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4.2. PopXL Custom Optimiser


We saw in the PopXL basics tutorial that we can easily create graphs with an internal state using the addons.Module class and we used modules to implement dense layers of a simple neural network.

In this tutorial, we will learn how to implement a custom optimiser: the Adam optimiser. Many optimisers, like layers, must manage some persistent variables. To manage this internal state we will re-use the same programming pattern with addons.Module as a base class to the optimiser.

Once you’ve finished this tutorial, you will:

  • be able to write your own custom optimiser for your models.

  • understand graph caching and how it helps subgraph reuse.

  • have used some of the built-in rules within the popxl.ops.var_updates module which are useful for a variety of tasks including updating your optimiser’s internal state variables.

If you are unfamiliar with PopXL, you may want to try out the tutorial covering the basic concepts. You may also want to refer to the PopXL user guide.


  1. Install a Poplar SDK (version 2.6 or later) and source the scripts for both PopART and Poplar as described in the Getting Started guide for your IPU system.

  2. Install system dependencies: apt-get install -y $(< required_apt_packages.txt)

  3. Create a Python virtual environment: python3 -m venv <virtual_env>.

  4. Activate the virtual environment: . <virtual_env>/bin/activate.

  5. Update pip: pip3 install --upgrade pip.

  6. Install requirements pip3 install -r requirements.txt (this will also install popxl.addons).

python3 -m venv virtual_env
. virtual_env/bin/activate
pip3 install --upgrade pip
pip3 install -r requirements.txt

To run the Jupyter Notebook version of this tutorial:

  1. Install a Poplar SDK (version 2.6 or later) and source the scripts for both PopART and Poplar as described in the Getting Started guide for your IPU system.

  2. Create a virtual environment.

  3. In the same virtual environment, install the Jupyter Notebook server: python -m pip install jupyter.

  4. Launch a Jupyter Server on a specific port: jupyter-notebook --no-browser --port <port number>. Be sure to be in the virtual environment.

  5. Connect via SSH to your remote machine, forwarding your chosen port: ssh -NL <port number>:localhost:<port number> <your username>@<remote machine>.

For more details about this process, or if you need troubleshooting, see our guide on using IPUs from Jupyter Notebooks.

If using VS Code, Intellisense can help you understand the tutorial code. It will show function and class descriptions when hovering over their names and lets you easily jump to their definitions. Consult the VS Code setup guide to use Intellisense for this tutorial.


We start by importing all the modules we will need for this tutorial:

from typing import Dict, Mapping, Optional, Union
from functools import partial

import numpy as np
import torch
import torchvision
from tqdm import tqdm

import popxl
import popxl_addons as addons
import popxl.ops as ops


Defining the Adam optimiser

Below, we implement the Adam optimiser by deriving from the addons.Module class. The Adam class defines the update step for a single variable in its build method. First we will see how to correctly deal with in-place operations, then we will define the update step using PopXL’s var_updates module, and, finally we will test and inspect our optimiser on a single variable.

Managing in-place ops

The update is performed in-place on the weight argument which contains the model variable updated by the Adam optimiser. Because this operation is in-place, we pass the argument as a TensorByRef and use the @popxl.in_sequence() decorator to prevent the operations from being rearranged by the compiler. The rest of the definition follows the same pattern used to add weights to our layers in the PopXL basics tutorial.

class Adam(addons.Module):
    # We need to specify `in_sequence` because many operations are in-place
    # and their order shouldn't be changed
    def build(
        weight: popxl.TensorByRef,
        grad: popxl.Tensor,
        lr: Union[float, popxl.Tensor],
        beta1: Union[float, popxl.Tensor] = 0.9,
        beta2: Union[float, popxl.Tensor] = 0.999,
        eps: Union[float, popxl.Tensor] = 1e-5,
        weight_decay: Union[float, popxl.Tensor] = 0.0,
        first_order_dtype: popxl.dtype = popxl.float16,
        bias_correction: bool = True,
        # Gradient estimator for the variable `weight` - same shape as the variable
        first_order = self.add_variable_input(
            partial(np.zeros, weight.shape),
        ops.var_updates.accumulate_moving_average_(first_order, grad, f=beta1)

