def _backward(PROBS, IDX, DPROBS, N, **meta):
BLOCK = meta['BLOCK']
row = tl.program_id(0)
cols = tl.arange(0, BLOCK)
idx = tl.load(IDX + row)
# pointers to probs
PROBS = PROBS + row * N + cols
# We know d(-log(p[i])/dlogit[k] = -id_mat[i,k] + p[k]
# and we have -log(p[k]) stored in PROBS, so this is easy
probs = -tl.load(PROBS, mask=cols < N, other=float('inf'))
probs = tl.exp(probs.to(tl.float32))
delta = cols == idx
# write result in-place in PROBS
dout = tl.load(DPROBS + row)
din = (probs - delta) * dout
tl.store(PROBS, din.to(tl.float16), mask=cols < N)
def _softmax(
Y, #output pointer
X, #input pointer
stride_xm, #stride tells us which to normalize
stride_ym, #
M, #num of rows
N, #num of cols
**meta):
## Note : Here stride is basically the amount you have to
## move in memory to get to the next element
## Ex: so if x is 2D, lets say x = torch.randn(10, 20),
## then x[1, 1].data_ptr() = x[1, 0].data_ptr() + x.stride(1)
# 1 program id per row index
m = tl.program_id(0)
# col indices
# here BLOCK is the smallest power of two greater than `N`
# In triton, the block size always has to be the power of 2
# in general powers of 2 nicely fit / align with cache lines
n = tl.arange(0, meta['BLOCK'])
# the memory address of all the elements
# that we want to load can be computed as follows
# m * stride_xm + n is how forward we want to move for the next pointer
X = X + m * stride_xm + n
# x is the vector of values loaded.
# - float(inf) is so we dont go over the edge
# if we do , penalize it and make it come back
x = tl.load(X, mask=n < N, other=-float('inf'))
# Substract maximum for numerical stability
z = x - tl.max(x, axis=0)
# Note that exponentials in Triton are fast
# but approximate (i.e., think __expf in CUDA)
num = tl.exp(z)
denom = tl.sum(num, axis=0)
y = num / denom
# Write back to Y
Y = Y + m * stride_ym + n
tl.store(Y, y, mask=n < N)
def _add(
X, # *Pointer* to first input vector
Y, # *Pointer* to second input vector
Z, # *Pointer* to output vector
N, # Size of the vector
**meta # Optional meta-parameters for the kernel
):
pid = tl.program_id(0)
# Create an offset for the blocks of pointers to be
# processed by this program instance
offsets = pid * meta['BLOCK'] + tl.arange(0, meta['BLOCK'])
# Create a mask to guard memory operations against
# out-of-bounds accesses
mask = offsets < N
# Load x
x = tl.load(X + offsets, mask=mask)
y = tl.load(Y + offsets, mask=mask)
# Write back x + y
z = x + y
tl.store(Z + offsets, z)
def _forward(LOGITS, PROBS, IDX, LOSS, N, **meta):
BLOCK = meta['BLOCK']
row = tl.program_id(0)
cols = tl.arange(0, BLOCK)
idx = tl.load(IDX + row)
# pointers to logit and probs
LOGITS = LOGITS + row * N + cols
WRIT_PROBS = PROBS + row * N + cols
READ_PROBS = PROBS + row * N + idx
# write-back negative log-probs
logits = tl.load(LOGITS, mask=cols < N, other=-float('inf'))
logits = logits.to(tl.float32)
logits = logits - tl.max(logits, 0)
probs = tl.log(tl.sum(tl.exp(logits), 0)) - logits
tl.store(WRIT_PROBS, probs, mask=cols < N)
