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Operasi Semantik

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

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

Lagipula

Lihat juga XlaBuilder::AfterAll .

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

AfterAll(operands)

Argumen Tipe Semantik
operands XlaOp jumlah token yang bervariasi

AllReduce

Lihat juga XlaBuilder::AllReduce .

Melakukan penghitungan khusus di seluruh replika.

AllReduce(operand, computation, replica_group_ids, channel_id)

Argumen Tipe Semantik
operand XlaOp Larik atau tupel larik yang tidak kosong untuk dikurangi di seluruh replika.
computation XlaComputation Perhitungan reduksi
replica_groups vektor vektor int64 Grup tempat pengurangan dilakukan
channel_id opsional int64 ID saluran opsional untuk komunikasi lintas modul
  • Jika operand adalah tupel array, pengurangan semua dilakukan pada setiap elemen tupel.
  • replica_groups adalah daftar grup replika di mana pengurangan dilakukan (replika id untuk replika saat ini dapat diambil menggunakan ReplicaId ). replica_groups harus kosong (dalam hal ini semua replika milik satu grup), atau berisi jumlah elemen yang sama dengan jumlah replika. Misalnya, replica_groups = {0, 2}, {1, 3} melakukan pengurangan antara replika 0 dan 2 , dan 1 dan 3 .
  • channel_id digunakan untuk komunikasi lintas modul: hanya operasi all-reduce dengan channel_id sama yang dapat berkomunikasi satu sama lain.

Bentuk keluarannya sama dengan bentuk masukan. Misalnya, jika ada dua replika dan operand memiliki nilai [1.0, 2.5] dan [3.0, 5.25] masing-masing pada dua replika, maka nilai output dari perhitungan operasi dan penjumlahan ini akan menjadi [4.0, 7.75] pada keduanya. replika. Jika inputnya adalah tupel, maka outputnya adalah tupel juga.

Menghitung hasil AllReduce membutuhkan satu input dari setiap replika, jadi jika satu replika mengeksekusi node AllReduce lebih sering daripada yang lain, replika sebelumnya akan menunggu selamanya. Karena semua replika menjalankan program yang sama, tidak banyak cara untuk itu terjadi, tetapi hal ini dimungkinkan ketika kondisi loop sementara bergantung pada data dari infeed dan data yang diinfeksi menyebabkan while loop berulang kali pada satu replika dari yang lain.

AllToAll

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, operand dipecah menjadi jumlah split_count blok sepanjang split_dimensions , dan blok tersebar ke semua inti, misalnya, blok ke-i dikirim ke inti ke-i.
  2. Fase mengumpulkan. Setiap inti menggabungkan blok yang diterima sepanjang concat_dimension .

Inti yang berpartisipasi dapat dikonfigurasi dengan:

  • replica_groups : setiap ReplicaGroup berisi daftar replika id yang berpartisipasi dalam komputasi (replika id 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 bahwa AllToAll akan diterapkan dalam replika {1, 2, 3} , dan dalam fase pengumpulan, dan blok yang diterima akan digabung dalam urutan yang sama yaitu 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 grup, dalam urutan rangkaian penampilan mereka.

Prasyarat:

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

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

Argumen Tipe Semantik
operand XlaOp n masukan dimensi array
split_dimension int64 Nilai dalam interval [0, n) yang memberi nama dimensi di mana operan dipisahkan
concat_dimension int64 sebuah nilai dalam interval [0, n) yang menamai dimensi di mana blok yang dipisahkan digabungkan
split_count int64 jumlah inti yang berpartisipasi dalam operasi ini. Jika replica_groups kosong, ini harus menjadi jumlah replika; jika tidak, ini harus sama dengan jumlah replika di setiap grup.
replica_groups ReplicaGroup vektor setiap grup berisi daftar replika id.

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 tiap core, operand dibagi menjadi 4 bagian sepanjang dimensi 0, sehingga tiap bagian memiliki bentuk f32 [4,4]. 4 bagian tersebar ke semua inti. Kemudian setiap inti menggabungkan bagian yang diterima sepanjang dimensi 1, dalam urutan atau inti 0-4. Jadi keluaran pada tiap inti berbentuk f32 [16,4].

BatchNormGrad

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

Menghitung gradien norma kelompok.

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

Argumen Tipe Semantik
operand XlaOp n array dimensi untuk dinormalisasi (x)
scale XlaOp Array 1 dimensi (\(\gamma\))
mean XlaOp Array 1 dimensi (\(\mu\))
variance XlaOp Array 1 dimensi (\(\sigma^2\))
grad_output XlaOp Gradien diteruskan ke BatchNormTraining (\( \nabla y\))
epsilon float Nilai Epsilon (\(\epsilon\))
feature_index int64 Indeks untuk menampilkan dimensi dalam operand

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 merupakan indeks yang valid untuk dimensi fitur dalam operand .

