5.2. Parameter Management
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The ultimate goal of training deep networks is to find good parameter values for a given architecture. When everything is standard, the nn.Sequential class is a perfectly good tool for it. However, very few models are entirely standard and most scientists want to build things that are novel. This section shows how to manipulate parameters. In particular we will cover the following aspects:

  • Accessing parameters for debugging, diagnostics, to visualize them or to save them is the first step to understanding how to work with custom models.

  • Second, we want to set them in specific ways, e.g., for initialization purposes. We discuss the structure of parameter initializers.

  • Last, we show how this knowledge can be put to good use by building networks that share some parameters.

As always, we start from our trusty Multilayer Perceptron with a hidden layer. This will serve as our choice for demonstrating the various features.

from mxnet import init, np, npx
from mxnet.gluon import nn

net = nn.Sequential()
net.add(nn.Dense(256, activation='relu'))
net.initialize()  # Use the default initialization method

x = np.random.uniform(size=(2, 20))
net(x)  # Forward computation
array([[ 0.06240272, -0.03268593,  0.02582653,  0.02254182, -0.03728798,
        -0.04253786,  0.00540613, -0.01364186, -0.09915452, -0.02272738],
       [ 0.02816677, -0.03341204,  0.03565666,  0.02506382, -0.04136416,
        -0.04941845,  0.01738528,  0.01081961, -0.09932579, -0.01176298]])

5.2.1. Parameter Access

In the case of a Sequential class we can access the parameters with ease, simply by indexing each of the layers in the network. The params variable then contains the required data. Let’s try this out in practice by inspecting the parameters of the first layer.

dense0_ (
  Parameter dense0_weight (shape=(256, 20), dtype=float32)
  Parameter dense0_bias (shape=(256,), dtype=float32)
dense1_ (
  Parameter dense1_weight (shape=(10, 256), dtype=float32)
  Parameter dense1_bias (shape=(10,), dtype=float32)

The output tells us a number of things. First, the layer consists of two sets of parameters: dense0_weight and dense0_bias, as we would expect. They are both single precision and they have the necessary shapes that we would expect from the first layer, given that the input dimension is 20 and the output dimension 256. In particular the names of the parameters are very useful since they allow us to identify parameters uniquely even in a network of hundreds of layers and with nontrivial structure. The second layer is structured accordingly. Targeted Parameters

In order to do something useful with the parameters we need to access them, though. There are several ways to do this, ranging from simple to general. Let’s look at some of them.

Parameter dense1_bias (shape=(10,), dtype=float32)
[0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]

The first returns the bias of the second layer. Since this is an object containing data, gradients, and additional information, we need to request the data explicitly. Note that the bias is all 0 since we initialized the bias to contain all zeros. Note that we can also access the parameters by name, such as dense0_weight. This is possible since each layer comes with its own parameter dictionary that can be accessed directly. Both methods are entirely equivalent but the first method leads to much more readable code.

Parameter dense0_weight (shape=(256, 20), dtype=float32)
[[ 0.06700657 -0.00369488  0.0418822  ... -0.05517294 -0.01194733
 [-0.03296221 -0.04391347  0.03839272 ...  0.05636378  0.02545484
  -0.007007  ]
 [-0.0196689   0.01582889 -0.00881553 ...  0.01509629 -0.01908049
 [-0.02055008 -0.02618652  0.06762936 ... -0.02315108 -0.06794678
 [ 0.02802853  0.06672969  0.05018687 ... -0.02206502 -0.01315478
 [-0.00638592  0.00914261  0.06667828 ... -0.00800052  0.03406764

Note that the weights are nonzero. This is by design since they were randomly initialized when we constructed the network. data is not the only function that we can invoke. For instance, we can compute the gradient with respect to the parameters. It has the same shape as the weight. However, since we did not invoke backpropagation yet, the values are all 0.

array([[0., 0., 0., ..., 0., 0., 0.],
       [0., 0., 0., ..., 0., 0., 0.],
       [0., 0., 0., ..., 0., 0., 0.],
       [0., 0., 0., ..., 0., 0., 0.],
       [0., 0., 0., ..., 0., 0., 0.],
       [0., 0., 0., ..., 0., 0., 0.]]) All Parameters at Once

Accessing parameters as described above can be a bit tedious, in particular if we have more complex blocks, or blocks of blocks (or even blocks of blocks of blocks), since we need to walk through the entire tree in reverse order to how the blocks were constructed. To avoid this, blocks come with a method collect_params which grabs all parameters of a network in one dictionary such that we can traverse it with ease. It does so by iterating over all constituents of a block and calls collect_params on subblocks as needed. To see the difference consider the following:

# parameters only for the first layer
# parameters of the entire network
dense0_ (
  Parameter dense0_weight (shape=(256, 20), dtype=float32)
  Parameter dense0_bias (shape=(256,), dtype=float32)
sequential0_ (
  Parameter dense0_weight (shape=(256, 20), dtype=float32)
  Parameter dense0_bias (shape=(256,), dtype=float32)
  Parameter dense1_weight (shape=(10, 256), dtype=float32)
  Parameter dense1_bias (shape=(10,), dtype=float32)

