An Operator Example¶

Let’s take the quadratic function as an example: f(x) = ax^2+bx+c. We want to implement an operator called quadratic taking x, which is a tensor, as an input and generating an output tensor y satisfying y.shape=x.shape and each element of y is calculated by feeding the corresponding element of x into the quadratic function f. Here variables a, b, and c are user input parameters. In frontend, the operator works like this:

x = [[1, 2], [3, 4]]
y = quadratic(data=x, a=1, b=2, c=3)
y = [[6, 11], [18, 27]]

To implement this, we first create three files: quadratic_op-inl.h, quadratic_op.cc, and quadratic_op.cu. The header file’s name is prefixed by the operator name and followed by op and -inl indicating that this is an operator implementation with inline functions shared by CPU and GPU computing. The CPU and GPU specific implementations reside in their own .cc and .cu files, respectively. We normally put pure tensor related operators (e.g. tile, repeat, etc.) under the directory src/operator/tensor, and neural network operators (e.g. Convolution, Pooling, etc.) under src/operator/nn. You may have noticed that many neural network operators including Convolution and Pooling are currently saved under src/operator. We plan to move them to src/operator/nn for better file organization and clearer hierarchy in the future.

Next, we are going to

1. Define the parameter struct for registering a, b, and c in quadratic_op-inl.h.
2. Define type and shape inference functions in quadratic_op-inl.h.
3. Define forward and backward functions in quadratic_op-inl.h.
4. Register the operator using nnvm in quadratic_op.cc and quadratic_op.cu for CPU and GPU computing, respectively.

Now let’s walk through the process step by step.

Parameter Registration¶

We first define struct QuadraticParam as a placeholder for the parameters a, b, and c in quadratic_op-inl.h. The struct inherits from a base template struct named dmlc::Parameter, where the template argument is the derived struct QuadraticParam. This technique, which is called curiously recurring template pattern, achieves static polymorphism. It is similar to using a virtual function, but without the cost associated with dynamic polymorphism.

struct QuadraticParam : public dmlc::Parameter<QuadraticParam> {
  float a, b, c;
  DMLC_DECLARE_PARAMETER(QuadraticParam) {
    DMLC_DECLARE_FIELD(a)
      .set_default(0.0)
      .describe("Coefficient of the quadratic term in the quadratic function.");
    DMLC_DECLARE_FIELD(b)
      .set_default(0.0)
      .describe("Coefficient of the linear term in the quadratic function.");
    DMLC_DECLARE_FIELD(c)
      .set_default(0.0)
      .describe("Constant term in the quadratic function.");
  }
};

The function calls in the above parameter struct are self-explanatory by their names. Note that for each parameter, we set the default value to 0.0 such that users can skip passing 0-value parameters through the quadratic operator interface. You can choose not to define the default value for a parameter if it is required at runtime. Meanwhile, adding brief descriptions to the parameters enables the documentation engine to display them on MXNet documentation web page.

Attribute Inference¶

Attribute inference is the process of deducing the properties of NDArrays in neural networks from user provided information. Two most common attributes of an NDArray are data shape and data type. Let’s take a look at the following example. Given an input NDArray called data, you invoke the quadratic operator like this: output = mx.nd.quadratic(data, a=1, b=2, c=3). Before calculating the output values, its shape and data type are inferred from the input data‘s shape and type following the rules you defined in order to allocate memory space for the output tensor.

One important thing to note that inference functions should be capable of performing mutual inference, i.e. inferring one argument’s attribute from another argument’s attribute if possible according to the definition of the operator. This is very useful for a computational graph to deduce unknown attributes for a neural network in symbolic programming. Users can view the computational graph as a symbol with every element initialized for running data throughout the neural network, including memory allocation for each tensor, device placement for each operator, etc. Users normally just need to provide minimum necessary information, such as input data shapes, etc., to the computational graph, and the graph will fill up the unknown attributes using the attribute inference functions defined in the operators building up the neural network.

Let’s consider the following example.

