Operation Semantics

The following describes the semantics of operations defined in the XlaBuilder 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.

AllToAll

See also XlaBuilder::AllToAll.

Alltoall is a collective operation that sends data from all cores to all cores. It has two phases:

  1. the scatter phase. On each core, the operand is split into split_count number of blocks along the split_dimensions, and the blocks are scattered to all cores, e.g., the ith block is send to the ith core.
  2. the gather phase. Each core concatenates the received blocks along the concat_dimension.

The participating cores can be configured by:

  • replica_groups: each ReplicaGroup contains a list of replica id. If empty, all replicas belong to one group in the order of 0 - (n-1). Alltoall will be applied within subgroups in the specified order. For example, replica groups = { {1,2,3},{4,5,0}} means, an Alltoall will be applied within replica 1, 2, 3, and in the gather phase, the received blocks will be concatenated in the order of 1, 2, 3; another Alltoall will be applied within replica 4, 5, 0, and the concatenation order is 4, 5, 0.

Prerequisites:

  • The dimension size of the operand on the split_dimension is divisible by split_count.
  • The operand's shape is not tuple.

AllToAll(operand, split_dimension, concat_dimension, split_count, replica_groups)

Arguments Type Semantics
operand XlaOp n dimensional input array
split_dimension int64 A value in the interval [0, n) that names the dimension along which the operand is split
concat_dimension int64 a value in the interval [0, n) that names the dimension along which the split blocks are concatenated
split_count int64 the number of cores that participate this operation. If replica_groups is empty, this should be the number of replicas; otherwise, this should be equal to the number of replicas in each group.
replica_groups ReplicaGroup vector each group contains a list of replica id.

Below shows an example of Alltoall.

XlaBuilder b("alltoall");
auto x = Parameter(&b, 0, ShapeUtil::MakeShape(F32, {4, 16}), "x");
AllToAll(x, /*split_dimension=*/1, /*concat_dimension=*/0, /*split_count=*/4);

In this example, there are 4 cores participating the Alltoall. On each core, the operand is split into 4 parts along dimension 0, so each part has shape f32[4,4]. The 4 parts are scattered to all cores. Then each core concatenates the received parts along dimension 1, in the order or core 0-4. So the output on each core has shape f32[16,4].

BatchNormGrad

See also XlaBuilder::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 XlaOp n dimensional array to be normalized (x)
scale XlaOp 1 dimensional array (\(\gamma\))
mean XlaOp 1 dimensional array (\(\mu\))
variance XlaOp 1 dimensional array (\(\sigma^2\))
grad_output XlaOp 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 array as operand and with feature dimension index \(l\), batch
size m and spatial sizes w and h):

\[ \begin{split} c_l&= \frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \left( \nabla y_{ijkl} \frac{x_{ijkl} - \mu_l}{\sigma^2_l+\epsilon} \right) \\\\ \nabla x_{ijkl} &= \frac{\gamma_{l}}{\sqrt{\sigma^2_{l}+\epsilon}} \left( \nabla y_{ijkl} - \mathrm{mean}(\nabla y) - c_l (x_{ijkl} - \mu_{l}) \right) \\\\ \nabla \gamma_l &= \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \left( \nabla y_{ijkl} \frac{x_{ijkl} - \mu_l}{\sqrt{\sigma^2_{l}+\epsilon}} \right) \\\\\ \nabla \beta_l &= \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \nabla y_{ijkl} \end{split} \]

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

The output type is a tuple of three handles:

Outputs Type Semantics
grad_operand XlaOp gradient with respect to input operand (\( \nabla x\))
grad_scale XlaOp gradient with respect to input scale (\( \nabla \gamma\))
grad_offset XlaOp gradient with respect to input offset(\( \nabla \beta\))

BatchNormInference

See also XlaBuilder::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 XlaOp n dimensional array to be normalized
scale XlaOp 1 dimensional array
offset XlaOp 1 dimensional array
mean XlaOp 1 dimensional array
variance XlaOp 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 XlaBuilder::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 XlaOp n dimensional array to be normalized (x)
scale XlaOp 1 dimensional array (\(\gamma\))
offset XlaOp 1 dimensional array (\(\beta\))
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 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.

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 XlaOps:

Outputs Type Semantics
output XlaOp n dimensional array with the same shape as input operand (y)
batch_mean XlaOp 1 dimensional array (\(\mu\))
batch_var XlaOp 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

See also XlaBuilder::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 XlaOp 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

See also XlaBuilder::Broadcast.

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

Broadcast(operand, broadcast_sizes)

Arguments Type Semantics
operand XlaOp 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

See also XlaBuilder::Call.

Invokes a computation with the given arguments.

Call(computation, args...)

Arguments Type Semantics
computation XlaComputation computation of type T_0, T_1, ..., T_N -> S with N parameters of arbitrary type
args sequence of N XlaOps N arguments of arbitrary type

The arity and types of the args must match the parameters of the computation. It is allowed to have no args.

Clamp

See also XlaBuilder::Clamp.

