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Two days in the past, I launched torch, an R package deal that gives the native performance that is dropped at Python customers by PyTorch. In that publish, I assumed fundamental familiarity with TensorFlow/Keras. Consequently, I portrayed torch in a method I figured can be useful to somebody who “grew up” with the Keras method of coaching a mannequin: Aiming to concentrate on variations, but not lose sight of the general course of.

This publish now adjustments perspective. We code a easy neural community “from scratch”, making use of simply certainly one of torch’s constructing blocks: tensors. This community will likely be as “uncooked” (low-level) as might be. (For the much less math-inclined individuals amongst us, it could function a refresher of what’s truly occurring beneath all these comfort instruments they constructed for us. However the true objective is for example what might be carried out with tensors alone.)

Subsequently, three posts will progressively present tips on how to cut back the hassle – noticeably proper from the beginning, enormously as soon as we end. On the finish of this mini-series, you’ll have seen how computerized differentiation works in torch, tips on how to use modules (layers, in keras converse, and compositions thereof), and optimizers. By then, you’ll have loads of the background fascinating when making use of torch to real-world duties.

This publish would be the longest, since there’s a lot to study tensors: Learn how to create them; tips on how to manipulate their contents and/or modify their shapes; tips on how to convert them to R arrays, matrices or vectors; and naturally, given the omnipresent want for velocity: tips on how to get all these operations executed on the GPU. As soon as we’ve cleared that agenda, we code the aforementioned little community, seeing all these elements in motion.

Tensors

Creation

Tensors could also be created by specifying particular person values. Right here we create two one-dimensional tensors (vectors), of sorts float and bool, respectively:

library(torch)
# a 1d vector of size 2
t <- torch_tensor(c(1, 2))
t

# additionally 1d, however of sort boolean
t <- torch_tensor(c(TRUE, FALSE))
t

torch_tensor 
 1
 2
[ CPUFloatType{2} ]

torch_tensor 
 1
 0
[ CPUBoolType{2} ]

And listed below are two methods to create two-dimensional tensors (matrices). Notice how within the second strategy, it is advisable to specify byrow = TRUE within the name to matrix() to get values organized in row-major order.

# a 3x3 tensor (matrix)
t <- torch_tensor(rbind(c(1,2,0), c(3,0,0), c(4,5,6)))
t

# additionally 3x3
t <- torch_tensor(matrix(1:9, ncol = 3, byrow = TRUE))
t

torch_tensor 
 1  2  0
 3  0  0
 4  5  6
[ CPUFloatType{3,3} ]

torch_tensor 
 1  2  3
 4  5  6
 7  8  9
[ CPULongType{3,3} ]

In increased dimensions particularly, it may be simpler to specify the kind of tensor abstractly, as in: “give me a tensor of <…> of form n1 x n2”, the place <…> could possibly be “zeros”; or “ones”; or, say, “values drawn from a normal regular distribution”:

# a 3x3 tensor of standard-normally distributed values
t <- torch_randn(3, 3)
t

# a 4x2x2 (3d) tensor of zeroes
t <- torch_zeros(4, 2, 2)
t

torch_tensor 
-2.1563  1.7085  0.5245
 0.8955 -0.6854  0.2418
 0.4193 -0.7742 -1.0399
[ CPUFloatType{3,3} ]

torch_tensor 
(1,.,.) = 
  0  0
  0  0

(2,.,.) = 
  0  0
  0  0

(3,.,.) = 
  0  0
  0  0

(4,.,.) = 
  0  0
  0  0
[ CPUFloatType{4,2,2} ]

Many comparable features exist, together with, e.g., torch_arange() to create a tensor holding a sequence of evenly spaced values, torch_eye() which returns an id matrix, and torch_logspace() which fills a specified vary with an inventory of values spaced logarithmically.

If no dtype argument is specified, torch will infer the info sort from the passed-in worth(s). For instance:

t <- torch_tensor(c(3, 5, 7))
t$dtype

t <- torch_tensor(1L)
t$dtype

torch_Float
torch_Long

However we will explicitly request a unique dtype if we would like:

t <- torch_tensor(2, dtype = torch_double())
t$dtype

torch_Double

torch tensors stay on a system. By default, this would be the CPU:

torch_device(sort='cpu')

However we may additionally outline a tensor to stay on the GPU:

t <- torch_tensor(2, system = "cuda")
t$system

torch_device(sort='cuda', index=0)

We’ll speak extra about gadgets beneath.

