操作的意味論

以下は、で定義されたオペレーションの意味を説明しXlaBuilderインターフェイスを。典型的には、これらの動作は、1対1でRPCインターフェースで定義されたオペレーションにマップxla_data.proto

命名法に関する注記:XLAが扱う一般化データ型は、いくつかの均一型(32ビット浮動小数点など)の要素を保持するN次元配列です。ドキュメント全体に、アレイは、任意の次元配列を示すために使用されます。便宜上、特殊なケースには、より具体的で馴染みのある名前が付けられています。例えば、ベクターは、1次元配列であり、マトリックスが2次元アレイです。

結局

参照してくださいXlaBuilder::AfterAll

AfterAllは、さまざまな数のトークンを受け取り、単一のトークンを生成します。トークンはプリミティブ型であり、副作用のある操作の間にスレッド化して順序付けを強制できます。 AfterAll集合演算後の動作を注文するためのトークンの結合として使用することができます。

AfterAll(operands)

引数タイプセマンティクス
operands XlaOpトークンの可変個引数の数

AllGather

参照してくださいXlaBuilder::AllGather

レプリカ間で連結を実行します。

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

引数タイプセマンティクス
operand XlaOpレプリカ間で連結する配列。
all_gather_dim int64連結ディメンション。
replica_groupsベクトルのベクトルint64連結が実行されるグループ。
channel_idオプションint64モジュール間通信用のオプションのチャネルID。
  • replica_groups連結が(現在のレプリカのレプリカIDを使用して取得することができる行われる間レプリカグループのリストであるReplicaId )。各グループのレプリカの順序によって、入力が結果に配置される順序が決まります。 replica_groupsいずれかの(すべてのレプリカから注文し、単一グループに属する場合には空でなければなりません0N - 1 )、又は複製の数と同じ数の要素を含みます。例えば、 replica_groups = {0, 2}, {1, 3}を実行レプリカとの間の連結0及び2 、及び1および3
  • shard_count各複製グループのサイズです。私たちは、場合によってはこれを必要とreplica_groups空です。
  • channel_idクロスモジュールの通信のために使用される。のみall-gather同じと操作channel_id互いに通信することができます。

出力形状を有する入力形状であるall_gather_dimなさshard_count倍大きいです。例えば、そこに2つのレプリカがあり、オペランド値がある場合[1.0, 2.5]及び[3.0, 5.25] 2つのレプリカのそれぞれに、このオペレーションから、出力値all_gather_dimある0になり[1.0, 2.5, 3.0, 5.25]の両方のレプリカ上。

AllReduce

参照してくださいXlaBuilder::AllReduce

レプリカ間でカスタム計算を実行します。

AllReduce(operand, computation, replica_group_ids, channel_id)

引数タイプセマンティクス
operand XlaOpレプリカ全体で削減する配列または配列の空でないタプル。
computation XlaComputation削減計算
replica_groupsベクトルのベクトルint64削減が実行されるグループ
channel_idオプションint64クロスモジュール通信用のオプションのチャネルID
  • 場合operandアレイのタプルで、全REDUCEタプルの各要素に対して行われます。
  • replica_groups還元は(現在のレプリカのレプリカIDを使用して取得することができる行われる間レプリカグループのリストであるReplicaId )。 replica_groupsいずれかの(すべてのレプリカが単一のグループに属する場合に)空である、または複製の数と同じ数の要素を含んでいなければなりません。例えば、 replica_groups = {0, 2}, {1, 3}レプリカ間行う還元0及び2 、及び1および3
  • channel_idクロスモジュールの通信のために使用される。のみall-reduce同じで動作をchannel_id互いに通信することができます。

出力形状は入力形状と同じです。例えば、2つのレプリカがある場合、オペランドは、値を有する[1.0, 2.5]及び[3.0, 5.25] 2つのレプリカ上に、それぞれ、このOPと加算演算の出力値になります[4.0, 7.75]の両方にレプリカ。入力がタプルの場合、出力もタプルです。

結果計算AllReduce 、各レプリカから1つの入力を有する必要がありそうである場合つのレプリカ実行A AllReduceノード別以上回、次いで元レプリカが永久に待機します。レプリカはすべて同じプログラムを実行しているため、これを行う方法は多くありませんが、whileループの状態がインフィードからのデータに依存し、入力されたデータによってwhileループがより多く繰り返される場合は可能です。あるレプリカで別のレプリカよりも。

AllToAll

参照してくださいXlaBuilder::AllToAll

AllToAllは、すべてのコアからすべてのコアにデータを送信する集合的な操作です。これには2つのフェーズがあります。

  1. 散布フェーズ。各コアに、オペランドに分割されsplit_count沿ったブロックの数split_dimensions 、及びブロックは、すべてのコアに散在している、例えば、i番目のブロックは、i番目のコアに送信されます。
  2. 収集フェーズ。各コアに沿って受信されたブロック連結concat_dimension

