Optimizing a NengoDL model

Optimizing Nengo models via deep learning training methods is one of the important features of NengoDL. This functionality is accessed via the Simulator.train() method. For example:

with nengo.Network() as net:
    <construct the model>

with nengo_dl.Simulator(net, ...) as sim:
    sim.train(<inputs>, <targets>, <optimizer>, n_epochs=10,
              objective=<objective>)

When the Simulator is first constructed, all the parameters in the model (e.g., encoders, decoders, connection weights, biases) are initialized based on the functions/distributions specified during model construction (see the Nengo documentation for more detail on how that works). What the Simulator.train() method does is then further optimize those parameters based on some inputs and desired outputs. We’ll go through each of those components in more detail below.

Simulator.train arguments

inputs

The first argument to the Simulator.train() function is the input data. We can think of a model as computing a function \(y = f(x, \theta)\), where \(f\) is the model, mapping inputs \(x\) to outputs \(y\) with parameters \(\theta\). This argument is specifying the values for \(x\).

In practice what that means is specifying values for the input Nodes in the model. A Node is a Nengo object that inserts values into a Network, usually used to define external inputs. Simulator.train() will override the normal Node values with the training data that is provided. This is specified as a dictionary {<node>: <array>, ...}, where <node> is the input node for which training data is being defined, and <array> is a numpy array containing the training values. This training array should have shape (n_inputs, n_steps, node.size_out), where n_inputs is the number of training examples, n_steps is the number of simulation steps to train across, and node.size_out is the dimensionality of the Node.

When training a NengoDL model the user must specify the minibatch_size to use during training, via the Simulator(..., minibatch_size=n) argument. This defines how many inputs (out of the total n_inputs defined above) will be used for each optimization step.

Here is an example illustrating how to define the input values for two input nodes:

with nengo.Network() as net:
    a = nengo.Node([0])
    b = nengo.Node([1, 2, 3])
    ...

n_inputs = 1000
minibatch_size = 20
n_steps = 10

with nengo_dl.Simulator(net, minibatch_size=minibatch_size) as sim:
    sim.train(inputs={a: np.random.randn(n_inputs, n_steps, 1),
                      b: np.random.randn(n_inputs, n_steps, 3)},
              ...)

Input values must be provided for at least one Node, but beyond that can be defined for as many Nodes as desired. Any Nodes that don’t have data provided will take on the values specified during model construction. Also note that inputs can only be defined for Nodes with no incoming connections (i.e., Nodes with size_in == 0).

targets

Returning to the network equation \(y = f(x, \theta)\), the goal in optimization is to find a set of parameter values such that given inputs \(x\) the actual network outputs \(y\) are as close as possible to some target values \(t\). This argument is specifying those desired outputs \(t\).

This works very similarly to defining inputs, except instead of assigning input values to Nodes it assigns target values to Probes. The structure of the argument is similar – a dictionary of {<probe>: <array>, ...}, where <array> has shape (n_inputs, n_steps, probe.size_in). Each entry in the target array defines the desired output for the corresponding entry in the input array.

For example:

with nengo.Network() as net:
    ...
    ens = nengo.Ensemble(10, 2)
    p = nengo.Probe(ens)

n_inputs = 1000
minibatch_size = 20
n_steps = 10

with nengo_dl.Simulator(net, minibatch_size=minibatch_size) as sim:
    sim.train(targets={p: np.random.randn(n_inputs, n_steps, 2)},
              ...)

Note that these examples use random inputs/targets, for the sake of simplicity. In practice we would do something like targets={p: my_func(inputs)}, where my_func is a function specifying what the ideal outputs are for the given inputs.

optimizer

The optimizer is the algorithm that defines how to update the network parameters during training. Any of the optimization methods implemented in TensorFlow can be used in NengoDL; more information can be found in the TensorFlow documentation.