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

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

Rata- mean input dan variance merepresentasikan nilai momen lintas dimensi batch dan spasial.

Jenis keluaran adalah tupel dari tiga pegangan:

Keluaran Tipe Semantik
grad_operand XlaOp gradien sehubungan dengan operand masukan (\( \nabla x\))
grad_scale XlaOp gradien sehubungan dengan scale masukan (\( \nabla \gamma\))
grad_offset XlaOp gradien sehubungan dengan masukan offset (\( \nabla \beta\))

BatchNormInference

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

Menormalkan larik di seluruh dimensi batch dan spasial.

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

Argumen Tipe Semantik
operand XlaOp n array dimensi untuk dinormalisasi
scale XlaOp Array 1 dimensi
offset XlaOp Array 1 dimensi
mean XlaOp Array 1 dimensi
variance XlaOp Array 1 dimensi
epsilon float Nilai Epsilon
feature_index int64 Indeks untuk menampilkan dimensi dalam operand

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

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

Keluarannya adalah larik berdimensi-n yang dinormalisasi dengan bentuk yang sama dengan operand masukan.

BatchNormTraining

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

Menormalkan larik di seluruh dimensi batch dan spasial.

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

Argumen Tipe Semantik
operand XlaOp n array dimensi untuk dinormalisasi (x)
scale XlaOp Array 1 dimensi (\(\gamma\))
offset XlaOp Array 1 dimensi (\(\beta\))
epsilon float Nilai Epsilon (\(\epsilon\))
feature_index int64 Indeks untuk menampilkan dimensi dalam operand

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

Algoritme berjalan sebagai berikut untuk setiap batch dalam operand \(x\) yang berisi elemen m 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 varian batch \(\sigma^2_l\): \(\sigma^2_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (x_{ijkl} - \mu_l)^2\)

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

Nilai epsilon, biasanya angka kecil, ditambahkan untuk menghindari kesalahan bagi-dengan-nol.

Jenis keluaran adalah tupel dari tiga XlaOp :

Keluaran Tipe Semantik
output XlaOp n array dimensi dengan bentuk yang sama dengan operand input (y)
batch_mean XlaOp Array 1 dimensi (\(\mu\))
batch_var XlaOp Array 1 dimensi (\(\sigma^2\))

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

BitcastConvertType

Lihat juga XlaBuilder::BitcastConvertType .

Mirip dengan tf.bitcast di TensorFlow, melakukan operasi bitcast berdasarkan elemen dari bentuk data ke bentuk target. Dimensi harus sesuai, dan konversinya adalah elemen yang bijaksana; misalnya elemen s32 menjadi elemen f32 melalui rutinitas bitcast. Bitcast diimplementasikan sebagai cast level rendah, sehingga mesin dengan representasi floating-point yang berbeda akan memberikan hasil yang berbeda.

BitcastConvertType(operand, new_element_type)

Argumen Tipe Semantik
operand XlaOp array tipe T dengan dims D
new_element_type PrimitiveType ketik U

Dimensi operan dan bentuk target harus sesuai. Lebar bit jenis elemen sumber dan tujuan harus sama. Jenis elemen sumber dan tujuan tidak boleh berupa tupel.

Siaran

Lihat juga XlaBuilder::Broadcast .

Menambahkan dimensi ke larik dengan menduplikasi data dalam larik.

Broadcast(operand, broadcast_sizes)

Argumen Tipe Semantik
operand XlaOp Larik yang akan diduplikasi
broadcast_sizes ArraySlice<int64> Ukuran dimensi baru

Dimensi baru disisipkan di sebelah kiri, yaitu jika broadcast_sizes memiliki nilai {a0, ..., aN} dan bentuk operan memiliki dimensi {b0, ..., bM} maka bentuk keluaran memiliki dimensi {a0, ..., aN, b0, ..., bM} .

Indeks dimensi baru menjadi salinan operan, yaitu

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 adalah larik dengan bentuk f32[2, 3] dan semua nilai dalam hasilnya adalah 2.0f .

BroadcastInDim

Lihat juga XlaBuilder::BroadcastInDim .

Memperluas ukuran dan peringkat array dengan menduplikasi data dalam array.

