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## Overview

This notebook shows how to compress a model using TensorFlow Compression.

In the example below, we compress the weights of an MNIST classifier to a much smaller size than their floating point representation, while retaining classification accuracy. This is done by a two step process, based on the paper Scalable Model Compression by Entropy Penalized Reparameterization:

Training a "compressible" model with an explicit

**entropy penalty**during training, which encourages compressibility of the model parameters. The weight on this penalty, \(\lambda\), enables continuously controlling the trade-off between the compressed model size and its accuracy.Encoding the compressible model into a compressed model using a coding scheme that is matched with the penalty, meaning that the penalty is a good predictor for model size. This ensures that the method doesn't require multiple iterations of training, compressing, and re-training the model for fine-tuning.

This method is strictly concerned with compressed model size, not with computational complexity. It can be combined with a technique like model pruning to reduce size and complexity.

Example compression results on various models:

Model (dataset) | Model size | Comp. ratio | Top-1 error comp. (uncomp.) |
---|---|---|---|

LeNet300-100 (MNIST) | 8.56 KB | 124x | 1.9% (1.6%) |

LeNet5-Caffe (MNIST) | 2.84 KB | 606x | 1.0% (0.7%) |

VGG-16 (CIFAR-10) | 101 KB | 590x | 10.0% (6.6%) |

ResNet-20-4 (CIFAR-10) | 128 KB | 134x | 8.8% (5.0%) |

ResNet-18 (ImageNet) | 1.97 MB | 24x | 30.0% (30.0%) |

ResNet-50 (ImageNet) | 5.49 MB | 19x | 26.0% (25.0%) |

Applications include:

- Deploying/broadcasting models to edge devices on a large scale, saving bandwidth in transit.
- Communicating global model state to clients in federated learning. The model architecture (number of hidden units, etc.) is unchanged from the initial model, and clients can continue learning on the decompressed model.
- Performing inference on extremely memory limited clients. During inference, the weights of each layer can be sequentially decompressed, and discarded right after the activations are computed.

## Setup

Install Tensorflow Compression via `pip`

.

`# Installs the latest version of TFC compatible with the installed TF version.`

`read MAJOR MINOR <<< "$(pip show tensorflow | perl -p -0777 -e 's/.*Version: (\d+)\.(\d+).*/\1 \2/sg')"`

`pip install "tensorflow-compression<$MAJOR.$(($MINOR+1))"`

ERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts. tf-keras 2.17.0 requires tensorflow<2.18,>=2.17, but you have tensorflow 2.14.1 which is incompatible.

Import library dependencies.

```
import matplotlib.pyplot as plt
import tensorflow as tf
import tensorflow_compression as tfc
import tensorflow_datasets as tfds
```

2024-08-16 01:51:15.348618: E tensorflow/compiler/xla/stream_executor/cuda/cuda_dnn.cc:9342] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered 2024-08-16 01:51:15.348666: E tensorflow/compiler/xla/stream_executor/cuda/cuda_fft.cc:609] Unable to register cuFFT factory: Attempting to register factory for plugin cuFFT when one has already been registered 2024-08-16 01:51:15.348703: E tensorflow/compiler/xla/stream_executor/cuda/cuda_blas.cc:1518] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered

## Define and train a basic MNIST classifier

In order to effectively compress dense and convolutional layers, we need to define custom layer classes. These are analogous to the layers under `tf.keras.layers`

, but we will subclass them later to effectively implement Entropy Penalized Reparameterization (EPR). For this purpose, we also add a copy constructor.

First, we define a standard dense layer:

```
class CustomDense(tf.keras.layers.Layer):
def __init__(self, filters, name="dense"):
super().__init__(name=name)
self.filters = filters
@classmethod
def copy(cls, other, **kwargs):
"""Returns an instantiated and built layer, initialized from `other`."""
self = cls(filters=other.filters, name=other.name, **kwargs)
self.build(None, other=other)
return self
def build(self, input_shape, other=None):
"""Instantiates weights, optionally initializing them from `other`."""
if other is None:
kernel_shape = (input_shape[-1], self.filters)
kernel = tf.keras.initializers.GlorotUniform()(shape=kernel_shape)
bias = tf.keras.initializers.Zeros()(shape=(self.filters,))
else:
kernel, bias = other.kernel, other.bias
self.kernel = tf.Variable(
tf.cast(kernel, self.variable_dtype), name="kernel")
self.bias = tf.Variable(
tf.cast(bias, self.variable_dtype), name="bias")
self.built = True
def call(self, inputs):
outputs = tf.linalg.matvec(self.kernel, inputs, transpose_a=True)
outputs = tf.nn.bias_add(outputs, self.bias)
return tf.nn.leaky_relu(outputs)
```