        # Variance estimator for the variable `weight` - same shape as the variable
        second_order = self.add_variable_input(
            "second_order", partial(np.zeros, weight.shape), popxl.float32, by_ref=True
        ops.var_updates.accumulate_moving_average_square_(second_order, grad, f=beta2)

        # Adam is a biased estimator: provide the step variable to correct bias
        step = None
        if bias_correction:
            step = self.add_variable_input(
                "step", partial(np.zeros, ()), popxl.float32, by_ref=True

        # Calculate the weight increment with an Adam heuristic
        # Here we use the built-in `adam_updater`, but you can write your own.
        dw = ops.var_updates.adam_updater(

        # in-place weight update: weight += (-lr)*dw
        ops.scaled_add_(weight, dw, b=-lr)

The Adam optimiser needs state to store the mean and uncentred variance (first and second moments) of the gradients. These need to be of type Variable, hence we add them with Module.add_variable_input, creating named inputs for them (first_order and second_order).

We used Module.add_variable_input in the PopXL introductory tutorial to add weights to our layers. However, in the Adam implementation you should notice a few differences.

  • We now have a @popxl.in_sequence() decorator on top of the build method. This forces all operations to be added in the exact order we define them, enforcing topological constraints between them. This is necessary here since most of the optimiser operations are in-place, hence their order of execution must be strictly preserved. Remember this whenever you have in-place operations.

  • The weight input is a popxl.TensorByRef: any change made to this variable will be automatically copied to the parent graph. See TensorByRef for more information.

  • Some parameters, such as learning rate or weight decay, are defined as Union[float, popxl.Tensor]. If the parameter was provided as a simple float, it would be “baked” into the graph, with no possibility of changing it at run-time. Instead, if the parameter is a Tensor (or TensorSpec) it will appear as an input to the graph, which needs to be provided when calling the graph. If you plan to change a parameter (for example, because you have a learning rate schedule), this is the way to go.

The rest of the logic is straightforward:

  • We update first_order, the mean estimator of the weight gradient.

  • We update second_order, the uncentred variance estimator of weight gradient.

  • We optionally correct the estimators, since they are biased.

  • We compute the increment delta-weight dw.

  • We update the variable weight with scaled_add_ to implement the equation weight -= lr * dw.

Using the var_updates module

The ops.var_updates module contains several useful update rules (you can also create your own). In this example, we will use three of the built-in rules:

  • ops.var_updates.accumulate_moving_average_(average, new_sample, coefficient) updates average in-place with an exponential moving average rule:

    average = (coefficient * average) + ((1 - coefficient) * new_sample)
  • accumulate_moving_average_square_(average, new_sample, coefficient) updates average in-place but uses the square of the sample.

  • ops.var_updates.adam_updater(...) returns the Adam increment dw required for the weight update. This is computed using the Adam internal state which comprises of the first and second moments.

Using our custom optimiser

Let’s inspect the optimiser graph and its use in a simple example.

ir = popxl.Ir(replication=1)

with ir.main_graph:
    var = popxl.variable(np.ones((2, 2)), popxl.float32)
    grad = popxl.variable(np.full((2, 2), 0.1), popxl.float32)

    # create graph and factories - float learning rate
    adam_facts, adam = Adam(cache=True).create_graph(var, var.spec, lr=1e-3)

    # create graph and factories - Tensor learning rate
    adam_facts_lr, adam_lr = Adam().create_graph(
        var, var.spec, lr=popxl.TensorSpec((), popxl.float32)

    print("Adam with float learning rate\n")
    print("\n Adam with tensor learning rate\n")

    # instantiate optimiser variables
    adam_state = adam_facts.init()
    adam_state_lr = adam_facts_lr.init()

    # optimization step for float lr: call the bound graph providing the
    # variable to update and the gradient
    adam.bind(adam_state).call(var, grad)

