Artificial Intelligence

NumPy-style broadcasting for R TensorFlow customers

August 12, 2024
roosho

NumPy-style broadcasting for R TensorFlow customers

We develop, prepare, and deploy TensorFlow fashions from R. However that doesn’t imply we don’t make use of documentation, weblog posts, and examples written in Python. We glance up particular performance within the official TensorFlow API docs; we get inspiration from different individuals’s code.

Relying on how comfy you might be with Python, there’s an issue. For instance: You’re purported to know the way broadcasting works. And maybe, you’d say you’re vaguely aware of it: So when arrays have completely different shapes, some components get duplicated till their shapes match and … and isn’t R vectorized anyway?

Whereas such a world notion may fit typically, like when skimming a weblog submit, it’s not sufficient to grasp, say, examples within the TensorFlow API docs. On this submit, we’ll attempt to arrive at a extra actual understanding, and verify it on concrete examples.

Talking of examples, listed here are two motivating ones.

Broadcasting in motion

The primary makes use of TensorFlow’s matmul to multiply two tensors. Would you prefer to guess the consequence – not the numbers, however the way it comes about typically? Does this even run with out error – shouldn’t matrices be two-dimensional (rank-2 tensors, in TensorFlow converse)?

a <- tf$fixed(keras::array_reshape(1:12, dim = c(2, 2, 3)))
a 
# tf.Tensor(
# [[[ 1.  2.  3.]
#   [ 4.  5.  6.]]
# 
#  [[ 7.  8.  9.]
#   [10. 11. 12.]]], form=(2, 2, 3), dtype=float64)

b <- tf$fixed(keras::array_reshape(101:106, dim = c(1, 3, 2)))
b  
# tf.Tensor(
# [[[101. 102.]
#   [103. 104.]
#   [105. 106.]]], form=(1, 3, 2), dtype=float64)

c <- tf$matmul(a, b)

Second, here’s a “actual instance” from a TensorFlow Chance (TFP) github challenge. (Translated to R, however protecting the semantics).
In TFP, we will have batches of distributions. That, per se, is no surprise. However have a look at this:

library(tfprobability)
d <- tfd_normal(loc = c(0, 1), scale = matrix(1.5:4.5, ncol = 2, byrow = TRUE))
d
# tfp.distributions.Regular("Regular", batch_shape=[2, 2], event_shape=[], dtype=float64)

We create a batch of 4 regular distributions: every with a unique scale (1.5, 2.5, 3.5, 4.5). However wait: there are solely two location parameters given. So what are their scales, respectively?
Fortunately, TFP builders Brian Patton and Chris Suter defined the way it works: TFP truly does broadcasting – with distributions – similar to with tensors!

We get again to each examples on the finish of this submit. Our principal focus can be to elucidate broadcasting as achieved in NumPy, as NumPy-style broadcasting is what quite a few different frameworks have adopted (e.g., TensorFlow).

Earlier than although, let’s rapidly overview just a few fundamentals about NumPy arrays: How one can index or slice them (indexing usually referring to single-element extraction, whereas slicing would yield – effectively – slices containing a number of components); tips on how to parse their shapes; some terminology and associated background.
Although not sophisticated per se, these are the sorts of issues that may be complicated to rare Python customers; but they’re typically a prerequisite to efficiently making use of Python documentation.

Said upfront, we’ll actually limit ourselves to the fundamentals right here; for instance, we received’t contact superior indexing which – similar to heaps extra –, will be regarded up intimately within the NumPy documentation.

Few information about NumPy

Fundamental slicing

For simplicity, we’ll use the phrases indexing and slicing roughly synonymously any longer. The fundamental gadget here’s a slice, specifically, a begin:cease construction indicating, for a single dimension, which vary of components to incorporate within the choice.