An instance of the desired TensorFlow optimizer is created (specifying any arguments required by that optimizer), and that instance is then passed to Simulator.train(). For example:

import tensorflow as tf

with nengo_dl.Simulator(net, ...) as sim:
    sim.train(optimizer=tf.train.MomentumOptimizer(
        learning_rate=1e-2, momentum=0.9, use_nesterov=True), ...)

objective

The goal in optimization is to minimize the error between the network’s actual outputs \(y\) and the targets \(t\). The objective is the function \(e = o(y, t)\) that computes an error value \(e\), given \(y\) and \(t\).

The default objective in NengoDL is the standard mean squared error. This will be used if the user doesn’t specify an objective.

Users can specify a custom objective by creating a function and passing that to the objective argument in Simulator.train(). Note that the objective is defined using TensorFlow operators. It should accept Tensors representing outputs and targets as input (each with shape (minibatch_size, n_steps, probe.size_in)) and return a scalar Tensor representing the error. This example manually computes mean squared error, rather than using the default:

import tensorflow as tf

def my_objective(outputs, targets):
    return tf.reduce_mean((targets - outputs) ** 2)

with nengo_dl.Simulator(net, ...) as sim:
    sim.train(objective=my_objective, ...)

Finally, it is also possible to pass None as the objective. This indicates that the error is being computed outside the simulation by the modeller. In this case the modeller should directly specify the output error gradient as the targets value. For example, we could apply the same mean squared error update this way:

with nengo_dl.Simulator(net, ...) as sim:
    sim.run(...)
    error = 2 * (sim.data[p] - my_targets)
    sim.train(targets=error, objective=None, ...)

If there are multiple output Probes defined in targets then by default the same objective will be used for all probes. This can be overridden by passing a dictionary with the form {my_probe0: my_objective0, my_probe1: my_objective1, ...} for the objective, specifying a different objective for each probe. In either case, the error will then be summed across the probes to produce an overall error value.

Note that Simulator.loss() can be used to check the loss (error) value for a given objective.

truncation

When optimizing a simulation over time we specify inputs and targets for all \(n\) steps of the simulation. The gradients are computed by running the simulation forward for \(n\) steps, comparing the outputs to the targets we specified, and then propagating the gradients backwards from \(n\) to 0. This is known as Backpropagation Through Time (BPTT).

However, in some cases we may not want to run BPTT over the full \(n\) steps (usually because it requires a lot of memory to store all the intermediate values for \(n\) steps of gradient calculation). In this case we choose some value \(m < n\), run the simulation for \(m\) steps, backpropagate the gradients over those \(m\) steps, then run the simulation for \(m\) more steps, and so on until we have run for the total \(n\) steps. This is known as Truncated BPTT.

The truncation argument is used to specify \(m\), i.e. sim.train(..., truncation=m). If no value is given then full un-truncated BPTT will be performed.

In general, truncated BPTT will result in worse performance than untruncated BPTT. Truncation limits the range of the temporal dynamics that the network is able to learn. For example, if we tried to learn a function where input \(x_t\) should influence the output at \(y_{t+m+1}\) that would not work well, because the errors from step \(t+m+1\) never make it back to step \(t\). More generally, a truncated system has less information about how outputs at \(t\) will affect future performance, which will limit how well that system can be optimized.

As mentioned, the main reason to use truncated BPTT is in order to reduce the memory demands during training. So if you find yourself running out of memory while training a model, consider using the truncation argument (while ensuring that the value of \(m\) is still large enough to capture the temporal dynamics in the task).

summaries

It is often useful to view information about how aspects of a model are changing over the course of training. TensorFlow has created TensorBoard to help visualize this kind of data, and the summaries argument can be used to specify the model data that you would like to export for TensorBoard.

It is specified as a list of objects for which we want to collect data. The data collected depends on the object: if it is a Connection then data will be collected about the distribution of the connection weights over the course of training; passing an Ensemble will collect data about the distribution of encoders, and Neurons will collect data about the distribution of biases. Additionally, the string "loss" can be passed, in which case the training error for the given objective will be collected over the course of training.