参加するコアは、次の方法で構成できます。

  • replica_groups :各ReplicaGroup(現在のレプリカのレプリカIDを使用して取得することができる計算に参加するレプリカIDのリストを含むReplicaId )。 AllToAllは、指定された順序でサブグループ内に適用されます。例えば、 replica_groups = { {1,2,3}, {4,5,0} } AllToAllがレプリカ内で適用されることを意味{1, 2, 3}及び位相、及び受信されたブロックは意志集まります0であれば、他のAllToAllは複製4、5、0内で適用され、連結順序も4、5 3、2、1の同じ順序で連結さreplica_groups空である、すべてのレプリカが1に属しグループ、それらの出現の連結順序で。

前提条件:

  • 上のオペランドの次元サイズsplit_dimensionで割り切れるsplit_count
  • オペランドの形状はタプルではありません。

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

引数タイプセマンティクス
operand XlaOp n次元の入力配列
split_dimension int64間隔の値[0, n)名称寸法がそれに沿ってオペランドが分割されていること
concat_dimension int64間隔の値[0, n)名前は寸法がそれに沿って分割されたブロックが連結されていること
split_count int64この操作に参加するコアの数。場合replica_groups空である、これはレプリカの数である必要があります。それ以外の場合、これは各グループのレプリカの数と同じである必要があります。
replica_groups ReplicaGroupベクトル各グループには、レプリカIDのリストが含まれています。

以下にAlltoallの例を示します。

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);

この例では、Alltoallに参加している4つのコアがあります。各コアで、オペランドは次元0に沿って4つの部分に分割されるため、各部分の形状はf32 [4,4]になります。 4つの部分がすべてのコアに分散しています。次に、各コアは、受け取ったパーツをディメンション1に沿って、コア0〜4の順序で連結します。したがって、各コアの出力の形状はf32 [16,4]です。

BatchNormGrad

参照してくださいXlaBuilder::BatchNormGrad、元のバッチ正規紙は、アルゴリズムの詳細については。

バッチノルムの勾配を計算します。

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

引数タイプセマンティクス
operand XlaOp正規化されるn次元配列(x)
scale XlaOp 1次元配列(
$$\gamma$$

)| | mean | XlaOp | 1次元配列(

$$\mu$$

)| | variance | XlaOp | 1次元配列(

$$\sigma^2$$

)| | grad_output | XlaOp |渡されたグラデーションBatchNormTraining

$$ | : : : \nabla y$$

):| epsilon | float |イプシロン値(

$$\epsilon$$

)| | feature_index | int64 |特徴次元へのインデックスoperand |

特徴次元における各機能の( feature_indexで特徴次元の指標であるoperand )、動作がに関して勾配計算operandoffsetおよびscaleすべての他の寸法を横切っ。 feature_indexにおける特徴次元の有効なインデックスでなければならないoperand

3段階の勾配は、以下の式(AS 4次元配列を想定して定義されるoperandおよび特徴次元インデックス付きl 、バッチサイズm及び空間サイズw及びh )。

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

入力meanvarianceバッチおよび空間次元を横切る瞬間値を表します。

出力タイプは、次の3つのハンドルのタプルです。

出力タイプセマンティクス
grad_operand XlaOp入力に対する勾配operand
$$ | : : : \nabla x$$

):| grad_scale | XlaOp |入力に対する勾配scale

$$ \nabla | : : : \gamma$$

):| grad_offset | XlaOp |入力に対する勾配offset

$$ \nabla | : : : \beta$$

):

BatchNormInference

参照してくださいXlaBuilder::BatchNormInference、元のバッチ正規紙は、アルゴリズムの詳細については。

バッチ次元と空間次元にわたって配列を正規化します。

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

引数タイプセマンティクス
operand XlaOp正規化されるn次元配列
scale XlaOp 1次元配列
offset XlaOp 1次元配列
mean XlaOp 1次元配列
variance XlaOp 1次元配列
epsilon floatイプシロン値
feature_index int64特徴次元へのインデックスoperand

特徴次元における各特徴(のためfeature_indexで特徴次元の指標であるoperand )、動作は、他のすべての次元にわたる平均及び分散を計算し、各要素正規化する平均および分散を使用するoperandfeature_indexにおける特徴次元の有効なインデックスでなければならないoperand

BatchNormInference呼び出すのと同じですBatchNormTraining計算せずにmeanvarianceバッチごとに。これは、入力された使用meanvariance推定値として代わりに。このオペアンプの目的は、推論にその名の待ち時間を減らすことですBatchNormInference

出力は、入力と同じ形状のn次元正規化アレイであるoperand

BatchNormTraining

参照してくださいXlaBuilder::BatchNormTrainingthe original batch normalization paperアルゴリズムの詳細については。

