用GAN生成二維樣本的小例子

本文轉載自查看原文 2017-06-15 09:17 3871 機器學習/ Python

同步自我的知乎專欄：https://zhuanlan.zhihu.com/p/27343585

本文完整代碼地址：Generative Adversarial Networks (GANs) with 2D Samples

50行GAN代碼的問題

Dev Nag寫的50行代碼的GAN，大概是網上流傳最廣的，關於GAN最簡單的小例子。這是一份用一維均勻樣本作為特征空間(latent space)樣本，經過生成網絡變換后，生成高斯分布樣本的代碼。結構非常清晰，卻有一個奇怪的問題，就是判別器(Discriminator)的輸入不是2維樣本，而是把整個mini-batch整體作為一個維度是batch size(代碼中batch size等於cardinality)那么大的樣本。也就是說判別網絡要判別的不是一個一維的目標分布，而是batch size那么大維度的分布：

 
          ...
d_input_size = 100   # Minibatch size - cardinality of distributions
...
class Discriminator(nn.Module):
    def __init__(self, input_size, hidden_size, output_size):
        super(Discriminator, self).__init__()
        self.map1 = nn.Linear(input_size, hidden_size)
        self.map2 = nn.Linear(hidden_size, hidden_size)
        self.map3 = nn.Linear(hidden_size, output_size)

    def forward(self, x):
        x = F.elu(self.map1(x))
        x = F.elu(self.map2(x))
        return F.sigmoid(self.map3(x))
...
D = Discriminator(input_size=d_input_func(d_input_size), hidden_size=d_hidden_size, output_size=d_output_size)
...
for epoch in range(num_epochs):
    for d_index in range(d_steps):
        # 1. Train D on real+fake
        D.zero_grad()

        #  1A: Train D on real
        d_real_data = Variable(d_sampler(d_input_size))
        d_real_decision = D(preprocess(d_real_data))
        d_real_error = criterion(d_real_decision, Variable(torch.ones(1)))  # ones = true
        d_real_error.backward()  # compute/store gradients, but don't change params

        #  1B: Train D on fake
        d_gen_input = Variable(gi_sampler(minibatch_size, g_input_size))
        d_fake_data = G(d_gen_input).detach()  # detach to avoid training G on these labels
        d_fake_decision = D(preprocess(d_fake_data.t()))
        d_fake_error = criterion(d_fake_decision, Variable(torch.zeros(1)))  # zeros = fake
        d_fake_error.backward()
        d_optimizer.step()  # Only optimizes D's parameters; changes based on stored gradients from backward()

    for g_index in range(g_steps):
        # 2. Train G on D's response (but DO NOT train D on these labels)
        G.zero_grad()

        gen_input = Variable(gi_sampler(minibatch_size, g_input_size))
        g_fake_data = G(gen_input)
        dg_fake_decision = D(preprocess(g_fake_data.t()))
        g_error = criterion(dg_fake_decision, Variable(torch.ones(1)))  # we want to fool, so pretend it's all genuine

        g_error.backward()
        g_optimizer.step()  # Only optimizes G's parameters

... 
         

不知作者是疏忽了還是有意為之，總之這么做的結果就是如此簡單的例子收斂都好。可能作者自己也察覺了收斂問題，就想把方差信息也放進來，於是又寫了個預處理函數(decorate_with_diffs)計算出每個樣本距離一批樣本中心的距離平方，作為給判別網絡的額外輸入，其實這樣還增加了輸入維度。結果當然是加不加這個方差信息都能勉強收斂，但是都不穩定。甚至作者自己貼出來的生成樣本分布（下圖）都不令人滿意：

如果直接把這份代碼改成二維的，就會發現除了簡單的對稱分布以外，其他分布基本都無法生成。

理論上講神經網絡作為一種通用的近似函數，只要capacity夠，學習多少維分布都不成問題，但是這樣寫法顯然極大增加了收斂難度。更自然的做法應該是：判別網絡只接受單個二維樣本，通過batch size或是多步迭代學習分布信息。