In distinction to R, Python indexing is zero-based, and the tip index is unique:

c(4L, 1L))
a
# tf.Tensor(
# [[1.]
#  [1.]
#  [1.]
#  [1.]], form=(4, 1), dtype=float32)

b <- tf$fixed(c(1, 2, 3, 4))
b
# tf.Tensor([1. 2. 3. 4.], form=(4,), dtype=float32)

a + b
# tf.Tensor(
# [[2. 3. 4. 5.]
# [2. 3. 4. 5.]
# [2. 3. 4. 5.]
# [2. 3. 4. 5.]], form=(4, 4), dtype=float32)

And second, after we add tensors with shapes (3, 3) and (3,), the 1-d tensor ought to get added to each row (not each column):

a <- tf$fixed(matrix(1:9, ncol = 3, byrow = TRUE), dtype = tf$float32)
a
# tf.Tensor(
# [[1. 2. 3.]
#  [4. 5. 6.]
#  [7. 8. 9.]], form=(3, 3), dtype=float32)

b <- tf$fixed(c(100, 200, 300))
b
# tf.Tensor([100. 200. 300.], form=(3,), dtype=float32)

a + b
# tf.Tensor(
# [[101. 202. 303.]
#  [104. 205. 306.]
#  [107. 208. 309.]], form=(3, 3), dtype=float32)

Now again to the preliminary matmul instance.

Again to the puzzles

The documentation for matmul says,

The inputs should, following any transpositions, be tensors of rank >= 2 the place the inside 2 dimensions specify legitimate matrix multiplication dimensions, and any additional outer dimensions specify matching batch measurement.

So right here (see code slightly below), the inside two dimensions look good – (2, 3) and (3, 2) – whereas the one (one and solely, on this case) batch dimension reveals mismatching values 2 and 1, respectively.
A case for broadcasting thus: Each “batches” of a get matrix-multiplied with b.

a <- tf$fixed(keras::array_reshape(1:12, dim = c(2, 2, 3)))
a 
# tf.Tensor(
# [[[ 1.  2.  3.]
#   [ 4.  5.  6.]]
# 
#  [[ 7.  8.  9.]
#   [10. 11. 12.]]], form=(2, 2, 3), dtype=float64)

b <- tf$fixed(keras::array_reshape(101:106, dim = c(1, 3, 2)))
b  
# tf.Tensor(
# [[[101. 102.]
#   [103. 104.]
#   [105. 106.]]], form=(1, 3, 2), dtype=float64)

c <- tf$matmul(a, b)
c
# tf.Tensor(
# [[[ 622.  628.]
#   [1549. 1564.]]
# 
#  [[2476. 2500.]
#   [3403. 3436.]]], form=(2, 2, 2), dtype=float64)

Let’s rapidly verify this actually is what occurs, by multiplying each batches individually:

tf$matmul(a[1, , ], b)
# tf.Tensor(
# [[[ 622.  628.]
#   [1549. 1564.]]], form=(1, 2, 2), dtype=float64)

tf$matmul(a[2, , ], b)
# tf.Tensor(
# [[[2476. 2500.]
#   [3403. 3436.]]], form=(1, 2, 2), dtype=float64)

Is it too bizarre to be questioning if broadcasting would additionally occur for matrix dimensions? E.g., may we attempt matmuling tensors of shapes (2, 4, 1) and (2, 3, 1), the place the 4 x 1 matrix can be broadcast to 4 x 3? – A fast check reveals that no.

To see how actually, when coping with TensorFlow operations, it pays off overcoming one’s preliminary reluctance and really seek the advice of the documentation, let’s attempt one other one.

Within the documentation for matvec, we’re informed:

Multiplies matrix a by vector b, producing a * b.
The matrix a should, following any transpositions, be a tensor of rank >= 2, with form(a)[-1] == form(b)[-1], and form(a)[:-2] capable of broadcast with form(b)[:-1].