バッチ次元と空間次元にわたって配列を正規化します。

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

引数タイプセマンティクス
operand XlaOp正規化されるn次元配列(x)
scale XlaOp 1次元配列(
$$\gamma$$

offset | XlaOp | 1次元配列(

$$\beta$$

epsilon | float |イプシロン値(

$$\epsilon$$

feature_index | int64 |特徴次元へのインデックスoperand

特徴次元における各特徴(のためfeature_indexで特徴次元の指標であるoperand )、動作は、他のすべての次元にわたる平均及び分散を計算し、各要素正規化する平均および分散を使用するoperandfeature_indexにおける特徴次元の有効なインデックスでなければならないoperand

次のようなアルゴリズムは、各バッチのために行くoperand

$$x$$

それに含まれるmの元素をwh空間次元の大きさとして(仮定operand 4次元配列です)。

  • バッチ平均を計算します
$$\mu_l$$

各特徴についてl特徴次元で:

$$\mu_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h x_{ijkl}$$
  • バッチ分散を計算します
$$\sigma^2_l$$

$$\sigma^2_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (x_{ijkl} - \mu_l)^2$$
  • 正規化、スケーリング、およびシフト:
$$y_{ijkl}=\frac{\gamma_l(x_{ijkl}-\mu_l)}{\sqrt[2]{\sigma^2_l+\epsilon} }+\beta_l$$

ゼロ除算エラーを回避するために、通常は小さい数値であるイプシロン値が追加されます。

出力タイプは、三のタプルXlaOp S。

出力タイプセマンティクス
output XlaOp入力と同じ形状のn次元配列operand (Y)
batch_mean XlaOp 1次元配列(
$$\mu$$

)| | batch_var | XlaOp | 1次元配列(

$$\sigma^2$$

)|

batch_meanbatch_var上記の式を使用して、バッチおよび空間次元にわたって計算モーメントです。

BitcastConvertType

参照してくださいXlaBuilder::BitcastConvertType

同様tf.bitcast TensorFlowにおいて、ターゲット形状へのデータの形状から要素毎bitcast動作を行います。寸法は一致する必要があり、変換は要素ごとに行われます。例えばs32要素はなっf32 bitcastルーチンを経由して要素。ビットキャストは低レベルのキャストとして実装されているため、浮動小数点表現が異なるマシンでは異なる結果が得られます。

BitcastConvertType(operand, new_element_type)

引数タイプセマンティクス
operand XlaOp薄暗いDを持つタイプTの配列
new_element_type PrimitiveTypeタイプU

オペランドの寸法とターゲットの形状は一致している必要があります。ソース要素タイプと宛先要素タイプのビット幅は等しくなければなりません。ソース要素と宛先要素のタイプはタプルであってはなりません。

放送

参照してくださいXlaBuilder::Broadcast

配列内のデータを複製することにより、配列に次元を追加します。

Broadcast(operand, broadcast_sizes)

引数タイプセマンティクス
operand XlaOp複製する配列
broadcast_sizes ArraySlice<int64>新しい寸法のサイズ

場合、新しい寸法は、すなわち、左側に挿入されbroadcast_sizes値有する{a0, ..., aN}とオペランド形状寸法有し{b0, ..., bM} 、出力の形状は寸法有し{a0, ..., aN, b0, ..., bM}

新しい次元は、オペランドのコピーにインデックスを付けます。

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

たとえば、 operandスカラーであるf32値と2.0f 、およびbroadcast_sizesある{2, 3}その結果は、形状と配列であろうf32[2, 3]及び結果の全ての値は、あろう2.0f

BroadcastInDim

参照してくださいXlaBuilder::BroadcastInDim

配列内のデータを複製することにより、配列のサイズとランクを拡張します。

BroadcastInDim(operand, out_dim_size, broadcast_dimensions)

引数タイプセマンティクス
operand XlaOp複製する配列
out_dim_size ArraySlice<int64>ターゲット形状の寸法のサイズ
broadcast_dimensions ArraySlice<int64>オペランド形状の各次元が対応するターゲット形状のどの次元

Broadcastと似ていますが、任意の場所にディメンションを追加し、サイズ1で既存のディメンションを拡張できます。

operandによって記述形状にブロードキャストされるout_dim_sizebroadcast_dimensions寸法マップoperand '[i]を出力形状の目寸法を目標形状の寸法に、オペランドのすなわちi番目の次元がbroadcast_dimensionにマッピングされます。寸法operandサイズ1を持っているか、それらがマッピングされる出力形状の寸法と同じ大きさでなければなりません。残りの次元はサイズ1の次元で埋められます。次に、縮退次元ブロードキャストは、これらの縮退次元に沿ってブロードキャストして、出力形状に到達します。セマンティクスは、上に詳細に記載されている放送ページ

コール

参照してくださいXlaBuilder::Call

指定された引数を使用して計算を呼び出します。

Call(computation, args...)

