Operasi Semantik

Berikut ini menjelaskan semantik operasi yang ditentukan dalam antarmuka XlaBuilder . Biasanya, operasi ini memetakan operasi satu-ke-satu yang ditentukan dalam antarmuka RPC di xla_data.proto .

Catatan tentang tata nama: tipe data umum yang ditangani XLA adalah array berdimensi N yang menampung elemen dari beberapa tipe seragam (seperti float 32-bit). Sepanjang dokumentasi, array digunakan untuk menunjukkan array berdimensi arbitrer. Untuk kenyamanan, kasus khusus memiliki nama yang lebih spesifik dan familiar; misalnya vektor adalah array 1 dimensi dan matriks adalah array 2 dimensi.

Lagipula

Lihat juga XlaBuilder::AfterAll .

AfterAll mengambil sejumlah token yang bervariasi dan menghasilkan satu token. Token adalah tipe primitif yang dapat dijalin di antara operasi dengan efek samping untuk menegakkan pemesanan. AfterAll dapat digunakan sebagai gabungan token untuk memesan operasi setelah operasi yang ditetapkan.

AfterAll(operands)

Semua Berkumpul

Lihat juga XlaBuilder::AllGather .

Melakukan penggabungan di seluruh replika.

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

• replica_groups adalah daftar grup replika tempat penggabungan dilakukan (id replika untuk replika saat ini dapat diambil menggunakan ReplicaId ). Urutan replika di setiap grup menentukan urutan penempatan masukannya dalam hasil. replica_groups harus kosong (dalam hal ini semua replika termasuk dalam satu grup, diurutkan dari 0 hingga N - 1 ), atau berisi jumlah elemen yang sama dengan jumlah replika. Misalnya, replica_groups = {0, 2}, {1, 3} melakukan penggabungan antara replika 0 dan 2 , serta 1 dan 3 .
• shard_count adalah ukuran setiap grup replika. Kami memerlukan ini jika replica_groups kosong.
• channel_id digunakan untuk komunikasi lintas modul: hanya operasi all-gather dengan channel_id yang sama yang dapat berkomunikasi satu sama lain.

Bentuk keluarannya adalah bentuk masukan dengan all_gather_dim yang dibuat shard_count kali lebih besar. Misalnya, jika ada dua replika dan operan memiliki nilai [1.0, 2.5] dan [3.0, 5.25] masing-masing pada dua replika tersebut, maka nilai keluaran dari operasi ini di mana all_gather_dim adalah 0 akan menjadi [1.0, 2.5, 3.0, 5.25] pada kedua replika.

Semua Kurangi

Lihat juga XlaBuilder::AllReduce .

Melakukan komputasi khusus di seluruh replika.

AllReduce(operand, computation, replica_group_ids, channel_id)

• Ketika operand adalah tupel dari array, pengurangan semua dilakukan pada setiap elemen tupel.
• replica_groups adalah daftar grup replika tempat pengurangan dilakukan (id replika untuk replika saat ini dapat diambil menggunakan ReplicaId ). replica_groups harus kosong (dalam hal ini semua replika termasuk dalam satu grup), atau berisi jumlah elemen yang sama dengan jumlah replika. Misalnya, replica_groups = {0, 2}, {1, 3} melakukan reduksi antara replika 0 dan 2 , serta 1 dan 3 .
• channel_id digunakan untuk komunikasi lintas modul: hanya operasi all-reduce dengan channel_id yang sama yang dapat berkomunikasi satu sama lain.

Bentuk keluaran sama dengan bentuk masukan. Misalnya, jika ada dua replika dan operan memiliki nilai [1.0, 2.5] dan [3.0, 5.25] masing-masing pada dua replika tersebut, maka nilai keluaran dari perhitungan operasi dan penjumlahan ini akan menjadi [4.0, 7.75] pada keduanya replika. Jika masukannya berupa tupel, maka keluarannya juga berupa tupel.

Menghitung hasil AllReduce memerlukan satu masukan dari setiap replika, jadi jika satu replika mengeksekusi simpul AllReduce lebih sering daripada yang lain, maka replika sebelumnya akan menunggu selamanya. Karena semua replika menjalankan program yang sama, tidak banyak cara untuk mewujudkannya, namun ada kemungkinan ketika kondisi perulangan while bergantung pada data dari infeed dan data yang dimasukkan menyebabkan perulangan while berulang kali lebih sering pada satu replika dari yang lain.

Semua Ke Semua

Lihat juga XlaBuilder::AllToAll .

AllToAll adalah operasi kolektif yang mengirimkan data dari semua inti ke semua inti. Ini memiliki dua fase:

1. Fase pencar. Pada setiap inti, operan dipecah menjadi split_count sejumlah blok di sepanjang split_dimensions , dan blok tersebut tersebar ke semua inti, misalnya, blok ke-i dikirim ke inti ke-i.
2. Fase berkumpul. Setiap inti menggabungkan blok yang diterima di sepanjang concat_dimension .