This provides us with a third way of accessing the parameters of the network. If we wanted to get the value of the bias term of the second layer we could simply use this:

array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])

Throughout the book we will see how various blocks name their subblocks (Sequential simply numbers them). This makes it very convenient to use regular expressions to filter out the required parameters.

sequential0_ (
  Parameter dense0_weight (shape=(256, 20), dtype=float32)
  Parameter dense1_weight (shape=(10, 256), dtype=float32)
sequential0_ (
  Parameter dense0_weight (shape=(256, 20), dtype=float32)
  Parameter dense0_bias (shape=(256,), dtype=float32)
) Rube Goldberg Striking Again

Let’s see how the parameter naming conventions work if we nest multiple blocks inside each other. For that we first define a function that produces blocks (a block factory, so to speak) and then we combine these inside yet larger blocks.

def block1():
    net = nn.Sequential()
    net.add(nn.Dense(32, activation='relu'))
    net.add(nn.Dense(16, activation='relu'))
    return net

def block2():
    net = nn.Sequential()
    for i in range(4):
    return net

rgnet = nn.Sequential()
array([[-4.1923025e-09,  1.9830502e-09,  8.9444063e-10,  6.2912990e-09,
        -3.3241778e-09,  5.4330038e-09,  1.6013515e-09, -3.7408681e-09,
         8.5468477e-09, -6.4805539e-09],
       [-3.7507064e-09,  1.4866974e-09,  6.8314709e-10,  5.6925784e-09,
        -2.6349172e-09,  4.8626667e-09,  1.4280275e-09, -3.4603027e-09,
         7.4127922e-09, -5.7896132e-09]])

Now that we are done designing the network, let’s see how it is organized. collect_params provides us with this information, both in terms of naming and in terms of logical structure.

<bound method Block.collect_params of Sequential(
  (0): Sequential(
    (0): Sequential(
      (0): Dense(20 -> 32, Activation(relu))
      (1): Dense(32 -> 16, Activation(relu))
    (1): Sequential(
      (0): Dense(16 -> 32, Activation(relu))
      (1): Dense(32 -> 16, Activation(relu))
    (2): Sequential(
      (0): Dense(16 -> 32, Activation(relu))
      (1): Dense(32 -> 16, Activation(relu))
    (3): Sequential(
      (0): Dense(16 -> 32, Activation(relu))
      (1): Dense(32 -> 16, Activation(relu))
  (1): Dense(16 -> 10, linear)
sequential1_ (
  Parameter dense2_weight (shape=(32, 20), dtype=float32)
  Parameter dense2_bias (shape=(32,), dtype=float32)
  Parameter dense3_weight (shape=(16, 32), dtype=float32)
  Parameter dense3_bias (shape=(16,), dtype=float32)
  Parameter dense4_weight (shape=(32, 16), dtype=float32)
  Parameter dense4_bias (shape=(32,), dtype=float32)
  Parameter dense5_weight (shape=(16, 32), dtype=float32)
  Parameter dense5_bias (shape=(16,), dtype=float32)
  Parameter dense6_weight (shape=(32, 16), dtype=float32)
  Parameter dense6_bias (shape=(32,), dtype=float32)
  Parameter dense7_weight (shape=(16, 32), dtype=float32)
  Parameter dense7_bias (shape=(16,), dtype=float32)
  Parameter dense8_weight (shape=(32, 16), dtype=float32)
  Parameter dense8_bias (shape=(32,), dtype=float32)
  Parameter dense9_weight (shape=(16, 32), dtype=float32)
  Parameter dense9_bias (shape=(16,), dtype=float32)
  Parameter dense10_weight (shape=(10, 16), dtype=float32)
  Parameter dense10_bias (shape=(10,), dtype=float32)

Since the layers are hierarchically generated, we can also access them accordingly. For instance, to access the first major block, within it the second subblock and then within it, in turn the bias of the first layer, we perform the following.

array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
       0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])

5.2.2. Parameter Initialization

Now that we know how to access the parameters, let’s look at how to initialize them properly. We discussed the need for initialization in Section 4.8. By default, MXNet initializes the weight matrices uniformly by drawing from \(U[-0.07, 0.07]\) and the bias parameters are all set to \(0\). However, we often need to use other methods to initialize the weights. MXNet’s init module provides a variety of preset initialization methods, but if we want something out of the ordinary, we need a bit of extra work. Built-in Initialization

Let’s begin with the built-in initializers. The code below initializes all parameters with Gaussian random variables.

# force_reinit ensures that the variables are initialized again, regardless of
# whether they were already initialized previously
net.initialize(init=init.Normal(sigma=0.01), force_reinit=True)
array([-9.8788980e-03,  5.3957910e-03, -7.0842835e-03, -7.4317548e-03,
       -1.4880489e-02,  6.4959107e-03, -8.2659349e-03,  1.8743129e-02,
        1.6201857e-02,  1.4534278e-03,  2.2331164e-03,  1.5926110e-02,
       -1.2915777e-02, -8.8592555e-05, -1.7293986e-03, -7.2338698e-03,
        8.7698260e-03, -4.9947016e-03, -9.6906107e-03,  2.0079101e-03])

If we wanted to initialize all parameters to 1, we could do this simply by changing the initializer to Constant(1).

net.initialize(init=init.Constant(1), force_reinit=True)
array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1.])

