知識蒸餾綜述：代碼整理

【GiantPandaCV導語】收集自RepDistiller中的蒸餾方法，儘可能簡單解釋蒸餾用到的策略，並提供了實現源碼。

1. KD: Knowledge Distillation

全稱：Distilling the Knowledge in a Neural Network

鏈接：https://arxiv.org/pdf/1503.02531.pdf

發表：NIPS14

最經典的，也是明確提出知識蒸餾概念的工作，通過使用帶溫度的softmax函數來軟化教師網絡的邏輯層輸出作爲學生網絡的監督信息，

\[q_{i}=\frac{\exp \left(z_{i} / T\right)}{\sum_{j} \exp \left(z_{j} / T\right)} \]

使用KL divergence來衡量學生網絡與教師網絡的差異，具體流程如下圖所示（來自Knowledge Distillation A Survey）

對學生網絡來說，一部分監督信息來自hard label標籤，另一部分來自教師網絡提供的soft label。

代碼實現：

class DistillKL(nn.Module):
    """Distilling the Knowledge in a Neural Network"""
    def __init__(self, T):
        super(DistillKL, self).__init__()
        self.T = T

    def forward(self, y_s, y_t):
        p_s = F.log_softmax(y_s/self.T, dim=1)
        p_t = F.softmax(y_t/self.T, dim=1)
        loss = F.kl_div(p_s, p_t, size_average=False) * (self.T**2) / y_s.shape[0]
        return loss

核心就是一個kl_div函數，用於計算學生網絡和教師網絡的分佈差異。

2. FitNet: Hints for thin deep nets

全稱：Fitnets: hints for thin deep nets

鏈接：https://arxiv.org/pdf/1412.6550.pdf

發表：ICLR 15 Poster

對中間層進行蒸餾的開山之作，通過將學生網絡的feature map擴展到與教師網絡的feature map相同尺寸以後，使用均方誤差MSE Loss來衡量兩者差異。

實現如下：

class HintLoss(nn.Module):
    """Fitnets: hints for thin deep nets, ICLR 2015"""
    def __init__(self):
        super(HintLoss, self).__init__()
        self.crit = nn.MSELoss()

    def forward(self, f_s, f_t):
        loss = self.crit(f_s, f_t)
        return loss

實現核心就是MSELoss

3. AT: Attention Transfer

全稱：Paying More Attention to Attention: Improving the Performance of Convolutional Neural Networks via Attention Transfer

鏈接：https://arxiv.org/pdf/1612.03928.pdf

發表：ICLR16

爲了提升學生模型性能提出使用注意力作爲知識載體進行遷移，文中提到了兩種注意力，一種是activation-based attention transfer，另一種是gradient-based attention transfer。實驗發現第一種方法既簡單效果又好。

實現如下：

class Attention(nn.Module):
    """Paying More Attention to Attention: Improving the Performance of Convolutional Neural Networks
    via Attention Transfer
    code: https://github.com/szagoruyko/attention-transfer"""
    def __init__(self, p=2):
        super(Attention, self).__init__()
        self.p = p

    def forward(self, g_s, g_t):
        return [self.at_loss(f_s, f_t) for f_s, f_t in zip(g_s, g_t)]

    def at_loss(self, f_s, f_t):
        s_H, t_H = f_s.shape[2], f_t.shape[2]
        if s_H > t_H:
            f_s = F.adaptive_avg_pool2d(f_s, (t_H, t_H))
        elif s_H < t_H:
            f_t = F.adaptive_avg_pool2d(f_t, (s_H, s_H))
        else:
            pass
        return (self.at(f_s) - self.at(f_t)).pow(2).mean()

    def at(self, f):
        return F.normalize(f.pow(self.p).mean(1).view(f.size(0), -1))

首先使用avgpool將尺寸調整一致，然後使用MSE Loss來衡量兩者差距。

4. SP: Similarity-Preserving

全稱：Similarity-Preserving Knowledge Distillation

鏈接：https://arxiv.org/pdf/1907.09682.pdf

發表：ICCV19

SP歸屬於基於關係的知識蒸餾方法。文章思想是提出相似性保留的知識，使得教師網絡和學生網絡會對相同的樣本產生相似的激活。可以從下圖看出處理流程，教師網絡和學生網絡對應feature map通過計算內積，得到bsxbs的相似度矩陣，然後使用均方誤差來衡量兩個相似度矩陣。