Inti yang berpartisipasi dapat dikonfigurasi dengan:

• replica_groups : setiap ReplicaGroup berisi daftar id replika yang berpartisipasi dalam komputasi (id replika untuk replika saat ini dapat diambil menggunakan ReplicaId ). AllToAll akan diterapkan dalam subgrup dalam urutan yang ditentukan. Misalnya, replica_groups = { {1,2,3}, {4,5,0} } berarti AllToAll akan diterapkan dalam replika {1, 2, 3} , dan dalam fase pengumpulan, dan blok yang diterima akan digabungkan dalam urutan yang sama 1, 2, 3. Kemudian, AllToAll lain akan diterapkan dalam replika 4, 5, 0, dan urutan penggabungan juga 4, 5, 0. Jika replica_groups kosong, semua replika menjadi milik satu kelompok, dalam urutan rangkaian kemunculannya.

Prasyarat:

• Ukuran dimensi operan pada split_dimension habis dibagi split_count .
• Bentuk operannya bukan tupel.

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

Di bawah ini menunjukkan contoh Alltoall.

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

Dalam contoh ini, ada 4 core yang berpartisipasi dalam Alltoall. Pada setiap inti operan dipecah menjadi 4 bagian sepanjang dimensi 1, sehingga setiap bagian berbentuk f32[4,4]. Keempat bagian tersebut tersebar ke seluruh inti. Kemudian setiap inti menggabungkan bagian-bagian yang diterima sepanjang dimensi 0, dalam urutan atau inti 0-4. Jadi output pada setiap inti berbentuk f32[16,4].

BatchNormGrad

Lihat juga XlaBuilder::BatchNormGrad dan makalah normalisasi batch asli untuk penjelasan rinci tentang algoritme.

Menghitung gradien norma batch.

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

Untuk setiap fitur dalam dimensi fitur ( feature_index adalah indeks untuk dimensi fitur dalam operand ), operasi menghitung gradien sehubungan dengan operand , offset , dan scale di semua dimensi lainnya. feature_index harus berupa indeks yang valid untuk dimensi fitur dalam operand .

Ketiga gradien ditentukan oleh rumus berikut (dengan asumsi array 4 dimensi sebagai operand dan dengan indeks dimensi fitur l , ukuran batch m dan ukuran spasial w dan h ):

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

mean dan variance masukan mewakili nilai momen di seluruh dimensi batch dan spasial.

Tipe keluarannya adalah tupel dari tiga pegangan:

Inferensi BatchNorm

Lihat juga XlaBuilder::BatchNormInference dan makalah normalisasi batch asli untuk penjelasan rinci tentang algoritme.

Menormalkan array di seluruh dimensi batch dan spasial.

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

Untuk setiap fitur dalam dimensi fitur ( feature_index adalah indeks untuk dimensi fitur dalam operand ), operasi menghitung mean dan varians di seluruh dimensi lainnya dan menggunakan mean dan varians untuk menormalkan setiap elemen dalam operand . feature_index harus berupa indeks yang valid untuk dimensi fitur dalam operand .

BatchNormInference setara dengan memanggil BatchNormTraining tanpa menghitung mean dan variance untuk setiap batch. Ini menggunakan mean masukan dan variance sebagai nilai perkiraan. Tujuan dari operasi ini adalah untuk mengurangi latensi dalam inferensi, oleh karena itu dinamakan BatchNormInference .

Outputnya adalah array ternormalisasi berdimensi n dengan bentuk yang sama dengan operand input.

Pelatihan BatchNorm

Lihat juga XlaBuilder::BatchNormTraining dan the original batch normalization paper untuk penjelasan rinci tentang algoritme.

Menormalkan array di seluruh dimensi batch dan spasial.

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

Untuk setiap fitur dalam dimensi fitur ( feature_index adalah indeks untuk dimensi fitur dalam operand ), operasi menghitung mean dan varians di seluruh dimensi lainnya dan menggunakan mean dan varians untuk menormalkan setiap elemen dalam operand . feature_index harus berupa indeks yang valid untuk dimensi fitur dalam operand .

Algoritmenya berjalan sebagai berikut untuk setiap batch dalam operand \(x\) yang berisi m elemen dengan w dan h sebagai ukuran dimensi spasial (dengan asumsi operand adalah array 4 dimensi):

• Menghitung rata-rata batch \(\mu_l\) untuk setiap fitur l dalam dimensi fitur:\(\mu_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h x_{ijkl}\)

• Menghitung varians batch \(\sigma^2_l\):\(\sigma^2_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (x_{ijkl} - \mu_l)^2\)

• Menormalkan, menskalakan, dan menggeser:\(y_{ijkl}=\frac{\gamma_l(x_{ijkl}-\mu_l)}{\sqrt[2]{\sigma^2_l+\epsilon} }+\beta_l\)

Nilai epsilon, biasanya sejumlah kecil, ditambahkan untuk menghindari kesalahan pembagian dengan nol.

Tipe outputnya adalah tuple dari tiga XlaOp s:

batch_mean dan batch_var adalah momen yang dihitung di seluruh dimensi batch dan spasial menggunakan rumus di atas.

Tipe Konversi Bitcast

Lihat juga XlaBuilder::BitcastConvertType .

Mirip dengan tf.bitcast di TensorFlow, melakukan operasi bitcast berdasarkan elemen dari bentuk data ke bentuk target. Ukuran input dan output harus sesuai: misalnya elemen s32 menjadi elemen f32 melalui rutin bitcast, dan satu elemen s32 akan menjadi empat elemen s8 . Bitcast diimplementasikan sebagai pemeran tingkat rendah, sehingga mesin dengan representasi floating-point berbeda akan memberikan hasil berbeda.

