AI/机器学习常用公式的LaTex代码汇总

在写AI/机器学习相关的论文或者博客的时候经常需要用到LaTex的公式，然而作为资深“伸手党”的我在网上搜索的时候，居然没有找到相关现成资源@-@

那么，我就把自己经常会遇到的公式整理如下，以NLP和一些通用指标函数为主。有需要的可以自取，当然发现有问题或者遗漏的也欢迎指正和补充。

Classical ML Equations in LaTeX

A collection of classical ML equations in Latex . Some of them are provided with simple notes and paper link. Hopes to help writings such as papers and blogs.

Better viewed at https://blmoistawinde.github.io/ml_equations_latex/

• Classical ML Equations in LaTeX
  • Model
    • RNNs(LSTM, GRU)
    • Attentional Seq2seq
      • Bahdanau Attention
      • Luong(Dot-Product) Attention
    • Transformer
      • Scaled Dot-Product attention
      • Multi-head attention
    • Generative Adversarial Networks(GAN)
      • Minmax game objective
    • Variational Auto-Encoder(VAE)
      • Reparameterization trick
  • Activations
    • Sigmoid
    • Softmax
    • Relu
  • Loss
    • Regression
      • Mean Absolute Error(MAE)
      • Mean Squared Error(MSE)
      • Huber loss
    • Classification
      • Cross Entropy
      • Negative Loglikelihood
      • Hinge loss
      • KL/JS divergence
    • Regularization
      • L1 regularization
      • L2 regularization
  • Metrics
    • Classification
      • Accuracy, Precision, Recall, F1
      • Sensitivity, Specificity and AUC
    • Regression
    • Clustering
      • (Normalized) Mutual Information (NMI)
    • Ranking
      • (Mean) Average Precision(MAP)
    • Similarity/Relevance
      • Cosine
      • Jaccard
      • Pointwise Mutual Information(PMI)
  • Notes
  • Reference

Model

RNNs(LSTM, GRU)

encoder hidden state hth_tht​ at time step ttt ht=RNNenc(xt,ht−1)h_t = RNN_{enc}(x_t, h_{t-1})ht​=RNNenc​(xt​,ht−1​)

decoder hidden state sts_tst​ at time step ttt

st=RNNdec(yt,st−1)s_t = RNN_{dec}(y_t, s_{t-1})st​=RNNdec​(yt​,st−1​)

h_t = RNN_{enc}(x_t, h_{t-1})
s_t = RNN_{dec}(y_t, s_{t-1})

The RNNencRNN_{enc}RNNenc​, RNNdecRNN_{dec}RNNdec​ are usually either

• LSTM (paper: Long short-term memory)
• GRU (paper: Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation).

Attentional Seq2seq

The attention weight αij\alpha_{ij}αij​, the iiith decoder step over the jjjth encoder step, resulting in context vector cic_ici​

ci=∑j=1Txαijhjc_i = \sum_{j=1}^{T_x} \alpha_{ij}h_jci​=j=1∑Tx​​αij​hj​

αij=exp⁡(eij)∑k=1Txexp⁡(eik) \alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k=1}^{T_x} \exp(e_{ik})} αij​=∑k=1Tx​​exp(eik​)exp(eij​)​

eik=a(si−1,hj) e_{ik} = a(s_{i-1}, h_j) eik​=a(si−1​,hj​)

c_i = \sum_{j=1}^{T_x} \alpha_{ij}h_j

\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k=1}^{T_x} \exp(e_{ik})}

e_{ik} = a(s_{i-1}, h_j)

aaa is an specific attention function, which can be

Bahdanau Attention

Paper: Neural Machine Translation by Jointly Learning to Align and Translate

eik=vTtanh(W[si−1;hj])e_{ik} = v^T tanh(W[s_{i-1}; h_j])eik​=vTtanh(W[si−1​;hj​])

e_{ik} = v^T tanh(W[s_{i-1}; h_j])

Luong(Dot-Product) Attention

Paper: Effective Approaches to Attention-based Neural Machine Translation

If sis_isi​ and hjh_jhj​ has same number of dimension.