最終Loss爲：

\[\mathcal{L}=\mathcal{L}_{\mathrm{CE}}\left(\mathbf{y}, \sigma\left(\mathbf{z}_{S}\right)\right)+\gamma \mathcal{L}_{\mathrm{SP}}\left(G_{T}, G_{S}\right) \]

\[\mathcal{L}_{\mathrm{SP}}\left(G_{T}, G_{S}\right)=\frac{1}{b^{2}} \sum_{\left(l, l^{\prime}\right) \in \mathcal{I}}\left\|G_{T}^{(l)}-G_{S}^{\left(l^{\prime}\right)}\right\|_{F}^{2} \]

G代表的就是bsxbs的矩陣。

實現如下：

class Similarity(nn.Module):
    """Similarity-Preserving Knowledge Distillation, ICCV2019, verified by original author"""
    def __init__(self):
        super(Similarity, self).__init__()

    def forward(self, g_s, g_t):
        return [self.similarity_loss(f_s, f_t) for f_s, f_t in zip(g_s, g_t)]

    def similarity_loss(self, f_s, f_t):
        bsz = f_s.shape[0]
        f_s = f_s.view(bsz, -1)
        f_t = f_t.view(bsz, -1)

        G_s = torch.mm(f_s, torch.t(f_s))
        # G_s = G_s / G_s.norm(2)
        G_s = torch.nn.functional.normalize(G_s)
        G_t = torch.mm(f_t, torch.t(f_t))
        # G_t = G_t / G_t.norm(2)
        G_t = torch.nn.functional.normalize(G_t)

        G_diff = G_t - G_s
        loss = (G_diff * G_diff).view(-1, 1).sum(0) / (bsz * bsz)
        return loss

5. CC: Correlation Congruence

全稱：Correlation Congruence for Knowledge Distillation

鏈接：https://arxiv.org/pdf/1904.01802.pdf

發表：ICCV19

CC也歸屬於基於關係的知識蒸餾方法。不應該僅僅引導教師網絡和學生網絡單個樣本向量之間的差異，還應該學習兩個樣本之間的相關性，而這個相關性使用的是Correlation Congruence 教師網絡雨學生網絡相關性之間的歐氏距離。

\[\begin{aligned} L_{C C} &=\frac{1}{n^{2}}\left\|\psi\left(\boldsymbol{F}_{t}\right)-\psi\left(\boldsymbol{F}_{s}\right)\right\|_{2}^{2} \\ &=\frac{1}{n^{2}} \sum_{i, j}\left(\varphi\left(\boldsymbol{f}_{i}^{s}, \boldsymbol{f}_{j}^{s}\right)-\varphi\left(\boldsymbol{f}_{i}^{t}, \boldsymbol{f}_{j}^{t}\right)\right)^{2} \end{aligned} \]

整體Loss如下：

\[L_{C C K D}=\alpha L_{\mathrm{CE}}+(1-\alpha) L_{\mathrm{KD}}+\beta L_{\mathrm{CC}} \]

實現如下：

class Correlation(nn.Module):
    """Similarity-preserving loss. My origianl own reimplementation 
    based on the paper before emailing the original authors."""
    def __init__(self):
        super(Correlation, self).__init__()

    def forward(self, f_s, f_t):
        return self.similarity_loss(f_s, f_t)

    def similarity_loss(self, f_s, f_t):
        bsz = f_s.shape[0]
        f_s = f_s.view(bsz, -1)
        f_t = f_t.view(bsz, -1)

        G_s = torch.mm(f_s, torch.t(f_s))
        G_s = G_s / G_s.norm(2)
        G_t = torch.mm(f_t, torch.t(f_t))
        G_t = G_t / G_t.norm(2)

        G_diff = G_t - G_s
        loss = (G_diff * G_diff).view(-1, 1).sum(0) / (bsz * bsz)
        return loss

6. VID: Variational Information Distillation

全稱：Variational Information Distillation for Knowledge Transfer

鏈接：https://arxiv.org/pdf/1904.05835.pdf

發表：CVPR19

利用互信息（Mutual Information）來衡量學生網絡和教師網絡差異。互信息可以表示出兩個變量的互相依賴程度，其值越大，表示變量之間的依賴程度越高。互信息計算如下：

\[\begin{aligned} I(\boldsymbol{t} ; \boldsymbol{s}) &=H(\boldsymbol{t})-H(\boldsymbol{t} \mid \boldsymbol{s}) \\ &=-\mathbb{E}_{\boldsymbol{t}}[\log p(\boldsymbol{t})]+\mathbb{E}_{\boldsymbol{t}, \boldsymbol{s}}[\log p(\boldsymbol{t} \mid \boldsymbol{s})] \end{aligned} \]