    # optimization step for tensor lr: call the bound graph providing the
    # variable to update, the gradient and the learning rate
    adam_lr.bind(adam_state_lr).call(var, grad, popxl.constant(1e-3))

ir.num_host_transfers = 1
session = popxl.Session(ir, "ipu_hw")
print("\n Before Adam update")
var_data = session.get_tensor_data(var)
state = session.get_tensors_data(adam_state.tensors)
print("Variable:\n", var)
print("Adam state:")
for name, data in state.items():
    print(name, "\n", state[name])

with session:

print("\n After Adam update")
var_data = session.get_tensor_data(var)
state = session.get_tensors_data(adam_state.tensors)
print("Variable:\n", var)
print("Adam state:")
for name, data in state.items():
    print(name, "\n", state[name])
Adam with float learning rate

Graph : Adam_subgraph(0)
  (%1, %2, first_order=%3, second_order=%4, step=%5) -> () {
    Accumulate.100 (%3 [(2, 2) float16], %2 [(2, 2) float32]) -> (%6 [(2, 2) float16])
    Accumulate.101 (%4 [(2, 2) float32], %2 [(2, 2) float32]) -> (%7 [(2, 2) float32])
    AdamUpdater.102 (%3 [(2, 2) float16], %4 [(2, 2) float32], %5 [() float32]) -> (%8 [(2, 2) float16])
    ScaledAddLhsInplace.103 (%1 [(2, 2) float32], %8 [(2, 2) float16]) -> (%9 [(2, 2) float32])

 Adam with tensor learning rate

Graph : Adam_subgraph(1)
  (%1, %2, %3, first_order=%4, second_order=%5, step=%6) -> () {
    Accumulate.104 (%4 [(2, 2) float16], %2 [(2, 2) float32]) -> (%7 [(2, 2) float16])
    Accumulate.105 (%5 [(2, 2) float32], %2 [(2, 2) float32]) -> (%8 [(2, 2) float32])
    AdamUpdater.106 (%4 [(2, 2) float16], %5 [(2, 2) float32], %6 [() float32]) -> (%9 [(2, 2) float16])
    Neg.107 (%3 [() float32]) -> (%10 [() float32])
    ScaledAddLhsInplace.108 (%1 [(2, 2) float32], %9 [(2, 2) float16], %10 [() float32]) -> (%11 [(2, 2) float32])

 Before Adam update
 Tensor[t popxl.dtypes.float32 (2, 2)]
Adam state:
Tensor[first_order popxl.dtypes.float16 (2, 2)]
 [[0. 0.]
 [0. 0.]]
Tensor[second_order popxl.dtypes.float32 (2, 2)]
 [[0. 0.]
 [0. 0.]]
Tensor[step popxl.dtypes.float32 ()]

 After Adam update
 Tensor[t popxl.dtypes.float32 (2, 2)]
Adam state:
Tensor[first_order popxl.dtypes.float16 (2, 2)]
 [[0.009995 0.009995]
 [0.009995 0.009995]]
Tensor[second_order popxl.dtypes.float32 (2, 2)]
 [[9.9998715e-06 9.9998715e-06]
 [9.9998715e-06 9.9998715e-06]]
Tensor[step popxl.dtypes.float32 ()]

MNIST with Adam

We can now refactor our MNIST example to incorporate the Adam optimiser. Note that we need an optimiser for each variable: we first define the optimiser_step function which creates the graph for each variable and performs a full weight update for all the variables. Since the Adam module uses cache=True, if two graphs happens to be the same, the same graph will be re-used.