>>> import mxnet as mx
>>> a = mx.sym.Variable('a', shape=(2, 0))
>>> b = mx.sym.Variable('b')
>>> c = mx.sym.Variable('c', shape=(0, 3))
>>> d = a * b + b * c
>>> print d.infer_shape()
([(2L, 3L), (2L, 3L), (2L, 3L)], [(2L, 3L)], [])

The last line of the above code snippet is a tuple of three lists returned by d.infer_shape(). The first list contains all the argument shapes of a, b, and c. The second contains the output shape of d. The third one represents the shapes of auxiliary states, which is not used in this case, and thus is empty. In this example, we only specified values for variable a‘s first dimension and c‘s second dimension. The 0 in shape (2, 0) indicates that the size of the second dimension is unknown, same meaning for shape (0, 3). However, the symbol d still successfully inferred the shapes for all the variables and final output. This is a result of mutual inference. In MXNet, the whole process can be interpreted as this:

1. a and b are combined via an element-wise multiplication operator, so the shapes of a and b are same and b‘s first dimension size is 2.
2. b and c are combined via an element-wise multiplication operator too, so the shapes of b and c are same and b‘s second dimension size is 3.
3. Now b‘s shape is completely known, so a and c missing dimension sizes are known as well.
4. d is a result from adding a * b and b * c, so d should also have the same shape as b.

The above four steps illustrate how shape inference logic works in MXNet. It is actually implemented in the shape inference functions of the operators for element-wise multiplication and addition.

For our quadratic operator, shape inference possesses quite similar logic.

inline bool QuadraticOpShape(const nnvm::NodeAttrs& attrs,
                             mxnet::ShapeVector* in_attrs,
                             mxnet::ShapeVector* out_attrs) {
  CHECK_EQ(in_attrs->size(), 1U);
  CHECK_EQ(out_attrs->size(), 1U);

  SHAPE_ASSIGN_CHECK(*out_attrs, 0, in_attrs->at(0));
  SHAPE_ASSIGN_CHECK(*in_attrs, 0, out_attrs->at(0));
  return out_attrs->at(0).ndim() != 0U && out_attrs->at(0).Size() != 0U;
}

Here are a few things to note about the above function:

1. attrs contains parameters a, b, and c from user input. It’s not used here since we don’t rely on that information for shape inference.
2. in_attrs is a vector containing all input shapes. Since there is only one input argument for operator quadratic, we used macro CHECK_EQ to assert when the vector’s size is wrong.
3. out_attrs is a vector containing all output shapes. We also used CHECK_EQ to verify the size of the vector since there is only one output.
4. We called macro SHAPE_ASSIGN_CHECK twice for mutual inference. One for inferring the output shape from the input shape, the other one is for inferring the input shape from the output shape. If there are any unequal non-zero values in the same dimension of two shapes, such as (2, 3) and (3, 3), the macro would throw an exception with an error message for shape inference.
5. At the end of the function body, we checked whether the output shape is completely known by testing whether the shape is not empty and the shape’s size is greater than 0. Note that in MXNet, an empty shape means that the shape is unknown, and a 0 in a shape means that the size of that dimension is unknown. In both situations, the missing shape information must be inferred from other shapes. If it cannot be inferred, the function should return false to notify the caller about shape inference failure.
6. MXNet provides a convenience function implementing the logic of mutual inference for general element-wise operators with the following interface. Users can instantiate this function with n_in=1 and n_out=1 to replace the above function QuadraticOpShape in operator registration (explained later). The function QuadraticOpShape posted here is for the purpose of illustration only.

template<int n_in, int n_out>
inline bool ElemwiseShape(const nnvm::NodeAttrs& attrs,
                          mxnet::ShapeVector *in_attrs,
                          mxnet::ShapeVector *out_attrs);

The same logic goes for data type inference. We will leave the analysis of the following code sample to users. Note that -1 means the data type is unknown and must be inferred from other input or output data types.

inline bool QuadraticOpType(const nnvm::NodeAttrs& attrs,
                            std::vector<int>* in_attrs,
                            std::vector<int>* out_attrs) {
  CHECK_EQ(in_attrs->size(), 1U);
  CHECK_EQ(out_attrs->size(), 1U);