互信息是教師模型的熵減去在已知學生模型條件下教師模型的熵。目標是最大化互信息，因爲互信息越大說明H(t|s)越小，即學生網絡確定的情況下，教師網絡的熵會變小，證明學生網絡已經學習的比較充分。

整體loss如下：

\[\mathcal{L}=\mathcal{L}_{\mathcal{S}}-\sum_{k=1}^{K} \lambda_{k} I\left(\boldsymbol{t}^{(k)}, \boldsymbol{s}^{(k)}\right) \]

由於p(t|s)很難計算，可以使用變分分佈q(t|s)去接近真實分佈。

\[\begin{aligned} &I(\boldsymbol{t} ; \boldsymbol{s})=H(\boldsymbol{t})-H(\boldsymbol{t} \mid \boldsymbol{s}) \\ &=H(\boldsymbol{t})+\mathbb{E}_{\boldsymbol{t}, \boldsymbol{s}}[\log p(\boldsymbol{t} \mid \boldsymbol{s})] \\ &=H(\boldsymbol{t})+\mathbb{E}_{\boldsymbol{t}, \boldsymbol{s}}[\log q(\boldsymbol{t} \mid \boldsymbol{s})]+\mathbb{E}_{\boldsymbol{s}}\left[D_{\mathrm{KL}}(p(\boldsymbol{t} \mid \boldsymbol{s}) \| q(\boldsymbol{t} \mid \boldsymbol{s}))\right] \\ &\geq H(\boldsymbol{t})+\mathbb{E}_{\boldsymbol{t}, \boldsymbol{s}}[\log q(\boldsymbol{t} \mid \boldsymbol{s})] \end{aligned} \]

其中q(t|s)是使用方差可學習的高斯分佈模擬（公式中的log_scale）：

\[\begin{aligned} -\log q(\boldsymbol{t} \mid \boldsymbol{s}) &=-\sum_{n=1}^{N} \log q\left(t_{n} \mid \boldsymbol{s}\right) \\ &=\sum_{n=1}^{N} \log \sigma_{n}+\frac{\left(t_{n}-\mu_{n}(\boldsymbol{s})\right)^{2}}{2 \sigma_{n}^{2}}+\text { constant } \end{aligned} \]

實現如下：

class VIDLoss(nn.Module):
    """Variational Information Distillation for Knowledge Transfer (CVPR 2019),
    code from author: https://github.com/ssahn0215/variational-information-distillation"""
    def __init__(self,
                 num_input_channels,
                 num_mid_channel,
                 num_target_channels,
                 init_pred_var=5.0,
                 eps=1e-5):
        super(VIDLoss, self).__init__()

        def conv1x1(in_channels, out_channels, stride=1):
            return nn.Conv2d(
                in_channels, out_channels,
                kernel_size=1, padding=0,
                bias=False, stride=stride)

        self.regressor = nn.Sequential(
            conv1x1(num_input_channels, num_mid_channel),
            nn.ReLU(),
            conv1x1(num_mid_channel, num_mid_channel),
            nn.ReLU(),
            conv1x1(num_mid_channel, num_target_channels),
        )
        self.log_scale = torch.nn.Parameter(
            np.log(np.exp(init_pred_var-eps)-1.0) * torch.ones(num_target_channels)
            )
        self.eps = eps

    def forward(self, input, target):
        # pool for dimentsion match
        s_H, t_H = input.shape[2], target.shape[2]
        if s_H > t_H:
            input = F.adaptive_avg_pool2d(input, (t_H, t_H))
        elif s_H < t_H:
            target = F.adaptive_avg_pool2d(target, (s_H, s_H))
        else:
            pass
        pred_mean = self.regressor(input)
        pred_var = torch.log(1.0+torch.exp(self.log_scale))+self.eps
        pred_var = pred_var.view(1, -1, 1, 1)
        neg_log_prob = 0.5*(
            (pred_mean-target)**2/pred_var+torch.log(pred_var)
            )
        loss = torch.mean(neg_log_prob)
        return loss