There’s one other crucial parameter to the tensor-creation features: requires_grad. Right here although, I must ask in your endurance: This one will prominently determine within the follow-up publish.

Conversion to built-in R information sorts

To transform torch tensors to R, use as_array():

t <- torch_tensor(matrix(1:9, ncol = 3, byrow = TRUE))
as_array(t)

     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    4    5    6
[3,]    7    8    9

Relying on whether or not the tensor is one-, two-, or three-dimensional, the ensuing R object will likely be a vector, a matrix, or an array:

t <- torch_tensor(c(1, 2, 3))
as_array(t) %>% class()

t <- torch_ones(c(2, 2))
as_array(t) %>% class()

t <- torch_ones(c(2, 2, 2))
as_array(t) %>% class()

[1] "numeric"

[1] "matrix" "array" 

[1] "array"

For one-dimensional and two-dimensional tensors, it’s also doable to make use of as.integer() / as.matrix(). (One motive you would possibly need to do that is to have extra self-documenting code.)

If a tensor presently lives on the GPU, it is advisable to transfer it to the CPU first:

t <- torch_tensor(2, system = "cuda")
as.integer(t$cpu())

[1] 2

Indexing and slicing tensors

Usually, we need to retrieve not a whole tensor, however solely a few of the values it holds, and even only a single worth. In these instances, we speak about slicing and indexing, respectively.

In R, these operations are 1-based, that means that once we specify offsets, we assume for the very first ingredient in an array to reside at offset 1. The identical conduct was applied for torch. Thus, loads of the performance described on this part ought to really feel intuitive.

The best way I’m organizing this part is the next. We’ll examine the intuitive components first, the place by intuitive I imply: intuitive to the R consumer who has not but labored with Python’s NumPy. Then come issues which, to this consumer, could look extra shocking, however will turn into fairly helpful.

Indexing and slicing: the R-like half

None of those needs to be overly shocking:

t <- torch_tensor(rbind(c(1,2,3), c(4,5,6)))
t

# a single worth
t[1, 1]

# first row, all columns
t[1, ]

# first row, a subset of columns
t[1, 1:2]

torch_tensor 
 1  2  3
 4  5  6
[ CPUFloatType{2,3} ]

torch_tensor 
1
[ CPUFloatType{} ]

torch_tensor 
 1
 2
 3
[ CPUFloatType{3} ]

torch_tensor 
 1
 2
[ CPUFloatType{2} ]

Notice how, simply as in R, singleton dimensions are dropped:

t <- torch_tensor(rbind(c(1,2,3), c(4,5,6)))

# 2x3
t$measurement() 

# only a single row: will likely be returned as a vector
t[1, 1:2]$measurement() 

# a single ingredient
t[1, 1]$measurement()

[1] 2 3

[1] 2

integer(0)

And similar to in R, you may specify drop = FALSE to maintain these dimensions:

t[1, 1:2, drop = FALSE]$measurement()

t[1, 1, drop = FALSE]$measurement()

[1] 1 2

[1] 1 1

Indexing and slicing: What to look out for

Whereas R makes use of unfavourable numbers to take away parts at specified positions, in torch unfavourable values point out that we begin counting from the top of a tensor – with -1 pointing to its final ingredient:

t <- torch_tensor(rbind(c(1,2,3), c(4,5,6)))

t[1, -1]

t[ , -2:-1] 

torch_tensor 
3
[ CPUFloatType{} ]

torch_tensor 
 2  3
 5  6
[ CPUFloatType{2,2} ]

This can be a function you would possibly know from NumPy. Similar with the next.

