Semántica de operaciones

A continuación se describe la semántica de las operaciones definidas en la interfaz XlaBuilder . Normalmente, estas operaciones se asignan uno a uno a las operaciones definidas en la interfaz RPC en xla_data.proto .

Una nota sobre la nomenclatura: el tipo de datos generalizado que trata XLA es una matriz N-dimensional que contiene elementos de algún tipo uniforme (como un flotante de 32 bits). En toda la documentación, matriz se utiliza para indicar una matriz de dimensiones arbitrarias. Por conveniencia, los casos especiales tienen nombres más específicos y familiares; por ejemplo, un vector es una matriz unidimensional y una matriz es una matriz bidimensional.

Después de todo

Véase también XlaBuilder::AfterAll .

AfterAll toma una cantidad variable de tokens y produce un solo token. Los tokens son tipos primitivos que se pueden interconectar entre operaciones de efectos secundarios para hacer cumplir los pedidos. AfterAll se puede utilizar como una combinación de tokens para ordenar una operación después de un conjunto de operaciones.

AfterAll(operands)

Todos reunidos

Véase también XlaBuilder::AllGather .

Realiza concatenación entre réplicas.

AllGather(operand, all_gather_dim, shard_count, replica_group_ids, channel_id)

• replica_groups es una lista de grupos de réplicas entre los cuales se realiza la concatenación (la identificación de réplica de la réplica actual se puede recuperar usando ReplicaId ). El orden de las réplicas en cada grupo determina el orden en que se ubican sus entradas en el resultado. replica_groups debe estar vacío (en cuyo caso todas las réplicas pertenecen a un solo grupo, ordenado de 0 a N - 1 ) o contener la misma cantidad de elementos que la cantidad de réplicas. Por ejemplo, replica_groups = {0, 2}, {1, 3} realiza una concatenación entre las réplicas 0 y 2 y 1 y 3 .
• shard_count es el tamaño de cada grupo de réplicas. Necesitamos esto en los casos en que replica_groups estén vacíos.
• channel_id se utiliza para la comunicación entre módulos: solo las operaciones all-gather con el mismo channel_id pueden comunicarse entre sí.

La forma de salida es la forma de entrada con all_gather_dim hecho shard_count veces más grande. Por ejemplo, si hay dos réplicas y el operando tiene el valor [1.0, 2.5] y [3.0, 5.25] respectivamente en las dos réplicas, entonces el valor de salida de esta operación donde all_gather_dim es 0 será [1.0, 2.5, 3.0, 5.25] en ambas réplicas.

TodoReducir

Véase también XlaBuilder::AllReduce .

Realiza un cálculo personalizado entre réplicas.

AllReduce(operand, computation, replica_group_ids, channel_id)

• Cuando operand es una tupla de matrices, la reducción total se realiza en cada elemento de la tupla.
• replica_groups es una lista de grupos de réplicas entre los cuales se realiza la reducción (la identificación de réplica de la réplica actual se puede recuperar usando ReplicaId ). replica_groups debe estar vacío (en cuyo caso todas las réplicas pertenecen a un solo grupo) o contener la misma cantidad de elementos que la cantidad de réplicas. Por ejemplo, replica_groups = {0, 2}, {1, 3} realiza una reducción entre las réplicas 0 y 2 y 1 y 3 .
• channel_id se utiliza para la comunicación entre módulos: solo las operaciones all-reduce con el mismo channel_id pueden comunicarse entre sí.

La forma de salida es la misma que la forma de entrada. Por ejemplo, si hay dos réplicas y el operando tiene el valor [1.0, 2.5] y [3.0, 5.25] respectivamente en las dos réplicas, entonces el valor de salida de esta operación y cálculo de suma será [4.0, 7.75] en ambas. réplicas. Si la entrada es una tupla, la salida también es una tupla.

Calcular el resultado de AllReduce requiere tener una entrada de cada réplica, por lo que si una réplica ejecuta un nodo AllReduce más veces que otra, entonces la réplica anterior esperará para siempre. Dado que todas las réplicas ejecutan el mismo programa, no hay muchas formas de que eso suceda, pero es posible cuando la condición de un bucle while depende de los datos de la alimentación y los datos que se introducen hacen que el bucle while se repita más veces. en una réplica que en otra.

Todos a todos

Consulte también XlaBuilder::AllToAll .

AllToAll es una operación colectiva que envía datos de todos los núcleos a todos los núcleos. Tiene dos fases:

1. La fase de dispersión. En cada núcleo, el operando se divide en un número de bloques split_count a lo largo de split_dimensions , y los bloques se distribuyen en todos los núcleos, por ejemplo, el i-ésimo bloque se envía al i-ésimo núcleo.
2. La fase de reunión. Cada núcleo concatena los bloques recibidos a lo largo de concat_dimension .

Los núcleos participantes se pueden configurar mediante:

• replica_groups : cada ReplicaGroup contiene una lista de ID de réplica que participan en el cálculo (el ID de réplica de la réplica actual se puede recuperar usando ReplicaId ). AllToAll se aplicará dentro de los subgrupos en el orden especificado. Por ejemplo, replica_groups = { {1,2,3}, {4,5,0} } significa que se aplicará un AllToAll dentro de las réplicas {1, 2, 3} y en la fase de recopilación, y los bloques recibidos se se concatenarán en el mismo orden de 1, 2, 3. Luego, se aplicará otro AllToAll dentro de las réplicas 4, 5, 0, y el orden de concatenación también es 4, 5, 0. Si replica_groups está vacío, todas las réplicas pertenecen a una grupo, en el orden de concatenación de su aparición.

Requisitos previos:

• El tamaño de la dimensión del operando en split_dimension es divisible por split_count .
• La forma del operando no es tupla.

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

A continuación se muestra un ejemplo de 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);

En este ejemplo, hay 4 núcleos que participan en Alltoall. En cada núcleo, el operando se divide en 4 partes a lo largo de la dimensión 1, por lo que cada parte tiene la forma f32[4,4]. Las 4 partes están repartidas en todos los núcleos. Luego, cada núcleo concatena las partes recibidas a lo largo de la dimensión 0, en el orden del núcleo 0-4. Entonces la salida de cada núcleo tiene la forma f32[16,4].

