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Operation Semantics

The following describes the semantics of operations defined in the ComputationBuilder interface. Typically, these operations map one-to-one to operations defined in the RPC interface in xla_data.proto.

A note on nomenclature: the generalized data type XLA deals with is an N-dimensional array holding elements of some uniform type (such as 32-bit float). Throughout the documentation, array is used to denote an arbitrary-dimensional array. For convenience, special cases have more specific and familiar names; for example a vector is a 1-dimensional array and a matrix is a 2-dimensional array.

BatchNormGrad

See also ComputationBuilder::BatchNormGrad and the original batch normalization paper for a detailed description of the algorithm.

Calculates gradients of batch norm.

BatchNormGrad(operand, scale, mean, variance, grad_output, epsilon, feature_index)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	n dimensional array to be normalized (x)
`scale`	`ComputationDataHandle`	1 dimensional array ($\gamma$)
`mean`	`ComputationDataHandle`	1 dimensional array ($\mu$)
`variance`	`ComputationDataHandle`	1 dimensional array ($\sigma^2$)
`grad_output`	`ComputationDataHandle`	Gradients passed to `BatchNormTraining` ($ \nabla y$)
`epsilon`	`float`	Epsilon value ($\epsilon$)
`feature_index`	`int64`	Index to feature dimension in `operand`

For each feature in the feature dimension (feature_index is the index for the feature dimension in operand), the operation calculates the gradients with respect to operand, offset and scale across all the other dimensions. The feature_index must be a valid index for the feature dimension in operand.

The three gradients are defined by the following formulas (Assuming a 4-dimensional tensor as operand and (l) is the index for feature dimension):

$ coef_l = \frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (\nabla y_{ijkl} * (x_{ijkl} - \mu_l) / (\sigma^2_{l}+\epsilon)) $

$ \nabla x_{ijkl} = \gamma_{l} * (1/\sqrt{\sigma^2_{l}+\epsilon}) * [\nabla y_{ijkl} - mean(\nabla y) - (x_{ijkl} - \mu_{l}) * coef_l] $

$ \nabla \beta_l = \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \nabla y_{ijkl} $

$ \nabla \gamma_l = \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \nabla y_{ijkl} * ((x_{ijkl} - \mu_l) / \sqrt{\sigma^2_{l}+\epsilon}) $

The inputs mean and variance represents moments value across batch and spatial dimensions.

The output type is a tuple of three handles:

Outputs	Type	Semantics
`grad_operand`	`ComputationDataHandle`	gradient with respect to input
`operand` ($ \nabla x$)	`grad_scale`	`ComputationDataHandle`
`grad_offset`	`ComputationDataHandle`	gradient with respect to input `offset`($ \nabla \beta$)

BatchNormInference

See also ComputationBuilder::BatchNormInference and the original batch normalization paper for a detailed description of the algorithm.

Normalizes an array across batch and spatial dimensions.

BatchNormInference(operand, scale, offset, mean, variance, epsilon, feature_index)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	n dimensional array to be normalized
`scale`	`ComputationDataHandle`	1 dimensional array
`offset`	`ComputationDataHandle`	1 dimensional array
`mean`	`ComputationDataHandle`	1 dimensional array
`variance`	`ComputationDataHandle`	1 dimensional array
`epsilon`	`float`	Epsilon value
`feature_index`	`int64`	Index to feature dimension in `operand`

For each feature in the feature dimension (feature_index is the index for the feature dimension in operand), the operation calculates the mean and variance across all the other dimensions and uses the mean and variance to normalize each element in operand. The feature_index must be a valid index for the feature dimension in operand.

BatchNormInference is equivalent to calling BatchNormTraining without computing mean and variance for each batch. It uses the input mean and variance instead as estimated values. The purpose of this op is to reduce latency in inference, hence the name BatchNormInference.

The output is an n-dimensional, normalized array with the same shape as input operand.

BatchNormTraining

See also ComputationBuilder::BatchNormTraining and the original batch normalization paper for a detailed description of the algorithm.

Normalizes an array across batch and spatial dimensions.

BatchNormTraining(operand, scale, offset, epsilon, feature_index)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	n dimensional array to be normalized (x)
`scale`	`ComputationDataHandle`	1 dimensional array ($\gamma$)
`offset`	`ComputationDataHandle`	1 dimensional array ($\beta$)
`epsilon`	`float`	Epsilon value ($\epsilon$)
`feature_index`	`int64`	Index to feature dimension in `operand`

The algorithm goes as follows for each batch in operand $x$ that contains m elements with w and h as the size of spatial dimensions (assuming operand is an 4 dimensional array):

Calculates batch mean $\mu_l$ for each feature l in feature dimension: $\mu_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h x_{ijkl}$
Calculates batch variance $\sigma^2_l$: $\sigma^2_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (x_{ijkl} - \mu_l)^2$
Normalizes, scales and shifts: $y_{ijkl}=\frac{\gamma_l(x_{ijkl}-\mu_l)}{\sqrt[2]{\sigma^2_l+\epsilon}}+\beta_l$

The epsilon value, usually a small number, is added to avoid divide-by-zero errors.

The output type is a tuple of three ComputationDataHandles:

Outputs	Type	Semantics
`output`	`ComputationDataHandle`	n dimensional array with the same shape as input `operand` (y)
`batch_mean`	`ComputationDataHandle`	1 dimensional array ($\mu$)
`batch_var`	`ComputationDataHandle`	1 dimensional array ($\sigma^2$)

The batch_mean and batch_var are moments calculated across the batch and spatial dimensions using the formulas above.

BitcastConvertType

Similar to a tf.bitcast in TensorFlow, performs an element-wise bitcast operation from a data shape to a target shape. The dimensions must match, and the conversion is an element-wise one; e.g. s32 elements become f32 elements via bitcast routine. Bitcast is implemented as a low-level cast, so machines with different floating-point representations will give different results.

BitcastConvertType(operand, new_element_type)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	array of type T with dims D
`new_element_type`	`PrimitiveType`	type U

The dimensions of the operand and the target shape must match. The bit-width of the source and destination element types must be equal. The source and destination element types must not be tuples.