BitcastConvertType(operand, new_element_type)

Dimensi operan dan bentuk target harus sesuai, kecuali dimensi terakhir yang akan berubah berdasarkan rasio ukuran primitif sebelum dan sesudah konversi.

Tipe elemen sumber dan tujuan tidak boleh berupa tupel.

Konversi Bitcast ke tipe primitif dengan lebar berbeda

Instruksi BitcastConvert HLO mendukung kasus di mana ukuran tipe elemen keluaran T' tidak sama dengan ukuran elemen masukan T . Karena keseluruhan operasi secara konseptual adalah bitcast dan tidak mengubah byte yang mendasarinya, bentuk elemen keluaran harus diubah. Untuk B = sizeof(T), B' = sizeof(T') , ada dua kemungkinan kasus.

Pertama, ketika B > B' , bentuk keluaran mendapat dimensi paling kecil baru dengan ukuran B/B' . Misalnya:

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

Aturannya tetap sama untuk skalar efektif:

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

Alternatifnya, untuk B' > B instruksi memerlukan dimensi logis terakhir dari bentuk masukan sama dengan B'/B , dan dimensi ini dihilangkan selama konversi:

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

Perhatikan bahwa konversi antara bitwidth yang berbeda tidak berdasarkan elemen.

Siaran

Lihat juga XlaBuilder::Broadcast .

Menambahkan dimensi ke array dengan menduplikasi data dalam array.

Broadcast(operand, broadcast_sizes)

Dimensi baru disisipkan di sebelah kiri, yaitu jika broadcast_sizes mempunyai nilai {a0, ..., aN} dan bentuk operannya berdimensi {b0, ..., bM} maka bentuk keluarannya berdimensi {a0, ..., aN, b0, ..., bM} .

Dimensi baru mengindeks ke dalam salinan operan, mis

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

Misalnya, jika operand adalah skalar f32 dengan nilai 2.0f , dan broadcast_sizes adalah {2, 3} , maka hasilnya akan berupa array dengan bentuk f32[2, 3] dan semua nilai pada hasilnya adalah 2.0f .

Siaran Dalam Dim

Lihat juga XlaBuilder::BroadcastInDim .

Memperluas ukuran dan peringkat array dengan menduplikasi data dalam array.

BroadcastInDim(operand, out_dim_size, broadcast_dimensions)

Mirip dengan Siaran, tetapi memungkinkan penambahan dimensi di mana saja dan memperluas dimensi yang ada dengan ukuran 1.

operand disiarkan ke bentuk yang dijelaskan oleh out_dim_size . broadcast_dimensions memetakan dimensi operand ke dimensi bentuk target, yaitu dimensi ke-i operan dipetakan ke dimensi ke-siaran[i] dari bentuk keluaran. Dimensi operand harus berukuran 1 atau berukuran sama dengan dimensi dalam bentuk keluaran yang dipetakan. Dimensi yang tersisa diisi dengan dimensi ukuran 1. Penyiaran dimensi yang merosot kemudian disiarkan sepanjang dimensi yang merosot tersebut untuk mencapai bentuk keluaran. Semantiknya dijelaskan secara rinci di halaman penyiaran .

Panggilan

Lihat juga XlaBuilder::Call .

Memanggil komputasi dengan argumen yang diberikan.

Call(computation, args...)

Arity dan tipe args harus sesuai dengan parameter computation . Tidak boleh ada args .

Cholesky

Lihat juga XlaBuilder::Cholesky .

Menghitung dekomposisi Cholesky dari sekumpulan matriks pasti positif simetris (Hermitian).

Cholesky(a, lower)

Jika lower bernilai true , hitung matriks segitiga bawah l sedemikian sehingga \( a = l . l^T \). Jika lower adalah false , hitung matriks segitiga atas u sedemikian rupa sehingga \( a = u^T . u \).

Data masukan hanya dibaca dari segitiga bawah/atas a , bergantung pada nilai lower . Nilai dari segitiga lainnya diabaikan. Data keluaran dikembalikan dalam segitiga yang sama; nilai-nilai dalam segitiga lainnya ditentukan oleh implementasi dan dapat berupa apa saja.

Jika pangkat a lebih besar dari 2, a diperlakukan sebagai kumpulan matriks, dimana semua dimensi kecuali 2 minor adalah dimensi kumpulan.

Jika a tidak pasti positif simetris (Hermitian), hasilnya ditentukan oleh implementasi.

Penjepit

Lihat juga XlaBuilder::Clamp .

Menjepit operan dalam rentang antara nilai minimum dan maksimum.

Clamp(min, operand, max)

Mengingat operan dan nilai minimum dan maksimum, kembalikan operan jika berada dalam rentang antara minimum dan maksimum, jika tidak, kembalikan nilai minimum jika operan berada di bawah rentang ini atau nilai maksimum jika operan berada di atas rentang ini. Artinya, clamp(a, x, b) = min(max(a, x), b) .

Ketiga array harus memiliki bentuk yang sama. Alternatifnya, sebagai bentuk penyiaran terbatas, min dan/atau max dapat berupa skalar bertipe T .

Contoh dengan skalar min dan max :

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

Runtuh

Lihat juga XlaBuilder::Collapse dan operasi tf.reshape .

Meruntuhkan dimensi array menjadi satu dimensi.