Graduación de norma por lotes

Consulte también XlaBuilder::BatchNormGrad y el documento de normalización de lotes original para obtener una descripción detallada del algoritmo.

Calcula los gradientes de la norma del lote.

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

Para cada característica en la dimensión de la característica ( feature_index es el índice de la dimensión de la característica en operand ), la operación calcula los gradientes con respecto al operand , offset y scale en todas las demás dimensiones. feature_index debe ser un índice válido para la dimensión de característica en operand .

Los tres gradientes se definen mediante las siguientes fórmulas (asumiendo una matriz de 4 dimensiones como operand y con índice de dimensión de característica l , tamaño de lote m y tamaños espaciales w y h ):

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

La mean y variance de las entradas representan el valor de los momentos en las dimensiones espaciales y de lote.

El tipo de salida es una tupla de tres identificadores:

Inferencia de norma por lotes

Consulte también XlaBuilder::BatchNormInference y el documento de normalización de lotes original para obtener una descripción detallada del algoritmo.

Normaliza una matriz en dimensiones espaciales y por lotes.

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

Para cada característica en la dimensión de característica ( feature_index es el índice de la dimensión de característica en operand ), la operación calcula la media y la varianza en todas las demás dimensiones y usa la media y la varianza para normalizar cada elemento en operand . feature_index debe ser un índice válido para la dimensión de característica en operand .

BatchNormInference equivale a llamar BatchNormTraining sin calcular mean y variance de cada lote. En su lugar, utiliza la mean y variance de entrada como valores estimados. El propósito de esta operación es reducir la latencia en la inferencia, de ahí el nombre BatchNormInference .

La salida es una matriz normalizada de n dimensiones con la misma forma que operand de entrada.

Entrenamiento de normas por lotes

Consulte también XlaBuilder::BatchNormTraining y the original batch normalization paper para obtener una descripción detallada del algoritmo.

Normaliza una matriz en dimensiones espaciales y por lotes.

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

Para cada característica en la dimensión de característica ( feature_index es el índice de la dimensión de característica en operand ), la operación calcula la media y la varianza en todas las demás dimensiones y usa la media y la varianza para normalizar cada elemento en operand . feature_index debe ser un índice válido para la dimensión de característica en operand .

El algoritmo es el siguiente para cada lote en operand \(x\) que contiene m elementos con w y h como tamaño de dimensiones espaciales (asumiendo que operand es una matriz de 4 dimensiones):

• Calcula la media del lote \(\mu_l\) para cada característica l en la dimensión de la característica:\(\mu_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h x_{ijkl}\)

• Calcula la variación del lote \(\sigma^2_l\):\(\sigma^2_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (x_{ijkl} - \mu_l)^2\)

• Normaliza, escala y cambia:\(y_{ijkl}=\frac{\gamma_l(x_{ijkl}-\mu_l)}{\sqrt[2]{\sigma^2_l+\epsilon} }+\beta_l\)

El valor épsilon, normalmente un número pequeño, se suma para evitar errores de división por cero.

El tipo de salida es una tupla de tres XlaOp s:

batch_mean y batch_var son momentos calculados en las dimensiones espaciales y de lote utilizando las fórmulas anteriores.

Tipo de conversión de Bitcast

Consulte también XlaBuilder::BitcastConvertType .

Similar a un tf.bitcast en TensorFlow, realiza una operación de difusión de bits por elementos desde una forma de datos a una forma de destino. El tamaño de entrada y salida debe coincidir: por ejemplo, los elementos s32 se convierten en elementos f32 mediante la rutina bitcast y un elemento s32 se convertirá en cuatro elementos s8 . Bitcast se implementa como una conversión de bajo nivel, por lo que las máquinas con diferentes representaciones de punto flotante darán resultados diferentes.

BitcastConvertType(operand, new_element_type)

Las dimensiones del operando y la forma objetivo deben coincidir, excepto la última dimensión que cambiará según la relación del tamaño primitivo antes y después de la conversión.

Los tipos de elementos de origen y destino no deben ser tuplas.

Conversión de Bitcast a tipo primitivo de diferente ancho

La instrucción BitcastConvert HLO admite el caso en el que el tamaño del elemento de salida tipo T' no es igual al tamaño del elemento de entrada T Como toda la operación es conceptualmente un bitcast y no cambia los bytes subyacentes, la forma del elemento de salida tiene que cambiar. Para B = sizeof(T), B' = sizeof(T') , hay dos casos posibles.

Primero, cuando B > B' , la forma de salida obtiene una nueva dimensión menor de tamaño B/B' . Por ejemplo:

  f16[10,2]{1,0} %output = f16[10,2]{1,0} bitcast-convert(f32[10]{0} %input)

La regla sigue siendo la misma para los escalares efectivos:

  f16[2]{0} %output = f16[2]{0} bitcast-convert(f32[] %input)

Alternativamente, para B' > B la instrucción requiere que la última dimensión lógica de la forma de entrada sea igual a B'/B y esta dimensión se elimina durante la conversión:

  f32[10]{0} %output = f32[10]{0} bitcast-convert(f16[10,2]{1,0} %input)

Tenga en cuenta que las conversiones entre diferentes anchos de bits no se realizan por elementos.

Transmisión

Consulte también XlaBuilder::Broadcast .

Agrega dimensiones a una matriz duplicando los datos de la matriz.

Broadcast(operand, broadcast_sizes)

Las nuevas dimensiones se insertan a la izquierda, es decir, si broadcast_sizes tiene valores {a0, ..., aN} y la forma del operando tiene dimensiones {b0, ..., bM} entonces la forma de la salida tiene dimensiones {a0, ..., aN, b0, ..., bM} .

Las nuevas dimensiones se indexan en copias del operando, es decir

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

Por ejemplo, si operand es un escalar f32 con valor 2.0f y broadcast_sizes es {2, 3} , entonces el resultado será una matriz con forma f32[2, 3] y todos los valores del resultado serán 2.0f .

