# Advanced Guide to Inception v3 on Cloud TPU

This document discusses aspects of the Inception model and how they come together to make the model run efficiently on Cloud TPU. It is an advanced view of the guide to running Inception v3 on Cloud TPU. Specific changes to the model that led to significant improvements are discussed in more detail. This document supplements the Inception v3 tutorial.

Inception v3 TPU training runs match accuracy curves produced by GPU jobs of similar configuration. The model has been successfully trained on v2-8, v2-128, and v2-512 configurations. The model has attained greater than 78.1% accuracy in about 170 epochs on each of these.

The code samples shown in this document are meant to be illustrative, a high-level picture of what happens in the actual implementation. Working code can be found on GitHub.

## Introduction

Inception v3 is a widely-used image recognition model that has been shown to attain greater than 78.1% accuracy on the ImageNet dataset. The model is the culmination of many ideas developed by multiple researchers over the years. It is based on the original paper: "Rethinking the Inception Architecture for Computer Vision" by Szegedy, et. al.

The model itself is made up of symmetric and asymmetric building blocks, including convolutions, average pooling, max pooling, concats, dropouts, and fully connected layers. Batchnorm is used extensively throughout the model and applied to activation inputs. Loss is computed via Softmax.

A high-level diagram of the model is shown below:

The Inception model README has more information about the Inception architecture.

## Estimator API

The TPU version of Inception v3 is written using TPUEstimator, an API designed to facilitate development, so that you can focus on the models themselves rather than on the details of the underlying hardware. The API does most of the low-level grunge work necessary for running models on TPUs behind the scenes, while automating common functions, such as saving and restoring checkpoints.

The Estimator API enforces separation of model and input portions of the code. You have to define model_fn and input_fn functions, corresponding to model definition and input pipeline / preprocessing stages of the TensorFlow graph, respectively. Below is a sample skeleton of these functions:

def model_fn(features, labels, mode, params):
…
return tpu_estimator.TPUEstimatorSpec(mode=mode, loss=loss, train_op=train_op)

def input_fn(params):
def parser(serialized_example):
…
return image, label

…
images, labels = dataset.make_one_shot_iterator().get_next()
return images, labels


Two key functions provided by the API are train() and evaluate() used to train and evaluate, respectively. These are usually called somewhere in the main function. An example of this is shown below:

def main(unused_argv):
…
run_config = tpu_config.RunConfig(
master=FLAGS.master,
model_dir=FLAGS.model_dir,
session_config=tf.ConfigProto(
allow_soft_placement=True, log_device_placement=True),
tpu_config=tpu_config.TPUConfig(FLAGS.iterations, FLAGS.num_shards),)

estimator = tpu_estimator.TPUEstimator(
model_fn=model_fn,
use_tpu=FLAGS.use_tpu,
train_batch_size=FLAGS.batch_size,
eval_batch_size=FLAGS.batch_size,
config=run_config)

estimator.train(input_fn=input_fn, max_steps=FLAGS.train_steps)

eval_results = inception_classifier.evaluate(
input_fn=imagenet_eval.input_fn, steps=eval_steps)


## ImageNet dataset

Before the model can be used to recognize images, it must be trained. This is usually done via supervised learning using a large set of labeled images. Although Inception v3 can be trained from many different labeled image sets, ImageNet is a common dataset of choice.

ImageNet has over ten million URLs of labeled images. About a million of the images also have bounding boxes specifying a more precise location for the labeled objects.

For this model, the ImageNet dataset is composed of 1,331,167 images which are split into training and evaluation datasets containing 1,281,167 and 50,000 images, respectively.

The training and evaluation datasets are kept separate intentionally. Only images from the training dataset are used to train the model and only images from the evaluation dataset are used to evaluate model accuracy.

The model expects images to be stored as TFRecords. To convert images from raw JPEG files into TFRecords, use the open source batch script: download_and_preprocess_imagenet.sh. The script should produce a series of files (for both training and validation) of the form:

${DATA_DIR}/train-00000-of-01024${DATA_DIR}/train-00001-of-01024
...
${DATA_DIR}/train-01023-of-01024 and${DATA_DIR}/validation-00000-of-00128
S{DATA_DIR}/validation-00001-of-00128
...
${DATA_DIR}/validation-00127-of-00128  where DATA_DIR is the location where the set resides, for example: DATA_DIR=$HOME/imagenet-data

The Getting Started section in the Inception model README includes detailed instructions on how to build and run this script.