Collapse(operand, dimensions)

Ciutkan menggantikan subset dimensi operan tertentu dengan satu dimensi. Argumen masukannya adalah larik sembarang bertipe T dan vektor indeks dimensi yang konstan waktu kompilasi. Indeks dimensi harus merupakan subset dimensi T yang berurutan (angka dimensi rendah hingga tinggi). Jadi, {0, 1, 2}, {0, 1}, atau {1, 2} semuanya merupakan himpunan dimensi yang valid, namun {1, 0} atau {0, 2} tidak. Dimensi tersebut digantikan oleh satu dimensi baru, yang posisinya sama dalam urutan dimensi dengan dimensi yang digantikannya, dengan ukuran dimensi baru yang sama dengan hasil kali ukuran dimensi asli. Nomor dimensi terendah dalam dimensions adalah dimensi yang paling lambat berubah (paling besar) dalam sarang loop yang meruntuhkan dimensi-dimensi ini, dan nomor dimensi tertinggi adalah yang paling cepat berubah (paling kecil). Lihat operator tf.reshape jika diperlukan pemesanan penciutan yang lebih umum.

Misalnya, v adalah array yang terdiri dari 24 elemen:

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} };

KolektifPermute

Lihat juga XlaBuilder::CollectivePermute .

CollectivePermute adalah operasi kolektif yang mengirim dan menerima data lintas replika.

CollectivePermute(operand, source_target_pairs)

Perhatikan bahwa ada batasan berikut pada source_target_pair :

• Dua pasangan mana pun tidak boleh memiliki id replika target yang sama, dan keduanya tidak boleh memiliki id replika sumber yang sama.
• Jika id replika bukan merupakan target pada pasangan mana pun, maka keluaran pada replika tersebut adalah tensor yang terdiri dari 0(s) dengan bentuk yang sama dengan masukan.

Menggabungkan

Lihat juga XlaBuilder::ConcatInDim .

Concatenate menyusun array dari beberapa operan array. Array tersebut memiliki peringkat yang sama dengan masing-masing operan array input (yang harus memiliki peringkat yang sama satu sama lain) dan berisi argumen sesuai urutan yang ditentukan.

Concatenate(operands..., dimension)

Kecuali dimension semua dimensi harus sama. Hal ini karena XLA tidak mendukung array "ragged". Perhatikan juga bahwa nilai peringkat-0 tidak dapat digabungkan (karena tidak mungkin memberi nama dimensi sepanjang terjadinya penggabungan).

Contoh 1 dimensi:

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

Contoh 2 dimensi:

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

Diagram:

Bersyarat

Lihat juga XlaBuilder::Conditional .

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

Jalankan true_computation jika pred adalah true , false_computation jika pred adalah false , dan mengembalikan hasilnya.

true_computation harus menggunakan satu argumen bertipe \(T_0\) dan akan dipanggil dengan true_operand yang harus bertipe sama. false_computation harus menggunakan satu argumen bertipe \(T_1\) dan akan dipanggil dengan false_operand yang harus bertipe sama. Jenis nilai yang dikembalikan dari true_computation dan false_computation harus sama.

Perhatikan bahwa hanya satu dari true_computation dan false_computation yang akan dieksekusi bergantung pada nilai pred .

Conditional(branch_index, branch_computations, branch_operands)

Jalankan branch_computations[branch_index] , dan kembalikan hasilnya. Jika branch_index adalah S32 yang < 0 atau >= N, maka branch_computations[N-1] dijalankan sebagai cabang default.

Setiap branch_computations[b] harus menggunakan satu argumen bertipe T_b dan akan dipanggil dengan branch_operands[b] yang harus bertipe sama. Tipe nilai yang dikembalikan dari setiap branch_computations[b] harus sama.

Perhatikan bahwa hanya satu dari branch_computations yang akan dieksekusi tergantung pada nilai branch_index .

Konv (konvolusi)

Lihat juga XlaBuilder::Conv .

Sebagai ConvWithGeneralPadding, tetapi paddingnya ditentukan secara singkat sebagai SAMA atau VALID. Padding yang SAMA mengisi input ( lhs ) dengan angka nol sehingga output memiliki bentuk yang sama dengan input ketika tidak memperhitungkan langkahnya. Padding VALID berarti tidak ada padding.

ConvWithGeneralPadding (konvolusi)

Lihat juga XlaBuilder::ConvWithGeneralPadding .

Menghitung konvolusi dari jenis yang digunakan dalam jaringan saraf. Di sini, konvolusi dapat dianggap sebagai jendela berdimensi n yang bergerak melintasi area dasar berdimensi n dan perhitungan dilakukan untuk setiap kemungkinan posisi jendela.

Misalkan n adalah banyaknya dimensi spasial. Argumen lhs adalah array peringkat n+2 yang menggambarkan area dasar. Ini disebut input, padahal tentu saja rhs juga merupakan input. Dalam jaringan saraf, ini adalah aktivasi masukan. Dimensi n+2 adalah, dengan urutan sebagai berikut:

• batch : Setiap koordinat dalam dimensi ini mewakili masukan independen yang konvolusinya dilakukan.
• z/depth/features : Setiap posisi (y,x) pada luas dasar memiliki vektor yang terkait dengannya, yang masuk ke dalam dimensi ini.
• spatial_dims : Menjelaskan n dimensi spasial yang menentukan area dasar yang dilewati jendela.