BroadcastInDim(operand, out_dim_size, broadcast_dimensions)

Argumen Tipe Semantik
operand XlaOp Larik yang akan diduplikasi
out_dim_size ArraySlice<int64> Ukuran dimensi bentuk target
broadcast_dimensions ArraySlice<int64> Dimensi mana dalam bentuk target yang sesuai dengan setiap dimensi bentuk operan

Mirip dengan Broadcast, 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 dari operan dipetakan ke dimensi broadcast_dimension [i] dari bentuk output. Dimensi operand harus berukuran 1 atau sama dengan dimensi dalam bentuk keluaran yang dipetakan. Dimensi yang tersisa diisi dengan dimensi ukuran 1. Penyiaran berdimensi-degenerasi kemudian disiarkan sepanjang dimensi-dimensi yang merosot ini untuk mencapai bentuk keluaran. Semantik dijelaskan secara rinci di halaman penyiaran .

Panggilan

Lihat juga XlaBuilder::Call .

Memanggil komputasi dengan argumen yang diberikan.

Call(computation, args...)

Argumen Tipe Semantik
computation XlaComputation komputasi tipe T_0, T_1, ..., T_N -> S dengan N parameter tipe sembarang
args urutan N XlaOp s Argumen N dengan tipe sembarang

Arity dan tipe args harus sesuai dengan parameter computation . Itu diperbolehkan untuk tidak args .

Cholesky

Lihat juga XlaBuilder::Cholesky .

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

Cholesky(a, lower)

Argumen Tipe Semantik
a XlaOp rank> 2 array dari tipe kompleks atau floating-point.
lower bool apakah akan menggunakan segitiga atas atau bawah dari a .

Jika lower adalah true , hitung matriks segitiga rendah l sedemikian rupa

$$ a = l . l^T $$

. Jika lower false , hitung matriks segitiga atas u sedemikian rupa

$$ a = u^T . u $$

.

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

Jika peringkat a lebih besar dari 2, a dianggap sebagai kumpulan matriks, di mana semua kecuali 2 dimensi minor adalah dimensi tumpukan.

Jika a tidak pasti positif simetris (Hermitian), hasilnya adalah definisi implementasi.

Penjepit

Lihat juga XlaBuilder::Clamp .

Menjepit operan ke dalam kisaran antara nilai minimum dan maksimum.

Clamp(min, operand, max)

Argumen Tipe Semantik
min XlaOp array tipe T
operand XlaOp array tipe T
max XlaOp array tipe T

Diberikan nilai operan dan minimum dan maksimum, mengembalikan operan jika berada dalam kisaran antara minimum dan maksimum, jika tidak mengembalikan nilai minimum jika operan di bawah kisaran ini atau nilai maksimum jika operan di atas kisaran ini. Yaitu, clamp(a, x, b) = min(max(a, x), b) .

Ketiga larik harus memiliki bentuk yang sama. Alternatifnya, sebagai bentuk penyiaran terbatas, min dan / atau max bisa menjadi skalar tipe 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};

Jatuh

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

Menciutkan dimensi larik menjadi satu dimensi.

Collapse(operand, dimensions)

Argumen Tipe Semantik
operand XlaOp array tipe T
dimensions vektor int64 secara berurutan, subset yang berurutan dari dimensi T.

Ciutkan menggantikan subset tertentu dari dimensi operan dengan satu dimensi. Argumen masukan adalah array sembarang tipe T dan vektor indeks dimensi konstanta waktu kompilasi. Indeks dimensi harus merupakan subset berurutan (nomor dimensi rendah ke tinggi), berturut-turut dari dimensi T. Jadi, {0, 1, 2}, {0, 1}, atau {1, 2} semua adalah kumpulan dimensi yang valid, tetapi {1, 0} atau {0, 2} bukan. Mereka diganti dengan satu dimensi baru, dalam posisi yang sama dalam urutan dimensi seperti yang mereka gantikan, dengan ukuran dimensi baru yang sama dengan produk ukuran dimensi asli. Nomor dimensi terendah dalam dimensions adalah variasi dimensi yang paling lambat (paling utama) di sarang loop yang menciutkan dimensi ini, dan nomor dimensi tertinggi paling cepat bervariasi (paling kecil). Lihat operator tf.reshape jika pengurutan collapse yang lebih umum diperlukan.

Misalnya, misalkan v adalah larik 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} };

CollectivePermute

Lihat juga XlaBuilder::CollectivePermute .

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

CollectivePermute(operand, source_target_pairs)

Argumen Tipe Semantik
operand XlaOp n masukan dimensi array
source_target_pairs <int64, int64> Daftar pasangan (source_replica_id, target_replica_id). Untuk setiap pasangan, operand dikirim dari replika sumber ke replika target.

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 sebuah replika id bukan merupakan target pada pasangan manapun, maka keluaran pada replika tersebut adalah tensor yang terdiri dari 0 (s) dengan bentuk yang sama dengan input.

Menggabungkan

Lihat juga XlaBuilder::ConcatInDim .

