15^{th}JANUARY2017

I. Goodfellow, Y. Bengio, A. Courville. *Deep Learning*. Chapter 10, MIT Press, 2016.

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In chapter 11, Goodfellow et al. give an introduction to recurrent neural networks as well as corresponding further developments like long short-term memory networks (LSTM). During their discussion they focus on three different schemes (Goodfellow et al. call it "patterns") of recurrent neural networks (of which the first two schemes are illustrated in Figure 1):

Figure 1 (

click to enlarge): Recurrent neural network where the hidden units are propagated through time (left) and where only the output is propagated through time (right). As detailed by Goodfellow et al., the second option represents strictly lower expressiveness in terms of which functions can be modeled.The key idea that makes recurrent neural networks interesting, is that the parameters (i.e. $W$, $U$, $V$ ...) are shared across time, allowing for variable length sequences to be processed. In addition, parameter sharing is important to generalize to unseen examples of different lengths. The general equations corresponding to the first scheme are as follows:

$a^{(t)} = b + Wh^{(t-1)} + U x^{(t)}$

$h^{(t)} = \text{tanh}(a^{(t)})$

$o^{(t)} = c + Vh^{(t)}$

$\hat{y}^{(t)} = \text{softmax}(o^{(t)})$

Relating to Figure 1, on top of the output $o^{(t)}$ a loss $L^{(t)}$ is applied which implicitly performs the softmax operation to compute $\hat{y}^{(t)}$ and computes the loss with regard to the true output $y^{(t)}$.

Recurrent neural networks are generally trained using error backpropagation through time, which describes error backpropagation applied to the individual networks starting from the last time step and going back to the first time step. As the parameters across time are shared, the gradients with respect to the involved parameters represent sums over time. For example, regarding $W$ the gradient is easily derived (by recursively applying the chain rule) as

$\nabla_W L = \sum_t \sum_i \left(\frac{\partial L}{\partial h_i^{(t)}}\right) \nabla_{W^{(t)}} h_i^{(t)}$

when considering the first model of Figure 1.

While the presented recurrent network is shallow - having only one hidden layer - deep recurrent neural networks can add multiple additional layers. Interestingly, one has many options of how these additional layers are connected through time. Goodfellow et al. illustrate this freedom using Figure 2. Note that the black square indicates a time delay of one time step for unfolding (see chapter 10.1 for details) the model.

Figure 2 (

click to enlarge): Examples of possible deep recurrent network architectures.Towards the end of the chapter, Goodfellow et al. focus on learning long-term dependencies. The described problem corresponds to exploding or vanishing gradients (with respect to time) when training recurrent neural networks for long sequences. Beneath simple techniques such as gradient clipping (also see chapter 8), several model modifications are discussed that simplify learning long-term dependencies. Among these models, Goodfellow et al. also discuss long short-term memory (LSTM) models. Other approaches include skip connections and leaky units.