Clamps an operand to within the range between a minimum and maximum value.

Clamp(min, operand, max)

Arguments Type Semantics
min XlaOp array of type T
operand XlaOp array of type T
max XlaOp array of type T

Given an operand and minimum and maximum values, returns the operand if it is in the range between the minimum and maximum, else returns the minimum value if the operand is below this range or the maximum value if the operand is above this range. That is, clamp(a, x, b) = min(max(a, x), b).

All three arrays must be the same shape. Alternatively, as a restricted form of broadcasting, min and/or max can be a scalar of type T.

Example with scalar min and max:

let operand: s32[3] = {-1, 5, 9};
let min: s32 = 0;
let max: s32 = 6;
==>
Clamp(min, operand, max) = s32[3]{0, 5, 6};

Collapse

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

Collapses dimensions of an array into one dimension.

Collapse(operand, dimensions)

Arguments Type Semantics
operand XlaOp 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

See also XlaBuilder::ConcatInDim.

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 XlaOp 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

See also XlaBuilder::Conditional.

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

Arguments Type Semantics
pred XlaOp Scalar of type PRED
true_operand XlaOp Argument of type T_0
true_computation XlaComputation XlaComputation of type T_0 -> S
false_operand XlaOp Argument of type T_1
false_computation XlaComputation XlaComputation 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)

See also XlaBuilder::Conv.

As ConvWithGeneralPadding, but the padding is specified in a short-hand way as either SAME or VALID. SAME padding pads the input (lhs) with zeroes so that the output has the same shape as the input when not taking striding into account. VALID padding simply means no padding.

ConvWithGeneralPadding (convolution)

See also XlaBuilder::ConvWithGeneralPadding.

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 XlaOp rank n+2 array of inputs
rhs XlaOp rank n+2 array of kernel weights
window_strides ArraySlice<int64> n-d array of kernel strides
padding ArraySlice< pair<int64, int64>> n-d array of (low, high) padding
lhs_dilation ArraySlice<int64> n-d lhs dilation factor array
rhs_dilation ArraySlice<int64> n-d rhs dilation factor array
feature_group_count int64 the number of feature groups

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 times feature_group_count 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 feature_group_count argument (default value 1) can be used for grouped convolutions. feature_group_count needs to be a divisor of both the input and the output feature dimension. If feature_group_count is greater than 1, it means that conceptually the input and output feature dimension and the rhs output feature dimension are split evenly into feature_group_count many groups, each group consisting of a consecutive subsequence of features. The input feature dimension of rhs needs to be equal to the lhs input feature dimension divided by feature_group_count (so it already has the size of a group of input features). The i-th groups are used together to compute feature_group_count many separate convolutions. The results of these convolutions are concatenated together in the output feature dimension.

For depthwise convolution the feature_group_count argument would be set to the input feature dimension, and the filter would be reshaped from [filter_height, filter_width, in_channels, channel_multiplier] to [filter_height, filter_width, 1, in_channels * channel_multiplier]. For more details, see tf.nn.depthwise_conv2d.

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

See also XlaBuilder::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 XlaOp 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.

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

See also XlaBuilder::CrossReplicaSum.

Computes a sum across replicas.

CrossReplicaSum(operand)

Arguments Type Semantics
operand XlaOp Array to sum across replicas.
replica_group_ids int64 vector Group ID for each replica.

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.

replica_group_ids identifies the group ID of each replica. The group ID must either be empty (all replicas belong to a single group), or contain the same number of elements as the number of replicas. For example, if replica_group_ids = {0, 1, 2, 3, 0, 1, 2, 3} has eight replicas, there are four subgroups of replica IDs: {0, 4}, {1, 5}, {2, 6}, and {3, 7}. The size of each subgroup must be identical, so, for example, using: replica_group_ids = {0, 1, 2, 0} for four replicas is invalid.

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

See also XlaBuilder::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 XlaOps 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.

Dot

See also XlaBuilder::Dot.

Dot(lhs, rhs)

Arguments Type Semantics
lhs XlaOp array of type T
rhs XlaOp array of type T

The exact semantics of this operation depend on the ranks of the operands:

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

The operation performs sum of products over the last dimension of lhs and the one-before-last dimension of rhs. These are the "contracted" dimensions. The contracted dimensions of lhs and rhs must be of the same size. In practice, it can be used to perform dot products between vectors, vector/matrix multiplications or matrix/matrix multiplications.

DotGeneral

See also XlaBuilder::DotGeneral.

DotGeneral(lhs, rhs, dimension_numbers)

Arguments Type Semantics
lhs XlaOp array of type T
rhs XlaOp 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

See also XlaBuilder::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.

DynamicSlice(operand, start_indices, size_indices)

Arguments Type Semantics
operand XlaOp N dimensional array of type T
start_indices XlaOp 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. :

The effective slice indices are computed by applying the following transformation for each index i in [1, N) before performing the slice:

start_indices[i] = clamp(start_indices[i], 0, operand.dimension_size[i] - size_indices[i])

This ensures that the extracted slice is always in-bounds with respect to the operand array. If the slice is in-bounds before the transformation is applied, the transformation has no effect.

