Symbol - Neural network graphs

In the previous tutorial, we introduced NDArray, the basic data structure for manipulating data in MXNet. Just using NDArray by itself, we can execute a wide range of mathematical operations. In fact, we could define and update a full neural network just by using NDArray.

NDArray allows you to write programs for scientific computation in an imperative fashion, making full use of the native control of any front-end language. Gluon API uses this approach under the hood (before hybridization) to allow for flexible and debugable networks. So you might wonder, why don’t we just use NDArray for all computation?

MXNet also provides the Symbol API, an interface for symbolic programming. With symbolic programming, rather than executing operations step by step, we first define a computation graph. This graph contains placeholders for inputs and designated outputs. We can then compile the graph, yielding a function that can be bound to NDArrays and run. MXNet’s Symbol API is similar to the network configurations used by Caffe and the symbolic programming in Theano. Gluon API takes advantage of this approach under the hood after the network has been hybridized.

Another advantage conferred by symbolic approach is that we can optimize our functions before using them. For example, when we execute mathematical computations in imperative fashion, we don’t know at the time that we run each operation, which values will be needed later on. But with symbolic programming, we declare the required outputs in advance. This means that we can recycle memory allocated in intermediate steps, as by performing operations in place. Symbolic API also uses less memory for the same network. Refer to How To and Architecture section to know more.

In our design notes, we present a more thorough discussion on the comparative strengths of imperative and symbolic programing. In this document, however, we’ll focus on explaining how to use MXNet’s Symbol API.

In MXNet, we can compose Symbols from other Symbols, using operators, such as simple matrix operations (e.g. +), or whole neural network layers (e.g. convolution layer). Operator can take multiple input variables, can produce multiple output symbols and can maintain internal state symbols. For a visual explanation of these concepts, see Symbolic Configuration and Execution in Pictures.

To make things concrete, let’s take a hands-on look at the Symbol API. There are a few different ways to compose a Symbol.


To complete this tutorial, we need:

  • MXNet. See the instructions for your operating system in Setup and Installation

  • Jupyter

    pip install jupyter
  • GPUs (optional). A section of this tutorial uses GPUs, if one is available. If not, the code will automatically switch to CPU.

Basic Symbol Composition

Basic Operators

The following example builds a simple expression: a + b. First, we create two placeholders with mx.sym.Variable, giving them the names a and b. We then construct the desired symbol by using the operator +. We don’t need to name our variables while creating them, MXNet will automatically generate a unique name for each. In the example below, c is assigned a unique name automatically.

import mxnet as mx
a = mx.sym.Variable('a')
b = mx.sym.Variable('b')
c = a + b
(a, b, c)

Most operators supported by NDArray are also supported by Symbol, for example:

# element-wise multiplication
d = a * b
# matrix multiplication
e =, b)
# reshape
f = mx.sym.reshape(d+e, shape=(1,4))
# broadcast
g = mx.sym.broadcast_to(f, shape=(2,4))
# plot
                    node_attrs={"shape": "oval", "fixedsize": "false"})

The computations declared in the above examples can be bound to the input data for evaluation by using bind method. We discuss this further in the Symbol Manipulation section.

Basic Neural Networks

Besides the basic operators, Symbol also supports a rich set of neural network layers. The following example constructs a two layer fully connected neural network and then visualizes the structure of that network given the input data shape.

net = mx.sym.Variable('data')
net = mx.sym.FullyConnected(data=net, name='fc1', num_hidden=128)
net = mx.sym.Activation(data=net, name='relu1', act_type="relu")
net = mx.sym.FullyConnected(data=net, name='fc2', num_hidden=10)
net = mx.sym.SoftmaxOutput(data=net, name='out')
                    shape={'data':(100, 200)},
                    node_attrs={"shape": "oval", "fixedsize": "false"})

Each Symbol takes a unique string name. NDArray and Symbol both represent a single tensor. Operators represent the computation between tensors. Operators take Symbol or NDArray as inputs and might also additionally accept other hyperparameters such as the number of hidden neurons (num_hidden) or the activation type (act_type) and produce the output.

We can view a Symbol simply as a function taking several arguments. And we can retrieve those arguments with the following method call:


These arguments are the parameters and inputs needed by each Symbol:

  • data: Input data needed by the variable data.
  • fc1_weight and fc1_bias: The weight and bias for the first fully connected layer fc1.
  • fc2_weight and fc2_bias: The weight and bias for the second fully connected layer fc2.
  • out_label: The label needed by the loss.

