操作的意味論

次に、 XlaBuilderインターフェースで定義された操作のセマンティクスについて説明します。通常、これらの操作は、 xla_data.protoのRPCインターフェイスで定義された操作に1対1でマップされます。

命名法に関する注意：XLAが扱う一般化されたデータ型は、いくつかの均一な型（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。ドキュメント全体を通して、配列は任意の次元の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。たとえば、ベクトルは1次元配列であり、行列は2次元配列です。

結局

XlaBuilder::AfterAllも参照してください。

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、副作用のある操作間でスレッド化して順序付けを強制できます。 AfterAllは、設定された操作の後に操作を順序付けるためのトークンの結合として使用できます。

AfterAll(operands)

AllGather

XlaBuilder::AllGatherも参照してください。

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)

• replica_groupsは、連結が実行されるレプリカグループのリストです（現在のレプリカのレプリカIDは、 ReplicaIdを使用して取得できます）。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は、空であるか（この場合、すべてのレプリカが0からN - 1の順序で単一のグループに属している）、またはレプリカの数と同じ数の要素を含んでいる必要があります。たとえば、 replica_groups = {0, 2}, {1, 3}は、レプリカ0と2 、および1と3の間の連結を実行します。
• shard_countは、各レプリカグループのサイズです。これは、 replica_groupsが空の場合に必要です。
• channel_idは、モジュール間の通信に使用されます。同じchannel_idを持つall-gather操作のみが相互に通信できます。

出力形状は、 all_gather_dimをshard_count倍大きくした入力形状です。たとえば、2つのレプリカがあり、オペランドの値が2つのレプリカでそれぞれ[1.0、2.5 [1.0, 2.5]と[3.0, 5.25] 3.0、5.25]の場合、 all_gather_dimが0のこのopからの出力値は[1.0, 2.5, 3.0, 5.25] 1.0、2.5、3.0]になります。 [1.0, 2.5, 3.0, 5.25]両方のレプリカで。

AllReduce

XlaBuilder::AllReduceも参照してください。

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)

• operandが配列のタプルである場合、all-reduceはタプルの各要素に対して実行されます。
• replica_groupsは、削減が実行されるレプリカグループのリストです（現在のレプリカのレプリカIDは、 ReplicaIdを使用して取得できます）。 replica_groupsは、空であるか（この場合、すべてのレプリカが1つのグループに属している）、またはレプリカの数と同じ数の要素を含んでいる必要があります。たとえば、 replica_groups = {0, 2}, {1, 3}は、レプリカ0と2 、および1と3の間でリダクションを実行します。
• channel_idは、モジュール間の通信に使用されます。同じchannel_idを持つall-reduce操作のみが相互に通信できます。

出力形状は入力形状と同じです。たとえば、2つのレプリカがあり、オペランドの値が2つのレプリカでそれぞれ[ [1.0, 2.5]と[3.0, 5.25]の場合、この操作と合計の計算からの出力値は、両方で[4.0, 7.75]になります。レプリカ。入力がタプルの場合、出力もタプルになります。

AllReduceの結果を計算するには、各レプリカから1つの入力が必要です。したがって、あるレプリカが別のレプリカよりもAllReduceノードを何度も実行すると、前のレプリカは永久に待機します。レプリカはすべて同じプログラムを実行しているため、これを実行する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、フィードされたデータによってwhileループがより多くの回数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

XlaBuilder::AllToAllも参照してください。

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。 2つのフェーズがあります。

1. 散布フェーズ。各コアで、オペランドはsplit_dimensions split_countブロック数に分割され、ブロックはすべてのコアに分散されます。たとえば、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。各コアは、 concat_dimensionに沿って受信したブロックを連結します。