另：這份代碼其實有130行。

從自定義的二維分布采樣

不管怎樣Dev Nag的代碼還是提供了一個用於理解和試驗GAN的很好的框架，做一些修改就可以得到一份更適合直觀演示，且更容易收斂的代碼，也就是本文的例子。

從可視化的角度二維顯然比一維更直觀，所以我們采用二維樣本。第一步，當然是要設定一個目標分布，作為二維的例子，分布的定義方式應該盡量自由，這個例子中我們的思路是通過灰度圖像定義的概率密度，進而來產生樣本，比如下面這樣：

二維情況下，這種采樣的一個實現方法是：求一個維度上的邊緣(marginal)概率+另一維度上近似的條件概率。比如把圖像中白色像素的值作為概率密度的相對大小，然后沿着x求和，然后在y軸上求出marginal probability density，接着再根據y的位置，近似得到對應x關於y的條件概率。采樣的時候先采y的值，再采x的值就能近似得到符合圖像描述的分布的樣本。具體細節就不展開講解了，看代碼：

from functools import partial
import numpy
from skimage import transform

EPS = 1e-6
RESOLUTION = 0.001
num_grids = int(1/RESOLUTION+0.5)

def generate_lut(img):
    """
    linear approximation of CDF & marginal
    :param density_img:
    :return: lut_y, lut_x
    """
    density_img = transform.resize(img, (num_grids, num_grids))
    x_accumlation = numpy.sum(density_img, axis=1)
    sum_xy = numpy.sum(x_accumlation)
    y_cdf_of_accumulated_x = [[0., 0.]]
    accumulated = 0
    for ir, i in enumerate(range(num_grids-1, -1, -1)):
        accumulated += x_accumlation[i]
        if accumulated == 0:
            y_cdf_of_accumulated_x[0][0] = float(ir+1)/float(num_grids)
        elif EPS < accumulated < sum_xy - EPS:
            y_cdf_of_accumulated_x.append([float(ir+1)/float(num_grids), accumulated/sum_xy])
        else:
            break
    y_cdf_of_accumulated_x.append([float(ir+1)/float(num_grids), 1.])
    y_cdf_of_accumulated_x = numpy.array(y_cdf_of_accumulated_x)

    x_cdfs = []
    for j in range(num_grids):
        x_freq = density_img[num_grids-j-1]
        sum_x = numpy.sum(x_freq)
        x_cdf = [[0., 0.]]
        accumulated = 0
        for i in range(num_grids):
            accumulated += x_freq[i]
            if accumulated == 0:
                x_cdf[0][0] = float(i+1) / float(num_grids)
            elif EPS < accumulated < sum_xy - EPS:
                x_cdf.append([float(i+1)/float(num_grids), accumulated/sum_x])
            else:
                break
        x_cdf.append([float(i+1)/float(num_grids), 1.])
        if accumulated > EPS:
            x_cdf = numpy.array(x_cdf)
            x_cdfs.append(x_cdf)
        else:
            x_cdfs.append(None)

    y_lut = partial(numpy.interp, xp=y_cdf_of_accumulated_x[:, 1], fp=y_cdf_of_accumulated_x[:, 0])
    x_luts = [partial(numpy.interp, xp=x_cdfs[i][:, 1], fp=x_cdfs[i][:, 0]) if x_cdfs[i] is not None else None for i in range(num_grids)]

    return y_lut, x_luts

def sample_2d(lut, N):
    y_lut, x_luts = lut
    u_rv = numpy.random.random((N, 2))
    samples = numpy.zeros(u_rv.shape)
    for i, (x, y) in enumerate(u_rv):
        ys = y_lut(y)
        x_bin = int(ys/RESOLUTION)
        xs = x_luts[x_bin](x)
        samples[i][0] = xs
        samples[i][1] = ys

    return samples

if __name__ == '__main__':
    from skimage import io
    density_img = io.imread('batman.jpg', True)
    lut_2d = generate_lut(density_img)
    samples = sample_2d(lut_2d, 10000)

    from matplotlib import pyplot
    fig, (ax0, ax1) = pyplot.subplots(ncols=2, figsize=(9, 4))
    fig.canvas.set_window_title('Test 2D Sampling')
    ax0.imshow(density_img, cmap='gray')
    ax0.xaxis.set_major_locator(pyplot.NullLocator())
    ax0.yaxis.set_major_locator(pyplot.NullLocator())

    ax1.axis('equal')
    ax1.axis([0, 1, 0, 1])
    ax1.plot(samples[:, 0], samples[:, 1], 'k,')
    pyplot.show()