When the slicing expression m:n is augmented by one other colon and a 3rd quantity – m:n:o –, we are going to take each oth merchandise from the vary specified by m and n:

t <- torch_tensor(1:10)
t[2:10:2]

torch_tensor 
  2
  4
  6
  8
 10
[ CPULongType{5} ]

Typically we don’t know what number of dimensions a tensor has, however we do know what to do with the ultimate dimension, or the primary one. To subsume all others, we will use ..:

t <- torch_randint(-7, 7, measurement = c(2, 2, 2))
t

t[.., 1]

t[2, ..]

torch_tensor 
(1,.,.) = 
  2 -2
 -5  4

(2,.,.) = 
  0  4
 -3 -1
[ CPUFloatType{2,2,2} ]

torch_tensor 
 2 -5
 0 -3
[ CPUFloatType{2,2} ]

torch_tensor 
 0  4
-3 -1
[ CPUFloatType{2,2} ]

Now we transfer on to a subject that, in follow, is simply as indispensable as slicing: altering tensor shapes.

Reshaping tensors

Modifications in form can happen in two essentially alternative ways. Seeing how “reshape” actually means: preserve the values however modify their structure, we may both alter how they’re organized bodily, or preserve the bodily construction as-is and simply change the “mapping” (a semantic change, because it had been).

Within the first case, storage should be allotted for 2 tensors, supply and goal, and parts will likely be copied from the latter to the previous. Within the second, bodily there will likely be only a single tensor, referenced by two logical entities with distinct metadata.

Not surprisingly, for efficiency causes, the second operation is most popular.

Zero-copy reshaping

We begin with zero-copy strategies, as we’ll need to use them every time we will.

A particular case typically seen in follow is including or eradicating a singleton dimension.

unsqueeze() provides a dimension of measurement 1 at a place specified by dim:

t1 <- torch_randint(low = 3, excessive = 7, measurement = c(3, 3, 3))
t1$measurement()

t2 <- t1$unsqueeze(dim = 1)
t2$measurement()

t3 <- t1$unsqueeze(dim = 2)
t3$measurement()

[1] 3 3 3

[1] 1 3 3 3

[1] 3 1 3 3

Conversely, squeeze() removes singleton dimensions:

t4 <- t3$squeeze()
t4$measurement()

[1] 3 3 3

The identical could possibly be achieved with view(). view(), nevertheless, is far more common, in that it lets you reshape the info to any legitimate dimensionality. (Legitimate that means: The variety of parts stays the identical.)

Right here we’ve a 3x2 tensor that’s reshaped to measurement 2x3:

t1 <- torch_tensor(rbind(c(1, 2), c(3, 4), c(5, 6)))
t1

t2 <- t1$view(c(2, 3))
t2

torch_tensor 
 1  2
 3  4
 5  6
[ CPUFloatType{3,2} ]

torch_tensor 
 1  2  3
 4  5  6
[ CPUFloatType{2,3} ]

(Notice how that is completely different from matrix transposition.)

As an alternative of going from two to a few dimensions, we will flatten the matrix to a vector.

t4 <- t1$view(c(-1, 6))

t4$measurement()

t4

[1] 1 6

torch_tensor 
 1  2  3  4  5  6
[ CPUFloatType{1,6} ]

In distinction to indexing operations, this doesn’t drop dimensions.

Like we stated above, operations like squeeze() or view() don’t make copies. Or, put otherwise: The output tensor shares storage with the enter tensor. We will actually confirm this ourselves:

t1$storage()$data_ptr()

t2$storage()$data_ptr()

[1] "0x5648d02ac800"

[1] "0x5648d02ac800"

What’s completely different is the storage metadata torch retains about each tensors. Right here, the related data is the stride:

A tensor’s stride() technique tracks, for each dimension, what number of parts must be traversed to reach at its subsequent ingredient (row or column, in two dimensions). For t1 above, of form 3x2, we’ve to skip over 2 objects to reach on the subsequent row. To reach on the subsequent column although, in each row we simply must skip a single entry:

[1] 2 1

For t2, of form 3x2, the space between column parts is identical, however the distance between rows is now 3:

[1] 3 1

Whereas zero-copy operations are optimum, there are instances the place they received’t work.