7. RKD: Relation Knowledge Distillation

全稱：Relational Knowledge Disitllation

鏈接：http://arxiv.org/pdf/1904.05068

發表：CVPR19

RKD也是基於關係的知識蒸餾方法，RKD提出了兩種損失函數，二階的距離損失和三階的角度損失。

Distance-wise Loss

\[\begin{aligned} &\mathcal{L}_{\mathrm{RKD}-\mathrm{D}}=\sum_{\left(x_{i}, x_{j}\right) \in \mathcal{X}^{2}} l_{\delta}\left(\psi_{\mathrm{D}}\left(t_{i}, t_{j}\right), \psi_{\mathrm{D}}\left(s_{i}, s_{j}\right)\right) \\ &\psi_{\mathrm{D}}\left(t_{i}, t_{j}\right)=\frac{1}{\mu}\left\|t_{i}-t_{j}\right\|_{2} \end{aligned} \]

Angle-wise Loss

\[\begin{aligned} &\mathcal{L}_{\mathrm{RKD}-\mathrm{A}}=\sum_{\left(x_{i}, x_{j}, x_{k}\right) \in \mathcal{X}^{3}} l_{\delta}\left(\psi_{\mathrm{A}}\left(t_{i}, t_{j}, t_{k}\right), \psi_{\mathrm{A}}\left(s_{i}, s_{j}, s_{k}\right)\right) \\ &\psi_{\mathrm{A}}\left(t_{i}, t_{j}, t_{k}\right)=\cos \angle t_{i} t_{j} t_{k}=\left\langle\mathbf{e}^{i j}, \mathbf{e}^{k j}\right\rangle \\ &\text { where } \mathbf{e}^{i j}=\frac{t_{i}-t_{j}}{\left\|t_{i}-t_{j}\right\|_{2}}, \mathbf{e}^{k j}=\frac{t_{k}-t_{j}}{\left\|t_{k}-t_{j}\right\|_{2}} \end{aligned} \]

實現如下：

class RKDLoss(nn.Module):
    """Relational Knowledge Disitllation, CVPR2019"""
    def __init__(self, w_d=25, w_a=50):
        super(RKDLoss, self).__init__()
        self.w_d = w_d
        self.w_a = w_a

    def forward(self, f_s, f_t):
        student = f_s.view(f_s.shape[0], -1)
        teacher = f_t.view(f_t.shape[0], -1)

        # RKD distance loss
        with torch.no_grad():
            t_d = self.pdist(teacher, squared=False)
            mean_td = t_d[t_d > 0].mean()
            t_d = t_d / mean_td

        d = self.pdist(student, squared=False)
        mean_d = d[d > 0].mean()
        d = d / mean_d

        loss_d = F.smooth_l1_loss(d, t_d)

        # RKD Angle loss
        with torch.no_grad():
            td = (teacher.unsqueeze(0) - teacher.unsqueeze(1))
            norm_td = F.normalize(td, p=2, dim=2)
            t_angle = torch.bmm(norm_td, norm_td.transpose(1, 2)).view(-1)

        sd = (student.unsqueeze(0) - student.unsqueeze(1))
        norm_sd = F.normalize(sd, p=2, dim=2)
        s_angle = torch.bmm(norm_sd, norm_sd.transpose(1, 2)).view(-1)

        loss_a = F.smooth_l1_loss(s_angle, t_angle)

        loss = self.w_d * loss_d + self.w_a * loss_a

        return loss

    @staticmethod
    def pdist(e, squared=False, eps=1e-12):
        e_square = e.pow(2).sum(dim=1)
        prod = e @ e.t()
        res = (e_square.unsqueeze(1) + e_square.unsqueeze(0) - 2 * prod).clamp(min=eps)

        if not squared:
            res = res.sqrt()

        res = res.clone()
        res[range(len(e)), range(len(e))] = 0
        return res

8. PKT:Probabilistic Knowledge Transfer

全稱：Probabilistic Knowledge Transfer for deep representation learning

鏈接：https://arxiv.org/abs/1803.10837

發表：CoRR18

提出一種概率知識轉移方法，引入了互信息來進行建模。該方法具有可跨模態知識轉移、無需考慮任務類型、可將手工特徵融入網絡等有點。

實現如下:

class PKT(nn.Module):
    """Probabilistic Knowledge Transfer for deep representation learning
    Code from author: https://github.com/passalis/probabilistic_kt"""
    def __init__(self):
        super(PKT, self).__init__()

    def forward(self, f_s, f_t):
        return self.cosine_similarity_loss(f_s, f_t)