Concatenate menyusun larik dari beberapa operan larik. Array memiliki peringkat yang sama dengan setiap operan larik masukan (yang harus memiliki peringkat yang sama satu sama lain) dan berisi argumen dalam urutan yang ditentukan.

Concatenate(operands..., dimension)

Argumen Tipe Semantik
operands urutan N XlaOp Array N tipe T dengan dimensi [L0, L1, ...]. Membutuhkan N> = 1.
dimension int64 Nilai dalam interval [0, N) yang memberi nama dimensi yang akan digabungkan antara operands .

Dengan pengecualian dimension semua dimensi harus sama. Ini karena XLA tidak mendukung array "compang-camping". Perhatikan juga bahwa nilai peringkat-0 tidak dapat digabungkan (karena tidak mungkin memberi nama dimensi di mana penggabungan terjadi).

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)

Argumen Tipe Semantik
pred XlaOp Skalar jenis PRED
true_operand XlaOp Argumen tipe \(T_0\)
true_computation XlaComputation XlaComputation tipe \(T_0 \to S\)
false_operand XlaOp Argumen tipe \(T_1\)
false_computation XlaComputation XlaComputation tipe \(T_1 \to S\)

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

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

Perhatikan bahwa hanya satu dari true_computation dan false_computation akan dijalankan tergantung pada nilai pred .

Conditional(branch_index, branch_computations, branch_operands)

Argumen Tipe Semantik
branch_index XlaOp Skalar tipe S32
branch_computations urutan N XlaComputation XlaComputations tipe \( T_0 \to S , T_1 \to S , ..., T_{N-1} \to S \)
branch_operands urutan N XlaOp Argumen tipe \( T_0 , T_1 , ..., T_{N-1} \)

branch_computations[branch_index] , dan mengembalikan 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 argumen tunggal berjenis T_b dan akan dipanggil dengan branch_operands[b] yang harus berjenis sama. Jenis nilai yang dikembalikan dari setiap branch_computations[b] harus sama.

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

Konv (konvolusi)

Lihat juga XlaBuilder::Conv .

Sebagai ConvWithGeneralPadding, tetapi padding ditentukan secara singkat sebagai SAME atau VALID. Padding yang sama mengisi input ( lhs ) dengan nol sehingga output memiliki bentuk yang sama dengan input saat tidak memperhitungkan langkah. Padding yang VALID berarti tidak ada padding.

ConvWithGeneralPadding (konvolusi)

Lihat juga XlaBuilder::ConvWithGeneralPadding .

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

Argumen Tipe Semantik
lhs XlaOp peringkat n + 2 larik input
rhs XlaOp peringkat n + 2 array bobot kernel
window_strides ArraySlice<int64> nd array langkah kernel
padding ArraySlice< pair<int64, int64>> dan larik padding (rendah, tinggi)
lhs_dilation ArraySlice<int64> nd lhs faktor dilasi array
rhs_dilation ArraySlice<int64> nd rhs array faktor dilasi
feature_group_count int64 jumlah grup fitur
batch_group_count int64 jumlah kelompok batch

Misalkan n adalah jumlah dimensi spasial. Argumen lhs adalah lhs rank n + 2 yang mendeskripsikan luas dasar. Ini disebut masukan, meskipun tentu saja rhs juga merupakan masukan. Dalam jaringan saraf, ini adalah aktivasi input. Dimensi n + 2 adalah, dalam urutan berikut:

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

Argumen rhs adalah larik peringkat n + 2 yang menjelaskan filter konvolusional / kernel / jendela. Dimensinya adalah, dalam urutan berikut:

  • output-z : Dimensi z dari output.
  • input-z : Ukuran kali dimensi ini feature_group_count harus sama dengan ukuran dimensi z dalam lhs.
  • spatial_dims : Menjelaskan n dimensi spasial yang menentukan jendela 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 di mana indeks spasial pertama habis dibagi 3.

Argumen padding menentukan jumlah padding nol yang akan diterapkan ke area dasar. Jumlah padding bisa negatif - nilai absolut padding negatif menunjukkan jumlah elemen yang akan dihapus dari dimensi yang ditentukan sebelum melakukan konvolusi. padding[0] menentukan padding untuk dimensi y dan padding[1] menentukan padding untuk dimensi x . Setiap pasangan memiliki bantalan rendah sebagai elemen pertama dan bantalan tinggi sebagai elemen kedua. Bantalan rendah diterapkan ke arah indeks lebih rendah sedangkan bantalan tinggi diterapkan ke arah indeks yang lebih tinggi. Misalnya 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 setara dengan memasukkan nilai nol yang sama ke dalam input ( lhs ) sebelum melakukan konvolusi.

