Examples

This section includes a complete example of a neural network in Torch, as well as links to additional examples.

Post-training quantization

In this example, we will train a fully-connected neural network on a synthetic 2D dataset with a checkerboard grid pattern of 100 x 100 points. The data is split into 9500 training and 500 test samples.

from torch import nn
import torch

N_FEAT = 2
class SimpleNet(nn.Module):
    """Simple MLP with torch"""

    def __init__(self, n_hidden=30):
        super().__init__()
        self.fc1 = nn.Linear(in_features=N_FEAT, out_features=n_hidden)
        self.fc2 = nn.Linear(in_features=n_hidden, out_features=n_hidden)
        self.fc3 = nn.Linear(in_features=n_hidden, out_features=2)


    def forward(self, x):
        """Forward pass."""
        x = torch.relu(self.fc1(x))
        x = torch.relu(self.fc2(x))
        x = self.fc3(x)
        return x

This network was trained using different numbers neurons in the hidden layers, and quantized using 3-bits weights and activations. The mean accumulator size shown below was extracted using the Virtual Library.

neurons	10	30	100
fp32 accuracy	68.70%	83.32%	88.06%
3bit accuracy	56.44%	55.54%	56.50%
mean accumulator size	6.6	6.9	7.4

This shows that the fp32 accuracy and accumulator size increases with the number of hidden neurons, while the 3-bit accuracy remains low irrespective of to the number of neurons. While all the configurations tried here were FHE compatible (accumulator < 8 bits), it is sometimes preferable to have lower accumulator size in order for the inference time to be faster.

The accumulator size is determined by Concrete Numpy as being the maximum bitwidth encountered anywhere in the encrypted circuit

Pruning using Torch

Considering that FHE only works with limited integer precision, there is a risk of overflowing in the accumulator, resulting in unpredictable results.

To understand how to overcome this limitation, consider a scenario where 2 bits are used for weights and layer inputs/outputs. The Linear layer computes a dot product between weights and inputs $y = \sum_i w_i x_i$. With 2 bits, no overflow can occur during the computation of the Linear layer as long the number of neurons does not exceed 14, i.e. the sum of 14 products of 2-bit numbers does not exceed 7 bits.

By default, Concrete-ML uses symmetric quantization for model weights, with values in the interval $\left[-2^{n_{bits}-1}, 2^{n_{bits}-1}-1\right]$. For example, for $n_{bits}=2$ the possible values are $[-2, -1, 0, 1]$, for $n_{bits}=3$ the values can be $[-4,-3,-2,-1,0,1,2,3]$.

However, in a typical setting, the weights will not all have the maximum or minimum value (e.g. $-2^{n_{bits}-1}$). Instead, weights typically have a normal distribution around 0, which is one of the motivating factors for their symmetric quantization. A symmetric distribution and many zero-valued weights are desirable because opposite sign weights can cancel each other out and zero weights do not increase the accumulator size.

This can be leveraged to train network with more neurons, while not overflowing the accumulator, using a technique called pruning, where the developer can impose a number of zero-valued weights. Torch provides support for pruning out of the box.

The following code shows how to use pruning in our previous example:

import torch.nn.utils.prune as prune

class PrunedSimpleNet(SimpleNet):
    """Simple MLP with torch"""

    def prune(self, max_non_zero, enable):
        # Linear layer weight has dimensions NumOutputs x NumInputs
        for layer in self.named_modules():
            if isinstance(layer, nn.Linear):
                num_zero_weights = (layer.weight.shape[1] - max_non_zero) * layer.weight.shape[0]
                if num_zero_weights <= 0:
                    continue

                if enable:
                    prune.l1_unstructured(layer, "weight", amount=num_zero_weights)
                else:
                    prune.remove(layer, "weight")

Results with PrunedSimpleNet, a pruned version of the SimpleNet with 100 neurons on the hidden layers are given below:

non-zero neurons	10	30
fp32 accuracy	82.50%	88.06%
3bit accuracy	57.74%	57.82%
mean accumulator size	6.6	6.8

This shows that the fp32 accuracy has been improved while maintaining constant mean accumulator size.

When pruning a larger neural network during training, it is easier to obtain a low a bitwidth accumulator while maintaining better final accuracy. Thus, pruning is more robust than training a similar smaller network.

Quantization-aware training (QAT)

While pruning helps maintain the post-quantization level of accuracy in low-precision settings, it does not help maintain accuracy when quantizing from floating point models. The best way to guarantee accuracy is to use quantization-aware training (read more in the quantization documentation).

In this example, QAT is done using Brevitas, changing Linear layers to QuantLinear and adding quantizers on the inputs of linear layers using QuantIdentity.

