RNTN

The Recursive Neural Tensor Network (RNTN) was state of the art for sentiment analysis in 2013.

This is an old (from 2015, before TensorFlow and Torch) GPU-implementation of RNTN described by Socher et al (2013) in Recursive Deep Models for Semantic Compositionality Over a Sentiment Treebank.

The model is trained using the Stanford Sentiment Treebank. Download extract extract train.txt and vocabulary.txt to ./data/. RNTN.py loads and trains the model.

Installation

The only dependencies are PyCUDA and NumPy.

Gradient derivations

Notation

• $d$ - Length of word vector
• $n$ - Node/layer
• $x$ - Activation/output of neuron $(x \in \mathbb{R}^{d}$; $\tanh z)$
• $z$ - Input to neuron $(z \in \mathbb{R}^{d}$; $z = Wx)$
• $t$ - Target vector $(t \in \mathbb{R}^5$; 0-1 coded)
• $y$ - Prediction $(y \in \mathbb{R}^5$; output of softmax layer - $softmax(z))$
• $W_s$ - Classification matrix $(W_s \in \mathbb{R}^{5 \times d})$
• $W$ - Weight matrix $(W \in \mathbb{R}^{d \times 2d})$
• $V$ - Weight tensor $(V^{1:d} \in \mathbb{R}^{2d \times 2d \times d} )$
• $L$ - Word embedding matrix $(L \in \mathbb{R}^{d \times |V|}$, $|V|$ is the size of the vocabulary)
• $\theta$ - All weight parameters $(\theta = (W_s, W, V, L))$
• $E$ - The cost as a function of $\theta$
• $\delta_l$ - Error going to the left child node $(\delta_r$ error to the right child node)

Softmax

$$y_{i} = \frac{e^{z_i}}{\sum\limits_{j}e^{z_j}}$$

$$\frac{\partial y_i}{\partial z_j} = y_{i}(\delta_{ij} - y_{j})$$

$\delta_{ij}$ is the Kronecker's delta:

$$ \delta_{ij} = \begin{cases} 0 &\text{if } i \neq j, \\ 1 &\text{if } i=j. \end{cases} $$

Cost function $E$

$$ E(\theta) = - \sum\limits_{i}\sum\limits_{j}{t_{j}^{i} \log{y_{j}^{i}} + \lambda||\theta||^2} $$ $$ \frac{\partial E}{\partial y_j} = \frac{t_j}{y_j} $$

Activation function

$$ x_i = \tanh{z_i} $$ $$ \frac{\partial x_i}{\partial z_i} = 1 - \tanh^2{z_i} $$

Derivative of $E$ w.r.t. the sentiment classification matrix $W_s$

$$ \frac{\partial E}{\partial W_s} = \sum\limits_{k}\frac{\partial E}{\partial y_k}{\frac{\partial y_k}{\partial z^{s}}}{\frac{\partial z^{s}}{\partial W_{s}}} $$

Derivative of the cost function:

$$ \frac{\partial E}{\partial y} = \frac{t}{y} $$

Derivative of the $softmax$ function:

$$ \frac{\partial y_k}{\partial z^{s}{i}} = y{i}(\delta_{ik} - y_{k}) $$

Derivative of the input:

$$ \frac{\partial z^{s}}{\partial W_s} = x $$

Combined:

$$ \begin{split} \frac{\partial E}{\partial W_s} = \sum\limits_{k}\frac{t_k}{y_k}y_{k}(\delta_{ik} - y_{i})x_j \\ = x_j \sum\limits_{k}{t_k (\delta_{ik}-y_i)} \\ = x_j(y_i - t_i) \end{split} $$

Derivative of $E$ w.r.t. the weight matrix $W$

For one training sentence:

$$ \frac{\partial E}{\partial W} = \sum\limits_{k}\frac{\partial E}{\partial y_k} \frac{\partial y_k}{\partial z_{s}} \frac{\partial z_{s}}{\partial x} \frac{\partial x}{\partial z} \frac{\partial z}{\partial W} $$

Derivative of input to $node_n$ w.r.t. activation of $node_{n-1}$:

$$ \frac{\partial z}{\partial x} = W $$

Derivative of a node's activation w.r.t. its input:

$$ \begin{split} \frac{\partial x}{\partial z} = 1 - \tanh^2z \\ f'(x) = 1 - x^2 \\ f' \bigg( \bigg[ \begin{array}{c} x^l \ x^r \end{array} \bigg] \bigg) = 1 - \bigg[ \begin{array}{c} x^l \ x^r \end{array} \bigg] \otimes \bigg[ \begin{array}{c} x^l \ x^r \end{array} \bigg] \end{split} $$

Derivative of a node's input w.r.t. its weight matrix $W$:

$$ \frac{\partial z}{\partial W} = x $$

Combined:

$$ \begin{split} \delta^s = W_s{^T}(y - t) \otimes f'(x_n) \\ \frac{\partial E}{\partial W} = W^T \delta^s \otimes f' \bigg( \bigg[ \begin{array}{c} x_{n-1}^l \ x_{n-1}^r \end{array} \bigg] \bigg) \bigg[ \begin{array}{c} x_{n-1}^l \ x^{r}{_{n-1}} \end{array} \bigg]^T\\ \end{split} $$

Derivative of $E$ w.r.t. the slice $k$ of the tensor layer $V^{[k]}$

Top node $(node_n)$:

$$ \begin{split} \delta^s = W_s{^T}(y - t) \otimes (1 - x{n}^2) \ \frac{\partial E_n}{\partial V^{[k]}} = \delta^s{k} \bigg[ \begin{array}{c} x^l{{n-1}} \ x^r{{n-1}} \end{array} \bigg] \bigg[ \begin{array}{c} x_{n-1}^l \ x^{r}{_{n-1}} \end{array} \bigg]^T \ \end{split} $$

Left child node $(node_{n-1})$:

$$ \begin{split} \delta_{n} = \delta^{s,n} \ \delta^{n-1}{k} = \big( W^T \delta^n + S \big) \otimes f' \bigg( \bigg[ \begin{array}{c} x^l{n-1}\ x^r_{n-1} \end{array} \bigg] \bigg) \ S = \sum\limits_{k = 1}^d \delta^n \bigg( V^{[k]} + \big(V^{[k]})^T \bigg) \bigg[ \begin{array}{c} x^l_{n-1}\ x^r_{n-1} \end{array} \bigg] \ \delta^{n-1}l = \delta_l^{s,n-1} + \delta^{n-1}[1:d] \ \frac{\partial E{n-1}}{\partial V^{[k]}} = \frac{\partial E_n}{\partial V^{[k]}} + \delta^{n-1}l \bigg[ \begin{array}{c} x{n-2}^l \ x^{r}{_{n-2}} \end{array} \bigg]^T \end{split} $$

Reference: R. Socher, A. Perelygin, J.Y. Wu, J. Chuang, C.D. Manning, A.Y. Ng and C. Potts. 2013. Recursive Deep Models for Semantic Compositionality Over a Sentiment Treebank. In EMNLP.