lhs_dilation dan rhs_dilation menentukan faktor dilasi yang akan diterapkan ke lhs dan rhs, masing-masing, di setiap dimensi spasial. Jika faktor dilasi dalam dimensi spasial adalah d, maka lubang d-1 secara implisit ditempatkan di antara setiap entri dalam dimensi tersebut, sehingga meningkatkan ukuran larik. Lubang diisi dengan nilai tanpa operasi, 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 menjadi pembagi dari dimensi fitur masukan dan keluaran. Jika feature_group_count lebih besar dari 1, artinya secara konseptual dimensi fitur input dan output serta dimensi fitur output rhs dibagi secara merata menjadi feature_group_count banyak grup, setiap grup terdiri dari rangkaian fitur yang berurutan. Dimensi fitur masukan rhs harus sama dengan dimensi fitur masukan lhs dibagi dengan feature_group_count (jadi sudah memiliki ukuran sekumpulan fitur masukan). Grup ke-i digunakan bersama untuk menghitung feature_group_count banyak konvolusi terpisah. Hasil dari konvolusi ini digabungkan bersama dalam dimensi fitur keluaran.

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

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

Bentuk keluaran memiliki dimensi ini, dalam urutan ini:

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

Penempatan yang valid dari jendela konvolusional ditentukan oleh langkah dan ukuran area alas setelah bantalan.

Untuk mendeskripsikan fungsi konvolusi, pertimbangkan konvolusi 2d, dan pilih beberapa koordinat batch , z , y , x pada keluaran. Kemudian (y,x) adalah posisi sudut jendela di dalam area dasar (misalnya, sudut kiri atas, tergantung bagaimana Anda menginterpretasikan dimensi spasial). Kami sekarang memiliki jendela 2d, diambil dari area dasar, di mana setiap titik 2d dikaitkan dengan vektor 1d, jadi kami mendapatkan kotak 3d. Dari kernel konvolusional, karena kami memperbaiki koordinat keluaran z , kami juga memiliki kotak 3d. Kedua kotak memiliki dimensi yang sama, jadi kita dapat mengambil jumlah dari hasil perkalian elemen antara dua kotak (mirip dengan perkalian titik). Itu adalah nilai keluaran.

Perhatikan bahwa jika output-z adalah misalnya 5, maka setiap posisi jendela menghasilkan 5 nilai dalam output ke dalam dimensi output z . 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-masingnya.

Berikut adalah pseudo-code 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;
}

ConvertElementType

Lihat juga XlaBuilder::ConvertElementType .

Mirip dengan static_cast berbasis elemen di C ++, melakukan operasi konversi berdasarkan elemen dari bentuk data ke bentuk target. Dimensi harus sesuai, dan konversinya adalah elemen yang bijaksana; misalnya elemen s32 menjadi elemen f32 melalui konversi rutin s32 f32 .

ConvertElementType(operand, new_element_type)

Argumen Tipe Semantik
operand XlaOp array tipe T dengan dims D
new_element_type PrimitiveType ketik U

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

Konversi seperti T=s32 ke U=f32 akan melakukan normalisasi konversi int-to-float seperti round-to-nearest-even.

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

CrossReplicaSum

Melakukan AllReduce dengan perhitungan penjumlahan.

CustomCall

Lihat juga XlaBuilder::CustomCall .

Panggil fungsi yang disediakan pengguna dalam komputasi.

CustomCall(target_name, args..., shape)

Argumen Tipe Semantik
target_name string Nama fungsinya. Instruksi panggilan akan dikeluarkan yang menargetkan nama simbol ini.
args urutan N XlaOp s N argumen tipe arbitrer, yang akan diteruskan ke fungsi.
shape Shape Bentuk keluaran dari fungsi

Tanda tangan fungsinya sama, terlepas dari arity atau tipe arg:

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 diberikan pengguna tidak boleh memiliki efek samping dan pelaksanaannya harus idempoten.

Dot

Lihat juga XlaBuilder::Dot .

Dot(lhs, rhs)

Argumen Tipe Semantik
lhs XlaOp array tipe T
rhs XlaOp array tipe T

Semantik yang tepat dari operasi ini bergantung pada jajaran operan:

Memasukkan Keluaran Semantik
vektor [n] dot vektor [n] skalar produk titik vektor
matriks [mxk] vektor dot [k] vektor [m] perkalian matriks-vektor
matriks [mxk] dot matriks [kxn] matriks [mxn] perkalian matriks-matriks

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

DotGeneral

Lihat juga XlaBuilder::DotGeneral .

DotGeneral(lhs, rhs, dimension_numbers)

Argumen Tipe Semantik
lhs XlaOp array tipe T
rhs XlaOp array tipe T
dimension_numbers DotDimensionNumbers nomor dimensi kontrak dan batch

Sebagai Dot, tetapi memungkinkan nomor dimensi kontrak dan batch ditentukan untuk 'lhs' dan 'rhs'.

