Implementing Sequential Algorithms on TPU | by Chaim Rand | Oct, 2024

It is a direct sequel to a previous post on the subject of implementing customized TPU operations with Pallas. Of specific curiosity are customized kernels that leverage the distinctive properties of the TPU structure in a fashion that optimizes runtime efficiency. On this publish, we’ll try to reveal this chance by making use of the ability of Pallas to the problem of operating sequential algorithms which might be interspersed inside a predominantly parallelizable deep studying (DL) workload.

We are going to deal with Non Maximum Suppression (NMS) of bounding-box proposals as a consultant algorithm, and discover methods to optimize its implementation. An vital part of laptop imaginative and prescient (CV) object detection options (e.g., Mask RCNN), NMS is usually used to filter out overlapping bounding bins, conserving solely the “greatest” ones. NMS receives a listing of bounding field proposals, an related checklist of scores, and an IOU threshold, and proceeds to greedily and iteratively select the remaining field with the best rating and disqualify all different bins with which it has an IOU that exceeds the given threshold. The truth that the field chosen on the n-th iteration will depend on the previous n-1 steps of the algorithm dictates the sequential nature of its implementation. Please see here and/or here for extra on the rational behind NMS and its implementation. Though we’ve chosen to deal with one particular algorithm, most of our dialogue ought to carry over to different sequential algorithms.

Offloading Sequential Algorithms to CPU

The presence of a sequential algorithm inside a predominantly parallelizable ML mannequin (e.g., Masks R-CNN) presents an fascinating problem. Whereas GPUs, generally used for such workloads, excel at executing parallel operations like matrix multiplication, they will considerably underperform in comparison with CPUs when dealing with sequential algorithms. This typically results in computation graphs that embrace crossovers between the GPU and CPU, the place the GPU handles the parallel operations and the CPU handles the sequential ones. NMS is a main instance of a sequential algorithm that’s generally offloaded onto the CPU. In truth, a detailed evaluation of torchvision’s “CUDA” implementation of NMS, reveals that even it runs a good portion of the algorithm on CPU.

Though offloading sequential operations to the CPU could result in improved runtime efficiency, there are a number of potential drawbacks to think about:

1. Cross-device execution between the CPU and GPU often requires a number of factors of synchronization between the gadgets which generally leads to idle time on the GPU whereas it waits for the CPU to finish its duties. Provided that the GPU is often the most costly part of the coaching platform our purpose is to reduce such idle time.
2. In normal ML workflows, the CPU is chargeable for making ready and feeding information to the mannequin, which resides on the GPU. If the info enter pipeline includes compute-intensive processing, this will pressure the CPU, resulting in “enter hunger” on the GPU. In such eventualities, offloading parts of the mannequin’s computation to the CPU might additional exacerbate this situation.

To keep away from these drawbacks you can take into account different approaches, reminiscent of changing the sequential algorithm with a comparable different (e.g., the one advised here), settling for a gradual/suboptimal GPU implementation of the sequential algorithm, or operating the workload on CPU — every of which include there personal potential trade-offs.

Sequential Algorithms on TPU

That is the place the distinctive structure of the TPU might current a possibility. Opposite to GPUs, TPUs are sequential processors. Whereas their skill to run extremely vectorized operations makes them aggressive with GPUs when operating parallelizable operations reminiscent of matrix multiplication, their sequential nature might make them uniquely fitted to operating ML workloads that embrace a mixture of each sequential and parallel parts. Armed with the Pallas extension to JAX, our newfound TPU kernel creation instrument, we’ll consider this chance by implementing and evaluating a customized implementation of NMS for TPU.

Disclaimers

The NMS implementations we’ll share under are supposed for demonstrative functions solely. We’ve not made any important effort to optimize them or to confirm their robustness, sturdiness, or accuracy. Please remember the fact that, as of the time of this writing, Pallas is an experimental function — nonetheless underneath energetic growth. The code we share (primarily based on JAX model 0.4.32) could change into outdated by the point you learn this. You should definitely seek advice from probably the most up-to-date APIs and sources accessible on your Pallas growth. Please don’t view our point out of any algorithm, library, or API as an endorsement for his or her use.