Transmisión en intensidad

Consulte también XlaBuilder::BroadcastInDim .

Expande el tamaño y la clasificación de una matriz duplicando los datos de la matriz.

BroadcastInDim(operand, out_dim_size, broadcast_dimensions)

Similar a Broadcast, pero permite agregar dimensiones en cualquier lugar y expandir las dimensiones existentes con tamaño 1.

El operand se transmite a la forma descrita por out_dim_size . broadcast_dimensions asigna las dimensiones del operand a las dimensiones de la forma de destino, es decir, la i-ésima dimensión del operando se asigna a la broadcast_dimension[i]'ésima dimensión de la forma de salida. Las dimensiones del operand deben tener tamaño 1 o ser del mismo tamaño que la dimensión en la forma de salida a la que están asignadas. Las dimensiones restantes se llenan con dimensiones de tamaño 1. La transmisión de dimensiones degeneradas luego transmite a lo largo de estas dimensiones degeneradas para alcanzar la forma de salida. La semántica se describe en detalle en la página de transmisión .

Llamar

Consulte también XlaBuilder::Call .

Invoca un cálculo con los argumentos dados.

Call(computation, args...)

La aridad y los tipos de los args deben coincidir con los parámetros del computation . Se permite no tener args .

cholesky

Véase también XlaBuilder::Cholesky .

Calcula la descomposición de Cholesky de un lote de matrices definidas positivas simétricas (hermitianas).

Cholesky(a, lower)

Si lower es true , calcula matrices triangulares inferiores l tales que \( a = l . l^T \). Si lower es false , calcula matrices triangulares superiores u tales que \( a = u^T . u \).

Los datos de entrada se leen solo desde el triángulo inferior/superior de a , dependiendo del valor de lower . Se ignoran los valores del otro triángulo. Los datos de salida se devuelven en el mismo triángulo; los valores en el otro triángulo están definidos por la implementación y pueden ser cualquier cosa.

Si el rango de a es mayor que 2, a se trata como un lote de matrices, donde todas, excepto las 2 dimensiones menores, son dimensiones de lote.

Si a no es definida positiva simétrica (hermitiana), el resultado está definido por la implementación.

Abrazadera

Consulte también XlaBuilder::Clamp .

Fija un operando dentro del rango entre un valor mínimo y máximo.

Clamp(min, operand, max)

Dado un operando y los valores mínimo y máximo, devuelve el operando si está en el rango entre el mínimo y el máximo; en caso contrario, devuelve el valor mínimo si el operando está por debajo de este rango o el valor máximo si el operando está por encima de este rango. Es decir, clamp(a, x, b) = min(max(a, x), b) .

Las tres matrices deben tener la misma forma. Alternativamente, como forma restringida de transmisión , min y/o max pueden ser un escalar de tipo T

Ejemplo con escalar min y 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};

Colapsar

Consulte también XlaBuilder::Collapse y la operación tf.reshape .

Contrae las dimensiones de una matriz en una dimensión.

Collapse(operand, dimensions)

Contraer reemplaza el subconjunto dado de dimensiones del operando por una única dimensión. Los argumentos de entrada son una matriz arbitraria de tipo T y un vector de índices de dimensión constante en tiempo de compilación. Los índices de dimensión deben ser un subconjunto consecutivo en orden (números de dimensión de menor a mayor) de las dimensiones de T. Por lo tanto, {0, 1, 2}, {0, 1} o {1, 2} son todos conjuntos de dimensiones válidos, pero {1, 0} o {0, 2} no lo son. Se reemplazan por una única dimensión nueva, en la misma posición en la secuencia de dimensiones que las que reemplazan, con el nuevo tamaño de dimensión igual al producto de los tamaños de dimensión originales. El número de dimensión más bajo en dimensions es la dimensión que varía más lentamente (la más importante) en el nido de bucle que colapsa estas dimensiones, y el número de dimensión más alto es la que varía más rápidamente (la más menor). Consulte el operador tf.reshape si necesita un orden de colapso más general.

Por ejemplo, sea v una matriz de 24 elementos:

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} };

ColectivoPermutar

Véase también XlaBuilder::CollectivePermute .

CollectivePermute es una operación colectiva que envía y recibe réplicas cruzadas de datos.

CollectivePermute(operand, source_target_pairs)

Tenga en cuenta que existen las siguientes restricciones en source_target_pair :

• Dos pares cualesquiera no deben tener el mismo ID de réplica de destino y no deben tener el mismo ID de réplica de origen.
• Si una identificación de réplica no es un objetivo en ningún par, entonces la salida de esa réplica es un tensor que consta de 0(s) con la misma forma que la entrada.

Concatenar

Consulte también XlaBuilder::ConcatInDim .

Concatenar compone una matriz a partir de múltiples operandos de matriz. La matriz tiene el mismo rango que cada uno de los operandos de la matriz de entrada (que deben tener el mismo rango entre sí) y contiene los argumentos en el orden en que fueron especificados.

Concatenate(operands..., dimension)

A excepción de dimension todas las dimensiones deben ser iguales. Esto se debe a que XLA no admite matrices "irregulares". Tenga en cuenta también que los valores de rango 0 no se pueden concatenar (ya que es imposible nombrar la dimensión a lo largo de la cual se produce la concatenación).

Ejemplo unidimensional:

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

Ejemplo bidimensional:

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

Diagrama:

Condicional

Consulte también XlaBuilder::Conditional .

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

Ejecuta true_computation si pred es true , false_computation si pred es false y devuelve el resultado.

true_computation debe aceptar un único argumento de tipo \(T_0\) y se invocará con true_operand , que debe ser del mismo tipo. El false_computation debe aceptar un único argumento de tipo \(T_1\) y se invocará con false_operand que debe ser del mismo tipo. El tipo de valor devuelto de true_computation y false_computation debe ser el mismo.

Tenga en cuenta que solo se ejecutará uno de true_computation y false_computation dependiendo del valor de pred .