## Input pipeline

Each Cloud TPU device has 8 cores and is connected to a host (CPU). Larger slices have multiple hosts. Other larger configurations interact with multiple hosts. For instance, v2-256 communicates with 16 hosts.

Hosts retrieve data from the file system or local memory, do whatever data preprocessing is required, and then transfer preprocessed data to the TPU cores. We consider these three phases of data handling done by the host individually and refer to the phases as: 1) Storage, 2) Preprocessing, 3) Transfer. A high level picture of the diagram is shown in the figure below

To yield good performance, the system should be balanced. Whatever amount of time a host CPU spends retrieving images, decoding them, and doing relevant preprocessing, should ideally be slightly less or about the same as that spent by the TPU doing computations. If the host CPU takes longer than the TPU to complete the three data handling phases, then execution will be host bound. (Note: because TPUs are so fast, this may be unavoidable for some very simple models.) Both cases are shown in the diagram below.

The current implementation of Inception v3 lives right at the edge of being input-bound. Images have to be retrieved from the file system, decoded, and then preprocessed. Different types of preprocessing stages are available, ranging from moderate to complex. If we use the most complex of preprocessing stages, the large number of expensive operations executed by the preprocessing stage will push the system over the edge and the training pipeline will be preprocessing bound. However, it is not necessary to resort to that level of complexity to attain greater than 78.1% accuracy, and we instead use a moderately complex preprocessing stage that tilts the scale in the other direction and keeps the model TPU-bound. This is discussed in more detail in the next section.

The model uses tf.data.Dataset to handle all input pipeline related needs. See the datasets performance guide for more information on how to optimize input pipelines using the tf.data API.

Although you can simply define a function and pass it to the Estimator API, in the case of Inception, create class InputPipeline encapsulates all required functionality and define a __call__ method instead.

The Estimator API makes it very straightforward to use this class. One simply has to pass it to the input_fn parameter of functions train() and evaluate(), as shown in the sample code snippet below:

def main(unused_argv):

…

inception_classifier = tpu_estimator.TPUEstimator(
model_fn=inception_model_fn,
use_tpu=FLAGS.use_tpu,
config=run_config,
params=params,
train_batch_size=FLAGS.train_batch_size,
eval_batch_size=eval_batch_size,
batch_axis=(batch_axis, 0))

…

for cycle in range(FLAGS.train_steps // FLAGS.train_steps_per_eval):
tf.logging.info('Starting training cycle %d.' % cycle)
inception_classifier.train(
input_fn=InputPipeline(True), steps=FLAGS.train_steps_per_eval)

tf.logging.info('Starting evaluation cycle %d .' % cycle)
eval_results = inception_classifier.evaluate(
input_fn=InputPipeline(False), steps=eval_steps, hooks=eval_hooks)
tf.logging.info('Evaluation results: %s' % eval_results)


The main elements of class InputPipeline are shown in the code snippet below, where we have highlighted the three phases with different colors. Method __call__ creates the dataset using tf.data.Dataset and then makes a series of API calls to utilize the built-in prefetch, interleave, and shuffling capabilities of the dataset.

class InputPipeline(object):

def __init__(self, is_training):
self.is_training = is_training

def __call__(self, params):
# Storage
file_pattern = os.path.join(
FLAGS.data_dir, 'train-*' if self.is_training else 'validation-*')
dataset = tf.data.Dataset.list_files(file_pattern)
if self.is_training and FLAGS.initial_shuffle_buffer_size > 0:
dataset = dataset.shuffle(
buffer_size=FLAGS.initial_shuffle_buffer_size)
if self.is_training:
dataset = dataset.repeat()

def prefetch_dataset(filename):
dataset = tf.data.TFRecordDataset(
filename, buffer_size=FLAGS.prefetch_dataset_buffer_size)
return dataset

dataset = dataset.apply(
tf.contrib.data.parallel_interleave(
prefetch_dataset,
cycle_length=FLAGS.num_files_infeed,
sloppy=True))
if FLAGS.followup_shuffle_buffer_size > 0:
dataset = dataset.shuffle(
buffer_size=FLAGS.followup_shuffle_buffer_size)