We start with a easy implementation of NMS in numpy that may function a baseline for efficiency comparability:

import numpy as np
def nms_cpu(bins, scores, max_output_size, threshold=0.1):
epsilon = 1e-5
# Convert bounding bins and scores to numpy
bins = np.array(bins)
scores = np.array(scores)
# coordinates of bounding bins
start_x = bins[:, 0]
start_y = bins[:, 1]
end_x = bins[:, 2]
end_y = bins[:, 3]
# Compute areas of bounding bins
areas = (end_x - start_x) * (end_y - start_y)
# Type by confidence rating of bounding bins
order = np.argsort(scores)
# Picked bounding bins
picked_boxes = []
# Iterate over bounding bins
whereas order.measurement > 0 and len(picked_boxes) < max_output_size:
# The index of the remaining field with the best rating
index = order[-1]
# Choose the bounding field with largest confidence rating
picked_boxes.append(index.merchandise())
# Compute coordinates of intersection
x1 = np.most(start_x[index], start_x[order[:-1]])
x2 = np.minimal(end_x[index], end_x[order[:-1]])
y1 = np.most(start_y[index], start_y[order[:-1]])
y2 = np.minimal(end_y[index], end_y[order[:-1]])
# Compute areas of intersection and union
w = np.most(x2 - x1, 0.0)
h = np.most(y2 - y1, 0.0)
intersection = w * h
union = areas[index] + areas[order[:-1]] - intersection
# Compute the ratio between intersection and union
ratio = intersection / np.clip(union, min=epsilon)
# discard bins above overlap threshold
hold = np.the place(ratio < threshold)
order = order[keep]
return picked_boxes

To guage the efficiency of our NMS perform, we generate a batch of random bins and scores (as JAX tensors) and run the script on a Google Cloud TPU v5e system utilizing the identical atmosphere and identical benchmarking utility as in our previous post. For this experiment, we specify the CPU because the JAX default device:

import jax
from jax import random
import jax.numpy as jnp
def generate_random_boxes(run_on_cpu = False):
if run_on_cpu:
jax.config.replace('jax_default_device', jax.gadgets('cpu')[0])
else:
jax.config.replace('jax_default_device', jax.gadgets('tpu')[0])
n_boxes = 1024
img_size = 1024
k1, k2, k3 = random.break up(random.key(0), 3)
# Randomly generate field sizes and positions
box_sizes = random.randint(k1,
form=(n_boxes, 2),
minval=1,
maxval=img_size)
top_left = random.randint(k2,
form=(n_boxes, 2),
minval=0,
maxval=img_size - 1)
bottom_right = jnp.clip(top_left + box_sizes, 0, img_size - 1)
# Concatenate top-left and bottom-right coordinates
rand_boxes = jnp.concatenate((top_left, bottom_right),
axis=1).astype(jnp.bfloat16)
rand_scores = jax.random.uniform(k3, 
form=(n_boxes,),
minval=0.0,
maxval=1.0)
return rand_boxes, rand_scores
rand_boxes, rand_scores = generate_random_boxes(run_on_cpu=True)
time = benchmark(nms_cpu)(rand_boxes, rand_scores, max_output_size=128)
print(f'nms_cpu: {time}')

The resultant common runtime is 2.99 milliseconds. Notice the belief that the enter and output tensors reside on the CPU. If they’re on the TPU, then the time to repeat them between the gadgets must also be considered.

If our NMS perform is a part inside a bigger computation graph operating on the TPU, we would desire a TPU-compatible implementation to keep away from the drawbacks of cross-device execution. The code block under accommodates a JAX implementation of NMS particularly designed to allow acceleration by way of JIT compilation. Denoting the variety of bins by N, we start by calculating the IOU between every of the N(N-1) pairs of bins and making ready an NxN boolean tensor (mask_threshold) the place the (i,j)-th entry signifies whether or not the IOU between bins i and j exceed the predefined threshold.

To simplify the iterative collection of bins, we create a duplicate of the masks tensor (mask_threshold2) the place the diagonal components are zeroed to forestall a field from suppressing itself. We additional outline two score-tracking tensors: out_scores, which retains the scores of the chosen bins (and zeros the scores of the eradicated ones), and remaining_scores, which maintains the scores of the bins nonetheless being thought-about. We then use the jax.lax.while_loop perform to iteratively select bins whereas updating the out_scores and remaining_scores tensors. Notice that the format of the output of this perform differs from the earlier perform and will have to be adjusted to suit into subsequent steps of the computation graph.

