Last week, we saw how to code a simple network from scratch , using nothing but torch tensors. Predictions, loss, gradients, weight updates – all these things we’ve been computing ourselves. Today, we make a significant change: Namely, we spare ourselves the cumbersome calculation of gradients, and have torch do it for us.

Prior to that though, let’s get some background.

Automatic differentiation with autograd#

torch uses a module called autograd to

  1. record operations performed on tensors, and

  2. store what will have to be done to obtain the corresponding gradients, once we’re entering the backward pass.

These prospective actions are stored internally as functions, and when it’s time to compute the gradients, these functions are applied in order: Application starts from the output node, and calculated gradients are successively propagated back through the network. This is a form of reverse mode automatic differentiation.

Autograd basics#

As users, we can see a bit of the implementation. As a prerequisite for this “recording” to happen, tensors have to be created with requires_grad = TRUE. For example:

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library(torch)

x <- torch_ones(2, 2, requires_grad = TRUE)

To be clear, x now is a tensor with respect to which gradients have to be calculated – normally, a tensor representing a weight or a bias, not the input data 1. If we subsequently perform some operation on that tensor, assigning the result to y,

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y <- x$mean()

we find that y now has a non-empty grad_fn that tells torch how to compute the gradient of y with respect to x:

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y$grad_fn

MeanBackward0

Actual computation of gradients is triggered by calling backward() on the output tensor.

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y$backward()

After backward() has been called, x has a non-null field termed grad that stores the gradient of y with respect to x:

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x$grad

torch_tensor 
 0.2500  0.2500
 0.2500  0.2500
[ CPUFloatType{2,2} ]

With longer chains of computations, we can take a glance at how torch builds up a graph of backward operations. Here is a slightly more complex example – feel free to skip if you’re not the type who just has to peek into things for them to make sense.

Digging deeper#

We build up a simple graph of tensors, with inputs x1 and x2 being connected to output out by intermediaries y and z.

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x1 <- torch_ones(2, 2, requires_grad = TRUE)
x2 <- torch_tensor(1.1, requires_grad = TRUE)

y <- x1 * (x2 + 2)

z <- y$pow(2) * 3

out <- z$mean()

To save memory, intermediate gradients are normally not being stored. Calling retain_grad() on a tensor allows one to deviate from this default. Let’s do this here, for the sake of demonstration:

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y$retain_grad()

z$retain_grad()

Now we can go backwards through the graph and inspect torch’s action plan for backprop, starting from out$grad_fn, like so:

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# how to compute the gradient for mean, the last operation executed
out$grad_fn

MeanBackward0

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# how to compute the gradient for the multiplication by 3 in z = y.pow(2) * 3
out$grad_fn$next_functions

[[1]]
MulBackward1

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# how to compute the gradient for pow in z = y.pow(2) * 3
out$grad_fn$next_functions[[1]]$next_functions

[[1]]
PowBackward0

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# how to compute the gradient for the multiplication in y = x * (x + 2)
out$grad_fn$next_functions[[1]]$next_functions[[1]]$next_functions

[[1]]
MulBackward0

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# how to compute the gradient for the two branches of y = x * (x + 2),
# where the left branch is a leaf node (AccumulateGrad for x1)
out$grad_fn$next_functions[[1]]$next_functions[[1]]$next_functions[[1]]$next_functions

[[1]]
torch::autograd::AccumulateGrad
[[2]]
AddBackward1

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# here we arrive at the other leaf node (AccumulateGrad for x2)
out$grad_fn$next_functions[[1]]$next_functions[[1]]$next_functions[[1]]$next_functions[[2]]$next_functions

[[1]]
torch::autograd::AccumulateGrad

If we now call out$backward(), all tensors in the graph will have their respective gradients calculated.

