diff --git a/flaml/automl.py b/flaml/automl.py
index a0315a625..08cd66519 100644
--- a/flaml/automl.py
+++ b/flaml/automl.py
@@ -104,6 +104,7 @@ class SearchState:
         self.trained_estimator = None
         self.sample_size = None
         self.trial_time = 0
+        self.best_n_iter = None
 
     def update(self, result, time_used, save_model_history=False):
         if result:
@@ -430,7 +431,7 @@ class AutoML:
 
     @property
     def time_to_find_best_model(self) -> float:
-        """time taken to find best model in seconds"""
+        """Time taken to find best model in seconds"""
         return self.__dict__.get("_time_taken_best_iter")
 
     def predict(self, X_test):
@@ -1768,6 +1769,17 @@ class AutoML:
         better = True  # whether we find a better model in one trial
         if self._ensemble:
             self.best_model = {}
+        if self._max_iter < 2 and self.estimator_list:
+            # when max_iter is 1, no need to search
+            self._max_iter = 0
+            self._best_estimator = estimator = self.estimator_list[0]
+            self._selected = state = self._search_states[estimator]
+            state.best_config_sample_size = self._state.data_size
+            state.best_config = (
+                state.init_config
+                if isinstance(state.init_config, dict)
+                else state.init_config[0]
+            )
         for self._track_iter in range(self._max_iter):
             if self._estimator_index is None:
                 estimator = self._active_estimators[0]
@@ -1844,9 +1856,9 @@ class AutoML:
                         metric="val_loss",
                         mode="min",
                         space=search_space,
-                        points_to_evaluate=points_to_evaluate
-                        if len(search_state.init_config) == len(search_space)
-                        else None,
+                        points_to_evaluate=[
+                            p for p in points_to_evaluate if len(p) == len(search_space)
+                        ],
                     )
                 search_state.search_alg = ConcurrencyLimiter(algo, max_concurrent=1)
                 # search_state.search_alg = algo
diff --git a/flaml/model.py b/flaml/model.py
index 385bef499..740515507 100644
--- a/flaml/model.py
+++ b/flaml/model.py
@@ -465,7 +465,7 @@ class XGBoostEstimator(SKLearnEstimator):
 
     def predict(self, X_test):
         import xgboost as xgb
-        
+
         if not issparse(X_test):
             X_test = self._preprocess(X_test)
         dtest = xgb.DMatrix(X_test)
diff --git a/test/test_automl.py b/test/test_automl.py
index 97297c8b3..7ccbda76b 100644
--- a/test/test_automl.py
+++ b/test/test_automl.py
@@ -31,7 +31,7 @@ class MyRegularizedGreedyForest(SKLearnEstimator):
             self.estimator_class = RGFClassifier
         else:
             from rgf.sklearn import RGFRegressor
-            
+
             self.estimator_class = RGFRegressor
 
     @classmethod
diff --git a/test/test_training_log.py b/test/test_training_log.py
index 73de45d25..20c728c01 100644
--- a/test/test_training_log.py
+++ b/test/test_training_log.py
@@ -25,23 +25,49 @@ class TestTrainingLog(unittest.TestCase):
                 "mem_thres": 1024 * 1024,
                 "n_jobs": 1,
                 "model_history": True,
-                "train_time_limit": 0.01,
+                "train_time_limit": 0.1,
                 "verbose": 3,
                 "ensemble": True,
                 "keep_search_state": True,
             }
             X_train, y_train = fetch_california_housing(return_X_y=True)
             automl.fit(X_train=X_train, y_train=y_train, **automl_settings)
-            automl._state._train_with_config(automl.best_estimator, automl.best_config)
-
             # Check if the training log file is populated.
             self.assertTrue(os.path.exists(filename))
-            with training_log_reader(filename) as reader:
-                count = 0
-                for record in reader.records():
-                    print(record)
-                    count += 1
-                self.assertGreater(count, 0)
+            if automl.best_estimator:
+                estimator, config = automl.best_estimator, automl.best_config
+                model0 = automl.best_model_for_estimator(estimator)
+                print(model0.estimator)
+
+                automl.time_budget = None
+                model, _ = automl._state._train_with_config(estimator, config)
+                # model0 and model are equivalent unless model0's n_estimator is out of search space range
+                assert (
+                    str(model0.estimator) == str(model.estimator)
+                    or model0["n_estimators"] < 4
+                )
+
+                # assuming estimator & config are saved and loaded as follows
+                automl = AutoML()
+                automl.fit(
+                    X_train=X_train,
+                    y_train=y_train,
+                    max_iter=0,
+                    task="regression",
+                    estimator_list=[estimator],
+                    n_jobs=1,
+                    starting_points={estimator: config},
+                )
+                # then the fitted model should be equivalent to model
+                # print(str(model.estimator), str(automl.model.estimator))
+                assert str(model.estimator) == str(automl.model.estimator)
+
+                with training_log_reader(filename) as reader:
+                    count = 0
+                    for record in reader.records():
+                        print(record)
+                        count += 1
+                    self.assertGreater(count, 0)
 
             automl_settings["log_file_name"] = None
             automl.fit(X_train=X_train, y_train=y_train, **automl_settings)