# There is a bug in the compiler, which fails to insert a barrier here.
# We add it explicitly for now. Will be fixed soon.
tl.debug_barrier()
# write-back loss
probs = tl.load(READ_PROBS)
tl.store(LOSS + row, probs)
def _matmul(A, B, C, M, N, K, stride_am, stride_ak, stride_bk, stride_bn,
stride_cm, stride_cn, **META):
# extract meta-parameters
BLOCK_M = META['BLOCK_M']
BLOCK_N = META['BLOCK_N']
BLOCK_K = META['BLOCK_K']
GROUP_M = 8
# matrix multiplication
pid = tl.program_id(0)
grid_m = (M + BLOCK_M - 1) // BLOCK_M
grid_n = (N + BLOCK_N - 1) // BLOCK_N
# re-order program ID for better L2 performance
width = GROUP_M * grid_n
group_id = pid // width
group_size = min(grid_m - group_id * GROUP_M, GROUP_M)
pid_m = group_id * GROUP_M + (pid % group_size)
pid_n = (pid % width) // (group_size)
# do matrix multiplication
rm = pid_m * BLOCK_M + tl.arange(0, BLOCK_M)
rn = pid_n * BLOCK_N + tl.arange(0, BLOCK_N)
rk = tl.arange(0, BLOCK_K)
A = A + (rm[:, None] * stride_am + rk[None, :] * stride_ak)
B = B + (rk[:, None] * stride_bk + rn[None, :] * stride_bn)
acc = tl.zeros((BLOCK_M, BLOCK_N), dtype=tl.float32)
for k in range(K, 0, -BLOCK_K):
a = tl.load(A)
b = tl.load(B)
acc += tl.dot(a, b)
A += BLOCK_K * stride_ak
B += BLOCK_K * stride_bk
# triton can accept arbitrary activation function
# via metaparameters!
if META['ACTIVATION']:
acc = META['ACTIVATION'](acc)
# rematerialize rm and rn to save registers
rm = pid_m * BLOCK_M + tl.arange(0, BLOCK_M)
rn = pid_n * BLOCK_N + tl.arange(0, BLOCK_N)
C = C + (rm[:, None] * stride_cm + rn[None, :] * stride_cn)
mask = (rm[:, None] < M) & (rn[None, :] < N)
tl.store(C, acc, mask=mask)
def _add(
X, # *pointer* to input vector 1
Y, # *pointer * to input vector 2
Z, # *pointer* to output vector
N, # Size of the vector
**meta #Optional meta parameters for the kernel
# In meta, block size and compile time constants etc parameters
):
#Roughly the start of the block
pid = tl.program_id(0)
#We dont worry about threads here. We are writing program for
#whole block for now
# Create offset for the block of pointers to be processed
# by the program instance
#Read all the block of pointers
offsets = pid * meta['BLOCK'] + tl.arange(0, meta['BLOCK'])
#Create a mask to guard memory operations against
#out-of-bound accesses
#mask defines the shape. It defines what to load and what to
#leave undefined
mask = offsets < N
#Load x
#scale + arange -> yields block of pointers
load_x_ptrs = X + offsets
load_y_ptrs = Y + offsets
x = tl.load(load_x_ptrs, mask=mask) # x <- tensor of data
y = tl.load(load_y_ptrs, mask=mask)
#Write x + y
z = x + y
tl.store(Z + offsets, z)
def softmax_kernel(output_ptr, input_ptr, input_row_stride, output_row_stride,
n_cols, **meta):
# The rows of the softmax are independent, so we parallelize across those
row_idx = tl.program_id(0)
BLOCK_SIZE = meta['BLOCK_SIZE']
# The stride represents how much we need to increase the pointer to advance 1 row
row_start_ptr = input_ptr + row_idx * input_row_stride
# The block size is the next power of two greater than n_cols, so we can fit each
# row in a single block
col_offsets = tl.arange(0, BLOCK_SIZE)
input_ptrs = row_start_ptr + col_offsets