Conditional(branch_index, branch_computations, branch_operands)

Ejecuta branch_computations[branch_index] y devuelve el resultado. Si branch_index es un S32 que es < 0 o >= N, entonces branch_computations[N-1] se ejecuta como la rama predeterminada.

Cada branch_computations[b] debe tomar un único argumento de tipo T_b y será invocado con branch_operands[b] que deben ser del mismo tipo. El tipo de valor devuelto de cada branch_computations[b] debe ser el mismo.

Tenga en cuenta que solo se ejecutará uno de los branch_computations dependiendo del valor de branch_index .

Conv (convolución)

Véase también XlaBuilder::Conv .

Como ConvWithGeneralPadding, pero el relleno se especifica de forma abreviada como SAME o VALID. El MISMO relleno rellena la entrada ( lhs ) con ceros para que la salida tenga la misma forma que la entrada cuando no se tienen en cuenta las zancadas. El relleno VÁLIDO simplemente significa que no hay relleno.

ConvWithGeneralPadding (convolución)

Consulte también XlaBuilder::ConvWithGeneralPadding .

Calcula una convolución del tipo utilizado en las redes neuronales. Aquí, se puede considerar una convolución como una ventana de n dimensiones que se mueve a través de un área base de n dimensiones y se realiza un cálculo para cada posición posible de la ventana.

Sea n el número de dimensiones espaciales. El argumento lhs es una matriz de rango n+2 que describe el área base. Esto se llama entrada, aunque, por supuesto, el lado derecho también es una entrada. En una red neuronal, estas son las activaciones de entrada. Las n+2 dimensiones son, en este orden:

• batch : cada coordenada en esta dimensión representa una entrada independiente para la cual se realiza la convolución.
• z/depth/features : Cada posición (y,x) en el área base tiene un vector asociado, que entra en esta dimensión.
• spatial_dims : Describe las n dimensiones espaciales que definen el área base por la que se mueve la ventana.

El argumento rhs es una matriz de rango n+2 que describe el filtro/núcleo/ventana convolucional. Las dimensiones son, en este orden:

• output-z : La dimensión z de la salida.
• input-z : El tamaño de esta dimensión multiplicado feature_group_count debe ser igual al tamaño de la dimensión z en lhs.
• spatial_dims : Describe las n dimensiones espaciales que definen la segunda ventana que se mueve a través del área base.

El argumento window_strides especifica el paso de la ventana convolucional en las dimensiones espaciales. Por ejemplo, si el paso en la primera dimensión espacial es 3, entonces la ventana solo se puede colocar en las coordenadas donde el primer índice espacial es divisible por 3.

El argumento padding especifica la cantidad de relleno de ceros que se aplicará al área base. La cantidad de relleno puede ser negativa: el valor absoluto del relleno negativo indica el número de elementos que se eliminarán de la dimensión especificada antes de realizar la convolución. padding[0] especifica el relleno para la dimensión y y padding[1] especifica el relleno para la dimensión x . Cada par tiene el relleno bajo como primer elemento y el relleno alto como segundo elemento. El relleno bajo se aplica en la dirección de los índices más bajos, mientras que el relleno alto se aplica en la dirección de los índices más altos. Por ejemplo, si padding[1] es (2,3) , habrá un relleno de 2 ceros a la izquierda y de 3 ceros a la derecha en la segunda dimensión espacial. Usar relleno equivale a insertar esos mismos valores cero en la entrada ( lhs ) antes de realizar la convolución.

Los argumentos lhs_dilation y rhs_dilation especifican el factor de dilatación que se aplicará a lhs y rhs, respectivamente, en cada dimensión espacial. Si el factor de dilatación en una dimensión espacial es d, entonces se colocan implícitamente d-1 agujeros entre cada una de las entradas en esa dimensión, aumentando el tamaño de la matriz. Los huecos se rellenan con un valor no operativo, que para convolución significa ceros.

La dilatación del rhs también se denomina convolución atroz. Para obtener más detalles, consulte tf.nn.atrous_conv2d . La dilatación del lado izquierdo también se llama convolución transpuesta. Para obtener más detalles, consulte tf.nn.conv2d_transpose .

El argumento feature_group_count (valor predeterminado 1) se puede utilizar para convoluciones agrupadas. feature_group_count debe ser un divisor de la dimensión de la característica de entrada y de salida. Si feature_group_count es mayor que 1, significa que conceptualmente la dimensión de la característica de entrada y salida y la dimensión de la característica de salida rhs se dividen uniformemente en muchos grupos feature_group_count , cada grupo consta de una subsecuencia consecutiva de características. La dimensión de la característica de entrada de rhs debe ser igual a la dimensión de la característica de entrada lhs dividida por feature_group_count (por lo que ya tiene el tamaño de un grupo de características de entrada). Los i-ésimos grupos se utilizan juntos para calcular feature_group_count muchas convoluciones separadas. Los resultados de estas convoluciones se concatenan juntos en la dimensión de la característica de salida.

Para una convolución en profundidad, el argumento feature_group_count se establecería en la dimensión de la característica de entrada y el filtro se reformaría de [filter_height, filter_width, in_channels, channel_multiplier] a [filter_height, filter_width, 1, in_channels * channel_multiplier] . Para obtener más detalles, consulte tf.nn.depthwise_conv2d .

El argumento batch_group_count (valor predeterminado 1) se puede utilizar para filtros agrupados durante la retropropagación. batch_group_count debe ser un divisor del tamaño de la dimensión del lhs izquierdo (entrada). Si batch_group_count es mayor que 1, significa que la dimensión del lote de salida debe ser del tamaño input batch / batch_group_count . batch_group_count debe ser un divisor del tamaño de la característica de salida.

La forma de salida tiene estas dimensiones, en este orden:

• batch : el tamaño de esta dimensión multiplicado batch_group_count debe ser igual al tamaño de la dimensión batch en lhs.
• z : Mismo tamaño que output-z en el kernel ( rhs ).
• spatial_dims : un valor para cada ubicación válida de la ventana convolucional.