With view(), this will occur when a tensor was obtained by way of an operation – apart from view() itself – that itself has already modified the stride. One instance can be transpose():

t1 <- torch_tensor(rbind(c(1, 2), c(3, 4), c(5, 6)))
t1
t1$stride()

t2 <- t1$t()
t2
t2$stride()

torch_tensor 
 1  2
 3  4
 5  6
[ CPUFloatType{3,2} ]

[1] 2 1

torch_tensor 
 1  3  5
 2  4  6
[ CPUFloatType{2,3} ]

[1] 1 2

In torch lingo, tensors – like t2 – that re-use current storage (and simply learn it otherwise), are stated to not be “contiguous”. One strategy to reshape them is to make use of contiguous() on them earlier than. We’ll see this within the subsequent subsection.

Reshape with copy

Within the following snippet, attempting to reshape t2 utilizing view() fails, because it already carries data indicating that the underlying information shouldn’t be learn in bodily order.

t1 <- torch_tensor(rbind(c(1, 2), c(3, 4), c(5, 6)))

t2 <- t1$t()

t2$view(6) # error!

Error in (operate (self, measurement)  : 
  view measurement just isn't suitable with enter tensor's measurement and stride (at the least one dimension spans throughout two contiguous subspaces).
  Use .reshape(...) as an alternative. (view at ../aten/src/ATen/native/TensorShape.cpp:1364)

Nevertheless, if we first name contiguous() on it, a new tensor is created, which can then be (just about) reshaped utilizing view().

t3 <- t2$contiguous()

t3$view(6)

torch_tensor 
 1
 3
 5
 2
 4
 6
[ CPUFloatType{6} ]

Alternatively, we will use reshape(). reshape() defaults to view()-like conduct if doable; in any other case it’ll create a bodily copy.

t2$storage()$data_ptr()

t4 <- t2$reshape(6)

t4$storage()$data_ptr()

[1] "0x5648d49b4f40"

[1] "0x5648d2752980"

Operations on tensors

Unsurprisingly, torch supplies a bunch of mathematical operations on tensors; we’ll see a few of them within the community code beneath, and also you’ll encounter heaps extra if you proceed your torch journey. Right here, we shortly check out the general tensor technique semantics.

Tensor strategies usually return references to new objects. Right here, we add to t1 a clone of itself:

t1 <- torch_tensor(rbind(c(1, 2), c(3, 4), c(5, 6)))
t2 <- t1$clone()

t1$add(t2)

torch_tensor 
  2   4
  6   8
 10  12
[ CPUFloatType{3,2} ]

On this course of, t1 has not been modified:

torch_tensor 
 1  2
 3  4
 5  6
[ CPUFloatType{3,2} ]

Many tensor strategies have variants for mutating operations. These all carry a trailing underscore:

t1$add_(t1)

# now t1 has been modified
t1

torch_tensor 
  4   8
 12  16
 20  24
[ CPUFloatType{3,2} ]

torch_tensor 
  4   8
 12  16
 20  24
[ CPUFloatType{3,2} ]

Alternatively, you may in fact assign the brand new object to a brand new reference variable:

torch_tensor 
  8  16
 24  32
 40  48
[ CPUFloatType{3,2} ]

There’s one factor we have to focus on earlier than we wrap up our introduction to tensors: How can we’ve all these operations executed on the GPU?

Operating on GPU

To examine in case your GPU(s) is/are seen to torch, run

cuda_is_available()

cuda_device_count()

[1] TRUE

[1] 1

Tensors could also be requested to stay on the GPU proper at creation:

system <- torch_device("cuda")

t <- torch_ones(c(2, 2), system = system) 

Alternatively, they are often moved between gadgets at any time:

torch_device(sort='cuda', index=0)

torch_device(sort='cpu')

That’s it for our dialogue on tensors — nearly. There’s one torch function that, though associated to tensor operations, deserves particular point out. It’s known as broadcasting, and “bilingual” (R + Python) customers will understand it from NumPy.

Broadcasting

We regularly must carry out operations on tensors with shapes that don’t match precisely.