# Load the row into SRAM, using a mask since BLOCK_SIZE may be > than n_cols
row = tl.load(input_ptrs, mask=col_offsets < n_cols, other=-float('inf'))
# Substract maximum for numerical stability
row_minus_max = row - tl.max(row, axis=0)
# Note that exponentials in Triton are fast but approximate (i.e., think __expf in CUDA)
numerator = tl.exp(row_minus_max)
denominator = tl.sum(numerator, axis=0)
softmax_output = numerator / denominator
# Write back output to DRAM
output_row_start_ptr = output_ptr + row_idx * output_row_stride
output_ptrs = output_row_start_ptr + col_offsets
tl.store(output_ptrs, softmax_output, mask=col_offsets < n_cols)
def _kernel(A, B, C, stride_za, stride_ha, stride_ma, stride_ka, stride_zb,
stride_hb, stride_kb, stride_nb, stride_zc, stride_hc, stride_mc,
stride_nc, DS0, DS1, SDD_K, SDD_off_width, lut, locks, nlocks,
**meta):
TM = meta['TM']
TN = meta['TN']
TK = meta['TK']
TZ = meta['TZ']
BLOCK = meta['BLOCK']
#------------#
#- Prologue -#
#------------#
pid0 = tl.program_id(0)
pid1 = tl.program_id(1)
pidz = tl.program_id(2)
if meta['SDD']:
pid1 = pid1 + SDD_off_width
blockidm = tl.arange(0, TM) // BLOCK
blockidn = tl.arange(0, TN) // BLOCK
offlutm = blockidm * (TN // BLOCK) * 4
offlutn = blockidn * 4
header = lut + pid1 * (TM // BLOCK) * (TN // BLOCK) * 4
z = tl.load(header + 0)
i = tl.load(header + 1 + offlutm)
j = tl.load(header + 2 + offlutn)
AS1 = SDD_K // TZ
lockid = tl.where(TZ > 1, 1, 0)
offka = pid0 * AS1
offkb = pid0 * AS1
offmc = 0
offnc = 0
offpa = 0
offpb = 0
maxid = TZ
offhc = 0
offha = z
offhb = z
ram = i * BLOCK + (tl.arange(0, TM) % BLOCK)
rbn = j * BLOCK + (tl.arange(0, TN) % BLOCK)
else:
header = lut + pid0 * 6
offset = tl.load(header + 0)
AS1 = tl.load(header + 1)
column = tl.load(header + 2)
depth = tl.load(header + 3)
lockid = tl.load(header + 4)
maxid = tl.load(header + 5)
pinc = lut + offset
offhc = depth
if meta['DSD']:
# output offset
offnc = pid1 * TN
offmc = column * TM
offpc = 0
# dense input offset
offnb = pid1 * TN
offkb = tl.load(pinc)
offkb = tl.multiple_of(offkb, 8) # compiler hint
offpb = 0
# sparse input offset
offma = 0
offka = 0
offpa = tl.load(pinc + 1)
offpa = tl.multiple_of(offpa, 8) # compiler hint
offpa = offpa * BLOCK * BLOCK
offha = 0
offhb = depth
else:
# output offset
offmc = pid1 * TM
offnc = column * TN
offpc = 0
# dense input offset
offma = pid1 * TM
offka = tl.load(pinc)
offka = tl.multiple_of(offka, 8) # compiler hint
offpa = 0
# sparse input offset
offnb = 0
offkb = 0
offpb = tl.load(pinc + 1)
offpb = tl.multiple_of(offpb, 8) # compiler hint
offpb = offpb * BLOCK * BLOCK
offha = depth
offhb = 0
ram = offma + tl.arange(0, TM)
rbn = offnb + tl.arange(0, TN)
# initialize a, b pointers
rka = offka + tl.arange(0, TK)
rkb = offkb + tl.arange(0, TK)
pa = A + pidz * stride_za + offha * stride_ha + offpa + ram[:,
None] * stride_ma + rka[
None, :] * stride_ka
pb = B + pidz * stride_zb + offhb * stride_hb + offpb + rbn[
None, :] * stride_nb + rkb[:, None] * stride_kb
if meta['DDS']:
checkam = ram[:, None] < DS0
else:
checkam = AS1 > 0
if meta['DSD']:
checkbn = rbn[None, :] < DS0
else:
checkbn = AS1 > 0
a = tl.load(pa, mask=checkam, other=0.)
b = tl.load(pb, mask=checkbn, other=0.)