And similarly, a 2D convolutional layer:

```
class CustomConv2D(tf.keras.layers.Layer):
def __init__(self, filters, kernel_size,
strides=1, padding="SAME", name="conv2d"):
super().__init__(name=name)
self.filters = filters
self.kernel_size = kernel_size
self.strides = strides
self.padding = padding
@classmethod
def copy(cls, other, **kwargs):
"""Returns an instantiated and built layer, initialized from `other`."""
self = cls(filters=other.filters, kernel_size=other.kernel_size,
strides=other.strides, padding=other.padding, name=other.name,
**kwargs)
self.build(None, other=other)
return self
def build(self, input_shape, other=None):
"""Instantiates weights, optionally initializing them from `other`."""
if other is None:
kernel_shape = 2 * (self.kernel_size,) + (input_shape[-1], self.filters)
kernel = tf.keras.initializers.GlorotUniform()(shape=kernel_shape)
bias = tf.keras.initializers.Zeros()(shape=(self.filters,))
else:
kernel, bias = other.kernel, other.bias
self.kernel = tf.Variable(
tf.cast(kernel, self.variable_dtype), name="kernel")
self.bias = tf.Variable(
tf.cast(bias, self.variable_dtype), name="bias")
self.built = True
def call(self, inputs):
outputs = tf.nn.convolution(
inputs, self.kernel, strides=self.strides, padding=self.padding)
outputs = tf.nn.bias_add(outputs, self.bias)
return tf.nn.leaky_relu(outputs)
```

Before we continue with model compression, let's check that we can successfully train a regular classifier.

Define the model architecture:

```
classifier = tf.keras.Sequential([
CustomConv2D(20, 5, strides=2, name="conv_1"),
CustomConv2D(50, 5, strides=2, name="conv_2"),
tf.keras.layers.Flatten(),
CustomDense(500, name="fc_1"),
CustomDense(10, name="fc_2"),
], name="classifier")
```

2024-08-16 01:51:18.645015: W tensorflow/core/common_runtime/gpu/gpu_device.cc:2211] Cannot dlopen some GPU libraries. Please make sure the missing libraries mentioned above are installed properly if you would like to use GPU. Follow the guide at https://www.tensorflow.org/install/gpu for how to download and setup the required libraries for your platform. Skipping registering GPU devices...

Load the training data:

```
def normalize_img(image, label):
"""Normalizes images: `uint8` -> `float32`."""
return tf.cast(image, tf.float32) / 255., label
training_dataset, validation_dataset = tfds.load(
"mnist",
split=["train", "test"],
shuffle_files=True,
as_supervised=True,
with_info=False,
)
training_dataset = training_dataset.map(normalize_img)
validation_dataset = validation_dataset.map(normalize_img)
```

Finally, train the model:

```
def train_model(model, training_data, validation_data, **kwargs):
model.compile(
optimizer=tf.keras.optimizers.Adam(learning_rate=1e-3),
loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),
metrics=[tf.keras.metrics.SparseCategoricalAccuracy()],
# Uncomment this to ease debugging:
# run_eagerly=True,
)
kwargs.setdefault("epochs", 5)
kwargs.setdefault("verbose", 1)
log = model.fit(
training_data.batch(128).prefetch(8),
validation_data=validation_data.batch(128).cache(),
validation_freq=1,
**kwargs,
)
return log.history["val_sparse_categorical_accuracy"][-1]
classifier_accuracy = train_model(
classifier, training_dataset, validation_dataset)
print(f"Accuracy: {classifier_accuracy:0.4f}")
```