# Preprocessing
dataset = dataset.map(
self.dataset_parser,
num_parallel_calls=FLAGS.num_parallel_calls)

dataset = dataset.prefetch(batch_size)
dataset = dataset.apply(
tf.contrib.data.batch_and_drop_remainder(batch_size))
dataset = dataset.prefetch(2)  # Prefetch overlaps in-feed with training
images, labels = dataset.make_one_shot_iterator().get_next()

# Transfer
return images, labels


The storage section begins with the creation of a dataset and includes the reading of TFRecords from storage (using tf.data.TFRecordDataset). Special purpose functions repeat() and shuffle() are used as needed. Function tf.contrib.data.parallel_interleave() maps function prefetch_dataset() across its input to produce nested datasets, and outputs their elements interleaved. It gets elements from cycle_length nested datasets in parallel, which increases throughput. The sloppy argument relaxes the requirement that the outputs be produced in a deterministic order, and allows the implementation to skip over nested datasets whose elements are not readily available when requested.

The preprocessing section calls dataset.map(parser) which in turn calls the parser function where images are preprocessed. The details of the preprocessing stage are discussed in the next section.

The transfer section (at the end of the function) includes the line return images, labels. TPUEstimator takes the returned values and automatically transfers them to the device.

The figure below shows a sample Cloud TPU performance trace of Inception v3. TPU compute time, discounting any infeeding stalls, is currently at 815 msecs or so.

Host storage is also seen on the trace and shown below:

Host preprocessing, which includes image decoding and a series of image distortion functions is shown below:

Host/TPU transfer can be seen here:

## Preprocessing Stage

Image preprocessing is a crucial part of the system and can heavily influence the maximum accuracy that the model attains during training. At a minimum, images need to be decoded and resized to fit the model. In the case of Inception, images need to be 299x299x3 pixels.

However, simply decoding and resizing will not be enough to get good accuracy. The ImageNet training dataset contains 1,281,167 images. One pass over the set of training images is referred to as an epoch. During training, the model will require several passes through the training dataset to improve its image recognition capabilities. In the case of Inception v3, the number of epochs needed will be somewhere in the 140 to 200 range depending on the global batchsize.

It is extremely beneficial to continuously alter the images before feeding them to the model and to do so in such a manner that a particular image is slightly different at every epoch. How to best do this preprocessing of images is as much art as it is science. On the one hand, a well designed preprocessing stage can significantly boost the recognition capabilities of a model. On the other, too simple a preprocessing stage may create an artificial ceiling on the maximum accuracy that the same model can attain during training.

Inception v3 offers different options for the preprocessing stage, ranging from relatively simple and computationally inexpensive to fairly complex and computationally expensive. Two distinct flavors of such can be found in files vgg_preprocessing.py and inception_preprocessing.py.

File vgg_preprocessing.py defines a preprocessing stage that has been used successfully to train resnet to 75% accuracy, but yields suboptimal results when applied on Inception v3.

File inception_preprocessing.py contains a multi-option preprocessing stage with different levels of complexity that has been used successfully to train Inception v3 to accuracies in the 78.1-78.5% range when run on TPUs. This section discusses the preprocessing pipeline.

Preprocessing differs depending on whether the model is undergoing training or being used for inference/evaluation.

At evaluation time, preprocessing is quite straightforward: crop a central region of the image and then resize it to the default 299x299 size. The snippet code that does this is shown below:

def preprocess_for_eval(image, height, width, central_fraction=0.875):
with tf.name_scope(scope, 'eval_image', [image, height, width]):
if image.dtype != tf.float32:
image = tf.image.convert_image_dtype(image, dtype=tf.float32)
image = tf.image.central_crop(image, central_fraction=central_fraction)
image = tf.expand_dims(image, 0)
image = tf.image.resize_bilinear(image, [height, width], align_corners=False)
image = tf.squeeze(image, [0])
image = tf.subtract(image, 0.5)
image = tf.multiply(image, 2.0)
image.set_shape([height, width, 3])
return image


