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Any-Precision Deep Neural Networks

2022.11.21

Deep dive into quantization Keywords: #Quantization

0. Abstract

  • github code: https://github.com/SHI-Labs/Any-Precision-DNNs
  • Proposal: Any-precision deep neural networks which are trained with a new method that allows the learned DNNs to be flexible in numerical precision during inference
  • How? By truncating the least significant bits.
  • Problem: Previous literature presents solutions to train models at fixed precision/accuracy trade-off point. $\rightarrow$ But how to produce a model flexible in runtime precision is largely unexplored

1. Introduction

  • Previous literature
    1. Mixed-precision models: models with some layers processing in ultra-low precision and some layers in high precision
  • Main Problem
    • efficiency/accuracy trade-off always exists, so we have to find the right trade-off point, but what if we demand flexibility?
    • That is, it would be favorable if we can dynamically change the efficiency/accuracy trad-off point given diffenert situations, without re-training or re-calibration
  • Contributions
    • In runtime, we can quantize its layers into various precision layers without fine-tuning or calibration and without any data . Accuracy changes smoothly w.r.t. its precision level without drastic performance degradation
    • Any-precision DNN trains better low-bit models with KD
  • Low-Precision Deep Neural Networks
    • Binarized Neural Networks, XNOR-Net
    • Quantization Operator
    • Straight Through Estimator(STE)

3. Any-Precision Deep Neural Networks

3.1 Overview

  • N-bit fixed-point integers to represent weights($w_Q$) and input activations($x_Q$)
  • Adding a layer-wise scaling factor $s$ helps reduce the output range variation and hence achieve better model accuracy.
  • The activations $y’$ are then quantized into N-bit fixed-point integers

3.2 Inference

  • Once training is finished, we can keep the weights at a higher precision level for storage e.g. 8 bits
  • We quantize the weights into lower bit-width by bit-shifting (truncate)

3.3 Training

3.3.1 Quantization-aware training (QAT)

“Train the model taking quantization into consideration”

  • A full precision copy of the weights W is maintained throught the training.
  • Small gradients are accumulated without loss of precision → gradients effect the ‘full precision copy of weights’
  • Once the model is trained, only the quantized weights are used for inference.
  • BN layer is important in low-precision DNN
  • $y’$ is first passed into the BN layer and THEN quantized into $y_Q$

3.3.2 Weights

  • Uniform quantization strategy:
    • Normalize weights to [0,1]:
      graph

    Where $\tanh(w)$ limits the range of weights to [-1,1] (normalize before quantization), and $\max|\tanh(w)|$ is the maximum taken over all weights in the layer. By construction, $\tanh(w)/\max|\tanh(w)|$ is still in the range of [-1,1]. Add 1 and half the value to keep the weights in a nice [0,1] range in order to proceed with the scaling step. (▲)

    • Quantize normalized value into N-bit integer

    Where $\text{MAX}_N$ denotes the upper-bound of N-bit integer. Because $w’$ is in range [0,1], we only need to multiply $\text{MAX}_N$ and round it to an integer to get $w_Q’$.

    • Values remapped to approximate the range of floating point values to obtain

    Do the reverse op. of (▲) so that $w_Q$ becomes the intended quantization. The rationale of multiplying expectation of the weights comes from XNOR-Net.

    • Approximate $w$ with $s*w_Q$ Keeping track of how $w$ was scaled throughout the quantization process,multiply $s$ to the final quantized weight.
  • Forward Pass
    • Execute feed-forward pass with quantized weights:

    Overload of notation $s$ - (1) quantization scaling factor for weights, (2) layer-wise scaling factor to reduce output range variation

  • Backward Pass
    • Gradients are computed w.r.t the underlying float-value variable $w$(because we use the quantized version $w_Q$, in the feed-forward pass) and updates are applied to $w$ as well.
    • Straight through estimator (STE): Used to approximate gradients
      (ex) round op. has zero derivatives everywhere (why? because it’s a step function). With STE, the gradient of round op. is 1.

3.3.3 Activations

  • Clipping: Clip acitvation values to be within [0,1]

3.3.4 Dynamic Model-wise Quantization

  • Previous works: Bit-width N is fixed during the training process → In runtime if we alter N, accuracy drops drastically
  • Proposal: Dynamically change N within the training stage to align the training and inference process
  • Problem 1: Distribution of activations varies under different bit-width N, especially when N is small, failing to converge
  • Solution 1: Adopt BN layer that calculates running average
  • Problem 2: Still fails
  • Solution 2: (Proposal) Adopt dynamically changed BN layer to work with different N in training. Parameters of all BatchNorm layers are kept after training and used in inference. Storage for each bitwidth BN layer is negligible.

3.3.5 Etc.

  • Knowledge distillation: KD gives ~$\pm$1% boost

4. Experiments

4.1 Implementation Details

  • For all models, first and last layer is real-valued.
  • In training, we train the networks with bit-width candidates {1,2,4,8,32}

4.2 Comparison to Dedicated Models

  • The proposed any-precision DNN generally achieved comparable performance to the competitive dedicated models.
  • Any-precision model is more compact

4.3 Post-Training Quantization Methods

  • Compared with three alternative post-training quantization method
    1. Directly quantizes dedicated models with bit-shifting (truncation)
      • Simply drop LSB.
      • Fails dramatically.
    2. Bit-shifting + BatchNorm calibration process
      • Calibration process: BN statistics is recalculated by feed-forwarding a number of training samples
      • Worked in 4-bit, but failed in 1, 2 bit settings
    3. ACIQ
      • Analytical weight clipping, adaptive bit allocation, bias correction
    4. (Proposed Method) Calibrate the remaining 3, 5, 6, 7-bit settings to get the missed copies of BN layer params.
      • In runtime we can freely choose any precision level from 1 to 8 bits.

4.4 Dynamically Change BN Layers

  • Keeping multiple copies of BN layer params for different bit-widths → Minimized input variations to the convolutional layers → Allow same set of convolutional layer params. to support any-precision in run time

4.5 Ablation Studies

Candidate bit-width List

How does the candidate bit-width list used during training influence test performance on other bit-widths

  • Training with more candidate bit-width generally leads to better generalization to the others.
    • i.e. 1,2,4,8-bits combination is better than rest.
  • Coverage of candidate bit-width matters.
    • i.e. 1,8-bits combination performs more stable across different runtime bit-width compared with 2,8-bits, 4,8-bits combination

Knowledge Distillation