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

See also XlaBuilder::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.

DynamicUpdateSlice(operand, update, start_indices)

Arguments Type Semantics
operand XlaOp N dimensional array of type T
update XlaOp 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 or equal to the operand size for each dimension to avoid generating out-of-bounds update indices.
start_indices XlaOp 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.

The effective slice indices are computed by applying the following transformation for each index i in [1, N) before performing the slice:

start_indices[i] = clamp(start_indices[i], 0, operand.dimension_size[i] - update.dimension_size[i])

This ensures that the updated slice is always in-bounds with respect to the operand array. If the slice is in-bounds before the transformation is applied, the transformation has no effect.

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

See also XlaBuilder::Add.

A set of element-wise binary arithmetic operations is supported.

Op(lhs, rhs)

Where Op is one of Add (addition), Sub (subtraction), Mul (multiplication), Div (division), Rem (remainder), Max (maximum), Min (minimum), LogicalAnd (logical AND), or LogicalOr (logical OR).

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

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

When Op is Rem, the sign of the result is taken from the dividend, and the absolute value of the result is always less than the divisor's absolute value.

Integer division overflow (signed/unsigned division/remainder by zero or signed divison/remainder of INT_SMIN with -1) produces an implementation defined value.

An alternative variant with different-rank broadcasting support exists for these operations:

Op(lhs, rhs, broadcast_dimensions)

Where Op is the same as above. This variant of the operation should be used for arithmetic operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers used to expand the rank of the lower-rank operand up to the rank of the higher-rank operand. broadcast_dimensions maps the dimensions of the lower-rank shape to the dimensions of the higher-rank shape. The unmapped dimensions of the expanded shape are filled with dimensions of size one. Degenerate-dimension broadcasting then broadcasts the shapes along these degenerate dimensions to equalize the shapes of both operands. The semantics are described in detail on the broadcasting page.

Element-wise comparison operations

See also XlaBuilder::Eq.

A set of standard element-wise binary comparison operations is supported. Note that standard IEEE 754 floating-point comparison semantics apply when comparing floating-point types.

Op(lhs, rhs)

Where Op is one of Eq (equal-to), Ne (not equal-to), Ge (greater-or-equal-than), Gt (greater-than), Le (less-or-equal-than), Lt (less-than).

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

The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays with the element type PRED. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.

An alternative variant with different-rank broadcasting support exists for these operations:

Op(lhs, rhs, broadcast_dimensions)

Where Op is the same as above. This variant of the operation should be used for comparison operations between arrays of different ranks (such as adding a matrix to a vector).

The additional broadcast_dimensions operand is a slice of integers specifying the dimensions to use for broadcasting the operands. The semantics are described in detail on the broadcasting page.

Element-wise unary functions

XlaBuilder 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 XlaOp 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 array.

General Semantics

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

gather(operand, start_indices, offset_dims, collapsed_slice_dims, slice_sizes, start_index_map)

Arguments Type Semantics
operand XlaOp The array we’re gathering from.
start_indices XlaOp Array containing the starting indices of the slices we gather.
index_vector_dim int64 The dimension in start_indices that "contains" the starting indices. See below for a detailed description.
offset_dims ArraySlice<int64> The set of dimensions in the output shape that offset into a array sliced from operand.
slice_sizes ArraySlice<int64> slice_sizes[i] is the bounds for the slice on dimension i.
collapsed_slice_dims ArraySlice<int64> The set of dimensions in each : slice that are collapsed away. These dimensions must have size 1.
start_index_map ArraySlice<int64> A map that describes how to map indices in start_indices to legal indices into operand.

For convenience, we label dimensions in the output array not in offset_dims as batch_dims.

The output is an array of rank batch_dims.size + operand.rank - collapsed_slice_dims.size.

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

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

  1. If i is present in batch_dims (i.e. is equal to batch_dims[k] for some k) then we pick the corresponding dimension bounds out of start_indices.shape, skipping index_vector_dim (i.e. pick start_indices.shape.dims[k] if k < index_vector_dim and start_indices.shape.dims[k+1] otherwise).

  2. If i is present in offset_dims (i.e. equal to offset_dims[k] for some k) then we pick the corresponding bound out of slice_sizes after accounting for collapsed_slice_dims (i.e. we pick adjusted_slice_sizes[k] where adjusted_slice_sizes is slice_sizes with the bounds at indices collapsed_slice_dims removed).

Formally, the operand index In corresponding to an output index Out is computed as follows:

  1. Let G = { Out[k] for k in batch_dims }. Use G to slice out vector S such that S[i] = start_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 = start_indices.