La figura anterior muestra cómo funciona el campo batch_group_count . Efectivamente, dividimos cada lote de lhs en grupos batch_group_count grupos de lotes y hacemos lo mismo con las funciones de salida. Luego, para cada uno de estos grupos hacemos convoluciones por pares y concatenamos la salida a lo largo de la dimensión de la característica de salida. La semántica operativa de todas las demás dimensiones (característica y espacial) sigue siendo la misma.

Las ubicaciones válidas de la ventana convolucional están determinadas por los pasos y el tamaño del área base después del relleno.

Para describir lo que hace una convolución, considere una convolución 2d y elija algunas coordenadas batch fijas, z , y , x en la salida. Entonces (y,x) es la posición de una esquina de la ventana dentro del área base (por ejemplo, la esquina superior izquierda, dependiendo de cómo interprete las dimensiones espaciales). Ahora tenemos una ventana 2D, tomada del área base, donde cada punto 2D está asociado a un vector 1D, por lo que obtenemos un cuadro 3D. Desde el núcleo convolucional, dado que fijamos la coordenada de salida z , también tenemos un cuadro 3d. Los dos cuadros tienen las mismas dimensiones, por lo que podemos tomar la suma de los productos de elementos entre los dos cuadros (similar a un producto escalar). Ese es el valor de salida.

Tenga en cuenta que si output-z es, por ejemplo, 5, entonces cada posición de la ventana produce 5 valores en la salida en la dimensión z de la salida. Estos valores difieren en la parte del núcleo convolucional que se utiliza: hay un cuadro 3D separado de valores que se utiliza para cada coordenada output-z . Entonces podrías considerarlo como 5 convoluciones separadas con un filtro diferente para cada una de ellas.

Aquí hay un pseudocódigo para una convolución 2D con relleno y zancadas:

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;
}

Convertir tipo de elemento

Consulte también XlaBuilder::ConvertElementType .

Similar a un static_cast por elementos en C++, realiza una operación de conversión por elementos de una forma de datos a una forma de destino. Las dimensiones deben coincidir y la conversión se realiza por elementos; por ejemplo, los elementos s32 se convierten en elementos f32 mediante una rutina de conversión s32 a f32 .

ConvertElementType(operand, new_element_type)

Las dimensiones del operando y la forma objetivo deben coincidir. Los tipos de elementos de origen y destino no deben ser tuplas.

Una conversión como T=s32 a U=f32 realizará una rutina de normalización de conversión de int a flotante, como redondear a par más cercano.

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

CruzReplicaSum

Realiza AllReduce con un cálculo de suma.

Llamada personalizada

Consulte también XlaBuilder::CustomCall .

Llame a una función proporcionada por el usuario dentro de un cálculo.

CustomCall(target_name, args..., shape)

La firma de la función es la misma, independientemente de la aridad o el tipo de argumentos:

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

Por ejemplo, si CustomCall se utiliza de la siguiente manera:

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

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

A continuación se muestra un ejemplo de una implementación de 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];
  // ...
}

La función proporcionada por el usuario no debe tener efectos secundarios y su ejecución debe ser idempotente.

Punto

Véase también XlaBuilder::Dot .

Dot(lhs, rhs)

La semántica exacta de esta operación depende de los rangos de los operandos:

La operación realiza suma de productos sobre la segunda dimensión de lhs (o la primera si tiene rango 1) y la primera dimensión de rhs . Estas son las dimensiones "contraídas". Las dimensiones contratadas de lhs y rhs deben ser del mismo tamaño. En la práctica, se puede utilizar para realizar productos escalares entre vectores, multiplicaciones de vector/matriz o multiplicaciones de matriz/matriz.

PuntoGeneral

Véase también XlaBuilder::DotGeneral .

DotGeneral(lhs, rhs, dimension_numbers)

Como Dot, pero permite especificar números de dimensión de lote y de contratación tanto para el 'izquierdo' como para el 'derecho'.

DotGeneral realiza la suma de productos sobre las dimensiones de contratación especificadas en 'dimension_numbers'.

Los números de dimensión de contratación asociados de 'izquierdo' y 'derecho' no necesitan ser los mismos, pero deben tener los mismos tamaños de dimensión.

Ejemplo con números de dimensión en contracción:

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} }

Los números de dimensión de lote asociados de 'lhs' y 'rhs' deben tener los mismos tamaños de dimensión.

Ejemplo con números de dimensiones de lote (tamaño de lote 2, matrices de 2x2):

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} } }

De ello se deduce que el número de dimensión resultante comienza con la dimensión de lote, luego la dimensión 'lhs' no contractual/no de lote y, finalmente, la dimensión 'rhs' no contractual/no de lote.

Rebanada dinámica

Consulte también XlaBuilder::DynamicSlice .

DynamicSlice extrae una submatriz de la matriz de entrada en Dynamic start_indices . El tamaño del segmento en cada dimensión se pasa en size_indices , que especifica el punto final de los intervalos de segmento exclusivos en cada dimensión: [inicio, inicio + tamaño). La forma de start_indices debe tener rango == 1, con un tamaño de dimensión igual al rango del operand .

DynamicSlice(operand, start_indices, size_indices)

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)

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), Atan2 (arctangent of y/x), LogicalAnd (logical AND), LogicalOr (logical OR), or LogicalXor (logical XOR).

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 division/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). Another set of operators, EqTotalOrder, NeTotalOrder, GeTotalOrder, GtTotalOrder, LeTotalOrder, and LtTotalOrder, provide the same functionalities, except that they additionally support a total order over the floating point numbers, by enforcing -NaN < -Inf < -Finite < -0 < +0 < +Finite < +Inf < +NaN.

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⌉ .

Clz(operand) Element-wise counting of the number of leading zeros x -> clz(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⌋ .

Imag(operand) Element-wise imaginary part of a complex (or real) shape. x -> imag(x) . If the operand is a floating point type, returns 0.

IsFinite(operand) Tests whether each element of operand is finite, ie, 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) .