Bidang DotDimensionNumbers Tipe Semantik
'lhs_contracting_dimensions' berulang int64 Nomor dimensi kontrak 'lhs'
'rhs_contracting_dimensions' berulang int64 Nomor dimensi kontrak 'rhs'
'lhs_batch_dimensions' berulang int64 Nomor dimensi batch 'lhs'
'rhs_batch_dimensions' berulang int64 nomor dimensi batch 'rhs'

DotGeneral melakukan penjumlahan produk di atas dimensi kontrak yang ditentukan dalam 'nomor_dimensi'.

Nomor dimensi kontrak terkait dari 'lhs' dan 'rhs' tidak harus sama tetapi harus memiliki 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 tumpukan (ukuran tumpukan 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} } }
Memasukkan Keluaran Semantik
[b0, m, k] dot [b0, k, n] [b0, m, n] batch matmul
[b0, b1, m, k] dot [b0, b1, k, n] [b0, b1, m, n] batch matmul

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'.

DynamicSlice

Lihat juga XlaBuilder::DynamicSlice .

DynamicSlice mengekstrak sub-larik dari larik input pada start_indices dinamis. Ukuran potongan di setiap dimensi diteruskan dalam size_indices , yang menentukan titik akhir interval potongan 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)

Argumen Tipe Semantik
operand XlaOp Array dimensi N tipe T
start_indices urutan N XlaOp Daftar bilangan bulat skalar N yang berisi indeks awal irisan untuk setiap dimensi. Nilai harus lebih besar dari atau sama dengan nol.
size_indices ArraySlice<int64> Daftar bilangan bulat N yang berisi ukuran potongan untuk setiap dimensi. Setiap nilai harus lebih besar dari nol, dan ukuran + awal harus kurang dari atau sama dengan ukuran dimensi untuk menghindari penggabungan ukuran dimensi modulo.

Indeks irisan efektif dihitung dengan menerapkan transformasi berikut untuk setiap indeks i di [1, N) sebelum melakukan irisan:

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

Ini memastikan bahwa potongan yang diekstraksi selalu berada dalam batas sehubungan dengan array operan. Jika potongan berada di dalam batas sebelum transformasi diterapkan, transformasi tidak berpengaruh.

Contoh 1 dimensi:

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

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

Contoh 2 dimensi:

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

Lihat juga XlaBuilder::DynamicUpdateSlice .

DynamicUpdateSlice menghasilkan hasil yang merupakan nilai operand array input, dengan update slice ditimpa di start_indices . Bentuk update menentukan bentuk sub-larik dari hasil yang diperbarui. Bentuk start_indices harus rank == 1, dengan ukuran dimensi sama dengan rank operand .

DynamicUpdateSlice(operand, update, start_indices)

Argumen Tipe Semantik
operand XlaOp Array dimensi N tipe T
update XlaOp Array berdimensi N tipe T berisi update slice. Setiap dimensi bentuk pembaruan harus lebih besar dari nol, dan start + update harus kurang dari atau sama dengan ukuran operan untuk setiap dimensi untuk menghindari pembuatan indeks pembaruan di luar batas.
start_indices urutan N XlaOp Daftar bilangan bulat skalar N yang berisi indeks awal irisan untuk setiap dimensi. Nilai harus lebih besar dari atau sama dengan nol.

Indeks irisan efektif dihitung dengan menerapkan transformasi berikut untuk setiap indeks i di [1, N) sebelum melakukan irisan:

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

Ini memastikan bahwa slice yang diperbarui selalu terikat dengan array operan. Jika potongan berada di dalam batas sebelum transformasi diterapkan, transformasi tidak berpengaruh.

Contoh 1 dimensi:

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}

Contoh 2 dimensi:

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} }

Operasi aritmatika biner berbasis elemen

Lihat juga XlaBuilder::Add .

Seperangkat operasi aritmatika biner yang bijaksana didukung.

Op(lhs, rhs)

Dimana Op adalah salah satu Add (Selain itu), Sub (pengurangan), Mul (perkalian), Div (divisi), Rem (sisanya), Max (maksimum), Min (minimum), LogicalAnd (logika AND), atau LogicalOr (logical ATAU).

Argumen Tipe Semantik
lhs XlaOp operan sisi kiri: larik tipe T.
rhs XlaOp operan sisi kanan: larik tipe T

Bentuk argumen harus serupa atau kompatibel. Lihat dokumentasi penyiaran tentang apa artinya bentuk agar kompatibel. Hasil operasi memiliki bentuk yang merupakan hasil penyiaran dua larik input. Dalam varian ini, operasi antara array dengan rank yang berbeda tidak didukung, kecuali salah satu operannya adalah skalar.