参加するコアは、次の方法で構成できます。

• replica_groups ：各ReplicaGroupには、計算に参加しているレプリカIDのリストが含まれています（現在のレプリカのレプリカIDは、 ReplicaIdを使用して取得できます）。 AllToAllは、指定された順序でサブグループ内に適用されます。たとえば、 replica_groups = { {1,2,3}, {4,5,0} }は、AllToAllがレプリカ{1, 2, 3} 1、2、3}内で、収集フェーズで適用され、受信したブロックが適用されることを意味します。 1、2、3の同じ順序で連結されます。次に、別のAllToAllがレプリカ4、5、0内に適用され、連結順序も4、5、0になりますreplica_groupsが空の場合、すべてのレプリカは1つに属します。グループ、それらの出現の連結順序で。

前提条件：

• split_dimensionのオペランドの次元サイズは、 split_dimensionで割り切れ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)

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ているため、各部分の形状はf32[4,4]です。 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

アルゴリズムの詳細な説明については、 XlaBuilder::BatchNormGradおよび元のバッチ正規化ペーパーも参照してください。

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)

特徴次元の各特徴（ feature_indexはoperandの特徴次元のインデックス）について、この操作は、 operand 、 offset 、および他のすべての次元にわたるscaleに関する勾配を計算します。 feature_indexは、 operandの機能ディメンションの有効なインデックスである必要があります。

3つの勾配は、次の式で定義されます（4次元配列をoperandとして想定し、特徴次元インデックスl 、バッチサイズm 、空間サイズwおよびhを使用）。

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

入力のmeanとvarianceは、バッチ次元と空間次元にわたるモーメント値を表します。

出力タイプは、次の3つのハンドルのタプルです。

BatchNormInference

アルゴリズムの詳細な説明については、 XlaBuilder::BatchNormInferenceおよび元のバッチ正規化ペーパーも参照してください。

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)

特徴次元の各特徴（ feature_indexはoperandの特徴次元のインデックス）について、操作は他のすべての次元の平均と分散を計算し、平均と分散を使用してoperandの各要素を正規化します。 feature_indexは、 operandの機能ディメンションの有効なインデックスである必要があります。

BatchNormInferenceは、各バッチのmeanとvarianceを計算せずにBatchNormTrainingを呼び出すことと同じです。代わりに、推定値として入力meanとvarianceを使用します。この操作の目的は、推論の待ち時間を短縮することです。そのため、 BatchNormInferenceという名前が付けられています。

出力は、入力operandと同じ形状のn次元の正規化された配列です。

BatchNormTraining

アルゴリズムの詳細な説明については、 XlaBuilder::BatchNormTrainingおよびthe original batch normalization paperも参照してください。

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)

特徴次元の各特徴（ feature_indexはoperandの特徴次元のインデックス）について、操作は他のすべての次元の平均と分散を計算し、平均と分散を使用してoperandの各要素を正規化します。 feature_indexは、 operandの機能ディメンションの有効なインデックスである必要があります。

アルゴリズムは、空間次元のサイズとしてwとhを持つm要素を含むoperand \(x\) の各バッチに対して次のようになります（ operandが4次元配列であると想定）。

• フィーチャ次元の各フィーチャlのバッチ平均 \(\mu_l\) を計算します：\(\mu_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h x_{ijkl}\)

• 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、3つのXlaOpのタプルです。

batch_meanとbatch_varは、上記の式を使用してバッチ次元と空間次元にわたって計算されたモーメントです。

BitcastConvertType

XlaBuilder::BitcastConvertTypeも参照してください。

TensorFlowのtf.bitcastと同様に、データシェイプからターゲットシェイプへの要素ごとのビットキャスト操作を実行します。入力サイズと出力サイズは一致する必要があります。たとえば、 s32要素はビットキャストルーチンを介してf32要素になり、1つのs32要素は4つのs8要素になります。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)