import functools
# Given N bins, calculates mask_threshold an NxN boolean masks
# the place the (i,j) entry signifies whether or not the IOU of bins i and j
# exceed the brink. Returns mask_threshold, mask_threshold2
# which is equal to mask_threshold with zero diagonal and
# the scores modified so that every one values are higher than 0
def init_tensors(bins, scores, threshold=0.1):
epsilon = 1e-5
# Extract left, prime, proper, backside coordinates
left = bins[:, 0]
prime = bins[:, 1]
proper = bins[:, 2]
backside = bins[:, 3]
# Compute areas of bins
areas = (proper - left) * (backside - prime)
# Calculate intersection factors
inter_l = jnp.most(left[None, :], left[:, None])
inter_t = jnp.most(prime[None, :], prime[:, None])
inter_r = jnp.minimal(proper[None, :], proper[:, None])
inter_b = jnp.minimal(backside[None, :], backside[:, None])
# Width, peak, and space of the intersection
inter_w = jnp.clip(inter_r - inter_l, 0)
inter_h = jnp.clip(inter_b - inter_t, 0)
inter_area = inter_w * inter_h
# Union of the areas
union = areas[None, :] + areas[:, None] - inter_area
# IoU calculation
iou = inter_area / jnp.clip(union, epsilon)
# Shift scores to be higher than zero
out_scores = scores - jnp.min(scores) + epsilon
# Create masks primarily based on IoU threshold
mask_threshold = iou > threshold
# Create masks excluding diagonal (i.e., self IoU is ignored)
mask_threshold2 = mask_threshold * (1-jnp.eye(mask_threshold.form[0],
dtype=mask_threshold.dtype))
return mask_threshold, mask_threshold2, out_scores
@functools.partial(jax.jit, static_argnames=['max_output_size', 'threshold'])
def nms_jax(bins, scores, max_output_size, threshold=0.1):
# initialize masks and rating tensors
mask_threshold, mask_threshold2, out_scores = init_tensors(bins,
scores,
threshold)
# The out_scores tensor will retain the scores of the chosen bins
# and 0 the scores of the eradicated ones
# remaining_scores will preserve non-zero scores for bins that
# haven't been chosen or eradicated
remaining_scores = out_scores.copy()
def choose_box(state):
i, remaining_scores, out_scores = state
# select index of field with highest rating from remaining scores
index = jnp.argmax(remaining_scores)
# verify validity of chosen field
legitimate = remaining_scores[index] > 0
# If legitimate, zero all scores with IOU higher than threshold
# (together with the chosen index)
remaining_scores = jnp.the place(mask_threshold[index] *legitimate,
0,
remaining_scores)
# zero the scores of the eradicated tensors (not together with
# the chosen index)
out_scores = jnp.the place(mask_threshold2[index]*legitimate,
0,
out_scores)
i = i + 1
return i, remaining_scores, out_scores
def cond_fun(state):
i, _, _ = state
return (i < max_output_size)
i = 0
state = (i, remaining_scores, out_scores)
_, _, out_scores = jax.lax.while_loop(cond_fun, choose_box, state)
# Output the resultant scores. To extract the chosen bins,
# Take the max_output_size highest scores:
# min = jnp.minimal(jnp.count_nonzero(scores), max_output_size)
# indexes = jnp.argsort(out_scores, descending=True)[:min]
return out_scores
# nms_jax might be run on both the CPU the TPU
rand_boxes, rand_scores = generate_random_boxes(run_on_cpu=True)
time = benchmark(nms_jax)(rand_boxes, rand_scores, max_output_size=128)
print(f'nms_jax on CPU: {time}')
rand_boxes, rand_scores = generate_random_boxes(run_on_cpu=False)
time = benchmark(nms_jax)(rand_boxes, rand_scores, max_output_size=128)
print(f'nms_jax on TPU: {time}')

The runtimes of this implementation of NMS are 1.231 and 0.416 milliseconds on CPU and TPU, respectively.

We now current a customized implementation of NMS through which we explicitly leverage the truth that on TPUs Pallas kernels are executed in a sequential manner. Our implementation makes use of two boolean matrix masks and two score-keeping tensors, much like the strategy in our earlier perform.

We outline a kernel perform, choose_box, chargeable for choosing the following field and updating the score-keeping tensors, that are maintained in scratch reminiscence. We invoke the kernel throughout a one-dimensional grid the place the variety of steps (i.e., the grid-size) is decided by the max_output_size parameter.

Notice that attributable to some limitations (as of the time of this writing) on the operations supported by Pallas, some acrobatics are required to implement each the “argmax” perform and the validity verify for the chosen bins. For the sake of brevity, we omit the technical particulars and refer the reader to the feedback within the code under.