    @staticmethod
    def cosine_similarity_loss(output_net, target_net, eps=0.0000001):
        # Normalize each vector by its norm
        output_net_norm = torch.sqrt(torch.sum(output_net ** 2, dim=1, keepdim=True))
        output_net = output_net / (output_net_norm + eps)
        output_net[output_net != output_net] = 0

        target_net_norm = torch.sqrt(torch.sum(target_net ** 2, dim=1, keepdim=True))
        target_net = target_net / (target_net_norm + eps)
        target_net[target_net != target_net] = 0

        # Calculate the cosine similarity
        model_similarity = torch.mm(output_net, output_net.transpose(0, 1))
        target_similarity = torch.mm(target_net, target_net.transpose(0, 1))

        # Scale cosine similarity to 0..1
        model_similarity = (model_similarity + 1.0) / 2.0
        target_similarity = (target_similarity + 1.0) / 2.0

        # Transform them into probabilities
        model_similarity = model_similarity / torch.sum(model_similarity, dim=1, keepdim=True)
        target_similarity = target_similarity / torch.sum(target_similarity, dim=1, keepdim=True)

        # Calculate the KL-divergence
        loss = torch.mean(target_similarity * torch.log((target_similarity + eps) / (model_similarity + eps)))

        return loss

9. AB: Activation Boundaries

全稱：Knowledge Transfer via Distillation of Activation Boundaries Formed by Hidden Neurons

鏈接：https://arxiv.org/pdf/1811.03233.pdf

發表：AAAI18

目標：讓教師網絡層的神經元的激活邊界儘量和學生網絡的一樣。所謂的激活邊界指的是分離超平面（針對的是RELU這種激活函數），其決定了神經元的激活與失活。AB提出的激活轉移損失，讓教師網絡與學生網絡之間的分離邊界儘可能一致。

實現如下：

class ABLoss(nn.Module):
    """Knowledge Transfer via Distillation of Activation Boundaries Formed by Hidden Neurons
    code: https://github.com/bhheo/AB_distillation
    """
    def __init__(self, feat_num, margin=1.0):
        super(ABLoss, self).__init__()
        self.w = [2**(i-feat_num+1) for i in range(feat_num)]
        self.margin = margin

    def forward(self, g_s, g_t):
        bsz = g_s[0].shape[0]
        losses = [self.criterion_alternative_l2(s, t) for s, t in zip(g_s, g_t)]
        losses = [w * l for w, l in zip(self.w, losses)] 
        # loss = sum(losses) / bsz
        # loss = loss / 1000 * 3
        losses = [l / bsz for l in losses]
        losses = [l / 1000 * 3 for l in losses]
        return losses

    def criterion_alternative_l2(self, source, target):
        loss = ((source + self.margin) ** 2 * ((source > -self.margin) & (target <= 0)).float() +
                (source - self.margin) ** 2 * ((source <= self.margin) & (target > 0)).float())
        return torch.abs(loss).sum()

10. FT: Factor Transfer

全稱：Paraphrasing Complex Network: Network Compression via Factor Transfer

鏈接：https://arxiv.org/pdf/1802.04977.pdf

發表：NIPS18

提出的是factor transfer的方法。所謂的factor，其實是對模型最後的數據結果進行一個編解碼的過程，提取出的一個factor矩陣，用教師網絡的factor來指導學生網絡的factor。

FT計算公式爲：

\[\begin{gathered} \mathcal{L}_{\text {student }}=\mathcal{L}_{\text {cls }}+\beta \mathcal{L}_{F T} \\ \mathcal{L}_{\text {cls }}=\mathcal{C}\left(S\left(I_{x}\right), y\right) \\ \mathcal{L}_{F T}=\left\|\frac{F_{T}}{\left\|F_{T}\right\|_{2}}-\frac{F_{S}}{\left\|F_{S}\right\|_{2}}\right\|_{p} \end{gathered} \]