We can also specify the names explicitly:

net = mx.symbol.Variable('data')
w = mx.symbol.Variable('myweight')
net = mx.symbol.FullyConnected(data=net, weight=w, name='fc1', num_hidden=128)

In the above example, FullyConnected layer has 3 inputs: data, weight, bias. When any input is not specified, a variable will be automatically generated for it.

More Complicated Composition

MXNet provides well-optimized Symbols for layers commonly used in deep learning (see src/operator). We can also define new operators in Python. The following example first performs an element-wise add between two Symbols, then feeds them to the fully connected operator:

lhs = mx.symbol.Variable('data1')
rhs = mx.symbol.Variable('data2')
net = mx.symbol.FullyConnected(data=lhs + rhs, name='fc1', num_hidden=128)

We can also construct a Symbol in a more flexible way than the single forward composition depicted in the preceding example:

data = mx.symbol.Variable('data')
net1 = mx.symbol.FullyConnected(data=data, name='fc1', num_hidden=10)
net2 = mx.symbol.Variable('data2')
net2 = mx.symbol.FullyConnected(data=net2, name='fc2', num_hidden=10)
composed = net2(data2=net1, name='composed')

In this example, net2 is used as a function to apply to an existing Symbol net1, and the resulting composed Symbol will have all the attributes of net1 and net2.

Once you start building some bigger networks, you might want to name some symbols with a common prefix to outline the structure of your network. You can use the Prefix class as follows:

data = mx.sym.Variable("data")
net = data
n_layer = 2
for i in range(n_layer):
    with"layer%d_" % (i + 1)):
        net = mx.sym.FullyConnected(data=net, name="fc", num_hidden=100)

Modularized Construction for Deep Networks

Constructing a deep network layer by layer, (like the Google Inception network), can be tedious owing to the large number of layers. So, for such networks, we often modularize the construction.

For example, in Google Inception network, we can first define a factory function which chains the convolution, batch normalization and rectified linear unit (ReLU) activation layers together.

def ConvFactory(data, num_filter, kernel,
                stride=(1, 1), pad=(0, 0), name=None, suffix=''):
    conv = mx.sym.Convolution(data=data, num_filter=num_filter,
                              kernel=kernel, stride=stride, pad=pad,
                              name='conv_%s%s' % (name, suffix))

    bn = mx.sym.BatchNorm(data=conv, name='bn_%s%s' % (name, suffix))

    act = mx.sym.Activation(data=bn, act_type='relu',
                            name='relu_%s%s' % (name, suffix))
    return act

prev = mx.sym.Variable(name="Previous Output")
conv_comp = ConvFactory(data=prev, num_filter=64, kernel=(7, 7), stride=(2, 2))
shape = {"Previous Output" : (128, 3, 28, 28)}
mx.viz.plot_network(symbol=conv_comp, shape=shape,
                    node_attrs={"shape": "oval", "fixedsize": "false"})

Then we can define a function that constructs an inception module based on factory function ConvFactory.

def InceptionFactoryA(data, num_1x1, num_3x3red, num_3x3, num_d3x3red,
                      num_d3x3, pool, proj, name):
    # 1x1
    c1x1 = ConvFactory(data=data, num_filter=num_1x1, kernel=(1, 1),
                       name=('%s_1x1' % name))

    # 3x3 reduce + 3x3
    c3x3r = ConvFactory(data=data, num_filter=num_3x3red, kernel=(1, 1),
                        name=('%s_3x3' % name), suffix='_reduce')
    c3x3 = ConvFactory(data=c3x3r, num_filter=num_3x3, kernel=(3, 3),
                       pad=(1, 1), name=('%s_3x3' % name))

    # double 3x3 reduce + double 3x3
    cd3x3r = ConvFactory(data=data, num_filter=num_d3x3red, kernel=(1, 1),
                         name=('%s_double_3x3' % name), suffix='_reduce')
    cd3x3 = ConvFactory(data=cd3x3r, num_filter=num_d3x3, kernel=(3, 3),
                        pad=(1, 1), name=('%s_double_3x3_0' % name))
    cd3x3 = ConvFactory(data=cd3x3, num_filter=num_d3x3, kernel=(3, 3),
                        pad=(1, 1), name=('%s_double_3x3_1' % name))

    # pool + proj
    pooling = mx.sym.Pooling(data=data, kernel=(3, 3), stride=(1, 1),
                             pad=(1, 1), pool_type=pool,
                             name=('%s_pool_%s_pool' % (pool, name)))
    cproj = ConvFactory(data=pooling, num_filter=proj, kernel=(1, 1),
                        name=('%s_proj' %  name))

    # concat
    concat = mx.sym.Concat(*[c1x1, c3x3, cd3x3, cproj],
                           name='ch_concat_%s_chconcat' % name)
    return concat

prev = mx.sym.Variable(name="Previous Output")
in3a = InceptionFactoryA(prev, 64, 64, 64, 64, 96, "avg", 32, name="in3a")
mx.viz.plot_network(symbol=in3a, shape=shape,
                    node_attrs={"shape": "oval", "fixedsize": "false"})

Finally, we can obtain the whole network by chaining multiple inception modules. See a complete example here.