During training, the cropping is randomized: a bounding box is chosen randomly to select a region of the image which is then resized. The resized image is then optionally flipped and its colors are distorted. The snippet of code that does this is shown below:

def preprocess_for_train(image, height, width, bbox, fast_mode=True, scope=None):
with tf.name_scope(scope, 'distort_image', [image, height, width, bbox]):
if bbox is None:
bbox = tf.constant([0.0, 0.0, 1.0, 1.0], dtype=tf.float32, shape=[1, 1, 4])
if image.dtype != tf.float32:
image = tf.image.convert_image_dtype(image, dtype=tf.float32)

distorted_image, distorted_bbox = distorted_bounding_box_crop(image, bbox)
distorted_image.set_shape([None, None, 3])

num_resize_cases = 1 if fast_mode else 4
distorted_image = apply_with_random_selector(
distorted_image,
lambda x, method: tf.image.resize_images(x, [height, width], method),
num_cases=num_resize_cases)

distorted_image = tf.image.random_flip_left_right(distorted_image)

if FLAGS.use_fast_color_distort:
distorted_image = distort_color_fast(distorted_image)
else:
num_distort_cases = 1 if fast_mode else 4
distorted_image = apply_with_random_selector(
distorted_image,
lambda x, ordering: distort_color(x, ordering, fast_mode),
num_cases=num_distort_cases)

distorted_image = tf.subtract(distorted_image, 0.5)
distorted_image = tf.multiply(distorted_image, 2.0)
return distorted_image


Function distort_color is in charge of color alteration. It offers a fast mode where only brightness and saturation are modified. The full mode modifies brightness, saturation, and hue, and randomly alters the order in which these get modified. A code snippet of this function is shown below:

def distort_color(image, color_ordering=0, fast_mode=True, scope=None):
with tf.name_scope(scope, 'distort_color', [image]):
if fast_mode:
if color_ordering == 0:
image = tf.image.random_brightness(image, max_delta=32. / 255.)
image = tf.image.random_saturation(image, lower=0.5, upper=1.5)
else:
image = tf.image.random_saturation(image, lower=0.5, upper=1.5)
image = tf.image.random_brightness(image, max_delta=32. / 255.)
else:
if color_ordering == 0:
image = tf.image.random_brightness(image, max_delta=32. / 255.)
image = tf.image.random_saturation(image, lower=0.5, upper=1.5)
image = tf.image.random_hue(image, max_delta=0.2)
image = tf.image.random_contrast(image, lower=0.5, upper=1.5)
elif color_ordering == 1:
image = tf.image.random_saturation(image, lower=0.5, upper=1.5)
image = tf.image.random_brightness(image, max_delta=32. / 255.)
image = tf.image.random_contrast(image, lower=0.5, upper=1.5)
image = tf.image.random_hue(image, max_delta=0.2)
elif color_ordering == 2:
image = tf.image.random_contrast(image, lower=0.5, upper=1.5)
image = tf.image.random_hue(image, max_delta=0.2)
image = tf.image.random_brightness(image, max_delta=32. / 255.)
image = tf.image.random_saturation(image, lower=0.5, upper=1.5)
elif color_ordering == 3:
image = tf.image.random_hue(image, max_delta=0.2)
image = tf.image.random_saturation(image, lower=0.5, upper=1.5)
image = tf.image.random_contrast(image, lower=0.5, upper=1.5)
image = tf.image.random_brightness(image, max_delta=32. / 255.)

return tf.clip_by_value(image, 0.0, 1.0)


Function distort_color is computationally expensive, partly due to the nonlinear RGB to HSV and HSV to RGB conversions that are required in order to access hue and saturation. Both fast and full modes require these conversions and although fast mode is less computationally expensive, it still pushes the model to the CPU-compute-bound region, when enabled.

As an alternative, a new function distort_color_fast has been added to the list of options. This function maps the image from RGB to YCrCb using the JPEG conversion scheme and randomly alters brightness and the Cr/Cb chromas before mapping back to RGB. The function is shown in the code snippet below:

def distort_color_fast(image, scope=None):
with tf.name_scope(scope, 'distort_color', [image]):
br_delta = random_ops.random_uniform([], -32./255., 32./255., seed=None)
cb_factor = random_ops.random_uniform(
[], -FLAGS.cb_distortion_range, FLAGS.cb_distortion_range, seed=None)
cr_factor = random_ops.random_uniform(
[], -FLAGS.cr_distortion_range, FLAGS.cr_distortion_range, seed=None)

channels = tf.split(axis=2, num_or_size_splits=3, value=image)
red_offset = 1.402 * cr_factor + br_delta
green_offset = -0.344136 * cb_factor - 0.714136 * cr_factor + br_delta
blue_offset = 1.772 * cb_factor + br_delta
channels[0] += red_offset
channels[1] += green_offset
channels[2] += blue_offset
image = tf.concat(axis=2, values=channels)
image = tf.clip_by_value(image, 0., 1.)
return image


