A key challenge in deep learning is to automatically derive the backward pass given the forward pass as a program, which has been long studied in the field of [automatic differentiation](https://arxiv.org/pdf/1502.05767.pdf), or autodiff, before the prosperity of deep learning.
We refer to the trace of the execution of the forward pass program as a *tape* [[1]](http://www.bcl.hamilton.ie/~barak/papers/toplas-reverse.pdf). When we train a deep learning model, the tape changes every iteration as the input data change, so we'd have to re-derive the backward pass, which is time-consuming, but also eases the case that the forward program includes control flows like if-else and for/while. With these control flows, the execution trace might change with iterations. Such changes are known as *dynamic networks* in the field of deep learning.
Deep learning systems that utilize the idea of dynamic networks gained their popularities in recent years. This article surveys the following typical systems:
DyNet uses a linear data structure, a list, to represent the tape. During the execution of the above example, it is a list of operators: `matmul`, `matmul`, and `softmax`. The list also includes information needed to do the backward pass, such as pointers to the inputs and outputs. Then the tape is played in reverse order at `loss.backward().`
The graph is composed of `Variable`s and `Function`s. During the forward execution, a `Variable` records its creator function, e.g. `h.creator = matmul`. And a Function records its inputs' previous/dependent functions `prev_func` through `creator`, e.g. `matmul.prev_func = matmul1`. At `loss.backward()`, a topological sort is performed on all `prev_func`s. Then the grad op is performed by the sorted order. Please be aware that a `Function` might have more than one `prev_func`s.
The list of DyNet could be considered the result of the topological sort of the graph of PyTorch. Or, the graph is the raw representation of the tape, which gives us the chance to *prune* part of the graph that is irrelevant with the backward pass before the topological sort [[2]](https://openreview.net/pdf?id=BJJsrmfCZ). Consider the following example, PyTorch only does backward on `SmallNet` while DyNet does both `SmallNet` and `BigNet`:
Another difference between DyNet and PyTorch is that DyNet lazily evaluates the forward pass, whereas PyTorch executes it immediately. Consider the following example:
The computation of `lookup`, `concat`, `matmul` and `softmax` didn't happen until the call of `loss_sym.value()`. This defered execution is useful because it allows some graph-like optimization possible, e.g. kernel fusion.
PyTorch chooses immediate evaluation. It avoids ever materializing a "forward graph"/"tape" (no need to explicitly call `dy.renew_cg()` to reset the list), recording only what is necessary to differentiate the computation, i.e. `creator` and `prev_func`.
[Backward code](https://github.com/HIPS/autograd/blob/442205dfefe407beffb33550846434baa90c4de7/autograd/core.py#L8-L40). In the forward pass, a graph of VJPNode is constructed.
# (1) Function Set definition, creates FunctionNode
model = FunctionSet(
l1=F.Linear(784, 100),
l2=F.Linear(100, 100),
l3=F.Linear(100, 10)).to_gpu()
# (2) Optimizer Setup
opt = optimizers.SGD()
opt.setup(model)
# (3) Forward computation
def forward(x, t):
h1 = F.relu(model.l1(x))
h2 = F.relu(model.l2(h1))
y = model.l3(h2)
return F.softmax_cross_entropy(y, t)
# (4) Training loop
for epoch in xrange(n_epoch):
for i in xrange(0, N, b_size):
x = Variable(to_gpu(...))
t = Variable(to_gpu(...))
opt.zero_grads()
loss = forward(x, t)
loss.backward()
opt.update()
```
In `forward(x, t)`, a graph of [`VariableNode`](https://github.com/chainer/chainer/blob/master/chainer/variable.py#L110) and [`FunctionNode`](https://github.com/chainer/chainer/blob/a69103a4aa59d5b318f39b01dbcb858d465b89cf/chainer/function_node.py#L19) is constructed. Every output's `VariableNode.creator` is pointed to the `FunctionNode`.
```python
class FunctionNode(object):
...
def apply(self, inputs):
outputs = self.forward(inputs)
ret = tuple([variable.Variable(y, requires_grad=requires_grad)
for y in outputs])
# Topological ordering
self.rank = max([x.rank for x in inputs]) if input_vars else 0
# Add backward edges
for y in ret:
y.creator_node = self
self.inputs = tuple([x.node for x in input_vars])
self.outputs = tuple([y.node for y in ret])
return ret
```
`loss.backward()` will calculate the accumulated gradient of all variables. All the backward of `FunctionNode`s will be called based on the topological order.
```python
class VariableNode(object):
...
def backward(self, retain_grad, loss_scale):
if self.creator_node is None:
return
cand_funcs = []
seen_set = set()
grads = {}
# Initialize error by 1, if this is a loss variable
if self.data.size == 1 and self._grad_var is None:
self.grad = numpy.ones_like(self.data)
grads[self._node] = self._grad_var
def add_cand(cand):
if cand not in seen_set:
# Negate since heapq is min-heap. This is a global variable
[forward](https://github.com/clab/dynet/blob/740a9626a13a2732544de142e256ad0d0a166658/dynet/exec.cc#L84-L158), [backward](https://github.com/clab/dynet/blob/740a9626a13a2732544de142e256ad0d0a166658/dynet/exec.cc#L166-L284). The trace is done by creating a tape of expressions in every iteration. Backward is done by traverse the tape in the reverse order.