實現如下：

class FactorTransfer(nn.Module):
    """Paraphrasing Complex Network: Network Compression via Factor Transfer, NeurIPS 2018"""
    def __init__(self, p1=2, p2=1):
        super(FactorTransfer, self).__init__()
        self.p1 = p1
        self.p2 = p2

    def forward(self, f_s, f_t):
        return self.factor_loss(f_s, f_t)

    def factor_loss(self, f_s, f_t):
        s_H, t_H = f_s.shape[2], f_t.shape[2]
        if s_H > t_H:
            f_s = F.adaptive_avg_pool2d(f_s, (t_H, t_H))
        elif s_H < t_H:
            f_t = F.adaptive_avg_pool2d(f_t, (s_H, s_H))
        else:
            pass
        if self.p2 == 1:
            return (self.factor(f_s) - self.factor(f_t)).abs().mean()
        else:
            return (self.factor(f_s) - self.factor(f_t)).pow(self.p2).mean()

    def factor(self, f):
        return F.normalize(f.pow(self.p1).mean(1).view(f.size(0), -1))

11. FSP: Flow of Solution Procedure

全稱：A Gift from Knowledge Distillation: Fast Optimization, Network Minimization and Transfer Learning

鏈接：https://openaccess.thecvf.com/content_cvpr_2017/papers/Yim_A_Gift_From_CVPR_2017_paper.pdf

發表：CVPR17

FSP認爲教學生網絡不同層輸出的feature之間的關係比教學生網絡結果好

定義了FSP矩陣來定義網絡內部特徵層之間的關係，是一個Gram矩陣反映老師教學生的過程。

\[G_{i, j}(x ; W)=\sum_{s=1}^{h} \sum_{t=1}^{w} \frac{F_{s, t, i}^{1}(x ; W) \times F_{s, t, j}^{2}(x ; W)}{h \times w} \]

使用的是L2 Loss進行約束FSP矩陣。

實現如下：

class FSP(nn.Module):
    """A Gift from Knowledge Distillation:
    Fast Optimization, Network Minimization and Transfer Learning"""
    def __init__(self, s_shapes, t_shapes):
        super(FSP, self).__init__()
        assert len(s_shapes) == len(t_shapes), 'unequal length of feat list'
        s_c = [s[1] for s in s_shapes]
        t_c = [t[1] for t in t_shapes]
        if np.any(np.asarray(s_c) != np.asarray(t_c)):
            raise ValueError('num of channels not equal (error in FSP)')

    def forward(self, g_s, g_t):
        s_fsp = self.compute_fsp(g_s)
        t_fsp = self.compute_fsp(g_t)
        loss_group = [self.compute_loss(s, t) for s, t in zip(s_fsp, t_fsp)]
        return loss_group

    @staticmethod
    def compute_loss(s, t):
        return (s - t).pow(2).mean()

    @staticmethod
    def compute_fsp(g):
        fsp_list = []
        for i in range(len(g) - 1):
            bot, top = g[i], g[i + 1]
            b_H, t_H = bot.shape[2], top.shape[2]
            if b_H > t_H:
                bot = F.adaptive_avg_pool2d(bot, (t_H, t_H))
            elif b_H < t_H:
                top = F.adaptive_avg_pool2d(top, (b_H, b_H))
            else:
                pass
            bot = bot.unsqueeze(1)
            top = top.unsqueeze(2)
            bot = bot.view(bot.shape[0], bot.shape[1], bot.shape[2], -1)
            top = top.view(top.shape[0], top.shape[1], top.shape[2], -1)

            fsp = (bot * top).mean(-1)
            fsp_list.append(fsp)
        return fsp_list

12. NST: Neuron Selectivity Transfer

全稱：Like what you like: knowledge distill via neuron selectivity transfer

鏈接：https://arxiv.org/pdf/1707.01219.pdf

發表：CoRR17

使用新的損失函數最小化教師網絡與學生網絡之間的Maximum Mean Discrepancy（MMD), 文中選擇的是對其教師網絡與學生網絡之間神經元選擇樣式的分佈。

\[\mathcal{L}_{\mathrm{MMD}^{2}}(\mathcal{X}, \mathcal{Y})=\left\|\frac{1}{N} \sum_{i=1}^{N} \phi\left(\boldsymbol{x}^{i}\right)-\frac{1}{M} \sum_{j=1}^{M} \phi\left(\boldsymbol{y}^{j}\right)\right\|_{2}^{2} \]

