In the first tutorial, 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. Here, 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:
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,
y <- x$mean()
we find that y now has a non-empty grad_fn that tells torch how to
compute the gradient of y with respect to x:
y$grad_fn
## MeanBackward0
Actual computation of gradients is triggered by calling backward()
on the output tensor.
y$backward()
After backward() has been called, x has a non-null field termed
grad that stores the gradient of y with respect to x:
x$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.
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:
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$grad_fn, like so:
# how to compute the gradient for mean, the last operation executed
out$grad_fn
## MeanBackward0
# how to compute the gradient for the multiplication by 3 in z = y.pow(2) * 3
out$grad_fn$next_functions
## [[1]]
## MulBackward1
# how to compute the gradient for pow in z = y.pow(2) * 3
out$grad_fn$next_functions[[1]]$next_functions
## [[1]]
## PowBackward0
# how to compute the gradient for the multiplication in y = x * (x + 2)
out$grad_fn$next_functions[[1]]$next_functions[[1]]$next_functions
## [[1]]
## MulBackward0
# how to compute the gradient for the two branches of y = x * (x + 2),
# where the left branch is a leaf node (AccumulateGrad for x1)
out$grad_fn$next_functions[[1]]$next_functions[[1]]$next_functions[[1]]$next_functions
## [[1]]
## torch::autograd::AccumulateGrad
## [[2]]
## AddBackward1
# 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$backward(), all tensors in the graph will have
their respective gradients calculated.
out$backward()
z$grad
## torch_tensor
## 0.2500 0.2500
## 0.2500 0.2500
## [ CPUFloatType{2,2} ]
y$grad
## torch_tensor
## 4.6500 4.6500
## 4.6500 4.6500
## [ CPUFloatType{2,2} ]
x2$grad
## torch_tensor
## 18.6000
## [ CPUFloatType{1} ]
x1$grad
## 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$backward().
With torch keeping track of operations as required, we don’t even have
to explicitly name the intermediate tensors any more. We can code
forward pass, loss calculation, and backward pass in just three lines:
y_pred <- x$mm(w1)$add(b1)$clamp(min = 0)$mm(w2)$add(b2)
loss <- (y_pred - y)$pow(2)$sum()
loss$backward()
Here is the complete code. We’re at an intermediate stage: We still manually compute the forward pass and the loss, and we still manually update the weights. Due to the latter, there is something I need to explain. But I’ll let you check out the new version first:
library(torch)
### generate training data -----------------------------------------------------
# input dimensionality (number of input features)
d_in <- 3
# output dimensionality (number of predicted features)
d_out <- 1
# number of observations in training set
n <- 100
# create random data
x <- torch_randn(n, d_in)
y <- x[, 1, NULL] * 0.2 - x[, 2, NULL] * 1.3 - x[, 3, NULL] * 0.5 + torch_randn(n, 1)
### initialize weights ---------------------------------------------------------
# dimensionality of hidden layer
d_hidden <- 32
# weights connecting input to hidden layer
w1 <- torch_randn(d_in, d_hidden, requires_grad = TRUE)
# weights connecting hidden to output layer
w2 <- torch_randn(d_hidden, d_out, requires_grad = TRUE)
# hidden layer bias
b1 <- torch_zeros(1, d_hidden, requires_grad = TRUE)
# output layer bias
b2 <- torch_zeros(1, d_out, requires_grad = TRUE)
### network parameters ---------------------------------------------------------
learning_rate <- 1e-4
### training loop --------------------------------------------------------------
for (t in 1:200) {
### -------- Forward pass --------
y_pred <- x$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_()
})
}
## Epoch: 10 Loss: 276.425
## Epoch: 20 Loss: 149.8547
## Epoch: 30 Loss: 112.9031
## Epoch: 40 Loss: 98.43044
## Epoch: 50 Loss: 91.59359
## Epoch: 60 Loss: 87.51484
## Epoch: 70 Loss: 84.87402
## Epoch: 80 Loss: 82.77007
## Epoch: 90 Loss: 80.96922
## Epoch: 100 Loss: 79.46633
## Epoch: 110 Loss: 78.10928
## Epoch: 120 Loss: 76.89656
## Epoch: 130 Loss: 75.83997
## Epoch: 140 Loss: 74.86513
## Epoch: 150 Loss: 74.01006
## Epoch: 160 Loss: 73.22728
## Epoch: 170 Loss: 72.49596
## Epoch: 180 Loss: 71.81541
## Epoch: 190 Loss: 71.17912
## Epoch: 200 Loss: 70.57628
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 final section in this chapter, 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.
So where do we stand? We started out coding a network completely from
scratch, making use of nothing but torch tensors. In this section, 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 …? Stay tuned.