eik=si−1Thje_{ik} = s_{i-1}^T h_jeik​=si−1T​hj​

otherwise

eik=si−1TWhje_{ik} = s_{i-1}^T W h_jeik​=si−1T​Whj​

e_{ik} = s_{i-1}^T h_j

e_{ik} = s_{i-1}^T W h_j

Finally, the output oio_ioi​ is produced by:

st=tanh(W[st−1;yt;ct])s_t = tanh(W[s_{t-1};y_t;c_t])st​=tanh(W[st−1​;yt​;ct​]) ot=softmax(Vst)o_t = softmax(Vs_t)ot​=softmax(Vst​)

s_t = tanh(W[s_{t-1};y_t;c_t])
o_t = softmax(Vs_t)

Transformer

Paper: Attention Is All You Need

Scaled Dot-Product attention

Attention(Q,K,V)=softmax(QKTdk)V Attention(Q, K, V) = softmax(\frac{QK^T}{\sqrt{d_k}})VAttention(Q,K,V)=softmax(dk​​QKT​)V

Attention(Q, K, V) = softmax(\frac{QK^T}{\sqrt{d_k}})V

where dk\sqrt{d_k}dk​​ is the dimension of the key vector kkk and query vector qqq .

Multi-head attention

MultiHead(Q,K,V)=Concat(head1,...,headh)WOMultiHead(Q, K, V) = Concat(head_1, ..., head_h)W^OMultiHead(Q,K,V)=Concat(head1​,...,headh​)WO

where headi=Attention(QWiQ,KWiK,VWiV) head_i = Attention(Q W^Q_i, K W^K_i, V W^V_i) headi​=Attention(QWiQ​,KWiK​,VWiV​)

MultiHead(Q, K, V) = Concat(head_1, ..., head_h)W^O

head_i = Attention(Q W^Q_i, K W^K_i, V W^V_i)

Generative Adversarial Networks(GAN)

Paper: Generative Adversarial Networks

Minmax game objective

min⁡Gmax⁡DEx∼pdata(x)[log⁡D(x)]+Ez∼pgenerated(z)[1−log⁡D(G(z))] \min_{G}\max_{D}\mathbb{E}_{x\sim p_{\text{data}}(x)}[\log{D(x)}] + \mathbb{E}_{z\sim p_{\text{generated}}(z)}[1 - \log{D(G(z))}] Gmin​Dmax​Ex∼pdata​(x)​[logD(x)]+Ez∼pgenerated​(z)​[1−logD(G(z))]

\min_{G}\max_{D}\mathbb{E}_{x\sim p_{\text{data}}(x)}[\log{D(x)}] +  \mathbb{E}_{z\sim p_{\text{generated}}(z)}[1 - \log{D(G(z))}]

Variational Auto-Encoder(VAE)

Paper: Auto-Encoding Variational Bayes

Reparameterization trick

To produce a latent variable z such that z∼qμ,σ(z)=N(μ,σ2)z \sim q_{\mu, \sigma}(z) = \mathcal{N}(\mu, \sigma^2)z∼qμ,σ​(z)=N(μ,σ2), we sample ϵ∼N(0,1)\epsilon \sim \mathcal{N}(0,1)ϵ∼N(0,1), than z is produced by

z=μ+ϵ⋅σz = \mu + \epsilon \cdot \sigmaz=μ+ϵ⋅σ

z \sim q_{\mu, \sigma}(z) = \mathcal{N}(\mu, \sigma^2)
\epsilon \sim \mathcal{N}(0,1)
z = \mu + \epsilon \cdot \sigma

Above is for 1-D case. For a multi-dimensional (vector) case we use:

ϵ⃗∼N(0,I) \vec{\epsilon} \sim \mathcal{N}(0, \textbf{I}) ϵ∼N(0,I)

z⃗∼N(μ⃗,σ2I) \vec{z} \sim \mathcal{N}(\vec{\mu}, \sigma^2 \textbf{I}) z∼N(μ​,σ2I)

\epsilon \sim \mathcal{N}(0, \textbf{I})
\vec{z} \sim \mathcal{N}(\vec{\mu}, \sigma^2 \textbf{I})

Activations

Sigmoid

Related to Logistic Regression. For single-label/multi-label binary classification.