  TYPE_ASSIGN_CHECK(*out_attrs, 0, in_attrs->at(0));
  TYPE_ASSIGN_CHECK(*in_attrs, 0, out_attrs->at(0));
  return out_attrs->at(0) != -1;
}

Again, MXNet provides the following convenience function for mutual type inference of element-wise operators. Users can use that in operator registration (explained later).

template<int n_in, int n_out>
inline bool ElemwiseType(const nnvm::NodeAttrs& attrs,
                         std::vector<int>* in_attrs,
                         std::vector<int>* out_attrs);

Forward Function¶

Forward function defines the operator’s behavior in the forward pass of neural networks. For our quadratic operator, it simply implements the logic of running a tensor through the quadratic function by performing a few element-wise operations. The forward function’s signature is fixed in MXNet as follows:

void (const nnvm::NodeAttrs& attrs,
      const OpContext& ctx,
      const std::vector<TBlob>& inputs,
      const std::vector<OpReqType>& req,
      const std::vector<TBlob>& outputs);

We first paste the whole forward function code here and then go through it line by line.

template<typename xpu>                                                        // 1
void QuadraticOpForward(const nnvm::NodeAttrs& attrs,                         // 2
                        const OpContext& ctx,                                 // 3
                        const std::vector<TBlob>& inputs,                     // 4
                        const std::vector<OpReqType>& req,                    // 5
                        const std::vector<TBlob>& outputs) {                  // 6
  CHECK_EQ(inputs.size(), 1U);                                                // 7
  CHECK_EQ(outputs.size(), 1U);                                               // 8
  CHECK_EQ(req.size(), 1U);                                                   // 9
  mshadow::Stream<xpu> *s = ctx.get_stream<xpu>();                            // 10
  const TBlob& in_data = inputs[0];                                           // 11
  const TBlob& out_data = outputs[0];                                         // 12
  const QuadraticParam& param = nnvm::get<QuadraticParam>(attrs.parsed);      // 13
  using namespace mxnet_op;                                                   // 14
  MSHADOW_TYPE_SWITCH(out_data.type_flag_, DType, {                           // 15
    MXNET_ASSIGN_REQ_SWITCH(req[0], req_type, {                               // 16
      Kernel<quadratic_forward<req_type>, xpu>::Launch(                       // 17
          s, out_data.Size(), out_data.dptr<DType>(), in_data.dptr<DType>(),  // 18
          param.a, param.b, param.c);                                         // 19
    });                                                                       // 20
  });                                                                         // 21
}                                                                             // 22