変換前後のプリミティブサイズの比率によって変化する最後の次元を除いて、オペランドとターゲット形状の次元は一致する必要があります。

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。

ビットキャスト-異なる幅のプリミティブ型への変換

BitcastConvert HLO命令は、出力要素タイプT'のサイズが入力要素Tのサイズと等しくない場合をサポートします。操作全体は概念的にはビットキャストであり、基になるバイトは変更されないため、出力要素の形状を変更する必要があります。 B = sizeof(T), B' = sizeof(T')の場合、2つのケースが考えられます。

まず、 B > B'場合、出力形状はサイズB/B'の新しい最小寸法を取得します。例えば：

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

有効なスカラーのルールは同じです。

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

または、 B' > Bの場合、命令では入力形状の最後の論理次元がB'/Bに等しい必要があり、この次元は変換中に削除されます。

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

異なるビット幅間の変換は要素ごとではないことに注意してください。

ブロードキャスト

XlaBuilder::Broadcastも参照してください。

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)

新しい次元が左側に挿入されます。つまり、 broadcast_sizesの値が{a0, ..., aN}で、オペランドの形状の次元が{b0, ..., bM}の場合、出力の形状の次元は{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が値2.0fのスカラーf32で、 broadcast_sizesが{2, 3}の場合、結果は形状f32[2, 3]の配列になり、結果のすべての値は2.0fになります。

BroadcastInDim

XlaBuilder::BroadcastInDimも参照してください。

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)

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は、 out_dim_sizeで記述された形状にブロードキャストされます。 broadcast_dimensionsは、 operandの次元をターゲット形状の次元にマップします。つまり、オペランドのi番目の次元は出力形状のbroadcast_dimension[i]番目の次元にマップされます。 operandの次元は、サイズ1であるか、マップされる出力形状の次元と同じサイズである必要があります。残りの次元はサイズ1の次元で埋められます。次に、縮退次元のブロードキャストは、これらの縮退した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。セマンティクスについては、ブロードキャストページで詳しく説明されています。

電話

XlaBuilder::Callも参照してください。

指定された引数を使用して計算を呼び出します。

Call(computation, args...)

argsのアリティとタイプは、 computationのパラメーターと一致する必要があります。 argsを含めることはできません。

コレスキー

XlaBuilder::Choleskyも参照してください。

対称（エルミート）正定行列のバッチのコレスキー分解を計算します。

Cholesky(a, lower)

lowerがtrue場合、 \( a = l . l^T \)となるような下三角行列lを計算します。 lowerがfalseの場合、 \( a = u^T . u \)となるような上三角行列uを計算します。

入力データは、 lowerの値に応じて、 aの下/上三角形からのみ読み取られます。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。

aのランクが2より大きい場合a a行列のバッチとして扱われ、マイナー2次元を除くすべてがバッチ次元です。

aが対称（エルミート）正定値でない場合、結果は実装定義になります。

クランプ

XlaBuilder::Clampも参照してください。

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)

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。つまり、 clamp(a, x, b) = min(max(a, x), b)です。

3つのアレイはすべて同じ形状である必要があります。あるいは、ブロードキャストの制限された形式として、 minおよび/またはmaxはタイプTのスカラーにすることができます。

スカラーのmin値とmaxの例：

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

XlaBuilder::Collapseおよびtf.reshape操作も参照してください。

配列の次元を1つの次元に折りたたみます。

Collapse(operand, dimensions)

折りたたみは、オペランドのディメンションの指定されたサブセットを単一のディメンションに置き換えます。入力引数は、タイプTの任意の配列と、コンパイル時定数の次元インデックスのベクトルです。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}は有効ではありません。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。次元の最小のdimensions番号は、これらの次元を折りたたむループネスト内の最もゆっくりと変化する次元（最もメジャー）であり、最大の次元番号は最も速く変化する（最もマイナー）です。より一般的な折りたたみ順序が必要な場合は、 tf.reshape演算子を参照してください。

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

XlaBuilder::CollectivePermuteも参照してください。

CollectivePermuteは、データ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)

source_target_pairには次の制限があることに注意してください。

• 2つのペアのターゲットレプリカIDは同じであってはならず、ソースレプリカIDも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（s）で構成されるテンソルです。