  2. Create a starting index, Sin, into operand using S by scattering S using start_index_map. More precisely:

    1. Sin[start_index_map[k]] = S[k] if k < start_index_map.size.
    2. Sin[_] = 0 otherwise.
  3. Create an index Oin into operand by scattering the indices at the offset dimensions in Out according to the collapsed_slice_dims set. More precisely:

    1. Oin[expand_offset_dims(k)] = Out[offset_dims[k]] if k < offset_dims.size (expand_offset_dims is defined below).
    2. Oin[_] = 0 otherwise.
  4. In is Oin + Sin where + is element-wise addition.

expand_offset_dims is the monotonic function with domain [0, offset.size) and range [0, operand.rank) \ collapsed_slice_dims. So if, e.g., offset.size is 4, operand.rank is 6 and collapsed_slice_dims is {0, 2} then expand_offset_dims is {01, 13, 24, 35}.

Informal Description and Examples

Informally, every index Out in the output array corresponds to an element E in the operand array, computed as follows:

  • We use the batch dimensions in Out to look up a starting index from start_indices.

  • We use start_index_map to map the starting index (which may have size less than operand.rank) to a "full" starting index into operand.

  • We dynamic-slice out a slice with size slice_sizes using the full starting index.

  • We reshape the slice by collapsing the collapsed_slice_dims dimensions. Since all collapsed slice dimensions have to have bound 1 this reshape is always legal.

  • We use the offset dimensions in Out to index into this slice to get the input element, E, corresponding to output index Out.

index_vector_dim is set to start_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] array. The position of a slice into the [16,11] array 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] array.

The behavior of the gather operation can then be depicted as an index transformation that takes [G,O0,O1], an index in the output shape, and maps it to an element in the input array in the following way:

We first select an (X,Y) vector from the gather indices array using G. The element in the output array at index [G,O0,O1] is then the element in the input array at index [X+O0,Y+O1].

slice_sizes 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" array 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 slice size is still [8,6].

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

  1. We can configure which dimensions in the output shape are the offset dimensions (dimensions containing O0, O1 in the last example). The output batch dimensions (dimensions containing G0, G1 in the last example) are defined to be the output dimensions that are not offset dimensions.

  2. The number of output offset dimensions explicitly present in the output shape may be smaller than the input rank. These "missing" dimensions, which are listed explicitly as collapsed_slice_dims, must have a slice size of 1. Since they have a slice size of 1 the only valid index for them is 0 and eliding them does not introduce ambiguity.

  3. The slice extracted from the "Gather Indices" array ((X, Y) in the last example) may have fewer elements than the input array 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 array as usual, except the starting index has only one element, X. Similarly, there is only one output offset index with the value O0. However, before being used as indices into the input array, these are expanded in accordance to "Gather Index Mapping" (start_index_map in the formal description) and "Offset Mapping" (expand_offset_dims in the formal description) into [X,0] and [0,O0] respectively, adding up to [X,O0]. In other words, the output index [G0,G1,O0] maps to the input index [GatherIndices[G0,G1,0],X] which gives us the semantics for tf.gather_nd.

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

GetDimensionSize

See also XlaBuilder::GetDimensionSize.

Returns the size of the given dimension of the operand. The operand must be array shaped.

GetDimensionSize(operand, dimension)

Arguments Type Semantics
operand XlaOp n dimensional input array
dimension int64 A value in the interval [0, n) that specifies the dimension

GetTupleElement

See also XlaBuilder::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.

See also tf.tuple.

Infeed

See also XlaBuilder::Infeed.

Infeed(shape)

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.

Reads a single data item from the implicit Infeed streaming interface of the device, interpreting the data as the given shape and its layout, and returns a XlaOp of the data. Multiple Infeed operations are allowed in a computation, but there must be a total order among the Infeed operations. For example, two Infeeds in the code below have a total order since there is a dependency between the while loops.

result1 = while (condition, init = init_value) {
  Infeed(shape)
}

result2 = while (condition, init = result1) {
  Infeed(shape)
}

Nested tuple shapes are not supported. For an empty tuple shape, the Infeed operation is effectively a no-op and proceeds without reading any data from the Infeed of the device.

Iota

Iota()

Builds a constant literal on device rather than a potentially large host transfer. Creates a rank 1 array of values starting at zero and incrementing by one.

Arguments Type Semantics
type PrimitiveType type U
size int64 The number of elements in the array.

Map

See also XlaBuilder::Map.

Map(operands..., computation)

Arguments Type Semantics
operands sequence of N XlaOps N arrays of types T0..T{N-1}
computation XlaComputation 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 |

Applies a scalar function over the given operands arrays, producing an array of the same dimensions where each element is the result of the mapped function applied to the corresponding elements in the input arrays.

The mapped function is an arbitrary computation with the restriction that it has N inputs of scalar type T and a single output with type S. The output has the same dimensions as the operands except that the element type T is replaced with S.

For example: Map(op1, op2, op3, computation, par1) maps elem_out <- computation(elem1, elem2, elem3, par1) at each (multi-dimensional) index in the input arrays to produce the output array.

Pad

See also XlaBuilder::Pad.

Pad(operand, padding_value, padding_config)

Arguments Type Semantics
operand XlaOp array of type T
padding_value XlaOp 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

Expands the given operand array by padding around the array as well as between the elements of the array with the given padding_value. padding_config specifies the amount of edge padding and the interior padding for each dimension.

