[POC] Experimental support for l1 error. (#7812)

Support adaptive tree, a feature supported by both sklearn and lightgbm.  The tree leaf is recomputed based on residue of labels and predictions after construction.

For l1 error, the optimal value is the median (50 percentile).

This is marked as experimental support for the following reasons:
- The value is not well defined for distributed training, where we might have empty leaves for local workers. Right now I just use the original leaf value for computing the average with other workers, which might cause significant errors.
- Some follow-ups are required, for exact, pruner, and optimization for quantile function. Also, we need to calculate the initial estimation.

tests/python/test_with_dask.py

@@ -35,6 +35,7 @@ import dask.dataframe as dd
 import dask.array as da
 from xgboost.dask import DaskDMatrix
 
+dask.config.set({"distributed.scheduler.allowed-failures": False})
 
 if hasattr(HealthCheck, 'function_scoped_fixture'):
     suppress = [HealthCheck.function_scoped_fixture]
@@ -673,7 +674,8 @@ def test_empty_dmatrix_training_continuation(client: "Client") -> None:
 def run_empty_dmatrix_reg(client: "Client", parameters: dict) -> None:
     def _check_outputs(out: xgb.dask.TrainReturnT, predictions: np.ndarray) -> None:
         assert isinstance(out['booster'], xgb.dask.Booster)
-        assert len(out['history']['validation']['rmse']) == 2
+        for _, v in out['history']['validation'].items():
+            assert len(v) == 2
         assert isinstance(predictions, np.ndarray)
         assert predictions.shape[0] == 1
 
@@ -866,6 +868,8 @@ def test_empty_dmatrix(tree_method) -> None:
             parameters = {'tree_method': tree_method}
             run_empty_dmatrix_reg(client, parameters)
             run_empty_dmatrix_cls(client, parameters)
+            parameters = {'tree_method': tree_method, "objective": "reg:absoluteerror"}
+            run_empty_dmatrix_reg(client, parameters)
 
 
 async def run_from_dask_array_asyncio(scheduler_address: str) -> xgb.dask.TrainReturnT:
@@ -1284,7 +1288,12 @@ class TestWithDask:
         def minimum_bin():
             return "max_bin" in params and params["max_bin"] == 2
 
-        if minimum_bin() and is_stump():
+        # See note on `ObjFunction::UpdateTreeLeaf`.
+        update_leaf = dataset.name.endswith("-l1")
+        if update_leaf and len(history) >= 2:
+            assert history[0] >= history[-1]
+            return
+        elif minimum_bin() and is_stump():
             assert tm.non_increasing(history, tolerance=1e-3)
         else:
             assert tm.non_increasing(history)
@@ -1304,7 +1313,7 @@ class TestWithDask:
            dataset=tm.dataset_strategy)
     @settings(deadline=None, suppress_health_check=suppress, print_blob=True)
     def test_approx(
-            self, client: "Client", params: Dict, dataset: tm.TestDataset
+        self, client: "Client", params: Dict, dataset: tm.TestDataset
     ) -> None:
         num_rounds = 30
         self.run_updater_test(client, params, num_rounds, dataset, 'approx')