使用核技巧(對應下面poly kernel)並進一步展開以後可得：

\[\begin{aligned} \mathcal{L}_{\mathrm{MMD}^{2}}(\mathcal{X}, \mathcal{Y}) &=\frac{1}{N^{2}} \sum_{i=1}^{N} \sum_{i^{\prime}=1}^{N} k\left(\boldsymbol{x}^{i}, \boldsymbol{x}^{i^{\prime}}\right) \\ &+\frac{1}{M^{2}} \sum_{j=1}^{M} \sum_{j^{\prime}=1}^{M} k\left(\boldsymbol{y}^{i}, \boldsymbol{y}^{i^{\prime}}\right) \\ &-\frac{2}{M N} \sum_{i=1}^{N} \sum_{j=1}^{M} k\left(\boldsymbol{x}^{i}, \boldsymbol{y}^{j}\right) \end{aligned} \]

實際上提供了Linear Kernel、Poly Kernel、Gaussian Kernel三種，這裏實現只給了Poly這種，這是因爲Poly這種方法可以與KD進行互補，這樣整體效果會非常好。

實現如下：

class NSTLoss(nn.Module):
    """like what you like: knowledge distill via neuron selectivity transfer"""
    def __init__(self):
        super(NSTLoss, self).__init__()
        pass

    def forward(self, g_s, g_t):
        return [self.nst_loss(f_s, f_t) for f_s, f_t in zip(g_s, g_t)]

    def nst_loss(self, f_s, f_t):
        s_H, t_H = f_s.shape[2], f_t.shape[2]
        if s_H > t_H:
            f_s = F.adaptive_avg_pool2d(f_s, (t_H, t_H))
        elif s_H < t_H:
            f_t = F.adaptive_avg_pool2d(f_t, (s_H, s_H))
        else:
            pass

        f_s = f_s.view(f_s.shape[0], f_s.shape[1], -1)
        f_s = F.normalize(f_s, dim=2)
        f_t = f_t.view(f_t.shape[0], f_t.shape[1], -1)
        f_t = F.normalize(f_t, dim=2)

        # set full_loss as False to avoid unnecessary computation
        full_loss = True
        if full_loss:
            return (self.poly_kernel(f_t, f_t).mean().detach() + self.poly_kernel(f_s, f_s).mean()
                    - 2 * self.poly_kernel(f_s, f_t).mean())
        else:
            return self.poly_kernel(f_s, f_s).mean() - 2 * self.poly_kernel(f_s, f_t).mean()

    def poly_kernel(self, a, b):
        a = a.unsqueeze(1)
        b = b.unsqueeze(2)
        res = (a * b).sum(-1).pow(2)
        return res

13. CRD: Contrastive Representation Distillation

全稱：Contrastive Representation Distillation

鏈接：https://arxiv.org/abs/1910.10699v2

發表：ICLR20

將對比學習引入知識蒸餾中，其目標修正爲：學習一個表徵，讓正樣本對的教師網絡與學生網絡儘可能接近，負樣本對教師網絡與學生網絡儘可能遠離。

構建的對比學習問題表示如下：

\[\begin{aligned} f^{S *} &=\underset{f^{S}}{\arg \max } \max _{h} \mathcal{L}_{\text {critic }}(h) \\ &=\underset{f^{S}}{\arg \max } \max _{h} \mathbb{E}_{q(T, S \mid C=1)}[\log h(T, S)]+N \mathbb{E}_{q(T, S \mid C=0)}[\log (1-h(T, S))] \end{aligned} \]

整體的蒸餾Loss表示如下：

\[\mathcal{L}_{K D}=(1-\alpha) H\left(y, y^{S}\right)+\alpha \rho^{2} H\left(\sigma\left(z^{T} / \rho\right), \sigma\left(z^{S} / \rho\right)\right) \]

實現如下：https://github.com/HobbitLong/RepDistiller

class ContrastLoss(nn.Module):
    """
    contrastive loss, corresponding to Eq (18)
    """
    def __init__(self, n_data):
        super(ContrastLoss, self).__init__()
        self.n_data = n_data

    def forward(self, x):
        bsz = x.shape[0]
        m = x.size(1) - 1

        # noise distribution
        Pn = 1 / float(self.n_data)

        # loss for positive pair
        P_pos = x.select(1, 0)
        log_D1 = torch.div(P_pos, P_pos.add(m * Pn + eps)).log_()