Log1p(operand) Element-wise natural logarithm of a number plus one x -> ln(x + 1)

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

Logistic(operand) Element-wise logistic function computation x -> logistic(x) .

PopulationCount(operand) Computes the number of bits set in each element of operand .

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

Real(operand) Element-wise real part of a complex (or real) shape. x -> real(x) . If the operand is a floating point type, returns the same value.

Rsqrt(operand) Element-wise reciprocal of square root operation x -> 1.0 / sqrt(x) .

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

\[\text{sgn}(x) = \begin{cases} -1 & x < 0\\ -0 & x = -0\\ NaN & x = NaN\\ +0 & x = +0\\ 1 & x > 0 \end{cases}\]

using the comparison operator of the element type of operand .

Sqrt(operand) Element-wise square root operation x -> sqrt(x) .

Cbrt(operand) Element-wise cubic root operation x -> cbrt(x) .

Tan(operand) Element-wise tangent x -> tan(x) .

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

Round(operand) Element-wise rounding, ties away from zero.

RoundNearestEven(operand) Element-wise rounding, ties to nearest even.

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).

Fft

The XLA FFT operation implements the forward and inverse Fourier Transforms for real and complex inputs/outputs. Multidimensional FFTs on up to 3 axes are supported.

See also XlaBuilder::Fft .

Multidimensional FFT

When more than 1 fft_length is provided, this is equivalent to applying a cascade of FFT operations to each of the innermost axes. Note that for the real->complex and complex->real cases, the innermost axis transform is (effectively) performed first (RFFT; last for IRFFT), which is why the innermost axis is the one which changes size. Other axis transforms will then be complex->complex.

Implementation details

CPU FFT is backed by Eigen's TensorFFT. GPU FFT uses cuFFT.

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)

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 + offset_dims.size .

The operand.rank must equal the sum of offset_dims.size and collapsed_slice_dims.size . Also, slice_sizes.size has to be equal to operand.rank .

If index_vector_dim is equal to start_indices.rank we implicitly consider start_indices to have a trailing 1 dimension (ie 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 (ie 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 (ie 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 (ie equal to offset_dims [ k ] for some k ) then we pick the corresponding bound out of slice_sizes after accounting for collapsed_slice_dims (ie 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 a given output index Out is calculated as follows:

1. Let G = { Out [ k ] for k in batch_dims }. Use G to slice out a 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, S in , into operand using S by scattering S using start_index_map . More precisely:

   1. S in [ start_index_map [ k ]] = S [ k ] if k < start_index_map.size .

   2. S in [ _ ] = 0 otherwise.

3. Create an index O in into operand by scattering the indices at the offset dimensions in Out according to the collapsed_slice_dims set. More precisely:

   1. O in [ remapped_offset_dims ( k )] = Out [ offset_dims [ k ]] if k < offset_dims.size ( remapped_offset_dims is defined below).

   2. O in [ _ ] = 0 otherwise.

4. In is O in + S in where + is element-wise addition.

remapped_offset_dims is a monotonic function with domain [ 0 , offset_dims.size ) and range [ 0 , operand.rank ) \ collapsed_slice_dims . So if, eg, offset_dims.size is 4 , operand.rank is 6 and collapsed_slice_dims is { 0 , 2 } then remapped_offset_dims is { 0 → 1 , 1 → 3 , 2 → 4 , 3 → 5 }.

If indices_are_sorted is set to true then XLA can assume that start_indices are sorted (in ascending start_index_map order) by the user. If they are not then the semantics is implementation defined.

If unique_indices is set to true then XLA can assume that all element scattered to are unique. So XLA could use non-atomic operations. If unique_indices is set to true and the indices being scattered to are not unique then the semantics is implementation defined.

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 (whose size may be less than operand.rank) to a "full" starting index into the 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 must have a bound of 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 do not change the operation fundamentally, but make 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 , O 0 , O 1 ], 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 , O 0 , O 1 ] is then the element in the input array at index [ X + O 0 , Y + O 1 ].

slice_sizes is [8,6] , which decides the range of O ₀ and O ₁ , 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 G 0 and G 1 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 O 0 , O 1 in the last example). The output batch dimensions (dimensions containing G 0 , G 1 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 :

G 0 and G 1 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 O 0 . 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" ( remapped_offset_dims in the formal description) into [ X , 0 ] and [ 0 , O 0 ] respectively, adding up to [ X , O 0 ]. In other words, the output index [ G 0 , G 1 , O 0 ] maps to the input index [ GatherIndices [ G 0 , G 1 , 0 ], O 0 ] 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)

SetDimensionSize

See also XlaBuilder::SetDimensionSize .

Sets the dynamic size of XlaOp's given dimension. The operand must be array shaped.

SetDimensionSize(operand, size, dimension)

Pass through the operand as result, with dynamic dimension tracked by the compiler.

Padded values will be ignored by downstream reduction ops.

let v: f32[10] = f32[10]{1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
let five: s32 = 5;
let six: s32 = 6;

// Setting dynamic dimension size doesn't change the upper bound of the static
// shape.
let padded_v_five: f32[10] = set_dimension_size(v, five, /*dimension=*/0);
let padded_v_six: f32[10] = set_dimension_size(v, six, /*dimension=*/0);

// sum == 1 + 2 + 3 + 4 + 5
let sum:f32[] = reduce_sum(padded_v_five);
// product == 1 * 2 * 3 * 4 * 5
let product:f32[] = reduce_product(padded_v_five);

// Changing padding size will yield different result.
// sum == 1 + 2 + 3 + 4 + 5 + 6
let sum:f32[] = reduce_sum(padded_v_six);

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)

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

See also XlaBuilder::Iota .

Iota(shape, iota_dimension)

Builds a constant literal on device rather than a potentially large host transfer. Creates an array that has specified shape and holds values starting at zero and incrementing by one along the specified dimension. For floating-point types, the produced array is equivalent to ConvertElementType(Iota(...)) where the Iota is of integral type and the conversion is to the floating-point type.

