Initially,
we started learning about torch basics by coding a simple neural
network from scratch, making use of just a single of torch’s features:
tensors.
Then,
we immensely simplified the task, replacing manual backpropagation with
autograd. Today, we modularize the network – in both the habitual
and a very literal sense: Low-level matrix operations are swapped out
for torch modules.
Modules
From other frameworks (Keras, say), you may be used to distinguishing
between models and layers. In torch, both are instances of
nn_Module(), and thus, have some methods in common. For those thinking
in terms of “models” and “layers”, I’m artificially splitting up this
section into two parts. In reality though, there is no dichotomy: New
modules may be composed of existing ones up to arbitrary levels of
recursion.
Base modules (“layers”)
Instead of writing out an affine operation by hand – x$mm(w1) + b1,
say –, as we’ve been doing so far, we can create a linear module. The
following snippet instantiates a linear layer that expects three-feature
inputs and returns a single output per observation:
The module has two parameters, “weight” and “bias”. Both now come
pre-initialized:
$weight
torch_tensor
-0.0385 0.1412 -0.5436
[ CPUFloatType{1,3} ]
$bias
torch_tensor
-0.1950
[ CPUFloatType{1} ]Modules are callable; calling a module executes its forward() method,
which, for a linear layer, matrix-multiplies input and weights, and adds
the bias.
Let’s try this:
data <- torch_randn(10, 3)
out <- l(data)Unsurprisingly, out now holds some data:
torch_tensor
0.2711
-1.8151
-0.0073
0.1876
-0.0930
0.7498
-0.2332
-0.0428
0.3849
-0.2618
[ CPUFloatType{10,1} ]In addition though, this tensor knows what will need to be done, should
ever it be asked to calculate gradients:
AddmmBackwardNote the difference between tensors returned by modules and self-created
ones. When creating tensors ourselves, we need to pass
requires_grad = TRUE to trigger gradient calculation. With modules,
torch correctly assumes that we’ll want to perform backpropagation at
some point.
By now though, we haven’t called backward() yet. Thus, no gradients
have yet been computed:
l$weight$grad
l$bias$gradtorch_tensor
[ Tensor (undefined) ]
torch_tensor
[ Tensor (undefined) ]Let’s change this:
Error in (function (self, gradient, keep_graph, create_graph) :
grad can be implicitly created only for scalar outputs (_make_grads at ../torch/csrc/autograd/autograd.cpp:47)Why the error? Autograd expects the output tensor to be a scalar,
while in our example, we have a tensor of size (10, 1). This error
won’t often occur in practice, where we work with batches of inputs
(sometimes, just a single batch). But still, it’s interesting to see how
to resolve this.
To make the example work, we introduce a – virtual – final aggregation
step – taking the mean, say. Let’s call it avg. If such a mean were
taken, its gradient with respect to l$weight would be obtained via the
chain rule:
\[
\begin{equation*}
\frac{\partial \ avg}{\partial w} = \frac{\partial \ avg}{\partial \ out} \ \frac{\partial \ out}{\partial w}
\end{equation*}
\]
Of the quantities on the right side, we’re interested in the second. We
need to provide the first one, the way it would look if really we were
taking the mean:
d_avg_d_out <- torch_tensor(10)$`repeat`(10)$unsqueeze(1)$t()
out$backward(gradient = d_avg_d_out)Now, l$weight$grad and l$bias$grad do contain gradients:
l$weight$grad
l$bias$gradtorch_tensor
1.3410 6.4343 -30.7135
[ CPUFloatType{1,3} ]
torch_tensor
100
[ CPUFloatType{1} ]In addition to nn_linear() , torch provides pretty much all the
common layers you might hope for. But few tasks are solved by a single
layer. How do you combine them? Or, in the usual lingo: How do you build
models?
Container modules (“models”)
Now, models are just modules that contain other modules. For example,
if all inputs are supposed to flow through the same nodes and along the
same edges, then nn_sequential() can be used to build a simple graph.