Unsurprisingly, we will add a scalar to a tensor:

t1 <- torch_randn(c(3,5))

t1 + 22

torch_tensor 
 23.1097  21.4425  22.7732  22.2973  21.4128
 22.6936  21.8829  21.1463  21.6781  21.0827
 22.5672  21.2210  21.2344  23.1154  20.5004
[ CPUFloatType{3,5} ]

The identical will work if we add tensor of measurement 1:

t1 <- torch_randn(c(3,5))

t1 + torch_tensor(c(22))

Including tensors of various sizes usually received’t work:

t1 <- torch_randn(c(3,5))
t2 <- torch_randn(c(5,5))

t1$add(t2) # error

Error in (operate (self, different, alpha)  : 
  The dimensions of tensor a (2) should match the scale of tensor b (5) at non-singleton dimension 1 (infer_size at ../aten/src/ATen/ExpandUtils.cpp:24)

Nevertheless, below sure situations, one or each tensors could also be just about expanded so each tensors line up. This conduct is what is supposed by broadcasting. The best way it really works in torch isn’t just impressed by, however truly an identical to that of NumPy.

The principles are:

1. We align array shapes, ranging from the appropriate.

   Say we’ve two tensors, certainly one of measurement 8x1x6x1, the opposite of measurement 7x1x5.

   Right here they’re, right-aligned:

# t1, form:     8  1  6  1
# t2, form:        7  1  5

2. Beginning to look from the appropriate, the sizes alongside aligned axes both must match precisely, or certainly one of them must be equal to 1: wherein case the latter is broadcast to the bigger one.

   Within the above instance, that is the case for the second-from-last dimension. This now provides

# t1, form:     8  1  6  1
# t2, form:        7  6  5

, with broadcasting occurring in t2.

3. If on the left, one of many arrays has an extra axis (or a couple of), the opposite is just about expanded to have a measurement of 1 in that place, wherein case broadcasting will occur as said in (2).

   That is the case with t1’s leftmost dimension. First, there’s a digital growth

# t1, form:     8  1  6  1
# t2, form:     1  7  1  5

after which, broadcasting occurs:

# t1, form:     8  1  6  1
# t2, form:     8  7  1  5

In response to these guidelines, our above instance

t1 <- torch_randn(c(3,5))
t2 <- torch_randn(c(5,5))

t1$add(t2)

could possibly be modified in numerous ways in which would permit for including two tensors.

For instance, if t2 had been 1x5, it could solely must get broadcast to measurement 3x5 earlier than the addition operation:

t1 <- torch_randn(c(3,5))
t2 <- torch_randn(c(1,5))

t1$add(t2)

torch_tensor 
-1.0505  1.5811  1.1956 -0.0445  0.5373
 0.0779  2.4273  2.1518 -0.6136  2.6295
 0.1386 -0.6107 -1.2527 -1.3256 -0.1009
[ CPUFloatType{3,5} ]

If it had been of measurement 5, a digital main dimension can be added, after which, the identical broadcasting would happen as within the earlier case.

t1 <- torch_randn(c(3,5))
t2 <- torch_randn(c(5))

t1$add(t2)

torch_tensor 
-1.4123  2.1392 -0.9891  1.1636 -1.4960
 0.8147  1.0368 -2.6144  0.6075 -2.0776
-2.3502  1.4165  0.4651 -0.8816 -1.0685
[ CPUFloatType{3,5} ]

Here’s a extra advanced instance. Broadcasting how occurs each in t1 and in t2:

t1 <- torch_randn(c(1,5))
t2 <- torch_randn(c(3,1))

t1$add(t2)

torch_tensor 
 1.2274  1.1880  0.8531  1.8511 -0.0627
 0.2639  0.2246 -0.1103  0.8877 -1.0262
-1.5951 -1.6344 -1.9693 -0.9713 -2.8852
[ CPUFloatType{3,5} ]

As a pleasant concluding instance, by means of broadcasting an outer product might be computed like so:

t1 <- torch_tensor(c(0, 10, 20, 30))

t2 <- torch_tensor(c(1, 2, 3))

t1$view(c(4,1)) * t2

torch_tensor 
  0   0   0
 10  20  30
 20  40  60
 30  60  90
[ CPUFloatType{4,3} ]

And now, we actually get to implementing that neural community!