Group Multiple Symbols

To construct neural networks with multiple loss layers, we can use mxnet.sym.Group to group multiple Symbols together. The following example groups two outputs:

net = mx.sym.Variable('data')
fc1 = mx.sym.FullyConnected(data=net, name='fc1', num_hidden=128)
net = mx.sym.Activation(data=fc1, name='relu1', act_type="relu")
out1 = mx.sym.SoftmaxOutput(data=net, name='softmax')
out2 = mx.sym.LinearRegressionOutput(data=net, name='regression')
group = mx.sym.Group([out1, out2])

Relations to NDArray

As you can see now, both Symbol and NDArray provide multi-dimensional array operations, such as c = a + b in MXNet. We briefly clarify the differences here.

The NDArray provides an imperative programming alike interface, in which the computations are evaluated sentence by sentence. While Symbol is closer to declarative programming, in which we first declare the computation and then evaluate with data. Examples in this category include regular expressions and SQL.

The pros for NDArray:

  • Straightforward.
  • Easy to work with native language features (for loop, if-else condition, ..) and libraries (numpy, ..).
  • Easy step-by-step code debugging.

The pros for Symbol:

  • Provides almost all functionalities of NDArray, such as +, *, sin, reshape etc.
  • Easy to save, load and visualize.
  • Easy for the backend to optimize the computation and memory usage.

Symbol Manipulation

One important difference of Symbol compared to NDArray is that we first declare the computation and then bind the computation with data to run. In this section, we introduce the functions to manipulate a Symbol directly. But note that, most of them are wrapped by the high-level packages: Module and Gluon.

Shape and Type Inference

For each Symbol, we can query its arguments, auxiliary states and outputs. We can also infer the output shape and type of the Symbol given the known input shape or type of some arguments, which facilitates memory allocation.

arg_name = c.list_arguments()  # get the names of the inputs
out_name = c.list_outputs()    # get the names of the outputs
# infers output shape given the shape of input arguments
arg_shape, out_shape, _ = c.infer_shape(a=(2, 3), b=(2, 3))
# infers output type given the type of input arguments
arg_type, out_type, _ = c.infer_type(a='float32', b='float32')
{'input' : dict(zip(arg_name, arg_shape)),
 'output' : dict(zip(out_name, out_shape))}
{'input' : dict(zip(arg_name, arg_type)),
 'output' : dict(zip(out_name, out_type))}

Bind with Data and Evaluate

The Symbol c constructed above declares what computation should be run. To evaluate it, we first need to feed the arguments, namely free variables, with data.

We can do it by using the bind method, which accepts device context and a dict mapping free variable names to NDArrays as arguments and returns an Executor. The Executor provides forward method for evaluation and an attribute outputs to get all the results.

ex = c.bind(ctx=mx.cpu(), args={'a' : mx.nd.ones([2,3]),
                                'b' : mx.nd.ones([2,3])})
print('number of outputs = %d\nthe first output = \n%s' % (
           len(ex.outputs), ex.outputs[0].asnumpy()))

We can evaluate the same Symbol on GPU with different data.

gpu_device = mx.gpu() if mx.test_utils.list_gpus() else mx.cpu()

ex_gpu = c.bind(ctx=gpu_device, args={'a' : mx.nd.ones([3,4], gpu_device)*2,
                                      'b' : mx.nd.ones([3,4], gpu_device)*3})

We can also use eval method to evaluate the Symbol. It combines calls to bind and forward methods.

ex = c.eval(ctx = mx.cpu(), a = mx.nd.ones([2,3]), b = mx.nd.ones([2,3]))
print('number of outputs = %d\nthe first output = \n%s' % (
            len(ex), ex[0].asnumpy()))

For neural nets, a more commonly used pattern is simple_bind, which creates all of the argument arrays for you. Then you can call forward, and backward to get gradients if needed.

Load and Save

Logically Symbols correspond to NDArrays. They both represent a tensor. They both are inputs/outputs of operators. We can either serialize a Symbol object by using pickle, or by using save and load methods directly as it is explained in this NDArray tutorial.

