前言

这是对PPQ库中KLD算法实现代码解析，关于PPQ库安装与使用详情见专栏上一篇博客。

PPQ算法实现

nvidia发布的PPT：8-bit Inference with TensorRT，百度可下载。下两图是KLD算法的实现伪代码：


下图是PPQ算法的实现过程：见https://github.com/openppl-public/ppq/blob/master/ppq/quantization/observer/range.py

def hist_to_scale_offset(
        self, histogram: torch.Tensor, hist_bins: int, hist_scale: float,
        config: TensorQuantizationConfig, computing_device: str = OBSERVER_KL_COMPUTING_DEVICE,
        scale_threshold: float=OBSERVER_MIN_SCALE
    ) -> Tuple[float, int]:
        """
        PPQ core quant parameter computing method - Histogram to scale & offset
        With a pre-defined histogram,
        this function will automatically search best clip value
        to minimize KL divergence between quantized result and fp32 input.
        only work for per-tensor symmetrical quantization policy for now.
        see also https://on-demand.gputechconf.com/gtc/2017/presentation/s7310-8-bit-inference-with-tensorrt.pdf
        Args:
            histogram (torch.Tensor): histogram records activation's statistics.
            hist_bins (int): how many bins are included in histogram(also known as histogram length)
            hist_scale (float): histogram step size. it can be solved by histogram.max_val / histogram.bins
            config (TensorQuantizationConfig): quantization config.
            computing_device (str, optional): computing device. Defaults to 'cpu'.
        Raises:
            ValueError: given quantization config is invalid.
        Returns:
            Tuple[float, int]: scale(fp32) and offset(int).
        """
        if config.policy.has_property(QuantizationProperty.ASYMMETRICAL):
            raise PermissionError('KL observer is not designed for ASYMMETRICAL quantization')
        
        if OBSERVER_MIN_SCALE_MANUL_OVERRIDE in config.detail:
            scale_threshold = config.detail[OBSERVER_MIN_SCALE_MANUL_OVERRIDE]

        # move histogram to cpu, speedup computation.
        histogram = histogram.to(computing_device).float()

        # compute symmtrical kl-divergence.
        # Here is a simple example: reference distribution P consisting of 8 bins, we want to quantize into 2 bins:
        # P = [ 1, 0, 2, 3, 5, 3, 1, 7]
        # we merge into 2 bins (8 / 2 = 4 consecutive bins are merged into one bin)
        # [1 + 0 + 2 + 3 , 5 + 3 + 1 + 7] = [6, 16]
        # then proportionally expand back to 8 bins, we preserve empty bins from the original distribution P:
        # Q = [ 6/3, 0, 6/3, 6/3, 16/4, 16/4, 16/4, 16/4] = [ 2, 0, 2, 2, 4, 4, 4, 4]
        # now we should normalize both distributions, after that we can compute KL_divergence
        # P /= sum(P) Q /= sum(Q)
        # result = KL_divergence(P, Q)
        # see also
        # https://github.com/NVIDIA/TensorRT/blob/3835424af081db4dc8cfa3ff3c9f4a8b89844421/tools/pytorch-quantization/pytorch_quantization/calib/histogram.py#L147

        losses, quant_bins = [], 2 ** (config.num_of_bits - 1)

        # following code is curcial, do not move
        histogram[: int(hist_bins * .002)] = 0
        histogram[int(hist_bins * .002)] = 1

        hist_sum = torch.sum(histogram)
        for bin_range in range(quant_bins, hist_bins + quant_bins - 1, quant_bins):
            p_hist = torch.zeros(size=(bin_range, ), dtype=torch.float, device=computing_device)
            p_hist[: bin_range].copy_(histogram[: bin_range])
            p_hist[bin_range - 1] += torch.sum(histogram[bin_range: ])
            p_hist = p_hist / hist_sum

            expand_ratio = int(bin_range / quant_bins)
            q_hist = histogram[: bin_range].clone()
            q_hist = q_hist.reshape((quant_bins, expand_ratio))
            positive_map = q_hist > 0
            positive_cnt = positive_map.sum(axis=1, keepdim=True)
            positive_cnt[positive_cnt == 0] = 1
            q_hist = torch.div(q_hist.sum(axis=1, keepdim=True), positive_cnt)
            q_hist = q_hist.repeat([1, expand_ratio])
            q_hist = q_hist * positive_map
            q_hist = q_hist / torch.sum(q_hist)
            q_hist = q_hist.flatten()

            losses.append({
                'kl': torch_KL_divergence(p_hist, q_hist),
                'bin_range': bin_range
            })

        best_bin_range = sorted(losses, key=lambda x: x['kl'])[0]['bin_range']
        scale, offset = (best_bin_range / self._hist_bins) * hist_scale * (self._hist_bins / quant_bins), 0
        
        if scale < scale_threshold and OBSERVER_WARNING: 
            ppq_warning('Numeric instability detected: '
                        'ppq find there is a scale value < 1e-7, '
                        'which probably cause numeric underflow in further computation.')
        scale = max(scale, scale_threshold)

        if config.policy.has_property(QuantizationProperty.POWER_OF_2):
            scale = ppq_round_to_power_of_2(scale, policy=RoundingPolicy.ROUND_HALF_UP)
        return scale, offset

NVIDIA的PPT中KLD算法流程

整个过程：从128循环到2048，i为截断阈值将bin截断（第i个条形图也会被舍弃），生成P和Q，计算每组P和Q的KL散度，最小散度对应阈值即为所求
输入Input：一个有2048个统计条条形图bin
输出：截断阈值threshhold，浮点数

KLD算法PPQ实现版本

算法流程：

具体代码分析：

PPQ与NVIDIA的区别：

1.原始histogram条形图舍弃
NVIDIA是：不进行预处理
PPQ：前其千分之二置为零，第千分之二个条形置为1
2.for循环找截断阈值
NVIDIA是：for i in range（102,2048）
PPQ库是：for bin_range in range(quant_bins, hist_bins + quant_bins - 1, quant_bins):
3.阈值m转为实际浮点数
NVIDIA是：threshold = ( m + 0.5 ) * ( width of a bin )
PPQ库是：