## ---------------- ##
## Inner Loop ##
## ---------------- ##
acc = tl.zeros((TM, TN), dtype=tl.float32)
for k in range(AS1, 0, -TK):
acc += tl.dot(a, b)
if meta['SDD']:
inc_a = TK * stride_ka
inc_b = TK * stride_kb
else:
pinc += 2
if meta['DSD']:
inc_b = tl.load(pinc)
inc_a = tl.load(pinc + 1)
inc_b = tl.multiple_of(inc_b, 8)
inc_a = tl.multiple_of(inc_a, 8)
inc_b = inc_b * stride_kb
if meta['DDS']:
inc_a = tl.load(pinc)
inc_b = tl.load(pinc + 1)
inc_a = tl.multiple_of(inc_a, 8)
inc_b = tl.multiple_of(inc_b, 8)
inc_a = inc_a * stride_ka
pa += inc_a
pb += inc_b
# pre-fetch
checkak = k > TK
checkbk = k > TK
checka = checkam & checkak
checkb = checkbn & checkbk
a = tl.load(pa, mask=checka)
b = tl.load(pb, mask=checkb)
c = acc.to(C.dtype.element_ty)
if meta['SDD']:
checkc = True
rr_blockidm = tl.arange(0, TM) // BLOCK
rr_blockidn = tl.arange(0, TN) // BLOCK
rr_offlutm = rr_blockidm * (TN // BLOCK) * 4
rr_offlutn = rr_blockidn * 4
off_bkid = 3 + rr_offlutm[:, None] + rr_offlutn[None, :]
bkid = tl.load(header + off_bkid)
offpc = bkid * BLOCK * BLOCK
rcm = tl.arange(0, TM) % BLOCK
rcn = tl.arange(0, TN) % BLOCK
else:
rcm = offmc + tl.arange(0, TM)
rcn = offnc + tl.arange(0, TN)
if meta['DSD']:
checkc = rcn[None, :] < DS0
if meta['DDS']:
checkc = rcm[:, None] < DS0
pc = C + offpc + offhc * stride_hc + pidz * stride_zc + rcm[:,
None] * stride_mc + rcn[
None, :] * stride_nc
# write-back directly
if lockid == 0:
tl.store(pc, c, mask=checkc)
# accumulate partial results using spin-locks
else:
plock = locks + tl.program_id(2) * nlocks * tl.num_programs(
1) + tl.program_id(1) * nlocks + lockid - 1
pcount = plock + tl.num_programs(2) * tl.num_programs(1) * nlocks
while tl.atomic_cas(plock, 0, 1) == 1:
pass
count = tl.load(pcount)
if count == 0:
tl.store(pc, c, mask=checkc)
else:
d = tl.load(pc, mask=checkc)
tl.store(pc, d + c, mask=checkc)
tl.atomic_xchg(pcount, (count + 1) % maxid)
tl.atomic_xchg(plock, 0)
def _kernel(A, B, C, M, N, K, stride_am, stride_ak, stride_bk, stride_bn,
stride_cm, stride_cn, LOCKS, **META):
# extract meta-parameters
BLOCK_M = META['BLOCK_M']
BLOCK_N = META['BLOCK_N']
BLOCK_K = META['BLOCK_K']
GROUP_M = META['GROUP_M']
SPLIT_K = META['SPLIT_K']
# matrix multiplication
pid = tl.program_id(0)
pid_z = tl.program_id(1)
grid_m = (M + BLOCK_M - 1) // BLOCK_M
grid_n = (N + BLOCK_N - 1) // BLOCK_N
# re-order program ID for better L2 performance
width = GROUP_M * grid_n
group_id = pid // width
group_size = min(grid_m - group_id * GROUP_M, GROUP_M)
pid_m = group_id * GROUP_M + (pid % group_size)
pid_n = (pid % width) // (group_size)
# do matrix multiplication
rm = pid_m * BLOCK_M + tl.arange(0, BLOCK_M)
rn = pid_n * BLOCK_N + tl.arange(0, BLOCK_N)
rk = tl.arange(0, BLOCK_K)
# pointers
K = K // SPLIT_K
A = A + (pid_z * K * stride_ak + rm[:, None] * stride_am +
rk[None, :] * stride_ak)
B = B + (pid_z * K * stride_bk + rk[:, None] * stride_bk +
rn[None, :] * stride_bn)
acc = tl.zeros((BLOCK_M, BLOCK_N), dtype=tl.float32)
for k in range(K, 0, -BLOCK_K):
if META['EVEN_K']:
a = tl.load(A)
b = tl.load(B)
else:
a = tl.load(A, mask=rk[None, :] < k, other=0.)
b = tl.load(B, mask=rk[:, None] < k, other=0.)
acc += tl.dot(a, b)
A += BLOCK_K * stride_ak
B += BLOCK_K * stride_bk
acc = acc.to(tl.float16)
# rematerialize rm and rn to save registers
rm = pid_m * BLOCK_M + tl.arange(0, BLOCK_M)
rn = pid_n * BLOCK_N + tl.arange(0, BLOCK_N)
C = C + (rm[:, None] * stride_cm + rn[None, :] * stride_cn)
mask = (rm < M)[:, None] & (rn < N)[None, :]
# handles write-back with reduction-splitting
if SPLIT_K == 1:
tl.store(C, acc, mask=mask)
else:
LOCKS = LOCKS + tl.program_id(0)
COUNT = LOCKS + tl.num_programs(0)
while tl.atomic_cas(LOCKS, 0, 1) == 1:
pass
count = tl.load(COUNT)
if count == 0:
tl.store(C, acc, mask=mask)
else:
curr = tl.load(C, mask=mask, other=0.)