引数タイプセマンティクス
computation XlaComputation型の計算T_0, T_1, ..., T_{N-1} -> S 、任意のタイプのNとパラメータ
args NのシーケンスXlaOp S任意の型のN個の引数

アリティとの種類のargsのパラメータを一致させる必要がありcomputation 。何もないために許可されているargs

コレスキー

参照してくださいXlaBuilder::Cholesky

計算コレスキー分解対称(エルミート)正定行列のバッチのは。

Cholesky(a, lower)

引数タイプセマンティクス
a XlaOp複合型または浮動小数点型のランク> 2の配列。
lower boolの上または下三角使用するかどうか。 a

場合lowerあるtrue計算するには、下三角行列lように

$$ a = l . l^T $$

。場合はlowerあるfalse 、上三角行列計算u 、このようなを

$$ a = u^T . u $$

入力データは、唯一の下部/上部の三角形から読み出された値に応じて、 a lower 。他の三角形の値は無視されます。出力データは同じ三角形で返されます。もう一方の三角形の値は実装によって定義されており、何でもかまいません。

ランク場合は2よりも大きい場合、すべてのマイナー除く2つの寸法は、バッチ寸法は行列のバッチとして扱われます。 a a

場合対称(エルミート)正定値でない、結果は実装定義です。 a

クランプ

参照してくださいXlaBuilder::Clamp

オペランドを最小値と最大値の間の範囲内にクランプします。

Clamp(min, operand, max)

引数タイプセマンティクス
min XlaOpタイプTの配列
operand XlaOpタイプTの配列
max XlaOpタイプTの配列

オペランドと最小値および最大値を指定すると、最小値と最大値の間の範囲にある場合はオペランドを返し、それ以外の場合は、オペランドがこの範囲を下回る場合は最小値を返し、オペランドがこの範囲を超える場合は最大値を返します。すなわち、 clamp(a, x, b) = min(max(a, x), b)

3つのアレイはすべて同じ形状である必要があります。代替的に、制限の形態として放送min及び/又はmax型のスカラーであり得るT

スカラーと例minmax

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

崩壊

参照してくださいXlaBuilder::Collapsetf.reshape操作を。

配列の次元を1つの次元に折りたたみます。

Collapse(operand, dimensions)

引数タイプセマンティクス
operand XlaOpタイプTの配列
dimensions int64ベクトルTの次元の順序どおりの連続したサブセット。

折りたたみは、オペランドの次元の指定されたサブセットを単一の次元に置き換えます。入力引数は、タイプTの任意の配列と、次元インデックスのコンパイル時定数ベクトルです。ディメンションインデックスは、Tのディメンションの順序どおり(低いディメンション番号から高いディメンション番号)の連続したサブセットである必要があります。したがって、{0、1、2}、{0、1}、または{1、2}はすべて有効なディメンションセットですが、{1、0}または{0、2}はそうではありません。それらは、元の寸法サイズの積に等しい新しい寸法サイズで、それらが置き換えるものと同じ寸法シーケンスの位置にある単一の新しい寸法に置き換えられます。で最も低い次元数dimensionsこれらの寸法を崩壊ループの入れ子、および最速(ほとんどマイナー)を変化させている最大の次元数で最も遅い変化寸法(ほとんどの主要な)です。参照してくださいtf.reshapeより一般的な崩壊の順序付けが必要な場合は、オペレータが。

たとえば、vを24個の要素の配列とします。

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} };

CollectivePermute

参照してくださいXlaBuilder::CollectivePermute

CollectivePermuteは、データのクロスレプリカを送受信する集合的な操作です。

CollectivePermute(operand, source_target_pairs)

引数タイプセマンティクス
operand XlaOp n次元の入力配列
source_target_pairs <int64, int64>ベクトル(source_replica_id、target_replica_id)ペアのリスト。ペアごとに、オペランドはソースレプリカからターゲットレプリカに送信されます。

上の次のような制限があることに注意してくださいsource_target_pair

  • 2つのペアのターゲットレプリカIDが同じであってはならず、ソースレプリカIDが同じであってはなりません。
  • レプリカIDがどのペアのターゲットでもない場合、そのレプリカの出力は、入力と同じ形状の0で構成されるテンソルです。

連結する

参照してくださいXlaBuilder::ConcatInDim

Concatenateは、複数の配列オペランドから配列を構成します。配列は、各入力配列オペランドと同じランクであり(互いに同じランクである必要があります)、指定された順序で引数が含まれています。

Concatenate(operands..., dimension)

引数タイプセマンティクス
operands NのシーケンスXlaOp次元が[L0、L1、...]のタイプTのN個の配列。 N> = 1が必要です。
dimension int64間隔の値[0, N)名寸法の間に連結されることoperands

除いて、 dimensionのすべての寸法が同じでなければなりません。これは、XLAが「不規則な」配列をサポートしていないためです。また、ランク0の値は連結できないことに注意してください(連結が発生するディメンションに名前を付けることはできないため)。

1次元の例:

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

2次元の例:

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

図:

条件付き

参照してくださいXlaBuilder::Conditional

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

引数タイプセマンティクス
pred XlaOpタイプのスカラPRED
true_operand XlaOp型の引数 \(T_0\)
true_computation XlaComputation型のXlaComputation \(T_0 \to S\)
false_operand XlaOp型の引数 \(T_1\)
false_computation XlaComputation型のXlaComputation \(T_1 \to S\)