Here's a sample image that has undergone preprocessing. A randomly chosen region of the image has been selected and colors altered using the distort_color_fast function.

Function distort_color_fast is computationally efficient and still allows training to be TPU execution time bound. In addition, it yields good results and has been used successfully to train the Inception v3 model to greater than 78.1% accuracy using batchsizes in the 1,024 to 16,384 range. It is used as the default choice for Inception v3.

## Optimizer

The current model showcases three flavors of optimizers: SGD, momentum, and RMSProp.

Stochastic gradient descent (SGD) is the simplest kind of update: the weights are nudged in the negative gradient direction. Despite its simplicity, good results can still be obtained on some models. The updates dynamics can be written as:

$$w_{k+1}=w_k-\alpha∇f(w_k)$$

Momentum is a popular optimizer that frequently leads to faster convergence than can be attained by SGD. This optimizer updates weights much like SGD but also adds a component in the direction of the previous update. The dynamics of the update are given by:

$$z_{k+1}=\beta z_k+∇f(w_k)$$
$$w_{k+1}=w_k-\alpha z_{k+1}$$

which can be written as:

$$w_{k+1}=w_k-\alpha ∇f(w_k)+\beta \left(w_k-w_{k-1}\right)$$

The last term is the component in the direction of the previous update. This is shown graphically in the figure below:

For the momentum $${\beta}$$, we use the commonly used value of 0.9.

RMSprop is a popular optimizer first proposed by Geoff Hinton in one of his lectures. The update dynamics are given by:

$$g_{k+1}^{-2} = \alpha g_{k}^{-2} + (1-\alpha) g_{k}^2$$ $$w_{k+1}=\beta w_k + \frac{\eta}{\sqrt {g_{k+1^{\mathbf{+{\epsilon}}}}^{-2}}} ∇f(w_k)$$

For Inception v3, tests show RMSProp giving the best results in terms of maximum accuracy and time to attain it, with momentum a close second. Thus RMSprop is set as the default optimizer. The parameters used are: decay $${\alpha}$$ = 0.9, momentum $${\beta}$$ = 0.9, and $${\epsilon}$$ = 1.0.

The code snippet with the optimizer choices is shown below

if FLAGS.optimizer == 'sgd':
tf.logging.info('Using SGD optimizer')
optimizer = tf.train.GradientDescentOptimizer(
learning_rate=learning_rate)
elif FLAGS.optimizer == 'momentum':
tf.logging.info('Using Momentum optimizer')
optimizer = tf.train.MomentumOptimizer(
learning_rate=learning_rate, momentum=0.9)
elif FLAGS.optimizer == 'RMS':
tf.logging.info('Using RMS optimizer')
optimizer = tf.train.RMSPropOptimizer(
learning_rate,
RMSPROP_DECAY,
momentum=RMSPROP_MOMENTUM,
epsilon=RMSPROP_EPSILON)
else:
tf.logging.fatal('Unknown optimizer:', FLAGS.optimizer)


When running on TPUs and using the Estimator API, the optimizer needs to be wrapped in a CrossShardOptimizer function in order to ensure synchronization among the replicas (along with any necessary cross communication). The snippet of code where this is done in Inception v3 is shown below:

if FLAGS.use_tpu:
optimizer = tpu_optimizer.CrossShardOptimizer(optimizer)

update_ops = tf.get_collection(tf.GraphKeys.UPDATE_OPS)
with tf.control_dependencies(update_ops):
train_op = optimizer.minimize(loss, global_step=global_step)