tl.store(C, acc + curr, mask=mask)
tl.atomic_xchg(COUNT, (count + 1) % SPLIT_K)
tl.atomic_xchg(LOCKS, 0)
def kernel(X, Z, **meta):
pid = tl.program_id(0)
old = tl.atomic_add(X, pid)
tl.store(Z + pid, old)
def matmul_kernel(
# Pointers to matrices
a_ptr, b_ptr, c_ptr,
# Matrix dimensions
M, N, K,
# The stride variables represent how much to increase the ptr by when moving by 1
# element in a particular dimension. E.g. stride_am is how much to increase a_ptr
# by to get the element one row down (A has M rows)
stride_am, stride_ak,
stride_bk, stride_bn,
stride_cm, stride_cn,
# Meta-parameters
**meta,
):
"""Kernel for computing the matmul C = A x B.
A has shape (M, K), B has shape (K, N) and C has shape (M, N)
"""
# extract meta-parameters
BLOCK_SIZE_M = meta['BLOCK_SIZE_M']
BLOCK_SIZE_N = meta['BLOCK_SIZE_N']
BLOCK_SIZE_K = meta['BLOCK_SIZE_K']
GROUP_SIZE_M = 8
# -----------------------------------------------------------
# Map program ids `pid` to the block of C it should compute.
# This is done in a grouped ordering to promote L2 data reuse
# See above `L2 Cache Optimizations` section for details
pid = tl.program_id(axis=0)
num_pid_m = tl.cdiv(M, BLOCK_SIZE_M)
num_pid_n = tl.cdiv(N, BLOCK_SIZE_N)
num_pid_in_group = GROUP_SIZE_M * num_pid_n
group_id = pid // num_pid_in_group
first_pid_m = group_id * GROUP_SIZE_M
group_size_m = min(num_pid_m - first_pid_m, GROUP_SIZE_M)
pid_m = first_pid_m + (pid % group_size_m)
pid_n = (pid % num_pid_in_group) // group_size_m
# ----------------------------------------------------------
# Create pointers for the first blocks of A and B.
# We will advance this pointer as we move in the K direction
# and accumulate
# a_ptrs is a block of [BLOCK_SIZE_M, BLOCK_SIZE_K] pointers
# b_ptrs is a block of [BLOCK_SIZE_K, BLOCK_SIZE_n] pointers
# see above `Pointer Arithmetics` section for details
offs_am = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M)
offs_bn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)
offs_k = tl.arange(0, BLOCK_SIZE_K)
a_ptrs = a_ptr + (offs_am[:, None]*stride_am + offs_k [None, :]*stride_ak)
b_ptrs = b_ptr + (offs_k [:, None]*stride_bk + offs_bn[None, :]*stride_bn)
# -----------------------------------------------------------
# Iterate to compute a block of the C matrix
# We accumulate into a `[BLOCK_SIZE_M, BLOCK_SIZE_N]` block
# of fp32 values for higher accuracy.
# `accumulator` will be converted back to fp16 after the loop
accumulator = tl.zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=tl.float32)
for k in range(0, K, BLOCK_SIZE_K):
# Note that for simplicity, we don't apply a mask here.
# This means that if K is not a multiple of BLOCK_SIZE_K,
# this will access out-of-bounds memory and produce an
# error or (worse!) incorrect results.
a = tl.load(a_ptrs)
b = tl.load(b_ptrs)
# We accumulate along the K dimension
accumulator += tl.dot(a, b)
# Advance the ptrs to the next K block
a_ptrs += BLOCK_SIZE_K * stride_ak
b_ptrs += BLOCK_SIZE_K * stride_bk
# you can fuse arbitrary activation functions here
# while the accumulator is still in FP32 !
if meta['ACTIVATION']:
accumulator = meta['ACTIVATION'](accumulator)
c = accumulator.to(tl.float16)
# -----------------------------------------------------------
# Write back the block of the output matrix C
offs_cm = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M)
offs_cn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)
c_ptrs = c_ptr + stride_cm * offs_cm[:, None] + stride_cn * offs_cn[None, :]
c_mask = (offs_cm[:, None] < M) & (offs_cn[None, :] < N)
tl.store(c_ptrs, c, mask=c_mask)
def kernel(X, Z, **meta):
pid = tl.program_id(0)
x = tl.load(X + pid)
old = GENERATE_TEST_HERE