百度飞桨黑客马拉松第三期–Laplace分布算子开发经验分享

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同时在paddle/distribution/kl.py 中注册_kl_laplace_laplace函数，使用时可直接调用kl_divergence计算laplace分布之间的kl散度。

3.2 测试和验收的考量

  在我们开发完对应的代码后，我们应该如何证明我们所开发出来的代码是正确的呢？这时候就需要单元测试的代码来证明我们的代码是正确的。那么什么是单元测试呢？单元测试的用例其实是一个“输入数据”和“预计输出”的集合。你需要跟你输入数据，根据逻辑功能给出预计输出，这里所说的根据逻辑功能是指，通过需求文档就能给出的预计输出。而非我们通过已经实现的代码去推导出的预计输出。这也是最容易被忽视的一点。你要去做单元测试，然后还要通过代码去推断出预计输出，如果你的代码逻辑本来就实现错了，给出的预计输出也是错的，那么你的单元测试将没有意义。实际上，这部分可以说是整个工作中最重要的部分也是比较难的部分，我们需要想出预计输出，并且如何通过已经实现的代码去推导出预计输出，只有单元测试通过了，我们的开发任务才算基本完成了。

  根据api类各个方法及特性传参的不同，把单测分成三个部分：测试分布的特性（无需额外参数）、测试分布的概率密度函数（需要传值）以及测试KL散度（需要传入一个实例）。

1. 测试Lapalce分布的特性

• 测试方法：该部分主要测试分布的均值、方差、熵等特征。类TestLaplace继承unittest.TestCase，分别实现方法setUp（初始化），test_mean（mean单测），test_variance（variance单测），test_stddev（stddev单测），test_entropy（entropy单测），test_sample（sample单测）。

  • 均值、方差、标准差通过Numpy计算相应值，对比Laplace类中相应property的返回值，若一致即正确；

  • 采样方法除验证其返回的数据类型及数据形状是否合法外，还需证明采样结果符合laplace分布。验证策略如下：随机采样30000个laplace分布下的样本值，计算采样样本的均值和方差，并比较同分布下scipy.stats.laplace返回的均值与方差，检查是否在合理误差范围内；同时通过Kolmogorov-Smirnov test进一步验证采样是否属于laplace分布，若计算所得ks值小于0.02，则拒绝不一致假设，两者属于同一分布；

  • 熵计算通过对比scipy.stats.laplace.entropy的值是否与类方法返回值一致验证结果的正确性。

• 测试用例：单测需要覆盖单一维度的Laplace分布和多维度分布情况，因此使用两种初始化参数

  • ‘one-dim’: loc=parameterize.xrand((2, )), scale=parameterize.xrand((2, ));
  • ‘multi-dim’: loc=parameterize.xrand((5, 5)), scale=parameterize.xrand((5, 5))。

2. 测试Lapalce分布的概率密度函数

• 测试方法：该部分主要测试分布各种概率密度函数。类TestLaplacePDF继承unittest.TestCase，分别实现方法setUp（初始化），test_prob（prob单测），test_log_prob（log_prob单测），test_cdf（cdf单测），test_icdf（icdf）。以上分布在scipy.stats.laplace中均有实现，因此给定某个输入value，对比相同参数下Laplace分布的scipy实现以及paddle实现的结果，若误差在容忍度范围内则证明实现正确。

• 测试用例：为不失一般性，测试使用多维位置参数和尺度参数初始化Laplace类，并覆盖int型输入及float型输入。

  • ‘value-float’: loc=np.array([0.2, 0.3]), scale=np.array([2, 3]), value=np.array([2., 5.]); * ‘value-int’: loc=np.array([0.2, 0.3]), scale=np.array([2, 3]), value=np.array([2, 5]);
  • ‘value-multi-dim’: loc=np.array([0.2, 0.3]), scale=np.array([2, 3]), value=np.array([[4., 6], [8, 2]])。

3. 测试Lapalce分布之间的KL散度

• 测试方法：该部分测试两个Laplace分布之间的KL散度。类TestLaplaceAndLaplaceKL继承unittest.TestCase，分别实现setUp（初始化），test_kl_divergence（kl_divergence）。在scipy中scipy.stats.entropy可用来计算两个分布之间的散度。因此对比两个Laplace分布在paddle.distribution.kl_divergence下和在scipy.stats.laplace下计算的散度，若结果在误差范围内，则证明该方法实现正确。

• 测试用例：分布1：loc=np.array([0.0]), scale=np.array([1.0]), 分布2: loc=np.array([1.0]), scale=np.array([0.5])

4、代码开发

  代码的开发主要参考Pytorch，此处涉及到单元测试代码的开发，kl散度注册等代码，需要仔细阅读PaddlePaddle中其他分布代码的实现形式。

import numbers

import numpy as np
import paddle
from paddle.distribution import distribution
from paddle.fluid import framework as framework


class Laplace(distribution.Distribution):
    r"""
    Creates a Laplace distribution parameterized by :attr:`loc` and :attr:`scale`.
    Mathematical details
    The probability density function (pdf) is
    .. math::
        pdf(x; mu, sigma) = frac{1}{2 * sigma} * e^{frac {-|x - mu|}{sigma}}
    In the above equation:
    * :math:`loc = mu`: is the location parameter.
    * :math:`scale = sigma`: is the scale parameter.
    Args:
        loc (scalar|Tensor): The mean of the distribution.
        scale (scalar|Tensor): The scale of the distribution.
        name(str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`.
    