Epoch 1/5 469/469 [==============================] - 51s 104ms/step - loss: 0.2070 - sparse_categorical_accuracy: 0.9377 - val_loss: 0.0735 - val_sparse_categorical_accuracy: 0.9768 Epoch 2/5 469/469 [==============================] - 48s 103ms/step - loss: 0.0626 - sparse_categorical_accuracy: 0.9809 - val_loss: 0.0611 - val_sparse_categorical_accuracy: 0.9814 Epoch 3/5 469/469 [==============================] - 48s 103ms/step - loss: 0.0428 - sparse_categorical_accuracy: 0.9869 - val_loss: 0.0580 - val_sparse_categorical_accuracy: 0.9815 Epoch 4/5 469/469 [==============================] - 49s 103ms/step - loss: 0.0318 - sparse_categorical_accuracy: 0.9903 - val_loss: 0.0626 - val_sparse_categorical_accuracy: 0.9813 Epoch 5/5 469/469 [==============================] - 49s 104ms/step - loss: 0.0247 - sparse_categorical_accuracy: 0.9923 - val_loss: 0.0546 - val_sparse_categorical_accuracy: 0.9846 Accuracy: 0.9846

Success! The model trained fine, and reached an accuracy of over 98% on the validation set within 5 epochs.

## Train a compressible classifier

Entropy Penalized Reparameterization (EPR) has two main ingredients:

Applying a

**penalty**to the model weights during training which corresponds to their entropy under a probabilistic model, which is matched with the encoding scheme of the weights. Below, we define a Keras`Regularizer`

which implements this penalty.**Reparameterizing**the weights, i.e. bringing them into a latent representation which is more compressible (yields a better trade-off between compressibility and model performance). For convolutional kernels, it has been shown that the Fourier domain is a good representation. For other parameters, the below example simply uses scalar quantization (rounding) with a varying quantization step size.

First, define the penalty.

The example below uses a code/probabilistic model implemented in the `tfc.PowerLawEntropyModel`

class, inspired by the paper Optimizing the Communication-Accuracy Trade-off in Federated Learning with Rate-Distortion Theory. The penalty is defined as:

\[ \log \Bigl(\frac {|x| + \alpha} \alpha\Bigr), \]

where \(x\) is one element of the model parameter or its latent representation, and \(\alpha\) is a small constant for numerical stability around values of 0.

```
_ = tf.linspace(-5., 5., 501)
plt.plot(_, tfc.PowerLawEntropyModel(0).penalty(_));
```

The penalty is effectively a regularization loss (sometimes called "weight loss"). The fact that it is concave with a cusp at zero encourages weight sparsity. The coding scheme applied for compressing the weights, an Elias gamma code, produces codes of length \( 1 + \lfloor \log_2 |x| \rfloor \) bits for the magnitude of the element. That is, it is matched to the penalty, and applying the penalty thus minimizes the expected code length.

```
class PowerLawRegularizer(tf.keras.regularizers.Regularizer):
def __init__(self, lmbda):
super().__init__()
self.lmbda = lmbda
def __call__(self, variable):
em = tfc.PowerLawEntropyModel(coding_rank=variable.shape.rank)
return self.lmbda * em.penalty(variable)
# Normalizing the weight of the penalty by the number of model parameters is a
# good rule of thumb to produce comparable results across models.
regularizer = PowerLawRegularizer(lmbda=2./classifier.count_params())
```

Second, define subclasses of `CustomDense`

and `CustomConv2D`

which have the following additional functionality:

- They take an instance of the above regularizer and apply it to the kernels and biases during training.
- They define kernel and bias as a
`@property`

, which perform quantization with straight-through gradients whenever the variables are accessed. This accurately reflects the computation that is carried out later in the compressed model. - They define additional
`log_step`

variables, which represent the logarithm of the quantization step size. The coarser the quantization, the smaller the model size, but the lower the accuracy. The quantization step sizes are trainable for each model parameter, so that performing optimization on the penalized loss function will determine what quantization step size is best.

The quantization step is defined as follows:

```
def quantize(latent, log_step):
step = tf.exp(log_step)
return tfc.round_st(latent / step) * step
```

With that, we can define the dense layer:

```
class CompressibleDense(CustomDense):
def __init__(self, regularizer, *args, **kwargs):
super().__init__(*args, **kwargs)
self.regularizer = regularizer
def build(self, input_shape, other=None):
"""Instantiates weights, optionally initializing them from `other`."""
super().build(input_shape, other=other)
if other is not None and hasattr(other, "kernel_log_step"):
kernel_log_step = other.kernel_log_step
bias_log_step = other.bias_log_step
else:
kernel_log_step = bias_log_step = -4.
self.kernel_log_step = tf.Variable(
tf.cast(kernel_log_step, self.variable_dtype), name="kernel_log_step")
self.bias_log_step = tf.Variable(
tf.cast(bias_log_step, self.variable_dtype), name="bias_log_step")
self.add_loss(lambda: self.regularizer(
self.kernel_latent / tf.exp(self.kernel_log_step)))
self.add_loss(lambda: self.regularizer(
self.bias_latent / tf.exp(self.bias_log_step)))
@property
def kernel(self):
return quantize(self.kernel_latent, self.kernel_log_step)
@kernel.setter
def kernel(self, kernel):
self.kernel_latent = tf.Variable(kernel, name="kernel_latent")
@property
def bias(self):
return quantize(self.bias_latent, self.bias_log_step)
@bias.setter
def bias(self, bias):
self.bias_latent = tf.Variable(bias, name="bias_latent")
```