二維GAN的小例子

雖然網上到處都有，這里還是貼一下GAN的公式：

$\min_{G} \max_{D} V(D,G)=E_{\bm{x} \sim p_{data}(\bm{x})} [\log D(\bm{x})]+E_{\bm{z} \sim p_{\bm{z}}(\bm{z})} [\log (1-D(G(\bm{z})))]$

就是一個你追我趕的零和博弈，這在Dev Nag的代碼里體現得很清晰：判別網絡訓一撥，然后生成網絡訓一撥，不斷往復。按照上節所述，本文例子在Dev Nag代碼的基礎上，把判別網絡每次接受一個batch作為輸入的方式變成了：每次接受一個二維樣本，通過每個batch的多個樣本計算loss。GAN部分的訓練代碼如下：

DIMENSION = 2

...

generator = SimpleMLP(input_size=z_dim, hidden_size=args.g_hidden_size, output_size=DIMENSION)
discriminator = SimpleMLP(input_size=DIMENSION, hidden_size=args.d_hidden_size, output_size=1)

...

for train_iter in range(args.iterations):
    for d_index in range(args.d_steps):
        # 1. Train D on real+fake
        discriminator.zero_grad()

        #  1A: Train D on real
        real_samples = sample_2d(lut_2d, bs)
        d_real_data = Variable(torch.Tensor(real_samples))
        d_real_decision = discriminator(d_real_data)
        labels = Variable(torch.ones(bs))
        d_real_loss = criterion(d_real_decision, labels)  # ones = true

        #  1B: Train D on fake
        latent_samples = torch.randn(bs, z_dim)
        d_gen_input = Variable(latent_samples)
        d_fake_data = generator(d_gen_input).detach()  # detach to avoid training G on these labels
        d_fake_decision = discriminator(d_fake_data)
        labels = Variable(torch.zeros(bs))
        d_fake_loss = criterion(d_fake_decision, labels)  # zeros = fake

        d_loss = d_real_loss + d_fake_loss
        d_loss.backward()

        d_optimizer.step()     # Only optimizes D's parameters; changes based on stored gradients from backward()

    for g_index in range(args.g_steps):
        # 2. Train G on D's response (but DO NOT train D on these labels)
        generator.zero_grad()

        latent_samples = torch.randn(bs, z_dim)
        g_gen_input = Variable(latent_samples)
        g_fake_data = generator(g_gen_input)
        g_fake_decision = discriminator(g_fake_data)
        labels = Variable(torch.ones(bs))
        g_loss = criterion(g_fake_decision, labels)  # we want to fool, so pretend it's all genuine

        g_loss.backward()
        g_optimizer.step()  # Only optimizes G's parameters

    ...

...

和Dev Nag的版本比起來除了上面提到的判別網絡，和樣本維度的修改，還加了可視化方便直觀演示和理解，比如用一個二維高斯分布產生一個折線形狀的分布，執行：

python gan_demo.py inputs/zig.jpg

訓練過程的可視化如下：

更多可視化例子可以參考這里。

Conditional GAN

對於一些復雜的分布，原始的GAN就會很吃力，比如用一個二維高斯分布產生兩坨圓形的分布：

因為latent space的分布就是一坨二維的樣本，所以即使模型有很強的非線性，也難以把這個分布“切開”並變換成兩個很好的圓形分布。因此在上面的動圖里能看到生成的兩坨樣本中間總是有一些殘存的樣本，像是兩個天體在交換物質。要改進這種情況，比較直接的想法是增加模型復雜度，或是提高latent space維度。也許模型可以學習到用其中部分維度產生一個圓形，用另一部分維度產生另一個圓形。不過我自己試了下，效果都不好。

其實這個例子人眼一看就知道是兩個分布在一個圖里，假設我們已經知道這個信息，那么生成依據的就是個條件概率。把這個條件加到GAN里，就是Conditional GAN，公式如下：

$\min_{G} \max_{D} V(D,G)=E_{\bm{x} \sim p_{data}(\bm{x} | \bm{y})} [\log D(\bm{x}|\bm{y})]+E_{\bm{z} \sim p_{z}(\bm{z})} [\log (1-D(G(\bm{z}|\bm{y})|\bm{y}))]$