実行のtrue_computation場合predあるtruefalse_computation場合predありfalse 、その結果を返します。

true_computation型の単一の引数に取る必要があります \(T_0\) とで呼び出されますtrue_operand同じタイプでなければなりません。 false_computation型の単一の引数に取る必要があります \(T_1\) とで呼び出されますfalse_operand同じタイプでなければなりません。戻り値の型true_computationfalse_computation同じでなければなりません。

一つだけのことに注意true_computationfalse_computationの値に応じて実行されるpred

Conditional(branch_index, branch_computations, branch_operands)

引数タイプセマンティクス
branch_index XlaOpタイプのスカラS32
branch_computations NのシーケンスXlaComputation型のXlaComputations \( T_0 \to S , T_1 \to S , ..., T_{N-1} \to S \)
branch_operands NのシーケンスXlaOp型の引数 \( T_0 , T_1 , ..., T_{N-1} \)

実行branch_computations[branch_index]その結果を返します。場合branch_indexあるS32 <0または> = Nであり、次いでbranch_computations[N-1] 、デフォルトのブランチとして実行されます。

branch_computations[b]タイプの単一の引数でなければならないT_bとで起動するbranch_operands[b]同じタイプのものでなければなりません。それぞれの戻り値の種類branch_computations[b]同じでなければなりません。

一方のみように注意branch_computationsの値に応じて実行されるbranch_index

畳み込み(畳み込み)

参照してくださいXlaBuilder::Conv

ConvWithGeneralPaddingと同じですが、パディングはSAMEまたはVALIDのいずれかとして簡単に指定されます。 SAMEパディングパッド入力( lhsアカウントにストライドを取っていないときに出力が入力と同じ形状を有するようにゼロで)。有効なパディングは、単にパディングがないことを意味します。

ConvWithGeneralPadding(畳み込み)

参照してくださいXlaBuilder::ConvWithGeneralPadding

ニューラルネットワークで使用される種類の畳み込みを計算します。ここで、畳み込みは、n次元のベース領域を横切って移動するn次元のウィンドウと考えることができ、ウィンドウの可能な位置ごとに計算が実行されます。

引数タイプセマンティクス
lhs XlaOp入力のランクn + 2配列
rhs XlaOpカーネルの重みのランクn + 2配列
window_strides ArraySlice<int64>カーネルストライドのnd配列
padding ArraySlice< pair<int64, int64>> (低、高)パディングのnd配列
lhs_dilation ArraySlice<int64> ndlhs拡張係数配列
rhs_dilation ArraySlice<int64> ndrhs拡張係数配列
feature_group_count int64機能グループの数
batch_group_count int64バッチグループの数

nを空間次元の数とします。 lhs引数は、ランクベース領域を記述するN + 2アレイです。もちろんrhsも入力ですが、これは入力と呼ばれます。ニューラルネットワークでは、これらは入力アクティベーションです。 n + 2次元は、この順序で次のとおりです。

  • batch :それぞれのこの次元の座標は、畳み込みが行われるための独立した入力を表します。
  • z/depth/features :ベース領域の各(X、Y)の位置は、この寸法に入り、それに関連するベクトルを有しています。
  • spatial_dims :記述しnを横切るウィンドウを移動することを基本領域を定義する空間次元を。

rhs引数は、畳み込みフィルタ/カーネル/ウィンドウを記述するランクn + 2アレイです。寸法は、この順序で次のとおりです。

  • output-z z出力の寸法。
  • input-zこの寸法倍のサイズfeature_group_countの大きさと等しくなければならないz LHSの寸法を。
  • spatial_dims :説明しn番目のウィンドウを定義する空間次元をそのベース領域を横切って移動します。

window_strides引数は、空間的な次元での畳み込み窓のストライドを指定します。たとえば、最初の空間次元のストライドが3の場合、ウィンドウは、最初の空間インデックスが3で割り切れる座標にのみ配置できます。

padding引数は、ベース領域に適用されるゼロパディングの量を指定します。パディングの量は負にすることができます。負のパディングの絶対値は、畳み込みを実行する前に指定された次元から削除する要素の数を示します。 padding[0] 、ディメンション用のパディング指定y及びpadding[1]の寸法のためにパディングを指定x 。各ペアには、最初の要素として低いパディングがあり、2番目の要素として高いパディングがあります。低いパディングは低いインデックスの方向に適用され、高いパディングは高いインデックスの方向に適用されます。例えば、 padding[1]である(2,3)左に2ゼロによって、および第二の空間次元における右側の3ゼロによってパディングが存在することになります。パディングを使用して入力(に、同じゼロ値を挿入することと等価であるlhs畳み込みを実行する前に)。

lhs_dilationrhs_dilation引数は各空間次元で、それぞれ、LHSとRHSに適用される拡張係数を指定します。空間次元の膨張係数がdの場合、d-1の穴がその次元の各エントリの間に暗黙的に配置され、配列のサイズが大きくなります。穴はno-op値で埋められます。これは、畳み込みの場合はゼロを意味します。