The convolutional layer is analogous. In addition, the convolution kernel is stored as its real-valued discrete Fourier transform (RDFT) whenever the kernel is set, and the transform is inverted whenever the kernel is used. Since the different frequency components of the kernel tend to be more or less compressible, each of them gets its own quantization step size assigned.

Define the Fourier transform and its inverse as follows:

```
def to_rdft(kernel, kernel_size):
# The kernel has shape (H, W, I, O) -> transpose to take DFT over last two
# dimensions.
kernel = tf.transpose(kernel, (2, 3, 0, 1))
# The RDFT has type complex64 and shape (I, O, FH, FW).
kernel_rdft = tf.signal.rfft2d(kernel)
# Map real and imaginary parts into regular floats. The result is float32
# and has shape (I, O, FH, FW, 2).
kernel_rdft = tf.stack(
[tf.math.real(kernel_rdft), tf.math.imag(kernel_rdft)], axis=-1)
# Divide by kernel size to make the DFT orthonormal (length-preserving).
return kernel_rdft / kernel_size
def from_rdft(kernel_rdft, kernel_size):
# Undoes the transformations in to_rdft.
kernel_rdft *= kernel_size
kernel_rdft = tf.dtypes.complex(*tf.unstack(kernel_rdft, axis=-1))
kernel = tf.signal.irfft2d(kernel_rdft, fft_length=2 * (kernel_size,))
return tf.transpose(kernel, (2, 3, 0, 1))
```

With that, define the convolutional layer as:

```
class CompressibleConv2D(CustomConv2D):
def __init__(self, regularizer, *args, **kwargs):
super().__init__(*args, **kwargs)
self.regularizer = regularizer
def build(self, input_shape, other=None):
"""Instantiates weights, optionally initializing them from `other`."""
super().build(input_shape, other=other)
if other is not None and hasattr(other, "kernel_log_step"):
kernel_log_step = other.kernel_log_step
bias_log_step = other.bias_log_step
else:
kernel_log_step = tf.fill(self.kernel_latent.shape[2:], -4.)
bias_log_step = -4.
self.kernel_log_step = tf.Variable(
tf.cast(kernel_log_step, self.variable_dtype), name="kernel_log_step")
self.bias_log_step = tf.Variable(
tf.cast(bias_log_step, self.variable_dtype), name="bias_log_step")
self.add_loss(lambda: self.regularizer(
self.kernel_latent / tf.exp(self.kernel_log_step)))
self.add_loss(lambda: self.regularizer(
self.bias_latent / tf.exp(self.bias_log_step)))
@property
def kernel(self):
kernel_rdft = quantize(self.kernel_latent, self.kernel_log_step)
return from_rdft(kernel_rdft, self.kernel_size)
@kernel.setter
def kernel(self, kernel):
kernel_rdft = to_rdft(kernel, self.kernel_size)
self.kernel_latent = tf.Variable(kernel_rdft, name="kernel_latent")
@property
def bias(self):
return quantize(self.bias_latent, self.bias_log_step)
@bias.setter
def bias(self, bias):
self.bias_latent = tf.Variable(bias, name="bias_latent")
```

Define a classifier model with the same architecture as above, but using these modified layers:

```
def make_mnist_classifier(regularizer):
return tf.keras.Sequential([
CompressibleConv2D(regularizer, 20, 5, strides=2, name="conv_1"),
CompressibleConv2D(regularizer, 50, 5, strides=2, name="conv_2"),
tf.keras.layers.Flatten(),
CompressibleDense(regularizer, 500, name="fc_1"),
CompressibleDense(regularizer, 10, name="fc_2"),
], name="classifier")
compressible_classifier = make_mnist_classifier(regularizer)
```

And train the model:

```
penalized_accuracy = train_model(
compressible_classifier, training_dataset, validation_dataset)
print(f"Accuracy: {penalized_accuracy:0.4f}")
```

