[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/cpp/objective/test_regression_obj.cc

@@ -1,11 +1,14 @@
 /*!
- * Copyright 2017-2021 XGBoost contributors
+ * Copyright 2017-2022 XGBoost contributors
  */
 #include <gtest/gtest.h>
-#include <xgboost/objective.h>
 #include <xgboost/generic_parameters.h>
 #include <xgboost/json.h>
+#include <xgboost/objective.h>
+
+#include "../../../src/objective/adaptive.h"
 #include "../helpers.h"
+
 namespace xgboost {
 
 TEST(Objective, DeclareUnifiedTest(LinearRegressionGPair)) {
@@ -378,4 +381,113 @@ TEST(Objective, CoxRegressionGPair) {
                    { 0,    0,    0,  0.160f,  0.186f,  0.348f, 0.610f,  0.639f});
 }
 #endif
+
+TEST(Objective, DeclareUnifiedTest(AbsoluteError)) {
+  Context ctx = CreateEmptyGenericParam(GPUIDX);
+  std::unique_ptr<ObjFunction> obj{ObjFunction::Create("reg:absoluteerror", &ctx)};
+  obj->Configure({});
+  CheckConfigReload(obj, "reg:absoluteerror");
+
+  MetaInfo info;
+  std::vector<float> labels{0.f, 3.f, 2.f, 5.f, 4.f, 7.f};
+  info.labels.Reshape(6, 1);
+  info.labels.Data()->HostVector() = labels;
+  info.num_row_ = labels.size();
+  HostDeviceVector<float> predt{1.f, 2.f, 3.f, 4.f, 5.f, 6.f};
+  info.weights_.HostVector() = {1.f, 1.f, 1.f, 1.f, 1.f, 1.f};
+
+  CheckObjFunction(obj, predt.HostVector(), labels, info.weights_.HostVector(),
+                   {1.f, -1.f, 1.f, -1.f, 1.f, -1.f}, info.weights_.HostVector());
+
+  RegTree tree;
+  tree.ExpandNode(0, /*split_index=*/1, 2, true, 0.0f, 2.f, 3.f, 4.f, 2.f, 1.f, 1.f);
+
+  HostDeviceVector<bst_node_t> position(labels.size(), 0);
+  auto& h_position = position.HostVector();
+  for (size_t i = 0; i < labels.size(); ++i) {
+    if (i < labels.size() / 2) {
+      h_position[i] = 1;  // left
+    } else {
+      h_position[i] = 2;  // right
+    }
+  }
+
+  auto& h_predt = predt.HostVector();
+  for (size_t i = 0; i < h_predt.size(); ++i) {
+    h_predt[i] = labels[i] + i;
+  }
+
+  obj->UpdateTreeLeaf(position, info, predt, &tree);
+  ASSERT_EQ(tree[1].LeafValue(), -1);
+  ASSERT_EQ(tree[2].LeafValue(), -4);
+}
+
+TEST(Objective, DeclareUnifiedTest(AbsoluteErrorLeaf)) {
+  Context ctx = CreateEmptyGenericParam(GPUIDX);
+  std::unique_ptr<ObjFunction> obj{ObjFunction::Create("reg:absoluteerror", &ctx)};
+  obj->Configure({});
+
+  MetaInfo info;
+  info.labels.Reshape(16, 1);
+  info.num_row_ = info.labels.Size();
+  CHECK_EQ(info.num_row_, 16);
+  auto h_labels = info.labels.HostView().Values();
+  std::iota(h_labels.begin(), h_labels.end(), 0);
+  HostDeviceVector<float> predt(h_labels.size());
+  auto& h_predt = predt.HostVector();
+  for (size_t i = 0; i < h_predt.size(); ++i) {
+    h_predt[i] = h_labels[i] + i;
+  }
+
+  HostDeviceVector<bst_node_t> position(info.labels.Size(), 0);
+  auto& h_position = position.HostVector();
+  for (int32_t i = 0; i < 3; ++i) {
+    h_position[i] = ~i;  // negation for sampled nodes.
+  }
+  for (size_t i = 3; i < 8; ++i) {
+    h_position[i] = 3;
+  }
+  // empty leaf for node 4
+  for (size_t i = 8; i < 13; ++i) {
+    h_position[i] = 5;
+  }
+  for (size_t i = 13; i < h_labels.size(); ++i) {
+    h_position[i] = 6;
+  }
+
+  RegTree tree;
+  tree.ExpandNode(0, /*split_index=*/1, 2, true, 0.0f, 2.f, 3.f, 4.f, 2.f, 1.f, 1.f);
+  tree.ExpandNode(1, /*split_index=*/1, 2, true, 0.0f, 2.f, 3.f, 4.f, 2.f, 1.f, 1.f);
+  tree.ExpandNode(2, /*split_index=*/1, 2, true, 0.0f, 2.f, 3.f, 4.f, 2.f, 1.f, 1.f);
+  ASSERT_EQ(tree.GetNumLeaves(), 4);
+
+  auto empty_leaf = tree[4].LeafValue();
+  obj->UpdateTreeLeaf(position, info, predt, &tree);
+  ASSERT_EQ(tree[3].LeafValue(), -5);
+  ASSERT_EQ(tree[4].LeafValue(), empty_leaf);
+  ASSERT_EQ(tree[5].LeafValue(), -10);
+  ASSERT_EQ(tree[6].LeafValue(), -14);
+}
+
+TEST(Adaptive, DeclareUnifiedTest(MissingLeaf)) {
+  std::vector<bst_node_t> missing{1, 3};
+
+  std::vector<bst_node_t> h_nidx = {2, 4, 5};
+  std::vector<size_t> h_nptr = {0, 4, 8, 16};
+
+  obj::detail::FillMissingLeaf(missing, &h_nidx, &h_nptr);
+
+  ASSERT_EQ(h_nidx[0], missing[0]);
+  ASSERT_EQ(h_nidx[2], missing[1]);
+  ASSERT_EQ(h_nidx[1], 2);
+  ASSERT_EQ(h_nidx[3], 4);
+  ASSERT_EQ(h_nidx[4], 5);
+
+  ASSERT_EQ(h_nptr[0], 0);
+  ASSERT_EQ(h_nptr[1], 0);  // empty
+  ASSERT_EQ(h_nptr[2], 4);
+  ASSERT_EQ(h_nptr[3], 4);  // empty
+  ASSERT_EQ(h_nptr[4], 8);
+  ASSERT_EQ(h_nptr[5], 16);
+}
 }  // namespace xgboost