rhsの拡張は、atrousconvolutionとも呼ばれます。詳細については、を参照してくださいtf.nn.atrous_conv2d 。 lhsの拡張は、転置畳み込みとも呼ばれます。詳細については、を参照してくださいtf.nn.conv2d_transpose

feature_group_count引数(デフォルト値1)がグループ化されたコンボリューションのために使用することができます。 feature_group_count入力と出力の特徴次元の両方の除数である必要があります。場合feature_group_count 1より大きい場合、それは概念的入出力特徴寸法とすることを意味rhs出力フィーチャ寸法が均等に分割されfeature_group_count 、多くのグループ機能の連続サブ配列からなる各群。入力特徴寸法rhsニーズに等しくなるようにlhsで割った入力特徴寸法feature_group_count (既に入力フィーチャのグループの大きさを有するように)。 i番目のグループは、計算するために一緒に使用されているfeature_group_count多くの別々の畳み込みを。これらの畳み込みの結果は、出力フィーチャディメンションで連結されます。

奥行き畳み込みためfeature_group_count引数は、入力された特徴寸法に設定されるであろう、そしてフィルタから再形成される[filter_height, filter_width, in_channels, channel_multiplier][filter_height, filter_width, 1, in_channels * channel_multiplier]詳細については、を参照してくださいtf.nn.depthwise_conv2d

batch_group_count (デフォルト値1)の引数は、バックプロパゲーション中にグループ化されたフィルタに使用することができます。 batch_group_countのサイズの約数である必要がlhs (入力)バッチ寸法。場合batch_group_count 1よりも大きい場合、出力バッチ寸法のサイズでなければならないことを意味input batch / batch_group_countbatch_group_count出力フィーチャサイズの約数でなければなりません。

出力形状には、次の順序で次の寸法があります。

  • batch :この次元の倍の大きさbatch_group_countの大きさと等しくなければならないbatch LHSで次元を。
  • z :と同じサイズoutput-zカーネル(上rhs )。
  • spatial_dims :畳み込み窓のそれぞれの有効な配置のための一つの値。

畳み込みウィンドウの有効な配置は、パディング後のストライドとベース領域のサイズによって決まります。

畳み込みが何を記述するために、2D畳み込みを考慮し、いくつかの固定された選択batchzyx出力の座標を。次に(y,x) (あなたが空間次元をどう解釈するかに応じて、例えば、左上隅、)ベース領域内のウィンドウの隅の位置です。これで、ベース領域から取得した2Dウィンドウができました。ここで、各2Dポイントが1Dベクトルに関連付けられているため、3Dボックスが得られます。私たちは、出力が固定座標以来、畳み込みカーネルから、 z 、我々はまた、3Dボックスを持っています。 2つのボックスの寸法は同じであるため、2つのボックス間の要素ごとの積の合計をとることができます(内積と同様)。それが出力値です。

場合、その音符output-z 、例えば、5である場合、ウィンドウの各位置はに出力における5つの値を生成しz出力の寸法。それぞれのために使用される値の別個の3Dボックスがある-これらの値は、畳み込みカーネルの一部が使用されるものでは異なるoutput-z座標は。したがって、それぞれに異なるフィルターを使用した5つの個別の畳み込みと考えることができます。

これは、パディングとストライドを使用した2D畳み込みの擬似コードです。

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

参照してくださいXlaBuilder::ConvertElementType

要素ごとに類似static_cast C ++で、ターゲット形状へのデータ形状から要素ごとの変換動作を行います。寸法は一致する必要があり、変換は要素ごとに行われます。例えばs32要素はなってf32を介した要素s32 -to- f32変換ルーチン。

ConvertElementType(operand, new_element_type)

引数タイプセマンティクス
operand XlaOp薄暗いDを持つタイプTの配列
new_element_type PrimitiveTypeタイプU

オペランドの寸法とターゲットの形状は一致している必要があります。ソース要素と宛先要素のタイプはタプルであってはなりません。

変換は、次のようなT=s32U=f32 、円形ツー最も近い偶数としてルーチン正規INTツーフロート変換を実行します。

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

実行AllReduce総和計算と。

CustomCall

参照してくださいXlaBuilder::CustomCall

計算内でユーザー提供の関数を呼び出します。

CustomCall(target_name, args..., shape)

引数タイプセマンティクス
target_name string関数の名前。このシンボル名を対象とする呼び出し命令が発行されます。
args NのシーケンスXlaOp S関数に渡される任意の型のN個の引数。
shape Shape関数の出力形状

関数のシグネチャは、引数のアリティやタイプに関係なく同じです。

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

たとえば、CustomCallが次のように使用されている場合:

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

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

ここでの実装例です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];
  // ...
}

ユーザー提供の関数には副作用があってはならず、その実行はべき等でなければなりません。

ドット

参照してくださいXlaBuilder::Dot

Dot(lhs, rhs)