Epoch 1/5 469/469 [==============================] - 54s 111ms/step - loss: 3.8076 - sparse_categorical_accuracy: 0.9307 - val_loss: 2.1789 - val_sparse_categorical_accuracy: 0.9741 Epoch 2/5 469/469 [==============================] - 52s 111ms/step - loss: 1.6671 - sparse_categorical_accuracy: 0.9768 - val_loss: 1.2815 - val_sparse_categorical_accuracy: 0.9822 Epoch 3/5 469/469 [==============================] - 52s 111ms/step - loss: 1.0644 - sparse_categorical_accuracy: 0.9835 - val_loss: 0.9371 - val_sparse_categorical_accuracy: 0.9846 Epoch 4/5 469/469 [==============================] - 52s 112ms/step - loss: 0.7908 - sparse_categorical_accuracy: 0.9865 - val_loss: 0.7463 - val_sparse_categorical_accuracy: 0.9853 Epoch 5/5 469/469 [==============================] - 52s 111ms/step - loss: 0.6484 - sparse_categorical_accuracy: 0.9879 - val_loss: 0.6469 - val_sparse_categorical_accuracy: 0.9857 Accuracy: 0.9857

The compressible model has reached a similar accuracy as the plain classifier.

However, the model is not actually compressed yet. To do this, we define another set of subclasses which store the kernels and biases in their compressed form – as a sequence of bits.

## Compress the classifier

The subclasses of `CustomDense`

and `CustomConv2D`

defined below convert the weights of a compressible dense layer into binary strings. In addition, they store the logarithm of the quantization step size at half precision to save space. Whenever the kernel or bias is accessed through the `@property`

, they are decompressed from their string representation and dequantized.

First, define functions to compress and decompress a model parameter:

```
def compress_latent(latent, log_step, name):
em = tfc.PowerLawEntropyModel(latent.shape.rank)
compressed = em.compress(latent / tf.exp(log_step))
compressed = tf.Variable(compressed, name=f"{name}_compressed")
log_step = tf.cast(log_step, tf.float16)
log_step = tf.Variable(log_step, name=f"{name}_log_step")
return compressed, log_step
def decompress_latent(compressed, shape, log_step):
latent = tfc.PowerLawEntropyModel(len(shape)).decompress(compressed, shape)
step = tf.exp(tf.cast(log_step, latent.dtype))
return latent * step
```

With these, we can define `CompressedDense`

:

```
class CompressedDense(CustomDense):
def build(self, input_shape, other=None):
assert isinstance(other, CompressibleDense)
self.input_channels = other.kernel.shape[0]
self.kernel_compressed, self.kernel_log_step = compress_latent(
other.kernel_latent, other.kernel_log_step, "kernel")
self.bias_compressed, self.bias_log_step = compress_latent(
other.bias_latent, other.bias_log_step, "bias")
self.built = True
@property
def kernel(self):
kernel_shape = (self.input_channels, self.filters)
return decompress_latent(
self.kernel_compressed, kernel_shape, self.kernel_log_step)
@property
def bias(self):
bias_shape = (self.filters,)
return decompress_latent(
self.bias_compressed, bias_shape, self.bias_log_step)
```

The convolutional layer class is analogous to the above.

```
class CompressedConv2D(CustomConv2D):
def build(self, input_shape, other=None):
assert isinstance(other, CompressibleConv2D)
self.input_channels = other.kernel.shape[2]
self.kernel_compressed, self.kernel_log_step = compress_latent(
other.kernel_latent, other.kernel_log_step, "kernel")
self.bias_compressed, self.bias_log_step = compress_latent(
other.bias_latent, other.bias_log_step, "bias")
self.built = True
@property
def kernel(self):
rdft_shape = (self.input_channels, self.filters,
self.kernel_size, self.kernel_size // 2 + 1, 2)
kernel_rdft = decompress_latent(
self.kernel_compressed, rdft_shape, self.kernel_log_step)
return from_rdft(kernel_rdft, self.kernel_size)
@property
def bias(self):
bias_shape = (self.filters,)
return decompress_latent(
self.bias_compressed, bias_shape, self.bias_log_step)
```

To turn the compressible model into a compressed one, we can conveniently use the `clone_model`

function. `compress_layer`

converts any compressible layer into a compressed one, and simply passes through any other types of layers (such as `Flatten`