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out$backward()

z$grad
y$grad
x2$grad
x1$grad

torch_tensor 
 0.2500  0.2500
 0.2500  0.2500
[ CPUFloatType{2,2} ]
torch_tensor 
 4.6500  4.6500
 4.6500  4.6500
[ CPUFloatType{2,2} ]
torch_tensor 
 18.6000
[ CPUFloatType{1} ]
torch_tensor 
 14.4150  14.4150
 14.4150  14.4150
[ CPUFloatType{2,2} ]

After this nerdy excursion, let’s see how autograd makes our network simpler.

The simple network, now using autograd#

Thanks to autograd, we say good-bye to the tedious, error-prone process of coding backpropagation ourselves. A single method call does it all: loss$backward().

With torch keeping track of operations as required, we don’t even have to explicitly name the intermediate tensors any more. We can code forward pass, loss calculation, and backward pass in just three lines:

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y_pred <- x$mm(w1)$add(b1)$clamp(min = 0)$mm(w2)$add(b2)
  
loss <- (y_pred - y)$pow(2)$sum()

loss$backward()

Here is the complete code. We’re at an intermediate stage: We still manually compute the forward pass and the loss, and we still manually update the weights. Due to the latter, there is something I need to explain. But I’ll let you check out the new version first:

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library(torch)

### 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)


### initialize weights ---------------------------------------------------------

# dimensionality of hidden layer
d_hidden <- 32
# weights connecting input to hidden layer
w1 <- torch_randn(d_in, d_hidden, requires_grad = TRUE)
# weights connecting hidden to output layer
w2 <- torch_randn(d_hidden, d_out, requires_grad = TRUE)

# hidden layer bias
b1 <- torch_zeros(1, d_hidden, requires_grad = TRUE)
# output layer bias
b2 <- torch_zeros(1, d_out, requires_grad = TRUE)

### network parameters ---------------------------------------------------------

learning_rate <- 1e-4

### training loop --------------------------------------------------------------

for (t in 1:200) {
  ### -------- Forward pass --------
  
  y_pred <- x$mm(w1)$add(b1)$clamp(min = 0)$mm(w2)$add(b2)
  
  ### -------- compute loss -------- 
  loss <- (y_pred - y)$pow(2)$sum()
  if (t %% 10 == 0)
    cat("Epoch: ", t, "   Loss: ", loss$item(), "\n")
  
  ### -------- Backpropagation --------
  
  # compute gradient of loss w.r.t. all tensors with requires_grad = TRUE
  loss$backward()
  
  ### -------- Update weights -------- 
  
  # Wrap in with_no_grad() because this is a part we DON'T 
  # want to record for automatic gradient computation
   with_no_grad({
     w1 <- w1$sub_(learning_rate * w1$grad)
     w2 <- w2$sub_(learning_rate * w2$grad)
     b1 <- b1$sub_(learning_rate * b1$grad)
     b2 <- b2$sub_(learning_rate * b2$grad)  
     
     # Zero gradients after every pass, as they'd accumulate otherwise
     w1$grad$zero_()
     w2$grad$zero_()
     b1$grad$zero_()
     b2$grad$zero_()  
   })

}

As explained above, after some_tensor$backward(), all tensors preceding it in the graph2 will have their grad fields populated. We make use of these fields to update the weights. But now that autograd is “on”, whenever we execute an operation we don’t want recorded for backprop, we need to explicitly exempt it: This is why we wrap the weight updates in a call to with_no_grad().

While this is something you may file under “nice to know” – after all, once we arrive at the last post in the series, this manual updating of weights will be gone – the idiom of zeroing gradients is here to stay: Values stored in grad fields accumulate; whenever we’re done using them, we need to zero them out before reuse.

Outlook#

So where do we stand? We started out coding a network completely from scratch, making use of nothing but torch tensors. Today, we got significant help from autograd.

But we’re still manually updating the weights, – and aren’t deep learning frameworks known to provide abstractions (“layers”, or: “modules”) on top of tensor computations …?

We address both issues in the follow-up installments. Thanks for reading!

  1. Unless we want to change the data, as when generating adversarial examples. ↩︎

  2. All that have requires_grad set to TRUE, to be precise. ↩︎