Argumen rhs adalah array peringkat n+2 yang mendeskripsikan filter/kernel/jendela konvolusional. Dimensinya adalah, dalam urutan ini:

• output-z : Dimensi z dari output.
• input-z : Ukuran dimensi ini dikalikan feature_group_count harus sama dengan ukuran dimensi z dalam lhs.
• spatial_dims : Menjelaskan n dimensi spasial yang menentukan jendela ke-nd yang bergerak melintasi area dasar.

Argumen window_strides menentukan langkah jendela konvolusional dalam dimensi spasial. Misalnya, jika langkah pada dimensi spasial pertama adalah 3, maka jendela hanya dapat ditempatkan pada koordinat yang indeks spasial pertama habis dibagi 3.

Argumen padding menentukan jumlah nol padding yang akan diterapkan pada area dasar. Jumlah padding bisa negatif -- nilai absolut padding negatif menunjukkan jumlah elemen yang akan dihapus dari dimensi tertentu sebelum melakukan konvolusi. padding[0] menentukan padding untuk dimensi y dan padding[1] menentukan padding untuk dimensi x . Setiap pasangan memiliki padding rendah sebagai elemen pertama dan padding tinggi sebagai elemen kedua. Padding rendah diterapkan ke arah indeks yang lebih rendah sedangkan padding tinggi diterapkan ke arah indeks yang lebih tinggi. Misalnya, jika padding[1] adalah (2,3) maka akan ada padding sebanyak 2 angka nol di kiri dan 3 angka nol di kanan pada dimensi spasial kedua. Menggunakan padding sama dengan memasukkan nilai nol yang sama ke dalam input ( lhs ) sebelum melakukan konvolusi.

Argumen lhs_dilation dan rhs_dilation menentukan faktor dilatasi yang akan diterapkan pada lhs dan rhs, masing-masing, di setiap dimensi spasial. Jika faktor dilatasi dalam dimensi spasial adalah d, maka lubang d-1 secara implisit ditempatkan di antara setiap entri dalam dimensi tersebut, sehingga meningkatkan ukuran larik. Lubang-lubang tersebut diisi dengan nilai no-op, yang untuk konvolusi berarti nol.

Pelebaran rhs juga disebut konvolusi atrous. Untuk detail selengkapnya, lihat tf.nn.atrous_conv2d . Pelebaran lhs juga disebut konvolusi transposisi. Untuk detail selengkapnya, lihat tf.nn.conv2d_transpose .

Argumen feature_group_count (nilai default 1) dapat digunakan untuk konvolusi yang dikelompokkan. feature_group_count harus berupa pembagi dimensi fitur masukan dan keluaran. Jika feature_group_count lebih besar dari 1, berarti secara konseptual dimensi fitur masukan dan keluaran serta dimensi fitur keluaran rhs dibagi rata menjadi banyak grup feature_group_count , masing-masing grup terdiri dari fitur-fitur yang berurutan. Dimensi fitur masukan rhs harus sama dengan dimensi fitur masukan lhs dibagi dengan feature_group_count (sehingga sudah memiliki ukuran sekelompok fitur masukan). Grup ke-i digunakan bersama untuk menghitung feature_group_count banyak konvolusi terpisah. Hasil konvolusi ini digabungkan menjadi satu dalam dimensi fitur keluaran.

Untuk konvolusi mendalam, argumen feature_group_count akan disetel ke dimensi fitur masukan, dan filter akan dibentuk ulang dari [filter_height, filter_width, in_channels, channel_multiplier] menjadi [filter_height, filter_width, 1, in_channels * channel_multiplier] . Untuk detail selengkapnya, lihat tf.nn.depthwise_conv2d .

Argumen batch_group_count (nilai default 1) dapat digunakan untuk filter yang dikelompokkan selama propagasi mundur. batch_group_count harus berupa pembagi ukuran dimensi batch lhs (input). Jika batch_group_count lebih besar dari 1, itu berarti dimensi batch keluaran harus berukuran input batch / batch_group_count . batch_group_count harus berupa pembagi ukuran fitur keluaran.

Bentuk keluaran memiliki dimensi berikut, dengan urutan sebagai berikut:

• batch : Ukuran dimensi ini dikali batch_group_count harus sama dengan ukuran dimensi batch dalam lhs.
• z : Ukurannya sama dengan output-z pada kernel ( rhs ).
• spatial_dims : Satu nilai untuk setiap penempatan jendela konvolusional yang valid.

Gambar di atas menunjukkan cara kerja bidang batch_group_count . Secara efektif, kami membagi setiap batch lhs menjadi grup batch_group_count , dan melakukan hal yang sama untuk fitur output. Kemudian, untuk masing-masing grup ini kami melakukan konvolusi berpasangan dan menggabungkan keluaran sepanjang dimensi fitur keluaran. Semantik operasional semua dimensi lainnya (fitur dan spasial) tetap sama.

Penempatan jendela konvolusional yang valid ditentukan oleh langkah dan ukuran area dasar setelah padding.

Untuk menjelaskan fungsi konvolusi, pertimbangkan konvolusi 2d, dan pilih beberapa koordinat batch tetap, z , y , x dalam output. Maka (y,x) adalah posisi sudut jendela di dalam area dasar (misalnya sudut kiri atas, tergantung bagaimana Anda menginterpretasikan dimensi spasial). Kita sekarang memiliki jendela 2d, diambil dari area dasar, dimana setiap titik 2d dikaitkan dengan vektor 1d, sehingga kita mendapatkan kotak 3d. Dari kernel konvolusional, karena kami memperbaiki koordinat keluaran z , kami juga memiliki kotak 3d. Kedua kotak tersebut memiliki dimensi yang sama, sehingga kita dapat mengambil jumlah perkalian elemen di antara kedua kotak tersebut (mirip dengan perkalian titik). Itu adalah nilai keluarannya.