• Line 1: xpu stands for a generic device type so that the function can be instantiated for both CPU and GPU computing using concrete types cpu and gpu. The instantiation happens at the time when the operator is registered in .cc and .cu files.
• Line 2: attrs is a node attribute containing the user input parameters a, b, and c. Here the node represents a placeholder for the operator in the whole computational graph for the neural network.
• Line 3: ctx holds something called stream for serializing asynchronous executions. Let’s consider this example for understanding the functionality of stream. We want to launch several GPU kernels with the same stream from CPU. Even though the launching operation is non-blocking, the stream guarantees that the kernels execute in the same order on GPU as they are launched from CPU.
• Line 4: inputs is a vector of input tensors (only one input tensor for the quadratic operator).
• Line 5: req is a vector of OpReqType values. Each value defines the way of writing calculated values to the output tensors. Therefore, the number of reqs must be the same as the number of output tensors. MXNet currently supports three types of req in frontend: null, write, and add. null means skipping calculating the corresponding output tensor, write means overwriting the values in the output tensor with the ones calculated by this operator, and add means adding the calculated values to the existing ones in the output tensor. Note that null and add are usually seen in backward passes. The former is for skipping calculating the gradients of un-learnable parameters (such as index arrays), and the latter is for accumulating gradients throughout networks.
• Line 6: outputs is a vector of output tensors (only one output tensor for the quadratic operator).
• Lines 7-9: Verify that the size of each vector is expected. Otherwise, stop moving forward and print error message.
• Line 10: Get the stream from the ctx for launching kernels.
• Lines 11-12: Define the references of the input and output tensors for later coding convenience. Note that TBlob can be understood as a uniform data structure for tensors of various dimensions, such that tensors of different dimensions can be put in a homogeneous container, such as std::vector and std::list. You can still get tensors of desired dimensions from a TBlob object through the interface get_with_shape.
• Line 13: Get user input parameters from the node attribute.
• Lines 15-21: This is the place where the mathematical formula of the operator is implemented. The macros MSHADOW_TYPE_SWITCH and MXNET_ASSIGN_REQ_SWITCH enable the code block to work for all the supported data types and req types in MXNet. Inside the inner-most macro, we launch the kernel for calculating the output tensor such that each thread takes an element from the input tensor, feeds it into the quadratic function, and assigns the output element to the output tensor based on req type. Note that Kernel::Launch serves as a universal interface for launching parallel computation on both CPU and GPU. This allows most of the simple operators to share the same piece of code for CPU and GPU as parallelization approaches are often identical on both types of devices. The kernel function is defined as the following, where the function Map is executed by each thread for each input element. The out_data.Size(), in the Kernel::Launch function corresponds to the factor by which the workload will get parallelized among the different threads, which here corresponds to the size of the output array. To explain a little bit more on the two macros used in the kernel struct: (1) MSHADOW_XINLINE is a consolidated macro for inlining functions compiled by both CPU and GPU compilers. It enables CPU and GPU computing to share the same piece of code. (2) KERNEL_ASSIGN is a macro for unifying the statements of different reqs into the same line of code. It’s named KERNEL_ASSIGN because we call the code blocks running parallel computation kernels. On CPUs, the kernels are normally wrapped by the OpenMP parallel directive; while on GPUs, they are the kernel functions launched by CUDA library.

template<int req>
struct quadratic_forward {
  template<typename DType>
  MSHADOW_XINLINE static void Map(int i, DType* out_data, const DType* in_data,
                                  const float a, const float b, const float c) {
    KERNEL_ASSIGN(out_data[i], req, in_data[i] * (a * in_data[i] + b) + c);
  }
};

Backward Function¶

Backward functions play the role of propagating derivatives of loss function with respect to the outputs of the last layer throughout the network to the first layer. The whole process is often known as backward propagation. We are not going to delineate the principle of backward propagation here since users can find great details covered in other resources, such as CS231n and How the backgropagation algorithm works. The problem we are going to solve here for the quadratic operator is that given a tensor representing the gradient of the loss function with respect to the output of the operator, calculate the gradient with respect to the input of the operator. There is no need to calculate the derivatives of loss function with respect to user input parameters a, b, and c since they are not learnable parameters in the network. To formulate the problem: given dL/dy and y = a*x^2 + b*x + c, where L represents the loss function and y stands for the output of the quadratic tensor, we need to solve for dL/dx. Using the chain-rule, it is obvious to find that

dL/dx = dL/dy * dy/dx = dL/dy * (2*a*x + b).

The above equation indicates that dL/dx depends on the gradient of the output tensor and value of the input tensor. The backward function’s signature is the same as the forward function’s. With the aforementioned information in mind, let’s breakdown the following backward function line by line.