PaddingConfig is a repeated field of PaddingConfigDimension, which contains three fields for each dimension: edge_padding_low, edge_padding_high, and interior_padding.

edge_padding_low and edge_padding_high specify the amount of padding added at the low-end (next to index 0) and the high-end (next to the highest index) of each dimension respectively. The amount of edge padding can be negative -- the absolute value of negative padding indicates the number of elements to remove from the specified dimension.

interior_padding specifies the amount of padding added between any two elements in each dimension; it may not be negative. Interior padding occurs logically before edge padding, so in the case of negative edge padding, elements are removed from the interior-padded operand.

This operation is a no-op if the edge padding pairs are all (0, 0) and the interior padding values are all 0. The figure below shows examples of different edge_padding and interior_padding values for a two-dimensional array.

Recv

See also XlaBuilder::Recv.

Recv(shape, channel_handle)

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

Receives data of the given shape from a Send instruction in another computation that shares the same channel handle. Returns a XlaOp for the received data.

The client API of Recv operation represents synchronous communication. However, the instruction is internally decomposed into 2 HLO instructions (Recv and RecvDone) to enable asynchronous data transfers. See also HloInstruction::CreateRecv and HloInstruction::CreateRecvDone.

Recv(const Shape& shape, int64 channel_id)

Allocates resources required to receive data from a Send instruction with the same channel_id. Returns a context for the allocated resources, which is used by a following RecvDone instruction to wait for the completion of the data transfer. The context is a tuple of {receive buffer (shape), request identifier (U32)} and it can only be used by a RecvDone instruction.

RecvDone(HloInstruction context)

Given a context created by a Recv instruction, waits for the data transfer to complete and returns the received data.

Reduce

See also XlaBuilder::Reduce.

Applies a reduction function to one or more arrays in parallel.

Reduce(operands..., init_values..., computation, dimensions)

Arguments Type Semantics
operands Sequence of N XlaOp N arrays of types T_0, ..., T_N.
init_values Sequence of N XlaOp N scalars of types T_0, ..., T_N.
computation XlaComputation computation of type
          :                       : `T_0, ..., T_N, T_0, ..., T_N -> Collate(T_0, ..., T_N)`

dimensions | int64 array | unordered array of dimensions to reduce

Where: * N is required to be greater or equal to 1. * All input arrays must have the same dimensions. * If N = 1, Collate(T) is T. * If N > 1, Collate(T_0, ..., T_N) is a tuple of N elements of type T.

The output of the op is Collate(Q_0, ..., Q_N) where Q_i is an array of type T_i, the dimensions of which are described below.

This operation reduces one or more dimensions of each input array into scalars. The rank of each returned array is rank(operand) - len(dimensions). init_value is the initial value used for every reduction and may be inserted anywhere during computation by the back-end. In most cases, init_value is an identity of the reduction function (for example, 0 for addition). The applied computation is always passed the init_value on the left-hand side.

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.

Some reduction functions like addition are not strictly associative for floats. However, if the range of the data is limited, floating-point addition is close enough to being associative for most practical uses. It is possible to conceive of some completely non-associative reductions, however, and these will produce incorrect or unpredictable results in XLA reductions.

As an example, when reducing across one dimension in a single 1D array with values [10, 11, 12, 13], with reduction function f (this is computation) then that could be computed as

f(10, f(11, f(12, f(init_value, 13)))

but there are also many other possibilities, e.g.

f(init_value, f(f(10, f(init_value, 11)), f(f(init_value, 12), f(init_value, 13))))

The following is a rough pseudo-code example of how reduction could be implemented, using summation as the reduction computation with an initial value of 0.

result_shape <- remove all dims in dimensions from operand_shape

# Iterate over all elements in result_shape. The number of r's here is equal
# to the rank of the result
for r0 in range(result_shape[0]), r1 in range(result_shape[1]), ...:
  # Initialize this result element
  result[r0, r1...] <- 0

  # Iterate over all the reduction dimensions
  for d0 in range(dimensions[0]), d1 in range(dimensions[1]), ...:
    # Increment the result element with the value of the operand's element.
    # The index of the operand's element is constructed from all ri's and di's
    # in the right order (by construction ri's and di's together index over the
    # whole operand shape).
    result[r0, r1...] += operand[ri... di]

Here's an example of reducing a 2D array (matrix). The shape has rank 2, dimension 0 of size 2 and dimension 1 of size 3:

Results of reducing dimensions 0 or 1 with an "add" function:

Note that both reduction results are 1D arrays. The diagram shows one as column and another as row just for visual convenience.

For a more complex example, here is a 3D array. Its rank is 3, dimension 0 of size 4, dimension 1 of size 2 and dimension 2 of size 3. For simplicity, the values 1 to 6 are replicated across dimension 0.