For example:
model <- nn_sequential(
nn_linear(3, 16),
nn_relu(),
nn_linear(16, 1)
)We can use the same technique as above to get an overview of all model
parameters (two weight matrices and two bias vectors):
$`0.weight`
torch_tensor
-0.1968 -0.1127 -0.0504
0.0083 0.3125 0.0013
0.4784 -0.2757 0.2535
-0.0898 -0.4706 -0.0733
-0.0654 0.5016 0.0242
0.4855 -0.3980 -0.3434
-0.3609 0.1859 -0.4039
0.2851 0.2809 -0.3114
-0.0542 -0.0754 -0.2252
-0.3175 0.2107 -0.2954
-0.3733 0.3931 0.3466
0.5616 -0.3793 -0.4872
0.0062 0.4168 -0.5580
0.3174 -0.4867 0.0904
-0.0981 -0.0084 0.3580
0.3187 -0.2954 -0.5181
[ CPUFloatType{16,3} ]
$`0.bias`
torch_tensor
-0.3714
0.5603
-0.3791
0.4372
-0.1793
-0.3329
0.5588
0.1370
0.4467
0.2937
0.1436
0.1986
0.4967
0.1554
-0.3219
-0.0266
[ CPUFloatType{16} ]
$`2.weight`
torch_tensor
Columns 1 to 10-0.0908 -0.1786 0.0812 -0.0414 -0.0251 -0.1961 0.2326 0.0943 -0.0246 0.0748
Columns 11 to 16 0.2111 -0.1801 -0.0102 -0.0244 0.1223 -0.1958
[ CPUFloatType{1,16} ]
$`2.bias`
torch_tensor
0.2470
[ CPUFloatType{1} ]To inspect an individual parameter, make use of its position in the
sequential model. For example:
torch_tensor
-0.3714
0.5603
-0.3791
0.4372
-0.1793
-0.3329
0.5588
0.1370
0.4467
0.2937
0.1436
0.1986
0.4967
0.1554
-0.3219
-0.0266
[ CPUFloatType{16} ]And just like nn_linear() above, this module can be called directly on
data:
On a composite module like this one, calling backward() will
backpropagate through all the layers:
out$backward(gradient = torch_tensor(10)$`repeat`(10)$unsqueeze(1)$t())
# e.g.
model[[1]]$bias$gradtorch_tensor
0.0000
-17.8578
1.6246
-3.7258
-0.2515
-5.8825
23.2624
8.4903
-2.4604
6.7286
14.7760
-14.4064
-1.0206
-1.7058
0.0000
-9.7897
[ CPUFloatType{16} ]And placing the composite module on the GPU will move all tensors there:
model$cuda()
model[[1]]$bias$gradtorch_tensor
0.0000
-17.8578
1.6246
-3.7258
-0.2515
-5.8825
23.2624
8.4903
-2.4604
6.7286
14.7760
-14.4064
-1.0206
-1.7058
0.0000
-9.7897
[ CUDAFloatType{16} ]Now let’s see how using nn_sequential() can simplify our example
network.
Simple network using modules
### generate training data -----------------------------------------------------
# input dimensionality (number of input features)
d_in <- 3
# output dimensionality (number of predicted features)
d_out <- 1
# number of observations in training set
n <- 100
# create random data
x <- torch_randn(n, d_in)
y <- x[, 1, NULL] * 0.2 - x[, 2, NULL] * 1.3 - x[, 3, NULL] * 0.5 + torch_randn(n, 1)
### define the network ---------------------------------------------------------
# dimensionality of hidden layer
d_hidden <- 32
model <- nn_sequential(
nn_linear(d_in, d_hidden),
nn_relu(),
nn_linear(d_hidden, d_out)
)
### network parameters ---------------------------------------------------------
learning_rate <- 1e-4
### training loop --------------------------------------------------------------
for (t in 1:200) {
### -------- Forward pass --------
y_pred <- model(x)
### -------- compute loss --------
loss <- (y_pred - y)$pow(2)$sum()
if (t %% 10 == 0)
cat("Epoch: ", t, " Loss: ", loss$item(), "\n")
### -------- Backpropagation --------
# Zero the gradients before running the backward pass.
model$zero_grad()
# compute gradient of the loss w.r.t. all learnable parameters of the model
loss$backward()
### -------- Update weights --------
# Wrap in with_no_grad() because this is a part we DON'T want to record
# for automatic gradient computation
# Update each parameter by its `grad`
with_no_grad({
model$parameters %>% purrr::walk(function(param) param$sub_(learning_rate * param$grad))
})
}The forward pass looks a lot better now; however, we still loop through
the model’s parameters and update each one by hand. Furthermore, you may
be already be suspecting that torch provides abstractions for common
loss functions. In the next and last installment of this series, we’ll
address both points, making use of torch losses and optimizers. See
you then!