template<typename xpu>                                                       // 1
void QuadraticOpBackward(const nnvm::NodeAttrs& attrs,                       // 2
                         const OpContext& ctx,                               // 3
                         const std::vector<TBlob>& inputs,                   // 4
                         const std::vector<OpReqType>& req,                  // 5
                         const std::vector<TBlob>& outputs) {                // 6
  CHECK_EQ(inputs.size(), 2U);                                               // 7
  CHECK_EQ(outputs.size(), 1U);                                              // 8
  CHECK_EQ(req.size(), 1U);                                                  // 9
  mshadow::Stream<xpu> *s = ctx.get_stream<xpu>();                           // 10
  const TBlob& out_grad = inputs[0];                                         // 11
  const TBlob& in_data = inputs[1];                                          // 12
  const TBlob& in_grad = outputs[0];                                         // 13
  const QuadraticParam& param = nnvm::get<QuadraticParam>(attrs.parsed);     // 14
  using namespace mxnet_op;                                                  // 15
  MSHADOW_TYPE_SWITCH(out_grad.type_flag_, DType, {                          // 16
    MXNET_ASSIGN_REQ_SWITCH(req[0], req_type, {                              // 17
      Kernel<quadratic_backward<req_type>, xpu>::Launch(                     // 18
          s, in_grad.Size(), in_grad.dptr<DType>(), out_grad.dptr<DType>(),  // 19
          in_data.dptr<DType>(), param.a, param.b);                          // 20
    });                                                                      // 21
  });                                                                        // 22
}                                                                            // 23

• Lines 1-6: Backward function has the same signature as forward function.
• Lines 7-9: Check the sizes of the function arguments. One thing to note that since the gradient of the input depends on both the gradient of the output and the input tensor itself, inputs must contain two TBlob objects.
• Line 10: Get the stream of the context for serializing asynchronous executions.
• Lines 11-13: Convenience reference variables for later use. We name out_grad as the gradient of the operator output, in_data as the input of the operator, and in_grad as the gradient of the operator input.
• Line 14: Get the parameter object of QuadraticParam.
• Lines 16-22: Same as in the forward function, this is where parallel computation for in_grad happens. The struct quadratic_backward implements the formula of calculating each element of in_grad by one thread as the following.

template<int req>
struct quadratic_backward {
  template<typename DType>
  MSHADOW_XINLINE static void Map(int i, DType* in_grad, const DType* out_grad,
                                  const DType* in_data, const float a, const float b) {
    KERNEL_ASSIGN(in_grad[i], req, out_grad[i] * (2 * a * in_data[i] + b));
  }
};

Operator Registration¶

So far, we have implemented necessary data structure and functions for the operator quadratic. Now let’s register them using nnvm to expose the operator quadratic to frontend. Users can consider the registration process as creating the operator object instance, saving it in the operator manager (a singleton), and setting attributes for the operator instance.

The following code is from quadratic_op.cc, which is responsible for registering the operator working on CPU.

DMLC_REGISTER_PARAMETER(QuadraticParam);                                           // 1

NNVM_REGISTER_OP(quadratic)                                                        // 2
.describe(R"code(This operators implements the quadratic function:                 // 3
.. math::

    f(x) = ax^2+bx+c

where :math:`x` is an input tensor and all operations
in the function are element-wise.

Example:

  .. code-block:: python
     :emphasize-lines: 1,3
     x = [[1, 2], [3, 4]]
     y = quadratic(data=x, a=1, b=2, c=3)
     y = [[6, 11], [18, 27]]

)code" ADD_FILELINE)                                                               // 4
.set_attr_parser(ParamParser<QuadraticParam>)                                      // 5
.set_num_inputs(1)                                                                 // 6
.set_num_outputs(1)                                                                // 7
.set_attr<nnvm::FListInputNames>("FListInputNames",                                // 8
  [](const NodeAttrs& attrs) {                                                     // 9
    return std::vector<std::string>{"data"};                                       // 10
  })                                                                               // 11
.set_attr<nnvm::FInferShape>("FInferShape", QuadraticOpShape)                      // 12
.set_attr<nnvm::FInferType>("FInferType", QuadraticOpType)                         // 13
.set_attr<FCompute>("FCompute", QuadraticOpForward<cpu>)                      // 14
.set_attr<nnvm::FGradient>("FGradient", ElemwiseGradUseIn{"_backward_quadratic"})  // 15
.set_attr<nnvm::FInplaceOption>("FInplaceOption",                                  // 16
  [](const NodeAttrs& attrs) {                                                     // 17
    return std::vector<std::pair<int, int> >{{0, 0}};                              // 18
  })                                                                               // 19
.add_argument("data", "NDArray-or-Symbol", "Input ndarray")                        // 20
.add_arguments(QuadraticParam::__FIELDS__());                                      // 21