Perhatikan bahwa jika output-z misalnya 5, maka setiap posisi jendela menghasilkan 5 nilai keluaran ke dalam dimensi z keluaran. Nilai-nilai ini berbeda di bagian mana dari kernel konvolusional yang digunakan - ada kotak nilai 3d terpisah yang digunakan untuk setiap koordinat output-z . Jadi Anda dapat menganggapnya sebagai 5 konvolusi terpisah dengan filter berbeda untuk masing-masing konvolusi.

Berikut adalah kode semu untuk konvolusi 2d dengan padding dan striding:

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

KonversiElementType

Lihat juga XlaBuilder::ConvertElementType .

Mirip dengan static_cast berdasarkan elemen di C++, melakukan operasi konversi berdasarkan elemen dari bentuk data ke bentuk target. Dimensinya harus cocok, dan konversinya harus berdasarkan elemen; misalnya elemen s32 menjadi elemen f32 melalui rutin konversi s32 -ke- f32 .

ConvertElementType(operand, new_element_type)

Dimensi operan dan bentuk target harus sesuai. Tipe elemen sumber dan tujuan tidak boleh berupa tupel.

Konversi seperti T=s32 ke U=f32 akan melakukan normalisasi konversi int-to-float seperti pembulatan ke genap terdekat.

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

Jumlah Replika Silang

Melakukan AllReduce dengan perhitungan penjumlahan.

Panggilan Khusus

Lihat juga XlaBuilder::CustomCall .

Memanggil fungsi yang disediakan pengguna dalam komputasi.

CustomCall(target_name, args..., shape)

Tanda tangannya sama, apa pun arity atau tipe argumennya:

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

Misalnya jika CustomCall digunakan sebagai berikut:

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

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

Berikut adalah contoh implementasi 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];
  // ...
}

Fungsi yang disediakan pengguna tidak boleh memiliki efek samping dan pelaksanaannya harus idempoten.

Dot

Lihat juga XlaBuilder::Dot .

Dot(lhs, rhs)

Semantik yang tepat dari operasi ini bergantung pada jajaran operan:

Operasi ini melakukan penjumlahan produk pada dimensi kedua lhs (atau dimensi pertama jika memiliki peringkat 1) dan dimensi pertama rhs . Ini adalah dimensi yang "dikontrak". Dimensi kontrak lhs dan rhs harus berukuran sama. Dalam prakteknya dapat digunakan untuk melakukan perkalian titik antar vektor, perkalian vektor/matriks, atau perkalian matriks/matriks.

titikjenderal

Lihat juga XlaBuilder::DotGeneral .

DotGeneral(lhs, rhs, dimension_numbers)

Seperti Titik, tetapi memungkinkan nomor dimensi kontrak dan batch ditentukan untuk 'lhs' dan 'rhs'.

DotGeneral melakukan penjumlahan produk pada dimensi kontrak yang ditentukan dalam 'angka_dimensi'.

Nomor dimensi kontrak terkait dari 'lhs' dan 'rhs' tidak harus sama tetapi harus mempunyai ukuran dimensi yang sama.

Contoh dengan nomor dimensi kontrak:

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} }

Nomor dimensi batch terkait dari 'lhs' dan 'rhs' harus memiliki ukuran dimensi yang sama.

Contoh dengan nomor dimensi batch (ukuran batch 2, matriks 2x2):

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

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

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

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

Oleh karena itu, nomor dimensi yang dihasilkan dimulai dengan dimensi batch, kemudian dimensi non-kontrak/non-batch 'lhs', dan terakhir dimensi non-kontrak/non-batch 'rhs'.

Irisan Dinamis

Lihat juga XlaBuilder::DynamicSlice .

DynamicSlice mengekstrak sub-array dari array input pada start_indices dinamis. Ukuran irisan di setiap dimensi diteruskan dalam size_indices , yang menentukan titik akhir interval irisan eksklusif di setiap dimensi: [mulai, mulai + ukuran). Bentuk start_indices harus rank == 1, dengan ukuran dimensi sama dengan rank operand .

DynamicSlice(operand, start_indices, size_indices)

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

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

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

1-dimensional example:

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

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

2-dimensional example:

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

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

DynamicUpdateSlice

See also XlaBuilder::DynamicUpdateSlice .

DynamicUpdateSlice generates a result which is the value of the input array operand , with a slice update overwritten at start_indices . The shape of update determines the shape of the sub-array of the result which is updated. The shape of start_indices must be rank == 1, with dimension size equal to the rank of operand .

DynamicUpdateSlice(operand, update, start_indices)

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

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

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

1-dimensional example:

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

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

2-dimensional example:

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

let s = {1, 1}

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

Element-wise binary arithmetic operations

See also XlaBuilder::Add .

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

Op(lhs, rhs)

Where Op is one of Add (addition), Sub (subtraction), Mul (multiplication), Div (division), Rem (remainder), Max (maximum), Min (minimum), Atan2 (arctangent of y/x), LogicalAnd (logical AND), LogicalOr (logical OR), or LogicalXor (logical XOR).

