[MXNET-978] Support higher order gradient for log. (apache#14992)

* add higher order gradient support for log, log10, log2

* add tests

* address comments

* simplify NodeEntry creation.

* address comments

* update comment to avoid confusion.

src/operator/tensor/elemwise_unary_op_basic.cc

@@ -1069,13 +1069,73 @@ The storage type of ``log2`` output is always dense
 .set_attr<nnvm::FGradient>("FGradient", ElemwiseGradUseIn{"_backward_log2"});
 
 MXNET_OPERATOR_REGISTER_BINARY_WITH_SPARSE_CPU_DR(_backward_log,
-                                                  unary_bwd<mshadow_op::log_grad>);
+                                                  unary_bwd<mshadow_op::log_grad>)
+.set_attr<nnvm::FGradient>("FGradient",
+  [](const nnvm::NodePtr& n, const std::vector<nnvm::NodeEntry>& ograds) {
+    // For f(x) -> f = log
+    // f''(x) = -1 * (f'(x) * f'(x))
+    auto gx = nnvm::NodeEntry{n};
+    auto ggx_mid = MakeNode("elemwise_mul", n->attrs.name + "_backward_mid_grad_grad",
+                            {gx, gx}, nullptr, &n);
+    auto ggx = MakeNode("negative", n->attrs.name + "_backward_grad_grad",
+                        {nnvm::NodeEntry{ggx_mid}}, nullptr, &n);
+
+    std::vector<nnvm::NodeEntry> ret;
+
+    ret.emplace_back(MakeNode("elemwise_mul", n->attrs.name + "_backward_grad_grad",
+                             {ograds[0], gx}, nullptr, &n));
+    ret.emplace_back(MakeNode("elemwise_mul", n->attrs.name + "_backward_grad_grad_inp",
+                             {ograds[0], nnvm::NodeEntry{ggx}}, nullptr, &n));
+    return ret;
+  });
 
 MXNET_OPERATOR_REGISTER_BINARY_WITH_SPARSE_CPU_DR(_backward_log10,
-                                                  unary_bwd<mshadow_op::log10_grad>);
+                                                  unary_bwd<mshadow_op::log10_grad>)
+.set_attr<nnvm::FGradient>("FGradient",
+  [](const nnvm::NodePtr& n, const std::vector<nnvm::NodeEntry>& ograds) {
+    // For f(x) -> f = log10
+    // f'(x) = 1 / (log(10) * x)
+    // f''(x) = -1 * (f'(x) * 1/x)
+    auto gx = nnvm::NodeEntry{n, 0, 0};
+    auto g_lx = MakeNode("reciprocal", n->attrs.name + "_backward_log_grad",
+                            {n->inputs[1]}, nullptr, &n);
+    auto ggx_mid = MakeNode("elemwise_mul", n->attrs.name + "_backward_mid_grad_grad",
+                            {gx, nnvm::NodeEntry{g_lx}}, nullptr, &n);
+    auto ggx = MakeNode("negative", n->attrs.name + "_backward_grad_grad",
+                        {nnvm::NodeEntry{ggx_mid}}, nullptr, &n);
+
+    std::vector<nnvm::NodeEntry> ret;
+
+    ret.emplace_back(MakeNode("elemwise_mul", n->attrs.name + "_backward_grad_grad",
+                             {ograds[0], gx}, nullptr, &n));
+    ret.emplace_back(MakeNode("elemwise_mul", n->attrs.name + "_backward_grad_grad_inp",
+                             {ograds[0], nnvm::NodeEntry{ggx}}, nullptr, &n));
+    return ret;
+  });
 