Similarly to the 2D example, we can reduce just one dimension. If we reduce dimension 0, for example, we get a rank-2 array where all values across dimension 0 were folded into a scalar:

|  4   8  12 |
| 16  20  24 |

If we reduce dimension 2, we also get a rank-2 array where all values across dimension 2 were folded into a scalar:

| 6  15 |
| 6  15 |
| 6  15 |
| 6  15 |

Note that the relative order between the remaining dimensions in the input is preserved in the output, but some dimensions may get assigned new numbers (since the rank changes).

We can also reduce multiple dimensions. Add-reducing dimensions 0 and 1 produces the 1D array | 20 28 36 |.

Reducing the 3D array over all its dimensions produces the scalar 84.

When N > 1, reduce function application is slightly more complex, as it is applied simultaneously to all inputs. For example, consider the following reduction function, which can be used to compute the max and the argmax of a a 1-D array in parallel:

f: (Float, Int, Float, Int) -> Float, Int
f(max, argmax, value, index):
  if value >= argmax:
    return (value, index)
  else:
    return (max, argmax)

For 1-D Input arrays V = Float[N], K = Int[N], and init values I_V = Float, I_K = Int, the result f_(N-1) of reducing across the only input dimension is equivalent to the following recursive application:

f_0 = f(I_V, I_K, V_0, K_0)
f_1 = f(f_0.first, f_0.second, V_1, K_1)
...
f_(N-1) = f(f_(N-2).first, f_(N-2).second, V_(N-1), K_(N-1))

Applying this reduction to an array of values, and an array of sequential indices (i.e. iota), will co-iterate over the arrays, and return a tuple containing the maximal value and the matching index.

ReducePrecision

See also XlaBuilder::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 XlaOp 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

See also XlaBuilder::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. Similar to Reduce, the applied computation is always passed the init_value on the left-hand side.

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

Arguments Type Semantics
operand XlaOp N dimensional array containing elements of type T. This is the base area on which the window is placed.
init_value XlaOp Starting value for the reduction. See Reduce for details.
computation XlaComputation 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
base_dilations ArraySlice<int64> array of integers for base dilation values
window_dilations ArraySlice<int64> array of integers for window dilation 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).
XlaComputation max;
{
  XlaBuilder 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.
XlaBuilder 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 XlaBuilder::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 XlaOp 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)

See also XlaBuilder::Rev.

Rev(operand, dimensions)

Arguments Type Semantics
operand XlaOp array of type T
dimensions ArraySlice<int64> dimensions to reverse

Reverses the order of elements in the operand array along the specified dimensions, generating an output array of the same shape. Each element of the operand array at a multidimensional index is stored into the output array at a transformed index. The multidimensional index is transformed by reversing the index in each dimension to be reversed (i.e., if a dimension of size N is one of the reversing dimensions, its index i is transformed into N - 1 - i).

One use for the Rev operation is to reverse the convolution weight array along the two window dimensions during the gradient computation in neural networks.

RngNormal

See also XlaBuilder::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 a floating point elemental type. The
parameters furthermore have to be scalar valued.

RngNormal(mu, sigma, shape)

Arguments Type Semantics
mu XlaOp Scalar of type T specifying mean of generated numbers
sigma XlaOp Scalar of type T specifying standard deviation of generated numbers
shape Shape Output shape of type T

RngUniform

See also XlaBuilder::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
element type have to be a boolean type, an integral type or a floating point types, and the types have to be consistent. The CPU and GPU backends currently only support F64, F32, F16, BF16, S64, U64, S32 and U32. Furthermore, the
parameters need to be scalar valued. If \(b <= a\) the result is
implementation-defined.

RngUniform(a, b, shape)

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

Scatter

The XLA scatter operation generates a result which is the value of the input array operand, with several slices (at indices specified by scatter_indices) updated with the values in updates using update_computation.

See also XlaBuilder::Scatter.

scatter(operand, scatter_indices, updates, update_computation, index_vector_dim, update_window_dims, inserted_window_dims, scatter_dims_to_operand_dims)

Arguments Type Semantics
operand XlaOp Array to be scattered into.
scatter_indices XlaOp Array containing the starting indices of the slices that must be scattered to.
updates XlaOp Array containing the values that must be used for scattering.
update_computation XlaComputation Computation to be used for combining the existing values in the input array and the updates during scatter. This computation should be of type T, T -> T.
index_vector_dim int64 The dimension in scatter_indices that contains the starting indices.
update_window_dims ArraySlice<int64> The set of dimensions in updates shape that are window dimensions.
inserted_window_dims ArraySlice<int64> The set of window dimensions that must be inserted into updates shape.
scatter_dims_to_operand_dims ArraySlice<int64> A dimensions map from the scatter indices to the operand index space. This array is interpreted as mapping i to scatter_dims_to_operand_dims[i] . It has to be one-to-one and total.

If index_vector_dim is equal to scatter_indices.rank we implicitly consider scatter_indices to have a trailing 1 dimension.

We define update_scatter_dims of type ArraySlice<int64> as the set of dimensions in updates shape that are not in update_window_dims, in ascending order.

The arguments of scatter should follow these constraints:

  • updates array must be of rank update_window_dims.size + scatter_indices.rank - 1.