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

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

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

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

Op(lhs, rhs, broadcast_dimensions)

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

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

Element-wise comparison operations

See also XlaBuilder::Eq .

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

Op(lhs, rhs)

Where Op is one of Eq (equal-to), Ne (not equal-to), Ge (greater-or-equal-than), Gt (greater-than), Le (less-or-equal-than), Lt (less-than). Another set of operators, EqTotalOrder, NeTotalOrder, GeTotalOrder, GtTotalOrder, LeTotalOrder, and LtTotalOrder, provide the same functionalities, except that they additionally support a total order over the floating point numbers, by enforcing -NaN < -Inf < -Finite < -0 < +0 < +Finite < +Inf < +NaN.

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

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

Op(lhs, rhs, broadcast_dimensions)

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

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

Element-wise unary functions

XlaBuilder supports these element-wise unary functions:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

using the comparison operator of the element type of operand .

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

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

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

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

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

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

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

Fft

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

See also XlaBuilder::Fft .

Multidimensional FFT

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

Implementation details

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

Gather

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

General Semantics

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

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

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

The output is an array of rank batch_dims.size + offset_dims.size .

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

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

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

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

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

Formally, the operand index In corresponding to a given output index Out is calculated as follows:

1. Let G = { Out [ k ] for k in batch_dims }. Use G to slice out a vector S such that S [ i ] = start_indices [Combine( G , i )] where Combine(A, b) inserts b at position index_vector_dim into A. Note that this is well defined even if G is empty -- if G is empty then S = start_indices .

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

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

   2. S in [ _ ] = 0 otherwise.

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

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

   2. O in [ _ ] = 0 otherwise.

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

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

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

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

Informal Description and Examples

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

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

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

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

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

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

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

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

The behavior of the gather operation can then be depicted as an index transformation that takes [ G , O 0 , O 1 ], an index in the output shape, and maps it to an element in the input array in the following way:

We first select an ( X , Y ) vector from the gather indices array using G . The element in the output array at index [ G , O 0 , O 1 ] is then the element in the input array at index [ X + O 0 , Y + O 1 ].

slice_sizes is [8,6] , which decides the range of O ₀ and O ₁ , and this in turn decides the bounds of the slice.

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

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

Again, this acts as a batch dynamic slice G 0 and G 1 as the batch dimensions. The slice size is still [8,6] .

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

1. We can configure which dimensions in the output shape are the offset dimensions (dimensions containing O 0 , O 1 in the last example). The output batch dimensions (dimensions containing G 0 , G 1 in the last example) are defined to be the output dimensions that are not offset dimensions.

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

3. The slice extracted from the "Gather Indices" array (( X , Y ) in the last example) may have fewer elements than the input array rank, and an explicit mapping dictates how the index should be expanded to have the same rank as the input.

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

G 0 and G 1 are used to slice out a starting index from the gather indices array as usual, except the starting index has only one element, X . Similarly, there is only one output offset index with the value O 0 . However, before being used as indices into the input array, these are expanded in accordance to "Gather Index Mapping" ( start_index_map in the formal description) and "Offset Mapping" ( remapped_offset_dims in the formal description) into [ X , 0 ] and [ 0 , O 0 ] respectively, adding up to [ X , O 0 ]. In other words, the output index [ G 0 , G 1 , O 0 ] maps to the input index [ GatherIndices [ G 0 , G 1 , 0 ], O 0 ] which gives us the semantics for tf.gather_nd .

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

GetDimensionSize

See also XlaBuilder::GetDimensionSize .

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

GetDimensionSize(operand, dimension)

SetDimensionSize

See also XlaBuilder::SetDimensionSize .

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

SetDimensionSize(operand, size, dimension)

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

Padded values will be ignored by downstream reduction ops.

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

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

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

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

GetTupleElement

See also XlaBuilder::GetTupleElement .

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

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

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

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

See also tf.tuple .

Infeed

See also XlaBuilder::Infeed .

Infeed(shape)

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

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

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

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

Iota

See also XlaBuilder::Iota .

Iota(shape, iota_dimension)

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

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

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

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

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

Map

See also XlaBuilder::Map .

Map(operands..., computation)

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

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

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

OptimizationBarrier

Blocks any optimization pass from moving computations across the barrier.

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

Pad

See also XlaBuilder::Pad .

Pad(operand, padding_value, padding_config)

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

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

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

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

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

Recv

See also XlaBuilder::Recv .

Recv(shape, channel_handle)

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

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

Recv(const Shape& shape, int64 channel_id)

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

RecvDone(HloInstruction context)

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

Reduce

See also XlaBuilder::Reduce .

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

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

Where:

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

This operation reduces one or more dimensions of each input array into scalars. The rank of each returned array is rank(operand) - len(dimensions) . The output of the op is Collate(Q_0, ..., Q_N) where Q_i is an array of type T_i , the dimensions of which are described below.

Different backends are allowed to reassociate the reduction computation. This can lead to numerical differences, as some reduction functions like addition are not associative for floats. However, if the range of the data is limited, floating-point addition is close enough to being associative for most practical uses.

Contoh

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

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

but there are also many other possibilities, eg

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

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

result_shape <- remove all dims in dimensions from operand_shape

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

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

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

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

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

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

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

|  4   8  12 |
| 16  20  24 |

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

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

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

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

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

Variadic Reduce

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

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

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

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

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

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

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

ReducePrecision

See also XlaBuilder::ReducePrecision .