NNVM_REGISTER_OP(_backward_quadratic)                                              // 22
.set_attr_parser(ParamParser<QuadraticParam>)                                      // 23
.set_num_inputs(2)                                                                 // 24
.set_num_outputs(1)                                                                // 25
.set_attr<nnvm::TIsBackward>("TIsBackward", true)                                  // 26
.set_attr<FCompute>("FCompute", QuadraticOpBackward<cpu>);                    // 27

• Line 1: Register the parameter struct.
• Line 2: Register an operator named quadratic by creating an instance of Op type and save it in the operator manager and return a reference of the just created operator object.
• Lines 3-4: Add description as an operator attribute including examples of the operator. The documentation engine will extract this description and display it on the documentation web page. emphasize-lines is optional. For more examples and troubleshooting with doc strings, refer to the MXNet developer wiki’s Documentation Guide.
• Line 5: Set parameter struct parser for the operator. It is used for parsing the parameters a, b, and c input from frontend.
• Line 6: Set the number of inputs for the operator.
• Line 7: Set the number of outputs for the operator.
• Lines 8-11: Defines a function generating a vector of names of the operator input arguments. This function is used to add missing arguments that users did not specify when creating a symbolic operator. For example, quad_func=mx.sym.quadratic() is still a valid symbol since we have added the attribute FListInputNames to the operator node in the computational graph. MXNet would add the missing argument with name quadratic0_data, where the prefix quadratic0 is the operator name appended with an index and the postfix data comes from the return value of the user defined FListInputName function. Users still can generate an executor for the quad_func like the following:

quad_exe = quad_func.simple_bind(ctx=mx.cpu(), quadratic0_data=(1,))

• Line 12: Register shape inference function.
• Line 13: Register type inference function.
• Line 14: Register forward function.
• Line 15: Register the function for creating the node of the operator in a backward pass. Note that we used a convenience functor struct ElemwiseGradUseIn. As you can tell from the name, the registered functor creates the node for gradient computation with dependencies on the output gradient node and input node. Similarly, there are other three functors defined as ElemwiseGradUseOut, ElemwiseGradUseInOut, and ElemwiseGradUseNone for developers’ convenience. In order to add this attribute, we also need to register a backward operator for quadratic with several basic attributes, as it can share attribute inference functions with the forward operator and is not exposed to frontend.
• Lines 16-19: This registered function implies that which output tensor can reuse which input tensor’s memory space instead of allocating a new memory space for the output. In the operator quadratic, there is only one input and output, and the output can reuse the input memory space, so we store a pair of zeros in the function return vector indicating that inputs[0]‘s memory space can be reused by outputs[0]. Note that this function just provides a hint to the computational graph initializer. If there are other nodes depending on the input tensor, the memory space of the input tensor will not be overwritten by the output.
• Line 20: Define the input argument name as data for the operator.
• Line 21: Add user input parameters a, b, and c as the attributes of the operator.
• Line 22: Register an operator named _backward_quadratic for backward pass of the operator quadratic. The underscore prefix in the operator name indicates that this is an operator not exposed to users. The convention of naming an internally used backward operator is prepending the prefix _backward_ to the corresponding forward operator name.
• Line 23: Set the parameter parser for the operator _backward_quadratic.
• Line 24: Set the number of inputs.
• Line 25: Set the number of outputs.
• Line 26: Add TIsBackward attribute for the operator. The shape and type inference passes use this attribute to determine whether a node in the graph is a forward or backward node.
• Line 27: Register backward function.

So far, we have acquired an operator working on CPU in frontend. In order to register the operator working on GPUs, we just need to add the following code to quadratic_op.cu. Note that forward and backward functions are registered with attribute key FCompute, rather than FCompute.