連結

XlaBuilder::ConcatInDimも参照してください。

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)

寸法を除いて、すべてのdimensionは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

XlaBuilder::Conditionalも参照してください。

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

predがtrue場合はtrue_computationを実行し、 predがfalseの場合はfalse_computationを実行して、結果を返します。

true_computationは、タイプ \(T_0\) の単一の引数を受け取る必要があり、同じタイプである必要があるtrue_operandで呼び出されます。 false_computationは、タイプ \(T_1\) の単一の引数を受け取る必要があり、同じタイプである必要があるfalse_operandで呼び出されます。 true_computationとfalse_computationの戻り値のタイプは同じである必要があります。

predの値に応じて、 true_computationとfalse_computationのいずれか1つだけが実行されることに注意してください。

Conditional(branch_index, branch_computations, branch_operands)

branch_computations[branch_index]を実行し、結果を返します。 branch_indexが<0または>=NのS32の場合、 branch_computations[N-1]がデフォルトのブランチとして実行されます。

各branch_computations[b]は、タイプT_bの単一の引数を受け取る必要があり、同じタイプである必要があるbranch_operands[b]で呼び出されます。各branch_computations[b]の戻り値のタイプは同じである必要があります。

branch_indexの値に応じて、 branch_computationsの1つだけが実行されることに注意してください。

変換（畳み込み）

XlaBuilder::Convも参照してください。

ConvWithGeneralPaddingと同じですが、パディングは簡単にSAMEまたはVALIDとして指定されます。同じパディングにより、入力（ lhs ）にゼロが埋め込まれ、ストライドを考慮しない場合に出力が入力と同じ形状になるようにします。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）

XlaBuilder::ConvWithGeneralPaddingも参照してください。

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。

nを空間次元の数とします。 lhs引数は、ベース領域を記述するランクn+2の配列です。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 ：この次元の各座標は、畳み込みが実行される独立した入力を表します。
• z/depth/features ：ベースエリアの各（y、x）位置には、この次元に入るベクトルが関連付けられています。
• spatial_dims ：ウィンドウが移動するベースエリアを定義するnの空間次元を記述します。

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

• output-z ：出力のz次元。
• input-z ：このディメンションのサイズにfeature_group_countを掛けたものは、lhs単位のzディメンションのサイズと等しくなければなりません。
• spatial_dims ：ベースエリアを横切って移動するndウィンドウを定義するnの空間次元を記述します。

window_strides引数は、空間次元での畳み込みウィンドウ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]は次元yのパディングを指定し、 padding[1]は次元xのパディングを指定します。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。たとえば、 padding[1]が(2,3)の場合、2番目の空間次元で左側に2つのゼロ、右側に3つのゼロによるパディングがあります。パディングを使用することは、畳み込みを実行する前に、同じゼロ値を入力（ lhs ）に挿入することと同じです。

lhs_dilation引数とrhs_dilation引数は、各空間次元でそれぞれlhsとrhsに適用される膨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、 tf.nn.atrous_conv2dを参照してください。 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、 tf.nn.conv2d_transposeを参照してください。

feature_group_count引数（デフォルト値1）は、グループ化された畳み込みに使用できます。 feature_group_countは、入力と出力の両方の特徴次元の除数である必要があります。 feature_group_countが1より大きい場合、概念的には、入力および出力の特徴次元とrhs出力の特徴次元がfeature_group_countの多くのグループに均等に分割され、各グループは特徴の連続したサブシーケンスで構成されます。 rhsの入力フィーチャの次元は、 lhsの入力フィーチャの次元をfeature_group_countで割ったものに等しい必要があります（したがって、すでに入力フィーチャのグループのサイズになっています）。 i番目のグループは、 feature_group_countの多くの個別の畳み込みを計算するために一緒に使用されます。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に連結されます。