σ(z)=11+e−z\sigma(z) = \frac{1} {1 + e^{-z}}σ(z)=1+e−z1​

\sigma(z) = \frac{1} {1 + e^{-z}}

Softmax

For multi-class single label classification.

σ(zi)=ezi∑j=1Kezj for i=1,2,…,K\sigma(z_i) = \frac{e^{z_{i}}}{\sum_{j=1}^K e^{z_{j}}} \ \ \ for\ i=1,2,\dots,Kσ(zi​)=∑j=1K​ezj​ezi​​ for i=1,2,…,K

\sigma(z_i) = \frac{e^{z_{i}}}{\sum_{j=1}^K e^{z_{j}}} \ \ \ for\ i=1,2,\dots,K

Relu

Relu(z)=max(0,z)Relu(z) = max(0, z)Relu(z)=max(0,z)

Relu(z) = max(0, z)

Loss

Regression

Below xxx and yyy are DDD dimensional vectors, and xix_ixi​ denotes the value on the iiith dimension of xxx.

Mean Absolute Error(MAE)

∑i=1D∣xi−yi∣\sum_{i=1}^{D}|x_i-y_i|i=1∑D​∣xi​−yi​∣

\sum_{i=1}^{D}|x_i-y_i|

Mean Squared Error(MSE)

∑i=1D(xi−yi)2\sum_{i=1}^{D}(x_i-y_i)^2i=1∑D​(xi​−yi​)2

\sum_{i=1}^{D}(x_i-y_i)^2

Huber loss

It’s less sensitive to outliers than the MSE as it treats error as square only inside an interval.

Lδ={12(y−y^)2if∣(y−y^)∣<δδ((y−y^)−12δ)otherwise L_{\delta}= \left\{\begin{matrix} \frac{1}{2}(y - \hat{y})^{2} & if \left | (y - \hat{y}) \right | < \delta\\ \delta ((y - \hat{y}) - \frac1 2 \delta) & otherwise \end{matrix}\right. Lδ​={21​(y−y^​)2δ((y−y^​)−21​δ)​if∣(y−y^​)∣<δotherwise​

L_{\delta}=
    \left\{\begin{matrix}
        \frac{1}{2}(y - \hat{y})^{2} & if \left | (y - \hat{y})  \right | < \delta\\
        \delta ((y - \hat{y}) - \frac1 2 \delta) & otherwise
    \end{matrix}\right.

Classification

Cross Entropy

• In binary classification, where the number of classes MMM equals 2, Binary Cross-Entropy(BCE) can be calculated as:

−(ylog⁡(p)+(1−y)log⁡(1−p))-{(y\log(p) + (1 - y)\log(1 - p))}−(ylog(p)+(1−y)log(1−p))

• If M>2M > 2M>2 (i.e. multiclass classification), we calculate a separate loss for each class label per observation and sum the result.

−∑c=1Myo,clog⁡(po,c)-\sum_{c=1}^My_{o,c}\log(p_{o,c})−c=1∑M​yo,c​log(po,c​)

-{(y\log(p) + (1 - y)\log(1 - p))}

-\sum_{c=1}^My_{o,c}\log(p_{o,c})

M - number of classes log - the natural log y - binary indicator (0 or 1) if class label c is the correct classification for observation o p - predicted probability observation o is of class c

Negative Loglikelihood

NLL(y)=−log⁡(p(y))NLL(y) = -{\log(p(y))}NLL(y)=−log(p(y))

Minimizing negative loglikelihood

min⁡θ∑y−log⁡(p(y;θ))\min_{\theta} \sum_y {-\log(p(y;\theta))}θmin​y∑​−log(p(y;θ))

is equivalent to Maximum Likelihood Estimation(MLE).

max⁡θ∏yp(y;θ)\max_{\theta} \prod_y p(y;\theta)θmax​y∏​p(y;θ)

Here p(y)p(y)p(y) is a scaler instead of vector. It is the value of the single dimension where the ground truth yyy lies. It is thus equivalent to cross entropy (See wiki).\

