
             y_2class,
             gbdt_04.predict(
-                dtrain_2class, iteration_range=(0, gbdt_04.best_iteration + 1)
+                dtrain_2class, iteration_range=(0, gbdt_04.num_boosted_rounds())
             )
         )
         assert res1 == res2
@@ -126,7 +126,7 @@ class TestTrainingContinuation:
 
         res1 = gbdt_05.predict(dtrain_5class)
         res2 = gbdt_05.predict(
-            dtrain_5class, iteration_range=(0, gbdt_05.best_iteration + 1)
+            dtrain_5class, iteration_range=(0, gbdt_05.num_boosted_rounds())
         )
         np.testing.assert_almost_equal(res1, res2)
 
@@ -138,15 +138,16 @@ class TestTrainingContinuation:
     @pytest.mark.skipif(**tm.no_sklearn())
     def test_training_continuation_updaters_json(self):
         # Picked up from R tests.
-        updaters = 'grow_colmaker,prune,refresh'
+        updaters = "grow_colmaker,prune,refresh"
         params = self.generate_parameters()
         for p in params:
-            p['updater'] = updaters
+            p["updater"] = updaters
         self.run_training_continuation(params[0], params[1], params[2])
 
     @pytest.mark.skipif(**tm.no_sklearn())
     def test_changed_parameter(self):
         from sklearn.datasets import load_breast_cancer
+
         X, y = load_breast_cancer(return_X_y=True)
         clf = xgb.XGBClassifier(n_estimators=2)
         clf.fit(X, y, eval_set=[(X, y)], eval_metric="logloss")