    Examples:
        .. code-block:: python
                        import paddle
                        m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                        m.sample()  # Laplace distributed with loc=0, scale=1
                        # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True, 
                        # [3.68546247])
    """

    def __init__(self, loc, scale):
        if not isinstance(loc, (numbers.Real, framework.Variable)):
            raise TypeError(
                f"Expected type of loc is Real|Variable, but got {type(loc)}")

        if not isinstance(scale, (numbers.Real, framework.Variable)):
            raise TypeError(
                f"Expected type of scale is Real|Variable, but got {type(scale)}"
            )

        if isinstance(loc, numbers.Real):
            loc = paddle.full(shape=(), fill_value=loc)

        if isinstance(scale, numbers.Real):
            scale = paddle.full(shape=(), fill_value=scale)

        if (len(scale.shape) > 0 or len(loc.shape) > 0) and (loc.dtype
                                                             == scale.dtype):
            self.loc, self.scale = paddle.broadcast_tensors([loc, scale])
        else:
            self.loc, self.scale = loc, scale

        super(Laplace, self).__init__(self.loc.shape)

    @property
    def mean(self):
        """Mean of distribution.
        Returns:
            Tensor: The mean value.
        """
        return self.loc

    @property
    def stddev(self):
        """Standard deviation.
        The stddev is 
        .. math::
            stddev = sqrt{2} * sigma
        In the above equation:
        * :math:`scale = sigma`: is the scale parameter.
        
        Returns:
            Tensor: The std value.
        """
        return (2**0.5) * self.scale

    @property
    def variance(self):
        """Variance of distribution.
        The variance is 
        .. math::
            variance = 2 * sigma^2
        
        In the above equation:
        * :math:`scale = sigma`: is the scale parameter.
        Returns:
            Tensor: The variance value.
        """
        return self.stddev.pow(2)

    def _validate_value(self, value):
        """Argument dimension check for distribution methods such as `log_prob`,
        `cdf` and `icdf`. 
        Args:
          value (Tensor|Scalar): The input value, which can be a scalar or a tensor.
        
        Returns:
          loc, scale, value: The broadcasted loc, scale and value, with the same dimension and data type.
        """
        if isinstance(value, numbers.Real):
            value = paddle.full(shape=(), fill_value=value)
        if value.dtype != self.scale.dtype:
            value = paddle.cast(value, self.scale.dtype)
        if len(self.scale.shape) > 0 or len(self.loc.shape) > 0 or len(
                value.shape) > 0:
            loc, scale, value = paddle.broadcast_tensors(
                [self.loc, self.scale, value])
        else:
            loc, scale = self.loc, self.scale

        return loc, scale, value

    def log_prob(self, value):
        """Log probability density/mass function.
        The log_prob is
        .. math::
            log_prob(value) = frac{-log(2 * sigma) - |value - mu|}{sigma}
        
        In the above equation:
        * :math:`loc = mu`: is the location parameter.
        * :math:`scale = sigma`: is the scale parameter.
        Args:
          value (Tensor|Scalar): The input value, can be a scalar or a tensor.
        Returns:
          Tensor: The log probability, whose data type is same with value.
        Examples:
            .. code-block:: python
                            import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            value = paddle.to_tensor([0.1])
                            m.log_prob(value) 
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [-0.79314721])
        """
        loc, scale, value = self._validate_value(value)
        log_scale = -paddle.log(2 * scale)

        return (log_scale - paddle.abs(value - loc) / scale)

    def entropy(self):
        """Entropy of Laplace distribution.
        The entropy is:
        .. math::
            entropy() = 1 + log(2 * sigma)
        In the above equation:
        * :math:`scale = sigma`: is the scale parameter.
        Returns:
            The entropy of distribution.
        Examples:
            .. code-block:: python
                            import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            m.entropy()
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [1.69314718])
        """
        return 1 + paddle.log(2 * self.scale)

    def cdf(self, value):
        """Cumulative distribution function.
        The cdf is
        .. math::
            cdf(value) = 0.5 - 0.5 * sign(value - mu) * e^frac{-|(mu - sigma)|}{sigma}
        In the above equation:
        * :math:`loc = mu`: is the location parameter.
        * :math:`scale = sigma`: is the scale parameter.
        Args:
            value (Tensor): The value to be evaluated.
        Returns:
            Tensor: The cumulative probability of value.
        