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

ReducePrecision(operand, mantissa_bits, exponent_bits)

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

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

ReduceScatter

See also XlaBuilder::ReduceScatter .

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

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

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

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

ReduceWindow

See also XlaBuilder::ReduceWindow .

Applies a reduction function to all elements in each window of a sequence of N multi-dimensional arrays, producing a single or a tuple of N multi-dimensional arrays as output. Each output array has the same number of elements as the number of valid positions of the window. A pooling layer can be expressed as a ReduceWindow . Similar to Reduce , the applied computation is always passed the init_values on the left-hand side.

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

Where:

• N is required to be greater or equal to 1.
• All input arrays must have the same dimensions.
• If N = 1 , Collate(T) is T .
• If N > 1 , Collate(T_0, ..., T_{N-1}) is a tuple of N elements of type (T0,...T{N-1}) .

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

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

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

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

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

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

ReplicaId

See also XlaBuilder::ReplicaId .

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

ReplicaId()

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

Reshape

See also XlaBuilder::Reshape and the Collapse operation.

Reshapes the dimensions of an array into a new configuration.

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

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

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

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

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

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

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

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


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

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

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

Rev (reverse)

See also XlaBuilder::Rev .

Rev(operand, dimensions)

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

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

RngNormal

See also XlaBuilder::RngNormal .

Constructs an output of a given shape with random numbers generated following the \(N(\mu, \sigma)\) normal distribution. The parameters \(\mu\) and\(\sigma\), and output shape have to have a floating point elemental type. The parameters furthermore have to be scalar valued.

RngNormal(mu, sigma, shape)

RngUniform

See also XlaBuilder::RngUniform .

Constructs an output of a given shape with random numbers generated following the uniform distribution over the interval \([a,b)\). The parameters and output element type have to be a boolean type, an integral type or a floating point types, and the types have to be consistent. The CPU and GPU backends currently only support F64, F32, F16, BF16, S64, U64, S32 and U32. Furthermore, the parameters need to be scalar valued. If \(b <= a\) the result is implementation-defined.

RngUniform(a, b, shape)

RngBitGenerator

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

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

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

RngBitGenerator(algorithm, key, shape)

Available values for algorithm :

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

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

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

Scatter

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

See also XlaBuilder::Scatter .

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

Where:

• N is required to be greater or equal to 1.
• operands [ 0 ], ..., operands [ N-1 ] must all have the same dimensions.
• updates [ 0 ], ..., updates [ N-1 ] must all have the same dimensions.
• If N = 1 , Collate(T) is T .
• If N > 1 , Collate(T_0, ..., T_N) is a tuple of N elements of type T .

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

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

The arguments of scatter should follow these constraints:

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

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

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

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

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

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

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

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

1. Let G = { U [ k ] for k in update_scatter_dims }. Use G to look up an index vector S in the scatter_indices array such that S [ i ] = scatter_indices [Combine( G , i )] where Combine(A, b) inserts b at positions index_vector_dim into A.
2. Create an index S in into operand using S by scattering S using the scatter_dims_to_operand_dims map. More formally:
   1. S in [ scatter_dims_to_operand_dims [ k ]] = S [ k ] if k < scatter_dims_to_operand_dims.size .
   2. S in [ _ ] = 0 otherwise.
3. Create an index W in into each operands array by scattering the indices at update_window_dims in U according to inserted_window_dims . More formally:
   1. W in [ window_dims_to_operand_dims ( k )] = U [ k ] if k is in update_window_dims , where window_dims_to_operand_dims is the monotonic function with domain [ 0 , update_window_dims.size ) and range [ 0 , operand.rank ) \ inserted_window_dims . (For example, if update_window_dims.size is 4 , operand.rank is 6 , and inserted_window_dims is { 0 , 2 } then window_dims_to_operand_dims is { 0 → 1 , 1 → 3 , 2 → 4 , 3 → 5 }).
   2. W in [ _ ] = 0 otherwise.
4. I is W in + S in where + is element-wise addition.

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

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

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

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

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

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

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

Select

See also XlaBuilder::Select .

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

Select(pred, on_true, on_false)

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

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

Example with non-scalar pred :

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

Example with scalar pred :

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

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

SelectAndScatter

See also XlaBuilder::SelectAndScatter .

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

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

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

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

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

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

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

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

Send

See also XlaBuilder::Send .

Send(operand, channel_handle)

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

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

Send(HloInstruction operand, int64 channel_id)

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

SendDone(HloInstruction context)

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

Scheduling of channel instructions

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

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

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

Slice

See also XlaBuilder::Slice .

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

Slice(operand, start_indices, limit_indices, strides)

1-dimensional example:

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

2-dimensional example:

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

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

Sort

See also XlaBuilder::Sort .

Sort(operands, comparator, dimension, is_stable)

If only one operand is provided:

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

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

If n > 1 operands are provided:

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

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

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

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

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

Top-K

See also the jax.lax.top_k operation.

TopK(operand)

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

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

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

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

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

Transpose

See also the tf.reshape operation.

Transpose(operand)

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

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

TriangularSolve

See also XlaBuilder::TriangularSolve .

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

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

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

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

Tuple

See also XlaBuilder::Tuple .

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

This is analogous to std::tuple in C++. Conceptually:

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

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

While

See also XlaBuilder::While .

While(condition, body, init)

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

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

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

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

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