If we want to initialize only a specific parameter in a different manner, we can simply set the initializer only for the appropriate subblock (or parameter) for that matter. For instance, below we initialize the second layer to a constant value of 42 and we use the Xavier initializer for the weights of the first layer.

net[1].initialize(init=init.Constant(42), force_reinit=True)
net[0].weight.initialize(init=init.Xavier(), force_reinit=True)
print(net[1].weight.data()[0, 0])
[-0.06319056 -0.10960881  0.11757872 -0.07595599 -0.0849717   0.0851637
  0.08330765  0.04028694 -0.0305525   0.02012795 -0.03856885  0.1375024
  0.10155623 -0.05016676 -0.02575382 -0.14205234  0.14225402  0.02719662
 -0.0888046  -0.00962897] Custom Initialization

Sometimes, the initialization methods we need are not provided in the init module. At this point, we can implement a subclass of the Initializer class so that we can use it like any other initialization method. Usually, we only need to implement the _init_weight function and modify the incoming ndarray according to the initial result. In the example below, we pick a decidedly bizarre and nontrivial distribution, just to prove the point. We draw the coefficients from the following distribution:

(5.2.1)\[\begin{split}\begin{aligned} w \sim \begin{cases} U[5, 10] & \text{ with probability } \frac{1}{4} \\ 0 & \text{ with probability } \frac{1}{2} \\ U[-10, -5] & \text{ with probability } \frac{1}{4} \end{cases} \end{aligned}\end{split}\]
class MyInit(init.Initializer):
    def _init_weight(self, name, data):
        print('Init', name, data.shape)
        data[:] = np.random.uniform(-10, 10, data.shape)
        data *= np.abs(data) >= 5

net.initialize(MyInit(), force_reinit=True)
Init dense0_weight (256, 20)
Init dense1_weight (10, 256)
array([-5.172625 , -7.0209026,  5.1446533, -9.844563 ,  8.545956 ,
       -0.       ,  0.       , -0.       ,  5.107664 ,  9.658335 ,
        5.8564453,  7.4483128,  0.       ,  0.       , -0.       ,
        7.9034443,  0.       ,  5.4223766,  8.5655575,  5.1224785])

If even this functionality is insufficient, we can set parameters directly. Since data() returns an ndarray we can access it just like any other matrix. A note for advanced users: if you want to adjust parameters within an autograd scope you need to use set_data to avoid confusing the automatic differentiation mechanics.

net[0].weight.data()[:] += 1
net[0].weight.data()[0, 0] = 42
array([42.       , -6.0209026,  6.1446533, -8.844563 ,  9.545956 ,
        1.       ,  1.       ,  1.       ,  6.107664 , 10.658335 ,
        6.8564453,  8.448313 ,  1.       ,  1.       ,  1.       ,
        8.903444 ,  1.       ,  6.4223766,  9.5655575,  6.1224785])

5.2.3. Tied Parameters

In some cases, we want to share model parameters across multiple layers. For instance when we want to find good word embeddings we may decide to use the same parameters both for encoding and decoding of words. We discussed one such case when we introduced Section 5.1. Let’s see how to do this a bit more elegantly. In the following we allocate a dense layer and then use its parameters specifically to set those of another layer.

net = nn.Sequential()
# We need to give the shared layer a name such that we can reference its
# parameters
shared = nn.Dense(8, activation='relu')
net.add(nn.Dense(8, activation='relu'),
        nn.Dense(8, activation='relu', params=shared.params),

x = np.random.uniform(size=(2, 20))

# Check whether the parameters are the same
print(net[1].weight.data()[0] == net[2].weight.data()[0])
net[1].weight.data()[0, 0] = 100
# Make sure that they are actually the same object rather than just having the
# same value
print(net[1].weight.data()[0] == net[2].weight.data()[0])
[ True  True  True  True  True  True  True  True]
[ True  True  True  True  True  True  True  True]

The above example shows that the parameters of the second and third layer are tied. They are identical rather than just being equal. That is, by changing one of the parameters the other one changes, too. What happens to the gradients is quite ingenious. Since the model parameters contain gradients, the gradients of the second hidden layer and the third hidden layer are accumulated in the shared.params.grad( ) during backpropagation.

5.2.4. Summary

  • We have several ways to access, initialize, and tie model parameters.

  • We can use custom initialization.

  • Gluon has a sophisticated mechanism for accessing parameters in a unique and hierarchical manner.

5.2.5. Exercises

  1. Use the FancyMLP defined in Section 5.1 and access the parameters of the various layers.

  2. Look at the MXNet documentation and explore different initializers.

  3. Try accessing the model parameters after net.initialize() and before net(x) to observe the shape of the model parameters. What changes? Why?

  4. Construct a multilayer perceptron containing a shared parameter layer and train it. During the training process, observe the model parameters and gradients of each layer.

  5. Why is sharing parameters a good idea?

5.2.6. Discussions