引数タイプセマンティクス
lhs XlaOpタイプTの配列
rhs XlaOpタイプTの配列

この演算の正確なセマンティクスは、オペランドのランクによって異なります。

入力出力セマンティクス
ベクトル[n]はdotベクトル[N]スカラーベクトル内積
行列[MXK】 dotベクトル[K]ベクトル[m]行列-ベクトル乗算
行列[MXK】 dotマトリクス[kxn]行列[mxn]行列-行列の乗算

操作を行うには、第二の次元にわたって積の和lhs (またはそれがランク1を持っている場合、最初)との最初の寸法rhs 。これらは「契約済み」の寸法です。契約寸法lhsrhs 、同じサイズでなければなりません。実際には、ベクトル間の内積、ベクトル/行列の乗算、または行列/行列の乗算を実行するために使用できます。

DotGeneral

参照してくださいXlaBuilder::DotGeneral

DotGeneral(lhs, rhs, dimension_numbers)

引数タイプセマンティクス
lhs XlaOpタイプTの配列
rhs XlaOpタイプTの配列
dimension_numbers DotDimensionNumbers契約およびバッチ寸法番号

ドットと同じですが、「lhs」と「rhs」の両方に契約およびバッチの次元番号を指定できます。

DotDimensionNumbersフィールドタイプセマンティクス
'lhs_contracting_dimensions'繰り返されるint64 'lhs'縮小寸法番号
'rhs_contracting_dimensions'繰り返されるint64 「rhs」収縮次元番号
'lhs_batch_dimensions'繰り返されるint64 'lhs'バッチ寸法番号
'rhs_batch_dimensions'繰り返されるint64 「rhs」バッチ寸法番号

DotGeneralは、「dimension_numbers」で指定された縮小ディメンションに対して積の合計を実行します。

'lhs'と 'rhs'からの関連する縮小寸法番号は同じである必要はありませんが、同じ寸法サイズである必要があります。

寸法番号を縮小する例:

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} }

'lhs'と 'rhs'の関連するバッチ次元番号は、同じ次元サイズである必要があります。

バッチ次元番号の例(バッチサイズ2、2x2マトリックス):

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} } }
入力出力セマンティクス
[B0、M、K] dot [B0、K、N] [b0、m、n]バッチmatmul
[B0、B1、M、K] dot [B0、B1、K、N] [b0、b1、m、n]バッチmatmul

したがって、結果のディメンション番号はバッチディメンションで始まり、次に「lhs」非契約/非バッチディメンション、最後に「rhs」非契約/非バッチディメンションになります。

DynamicSlice

参照してくださいXlaBuilder::DynamicSlice

DynamicSliceは、動的に入力配列からのサブ配列抽出start_indices 。各次元のスライスのサイズが渡されsize_indices各次元で排他的スライス間隔の終了点を指定し、[+サイズ開始、開始します)。形状start_indicesの階数に等しい寸法サイズのランク== 1でなければならないoperand

DynamicSlice(operand, start_indices, size_indices)

引数タイプセマンティクス
operand XlaOpタイプTのN次元配列
start_indices NのシーケンスXlaOp各次元のスライスの開始インデックスを含むN個のスカラー整数のリスト。値はゼロ以上である必要があります。
size_indices ArraySlice<int64>各次元のスライスサイズを含むN個の整数のリスト。各値は厳密にゼロより大きくなければならず、開始+サイズは、モジュロ次元サイズの折り返しを避けるために、次元のサイズ以下でなければなりません。

有効なスライスインデックスは、各インデックスは、次の変換を適用することによって計算されるi[1, N)スライスを実行する前に:

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

これにより、抽出されたスライスが常にオペランド配列に対してインバウンドになります。変換が適用される前にスライスがインバウンドである場合、変換は効果がありません。

1次元の例:

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次元の例:

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

参照してくださいXlaBuilder::DynamicUpdateSlice

DynamicUpdateSliceは、入力配列の値である結果を生成operandスライスして、 updateで上書きstart_indices 。形状update更新され、結果のサブ配列の形状を決定します。形状start_indicesの階数に等しい寸法サイズのランク== 1でなければならないoperand

DynamicUpdateSlice(operand, update, start_indices)

引数タイプセマンティクス
operand XlaOpタイプTのN次元配列
update XlaOpスライスの更新を含むタイプTのN次元配列。更新形状の各次元は厳密にゼロより大きくなければならず、開始+更新は、範囲外の更新インデックスの生成を回避するために、各次元のオペランドサイズ以下でなければなりません。
start_indices NのシーケンスXlaOp List of N scalar 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 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.

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⌋ .

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) .

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) .

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).

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 .

