Fixed Point Quantization

Quantization techniques store and calculate numbers in more compact formats. TensorFlow Lite adds quantization that uses an 8-bit fixed point representation.

Since a challenge for modern neural networks is optimizing for high accuracy, the priority has been improving accuracy and speed during training. Using floating point arithmetic is an easy way to preserve accuracy and GPUs are designed to accelerate these calculations.

However, as more machine learning models are deployed to mobile devices, inference efficiency has become a critical issue. Where the computational demand for training grows with the amount of models trained on different architectures, the computational demand for inference grows in proportion to the amount of users.

Quantization benefits

Using 8-bit calculations help your models run faster and use less power. This is especially important for mobile devices and embedded applications that can't run floating point code efficiently, for example, Internet of Things (IoT) and robotics devices. There are additional opportunities to extend this support to more backends and research lower precision networks.

Smaller file sizes

Neural network models require a lot of space on disk. For example, the original AlexNet requires over 200 MB for the float format—almost all of that for the model's millions of weights. Because the weights are slightly different floating point numbers, simple compression formats perform poorly (like zip).

Weights fall in large layers of numerical values. For each layer, weights tend to be normally distributed within a range. Quantization can shrink file sizes by storing the minimum and maximum weight for each layer, then compress each weight's float value to an 8-bit integer representing the closest real number in a linear set of 256 within the range.

Faster inference

Since calculations are run entirely on 8-bit inputs and outputs, quantization reduces the computational resources needed for inference calculations. This is more involved, requiring changes to all floating point calculations, but results in a large speed-up for inference time.

Memory efficiency

Since fetching 8-bit values only requires 25% of the memory bandwidth of floats, more efficient caches avoid bottlenecks for RAM access. In many cases, the power consumption for running a neural network is dominated by memory access. The savings from using fixed-point 8-bit weights and activations are significant.

Typically, SIMD operations are available that run more operations per clock cycle. In some cases, a DSP chip is available that accelerates 8-bit calculations resulting in a massive speedup.

Fixed point quantization techniques

The goal is to use the same precision for weights and activations during both training and inference. But an important difference is that training consists of a forward pass and a backward pass, while inference only uses a forward pass. When we train the model with quantization in the loop, we ensure that the forward pass matches precision for both training and inference.

To minimize the loss in accuracy for fully fixed point models (weights and activations), train the model with quantization in the loop. This simulates quantization in the forward pass of a model so weights tend towards values that perform better during quantized inference. The backward pass uses quantized weights and activations and models quantization as a straight through estimator. (See Bengio et al., 2013)

Additionally, the minimum and maximum values for activations are determined during training. This allows a model trained with quantization in the loop to be converted to a fixed point inference model with little effort, eliminating the need for a separate calibration step.

Quantization training with TensorFlow

TensorFlow can train models with quantization in the loop. Because training requires small gradient adjustments, floating point values are still used. To keep models as floating point while adding the quantization error in the training loop, fake quantization nodes simulate the effect of quantization in the forward and backward passes.

Since it's difficult to add these fake quantization operations to all the required locations in the model, there's a function available that rewrites the training graph. To create a fake quantized training graph:

# Build forward pass of model.
loss = tf.losses.get_total_loss()

# Call the training rewrite which rewrites the graph in-place with
# FakeQuantization nodes and folds batchnorm for training. It is
# often needed to fine tune a floating point model for quantization
# with this training tool. When training from scratch, quant_delay
# can be used to activate quantization after training to converge
# with the float graph, effectively fine-tuning the model.

# Call backward pass optimizer as usual.
optimizer = tf.train.GradientDescentOptimizer(learning_rate)

The rewritten eval graph is non-trivially different from the training graph since the quantization ops affect the batch normalization step. Because of this, we've added a separate rewrite for the eval graph:

# Build eval model
logits = tf.nn.softmax_cross_entropy_with_logits_v2(...)

# Call the eval rewrite which rewrites the graph in-place with
# FakeQuantization nodes and fold batchnorm for eval.

# Save the checkpoint and eval graph proto to disk for freezing
# and providing to TFLite.
with open(eval_graph_file, ‘w’) as f:
saver = tf.train.Saver(), checkpoint_name)

Methods to rewrite the training and eval graphs are an active area of research and experimentation. Although rewrites and quantized training might not work or improve performance for all models, we are working to generalize these techniques.

Generating fully quantized models

The previously demonstrated after-rewrite eval graph only simulates quantization. To generate real fixed point computations from a trained quantization model, convert it to a fixed point kernel. Tensorflow Lite supports this conversion from the graph resulting from create_eval_graph.

First, create a frozen graph that will be the input for the TensorFlow Lite toolchain:

bazel build tensorflow/python/tools:freeze_graph && \
  bazel-bin/tensorflow/python/tools/freeze_graph \
  --input_graph=eval_graph_def.pb \
  --input_checkpoint=checkpoint \
  --output_graph=frozen_eval_graph.pb --output_node_names=outputs

Provide this to the TensorFlow Lite Optimizing Converter (TOCO) to get a fully quantized TensorFLow Lite model:

bazel build tensorflow/contrib/lite/toco:toco && \
  ./bazel-bin/third_party/tensorflow/contrib/lite/toco/toco \
  --input_file=frozen_eval_graph.pb \
  --output_file=tflite_model.tflite \
  --input_format=TENSORFLOW_GRAPHDEF --output_format=TFLITE \
  --inference_type=QUANTIZED_UINT8 \
  --input_shape="1,224, 224,3" \
  --input_array=input \
  --output_array=outputs \
  --std_value=127.5 --mean_value=127.5

See the documentation for tf.contrib.quantize and TensorFlow Lite.

Quantized accuracy

Fixed point MobileNet models are released with 8-bit weights and activations. Using the rewriters, these models achieve the Top-1 accuracies listed in Table 1. For comparison, the floating point accuracies are listed for the same models. The code used to generate these models is available along with links to all of the pretrained mobilenet_v1 models.

Image Size Depth Top-1 Accuracy:
Floating point
Top-1 Accuracy:
Fixed point: 8 bit weights and activations
Table 1: MobileNet Top-1 accuracy on Imagenet Validation dataset.

Representation for quantized tensors

TensorFlow approaches the conversion of floating-point arrays of numbers into 8-bit representations as a compression problem. Since the weights and activation tensors in trained neural network models tend to have values that are distributed across comparatively small ranges (for example, -15 to +15 for weights or -500 to 1000 for image model activations). And since neural nets tend to be robust handling noise, the error introduced by quantizing to a small set of values maintains the precision of the overall results within an acceptable threshold. A chosen representation must perform fast calculations, especially the large matrix multiplications that comprise the bulk of the computations while running a model.

This is represented with two floats that store the overall minimum and maximum values corresponding to the lowest and highest quantized value. Each entry in the quantized array represents a float value in that range, distributed linearly between the minimum and maximum. For example, with a minimum of -10.0 and maximum of 30.0f, and an 8-bit array, the quantized values represent the following:

Table 2: Example quantized value range

The advantages of this representation format are:

  • It efficiently represents an arbitrary magnitude of ranges.
  • The values don't have to be symmetrical.
  • The format represents both signed and unsigned values.
  • The linear spread makes multiplications straightforward.

Alternative techniques use lower bit depths by non-linearly distributing the float values across the representation, but currently are more expensive in terms of computation time. (See Han et al., 2016.)

The advantage of having a clear definition of the quantized format is that it's always possible to convert back and forth from fixed-point to floating-point for operations that aren't quantization-ready, or to inspect the tensors for debugging.