Broadcast

Adds dimensions to an array by duplicating the data in the array.

Broadcast(operand, broadcast_sizes)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	The array to duplicate
`broadcast_sizes`	`ArraySlice<int64>`	The sizes of the new dimensions

The new dimensions are inserted on the left, i.e. if broadcast_sizes has values {a0, ..., aN} and the operand shape has dimensions {b0, ..., bM} then the shape of the output has dimensions {a0, ..., aN, b0, ..., bM}.

The new dimensions index into copies of the operand, i.e.

output[i0, ..., iN, j0, ..., jM] = operand[j0, ..., jM]

For example, if operand is a scalar f32 with value 2.0f, and broadcast_sizes is {2, 3}, then the result will be an array with shape f32[2, 3] and all the values in the result will be 2.0f.

Call

Arguments	Type	Semantics
`computation`	`Computation`	computation of type `T_0, T_1, ..., T_N -> S` with N parameters of arbitrary type
`args`	sequence of N `ComputationDataHandle`s	N arguments of arbitrary type

Clamp

Arguments	Type	Semantics
`min`	`ComputationDataHandle`	array of type T
`operand`	`ComputationDataHandle`	array of type T
`max`	`ComputationDataHandle`	array of type T

Collapse

See also ComputationBuilder::Collapse and the tf.reshape operation.

Collapses dimensions of an array into one dimension.

Collapse(operand, dimensions)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	array of type T
`dimensions`	`int64` vector	in-order, consecutive subset of T's dimensions.

Collapse replaces the given subset of the operand's dimensions by a single dimension. The input arguments are an arbitrary array of type T and a compile-time-constant vector of dimension indices. The dimension indices must be an in-order (low to high dimension numbers), consecutive subset of T's dimensions. Thus, {0, 1, 2}, {0, 1}, or {1, 2} are all valid dimension sets, but {1, 0} or {0, 2} are not. They are replaced by a single new dimension, in the same position in the dimension sequence as those they replace, with the new dimension size equal to the product of original dimension sizes. The lowest dimension number in dimensions is the slowest varying dimension (most major) in the loop nest which collapses these dimension, and the highest dimension number is fastest varying (most minor). See the tf.reshape operator if more general collapse ordering is needed.

For example, let v be an array of 24 elements:

let v = f32[4x2x3] { { {10, 11, 12},  {15, 16, 17}},
                    { {20, 21, 22},  {25, 26, 27}},
                    { {30, 31, 32},  {35, 36, 37}},
                    { {40, 41, 42},  {45, 46, 47}}};

// Collapse to a single dimension, leaving one dimension.
let v012 = Collapse(v, {0,1,2});
then v012 == f32[24] {10, 11, 12, 15, 16, 17,
                      20, 21, 22, 25, 26, 27,
                      30, 31, 32, 35, 36, 37,
                      40, 41, 42, 45, 46, 47};

// Collapse the two lower dimensions, leaving two dimensions.
let v01 = Collapse(v, {0,1});
then v01 == f32[4x6] { {10, 11, 12, 15, 16, 17},
                      {20, 21, 22, 25, 26, 27},
                      {30, 31, 32, 35, 36, 37},
                      {40, 41, 42, 45, 46, 47}};

// Collapse the two higher dimensions, leaving two dimensions.
let v12 = Collapse(v, {1,2});
then v12 == f32[8x3] { {10, 11, 12},
                      {15, 16, 17},
                      {20, 21, 22},
                      {25, 26, 27},
                      {30, 31, 32},
                      {35, 36, 37},
                      {40, 41, 42},
                      {45, 46, 47}};

Concatenate

Concatenate composes an array from multiple array operands. The array is of the same rank as each of the input array operands (which must be of the same rank as each other) and contains the arguments in the order that they were specified.

Concatenate(operands..., dimension)

Arguments	Type	Semantics
`operands`	sequence of N `ComputationDataHandle`	N arrays of type T with dimensions [L0, L1, ...]. Requires N >= 1.
`dimension`	`int64`	A value in the interval `[0, N)` that names the dimension to be concatenated between the `operands`.

With the exception of dimension all dimensions must be the same. This is because XLA does not support "ragged" arrays. Also note that rank-0 values cannot be concatenated (as it's impossible to name the dimension along which the concatenation occurs).

1-dimensional example:

Concat({ {2, 3}, {4, 5}, {6, 7}}, 0)
>>> {2, 3, 4, 5, 6, 7}

2-dimensional example:

let a = {
  {1, 2},
  {3, 4},
  {5, 6},
};
let b = {
  {7, 8},
};
Concat({a, b}, 0)
>>> {
  {1, 2},
  {3, 4},
  {5, 6},
  {7, 8},
}

Diagram:

Conditional

Conditional(pred, true_operand, true_computation, false_operand, false_computation)

Arguments	Type	Semantics
`pred`	`ComputationDataHandle`	Scalar of type `PRED`
`true_operand`	`ComputationDataHandle`	Argument of type `T_0`
`true_computation`	`Computation`	Computation of type `T_0 -> S`
`false_operand`	`ComputationDataHandle`	Argument of type `T_1`
`false_computation`	`Computation`	Computation of type `T_1 -> S`

Executes true_computation if pred is true, false_computation if pred is false, and returns the result.

The true_computation must take in a single argument of type T_0 and will be invoked with true_operand which must be of the same type. The false_computation must take in a single argument of type T_1 and will be invoked with false_operand which must be of the same type. The type of the returned value of true_computation and false_computation must be the same.

Note that only one of true_computation and false_computation will be executed depending on the value of pred.

Conv (convolution)

ConvWithGeneralPadding (convolution)

Computes a convolution of the kind used in neural networks. Here, a convolution can be thought of as a n-dimensional window moving across a n-dimensional base area and a computation is performed for each possible position of the window.