Arguments Type Semantics
operand XlaOp The array we are Fourier transforming.
fft_type FftType See the table below.
fft_length ArraySlice<int64> The time-domain lengths of the axes being transformed. This is needed in particular for IRFFT to right-size the innermost axis, since RFFT(fft_length=[16]) has the same output shape as RFFT(fft_length=[17]) .
FftType Semantics
FFT Forward complex-to-complex FFT. Shape is unchanged.
IFFT Inverse complex-to-complex FFT. Shape is unchanged.
RFFT Forward real-to-complex FFT. Shape of the innermost axis is reduced to fft_length[-1] // 2 + 1 if fft_length[-1] is a non-zero value, omitting the reversed conjugate part of the transformed signal beyond the Nyquist frequency.
IRFFT Inverse real-to-complex FFT (ie takes complex, returns real). Shape of the innermost axis is expanded to fft_length[-1] if fft_length[-1] is a non-zero value, inferring the part of the transformed signal beyond the Nyquist frequency from the reverse conjugate of the 1 to fft_length[-1] // 2 + 1 entries.

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)

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 an 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.
indices_are_sorted bool Whether the indices are guaranteed to be sorted by the caller.
unique_indices bool Whether the indices are guaranteed to be unique by the caller.

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 . 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.size ) and range [ 0 , operand.rank ) \ collapsed_slice_dims . So if, eg, offset.size is 4 , operand.rank is 6 and collapsed_slice_dims is { 0 , 2 } then remapped_offset_dims is { 01 , 13 , 24 , 35 }.

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 ], 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

SetDimensionSize

See also XlaBuilder::SetDimensionSize .

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

SetDimensionSize(operand, size, dimension)

Arguments Type Semantics
operand XlaOp n dimensional input array.
size XlaOp int32 representing the runtime dynamic size.
dimension int64 A value in the interval [0, n) that specifies the 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)

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. 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.

Arguments Type Semantics
type PrimitiveType type U
size int64 The number of elements in the array.
iota_dimension int64 The dimension to increment along.

Map

See also XlaBuilder::Map .

Map(operands..., computation)

Arguments Type Semantics
operands sequence of N XlaOp s N arrays of types T 0..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-1} .
init_values Sequence of N XlaOp N scalars of types T_0, ..., T_{N-1} .
computation XlaComputation computation of type T_0, ..., T_{N-1}, T_0, ..., T_{N-1} -> Collate(T_0, ..., T_{N-1}) .
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-1}) 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) . The initial value used for every reduction is init_value , and it may be inserted anywhere during computation by the back-end. It is required that init_value is an identity of the reduction function (for example, 0 for addition) or undefined behavior will occur. 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.

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, 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)

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.

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)

Arguments Type Semantics
operand XlaOp Array or a non-empty tuple of arrays to reduce across replicas.
computation XlaComputation Reduction computation
replica_groups vector of vectors of int64 Groups between which the reductions are performed
channel_id optional int64 Optional channel ID for cross-module communication
  • 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_dim 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)

Arguments Type Semantics
operands N XlaOps A sequence of N multi-dimensional arrays of types T_0,..., T_{N-1} , each representing the base area on which the window is placed.
init_values N XlaOps The N starting values for the reduction, one for each of the N operands. See Reduce for details.
computation XlaComputation Reduction function of type T_0, ..., T_{N-1}, T_0, ..., T_{N-1} -> Collate(T_0, ..., T_{N-1}) , to apply to elements in each window of all the input operands.
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, which pads so as to have the same output shape as input if the stride is 1, or Padding::kValid, which uses no padding and "stops" the window once it no longer fits)

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().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,
    /*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)

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 (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)

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

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)

Arguments Type Semantics
algorithm RandomAlgorithm PRNG algorithm to be used.
initial_state XlaOp Initial state for the PRNG algorithm.
shape Shape Output shape for generated data.

Available values for algorithm :

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.
indices_are_sorted bool Whether the indices are guaranteed to be sorted by the caller.

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 (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 [ 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 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 operand 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 { 01 , 13 , 24 , 35 }).
    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 operand , ie 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, if O is a valid index for output :
    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 .

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)

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 (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)

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.
strides ArraySlice<int64> List of N integers that decides the input stride of the slice. The slice picks every strides[d] element in dimension d .

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)

Arguments Type Semantics
operands ArraySlice<XlaOp> The operands to sort.
comparator XlaComputation The comparator computation to use.
dimension int64 The dimension along which to sort.
is_stable bool Whether stable sorting should be used.

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. By default, is_stable is set to false.

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)).

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)

Arguments Type Semantics
a XlaOp a rank > 2 array of a complex or floating-point type with shape [..., M, M] .
b XlaOp a rank > 2 array of the same type with shape [..., M, K] if left_side is true, [..., K, M] otherwise.
left_side bool indicates whether to solve a system of the form op(a) * x = b ( true ) or x * op(a) = b ( false ).
lower bool whether to use the upper or lower triangle of a .
unit_diagonal bool if true , the diagonal elements of a are assumed to be 1 and not accessed.
transpose_a Transpose whether to use a as is, transpose it or take its conjugate transpose.

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)

Arguments Type Semantics
condition XlaComputation XlaComputation of type T -> PRED which defines the termination condition of the loop.
body XlaComputation XlaComputation of type T -> T which defines the body of the loop.
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};
}