## Exponential moving average

The normal course of action while training is for trainable parameters to get updated during backpropagation in accordance to the optimizer update rules. These were discussed in the previous section and repeated here for convenience:

$${\theta_{k+1}} = {\theta_k}-{\alpha ∇f(\theta_k)} \qquad(SGD)$$
$${\theta_{k+1}}={\theta_k}-{\alpha z_{k+1}} \qquad(momentum)$$
$${\theta_{k+1}}= {\beta \theta_k}+\frac{\eta}{\sqrt {g_{k+1^{\mathbf+{\epsilon}}}}^{-2}} ∇f(\theta_k) \qquad(RMSprop)$$

Exponential moving average (also known as exponential smoothing) is an optional post-processing step that is applied to the updated weights and can sometimes leads to noticeable improvements in performance. Inception v3 benefits tremendously from having this additional step. TensorFlow provides the function tf.train.ExponentialMovingAverage that computes the ema $${\hat{\theta}}$$ of weight $${\theta}$$ using the formula:

$${\hat{\theta_t}}={\alpha {\hat{\theta}{_{t-1}}}}+{(1-\alpha)}{\theta_t}$$

where $${\alpha}$$ is a decay factor (close to 1.0). In the case of Inception v3, $${\alpha}$$ is set to 0.995.

Even though this is an Infinite impulse response (IIR) filter, the decay factor establishes an effective window where most of the energy (or relevant samples) reside, as shown in the following diagram:

To see this more clearly, we rewrite the filter equation, as follows:

$${\hat{\theta}_{t+T+1}}={\alpha(1-\alpha)}({\theta_{t+T}}+{\alpha \theta_{t+T-1}}+...+{\alpha^{t+T}}{\theta_0})$$

where we used $${\hat\theta_{-1}}=0$$.

The $${\alpha}^k$$ values decay with increasing k, thus effectively only a subset of the samples will have a sizable influence on $$\hat{\theta}_{t+T+1}$$. The rule of thumb for the duration of said window is: $$\frac {1} {1-\alpha}$$, which corresponds to $${\alpha}$$ = 200 for =0.995.

We first get a collection of trainable variables and then use the apply() method to create shadow variables for each trained variable (and add corresponding ops to maintain moving averages for these in their shadow copies). A snippet of the code that does this on Inception v3 is shown below:

update_ops = tf.get_collection(tf.GraphKeys.UPDATE_OPS)
with tf.control_dependencies(update_ops):
train_op = optimizer.minimize(loss, global_step=global_step)

if FLAGS.moving_average:
ema = tf.train.ExponentialMovingAverage(
decay=MOVING_AVERAGE_DECAY, num_updates=global_step)
variables_to_average = (tf.trainable_variables() +
tf.moving_average_variables())
with tf.control_dependencies([train_op]), tf.name_scope('moving_average'):
train_op = ema.apply(variables_to_average)


We'd like to use the ema variables during evaluation. To accomplish this, we define class LoadEMAHook that applies method variables_to_restore() to the checkpoint file to evaluate using the shadow variable names:

class LoadEMAHook(tf.train.SessionRunHook):
def __init__(self, model_dir):
super(LoadEMAHook, self).__init__()
self._model_dir = model_dir

def begin(self):
ema = tf.train.ExponentialMovingAverage(MOVING_AVERAGE_DECAY)
variables_to_restore = ema.variables_to_restore()
self._load_ema = tf.contrib.framework.assign_from_checkpoint_fn(
tf.train.latest_checkpoint(self._model_dir), variables_to_restore)

def after_create_session(self, sess, coord):
tf.logging.info('Reloading EMA...')
self._load_ema(sess)


The hooks function is passed to evaluate() as shown in the snippet below:

if FLAGS.moving_average:
eval_hooks = [LoadEMAHook(FLAGS.model_dir)]
else:
eval_hooks = []

…

eval_results = inception_classifier.evaluate(
input_fn=InputPipeline(False), steps=eval_steps, hooks=eval_hooks)


## Batch normalization

Batch normalization is a widely used technique for normalizing input features on models that can lead to substantial reduction in convergence time. It is one of the more popular and useful algorithmic improvements in machine learning of recent years and is used across a wide range of models, including Inception v3.

Activation inputs are first normalized by subtracting batch mean and dividing by the batch standard deviation, but batch normalization does more than that. To keep things balanced in the presence of back propagation, two trainable parameters are introduced in every layer. Normalized outputs $${\hat{x}}$$ undergo a subsequent operation $${\gamma\hat{x}}+\beta$$, where $${\gamma}$$ and $${\beta}$$ are a sort of standard deviation and mean, but these are learned by the model itself.