A easy neural community utilizing torch tensors

Our activity, which we strategy in a low-level method as we speak however significantly simplify in upcoming installments, consists of regressing a single goal datum based mostly on three enter variables.

We instantly use torch to simulate some information.

Toy information

library(torch)

# enter dimensionality (variety of enter options)
d_in <- 3
# output dimensionality (variety of predicted options)
d_out <- 1
# variety of observations in coaching set
n <- 100


# create random information
# enter
x <- torch_randn(n, d_in)
# goal
y <- x[, 1, drop = FALSE] * 0.2 -
  x[, 2, drop = FALSE] * 1.3 -
  x[, 3, drop = FALSE] * 0.5 +
  torch_randn(n, 1)

Subsequent, we have to initialize the community’s weights. We’ll have one hidden layer, with 32 items. The output layer’s measurement, being decided by the duty, is the same as 1.

Initialize weights

# dimensionality of hidden layer
d_hidden <- 32

# weights connecting enter to hidden layer
w1 <- torch_randn(d_in, d_hidden)
# weights connecting hidden to output layer
w2 <- torch_randn(d_hidden, d_out)

# hidden layer bias
b1 <- torch_zeros(1, d_hidden)
# output layer bias
b2 <- torch_zeros(1, d_out)

Now for the coaching loop correct. The coaching loop right here actually is the community.

Coaching loop

In every iteration (“epoch”), the coaching loop does 4 issues:

• runs by means of the community, computing predictions (ahead go)

• compares these predictions to the bottom reality and quantify the loss

• runs backwards by means of the community, computing the gradients that point out how the weights needs to be modified

• updates the weights, making use of the requested studying fee.

Right here is the template we’re going to fill:

for (t in 1:200) {
    
    ### -------- Ahead go -------- 
    
    # right here we'll compute the prediction
    
    
    ### -------- compute loss -------- 
    
    # right here we'll compute the sum of squared errors
    

    ### -------- Backpropagation -------- 
    
    # right here we'll go by means of the community, calculating the required gradients
    

    ### -------- Replace weights -------- 
    
    # right here we'll replace the weights, subtracting portion of the gradients 
}

The ahead go effectuates two affine transformations, one every for the hidden and output layers. In-between, ReLU activation is utilized:

  # compute pre-activations of hidden layers (dim: 100 x 32)
  # torch_mm does matrix multiplication
  h <- x$mm(w1) + b1
  
  # apply activation operate (dim: 100 x 32)
  # torch_clamp cuts off values beneath/above given thresholds
  h_relu <- h$clamp(min = 0)
  
  # compute output (dim: 100 x 1)
  y_pred <- h_relu$mm(w2) + b2

Our loss right here is imply squared error:

Calculating gradients the guide method is a bit tedious, however it may be carried out:

  # gradient of loss w.r.t. prediction (dim: 100 x 1)
  grad_y_pred <- 2 * (y_pred - y)
  # gradient of loss w.r.t. w2 (dim: 32 x 1)
  grad_w2 <- h_relu$t()$mm(grad_y_pred)
  # gradient of loss w.r.t. hidden activation (dim: 100 x 32)
  grad_h_relu <- grad_y_pred$mm(w2$t())
  # gradient of loss w.r.t. hidden pre-activation (dim: 100 x 32)
  grad_h <- grad_h_relu$clone()
  
  grad_h[h < 0] <- 0
  
  # gradient of loss w.r.t. b2 (form: ())
  grad_b2 <- grad_y_pred$sum()
  
  # gradient of loss w.r.t. w1 (dim: 3 x 32)
  grad_w1 <- x$t()$mm(grad_h)
  # gradient of loss w.r.t. b1 (form: (32, ))
  grad_b1 <- grad_h$sum(dim = 1)

The ultimate step then makes use of the calculated gradients to replace the weights:

  learning_rate <- 1e-4
  
  w2 <- w2 - learning_rate * grad_w2
  b2 <- b2 - learning_rate * grad_b2
  w1 <- w1 - learning_rate * grad_w1
  b1 <- b1 - learning_rate * grad_b1

Let’s use these snippets to fill within the gaps within the above template, and provides it a attempt!