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

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

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

Op(lhs, rhs, broadcast_dimensions)

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

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

Element-wise comparison operations

See also XlaBuilder::Eq .

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

Op(lhs, rhs)

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

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

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

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

Op(lhs, rhs, broadcast_dimensions)

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

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

Element-wise unary functions

XlaBuilder supports these element-wise unary functions:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

using the comparison operator of the element type of operand .

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

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

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

Arguments Type Semantics
operand XlaOp The operand to the function

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

Fft

The XLA FFT operation implements the forward and inverse Fourier Transforms for real and complex inputs/outputs. Multidimensional FFTs on up to 3 axes are supported, except on TPU, where only a single axis is supported (please file a GitHub issue if you require higher order).

See also XlaBuilder::Fft .

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

Multidimensional FFT

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

Implementation details

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

Gather

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

General Semantics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    2. S in [ _ ] = 0 otherwise.

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

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

    2. O in [ _ ] = 0 otherwise.

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

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

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

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

Informal Description and Examples

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

GetDimensionSize

See also XlaBuilder::GetDimensionSize .

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

GetDimensionSize(operand, dimension)

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

SetDimensionSize

See also XlaBuilder::SetDimensionSize .

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

SetDimensionSize(operand, size, dimension)

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

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

Padded values will be ignored by downstream reduction ops.

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

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

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

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

GetTupleElement

See also XlaBuilder::GetTupleElement .

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

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

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

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

See also tf.tuple .

Infeed

See also XlaBuilder::Infeed .

Infeed(shape)

Argument Type Semantics
shape Shape Shape of the data read from the Infeed interface. The layout field of the shape must be set to match the layout of the data sent to the device; otherwise its behavior is undefined.

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

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

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

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

Iota

Iota()

Builds a constant literal on device rather than a potentially large host transfer. Creates a rank 1 array of values starting at zero and incrementing by one. For floating-point types, the produced array is equivalent to ConvertElementType(Iota(...)) where the Iota is of integral type and the conversion is to the floating-point type.

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

Map

See also XlaBuilder::Map .

Map(operands..., computation)

Arguments Type Semantics
operands sequence of N XlaOp s N arrays of types T 0..T {N-1}
computation XlaComputation computation of type T_0, T_1, ..., T_{N + M -1} -> S with N parameters of type T and M of arbitrary type
dimensions int64 array array of map dimensions

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

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

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

Pad

See also XlaBuilder::Pad .

Pad(operand, padding_value, padding_config)

Arguments Type Semantics
operand XlaOp array of type T
padding_value XlaOp scalar of type T to fill in the added padding
padding_config PaddingConfig padding amount on both edges (low, high) and between the elements of each dimension

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

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

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

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

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

Recv

See also XlaBuilder::Recv .

Recv(shape, channel_handle)

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

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

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

Recv(const Shape& shape, int64 channel_id)

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

RecvDone(HloInstruction context)

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

Reduce

See also XlaBuilder::Reduce .

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

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

Arguments Type Semantics
operands Sequence of N XlaOp N arrays of types T_0, ..., T_N .
init_values Sequence of N XlaOp N scalars of types T_0, ..., T_N .
computation XlaComputation computation of type T_0, ..., T_N, T_0, ..., T_N -> Collate(T_0, ..., T_N) .
dimensions int64 array unordered array of dimensions to reduce.

Where:

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

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

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

The evaluation order of the reduction function is arbitrary and may be non-deterministic. Therefore, the reduction function should not be overly sensitive to reassociation.

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

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

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

but there are also many other possibilities, eg

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

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

result_shape <- remove all dims in dimensions from operand_shape

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

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

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

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

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

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

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

|  4   8  12 |
| 16  20  24 |

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

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

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

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

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

Variadic Reduce

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

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

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

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

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

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

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

ReducePrecision

See also XlaBuilder::ReducePrecision .

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

ReducePrecision(operand, mantissa_bits, exponent_bits)

Arguments Type Semantics
operand XlaOp array of floating-point type T .
exponent_bits int32 number of exponent bits in lower-precision format
mantissa_bits int32 number of mantissa bits in lower-precision format

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

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

ReduceWindow

See also XlaBuilder::ReduceWindow .

Applies a reduction function to all elements in each window of the input multi-dimensional array, producing an output multi-dimensional array with the same number of elements as the number of valid positions of the window. A pooling layer can be expressed as a ReduceWindow . Similar to Reduce , the applied computation is always passed the init_value on the left-hand side.