 MXNET_OPERATOR_REGISTER_BINARY_WITH_SPARSE_CPU_DR(_backward_log2,
-                                                  unary_bwd<mshadow_op::log2_grad>);
+                                                  unary_bwd<mshadow_op::log2_grad>)
+.set_attr<nnvm::FGradient>("FGradient",
+  [](const nnvm::NodePtr& n, const std::vector<nnvm::NodeEntry>& ograds) {
+    // For f(x) -> f = log2
+    // f'(x) = 1 / (log(2) * x)
+    // f''(x) = -1 * (f'(x) * 1/x)
+    auto gx = nnvm::NodeEntry{n};
+    auto g_lx = MakeNode("reciprocal", n->attrs.name + "_backward_log_grad",
+                            {n->inputs[1]}, nullptr, &n);
+    auto ggx_mid = MakeNode("elemwise_mul", n->attrs.name + "_backward_mid_grad_grad",
+                            {gx, nnvm::NodeEntry{g_lx}}, nullptr, &n);
+    auto ggx = MakeNode("negative", n->attrs.name + "_backward_grad_grad",
+                        {nnvm::NodeEntry{ggx_mid}}, nullptr, &n);
+
+    std::vector<nnvm::NodeEntry> ret;
+
+    ret.emplace_back(MakeNode("elemwise_mul", n->attrs.name + "_backward_grad_grad",
+                             {ograds[0], gx}, nullptr, &n));
+    ret.emplace_back(MakeNode("elemwise_mul", n->attrs.name + "_backward_grad_grad_inp",
+                             {ograds[0], nnvm::NodeEntry{ggx}}, nullptr, &n));
+    return ret;
+  });
 
 // log1p
 MXNET_OPERATOR_REGISTER_UNARY_WITH_RSP_CSR(log1p, cpu, mshadow_op::log1p)

tests/python/unittest/test_higher_order_grad.py

@@ -0,0 +1,80 @@
+# Licensed to the Apache Software Foundation (ASF) under one
+# or more contributor license agreements.  See the NOTICE file
+# distributed with this work for additional information
+# regarding copyright ownership.  The ASF licenses this file
+# to you under the Apache License, Version 2.0 (the
+# "License"); you may not use this file except in compliance
+# with the License.  You may obtain a copy of the License at
+#
+#   http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an
+# "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+# KIND, either express or implied.  See the License for the
+# specific language governing permissions and limitations
+# under the License.
+
+import math
+
+from mxnet import nd, autograd
+from mxnet.test_utils import assert_almost_equal, random_arrays
+from common import with_seed
+
+
+@with_seed()
+def test_log():
+    def log(x):
+        return nd.log(x)
+
+    def grad_grad_op(x):
+        return -1/(x**2)
+
+    arrays = random_arrays((2, 2), (2, 3), (4, 5, 2), (3, 1, 4, 5))
+
+    for array in arrays:
+        check_second_order_unary(array, log, grad_grad_op)
+
+
+@with_seed()
+def test_log2():
+    def log2(x):
+        return nd.log2(x)
+
+    def grad_grad_op(x):
+        return -1/((x**2) * math.log(2))
+
+    arrays = random_arrays((2, 2), (2, 3), (4, 5, 2), (3, 1, 4, 5))
+
+    for array in arrays:
+        check_second_order_unary(array, log2, grad_grad_op)
+
+
+@with_seed()
+def test_log10():
+    def log10(x):
+        return nd.log10(x)
+
+    def grad_grad_op(x):
+        return -1/((x**2) * math.log(10))
+
+    arrays = random_arrays((2, 2), (2, 3), (4, 5, 2), (3, 1, 4, 5))
+
+    for array in arrays:
+        check_second_order_unary(array, log10, grad_grad_op)
+
+
+def check_second_order_unary(x, op, grad_grad_op):
+    x = nd.array(x)
+    expect_grad_grad = grad_grad_op(x)
+    x.attach_grad()
+    with autograd.record():
+        y = op(x)
+        y_grad = autograd.grad(y, x, create_graph=True, retain_graph=True)[0]
+    y_grad.backward()
+    assert_almost_equal(expect_grad_grad.asnumpy(), x.grad.asnumpy())
+
+
+if __name__ == '__main__':
+    import nose
+    nose.runmodule()