        Examples:
            .. code-block:: python
                            import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            value = paddle.to_tensor([0.1])
                            m.cdf(value)
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [0.54758132])
        """
        loc, scale, value = self._validate_value(value)
        iterm = (0.5 * (value - loc).sign() *
                 paddle.expm1(-(value - loc).abs() / scale))

        return 0.5 - iterm

    def icdf(self, value):
        """Inverse Cumulative distribution function.
        The icdf is 
        .. math::
            cdf^{-1}(value)= mu - sigma * sign(value - 0.5) * ln(1 - 2 * |value-0.5|)
        In the above equation:
        * :math:`loc = mu`: is the location parameter.
        * :math:`scale = sigma`: is the scale parameter.
        Args:
            value (Tensor): The value to be evaluated.
        Returns:
            Tensor: The cumulative probability of value.
        Examples:
            .. code-block:: python
                            import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            value = paddle.to_tensor([0.1])
                            m.icdf(value)
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [-1.60943794])
        """
        loc, scale, value = self._validate_value(value)
        term = value - 0.5

        return (loc - scale * (term).sign() * paddle.log1p(-2 * term.abs()))

    def sample(self, shape=()):
        """Generate samples of the specified shape.
        Args:
            shape(tuple[int]): The shape of generated samples.
        Returns:
            Tensor: A sample tensor that fits the Laplace distribution.
        Examples:
            .. code-block:: python
                            import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            m.sample()  # Laplace distributed with loc=0, scale=1
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [3.68546247])
        """
        if not isinstance(shape, tuple):
            raise TypeError(
                f'Expected shape should be tuple[int], but got {type(shape)}')

        with paddle.no_grad():
            return self.rsample(shape)

    def rsample(self, shape):
        """Reparameterized sample.
        Args:
            shape(tuple[int]): The shape of generated samples.
        Returns:
            Tensor: A sample tensor that fits the Laplace distribution.
        Examples:
            .. code-block:: python
                            import paddle
                            m = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            m.rsample((1,))  # Laplace distributed with loc=0, scale=1
                            # Tensor(shape=[1, 1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [[0.04337667]])
        """

        eps = self._get_eps()
        shape = self._extend_shape(shape) or (1, )
        uniform = paddle.uniform(shape=shape,
                                 min=float(np.nextafter(-1, 1)) + eps / 2,
                                 max=1. - eps / 2,
                                 dtype=self.loc.dtype)

        if len(self.scale.shape) == 0 and len(self.loc.shape) == 0:
            loc, scale, uniform = paddle.broadcast_tensors(
                [self.loc, self.scale, uniform])
        else:
            loc, scale = self.loc, self.scale

        return (loc - scale * uniform.sign() * paddle.log1p(-uniform.abs()))

    def _get_eps(self):
        """
        Get the eps of certain data type.
        Note: 
            Since paddle.finfo is temporarily unavailable, we 
            use hard-coding style to get eps value.
        Returns:
            Float: An eps value by different data types.
        """
        eps = 1.19209e-07
        if (self.loc.dtype == paddle.float64
                or self.loc.dtype == paddle.complex128):
            eps = 2.22045e-16

        return eps

    def kl_divergence(self, other):
        """Calculate the KL divergence KL(self || other) with two Laplace instances.
        The kl_divergence between two Laplace distribution is
        .. math::
            KL_divergence(mu_0, sigma_0; mu_1, sigma_1) = 0.5 (ratio^2 + (frac{diff}{sigma_1})^2 - 1 - 2 ln {ratio})
        .. math::
            ratio = frac{sigma_0}{sigma_1}
        .. math::
            diff = mu_1 - mu_0
        In the above equation:
        * :math:`loc = mu`: is the location parameter of self.
        * :math:`scale = sigma`: is the scale parameter of self.
        * :math:`loc = mu_1`: is the location parameter of the reference Laplace distribution.
        * :math:`scale = sigma_1`: is the scale parameter of the reference Laplace distribution.
        * :math:`ratio`: is the ratio between the two distribution.
        * :math:`diff`: is the difference between the two distribution.
        Args:
            other (Laplace): An instance of Laplace.
        Returns:
            Tensor: The kl-divergence between two laplace distributions.
        Examples:
            .. code-block:: python
                            import paddle
                            
                            m1 = paddle.distribution.Laplace(paddle.to_tensor([0.0]), paddle.to_tensor([1.0]))
                            m2 = paddle.distribution.Laplace(paddle.to_tensor([1.0]), paddle.to_tensor([0.5]))
                            m1.kl_divergence(m2)
                            # Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
                            # [1.04261160])
        """

        var_ratio = other.scale / self.scale
        t = paddle.abs(self.loc - other.loc)
        term1 = ((self.scale * paddle.exp(-t / self.scale) + t) / other.scale)
        term2 = paddle.log(var_ratio)

        return term1 + term2 - 1
        

5、总结

  目前，该API已经锁定贡献。回顾API的开发过程，实际上该API的开发并不难，主要的问题在于如何进行单元测试，证明开发的API是正确的，并且还有一些相关的细节点，比如KL散度的注册等。还有就是最开始走了弯路，参照了Normal的开发风格，将API写成了2.0风格的，影响了一些时间，并且在最后的单测中，发现了Uniform实现方式的一些Bug，此处Debug花费了一些时间，整体来看，花时间的部分是在单测部分，对比奖金与花费的时间，综合看起来不太划算，对于想赚钱的来说；对于大部分学生来说，有必要多参加此类比赛，与日常工作内容差不多。