, etc.).

```
def compress_layer(layer):
if isinstance(layer, CompressibleDense):
return CompressedDense.copy(layer)
if isinstance(layer, CompressibleConv2D):
return CompressedConv2D.copy(layer)
return type(layer).from_config(layer.get_config())
compressed_classifier = tf.keras.models.clone_model(
compressible_classifier, clone_function=compress_layer)
```

Now, let's validate that the compressed model still performs as expected:

```
compressed_classifier.compile(metrics=[tf.keras.metrics.SparseCategoricalAccuracy()])
_, compressed_accuracy = compressed_classifier.evaluate(validation_dataset.batch(128))
print(f"Accuracy of the compressible classifier: {penalized_accuracy:0.4f}")
print(f"Accuracy of the compressed classifier: {compressed_accuracy:0.4f}")
```

79/79 [==============================] - 1s 10ms/step - loss: 0.0000e+00 - sparse_categorical_accuracy: 0.9857 Accuracy of the compressible classifier: 0.9857 Accuracy of the compressed classifier: 0.9857

The classification accuracy of the compressed model is identical to the one achieved during training!

In addition, the size of the compressed model weights is much smaller than the original model size:

```
def get_weight_size_in_bytes(weight):
if weight.dtype == tf.string:
return tf.reduce_sum(tf.strings.length(weight, unit="BYTE"))
else:
return tf.size(weight) * weight.dtype.size
original_size = sum(map(get_weight_size_in_bytes, classifier.weights))
compressed_size = sum(map(get_weight_size_in_bytes, compressed_classifier.weights))
print(f"Size of original model weights: {original_size} bytes")
print(f"Size of compressed model weights: {compressed_size} bytes")
print(f"Compression ratio: {(original_size/compressed_size):0.0f}x")
```

Size of original model weights: 5024320 bytes Size of compressed model weights: 19047 bytes Compression ratio: 264x

Storing the models on disk requires some overhead for storing the model architecture, function graphs, etc.

Lossless compression methods such as ZIP are good at compressing this type of data, but not the weights themselves. That is why there is still a significant benefit of EPR when counting model size inclusive of that overhead, after also applying ZIP compression:

```
import os
import shutil
def get_disk_size(model, path):
model.save(path)
zip_path = shutil.make_archive(path, "zip", path)
return os.path.getsize(zip_path)
original_zip_size = get_disk_size(classifier, "/tmp/classifier")
compressed_zip_size = get_disk_size(
compressed_classifier, "/tmp/compressed_classifier")
print(f"Original on-disk size (ZIP compressed): {original_zip_size} bytes")
print(f"Compressed on-disk size (ZIP compressed): {compressed_zip_size} bytes")
print(f"Compression ratio: {(original_zip_size/compressed_zip_size):0.0f}x")
```

INFO:tensorflow:Assets written to: /tmp/classifier/assets INFO:tensorflow:Assets written to: /tmp/classifier/assets INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets Original on-disk size (ZIP compressed): 13906833 bytes Compressed on-disk size (ZIP compressed): 61183 bytes Compression ratio: 227x

## Regularization effect and size–accuracy trade-off

Above, the \(\lambda\) hyperparameter was set to 2 (normalized by the number of parameters in the model). As we increase \(\lambda\), the model weights are more and more heavily penalized for compressibility.

For low values, the penalty can act like a weight regularizer. It actually has a beneficial effect on the generalization performance of the classifier, and can lead to a slightly higher accuracy on the validation dataset:

Accuracy of the vanilla classifier: 0.9846 Accuracy of the penalized classifier: 0.9857

For higher values, we see a smaller and smaller model size, but also a gradually diminishing accuracy. To see this, let's train a few models and plot their size vs. accuracy:

```
def compress_and_evaluate_model(lmbda):
print(f"lambda={lmbda:0.0f}: training...", flush=True)
regularizer = PowerLawRegularizer(lmbda=lmbda/classifier.count_params())
compressible_classifier = make_mnist_classifier(regularizer)
train_model(
compressible_classifier, training_dataset, validation_dataset, verbose=0)
print("compressing...", flush=True)
compressed_classifier = tf.keras.models.clone_model(
compressible_classifier, clone_function=compress_layer)
compressed_size = sum(map(
get_weight_size_in_bytes, compressed_classifier.weights))
compressed_zip_size = float(get_disk_size(
compressed_classifier, "/tmp/compressed_classifier"))
print("evaluating...", flush=True)
compressed_classifier = tf.keras.models.load_model(
"/tmp/compressed_classifier")
compressed_classifier.compile(
metrics=[tf.keras.metrics.SparseCategoricalAccuracy()])
_, compressed_accuracy = compressed_classifier.evaluate(
validation_dataset.batch(128), verbose=0)
print()
return compressed_size, compressed_zip_size, compressed_accuracy
lambdas = (2., 5., 10., 20., 50.)
metrics = [compress_and_evaluate_model(l) for l in lambdas]
metrics = tf.convert_to_tensor(metrics, tf.float32)
```

