Artificial Intelligence

NumPy-style broadcasting for R TensorFlow customers

23 January 2024
Bourahima



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 snug you might be with Python, there’s an issue. For instance: You’re imagined to understand how broadcasting works. And maybe, you’d say you’re vaguely accustomed to it: So when arrays have totally different shapes, some components get duplicated till their shapes match and … and isn’t R vectorized anyway?

Whereas such a world notion may go on the whole, like when skimming a weblog put up, it’s not sufficient to grasp, say, examples within the TensorFlow API docs. On this put up, 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 wish to guess the outcome – not the numbers, however the way it comes about on the whole? 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 concern. (Translated to R, however conserving 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 really does broadcasting – with distributions – identical to with tensors!

We get again to each examples on the finish of this put up. Our principal focus will probably be to clarify 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 shortly evaluation a couple of 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); methods to parse their shapes; some terminology and associated background.
Although not difficult 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 – identical to heaps extra –, may be seemed up intimately within the NumPy documentation.

Few details about NumPy

Fundamental slicing

For simplicity, we’ll use the phrases indexing and slicing kind of synonymously any longer. The essential machine 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 internal 2 dimensions specify legitimate matrix multiplication dimensions, and any additional outer dimensions specify matching batch measurement.

So right here (see code just under), the internal two dimensions look good – (2, 3) and (3, 2) – whereas the one (one and solely, on this case) batch dimension exhibits 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 shortly 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 strive 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 exhibits that no.

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

Within the documentation for matvec, we’re advised:

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 – up to now, there isn’t any 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 every single place

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 advised that there’s broadcasting occurring, we will cause like this: Aligning each shapes on the proper 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.

Which means: We have now two distributions with imply (0) (certainly one of scale (1.5), the opposite of scale (3.5)), and in addition two with imply (1) (corresponding scales being (2.5) and (4.5)).

Right here’s a extra direct technique 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 idea” (its guidelines are), however may have some training to get it proper. Particularly together with the truth that capabilities / operators do have their very own views on which components of its inputs ought to broadcast, and which shouldn’t. Actually, there isn’t any means round trying up the precise behaviors within the documentation.

Hopefully although, you’ve discovered this put up to be an excellent begin into the subject. Perhaps, just like the creator, you’re feeling such as you may see broadcasting occurring wherever on this planet now. Thanks for studying!