diff --git a/paddle/framework/backward.cc b/paddle/framework/backward.cc
index 11690342185ba6046df2b0f9121bb9bd8cd9a073..d8653b5dd681603b7261e58de02c6787bcdcebfe 100644
--- a/paddle/framework/backward.cc
+++ b/paddle/framework/backward.cc
@@ -29,10 +29,10 @@ static bool AllInSet(const std::vector<std::string>& names,
   return true;
 }
 
-static std::vector<int> InSetIdx(const std::vector<std::string>& names,
-                                 const std::string& suffix,
-                                 const std::unordered_set<std::string>& set) {
-  std::vector<int> ret_val;
+static std::vector<size_t> InSetIdx(
+    const std::vector<std::string>& names, const std::string& suffix,
+    const std::unordered_set<std::string>& set) {
+  std::vector<size_t> ret_val;
   ret_val.reserve(names.size());
   for (size_t i = 0; i < names.size(); ++i) {
     if (set.find(names[i] + suffix) != set.end()) {
@@ -78,7 +78,7 @@ static std::shared_ptr<OperatorBase> BackwardImpl(
 }
 
 extern std::shared_ptr<OperatorBase> Backward(
-    const std::shared_ptr<OperatorBase>& forwardOp,
+    const OperatorBase& forwardOp,
     const std::unordered_set<std::string>& no_grad_vars) {
   std::unordered_set<std::string> no_grad_names;
   no_grad_names.reserve(no_grad_vars.size());
@@ -87,7 +87,7 @@ extern std::shared_ptr<OperatorBase> Backward(
     no_grad_names.insert(name + OperatorBase::GRAD_VAR_SUFFIX());
   }
   int uid = 0;
-  return BackwardImpl(*forwardOp, no_grad_names, uid);
+  return BackwardImpl(forwardOp, no_grad_names, uid);
 }
 }  // namespace framework
 }  // namespace paddle
diff --git a/paddle/framework/backward.h b/paddle/framework/backward.h
index e835ef6351102686d68d657fdc5a6a2913ace3e6..d711c7bbb642781d50ccad0c249a38c939e5e31a 100644
--- a/paddle/framework/backward.h
+++ b/paddle/framework/backward.h
@@ -25,7 +25,7 @@ namespace framework {
  * @return
  */
 extern std::shared_ptr<OperatorBase> Backward(
-    const std::shared_ptr<OperatorBase>& forwardOp,
+    const OperatorBase& forwardOp,
     const std::unordered_set<std::string>& no_grad_vars);
 }  // namespace framework
 }  // namespace paddle