  • Bounds of dimension i in updates must conform to the following:

    • If i is present in update_window_dims (i.e. equal to update_window_dims[k] for some k), then the bound of dimension i in updates must not exceed the corresponding bound of operand after accounting for the inserted_window_dims (i.e. adjusted_window_bounds[k], where adjusted_window_bounds contains the bounds of operand with the bounds at indices inserted_window_dims removed).
    • If i is present in update_scatter_dims (i.e. equal to update_scatter_dims[k] for some k), then the bound of dimension i in updates must be equal to the corresponding bound of scatter_indices, skipping index_vector_dim (i.e. scatter_indices.shape.dims[k], if k < index_vector_dim and scatter_indices.shape.dims[k+1] otherwise).
  • update_window_dims must be in ascending order, not have any repeating dimension numbers, and be in the range [0, updates.rank).

  • inserted_window_dims must be in ascending order, not have any repeating dimension numbers, and be in the range [0, operand.rank).

  • scatter_dims_to_operand_dims.size must be equal to scatter_indices[index_vector_dim], and its values must be in the range [0, operand.rank).

For a given index U in the updates array, the corresponding index I in the operand array into which this update has to be applied is computed as follows:

  1. Let G = { U[k] for k in update_scatter_dims }. Use G to look up an index vector S in the scatter_indices array such that S[i] = scatter_indices[Combine(G, i)] where Combine(A, b) inserts b at positions index_vector_dim into A.
  2. Create an index Sin into operand using S by scattering S using the scatter_dims_to_operand_dims map. More formally:
    1. Sin[scatter_dims_to_operand_dims[k]] = S[k] if k < scatter_dims_to_operand_dims.size.
    2. Sin[_] = 0 otherwise.
  3. Create an index Win into operand by scattering the indices at update_window_dims in U according to inserted_window_dims. More formally:
    1. Win[window_dims_to_operand_dims(k)] = U[k] if k < update_window_dims.size, where window_dims_to_operand_dims is the monotonic function with domain [0, update_window_dims.size) and range [0, operand.rank) \ inserted_window_dims. (For example, if update_window_dims.size is 4, operand.rank is 6, and inserted_window_dims is {0, 2} then window_dims_to_operand_dims is {01, 13, 24, 35}).
    2. Win[_] = 0 otherwise.
  4. I is Win + Sin where + is element-wise addition.

In summary, the scatter operation can be defined as follows.

  • Initialize output with operand, i.e. for all indices O in the operand array:
    output[O] = operand[O]
  • For every index U in the updates array and the corresponding index O in the operand array:
    output[O] = update_computation(output[O], updates[U])

The order in which updates are applied is non-deterministic. So, when multiple indices in updates refer to the same index in operand, the corresponding value in output will be non-deterministic.

Note that the first parameter that is passed into the update_computation will always be the current value from the output array and the second parameter will always be the value from the updates array. This is important specifically for cases when the update_computation is not commutative.

Informally, the scatter op can be viewed as an inverse of the gather op, i.e. the scatter op updates the elements in the input that are extracted by the corresponding gather op.

For a detailed informal description and examples, refer to the "Informal Description" section under Gather.

Select

See also XlaBuilder::Select.

Constructs an output array from elements of two input arrays, based on the values of a predicate array.

Select(pred, on_true, on_false)

Arguments Type Semantics
pred XlaOp array of type PRED
on_true XlaOp array of type T
on_false XlaOp array of type T

The arrays on_true and on_false must have the same shape. This is also the shape of the output array. The array pred must have the same dimensionality as on_true and on_false, with the PRED element type.

For each element P of pred, the corresponding element of the output array is taken from on_true if the value of P is true, and from on_false if the value of P is false. As a restricted form of broadcasting, pred can be a scalar of type PRED. In this case, the output array is taken wholly from on_true if pred is true, and from on_false if pred is false.

Example with non-scalar pred:

let pred: PRED[4] = {true, false, false, true};
let v1: s32[4] = {1, 2, 3, 4};
let v2: s32[4] = {100, 200, 300, 400};
==>
Select(pred, v1, v2) = s32[4]{1, 200, 300, 4};

Example with scalar pred:

let pred: PRED = true;
let v1: s32[4] = {1, 2, 3, 4};
let v2: s32[4] = {100, 200, 300, 400};
==>
Select(pred, v1, v2) = s32[4]{1, 2, 3, 4};

Selections between tuples are supported. Tuples are considered to be scalar types for this purpose. If on_true and on_false are tuples (which must have the same shape!) then pred has to be a scalar of type PRED.

SelectAndScatter

See also XlaBuilder::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:

  1. Current value at the selected index in the output array
  2. 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 XlaOp array of type T over which the windows slide
select XlaComputation 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 XlaOp array of type T with the values to scatter
init_value XlaOp scalar value of type T for the initial value of the output array
scatter XlaComputation 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.

Send

See also XlaBuilder::Send.

Send(operand, channel_handle)

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

Sends the given operand data to a Recv instruction in another computation that shares the same channel handle. Does not return any data.