We will use a simple float learning rate (rather than Tensor), since we don’t plan to change its value during training.

def optimiser_step(
    grads: Dict[popxl.Tensor, popxl.Tensor],
    optimiser: addons.Module,
    learning_rate: popxl.float32 = 1e-3,
    Update all variables creating per-variable optimisers.
    for name, var in variables.named_tensors.items():
        # Create optimiser and state factories for the variable
        opt_facts, opt_graph = optimiser.create_graph(
            var, var.spec, lr=learning_rate, weight_decay=0.0, bias_correction=False
        state = opt_facts.init()

        # Bind the graph to its state and call it.
        # Both the state and the variables are updated in-place and are passed
        # by ref, hence after the graph is called they are updated.
        opt_graph.bind(state).call(var, grads[var])

We load the data, and define our network using exactly the same code as in the PopXL basics tutorial:

def get_mnist_data(test_batch_size: int, batch_size: int):
    training_data =
                    # mean and std computed on the training set.
                    torchvision.transforms.Normalize((0.1307,), (0.3081,)),

    validation_data =
                    torchvision.transforms.Normalize((0.1307,), (0.3081,)),

    return training_data, validation_data

class Linear(addons.Module):
    def __init__(self, out_features: int, bias: bool = True):
        self.out_features = out_features
        self.bias = bias

    def build(self, x: popxl.Tensor) -> popxl.Tensor:
        # add a state variable to the module
        w = self.add_variable_input(
            partial(np.random.normal, 0, 0.02, (x.shape[-1], self.out_features)),
        y = x @ w
        if self.bias:
            # add a state variable to the module
            b = self.add_variable_input("bias", partial(np.zeros, y.shape[-1]), x.dtype)
            y = y + b

        return y

class Net(addons.Module):
    def __init__(self, cache: Optional[addons.GraphCache] = None):
        self.fc1 = Linear(512)
        self.fc2 = Linear(512)
        self.fc3 = Linear(512)
        self.fc4 = Linear(10)

    def build(self, x: popxl.Tensor):
        x = x.reshape((-1, 28 * 28))
        x = ops.gelu(self.fc1(x))
        x = ops.gelu(self.fc2(x))
        x = ops.gelu(self.fc3(x))
        x = self.fc4(x)
        return x

The training code is almost unchanged from that of the PopXL basics tutorial, the only difference being that we now call our Adam class and optimiser_step function as the optimiser instead of the simple scaled_add_:

def train_program(batch_size, device, learning_rate):
    ir = popxl.Ir(replication=1)

    with ir.main_graph:
        # Create input streams from host to device
        img_stream = popxl.h2d_stream((batch_size, 28, 28), popxl.float32, "image")
        img_t = ops.host_load(img_stream)  # load data
        label_stream = popxl.h2d_stream((batch_size,), popxl.int32, "labels")
        labels = ops.host_load(label_stream, "labels")

        # Create forward graph
        facts, fwd_graph = Net().create_graph(img_t)

        # Create backward graph via autodiff transform
        bwd_graph = addons.autodiff(fwd_graph)

        # Initialise variables (weights)
        variables = facts.init()

        # Call the forward graph with call_with_info because we want to retrieve
        # information from the call site
        fwd_info = fwd_graph.bind(variables).call_with_info(img_t)
        x = fwd_info.outputs[0]  # forward output

        # Compute loss and starting gradient for backpropagation
        loss, dx = addons.ops.cross_entropy_with_grad(x, labels)

        # Setup a stream to retrieve loss values from the host
        loss_stream = popxl.d2h_stream(loss.shape, loss.dtype, "loss")
        ops.host_store(loss_stream, loss)

        # Retrieve activations from the forward graph
        activations = bwd_graph.grad_graph_info.inputs_dict(fwd_info)

        # Call the backward graph providing the starting value for
        # backpropagation and activations
        bwd_info = bwd_graph.call_with_info(dx, args=activations)

        # Adam optimiser, with cache
        grads_dict = bwd_graph.grad_graph_info.fwd_parent_ins_to_grad_parent_outs(
            fwd_info, bwd_info
        optimiser = Adam(cache=True)
        optimiser_step(variables, grads_dict, optimiser, learning_rate)

    ir.num_host_transfers = 1
    return popxl.Session(ir, device), [img_stream, label_stream], variables, loss_stream

You will notice above that we created the Adam module using cache=True. This will enable graph reuse, if possible, when calling optimiser.create_graph. For our optimiser this would be when there are multiple variables with the same shape and data type.