from jax.experimental import pallas as pl
from jax.experimental.pallas import tpu as pltpu
# argmax helper perform
def pallas_argmax(scores, n_boxes):
# we assume that the index of every field is saved within the
# least important bits of the rating (see under)
idx = jnp.max(scores.astype(float)).astype(int) % n_boxes
return idx
# Pallas kernel definition
def choose_box(scores, thresh_mask1, thresh_mask2, ret_scores,
scores_scratch, remaining_scores_scratch, *, nsteps, n_boxes):
# initialize scratch reminiscence on first step
@pl.when(pl.program_id(0) == 0)
def _():
scores_scratch[...] = scores[...]
remaining_scores_scratch[...] = scores[...]
remaining_scores = remaining_scores_scratch[...]
# select field
idx = pallas_argmax(remaining_scores, n_boxes)
# we use any to verfiy validity of the chosen field due
# to limitations on indexing in pallas
legitimate = (remaining_scores>0).any()
# updating rating tensors
remaining_scores_scratch[...] = jnp.the place(thresh_mask1[idx,...]*legitimate,
0,
remaining_scores)
scores_scratch[...] = jnp.the place(thresh_mask2[idx,...]*legitimate,
0,
scores_scratch[...])
# set return worth on ultimate step
@pl.when(pl.program_id(0) == nsteps - 1)
def _():
ret_scores[...] = scores_scratch[...]
@functools.partial(jax.jit, static_argnames=['max_output_size', 'threshold'])
def nms_pallas(bins, scores, max_output_size, threshold=0.1):
n_boxes = scores.measurement
mask_threshold, mask_threshold2, scores = init_tensors(bins, 
scores,
threshold)
# So as to work across the Pallas argsort limitation
# we create a brand new scores tensor with the identical ordering of
# the enter scores tensor through which the index of every rating
# within the ordering is encoded within the least important bits
sorted = jnp.argsort(scores, descending=True)
# descending integers: n_boxes-1, ..., 2, 1, 0
descending = jnp.flip(jnp.arange(n_boxes))
# new scores in descending with the least important
# bits carrying the argsort of the enter scores
ordered_scores = n_boxes * descending + sorted
# new scores with identical ordering as enter scores
scores = jnp.empty_like(ordered_scores
).at[sorted].set(ordered_scores)
grid = (max_output_size,)
return pl.pallas_call(
functools.partial(choose_box, 
nsteps=max_output_size,
n_boxes=n_boxes),
grid_spec=pltpu.PrefetchScalarGridSpec(
num_scalar_prefetch=0,
in_specs=[
pl.BlockSpec(block_shape=(n_boxes,)),
pl.BlockSpec(block_shape=(n_boxes, n_boxes)),
pl.BlockSpec(block_shape=(n_boxes, n_boxes)),
],
out_specs=pl.BlockSpec(block_shape=(n_boxes,)),
scratch_shapes=[pltpu.VMEM((n_boxes,), scores.dtype),
pltpu.VMEM((n_boxes,), scores.dtype)],
grid=grid,
),
out_shape=jax.ShapeDtypeStruct((n_boxes,), scores.dtype),
compiler_params=dict(mosaic=dict(
dimension_semantics=("arbitrary",)))
)(scores, mask_threshold, mask_threshold2)
rand_boxes, rand_scores = generate_random_boxes(run_on_cpu=False)
time = benchmark(nms_pallas)(rand_boxes, rand_scores, max_output_size=128)
print(f'nms_pallas: {time}')

The common runtime of our customized NMS operator is 0.139 milliseconds, making it roughly thrice sooner than our JAX-native implementation. This outcome highlights the potential of tailoring the implementation of sequential algorithms to the distinctive properties of the TPU structure.

Notice that in our Pallas kernel implementation, we load the complete enter tensors into TPU VMEM memory. Given the restricted the capability of VMEM, scaling up the enter measurement (i.e., enhance the variety of bounding bins) will probably result in reminiscence points. Usually, such limitations might be addressed by chunking the inputs with BlockSpecs. Sadly, making use of this strategy would break the present NMS implementation. Implementing NMS throughout enter chunks would require a special design, which is past the scope of this publish.

The outcomes of our experiments are summarized within the desk under:

These outcomes reveal the potential for operating full ML computation graphs on TPU, even after they embrace sequential parts. The efficiency enchancment demonstrated by our Pallas NMS operator, particularly, highlights the chance of customizing kernels in a manner that leverages the TPUs strengths.

In our previous post we realized of the chance for constructing customized TPU operators utilizing the Pallas extension for JAX. Maximizing this chance requires tailoring the kernel implementations to the precise properties of the TPU structure. On this publish, we centered on the sequential nature of the TPU processor and its use in optimizing a customized NMS kernel. Whereas scaling the answer to assist an unrestricted variety of bounding bins would require additional work, the core ideas we’ve mentioned stay relevant.

Nonetheless within the experimental part of its growth, there stay some limitations in Pallas which will require artistic workarounds. However the energy and potential are clearly evident and we anticipate that they may solely enhance because the framework matures.