深さ方向の畳み込みの場合、 feature_group_count引数は入力フィーチャ次元に設定され、フィルターは[filter_height, filter_width, in_channels, channel_multiplier]から[filter_height, filter_width, 1, in_channels * channel_multiplier]に再形成されます。詳細については、 tf.nn.depthwise_conv2dを参照してください。

batch_group_count （デフォルト値1）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ーに使用できます。 batch_group_countは、 lhs （入力）バッチディメンションのサイズの除数である必要があります。 batch_group_countが1より大きい場合は、出力バッチディメンションのサイズがinput batch / batch_group_countであることを意味します。 batch_group_countは、出力フィーチャサイズの除数である必要があります。

出力形状には、次の順序で次の寸法があります。

• batch ：このディメンションのサイズにbatch_group_countを掛けたものは、 batchディメンションのサイズ（lhs）と同じである必要があります。
• z ：カーネルのoutput-zと同じサイズ（ rhs ）。
• spatial_dims ：畳み込みウィンドウの有効な配置ごとに1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。

畳み込みの機能を説明するために、2次元の畳み込みを検討し、出力で固定batch 、 z 、 y 、 x座標を選択します。次に(y,x)は、ベース領域内のウィンドウのコーナーの位置です（たとえば、空間寸法の解釈方法に応じて、左上のコーナー）。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ポイントは1Dベクトルに関連付けられているため、3Dボックスを取得します。畳み込みカーネルから、出力座標zを固定したので、3Dボックスもあります。 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（ドット積と同様）。それが出力値です。

output-zがたとえば5の場合、ウィンドウの各位置は、出力のz次元への出力に5つの値を生成することに注意してください。これらの値は、畳み込みカーネルのどの部分が使用されるかによって異なります。 output-z座標ごとに使用される値の個別の3Dボックスがあります。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

XlaBuilder::ConvertElementTypeも参照してください。

C ++の要素ごとのstatic_castと同様に、データ形状からターゲット形状への要素ごとの変換操作を実行します。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。たとえば、 s32要素はs32からf32への変換ルーチンを介してf32要素になります。

ConvertElementType(operand, new_element_type)

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。

T=s32からU=f32などの変換は、round-to-nearest-evenなどの正規化intからfloatへの変換ルーチンを実行します。

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

合計計算を使用してAllReduceを実行します。

CustomCall

XlaBuilder::CustomCallも参照してください。

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)

引数のアリティやタイプに関係なく、関数のシグネチャは同じです。

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])

myfuncの実装例を次に示します。

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

XlaBuilder::Dotも参照してください。

Dot(lhs, rhs)

この操作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。

この操作は、 lhsの2番目の次元（ランク1の場合は1番目）とrhsの1番目の次元で積の合計を実行します。これらは「契約された」寸法です。 lhsとrhsの収縮寸法は同じサイズでなければなりません。実際には、ベクトル間のドット積、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

XlaBuilder::DotGeneralも参照してください。

DotGeneral(lhs, rhs, dimension_numbers)

ドットと同じですが、「lhs」と「rhs」の両方に縮小およびバッチ次元番号を指定できます。

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次元番号は同じである必要はありませんが、同じ次元サイズである必要があります。

縮小する次元番号の例：

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'および'rhs'から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} } }

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」非収縮/非バッチディメンション、最後に「rhs」非収縮/非バッチディメンションになります。

DynamicSlice

XlaBuilder::DynamicSliceも参照してください。

DynamicSliceは、動的start_indicesの入力配列からサブ配列を抽出します。各次元のスライスのサイズはsize_indicesで渡されます。これは、各次元の排他的スライス間隔の終点を指定します：[start、start + size）。 start_indicesの形状はrank==1である必要があり、次元サイズはoperandのランクと同じです。

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), LogicalAnd (logical AND), or LogicalOr (logical OR).

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⌉ .

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) .

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 ], 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)

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 [ 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. By default, is_stable is set to false.

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};
}