NLL(y) = -{\log(p(y))}

\min_{\theta} \sum_y {-\log(p(y;\theta))}

\max_{\theta} \prod_y p(y;\theta)

Hinge loss

Used in Support Vector Machine(SVM).

max(0,1−y⋅y^)max(0, 1 - y \cdot \hat{y})max(0,1−y⋅y^​)

max(0, 1 - y \cdot \hat{y})

KL/JS divergence

KL(y^∣∣y)=∑c=1My^clog⁡y^cycKL(\hat{y} || y) = \sum_{c=1}^{M}\hat{y}_c \log{\frac{\hat{y}_c}{y_c}}KL(y^​∣∣y)=c=1∑M​y^​c​logyc​y^​c​​

JS(y^∣∣y)=12(KL(y∣∣y+y^2)+KL(y^∣∣y+y^2))JS(\hat{y} || y) = \frac{1}{2}(KL(y||\frac{y+\hat{y}}{2}) + KL(\hat{y}||\frac{y+\hat{y}}{2}))JS(y^​∣∣y)=21​(KL(y∣∣2y+y^​​)+KL(y^​∣∣2y+y^​​))

KL(\hat{y} || y) = \sum_{c=1}^{M}\hat{y}_c \log{\frac{\hat{y}_c}{y_c}}

v

Regularization

The ErrorErrorError below can be any of the above loss.

L1 regularization

A regression model that uses L1 regularization technique is called Lasso Regression.

Loss=Error(Y−Y^)+λ∑1n∣wi∣Loss = Error(Y - \widehat{Y}) + \lambda \sum_1^n |w_i|Loss=Error(Y−Y)+λ1∑n​∣wi​∣

Loss = Error(Y - \widehat{Y}) + \lambda \sum_1^n |w_i|

L2 regularization

A regression model that uses L1 regularization technique is called Ridge Regression.

Loss=Error(Y−Y^)+λ∑1nwi2Loss = Error(Y - \widehat{Y}) + \lambda \sum_1^n w_i^{2}Loss=Error(Y−Y)+λ1∑n​wi2​

Loss = Error(Y - \widehat{Y}) +  \lambda \sum_1^n w_i^{2}

Metrics

Some of them overlaps with loss, like MAE, KL-divergence.

Classification

Accuracy, Precision, Recall, F1

Accuracy=TP+TFTP+TF+FP+FNAccuracy = \frac{TP+TF}{TP+TF+FP+FN}Accuracy=TP+TF+FP+FNTP+TF​

Precision=TPTP+FPPrecision = \frac{TP}{TP+FP}Precision=TP+FPTP​

Recall=TPTP+FNRecall = \frac{TP}{TP+FN}Recall=TP+FNTP​

F1=2∗Precision∗RecallPrecision+Recall=2∗TP2∗TP+FP+FNF1 = \frac{2*Precision*Recall}{Precision+Recall} = \frac{2*TP}{2*TP+FP+FN}F1=Precision+Recall2∗Precision∗Recall​=2∗TP+FP+FN2∗TP​

Accuracy = \frac{TP+TF}{TP+TF+FP+FN}
Precision = \frac{TP}{TP+FP}
Recall = \frac{TP}{TP+FN}
F1 = \frac{2*Precision*Recall}{Precision+Recall} = \frac{2*TP}{2*TP+FP+FN}

Sensitivity, Specificity and AUC

Sensitivity=Recall=TPTP+FNSensitivity = Recall = \frac{TP}{TP+FN}Sensitivity=Recall=TP+FNTP​

Specificity=TNFP+TNSpecificity = \frac{TN}{FP+TN}Specificity=FP+TNTN​

Sensitivity = Recall = \frac{TP}{TP+FN}
Specificity = \frac{TN}{FP+TN}

AUC is calculated as the Area Under the SensitivitySensitivitySensitivity(TPR)-(1−Specificity)(1-Specificity)(1−Specificity)(FPR) Curve.

Regression

MAE, MSE, equation above.