The full set of equations is in the paper and is repeated here for convenience:

Input: Values of x over a mini-batch: $$\Phi=$$ { $${x_{1..m}\\}$$ } Parameters to be learned: $${\gamma}$$,$${\beta}$$

Output: { $${y_i}=BN_{\gamma,\beta}{(x_i)}$$ }

${\mu_\phi} \leftarrow {\frac{1}{m}}{\sum_{i=1}^m}x_i \qquad \mathsf(mini-batch\ mean)$

${\sigma_\phi}^2 \leftarrow {\frac{1}{m}}{\sum_{i=1}^m} {(x_i - {\mu_\phi})^2} \qquad \mathbf(mini-batch\ variance)$

${\hat{x_i}} \leftarrow {\frac{x_i-{\mu_\phi}}{\sqrt {\sigma^2_\phi}+{\epsilon}}}\qquad \mathbf(normalize)$

${y_i}\leftarrow {\gamma \hat{x_i}} + \beta \equiv BN_{\gamma,\beta}{(x_i)}\qquad \mathbf(scale \ and \ shift)$

Normalization happens happily during training, but come evaluation time, we'd like the model to behave in a deterministic fashion: the classification result of an image should depend solely on the input image and not the set of images that are being fed to the model. Thus, we need to fix $${\mu}$$ and $${\sigma}^2$$ and these need to represent the image population statistics.

To accomplish this the model computes moving averages of the mean and variance over the minibatches:

${\hat\mu_i} = {\alpha \hat\mu_{t-1}}+{(1-\alpha)\mu_t}$

${\hat\sigma_t}^2 = {\alpha{\hat\sigma^2_{t-1}}} + {(1-\alpha) {\sigma_t}^2}$

In the specific case of Inception v3, a sensible decay factor had been obtained (via hyperparameter tuning) for use in GPUs. We would like to use this value on TPUs as well, but in order to do that, we need to make some adjustments.

Batchnorm moving mean and variance are both calculated using a loss pass filter, whose canonical equation is shown below (here, $${y_t}$$ represents moving mean or variance):

${y_t}={\alpha y_{t-1}}+{(1-\alpha)}{x_t}$

(1)

In an 8x1 GPU (synchronous) job each replica reads the current moving mean and updates it. The updates are sequential, in the sense that the new moving variable must first be written by the current replica before the next one can read it.

When there are 8 replicas, the set of operations for an ensemble update is as follows:

${y_t}={\alpha y_{t-1}}+{(1-\alpha)}{x_t}$

${y_{t+1}}={\alpha y_{t}}+{(1-\alpha)}{x_{t+1}}$

${y_{t+2}}={\alpha y_{t+1}}+{(1-\alpha)}{x_{t+2}}$

${y_{t+3}}={\alpha y_{t+2}}+{(1-\alpha)}{x_{t+3}}$

${y_{t+4}}={\alpha y_{t+3}}+{(1-\alpha)}{x_{t+4}}$

${y_{t+5}}={\alpha y_{t+4}}+{(1-\alpha)}{x_{t+5}}$

${y_{t+6}}={\alpha y_{t+5}}+{(1-\alpha)}{x_{t+6}}$

${y_{t+7}}={\alpha y_{t+6}}+{(1-\alpha)}{x_{t+7}}$

This set of 8 sequential updates can be written as:

${y_{t+7}}={\alpha^8y_{t-1}}+(1-\alpha){\sum_{k=0}^7} {\alpha^{7-k}}{x_{t+k}}$

(2)

In the current moving moment calculation implementation on TPUs, each shard performs calculations independently and there's no cross-shard communication. Batches are distributed to every shard and each of them processes 1/8th of the total number of batches (when there are 8 shards).

Although each shard goes through the motions and computes the moving moments (that is, mean and variance), only the results from shard 0 are communicated back to the host CPU. So, effectively, only one replica is doing the moving mean/variance update:

${z_t}={\beta {z_{t-1}}}+{(1-\beta)u_t}$

(3)

and this update is happening at 1/8th the rate of its sequential counterpart.