Placing all of it collectively

library(torch)

### generate coaching information -----------------------------------------------------

# enter dimensionality (variety of enter options)
d_in <- 3
# output dimensionality (variety of predicted options)
d_out <- 1
# variety of observations in coaching set
n <- 100


# create random information
x <- torch_randn(n, d_in)
y <-
  x[, 1, NULL] * 0.2 - x[, 2, NULL] * 1.3 - x[, 3, NULL] * 0.5 + torch_randn(n, 1)


### initialize weights ---------------------------------------------------------

# dimensionality of hidden layer
d_hidden <- 32
# weights connecting enter to hidden layer
w1 <- torch_randn(d_in, d_hidden)
# weights connecting hidden to output layer
w2 <- torch_randn(d_hidden, d_out)

# hidden layer bias
b1 <- torch_zeros(1, d_hidden)
# output layer bias
b2 <- torch_zeros(1, d_out)

### community parameters ---------------------------------------------------------

learning_rate <- 1e-4

### coaching loop --------------------------------------------------------------

for (t in 1:200) {
  ### -------- Ahead go --------
  
  # compute pre-activations of hidden layers (dim: 100 x 32)
  h <- x$mm(w1) + b1
  # apply activation operate (dim: 100 x 32)
  h_relu <- h$clamp(min = 0)
  # compute output (dim: 100 x 1)
  y_pred <- h_relu$mm(w2) + b2
  
  ### -------- compute loss --------

  loss <- as.numeric((y_pred - y)$pow(2)$sum())
  
  if (t %% 10 == 0)
    cat("Epoch: ", t, "   Loss: ", loss, "n")
  
  ### -------- Backpropagation --------
  
  # gradient of loss w.r.t. prediction (dim: 100 x 1)
  grad_y_pred <- 2 * (y_pred - y)
  # gradient of loss w.r.t. w2 (dim: 32 x 1)
  grad_w2 <- h_relu$t()$mm(grad_y_pred)
  # gradient of loss w.r.t. hidden activation (dim: 100 x 32)
  grad_h_relu <- grad_y_pred$mm(
    w2$t())
  # gradient of loss w.r.t. hidden pre-activation (dim: 100 x 32)
  grad_h <- grad_h_relu$clone()
  
  grad_h[h < 0] <- 0
  
  # gradient of loss w.r.t. b2 (form: ())
  grad_b2 <- grad_y_pred$sum()
  
  # gradient of loss w.r.t. w1 (dim: 3 x 32)
  grad_w1 <- x$t()$mm(grad_h)
  # gradient of loss w.r.t. b1 (form: (32, ))
  grad_b1 <- grad_h$sum(dim = 1)
  
  ### -------- Replace weights --------
  
  w2 <- w2 - learning_rate * grad_w2
  b2 <- b2 - learning_rate * grad_b2
  w1 <- w1 - learning_rate * grad_w1
  b1 <- b1 - learning_rate * grad_b1
  
}

Epoch:  10     Loss:  352.3585 
Epoch:  20     Loss:  219.3624 
Epoch:  30     Loss:  155.2307 
Epoch:  40     Loss:  124.5716 
Epoch:  50     Loss:  109.2687 
Epoch:  60     Loss:  100.1543 
Epoch:  70     Loss:  94.77817 
Epoch:  80     Loss:  91.57003 
Epoch:  90     Loss:  89.37974 
Epoch:  100    Loss:  87.64617 
Epoch:  110    Loss:  86.3077 
Epoch:  120    Loss:  85.25118 
Epoch:  130    Loss:  84.37959 
Epoch:  140    Loss:  83.44133 
Epoch:  150    Loss:  82.60386 
Epoch:  160    Loss:  81.85324 
Epoch:  170    Loss:  81.23454 
Epoch:  180    Loss:  80.68679 
Epoch:  190    Loss:  80.16555 
Epoch:  200    Loss:  79.67953 

This appears to be like prefer it labored fairly properly! It additionally ought to have fulfilled its objective: Displaying what you may obtain utilizing torch tensors alone. In case you didn’t really feel like going by means of the backprop logic with an excessive amount of enthusiasm, don’t fear: Within the subsequent installment, this can get considerably much less cumbersome. See you then!