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

Arguments Type Semantics
operand XlaOp N dimensional array containing elements of type T. This is the base area on which the window is placed.
init_value XlaOp Starting value for the reduction. See Reduce for details.
computation XlaComputation Reduction function of type T, T -> T , to apply to all elements in each window
window_dimensions ArraySlice<int64> array of integers for window dimension values
window_strides ArraySlice<int64> array of integers for window stride values
base_dilations ArraySlice<int64> array of integers for base dilation values
window_dilations ArraySlice<int64> array of integers for window dilation values
padding Padding padding type for window (Padding::kSame, which pads so as to have the same output shape as input if the stride is 1, or Padding::kValid, which uses no padding and "stops" the window once it no longer fits)

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

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

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

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

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

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

ReplicaId

See also XlaBuilder::ReplicaId .

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

ReplicaId()

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

Reshape

See also XlaBuilder::Reshape and the Collapse operation.

Reshapes the dimensions of an array into a new configuration.

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

Arguments Type Semantics
operand XlaOp array of type T
dimensions int64 vector order in which dimensions are collapsed
new_sizes int64 vector vector of sizes of new dimensions

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

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

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

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

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

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

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


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

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

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

Rev (reverse)

See also XlaBuilder::Rev .

Rev(operand, dimensions)

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

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

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

RngNormal

See also XlaBuilder::RngNormal .

Constructs an output of a given shape with random numbers generated following the

$$N(\mu, \sigma)$$

normal distribution. The parameters

$$\mu$$

and

$$\sigma$$

, and output shape have to have a floating point elemental type. The parameters furthermore have to be scalar valued.

RngNormal(mu, sigma, shape)

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

RngUniform

See also XlaBuilder::RngUniform .

Constructs an output of a given shape with random numbers generated following the uniform distribution over the interval

$$[a,b)$$

. The parameters and output element type have to be a boolean type, an integral type or a floating point types, and the types have to be consistent. The CPU and GPU backends currently only support F64, F32, F16, BF16, S64, U64, S32 and U32. Furthermore, the parameters need to be scalar valued. If

$$b <= a$$

the result is implementation-defined.

RngUniform(a, b, shape)

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

RngBitGenerator

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

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

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

RngBitGenerator(algorithm, key, shape)

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

Available values for algorithm :

Scatter

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

See also XlaBuilder::Scatter .

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

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

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

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

The arguments of scatter should follow these constraints:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Select

See also XlaBuilder::Select .

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

Select(pred, on_true, on_false)

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

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

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

Example with non-scalar pred :

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

Example with scalar pred :

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

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

SelectAndScatter

See also XlaBuilder::SelectAndScatter .

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

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

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

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

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

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

Arguments Type Semantics
operand XlaOp array of type T over which the windows slide
select XlaComputation binary computation of type T, T -> PRED , to apply to all elements in each window; returns true if the first parameter is selected and returns false if the second parameter is selected
window_dimensions ArraySlice<int64> array of integers for window dimension values
window_strides ArraySlice<int64> array of integers for window stride values
padding Padding padding type for window (Padding::kSame or Padding::kValid)
source XlaOp array of type T with the values to scatter
init_value XlaOp scalar value of type T for the initial value of the output array
scatter XlaComputation binary computation of type T, T -> T , to apply each scatter source element with its destination element

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

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

Send

See also XlaBuilder::Send .

Send(operand, channel_handle)

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

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

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

Send(HloInstruction operand, int64 channel_id)

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

SendDone(HloInstruction context)

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

Scheduling of channel instructions

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

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

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

Slice

See also XlaBuilder::Slice .

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

Slice(operand, start_indices, limit_indices)

Arguments Type Semantics
operand XlaOp N dimensional array of type T
start_indices ArraySlice<int64> List of N integers containing the starting indices of the slice for each dimension. Values must be greater than or equal to zero.
limit_indices ArraySlice<int64> List of N integers containing the ending indices (exclusive) for the slice for each dimension. Each value must be greater than or equal to the respective start_indices value for the dimension and less than or equal to the size of the dimension.
strides ArraySlice<int64> List of N integers that decides the input stride of the slice. The slice picks every strides[d] element in dimension d .

1-dimensional example:

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

2-dimensional example:

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

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

Sort

See also XlaBuilder::Sort .

Sort(operands, comparator, dimension, is_stable)

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

If only one operand is provided:

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

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

If n > 1 operands are provided:

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

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

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

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

If is_stable is set to true, the sort is guaranteed to be stable, that is, if there are elements which are considered to be equal by the comparator, the relative order of the equal values is preserved. By default, is_stable is set to false.

Transpose

See also the tf.reshape operation.

Transpose(operand)

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

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

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

TriangularSolve

See also XlaBuilder::TriangularSolve .

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

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

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

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

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

Tuple

See also XlaBuilder::Tuple .

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

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

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

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

While

See also XlaBuilder::While .

While(condition, body, init)

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

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

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

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

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

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