In the first tutorial, 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. In this section, 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:

library(torch)
l <- nn_linear(3, 1)

The module has two parameters, “weight” and “bias”. Both now come pre-initialized:

l$parameters

## $weight
## torch_tensor
##  0.2746  0.1806  0.5658
## [ CPUFloatType{1,3} ]
## 
## $bias
## torch_tensor
##  0.1348
## [ 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:

out$data()

## torch_tensor
##  0.0457
## -0.2944
##  0.1046
##  0.8796
## -0.3529
## -0.2735
## -0.3491
## -1.2709
## -0.2419
##  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:

out$grad_fn

## AddmmBackward

Note 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

## torch_tensor
## [ Tensor (undefined) ]

l$bias$grad

## torch_tensor
## [ Tensor (undefined) ]

Let’s change this:

# out$backward() # error

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

## torch_tensor
## -64.5866  -4.5878 -17.3627
## [ CPUFloatType{1,3} ]

l$bias$grad

## 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):

model$parameters

## $`0.weight`
## torch_tensor
##  0.5432 -0.1786  0.2262
## -0.4251 -0.2114 -0.3005
##  0.5404  0.5167  0.4031
## -0.2928  0.2713 -0.5621
## -0.1538  0.3970  0.3493
## -0.4646  0.3377  0.0653
##  0.3810 -0.0525  0.0309
## -0.0062 -0.2525 -0.2091
## -0.0372 -0.2401 -0.0344
## -0.0682  0.0348 -0.3300
##  0.5520  0.3174 -0.5156
## -0.0116  0.1722  0.3682
## -0.1925 -0.3215  0.3936
## -0.1558 -0.3241  0.1210
##  0.0903 -0.3383  0.3782
##  0.2078 -0.3504  0.2801
## [ CPUFloatType{16,3} ]
## 
## $`0.bias`
## torch_tensor
## -0.2698
## -0.1447
## -0.2124
## -0.2239
##  0.1827
## -0.1008
## -0.0195
## -0.3044
##  0.1232
## -0.3131
## -0.2504
##  0.3246
## -0.2733
## -0.5319
## -0.4463
## -0.5719
## [ CPUFloatType{16} ]
## 
## $`2.weight`
## torch_tensor
## Columns 1 to 10-0.0515 -0.1148 -0.0378 -0.0299 -0.1381  0.1993 -0.0235 -0.0346 -0.0807 -0.2410
## 
## Columns 11 to 16-0.1232 -0.1432  0.2147  0.2096  0.0261  0.2043
## [ CPUFloatType{1,16} ]
## 
## $`2.bias`
## torch_tensor
## -0.2238
## [ CPUFloatType{1} ]

To inspect an individual parameter, make use of its position in the sequential model. For example:

model[[1]]$bias

## torch_tensor
## -0.2698
## -0.1447
## -0.2124
## -0.2239
##  0.1827
## -0.1008
## -0.0195
## -0.3044
##  0.1232
## -0.3131
## -0.2504
##  0.3246
## -0.2733
## -0.5319
## -0.4463
## -0.5719
## [ CPUFloatType{16} ]

And just like nn_linear() above, this module can be called directly on data:

out <- model(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]]$bias$grad

## torch_tensor
##  -1.0297
##  -9.1879
##  -0.3781
##  -1.1946
##  -9.6673
##  11.9571
##  -0.4693
##  -0.6913
##  -5.6459
##  -2.4103
##  -3.6958
## -11.4523
##   8.5864
##   4.1915
##   0.0000
##   0.0000
## [ CPUFloatType{16} ]

And placing the composite module on the GPU will move all tensors there:

model$cuda()
model[[1]]$bias$grad

## torch_tensor
##  -1.0297
##  -9.1879
##  -0.3781
##  -1.1946
##  -9.6673
##  11.9571
##  -0.4693
##  -0.6913
##  -5.6459
##  -2.4103
##  -3.6958
## -11.4523
##   8.5864
##   4.1915
##   0.0000
##   0.0000
## [ 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))
  })
  
}

## Epoch:  10    Loss:  180.9211 
## Epoch:  20    Loss:  142.5852 
## Epoch:  30    Loss:  118.842 
## Epoch:  40    Loss:  104.6096 
## Epoch:  50    Loss:  96.36818 
## Epoch:  60    Loss:  91.74362 
## Epoch:  70    Loss:  89.16847 
## Epoch:  80    Loss:  87.71291 
## Epoch:  90    Loss:  86.8581 
## Epoch:  100    Loss:  86.31103 
## Epoch:  110    Loss:  85.92104 
## Epoch:  120    Loss:  85.61875 
## Epoch:  130    Loss:  85.36913 
## Epoch:  140    Loss:  85.14122 
## Epoch:  150    Loss:  84.93426 
## Epoch:  160    Loss:  84.73933 
## Epoch:  170    Loss:  84.55376 
## Epoch:  180    Loss:  84.37543 
## Epoch:  190    Loss:  84.2056 
## Epoch:  200    Loss:  84.03828

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 final tutorial, we’ll address both points, making use of torch losses and optimizers.