Arguments	Type	Semantics
`lhs`	`ComputationDataHandle`	rank n+2 array of inputs
`rhs`	`ComputationDataHandle`	rank n+2 array of kernel weights
`window_strides`	`ArraySlice<int64>`	size n array of kernel strides
`padding`	`ArraySlice<pair<int64, int64>>`	size n array of (low, high) padding
`lhs_dilation`	`ArraySlice<int64>`	size n lhs dilation factor
: : : array
`rhs_dilation`	`ArraySlice<int64>`	size n rhs dilation factor
: : : array

Let n be the number of spatial dimensions. The lhs argument is a rank n+2 array describing the base area. This is called the input, even though of course the rhs is also an input. In a neural network, these are the input activations. The n+2 dimensions are, in this order:

batch: Each coordinate in this dimension represents an independent input for which convolution is carried out.
z/depth/features: Each (y,x) position in the base area has a vector associated to it, which goes into this dimension.
spatial_dims: Describes the n spatial dimensions that define the base area that the window moves across.

The rhs argument is a rank n+2 array describing the convolutional filter/kernel/window. The dimensions are, in this order:

output-z: The z dimension of the output.
input-z: The size of this dimension should equal the size of the z dimension in lhs.
spatial_dims: Describes the n spatial dimensions that define the n-d window that moves across the base area.

The window_strides argument specifies the stride of the convolutional window in the spatial dimensions. For example, if the stride in the first spatial dimension is 3, then the window can only be placed at coordinates where the first spatial index is divisible by 3.

The padding argument specifies the amount of zero padding to be applied to the base area. The amount of padding can be negative -- the absolute value of negative padding indicates the number of elements to remove from the specified dimension before doing the convolution. padding[0] specifies the padding for dimension y and padding[1] specifies the padding for dimension x. Each pair has the low padding as the first element and the high padding as the second element. The low padding is applied in the direction of lower indices while the high padding is applied in the direction of higher indices. For example, if padding[1] is (2,3) then there will be a padding by 2 zeroes on the left and by 3 zeroes on the right in the second spatial dimension. Using padding is equivalent to inserting those same zero values into the input (lhs) before doing the convolution.

The lhs_dilation and rhs_dilation arguments specify the dilation factor to be applied to the lhs and rhs, respectively, in each spatial dimension. If the dilation factor in a spatial dimension is d, then d-1 holes are implicitly placed between each of the entries in that dimension, increasing the size of the array. The holes are filled with a no-op value, which for convolution means zeroes.

Dilation of the rhs is also called atrous convolution. For more details, see tf.nn.atrous_conv2d. Dilation of the lhs is also called transposed convolution. For more details, see tf.nn.conv2d_transpose.

The output shape has these dimensions, in this order:

batch: Same size as batch on the input (lhs).
z: Same size as output-z on the kernel (rhs).
spatial_dims: One value for each valid placement of the convolutional window.

The valid placements of the convolutional window are determined by the strides and the size of the base area after padding.

To describe what a convolution does, consider a 2d convolution, and pick some fixed batch, z, y, x coordinates in the output. Then (y,x) is a position of a corner of the window within the base area (e.g. the upper left corner, depending on how you interpret the spatial dimensions). We now have a 2d window, taken from the base area, where each 2d point is associated to a 1d vector, so we get a 3d box. From the convolutional kernel, since we fixed the output coordinate z, we also have a 3d box. The two boxes have the same dimensions, so we can take the sum of the element-wise products between the two boxes (similar to a dot product). That is the output value.

Note that if output-z is e.g., 5, then each position of the window produces 5 values in the output into the z dimension of the output. These values differ in what part of the convolutional kernel is used - there is a separate 3d box of values used for each output-z coordinate. So you could think of it as 5 separate convolutions with a different filter for each of them.

Here is pseudo-code for a 2d convolution with padding and striding:

for (b, oz, oy, ox) {  // output coordinates
  value = 0;
  for (iz, ky, kx) {  // kernel coordinates and input z
    iy = oy*stride_y + ky - pad_low_y;
    ix = ox*stride_x + kx - pad_low_x;
    if ((iy, ix) inside the base area considered without padding) {
      value += input(b, iz, iy, ix) * kernel(oz, iz, ky, kx);
    }
  }
  output(b, oz, oy, ox) = value;
}

ConvertElementType

Similar to an element-wise static_cast in C++, performs an element-wise conversion operation from a data shape to a target shape. The dimensions must match, and the conversion is an element-wise one; e.g. s32 elements become f32 elements via an s32-to-f32 conversion routine.

ConvertElementType(operand, new_element_type)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	array of type T with dims D
`new_element_type`	`PrimitiveType`	type U

The dimensions of the operand and the target shape must match. The source and destination element types must not be tuples.

A conversion such as T=s32 to U=f32 will perform a normalizing int-to-float conversion routine such as round-to-nearest-even.

Note: The precise float-to-int and visa-versa conversions are currently unspecified, but may become additional arguments to the convert operation in the future. Not all possible conversions have been implemented for all targets.

let a: s32[3] = {0, 1, 2};
let b: f32[3] = convert(a, f32);
then b == f32[3]{0.0, 1.0, 2.0}

CrossReplicaSum

Computes a sum across replicas.

CrossReplicaSum(operand)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	Array to sum across replicas.

The output shape is the same as the input shape. For example, if there are two replicas and the operand has the value (1.0, 2.5) and (3.0, 5.25) respectively on the two replicas, then the output value from this op will be (4.0, 7.75) on both replicas.

Computing the result of CrossReplicaSum requires having one input from each replica, so if one replica executes a CrossReplicaSum node more times than another, then the former replica will wait forever. Since the replicas are all running the same program, there are not a lot of ways for that to happen, but it is possible when a while loop's condition depends on data from infeed and the data that is infed causes the while loop to iterate more times on one replica than another.

CustomCall

Call a user-provided function within a computation.