Similar to the Recv operation, the client API of Send operation represents synchronous communication, and is internally decomposed into 2 HLO instructions (Send and SendDone) to enable asynchronous data transfers. See also HloInstruction::CreateSend and HloInstruction::CreateSendDone.

Send(HloInstruction operand, int64 channel_id)

Initiates an asynchronous transfer of the operand to the resources allocated by the Recv instruction with the same channel id. Returns a context, which is used by a following SendDone instruction to wait for the completion of the data transfer. The context is a tuple of {operand (shape), request identifier (U32)} and it can only be used by a SendDone instruction.

SendDone(HloInstruction context)

Given a context created by a Send instruction, waits for the data transfer to complete. The instruction does not return any data.

Scheduling of channel instructions

The execution order of the 4 instructions for each channel (Recv, RecvDone, Send, SendDone) is as below.

  • Recv happens before Send
  • Send happens before RecvDone
  • Recv happens before RecvDone
  • Send happens before SendDone

When the backend compilers generate a linear schedule for each computation that communicates via channel instructions, there must not be cycles across the computations. For example, below schedules lead to deadlocks.

Slice

See also XlaBuilder::Slice.

Slicing extracts a sub-array from the input array. The sub-array is of the same rank as the input and contains the values inside a bounding box within the input array where the dimensions and indices of the bounding box are given as arguments to the slice operation.

Slice(operand, start_indices, limit_indices)

Arguments Type Semantics
operand XlaOp N dimensional array of type T
start_indices ArraySlice<int64> List of N integers containing the starting indices of the slice for each dimension. Values must be greater than or equal to zero.
limit_indices ArraySlice<int64> List of N integers containing the ending indices (exclusive) for the slice for each dimension. Each value must be greater than or equal to the respective start_indices value for the dimension and less than or equal to the size of the dimension.

1-dimensional example:

let a = {0.0, 1.0, 2.0, 3.0, 4.0}
Slice(a, {2}, {4}) 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} }

Slice(b, {2, 1}, {4, 3}) produces:
  { { 7.0,  8.0},
    {10.0, 11.0} }

Sort

See also XlaBuilder::Sort.

There are two versions of the Sort instruction: a single-operand and a multi-operand version.

Sort(operand, dimension)

Arguments Type Semantics
operand XlaOp The operand to sort.
dimension int64 The dimension along which to sort.

Sorts the elements in the operand in ascending order along the provided dimension. For example, for a rank-2 (matrix) operand, a dimension value of 0 will sort each column independently, and a dimension value of 1 will sort each row independently. If the operand's elements have floating point type, and the operand contains NaN elements, the order of elements in the output is implementation-defined.

Sort(keys, values, ... values, dimension)

Sorts both the key and one or more value operands. The keys are sorted as in the single-operand version. Each of the values inputs is sorted according to the order of the corresponding keys. For example, if the three inputs are keys = [3, 1], values0 = [42, 50], values1 = [-3.0, 1.1], then the output of the sort is the tuple {[1, 3], [50, 42], [1.1, -3.0]}.

The sort is not guaranteed to be stable, that is, if the keys array contains duplicates, the order of values corresponding to these keys may not be preserved.

Arguments Type Semantics
keys XlaOp The sort keys.
values Sequence of N XlaOps The values to sort.
dimension int64 The dimension along which to sort.

The keys and each of the values inputs must have the same dimensions, but may have different element types.

Transpose

See also the tf.reshape operation.

Transpose(operand)

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

Permutes the operand dimensions with the given permutation, so ∀ i . 0 ≤ i < rank ⇒ input_dimensions[permutation[i]] = output_dimensions[i].

This is the same as Reshape(operand, permutation, Permute(permutation, operand.shape.dimensions)).

Tuple

See also XlaBuilder::Tuple.

A tuple containing a variable number of data handles, each of which has its own shape.

This is analogous to std::tuple 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);

Tuples can be deconstructed (accessed) via the GetTupleElement operation.

While

See also XlaBuilder::While.

While(condition, body, init)

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

Sequentially executes the body until the condition fails. This is similar to a typical while loop in many other languages except for the differences and restrictions listed below.

  • A While node returns a value of type T, which is the result from the last execution of the body.
  • The shape of the type T is statically determined and must be the same across all iterations.

The T parameters of the computations are initialized with the init value in the first iteration and are automatically updated to the new result from body in each subsequent iteration.

One main use case of the While node is to implement the repeated execution of training in neural networks. Simplified pseudocode is shown below with a graph that represents the computation. The code can be found in while_test.cc. The type T in this example is a Tuple consisting of an int32 for the iteration count and a vector[10] for the accumulator. For 1000 iterations, the loop keeps adding a constant vector to the accumulator.

// Pseudocode for the computation.
init = {0, zero_vector[10]} // Tuple of int32 and float[10].
result = init;
while (result(0) < 1000) {
  iteration = result(0) + 1;
  new_vector = result(1) + constant_vector[10];
  result = {iteration, new_vector};
}