Alternatively, you can manually create summaries using tf.summary.* ops for any Tensors you would like to track (see the TensorFlow documentation), and include those in the summaries list.

TensorBoard can be used to view the exported data via the command

tensorboard --logdir <tensorboard_dir>

where tensorboard_dir is the value specified on Simulator creation via nengo_dl.Simulator(..., tensorboard=tensorboard_dir). After TensorBoard is running you can view the data by opening a web browser and navigating to http://localhost:6006.

For details on the usage of TensorBoard, consult the TensorFlow documentation. However, as a brief summary, you will find plots showing the loss values over the course of training in the Scalars tab at the top, and plots showing the distributions of weights/encoders/biases over time in the Distributions or Histograms tabs. If you call sim.train several times with the same summaries, each call will result in its own set of plots, with a suffix added to the label indicating the call number (e.g. label, label_1, label_2, ...). If you run your code multiple times with the same tensorboard_dir, data will be organized according to run number; you can turn on/off the plots for different runs using the checkboxes in the bottom left.

Other parameters

  • n_epochs (int): run training for this many passes through the input data
  • shuffle (bool): if True (default), randomly assign data to different minibatches each epoch
  • profile (bool or dict): collect profiling information (as in Simulator.run)

Choosing which elements to optimize

By default, NengoDL will optimize the following elements in a model:

  1. Connection weights (neuron–neuron weight matrices or decoders)
  2. Ensemble encoders
  3. Neuron biases

These elements will not be optimized if they are targeted by an online learning rule. For example, nengo.PES modifies connection weights as a model is running. If we also tried to optimize those weights with some offline training method then those two processes would conflict with each other, likely resulting in unintended effects. So NengoDL will assume that those elements should not be optimized.

Any of these default behaviours can be overridden using Nengo’s config system. Specifically, setting the trainable config attribute for an object will control whether or not it will be optimized.

configure_settings() is a utility function that can be used to add a configurable trainable attribute to the objects in a network. Setting trainable=None will use the defaults described above, or True/False can be passed to override the default for all objects in a model.

For example, suppose we only want to optimize one connection in our network, while leaving everything else unchanged. This could be achieved via

with nengo.Network() as net:
    # this adds the `trainable` attribute to all the trainable objects
    # in the network, and initializes it to `False`
    nengo_dl.configure_settings(trainable=False)

    a = nengo.Node([0])
    b = nengo.Ensemble(10, 1)
    c = nengo.Node(size_in=1)

    nengo.Connection(a, b)

    # make this specific connection trainable
    conn = nengo.Connection(b, c)
    net.config[conn].trainable = True

Or if we wanted to disable training for some subnetwork:

with nengo.Network() as net:
    nengo_dl.configure_settings(trainable=None)
    ...
    with nengo.Network() as subnet:
        net.config[subnet].trainable = False
        ...

Note that config[nengo.Ensemble].trainable controls both encoders and biases, as both are properties of an Ensemble. However, it is possible to separately control the biases via config[nengo.ensemble.Neurons].trainable or config[my_ensemble.neurons].trainable.

There are two important caveats to keep in mind when configuring trainable, which differ from the standard config behaviour:

  1. trainable applies to all objects in a network, regardless of whether they were created before or after trainable is set. For example,

    with nengo.Network() as net:
    ...
    net.config[nengo.Ensemble].trainable = False
    a = nengo.Ensemble(10, 1)
    ...

    is the same as

    with nengo.Network() as net:
    ...
    a = nengo.Ensemble(10, 1)
    net.config[nengo.Ensemble].trainable = False
    ...

  2. trainable can only be set on the config of the top-level network. For example,

    with nengo.Network() as net:
    nengo_dl.configure_settings(trainable=None)

    with nengo.Network() as subnet:
        my_ens = nengo.Ensemble(...)

        # incorrect
        subnet.config[my_ens].trainable = False

        # correct
        net.config[my_ens].trainable = False