For example, Iota(s32[4, 8], 0) returns

  [[0, 0, 0, 0, 0, 0, 0, 0 ],
   [1, 1, 1, 1, 1, 1, 1, 1 ],
   [2, 2, 2, 2, 2, 2, 2, 2 ],
   [3, 3, 3, 3, 3, 3, 3, 3 ]]

Iota(s32[4, 8], 1) returns

  [[0, 1, 2, 3, 4, 5, 6, 7 ],
   [0, 1, 2, 3, 4, 5, 6, 7 ],
   [0, 1, 2, 3, 4, 5, 6, 7 ],
   [0, 1, 2, 3, 4, 5, 6, 7 ]]

Map

See also XlaBuilder::Map .

Map(operands..., computation)

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.

OptimizationBarrier

Blocks any optimization pass from moving computations across the barrier.

Ensures that all inputs are evaluated before any operators that depend on the barrier's outputs.

Pad

See also XlaBuilder::Pad .

Pad(operand, padding_value, padding_config)

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)

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)

Where:

• N is required to be greater or equal to 1.
• The computation has to be "roughly" associative (see below).
• All input arrays must have the same dimensions.
• All initial values have to form an identity under computation .
• If N = 1 , Collate(T) is T .
• If N > 1 , Collate(T_0, ..., T_{N-1}) is a tuple of N elements of type T .

This operation reduces one or more dimensions of each input array into scalars. The rank of each returned array is rank(operand) - len(dimensions) . 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.

Different backends are allowed to reassociate the reduction computation. This can lead to numerical differences, as some reduction functions like addition are not 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.

Examples

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, eg

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 .

Variadic Reduce

When N > 1 , reduce function application is slightly more complex, as it is applied simultaneously to all inputs. The operands are supplied to the computation in the following order:

• Running reduced value for the first operand
• ...
• Running reduced value for the N'th operand
• Input value for the first operand
• ...
• Input value for the N'th operand

For example, consider the following reduction function, which can be used to compute the max and the argmax of a 1-D array in parallel:

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

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.

ReduceScatter

See also XlaBuilder::ReduceScatter .

ReduceScatter is a collective operation that effectively does an AllReduce and then scatters the result by splitting it into shard_count blocks along the scatter_dimension and replica i in the replica group receives the ith shard.

ReduceScatter(operand, computation, scatter_dim, shard_count, replica_group_ids, channel_id)

• When operand is a tuple of arrays, the reduce-scatter is performed on each element of the tuple.
• replica_groups is a list of replica groups between which the reduction is performed (replica id for the current replica can be retrieved using ReplicaId ). The order of replicas in each group determines the order in which the all-reduce result will be scattered. replica_groups must either be empty (in which case all replicas belong to a single group), or contain the same number of elements as the number of replicas. When there are more than one replica groups, they all must be of the same size. For example, replica_groups = {0, 2}, {1, 3} performs reduction between the replicas 0 and 2 , and 1 and 3 and then scatters the result.
• shard_count is the size of each replica group. We need this in cases where replica_groups are empty. If replica_groups is not empty, shard_count must be equal to the size of each replica group.
• channel_id is used for cross-module communication: only reduce-scatter operations with the same channel_id can communicate with each other.

The output shape is the input shape with the scatter_dimension made shard_count times smaller. For example, if there are two replicas and the operand has the value [1.0, 2.25] and [3.0, 5.25] respectively on the two replicas, then the output value from this op where scatter_dim is 0 will be [4.0] for the first replica and [7.5] for the second replica.

ReduceWindow

See also XlaBuilder::ReduceWindow .

Applies a reduction function to all elements in each window of a sequence of N multi-dimensional arrays, producing a single or a tuple of N multi-dimensional arrays as output. Each output array has 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_values on the left-hand side.

ReduceWindow(operands..., init_values..., computation, window_dimensions, window_strides, padding)

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-1}) is a tuple of N elements of type (T0,...T{N-1}) .

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().value();
}

// 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,
    /*init_val=*/builder.ConstantLiteral(LiteralUtil::MinValue(F32)),
    *max,
    /*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.

For a non-trivial padding example, consider computing reduce-window minimum (initial value is MAX_FLOAT ) with dimension 3 and stride 2 over the input array [10000, 1000, 100, 10, 1] . Padding kValid computes minimums over two valid windows: [10000, 1000, 100] and [100, 10, 1] , resulting in the output [100, 1] . Padding kSame first pads the array so that the shape after the reduce-window would be the same as input for stride one by adding initial elements on both sides, getting [MAX_VALUE, 10000, 1000, 100, 10, 1, MAX_VALUE] . Running reduce-window over the padded array operates on three windows [MAX_VALUE, 10000, 1000] , [1000, 100, 10] , [10, 1, MAX_VALUE] , and yields [1000, 10, 1] .

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.

ReplicaId

See also XlaBuilder::ReplicaId .

Returns the unique ID (U32 scalar) of the replica.

ReplicaId()

The unique ID of each replica is an unsigned integer in the interval [0, N) , where N is the number of replicas. Since all the replicas are running the same program, a ReplicaId() call in the program will return a different value on each replica.

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)

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)

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 (ie, 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)

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)

RngBitGenerator

Generates an output with a given shape filled with uniform random bits using the specified algorithm (or backend default) and returns an updated state (with the same shape as initial state) and the generated random data.

Initial state is the initial state of the current random number generation. It and the required shape and valid values are dependent on the algorithm used.

The output is guaranteed to be a deterministic function of the initial state but it is not guaranteed to be deterministic between backends and different compiler versions.

RngBitGenerator(algorithm, key, shape)

Available values for algorithm :

• rng_default : Backend specific algorithm with backend specific shape requirements.

• rng_three_fry : ThreeFry counter-based PRNG algorithm. The initial_state shape is u64[2] with arbitrary values. Salmon et al. SC 2011. Parallel random numbers: as easy as 1, 2, 3.

• rng_philox : Philox algorithm to generate random numbers in parallel. The initial_state shape is u64[3] with arbitrary values. Salmon et al. SC 2011. Parallel random numbers: as easy as 1, 2, 3.