In our understanding, given enter tensors of shapes (2, 2, 3) and (2, 3), matvec ought to carry out two matrix-vector multiplications: as soon as for every batch, as listed by every enter’s leftmost dimension. Let’s verify this – thus far, there isn’t a broadcasting concerned:

# two matrices
a <- tf$fixed(keras::array_reshape(1:12, dim = c(2, 2, 3)))
a
# tf.Tensor(
# [[[ 1.  2.  3.]
#   [ 4.  5.  6.]]
# 
#  [[ 7.  8.  9.]
#   [10. 11. 12.]]], form=(2, 2, 3), dtype=float64)

b = tf$fixed(keras::array_reshape(101:106, dim = c(2, 3)))
b
# tf.Tensor(
# [[101. 102. 103.]
#  [104. 105. 106.]], form=(2, 3), dtype=float64)

c <- tf$linalg$matvec(a, b)
c
# tf.Tensor(
# [[ 614. 1532.]
#  [2522. 3467.]], form=(2, 2), dtype=float64)

Doublechecking, we manually multiply the corresponding matrices and vectors, and get:

tf$linalg$matvec(a[1,  , ], b[1, ])
# tf.Tensor([ 614. 1532.], form=(2,), dtype=float64)

tf$linalg$matvec(a[2,  , ], b[2, ])
# tf.Tensor([2522. 3467.], form=(2,), dtype=float64)

The identical. Now, will we see broadcasting if b has only a single batch?

b = tf$fixed(keras::array_reshape(101:103, dim = c(1, 3)))
b
# tf.Tensor([[101. 102. 103.]], form=(1, 3), dtype=float64)

c <- tf$linalg$matvec(a, b)
c
# tf.Tensor(
# [[ 614. 1532.]
#  [2450. 3368.]], form=(2, 2), dtype=float64)

Multiplying each batch of a with b, for comparability:

tf$linalg$matvec(a[1,  , ], b)
# tf.Tensor([ 614. 1532.], form=(2,), dtype=float64)

tf$linalg$matvec(a[2,  , ], b)
# tf.Tensor([[2450. 3368.]], form=(1, 2), dtype=float64)

It labored!

Now, on to the opposite motivating instance, utilizing tfprobability.

Broadcasting in all places

Right here once more is the setup:

library(tfprobability)
d <- tfd_normal(loc = c(0, 1), scale = matrix(1.5:4.5, ncol = 2, byrow = TRUE))
d
# tfp.distributions.Regular("Regular", batch_shape=[2, 2], event_shape=[], dtype=float64)

What’s going on? Let’s examine location and scale individually:

d$loc
# tf.Tensor([0. 1.], form=(2,), dtype=float64)

d$scale
# tf.Tensor(
# [[1.5 2.5]
#  [3.5 4.5]], form=(2, 2), dtype=float64)

Simply specializing in these tensors and their shapes, and having been informed that there’s broadcasting occurring, we will motive like this: Aligning each shapes on the appropriate and increasing loc’s form by 1 (on the left), we have now (1, 2) which can be broadcast with (2,2) – in matrix-speak, loc is handled as a row and duplicated.

That means: We’ve got two distributions with imply (0) (one among scale (1.5), the opposite of scale (3.5)), and likewise two with imply (1) (corresponding scales being (2.5) and (4.5)).

Right here’s a extra direct approach to see this:

d$imply()
# tf.Tensor(
# [[0. 1.]
#  [0. 1.]], form=(2, 2), dtype=float64)

d$stddev()
# tf.Tensor(
# [[1.5 2.5]
#  [3.5 4.5]], form=(2, 2), dtype=float64)

Puzzle solved!

Summing up, broadcasting is easy “in principle” (its guidelines are), however might have some training to get it proper. Particularly along side the truth that capabilities / operators do have their very own views on which elements of its inputs ought to broadcast, and which shouldn’t. Actually, there isn’t a manner round wanting up the precise behaviors within the documentation.

Hopefully although, you’ve discovered this submit to be a very good begin into the subject. Perhaps, just like the writer, you are feeling such as you would possibly see broadcasting occurring anyplace on the planet now. Thanks for studying!