kshitij12345 commented on a change in pull request #14992: [MXNET-978] Support
higher order gradient for `log`.
URL: https://github.com/apache/incubator-mxnet/pull/14992#discussion_r288657874
##########
File path: 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};
Review comment:
For natural `log`,
we have with us in `gradient` function, `gx` i.e. `1/x` as well as `x`.
Since, second derivative of log is `-(gx * gx)` = `-1/(x^2)`. We use the
pattern.
Considering `log2` (similar case for `log10`)
we have with us, `gx` i.e. `1/(log(2) * x)` as well as `x`.
Since second derivative is `-1/(log(2) * x * x)`
which we get in the code using `negative(gx * reciprocal(x))`, where
`gx=1/(log(2) * x`.
Another way to get that will be `negative(gx * gx * log(2.0))`.
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