NumPy-style broadcasting for R TensorFlow customers

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    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 folks’s code.

    Relying on how comfy you might be with Python, there’s an issue. For instance: You’re presupposed 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 parts get duplicated till their shapes match and … and isn’t R vectorized anyway?

    Whereas such a world notion may match typically, like when skimming a weblog publish, it’s not sufficient to know, say, examples within the TensorFlow API docs. On this publish, we’ll attempt to arrive at a extra precise understanding, and test it on concrete examples.

    Talking of examples, listed below 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 communicate)?

    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 problem. (Translated to R, however preserving the semantics). In TFP, we are able to 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 distinct 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 – identical to with tensors!

    We get again to each examples on the finish of this publish. Our foremost focus might be to clarify broadcasting as finished in NumPy, as NumPy-style broadcasting is what quite a few different frameworks have adopted (e.g., TensorFlow).

    Earlier than although, let’s rapidly assessment a couple of fundamentals about NumPy arrays: Learn how to index or slice them (indexing usually referring to single-element extraction, whereas slicing would yield – nicely – slices containing a number of parts); how you can 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 usually a prerequisite to efficiently making use of Python documentation.

    Acknowledged upfront, we’ll actually prohibit ourselves to the fundamentals right here; for instance, we gained’t contact superior indexing which – identical to tons extra –, might be appeared up intimately within the NumPy documentation.

    Few details about NumPy

    Primary slicing

    For simplicity, we’ll use the phrases indexing and slicing kind of synonymously any further. The essential machine here’s a slice, specifically, a begin:cease construction indicating, for a single dimension, which vary of parts to incorporate within the choice.

    In distinction to R, Python indexing is zero-based, and the top 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 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 rapidly test 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., might 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 take a look at 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 attempt one other one.

    Within the documentation for matvec, we’re instructed:

    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] in a position to 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 test this – to this point, there is no such thing as 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 all over the 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 instructed that there’s broadcasting occurring, we are able to motive like this: Aligning each shapes on the suitable and increasing loc’s form by 1 (on the left), we’ve got (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) (one among 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 method 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 concept” (its guidelines are), however might have some working towards to get it proper. Particularly along side the truth that features / operators do have their very own views on which components of its inputs ought to broadcast, and which shouldn’t. Actually, there is no such thing as a manner round trying up the precise behaviors within the documentation.

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

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