        # loss for K negative pair
        P_neg = x.narrow(1, 1, m)
        log_D0 = torch.div(P_neg.clone().fill_(m * Pn), P_neg.add(m * Pn + eps)).log_()

        loss = - (log_D1.sum(0) + log_D0.view(-1, 1).sum(0)) / bsz

        return loss
        
class CRDLoss(nn.Module):
    """CRD Loss function
    includes two symmetric parts:
    (a) using teacher as anchor, choose positive and negatives over the student side
    (b) using student as anchor, choose positive and negatives over the teacher side

    Args:
        opt.s_dim: the dimension of student's feature
        opt.t_dim: the dimension of teacher's feature
        opt.feat_dim: the dimension of the projection space
        opt.nce_k: number of negatives paired with each positive
        opt.nce_t: the temperature
        opt.nce_m: the momentum for updating the memory buffer
        opt.n_data: the number of samples in the training set, therefor the memory buffer is: opt.n_data x opt.feat_dim
    """
    def __init__(self, opt):
        super(CRDLoss, self).__init__()
        self.embed_s = Embed(opt.s_dim, opt.feat_dim)
        self.embed_t = Embed(opt.t_dim, opt.feat_dim)
        self.contrast = ContrastMemory(opt.feat_dim, opt.n_data, opt.nce_k, opt.nce_t, opt.nce_m)
        self.criterion_t = ContrastLoss(opt.n_data)
        self.criterion_s = ContrastLoss(opt.n_data)

    def forward(self, f_s, f_t, idx, contrast_idx=None):
        """
        Args:
            f_s: the feature of student network, size [batch_size, s_dim]
            f_t: the feature of teacher network, size [batch_size, t_dim]
            idx: the indices of these positive samples in the dataset, size [batch_size]
            contrast_idx: the indices of negative samples, size [batch_size, nce_k]

        Returns:
            The contrastive loss
        """
        f_s = self.embed_s(f_s)
        f_t = self.embed_t(f_t)
        out_s, out_t = self.contrast(f_s, f_t, idx, contrast_idx)
        s_loss = self.criterion_s(out_s)
        t_loss = self.criterion_t(out_t)
        loss = s_loss + t_loss
        return loss

14. Overhaul

全稱：A Comprehensive Overhaul of Feature Distillation

鏈接：http://openaccess.thecvf.com/content_ICCV_2019/papers/

發表：CVPR19

teacher transform中提出使用margin RELU激活函數。

student transform中提出使用1x1卷積。
distillation feature postion選擇Pre-ReLU。

distance function部分提出了Partial L2 損失函數。

\[d_{p}(\boldsymbol{T}, \boldsymbol{S})=\sum_{i}^{W H C} \begin{cases}0 & \text { if } S_{i} \leq T_{i} \leq 0 \\ \left(T_{i}-S_{i}\right)^{2} & \text { otherwise. }\end{cases} \]

部分實現如下：

class OFD(nn.Module):
  '''
  A Comprehensive Overhaul of Feature Distillation
  http://openaccess.thecvf.com/content_ICCV_2019/papers/
  Heo_A_Comprehensive_Overhaul_of_Feature_Distillation_ICCV_2019_paper.pdf
  '''
  def __init__(self, in_channels, out_channels):
    super(OFD, self).__init__()
    self.connector = nn.Sequential(*[
        nn.Conv2d(in_channels, out_channels, kernel_size=1, stride=1, padding=0, bias=False),
        nn.BatchNorm2d(out_channels)
      ])

    for m in self.modules():
      if isinstance(m, nn.Conv2d):
        nn.init.kaiming_normal_(m.weight, mode='fan_out', nonlinearity='relu')
        if m.bias is not None:
          nn.init.constant_(m.bias, 0)
      elif isinstance(m, nn.BatchNorm2d):
        nn.init.constant_(m.weight, 1)
        nn.init.constant_(m.bias, 0)

  def forward(self, fm_s, fm_t):
    margin = self.get_margin(fm_t)
    fm_t = torch.max(fm_t, margin)
    fm_s = self.connector(fm_s)

    mask = 1.0 - ((fm_s <= fm_t) & (fm_t <= 0.0)).float()
    loss = torch.mean((fm_s - fm_t)**2 * mask)

    return loss

  def get_margin(self, fm, eps=1e-6):
    mask = (fm < 0.0).float()
    masked_fm = fm * mask

    margin = masked_fm.sum(dim=(0,2,3), keepdim=True) / (mask.sum(dim=(0,2,3), keepdim=True)+eps)

    return margin