CustomCall(target_name, args..., shape)

Arguments	Type	Semantics
`target_name`	`string`	Name of the function. A call instruction will be emitted which targets this symbol name.
`args`	sequence of N `ComputationDataHandle`s	N arguments of arbitrary type, which will be passed to the function.
`shape`	`Shape`	Output shape of the function

The function signature is the same, regardless of the arity or type of args:

extern "C" void target_name(void* out, void** in);

For example, if CustomCall is used as follows:

let x = f32[2] {1,2};
let y = f32[2x3] { {10, 20, 30}, {40, 50, 60}};

CustomCall("myfunc", {x, y}, f32[3x3])

Here is an example of an implementation of myfunc:

extern "C" void myfunc(void* out, void** in) {
  float (&x)[2] = *static_cast<float(*)[2]>(in[0]);
  float (&y)[2][3] = *static_cast<float(*)[2][3]>(in[1]);
  EXPECT_EQ(1, x[0]);
  EXPECT_EQ(2, x[1]);
  EXPECT_EQ(10, y[0][0]);
  EXPECT_EQ(20, y[0][1]);
  EXPECT_EQ(30, y[0][2]);
  EXPECT_EQ(40, y[1][0]);
  EXPECT_EQ(50, y[1][1]);
  EXPECT_EQ(60, y[1][2]);
  float (&z)[3][3] = *static_cast<float(*)[3][3]>(out);
  z[0][0] = x[1] + y[1][0];
  // ...
}

The user-provided function must not have side-effects and its execution must be idempotent.

Note: The opaque nature of the user-provided function restricts optimization opportunities for the compiler. Try to express your computation in terms of native XLA ops whenever possible; only use CustomCall as a last resort.

Dot

Input	Output	Semantics
vector [n] `dot` vector [n]	scalar	vector dot product
matrix [m x k] `dot` vector [k]	vector [m]	matrix-vector multiplication
matrix [m x k] `dot` matrix [k x n]	matrix [m x n]	matrix-matrix multiplication

DotGeneral

DotGeneral(lhs, rhs, dimension_numbers)

Arguments	Type	Semantics
`lhs`	`ComputationDataHandle`	array of type T
`rhs`	`ComputationDataHandle`	array of type T
`dimension_numbers`	`DotDimensionNumbers`	array of type T

As Dot, but allows contracting and batch dimension numbers to be specified for both the 'lhs' and 'rhs'.

DotDimensionNumbers Fields	Type	Semantics
'lhs_contracting_dimensions'	repeated int64	'lhs' contracting dimension numbers
'rhs_contracting_dimensions'	repeated int64	'rhs' contracting dimension numbers
'lhs_batch_dimensions'	repeated int64	'lhs' batch dimension numbers
'rhs_batch_dimensions'	repeated int64	'rhs' batch dimension numbers

DotGeneral performs the sum of products over contracting dimensions specified in 'dimension_numbers'.

Associated contracting dimension numbers from the 'lhs' and 'rhs' do not need to be the same, but must be listed in the same order in both 'lhs/rhs_contracting_dimensions' arrays and have the same dimension sizes. There must be exactly one contracting dimension on both 'lhs' and 'rhs'.

Example with contracting dimension numbers:

lhs = { {1.0, 2.0, 3.0},
        {4.0, 5.0, 6.0} }

rhs = { {1.0, 1.0, 1.0},
        {2.0, 2.0, 2.0} }

DotDimensionNumbers dnums;
dnums.add_lhs_contracting_dimensions(1);
dnums.add_rhs_contracting_dimensions(1);

DotGeneral(lhs, rhs, dnums) -> { {6.0, 12.0},
                                 {15.0, 30.0} }

Associated batch dimension numbers from the 'lhs' and 'rhs' must have the same dimension number, must be listed in the same order in both arrays, must have the same dimension sizes, and must be ordered before contracting and non-contracting/non-batch dimension numbers.

Example with batch dimension numbers (batch size 2, 2x2 matrices):

lhs = { { {1.0, 2.0},
          {3.0, 4.0} },
        { {5.0, 6.0},
          {7.0, 8.0} } }

rhs = { { {1.0, 0.0},
          {0.0, 1.0} },
        { {1.0, 0.0},
          {0.0, 1.0} } }

DotDimensionNumbers dnums;
dnums.add_lhs_contracting_dimensions(2);
dnums.add_rhs_contracting_dimensions(1);
dnums.add_lhs_batch_dimensions(0);
dnums.add_rhs_batch_dimensions(0);

DotGeneral(lhs, rhs, dnums) -> { { {1.0, 2.0},
                                   {3.0, 4.0} },
                                 { {5.0, 6.0},
                                   {7.0, 8.0} } }

Input	Output	Semantics
[b0, m, k] `dot` [b0, k, n]	[b0, m, n]	batch matmul
[b0, b1, m, k] `dot` [b0, b1, k, n]	[b0, b1, m, n]	batch matmul

It follows that the resulting dimension number starts with the batch dimension, then the 'lhs' non-contracting/non-batch dimension, and finally the 'rhs' non-contracting/non-batch dimension.

DynamicSlice

DynamicSlice extracts a sub-array from the input array at dynamic start_indices. The size of the slice in each dimension is passed in size_indices, which specify the end point of exclusive slice intervals in each dimension: [start, start + size). The shape of start_indices must be rank == 1, with dimension size equal to the rank of operand. Note: handling of out-of-bounds slice indices (generated by incorrect runtime calculation of 'start_indices') is currently implementation-defined.

DynamicSlice(operand, start_indices, size_indices)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	N dimensional array of type T
`start_indices`	`ComputationDataHandle`	Rank 1 array of N integers containing the starting indices of the slice for each dimension. Value must be greater than or equal to zero.
`size_indices`	`ArraySlice<int64>`	List of N integers containing the slice size for each dimension. Each value must be strictly greater than zero, and start + size must be less than or equal to the size of the dimension to avoid wrapping modulo dimension size.

1-dimensional example:

let a = {0.0, 1.0, 2.0, 3.0, 4.0}
let s = {2}

DynamicSlice(a, s, {2}) produces:
  {2.0, 3.0}

2-dimensional example:

let b =
 { {0.0,  1.0,  2.0},
   {3.0,  4.0,  5.0},
   {6.0,  7.0,  8.0},
   {9.0, 10.0, 11.0} }
let s = {2, 1}

DynamicSlice(b, s, {2, 2}) produces:
  { { 7.0,  8.0},
    {10.0, 11.0} }

DynamicUpdateSlice

DynamicUpdateSlice generates a result which is the value of the input array operand, with a slice update overwritten at start_indices. The shape of update determines the shape of the sub-array of the result which is updated. The shape of start_indices must be rank == 1, with dimension size equal to the rank of operand. Note: handling of out-of-bounds slice indices (generated by incorrect runtime calculation of 'start_indices') is currently implementation-defined.