NNVM_REGISTER_OP(quadratic)
.set_attr<FCompute>("FCompute", QuadraticOpForward<gpu>);

NNVM_REGISTER_OP(_backward_quadratic)
.set_attr<FCompute>("FCompute", QuadraticOpBackward<gpu>);

Unit Test¶

Now we have finished implementing the operator quadratic in MXNet backend. If you use python, when you type import mxnet as mx, two python functions for invoking your backend implementation are generated on the fly: one is for imperative programming registered as mxnet.ndarray.quadratic or mx.nd.quadratic for short; the other one is for symbolic programming registered under module mxnet.symbol.quadratic or mx.sym.quadratic for short.

In order to unit test it in frontend, we need to add the following code to the python file test_operator.py. A typical operator implementation tests for both the symbol API and the ndarray API. The following test has both these tests. The imperative API test, tests for the ndarray API, mx.nd.contrib.quadratic. The symbol API test, tests for the complete functionality of the operator - the forward pass and the backward pass. To facilitate the testing of these functionalities we use three helper functions available in the mxnet.test_utils module:

• check_symbolic_forward
• check_symbolic_backward
• check_numeric_gradient

def test_quadratic_function():
    def f(x, a, b, c):
        return a * x**2 + b * x + c

    a = np.random.random_sample()
    b = np.random.random_sample()
    c = np.random.random_sample()
    data = mx.symbol.Variable('data')
    quad_sym = mx.sym.contrib.quadratic(data=data, a=a, b=b, c=c)
    for dtype in [np.float16, np.float32, np.float64]:
        for ndim in range(1, 6):
            shape = rand_shape_nd(ndim, 5)
            data_np = np.random.randn(*shape).astype(dtype)
            expected = f(data_np, a, b, c)
            backward_expected = 2 * a * data_np + b

            # check imperative forward
            output = mx.nd.contrib.quadratic(mx.nd.array(data_np), a=a, b=b, c=c)
            assert_almost_equal(output.asnumpy(),expected,
                                rtol=1e-2 if dtype is np.float16 else 1e-5,
                                atol=1e-2 if dtype is np.float16 else 1e-5)
            # check forward
            check_symbolic_forward(quad_sym, [data_np], [expected],
                                    rtol=1e-2 if dtype is np.float16 else 1e-5,
                                    atol=1e-2 if dtype is np.float16 else 1e-5)
            # check backward
            check_symbolic_backward(quad_sym, [data_np], [np.ones(expected.shape)],
                                        [backward_expected],
                                        rtol=1e-2 if dtype is np.float16 else 1e-5,
                                        atol=1e-2 if dtype is np.float16 else 1e-5)
            # check backward using finite difference
            check_numeric_gradient(quad_sym, [data_np], atol=0.001)

In the above test we create a quadratic symbol and feed it into the three utility functions. The check_symbolic_forward and check_symbolic_backward tests the computed values against the expected values that we pass as an argument to the function. The check_numeric_gradient utility function performs gradient checking to verify the implementation for the backward function of the operator. It will perform a perturbation on the input and calculate the response rate of the output using the finite difference method. Then it will compare the gradient from the backward pass with the values from the finite difference method. All three of these tests will be successful once the comparison satisfies user specified rtol and atol values. Here rtol and atol expand to relative tolerance and absolute tolerance respectively. They are used to specify how far the computed values can deviate from the expected values. They are defined as follows

abs(Expected_Value - Computed_Value) < RTOL * abs(Expected_Value) + ATOL

For example, if rtol is 1e-5 and atol is 1e-5 and the expected value is 1.5623145, then the computed value should lie within the range of (1.562288876855, 1.562340123145) else the test will fail. Make sure you tune the rtol and atol values accordingly. Giving very low values for rtol and atol will likely make the test very flaky. It is recommended that you use the flakiness checker tool to check if the test you have written is flaky or not. You can run the flakiness checker tool for the above test with the following command -

python tools/flakiness_checker.py test_operator.test_quadratic_function

Please note that for check_symbolic_forward and check_symbolic_backward we pass both the operator symbols and expected results for comparison, for check_numeric_gradient we only pass the operator symbol, as the check_numeric_gradient computes the expected value using finite difference method. Which is why it is highly recommended to add check_numeric_gradient test for every operator with backward function implemented as it eliminates the possibility of passing incorrect expected results into check_symbolic_backward.