lambda=2: training... compressing... WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets evaluating... WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. lambda=5: training... compressing... WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets evaluating... WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. lambda=10: training... compressing... WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets evaluating... WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. lambda=20: training... compressing... WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets evaluating... WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. lambda=50: training... compressing... WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. WARNING:tensorflow:Compiled the loaded model, but the compiled metrics have yet to be built. `model.compile_metrics` will be empty until you train or evaluate the model. INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets INFO:tensorflow:Assets written to: /tmp/compressed_classifier/assets evaluating... WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually. WARNING:tensorflow:No training configuration found in save file, so the model was *not* compiled. Compile it manually.

The plot should ideally show an elbow-shaped size–accuracy trade-off, but it is normal for accuracy metrics to be somewhat noisy. Depending on initialization, the curve can exhibit some kinks.

Due to the regularization effect, the EPR compressed model is more accurate on the test set than the original model for small values of \(\lambda\). The EPR compressed model is also many times smaller, even if we compare the sizes after additional ZIP compression.

## Decompress the classifier

`CompressedDense`

and `CompressedConv2D`

decompress their weights on every forward pass. This makes them ideal for memory-limited devices, but the decompression can be computationally expensive, especially for small batch sizes.

To decompress the model once, and use it for further training or inference, we can convert it back into a model using regular or compressible layers. This can be useful in model deployment or federated learning scenarios.

First, converting back into a plain model, we can do inference, and/or continue regular training without a compression penalty:

```
def decompress_layer(layer):
if isinstance(layer, CompressedDense):
return CustomDense.copy(layer)
if isinstance(layer, CompressedConv2D):
return CustomConv2D.copy(layer)
return type(layer).from_config(layer.get_config())
decompressed_classifier = tf.keras.models.clone_model(
compressed_classifier, clone_function=decompress_layer)
```

```
decompressed_accuracy = train_model(
decompressed_classifier, training_dataset, validation_dataset, epochs=1)
print(f"Accuracy of the compressed classifier: {compressed_accuracy:0.4f}")
print(f"Accuracy of the decompressed classifier after one more epoch of training: {decompressed_accuracy:0.4f}")
```

469/469 [==============================] - 50s 105ms/step - loss: 0.0788 - sparse_categorical_accuracy: 0.9760 - val_loss: 0.0601 - val_sparse_categorical_accuracy: 0.9797 Accuracy of the compressed classifier: 0.9857 Accuracy of the decompressed classifier after one more epoch of training: 0.9797

Note that the validation accuracy drops after training for an additional epoch, since the training is done without regularization.

Alternatively, we can convert the model back into a "compressible" one, for inference and/or further training with a compression penalty:

```
def decompress_layer_with_penalty(layer):
if isinstance(layer, CompressedDense):
return CompressibleDense.copy(layer, regularizer=regularizer)
if isinstance(layer, CompressedConv2D):
return CompressibleConv2D.copy(layer, regularizer=regularizer)
return type(layer).from_config(layer.get_config())
decompressed_classifier = tf.keras.models.clone_model(
compressed_classifier, clone_function=decompress_layer_with_penalty)
```

```
decompressed_accuracy = train_model(
decompressed_classifier, training_dataset, validation_dataset, epochs=1)
print(f"Accuracy of the compressed classifier: {compressed_accuracy:0.4f}")
print(f"Accuracy of the decompressed classifier after one more epoch of training: {decompressed_accuracy:0.4f}")
```

469/469 [==============================] - 55s 112ms/step - loss: 0.7432 - sparse_categorical_accuracy: 0.9900 - val_loss: 0.7575 - val_sparse_categorical_accuracy: 0.9872 Accuracy of the compressed classifier: 0.9857 Accuracy of the decompressed classifier after one more epoch of training: 0.9872

Here, the accuracy improves after training for an additional epoch.