DynamicUpdateSlice(operand, update, start_indices)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	N dimensional array of type T
`update`	`ComputationDataHandle`	N dimensional array of type T containing the slice update. Each dimension of update shape must be strictly greater than zero, and start + update must be less than operand size for each dimension to avoid generating out-of-bounds update indices.
`start_indices`	`ComputationDataHandle`	Rank 1 array of N integers containing the starting indices of the slice for each dimension. Value must be greater than or equal to zero.

1-dimensional example:

let a = {0.0, 1.0, 2.0, 3.0, 4.0}
let u = {5.0, 6.0}
let s = {2}

DynamicUpdateSlice(a, u, s) produces:
  {0.0, 1.0, 5.0, 6.0, 4.0}

2-dimensional example:

let b =
 { {0.0,  1.0,  2.0},
   {3.0,  4.0,  5.0},
   {6.0,  7.0,  8.0},
   {9.0, 10.0, 11.0} }
let u =
 { {12.0,  13.0},
   {14.0,  15.0},
   {16.0,  17.0} }

let s = {1, 1}

DynamicUpdateSlice(b, u, s) produces:
 { {0.0,  1.0,  2.0},
   {3.0, 12.0, 13.0},
   {6.0, 14.0, 15.0},
   {9.0, 16.0, 17.0} }

Element-wise binary arithmetic operations

Arguments	Type	Semantics
`lhs`	`ComputationDataHandle`	left-hand-side operand: array of type T
`rhs`	`ComputationDataHandle`	right-hand-side operand: array of type T

Element-wise comparison operations

Arguments	Type	Semantics
`lhs`	`ComputationDataHandle`	left-hand-side operand: array of type T
`rhs`	`ComputationDataHandle`	right-hand-side operand: array of type T

Element-wise unary functions

ComputationBuilder supports these element-wise unary functions:

Abs(operand) Element-wise abs x -> |x|.

Ceil(operand) Element-wise ceil x -> ⌈x⌉.

Cos(operand) Element-wise cosine x -> cos(x).

Exp(operand) Element-wise natural exponential x -> e^x.

Floor(operand) Element-wise floor x -> ⌊x⌋.

IsFinite(operand) Tests whether each element of operand is finite, i.e., is not positive or negative infinity, and is not NaN. Returns an array of PRED values with the same shape as the input, where each element is true if and only if the corresponding input element is finite.

Log(operand) Element-wise natural logarithm x -> ln(x).

LogicalNot(operand) Element-wise logical not x -> !(x).

Neg(operand) Element-wise negation x -> -x.

Sign(operand) Element-wise sign operation x -> sgn(x) where

$$\text{sgn}(x) = \begin{cases} -1 & x < 0\\ 0 & x = 0\\ 1 & x > 0 \end{cases}$$

using the comparison operator of the element type of operand.

Tanh(operand) Element-wise hyperbolic tangent x -> tanh(x).

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	The operand to the function

The function is applied to each element in the operand array, resulting in an array with the same shape. It is allowed for operand to be a scalar (rank 0).

Gather

The XLA gather operation stitches together several slices (each slice at a potentially different runtime offset) of an input tensor into an output tensor.

General Semantics

See also ComputationBuilder::Gather. For a more intuitive description, see the "Informal Description" section below.

gather(operand, gather_indices, output_window_dims, elided_window_dims, window_bounds, gather_dims_to_operand_dims)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	The tensor we’re gathering from.
`gather_indices`	`ComputationDataHandle`	Tensor containing the starting indices of the slices we're we're stitching together into the output tensor.
`index_vector_dim`	`int64`	The dimension in `gather_indices` that contains the starting indices.
`output_window_dims`	`ArraySlice<int64>`	The set of dimensions in the output shape that are window dimensions (defined below). Not all window dimensions may be present in the output shape.
`elided_window_dims`	`ArraySlice<int64>`	The set of window dimensions that are not present in the output shape. `window_bounds[i]` must be `1` for all `i` in `elided_window_dims`.
`window_bounds`	`ArraySlice<int64>`	`window_bounds[i]` is the bounds for window dimension `i`. This includes both the window dimensions that are explicitly part of the output shape (via `output_window_dims`) and the window dimensions that are elided (via `elided_window_dims`).
`gather_dims_to_operand_dims`	`ArraySlice<int64>`	A dimension map (the array is interpreted as mapping `i` to `gather_dims_to_operand_dims[i]`) from the gather indices in `gather_indices` to the operand index space. It has to be one-to-one and total.

For every index Out in the output tensor, we compute two things (more precisely described later):

An index into gather_indices.rank - 1 dimensions of gather_indices, which gives us a starting index of a slice, operand slice, in the operand tensor. These gather_indices.rank - 1 dimensions are all the dimensions in gather_indices except index_vector_dim.
A window index that has the same rank as the operand. This index is composed of the values in Out at dimensions output_window_dims, embedded with zeroes according to elided_window_dims.

The window index is the relative index of the element in operand slice that should be present in the output at index Out.

The output is a tensor of rank output_window_dims.size + gather_indices.rank - 1. Additionally, as a shorthand, we define output_gather_dims of type ArraySlice<int64> as the set of dimensions in the output shape but not in output_window_dims, in ascending order. E.g. if the output tensor has rank 5, output_window_dims is {2, 4} then output_gather_dims is {0, 1, 3}

If index_vector_dim is equal to gather_indices.rank we implicitly consider gather_indices to have a trailing 1 dimension (i.e. if gather_indices was of shape [6,7] and index_vector_dim is 2 then we implicitly consider the shape of gather_indices to be [6,7,1]).