When serializing NDArray, we serialize the tensor data in it and directly dump to disk in binary format. But Symbol uses a concept of graph. Graphs are composed by chaining operators. They are implicitly represented by output Symbols. So, when serializing a Symbol, we serialize the graph of which the Symbol is an output. While serialization, Symbol uses more readable json format for serialization. To convert Symbol to json string, use tojson method.

c2 = mx.sym.load('symbol-c.json')
c.tojson() == c2.tojson()

Customized Symbol

Most operators such as mx.sym.Convolution and mx.sym.Reshape are implemented in C++ for better performance. MXNet also allows users to write new operators using any front-end language such as Python. It often makes the developing and debugging much easier. To implement an operator in Python, refer to How to create new operators.

Advanced Usages

Type Cast

By default, MXNet uses 32-bit floats. But for better accuracy-performance, we can also use a lower precision data type. For example, The NVIDIA Tesla Pascal GPUs (e.g. P100) have improved 16-bit float performance, while GTX Pascal GPUs (e.g. GTX 1080) are fast on 8-bit integers.

To convert the data type as per the requirements, we can use mx.sym.cast operator as follows:

a = mx.sym.Variable('data')
b = mx.sym.cast(data=a, dtype='float16')
arg, out, _ = b.infer_type(data='float32')
print({'input':arg, 'output':out})

c = mx.sym.cast(data=a, dtype='uint8')
arg, out, _ = c.infer_type(data='int32')
print({'input':arg, 'output':out})

Variable Sharing

To share the contents between several Symbols, we can bind these Symbols with the same array as follows:

a = mx.sym.Variable('a')
b = mx.sym.Variable('b')
b = a + a * a

data = mx.nd.ones((2,3)) * 2
ex = b.bind(ctx=mx.cpu(), args={'a':data, 'b':data})

Weight tying

You can use same principle to tie weights of different layers. In the example below two FullyConnected layers share same weights and biases, but process different data. Let’s demonstrate how we can do it.

In this example we first create training and evaluation datasets. Both of them consist of two individual NDArrays. We are using NDArrayIter to iterate over all of them.

import numpy as np
import mxnet as mx

# Training data
train_data_size = 4
train_data1 = mx.random.uniform(shape=(train_data_size, 2))
train_data2 = mx.random.uniform(shape=(train_data_size, 2))
train_label = mx.nd.array([i % 2 for i in range(train_data_size)])
batch_size = 3

# Evaluation Data
eval_data_size = 3
eval_data1 = mx.random.uniform(shape=(eval_data_size, 2))
eval_data2 = mx.random.uniform(shape=(eval_data_size, 2))
eval_label = np.array([i % 2 for i in range(eval_data_size)])

train_iter ={'inputs_left': train_data1,
                                'inputs_right': train_data2},
                               train_label, batch_size, shuffle=True,
                               label_name='labels', last_batch_handle='pad')

eval_iter ={'inputs_left': eval_data1,
                               'inputs_right': eval_data2},
                              eval_label, batch_size, shuffle=False, 

We define a Symbol for both inputs_left and inputs_right variables, and separate symbols for shared_weight and shared_bias. We use shared_weight and shared_bias symbols in both FullyConnected layers, making sure that they are using the same data underlying the symbols. This is where weight tying is happening.

num_hidden_nodes = 2

# Assume the left and right inputs have the same shape as each other
inputs_left = mx.sym.var('inputs_left')
inputs_right = mx.sym.var('inputs_right')
labels = mx.symbol.Variable('labels')

shared_weight = mx.symbol.Variable('shared_weight')
shared_bias = mx.symbol.Variable('shared_bias')

fc_left_sym = mx.sym.FullyConnected(data=inputs_left, weight=shared_weight,
                                    num_hidden=num_hidden_nodes, name='fc_left')

fc_right_sym = mx.sym.FullyConnected(data=inputs_right, weight=shared_weight,
                                     num_hidden=num_hidden_nodes, name='fc_right')

combined = mx.sym.concat(fc_left_sym, fc_right_sym)
output = mx.sym.SoftmaxOutput(data=combined, label=labels, name='softmax')

In the next lines of the code, we use Module API to start the training. We first create a Module object and then call fit providing data iterators. To use trained model for prediction, we use predict method, providing evaluation data iterator.

model = mx.mod.Module(
    data_names=['inputs_left', 'inputs_right'],
), eval_iter,
          optimizer_params={'learning_rate': 0.01, 'momentum': 0.9},

result = model.predict(eval_iter).asnumpy()