Scatter

The XLA scatter operation generates a sequence of results which are the values of the input array operands , with several slices (at indices specified by scatter_indices ) updated with the sequence of values in updates using update_computation .

See also XlaBuilder::Scatter .

scatter(operands..., scatter_indices, updates..., update_computation, index_vector_dim, update_window_dims, inserted_window_dims, scatter_dims_to_operand_dims)

Where:

• N is required to be greater or equal to 1.
• operands [ 0 ], ..., operands [ N-1 ] must all have the same dimensions.
• updates [ 0 ], ..., updates [ N-1 ] must all 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 .

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:

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

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

  • If i is present in update_window_dims (ie 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 (ie 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 (ie 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 (ie 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) .

• operand.rank must equal the sum of update_window_dims.size and inserted_window_dims.size .

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

For a given index U in each updates array, the corresponding index I in the corresponding operands 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 S in into operand using S by scattering S using the scatter_dims_to_operand_dims map. More formally:
   1. S in [ scatter_dims_to_operand_dims [ k ]] = S [ k ] if k < scatter_dims_to_operand_dims.size .
   2. S in [ _ ] = 0 otherwise.
3. Create an index W in into each operands array by scattering the indices at update_window_dims in U according to inserted_window_dims . More formally:
   1. W in [ window_dims_to_operand_dims ( k )] = U [ k ] if k is in update_window_dims , 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 { 0 → 1 , 1 → 3 , 2 → 4 , 3 → 5 }).
   2. W in [ _ ] = 0 otherwise.
4. I is W in + S in where + is element-wise addition.

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

• Initialize output with operands , ie for all indices J , for all indices O in the operands [ J ] array:
  output [ J ][ O ] = operands [ J ][ O ]
• For every index U in the updates [ J ] array and the corresponding index O in the operand [ J ] array, if O is a valid index for output :
  (output [ 0 ][ O ], ..., output [ N-1 ][ O ]) = update_computation ( output [ 0 ][ O ], ..., , output [ N-1 ][ O ], updates [ 0 ][ U ], ..., updates [ N-1 ][ U ])

The order in which updates are applied is non-deterministic. So, when multiple indices in updates refer to the same index in operands , 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 .

If indices_are_sorted is set to true then XLA can assume that start_indices are sorted (in ascending start_index_map order) by the user. If they are not then the semantics is implementation defined.

Informally, the scatter op can be viewed as an inverse of the gather op, ie 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)

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 (ie, 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)

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)

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, strides)

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 .

Sort(operands, comparator, dimension, is_stable)

If only one operand is provided:

• If the operand is a rank-1 tensor (an array), the result is a sorted array. If you want to sort the array into ascending order, the comparator should perform a less-than comparison. Formally, after the array is sorted, it holds for all index positions i, j with i < j that either comparator(value[i], value[j]) = comparator(value[j], value[i]) = false or comparator(value[i], value[j]) = true .

• If the operand has higher rank, the operand is sorted along the provided dimension. For example, for a rank-2 tensor (a matrix), a dimension value of 0 will independently sort every column, and a dimension value of 1 will independently sort each row. If no dimension number is provided, then the last dimension is chosen by default. For the dimension which is sorted, the same sorting order applies as in the rank-1 case.

If n > 1 operands are provided:

• All n operands must be tensors with the same dimensions. The element types of the tensors may be different.

• All operands are sorted together, not individually. Conceptually the operands are treated as a tuple. When checking whether the elements of each operand at index positions i and j need to be swapped, the comparator is called with 2 * n scalar parameters, where parameter 2 * k corresponds to the value at position i from the k-th operand, and parameter 2 * k + 1 corresponds to the value at position j from the k-th operand. Usually, the comparator would thus compare parameters 2 * k and 2 * k + 1 with each other and possibly use other parameter pairs as tie breakers.

• The result is a tuple that consists of the operands in sorted order (along the provided dimension, as above). The i-th operand of the tuple corresponds to the i-th operand of Sort.

For example, if there are three operands operand0 = [3, 1] , operand1 = [42, 50] , operand2 = [-3.0, 1.1] , and the comparator compares only the values of operand0 with less-than, then the output of the sort is the tuple ([1, 3], [50, 42], [1.1, -3.0]) .

If is_stable is set to true, the sort is guaranteed to be stable, that is, if there are elements which are considered to be equal by the comparator, the relative order of the equal values is preserved. Two elements e1 and e2 are equal if and only if comparator(e1, e2) = comparator(e2, e1) = false . By default, is_stable is set to false.

Top-K

See also the jax.lax.top_k operation.

TopK(operand)

Returns top k values and their indices as a tuple, along the last dimension of the operand using the given comparator (for usual topk behavior, it should be strict-greater-than operation).

For example, given strict > operator, k=1 and the following operand of shape f32[2,3] :

[[0.1, 0.3, 0.1], [0.7, 0.2, -0.1]]

The TopK application returns the following tuple of shape (f32[2,1], s32[2,1]) :

([[0.3], [0.7]], [[1], [0]])

Transpose

See also the tf.reshape operation.

Transpose(operand)

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)).

TriangularSolve

See also XlaBuilder::TriangularSolve .

Solves systems of linear equations with lower or upper triangular coefficient matrices by forward- or back-substitution. Broadcasting along leading dimensions, this routine solves one of the matrix systems op(a) * x = b , or x * op(a) = b , for the variable x , given a and b , where op(a) is either op(a) = a , or op(a) = Transpose(a) , or op(a) = Conj(Transpose(a)) .

TriangularSolve(a, b, left_side, lower, unit_diagonal, transpose_a)

Input data is read only from the lower/upper triangle of a , depending on the value of lower . Values from the other triangle are ignored. Output data is returned in the same triangle; the values in the other triangle are implementation-defined and may be anything.

If the rank of a and b are greater than 2, they are treated as batches of matrices, where all except the minor 2 dimensions are batch dimensions. a and b must have equal batch 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)

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};
}