The bounds for the output tensor along dimension i is computed as follows:

If i is present in output_gather_dims (i.e. is equal to output_gather_dims[k] for some k) then we pick the corresponding dimension bounds out of gather_indices.shape, skipping index_vector_dim (i.e. pick gather_indices.shape.dims[k] if k < index_vector_dim and gather_indices.shape.dims[k+1] otherwise).
If i is present in output_window_dims (i.e. equal to output_window_dims[k] for some k) then we pick the corresponding bound out of window_bounds after accounting for elided_window_dims (i.e. we pick adjusted_window_bounds[k] where adjusted_window_bounds is window_bounds with the bounds at indices elided_window_dims removed).

The operand index In corresponding to an output index Out is computed as follows:

Let G = { Out[k] for k in output_gather_dims }. Use G to slice out vector S such that S[i] = gather_indices[Combine(G, i)] where Combine(A, b) inserts b at position index_vector_dim into A. Note that this is well defined even if G is empty -- if G is empty then S = gather_indices.
Create an index, S_in, into operand using S by scattering S using the gather_dims_to_operand_dims map (S_in is the starting indices for operand slice mentioned above). More precisely:
1. S_in[gather_dims_to_operand_dims[k]] = S[k] if k < gather_dims_to_operand_dims.size.
2. S_in[_] = 0 otherwise.
Create an index W_in into operand by scattering the indices at the output window dimensions in Out according to the elided_window_dims set (W_in is the window index mentioned above). More precisely:
1. W_in[window_dims_to_operand_dims(k)] = Out[k] if k < output_window_dims.size (window_dims_to_operand_dims is defined below).
2. W_in[_] = 0 otherwise.
In is W_in + S_in where + is element-wise addition.

window_dims_to_operand_dims is the monotonic function with domain [0, output_window_dims.size) and range [0, operand.rank) \ elided_window_dims. So if, e.g., output_window_dims.size is 4, operand.rank is 6 and elided_window_dims is {0, 2} then window_dims_to_operand_dims is {0→1, 1→3, 2→4, 3→5}.

Informal Description and Examples

index_vector_dim is set to gather_indices.rank - 1 in all of the examples that follow. More interesting values for index_vector_dim does not change the operation fundamentally, but makes the visual representation more cumbersome.

To get an intuition on how all of the above fits together, let's look at an example that gathers 5 slices of shape [8,6] from a [16,11] tensor. The position of a slice into the [16,11] tensor can be represented as an index vector of shape S64[2], so the set of 5 positions can be represented as a S64[5,2] tensor.

The behavior of the gather operation can then be depicted as an index transformation that takes [G,W₀,W₁], an index in the output shape, and maps it to an element in the input tensor in the following way:

We first select an (X,Y) vector from the gather indices tensor using G. The element in the output tensor at index [G,W₀,W₁] is then the element in the input tensor at index [X+W₀,Y+W₁].

window_bounds is [8,6], which decides the range of W₀ and W₁, and this in turn decides the bounds of the slice.

This gather operation acts as a batch dynamic slice with G as the batch dimension.

The gather indices may be multidimensional. For instance, a more general version of the example above using a "gather indices" tensor of shape [4,5,2] would translate indices like this:

Again, this acts as a batch dynamic slice G₀ and G₁ as the batch dimensions. The window bounds are still [8,6].

The gather operation in XLA generalizes the informal semantics outlined above in the following ways:

We can configure which dimensions in the output shape are the window dimensions (dimensions containing W₀, W₁ in the last example). The output gather dimensions (dimensions containing G₀, G₁ in the last example) are defined to be the output dimensions that are not window dimensions.
The number of output window dimensions explicitly present in the output shape may be smaller than the input rank. These "missing" dimensions, which are listed explicitly as elided_window_dims, must have a window bound of 1. Since they have a window bound of 1 the only valid index for them is 0 and eliding them does not introduce ambiguity.
The slice extracted from the "Gather Indices" tensor ((X, Y) in the last example) may have fewer elements than the input tensor rank, and an explicit mapping dictates how the index should be expanded to have the same rank as the input.

As a final example, we use (2) and (3) to implement tf.gather_nd:

G₀ and G₁ are used to slice out a starting index from the gather indices tensor as usual, except the starting index has only one element, X. Similarly, there is only one output window index with the value W₀. However, before being used as indices into the input tensor, these are expanded in accordance to "Gather Index Mapping" (gather_dims_to_operand_dims in the formal description) and "Window Mapping" (window_dims_to_operand_dims in the formal description) into [0,W₀] and [X,0] respectively, adding up to [X,W₀]. In other words, the output index [G₀,G₁,W₀] maps to the input index [GatherIndices[G₀,G₁,0],X] which gives us the semantics for tf.gather_nd.

window_bounds for this case is [1,11]. Intuitively this means that every index X in the gather indices tensor picks an entire row and the result is the concatenation of all these rows.

GetTupleElement

Indexes into a tuple with a compile-time-constant value.

The value must be a compile-time-constant so that shape inference can determine the type of the resulting value.

This is analogous to std::get<int N>(t) in C++. Conceptually:

let v: f32[10] = f32[10]{0, 1, 2, 3, 4, 5, 6, 7, 8, 9};
let s: s32 = 5;
let t: (f32[10], s32) = tuple(v, s);
let element_1: s32 = gettupleelement(t, 1);  // Inferred shape matches s32.

Infeed

Argument	Type	Semantics
`shape`	`Shape`	Shape of the data read from the Infeed interface. The layout field of the shape must be set to match the layout of the data sent to the device; otherwise its behavior is undefined.

Map

Arguments	Type	Semantics
`operands`	sequence of N `ComputationDataHandle`s	N arrays of types T_0..T_{N-1}
`computation`	`Computation`	computation of type `T_0, T_1, ..., T_{N + M -1} -> S` with N parameters of type T and M of arbitrary type
`dimensions`	`int64` array	array of map dimensions
`static_operands`	sequence of M `ComputationDataHandle`s	M arrays of arbitrary type

Pad

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	array of type `T`
`padding_value`	`ComputationDataHandle`	scalar of type `T` to fill in the added padding
`padding_config`	`PaddingConfig`	padding amount on both edges (low, high) and between the elements of each dimension

Recv

Arguments	Type	Semantics
`shape`	`Shape`	shape of the data to receive
`channel_handle`	`ChannelHandle`	unique identifier for each send/recv pair

Reduce

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	array of type `T`
`init_value`	`ComputationDataHandle`	scalar of type `T`
`computation`	`Computation`	computation of type `T, T -> T`
`dimensions`	`int64` array	unordered array of dimensions to reduce

ReducePrecision

Models the effect of converting floating-point values to a lower-precision format (such as IEEE-FP16) and back to the original format. The number of exponent and mantissa bits in the lower-precision format can be specified arbitrarily, although all bit sizes may not be supported on all hardware implementations.

