Add recipe for prototype vmap (#1209)

Test Plan:
- Run `vmap_recipe.py` locally

Co-authored-by: Brian Johnson <brianjo@fb.com>

Author: zou3519

prototype_source/vmap_recipe.py

@@ -0,0 +1,121 @@
+"""
+torch.vmap
+==========
+This tutorial introduces torch.vmap, an autovectorizer for PyTorch operations.
+torch.vmap is a prototype feature and cannot handle a number of use cases;
+however, we would like to gather use cases for it to inform the design. If you
+are considering using torch.vmap or think it would be really cool for something,
+please contact us at https://github.com/pytorch/pytorch/issues/42368.
+
+So, what is vmap?
+-----------------
+vmap is a higher-order function. It accepts a function `func` and returns a new
+function that maps `func` over some dimension of the inputs. It is highly
+inspired by JAX's vmap.
+
+Semantically, vmap pushes the "map" into PyTorch operations called by `func`,
+effectively vectorizing those operations.
+"""
+import torch
+# NB: vmap is only available on nightly builds of PyTorch.
+# You can download one at pytorch.org if you're interested in testing it out.
+from torch import vmap
+
+####################################################################
+# The first use case for vmap is making it easier to handle
+# batch dimensions in your code. One can write a function `func`
+# that runs on examples and then lift it to a function that can
+# take batches of examples with `vmap(func)`. `func` however
+# is subject to many restrictions:
+# - it must be functional (one cannot mutate a Python data structure
+#   inside of it), with teh exception of in-place PyTorch operations.
+# - batches of examples must be provided as Tensors. This means that
+#   vmap doesn't handle variable-length sequences out of the box.
+#
+# One example of using `vmap` is to compute batched dot products. PyTorch
+# doesn't provide a batched `torch.dot` API; instead of unsuccessfully
+# rummaging through docs, use `vmap` to construct a new function:
+
+torch.dot                            # [D], [D] -> []
+batched_dot = torch.vmap(torch.dot)  # [N, D], [N, D] -> [N]
+x, y = torch.randn(2, 5), torch.randn(2, 5)
+batched_dot(x, y)
+
+####################################################################
+# `vmap` can be helpful in hiding batch dimensions, leading to a simpler
+# model authoring experience.
+batch_size, feature_size = 3, 5
+weights = torch.randn(feature_size, requires_grad=True)
+
+# Note that model doesn't work with a batch of feature vectors because
+# torch.dot must take 1D tensors. It's pretty easy to rewrite this
+# to use `torch.matmul` instead, but if we didn't want to do that or if
+# the code is more complicated (e.g., does some advanced indexing
+# shenanigins), we can simply call `vmap`. `vmap` batches over ALL
+# inputs, unless otherwise specified (with the in_dims argument,
+# please see the documentation for more details).
+def model(feature_vec):
+    # Very simple linear model with activation
+    return feature_vec.dot(weights).relu()
+
+examples = torch.randn(batch_size, feature_size)
+result = torch.vmap(model)(examples)
+expected = torch.stack([model(example) for example in examples.unbind()])
+assert torch.allclose(result, expected)
+
+####################################################################
+# `vmap` can also help vectorize computations that were previously difficult
+# or impossible to batch. This bring us to our second use case: batched
+# gradient computation.
+# - https://github.com/pytorch/pytorch/issues/8304
+# - https://github.com/pytorch/pytorch/issues/23475
+#
+# The PyTorch autograd engine computes vjps (vector-Jacobian products).
+# Using vmap, we can compute (batched vector) - jacobian products.
+#
+# One example of this is computing a full Jacobian matrix (this can also be
+# applied to computing a full Hessian matrix).
+# Computing a full Jacobian matrix for some function f: R^N -> R^N usually
+# requires N calls to `autograd.grad`, one per Jacobian row.
+
+# Setup
+N = 5
+def f(x):
+    return x ** 2
+
+x = torch.randn(N, requires_grad=True)
+y = f(x)
+basis_vectors = torch.eye(N)
+
+# Sequential approach
+jacobian_rows = [torch.autograd.grad(y, x, v, retain_graph=True)[0]
+                 for v in basis_vectors.unbind()]
+jacobian = torch.stack(jacobian_rows)
+
+# Using `vmap`, we can vectorize the whole computation, computing the
+# Jacobian in a single call to `autograd.grad`.
+def get_vjp(v):
+    return torch.autograd.grad(y, x, v)[0]
+
+jacobian_vmap = vmap(get_vjp)(basis_vectors)
+assert torch.allclose(jacobian_vmap, jacobian)
+
+####################################################################
+# The third main use case for vmap is computing per-sample-gradients.
+# This is something that the vmap prototype cannot handle performantly
+# right now. We're not sure what the API for computing per-sample-gradients
+# should be, but if you have ideas, please comment in
+# https://github.com/pytorch/pytorch/issues/7786.
+
+def model(sample, weight):
+    # do something...    
+    return torch.dot(sample, weight)
+
+def grad_sample(sample):
+    return torch.autograd.functional.vjp(lambda weight: model(sample), weight)[1]
+
+# The following doesn't actually work in the vmap prototype. But it
+# could be an API for computing per-sample-gradients.
+
+# batch_of_samples = torch.randn(64, 5)
+# vmap(grad_sample)(batch_of_samples)