The quantization-aware training (QAT) import tool in Concrete-ML is a work in progress. While it has been tested with some networks built with Brevitas, it is possible to use other tools to obtain QAT networks.

import brevitas.nn as qnn


from brevitas.core.bit_width import BitWidthImplType
from brevitas.core.quant import QuantType
from brevitas.core.restrict_val import FloatToIntImplType, RestrictValueType
from brevitas.core.scaling import ScalingImplType
from brevitas.core.zero_point import ZeroZeroPoint
from brevitas.inject import ExtendedInjector
from brevitas.quant.solver import ActQuantSolver, WeightQuantSolver
from dependencies import value

# Configure quantization options
class CommonQuant(ExtendedInjector):
    bit_width_impl_type = BitWidthImplType.CONST
    scaling_impl_type = ScalingImplType.CONST
    restrict_scaling_type = RestrictValueType.FP
    zero_point_impl = ZeroZeroPoint
    float_to_int_impl_type = FloatToIntImplType.ROUND
    scaling_per_output_channel = False
    narrow_range = True
    signed = True

    @value
    def quant_type(bit_width):
        if bit_width is None:
            return QuantType.FP
        elif bit_width == 1:
            return QuantType.BINARY
        else:
            return QuantType.INT

# Quantization options for weights/activations
class CommonWeightQuant(CommonQuant, WeightQuantSolver):
    scaling_const = 1.0
    signed = True


class CommonActQuant(CommonQuant, ActQuantSolver):
    min_val = -1.0
    max_val = 1.0

class QATPrunedSimpleNet(nn.Module):
    def __init__(self, n_hidden):
        super(QATPrunedSimpleNet, self).__init__()

        n_bits = 3
        self.quant_inp = qnn.QuantIdentity(
            act_quant=CommonActQuant,
            bit_width=n_bits,
            return_quant_tensor=True,
        )

        self.fc1 = qnn.QuantLinear(
            N_FEAT,
            n_hidden,
            True,
            weight_quant=CommonWeightQuant,
            weight_bit_width=n_bits,
            bias_quant=None,
        )

        self.q1 = qnn.QuantIdentity(
            act_quant=CommonActQuant, bit_width=n_bits, return_quant_tensor=True
        )

        self.fc2 = qnn.QuantLinear(
            n_hidden,
            n_hidden,
            True,
            weight_quant=CommonWeightQuant,
            weight_bit_width=3,
            bias_quant=None
        )

        self.q2 = qnn.QuantIdentity(
            act_quant=CommonActQuant, bit_width=n_bits, return_quant_tensor=True
        )

        self.fc3 = qnn.QuantLinear(
            n_hidden,
            2,
            True,
            weight_quant=CommonWeightQuant,
            weight_bit_width=n_hidden,
            bias_quant=None,
        )

        for m in self.modules():
            if isinstance(m, qnn.QuantLinear):
                torch.nn.init.uniform_(m.weight.data, -1, 1)

    def forward(self, x):
        x = self.quant_inp(x)
        x = self.q1(torch.relu(self.fc1(x)))
        x = self.q2(torch.relu(self.fc2(x)))
        x = self.fc3(x)
        return x

    def prune(self, max_non_zero, enable):
        # Linear layer weight has dimensions NumOutputs x NumInputs
        for name, layer in self.named_modules():
            if isinstance(layer, nn.Linear):
                num_zero_weights = (layer.weight.shape[1] - max_non_zero) * layer.weight.shape[0]
                if num_zero_weights <= 0:
                    continue

                if enable:
                    print(f"Pruning layer {name} factor {num_zero_weights}")
                    prune.l1_unstructured(layer, "weight", amount=num_zero_weights)
                else:
                    prune.remove(layer, "weight")

Training this network with 30 non-zero neurons out of 100 total gives good accuracy while being FHE compatible (accumulator size < 8 bits).

non-zero neurons	30
3bit accuracy brevitas	94.4%
3bit accuracy in Concrete-ML	91.8%
accumulator size	7

The torch QAT training loop is the same as the standard floating point training loop, but hyperparameters such as learning rate might need to be adjusted.

Quantization Aware Training is somewhat slower thant normal training. QAT introduces quantization during both the forward and backward passes. The quantization process is inefficient on GPUs as its computational intensity is low with respect to data transfer time.

Additional examples

The following table summarizes the examples in this section.

Model	Dataset	Metric	Clear	Quantized	FHE
Fully Connected NN	Iris	accuracy	0.947	0.895	0.895
QAT Fully Connected NN	Synthetic (Checkerboard)	accuracy	0.94	0.94	0.94
Convolutional NN	Digits	accuracy	0.90	**	**

In this table, ** means that the accuracy is actually random-like, because the quantization we need to set to fullfil bitwidth constraints is too strong.

Examples

• FullyConnectedNeuralNetwork.ipynb

• QuantizationAwareTraining.ipynb

• ConvolutionalNeuralNetwork.ipynb