In order to compare GPU and TPU update equations, we need to align the respective timescales. Specifically, the set of operations that comprise a set of 8 sequential updates on the GPU should be compared against a single update on the TPU. This is illustrated in the diagram below:

Let us show the equations with the modified time indexes:

${y_t}={\alpha^8y_{t-1}}+(1-\alpha){\sum_{k=0}^7} {\alpha^{7-k}}{x_{t-k/8}} \qquad \mathsf(GPU)$

${z_t}={\beta {z_{t-1}}}+{(1-\beta)u_t}\qquad \mathsf(TPU)$

If we make the simplifying assumption that the 8 mini batches (normalized across all relevant dimensions) each yield similar values within the GPU 8-minibatch sequential update, then we can approximate these equations as follows:

${y_t}={\alpha^8y_{t-1}}+(1-\alpha){\sum_{k=0}^7} {\alpha^{7-k}}{\hat{x_t}}={\alpha^8y_{t-1}+(1-\alpha^8){\hat{x_t}}} \qquad \mathsf(GPU)$

${z_t}={\beta {z_{t-1}}}+{(1-\beta)u_t}\qquad \mathsf(TPU)$

Therefore, to match the effect of a given decay factor on the GPU, we need to modify the decay factor on the TPU accordingly. Specifically, we need to set $${\beta}$$=$${\alpha}^8$$.

For Inception v3, the decay value used in the GPU is $${\alpha}$$=0.9997, which translates to a TPU decay value of $${\beta}$$=0.9976.

## Learning rate adaptation

As batch sizes become larger, training becomes more difficult. Different techniques continue to be proposed to allow efficient training for large batch sizes (see here, here, and here, for example).

One of said techniques: gradual learning rate ramp-up, was used to train the model to greater than 78.1% accuracy for batch sizes ranging from 4,096 to 16,384. For the case of inception v3, the learning rate is first set to about 10% of what would normally be the starting learning rate. The learning rate remains constant at this low value for a specified (small) number of 'cold epochs', and then begins a linear increase for a specified number of 'warm-up epochs' at the end of which it intersects what would have been the learning rate should a normal exponential decay had been used. This is illustrated in the following picture.

The bit of code that does this is shown in the snippet below:

initial_learning_rate = FLAGS.learning_rate * FLAGS.train_batch_size / 256
if FLAGS.use_learning_rate_warmup:
warmup_decay = FLAGS.learning_rate_decay**(
(FLAGS.warmup_epochs + FLAGS.cold_epochs) /
FLAGS.learning_rate_decay_epochs)
adj_initial_learning_rate = initial_learning_rate * warmup_decay

final_learning_rate = 0.0001 * initial_learning_rate

train_op = None
if training_active:
batches_per_epoch = _NUM_TRAIN_IMAGES / FLAGS.train_batch_size
global_step = tf.train.get_or_create_global_step()
current_epoch = tf.cast(
(tf.cast(global_step, tf.float32) / batches_per_epoch), tf.int32)

learning_rate = tf.train.exponential_decay(
learning_rate=initial_learning_rate,
global_step=global_step,
decay_steps=int(FLAGS.learning_rate_decay_epochs * batches_per_epoch),
decay_rate=FLAGS.learning_rate_decay,
staircase=True)

if FLAGS.use_learning_rate_warmup:
wlr = 0.1 * adj_initial_learning_rate
wlr_height = tf.cast(
0.9 * adj_initial_learning_rate /
(FLAGS.warmup_epochs + FLAGS.learning_rate_decay_epochs - 1),
tf.float32)
epoch_offset = tf.cast(FLAGS.cold_epochs - 1, tf.int32)
exp_decay_start = (FLAGS.warmup_epochs + FLAGS.cold_epochs +
FLAGS.learning_rate_decay_epochs)
lin_inc_lr = tf.add(
wlr, tf.multiply(
tf.cast(tf.subtract(current_epoch, epoch_offset), tf.float32),
wlr_height))
learning_rate = tf.where(
tf.greater_equal(current_epoch, FLAGS.cold_epochs),
(tf.where(tf.greater_equal(current_epoch, exp_decay_start),
learning_rate, lin_inc_lr)),
wlr)

# Set a minimum boundary for the learning rate.
learning_rate = tf.maximum(
learning_rate, final_learning_rate, name='learning_rate')

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