Clustering

(Normalized) Mutual Information (NMI)

The Mutual Information is a measure of the similarity between two labels of the same data. Where ∣Ui∣|U_i|∣Ui​∣ is the number of the samples in cluster UiU_iUi​ and ∣Vi∣|V_i|∣Vi​∣ is the number of the samples in cluster ViV_iVi​ , the Mutual Information between cluster UUU and VVV is given as:

MI(U,V)=∑i=1∣U∣∑j=1∣V∣∣Ui∩Vj∣Nlog⁡N∣Ui∩Vj∣∣Ui∣∣Vj∣ MI(U,V)=\sum_{i=1}^{|U|} \sum_{j=1}^{|V|} \frac{|U_i\cap V_j|}{N} \log\frac{N|U_i \cap V_j|}{|U_i||V_j|} MI(U,V)=i=1∑∣U∣​j=1∑∣V∣​N∣Ui​∩Vj​∣​log∣Ui​∣∣Vj​∣N∣Ui​∩Vj​∣​

MI(U,V)=\sum_{i=1}^{|U|} \sum_{j=1}^{|V|} \frac{|U_i\cap V_j|}{N}
\log\frac{N|U_i \cap V_j|}{|U_i||V_j|}

Normalized Mutual Information (NMI) is a normalization of the Mutual Information (MI) score to scale the results between 0 (no mutual information) and 1 (perfect correlation). In this function, mutual information is normalized by some generalized mean of H(labels_true) and H(labels_pred)), See wiki.

Skip RI, ARI for complexity.

Also skip metrics for related tasks (e.g. modularity for community detection[graph clustering], coherence score for topic modeling[soft clustering]).

Ranking

Skip nDCG (Normalized Discounted Cumulative Gain) for its complexity.

(Mean) Average Precision(MAP)

Average Precision is calculated as:

AP=∑n(Rn−Rn−1)Pn\text{AP} = \sum_n (R_n - R_{n-1}) P_nAP=n∑​(Rn​−Rn−1​)Pn​

\text{AP} = \sum_n (R_n - R_{n-1}) P_n

where RnR_nRn​ and PnP_nPn​ are the precision and recall at the nnnth threshold,

MAP is the mean of AP over all the queries.

Similarity/Relevance

Cosine

Cosine(x,y)=x⋅y∣x∣∣y∣Cosine(x,y) = \frac{x \cdot y}{|x||y|}Cosine(x,y)=∣x∣∣y∣x⋅y​

Cosine(x,y) = \frac{x \cdot y}{|x||y|}

Jaccard

Similarity of two sets UUU and VVV.

Jaccard(U,V)=∣U∩V∣∣U∪V∣Jaccard(U,V) = \frac{|U \cap V|}{|U \cup V|}Jaccard(U,V)=∣U∪V∣∣U∩V∣​

Jaccard(U,V) = \frac{|U \cap V|}{|U \cup V|}

Pointwise Mutual Information(PMI)

Relevance of two events xxx and yyy.

PMI(x;y)=log⁡p(x,y)p(x)p(y)PMI(x;y) = \log{\frac{p(x,y)}{p(x)p(y)}}PMI(x;y)=logp(x)p(y)p(x,y)​

PMI(x;y) = \log{\frac{p(x,y)}{p(x)p(y)}}

For example, p(x)p(x)p(x) and p(y)p(y)p(y) is the frequency of word xxx and yyy appearing in corpus and p(x,y)p(x,y)p(x,y) is the frequency of the co-occurrence of the two.

Notes

This repository now only contains simple equations for ML. They are mainly about deep learning and NLP now due to personal research interests.

For time issues, elegant equations in traditional ML approaches like SVM, SVD, PCA, LDA are not included yet.

Moreover, there is a trend towards more complex metrics, which have to be calculated with complicated program (e.g. BLEU, ROUGE, METEOR), iterative algorithms (e.g. PageRank), optimization (e.g. Earth Mover Distance), or even learning based (e.g. BERTScore). They thus cannot be described using simple equations.

Reference

Pytorch Documentation

Scikit-learn Documentation

Machine Learning Glossary

Wikipedia

https://blog.floydhub.com/gans-story-so-far/

https://ermongroup.github.io/cs228-notes/extras/vae/

Thanks for a-rodin’s solution to show Latex in Github markdown, which I have wrapped into latex2pic.py.