ReducePrecision(operand, mantissa_bits, exponent_bits)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	array of floating-point type `T`.
`exponent_bits`	`int32`	number of exponent bits in lower-precision format
`mantissa_bits`	`int32`	number of mantissa bits in lower-precision format

The result is an array of type T. The input values are rounded to the nearest value representable with the given number of mantissa bits (using "ties to even" semantics), and any values that exceed the range specified by the number of exponent bits are clamped to positive or negative infinity. NaN values are retained, although they may be converted to canonical NaN values.

The lower-precision format must have at least one exponent bit (in order to distinguish a zero value from an infinity, since both have a zero mantissa), and must have a non-negative number of mantissa bits. The number of exponent or mantissa bits may exceed the corresponding value for type T; the corresponding portion of the conversion is then simply a no-op.

ReduceWindow

Applies a reduction function to all elements in each window of the input multi-dimensional array, producing an output multi-dimensional array with the same number of elements as the number of valid positions of the window. A pooling layer can be expressed as a ReduceWindow.

ReduceWindow(operand, init_value, computation, window_dimensions, window_strides, padding)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	N dimensional array containing elements of type T. This is the base area on which the window is placed.
`init_value`	`ComputationDataHandle`	Starting value for the reduction. See Reduce for details.
`computation`	`Computation`	Reduction function of type `T, T -> T`, to apply to all elements in each window
`window_dimensions`	`ArraySlice<int64>`	array of integers for window dimension values
`window_strides`	`ArraySlice<int64>`	array of integers for window stride values
`padding`	`Padding`	padding type for window (Padding\:\:kSame or Padding\:\:kValid)

Below code and figure shows an example of using ReduceWindow. Input is a matrix of size [4x6] and both window_dimensions and window_stride_dimensions are [2x3].

// Create a computation for the reduction (maximum).
Computation max;
{
  ComputationBuilder builder(client_, "max");
  auto y = builder.Parameter(0, ShapeUtil::MakeShape(F32, {}), "y");
  auto x = builder.Parameter(1, ShapeUtil::MakeShape(F32, {}), "x");
  builder.Max(y, x);
  max = builder.Build().ConsumeValueOrDie();
}

// Create a ReduceWindow computation with the max reduction computation.
ComputationBuilder builder(client_, "reduce_window_2x3");
auto shape = ShapeUtil::MakeShape(F32, {4, 6});
auto input = builder.Parameter(0, shape, "input");
builder.ReduceWindow(
    input, *max,
    /*init_val=*/builder.ConstantLiteral(LiteralUtil::MinValue(F32)),
    /*window_dimensions=*/{2, 3},
    /*window_stride_dimensions=*/{2, 3},
    Padding::kValid);

Stride of 1 in a dimension specifies that the position of a window in the dimension is 1 element away from its adjacent window. In order to specify that no windows overlap with each other, window_stride_dimensions should be equal to window_dimensions. The figure below illustrates the use of two different stride values. Padding is applied to each dimension of the input and the calculations are the same as though the input came in with the dimensions it has after padding.

The evaluation order of the reduction function is arbitrary and may be non-deterministic. Therefore, the reduction function should not be overly sensitive to reassociation. See the discussion about associativity in the context of Reduce for more details.

Reshape

See also ComputationBuilder::Reshape and the Collapse operation.

Reshapes the dimensions of an array into a new configuration.

Reshape(operand, new_sizes) Reshape(operand, dimensions, new_sizes)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	array of type T
`dimensions`	`int64` vector	order in which dimensions are collapsed
`new_sizes`	`int64` vector	vector of sizes of new dimensions

Conceptually, reshape first flattens an array into a one-dimensional vector of data values, and then refines this vector into a new shape. The input arguments are an arbitrary array of type T, a compile-time-constant vector of dimension indices, and a compile-time-constant vector of dimension sizes for the result. The values in the dimension vector, if given, must be a permutation of all of T's dimensions; the default if not given is {0, ..., rank - 1}. The order of the dimensions in dimensions is from slowest-varying dimension (most major) to fastest-varying dimension (most minor) in the loop nest which collapses the input array into a single dimension. The new_sizes vector determines the size of the output array. The value at index 0 in new_sizes is the size of dimension 0, the value at index 1 is the size of dimension 1, and so on. The product of the new_size dimensions must equal the product of the operand's dimension sizes. When refining the collapsed array into the multidimensional array defined by new_sizes, the dimensions in new_sizes are ordered from slowest varying (most major) and to fastest varying (most minor).

For example, let v be an array of 24 elements:

let v = f32[4x2x3] { { {10, 11, 12}, {15, 16, 17}},
                    { {20, 21, 22}, {25, 26, 27}},
                    { {30, 31, 32}, {35, 36, 37}},
                    { {40, 41, 42}, {45, 46, 47}}};

In-order collapse:
let v012_24 = Reshape(v, {0,1,2}, {24});
then v012_24 == f32[24] {10, 11, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27,
                         30, 31, 32, 35, 36, 37, 40, 41, 42, 45, 46, 47};

let v012_83 = Reshape(v, {0,1,2}, {8,3});
then v012_83 == f32[8x3] { {10, 11, 12}, {15, 16, 17},
                          {20, 21, 22}, {25, 26, 27},
                          {30, 31, 32}, {35, 36, 37},
                          {40, 41, 42}, {45, 46, 47}};