Now let’s run a training session.

Since we are using the Adam optimiser, we need to use a smaller learning rate than before.

train_batch_size = 8
test_batch_size = 80
device = "ipu_hw"
learning_rate = 1e-3
epochs = 1

training_data, test_data = get_mnist_data(test_batch_size, train_batch_size)
train_session, train_input_streams, train_variables, loss_stream = train_program(
    train_batch_size, device, learning_rate

num_batches = len(training_data)
with train_session:
    for epoch in range(1, epochs + 1):
        print(f"Epoch {epoch}/{epochs}")
        bar = tqdm(training_data, total=num_batches)
        for data, labels in bar:
            inputs: Mapping[popxl.HostToDeviceStream, np.ndarray] = dict(
                zip(train_input_streams, [data.squeeze().float(),])
            loss =[loss_stream]
Epoch 1/1
Loss:0.0191: 100%|██████████| 7500/7500 [00:26<00:00, 284.61it/s]
# Retrieve the trained weights to use during inference
train_vars_to_data = train_session.get_tensors_data(train_variables.tensors)


As we did previously, to test our model we need to create an inference-only program and run it on the test dataset.

def test_program(test_batch_size, device):
    ir = popxl.Ir(replication=1)

    with ir.main_graph:
        # Inputs
        in_stream = popxl.h2d_stream((test_batch_size, 28, 28), popxl.float32, "image")
        in_t = ops.host_load(in_stream)

        # Create graphs
        facts, graph = Net().create_graph(in_t)

        # Initialise variables
        variables = facts.init()

        # Forward
        (outputs,) = graph.bind(variables).call(in_t)
        out_stream = popxl.d2h_stream(outputs.shape, outputs.dtype, "outputs")
        ops.host_store(out_stream, outputs)

    ir.num_host_transfers = 1
    return popxl.Session(ir, device), [in_stream], variables, out_stream

We create the test program and copy the trained weights to it:

test_session, test_input_streams, test_variables, out_stream = test_program(
    test_batch_size, device

train_vars_to_test_vars = train_variables.to_mapping(test_variables)

test_vars_to_data = {
    test_var: train_vars_to_data[train_var].copy()
    for train_var, test_var in train_vars_to_test_vars.items()


Finally, let’s run the test session and measure the accuracy:

def accuracy(predictions: np.ndarray, labels: np.ndarray):
    ind = np.argmax(predictions, axis=-1).flatten()
    labels = labels.detach().numpy().flatten()
    return np.mean(ind == labels) * 100.0

num_batches = len(test_data)
sum_acc = 0.0
with test_session:
    for data, labels in tqdm(test_data, total=num_batches):
        inputs: Mapping[popxl.HostToDeviceStream, np.ndarray] = dict(
            zip(test_input_streams, [data.squeeze().float(),])
        output =
        sum_acc += accuracy(output[out_stream], labels)

test_set_accuracy = sum_acc / len(test_data)
print(f"Accuracy on test set: {test_set_accuracy:0.2f}%")
100%|██████████| 125/125 [00:01<00:00, 95.30it/s]
Accuracy on test set: 95.21%


In this tutorial we wrote a custom optimiser using the popxl.addons API. We achieved the following:

  • built an Adam Optimiser (by subclassing addons.Module) and ran it with an MNIST model.

  • used the popxl.in_sequence() function and learnt why it is needed, to prevent operations from being rearranged by the compiler.

  • used popxl.TensorByRef to pass variable updates back to the parent graph.

  • explored some of the functions within the popxl.ops.var_updates module.

  • became familiar with using Tensor parameters versus simpler (built-in) types in, when declaring dynamic parameters.

  • exploited graph caching in addons.Module: addons.Module(cache=True) to enable graph reuse.

To try out more features in PopXL look at our other tutorials.

You can also read our PopXL User Guide for more information.

As the PopXL API is still experimental, we would love to hear your feedback on it ( Your input could help drive its future direction.

Generated:2022-07-21T18:56 SDK:2.6.0+1074 SST:0.0.7