Out-of-order collapse:
let v021_24 = Reshape(v, {1,2,0}, {24});
then v012_24 == f32[24]  {10, 20, 30, 40, 11, 21, 31, 41, 12, 22, 32, 42,
                          15, 25, 35, 45, 16, 26, 36, 46, 17, 27, 37, 47};

let v021_83 = Reshape(v, {1,2,0}, {8,3});
then v021_83 == f32[8x3] { {10, 20, 30}, {40, 11, 21},
                          {31, 41, 12}, {22, 32, 42},
                          {15, 25, 35}, {45, 16, 26},
                          {36, 46, 17}, {27, 37, 47}};

let v021_262 = Reshape(v, {1,2,0}, {2,6,2});
then v021_262 == f32[2x6x2] { { {10, 20}, {30, 40},
                              {11, 21}, {31, 41},
                              {12, 22}, {32, 42}},
                             { {15, 25}, {35, 45},
                              {16, 26}, {36, 46},
                              {17, 27}, {37, 47}}};

As a special case, reshape can transform a single-element array to a scalar and vice versa. For example,

Reshape(f32[1x1] { {5}}, {0,1}, {}) == 5;
Reshape(5, {}, {1,1}) == f32[1x1] { {5}};

Rev (reverse)

RngNormal

Constructs an output of a given shape with random numbers generated following

the $$N(\mu, \sigma)$$ normal distribution. The parameters mu and sigma, and

output shape have to have elemental type F32. The parameters furthermore have to be scalar valued.

RngNormal(mean, sigma, shape)

Arguments	Type	Semantics
`mu`	`ComputationDataHandle`	Scalar of type F32 specifying mean of generated numbers
`sigma`	`ComputationDataHandle`	Scalar of type F32 specifying standard deviation of generated numbers
`shape`	`Shape`	Output shape of type F32

RngUniform

Constructs an output of a given shape with random numbers generated following

the uniform distribution over the interval $$[a,b)$$. The parameters and output

shape may be either F32, S32 or U32, but the types have to be consistent.

Furthermore, the parameters need to be scalar valued. If $$b <= a$$ the result

is implementation-defined.

RngUniform(a, b, shape)

Arguments	Type	Semantics
`a`	`ComputationDataHandle`	Scalar of type T specifying lower limit of interval
`b`	`ComputationDataHandle`	Scalar of type T specifying upper limit of interval
`shape`	`Shape`	Output shape of type T

Select

Arguments	Type	Semantics
`pred`	`ComputationDataHandle`	array of type PRED
`on_true`	`ComputationDataHandle`	array of type T
`on_false`	`ComputationDataHandle`	array of type T

SelectAndScatter

This operation can be considered as a composite operation that first computes ReduceWindow on the operand array to select an element from each window, and then scatters the source array to the indices of the selected elements to construct an output array with the same shape as the operand array. The binary select function is used to select an element from each window by applying it across each window, and it is called with the property that the first parameter's index vector is lexicographically less than the second parameter's index vector. The select function returns true if the first parameter is selected and returns false if the second parameter is selected, and the function must hold transitivity (i.e., if select(a, b) and select(b, c) are true, then select(a, c) is also true) so that the selected element does not depend on the order of the elements traversed for a given window.

The function scatter is applied at each selected index in the output array. It takes two scalar parameters:

Current value at the selected index in the output array
The scatter value from source that applies to the selected index

It combines the two parameters and returns a scalar value that's used to update the value at the selected index in the output array. Initially, all indices of the output array are set to init_value.

The output array has the same shape as the operand array and the source array must have the same shape as the result of applying a ReduceWindow operation on the operand array. SelectAndScatter can be used to backpropagate the gradient values for a pooling layer in a neural network.

SelectAndScatter(operand, select, window_dimensions, window_strides, padding, source, init_value, scatter)

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	array of type T over which the windows slide
`select`	`Computation`	binary computation of type `T, T -> PRED`, to apply to all elements in each window; returns `true` if the first parameter is selected and returns `false` if the second parameter is selected
`window_dimensions`	`ArraySlice<int64>`	array of integers for window dimension values
`window_strides`	`ArraySlice<int64>`	array of integers for window stride values
`padding`	`Padding`	padding type for window (Padding\:\:kSame or Padding\:\:kValid)
`source`	`ComputationDataHandle`	array of type T with the values to scatter
`init_value`	`ComputationDataHandle`	scalar value of type T for the initial value of the output array
`scatter`	`Computation`	binary computation of type `T, T -> T`, to apply each scatter source element with its destination element

The figure below shows examples of using SelectAndScatter, with the select function computing the maximal value among its parameters. Note that when the windows overlap, as in the figure (2) below, an index of the operand array may be selected multiple times by different windows. In the figure, the element of value 9 is selected by both of the top windows (blue and red) and the binary addition scatter function produces the output element of value 8 (2 + 6).

The evaluation order of the scatter function is arbitrary and may be non-deterministic. Therefore, the scatter function should not be overly sensitive to reassociation. See the discussion about associativity in the context of Reduce for more details.

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	data to send (array of type T)
`channel_handle`	`ChannelHandle`	unique identifier for each send/recv pair

Arguments	Type	Semantics
`operand`	`ComputationDataHandle`	The operand to transpose.
`permutation`	`ArraySlice<int64>`	How to permute the dimensions.

Arguments	Type	Semantics
`condition`	`Computation`	Computation of type `T -> PRED` which defines the termination condition of the loop.
`body`	`Computation`	Computation of type `T -> T` which defines the body of the loop.
`init`	`T`	Initial value for the parameter of `condition` and `body`.

Operation Semantics

BatchNormGrad

BatchNormInference

BatchNormTraining

BitcastConvertType

Broadcast

Call

Clamp

Collapse

Concatenate

Conditional

Conv (convolution)

ConvWithGeneralPadding (convolution)

ConvertElementType

CrossReplicaSum

CustomCall

Dot

DotGeneral

DynamicSlice

DynamicUpdateSlice

Element-wise binary arithmetic operations

Element-wise comparison operations

Element-wise unary functions

Gather

General Semantics

Informal Description and Examples

GetTupleElement

Infeed

Map

Pad

Recv

Reduce

ReducePrecision

ReduceWindow

Reshape

Rev (reverse)

RngNormal

RngUniform

Select

SelectAndScatter

Send

Slice

Sort

Transpose

Tuple

While