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Going back to our math-y definitions, we see that it basically fits in to the same framework, except we have q n-long vectors going into f and coming out of g . So q n -long vectors go in to f , and a single m -long vector comes out. We then give this m -long vector back to g , which spits out q n -long vectors.

That was a lot of letters, but you get the idea (I hope).

Like much of deep learning, the concept itself is pretty simple, but the implications are pretty cool. We can take any sequence — a variable-length sequence, mind you — and convert it into a fixed-size vector. And then convert that back to a variable-length sequence.

It turns out this model is actually incredibly powerful, so let’s take a look at one particularly useful (and successful) application: machine translation.

Let’s take these ideas we just learned about sequence-to-sequence (or seq2seq, for short) RNNs and apply them to machine translation. We throw in a sequence of words in one language, and it outputs a sequence of words in another. Simple enough, right?

The model we’re going to look at specifically is Google’s implementation of NMT. You can read all the gory details Carolbar Fashion Womens Bows Pointed Toe Patent Leather Sexy Evenings High Stiletto Heel Dress Shoes Red QeuOh
, but for now why don’t I give you the watered-down version.

At it’s core, the GNMT architecture is just another seq2seq model. We have an encoder, consisting of 8 LSTM layers with skip connections (the first layer is bidirectional). We also have a decoder, once again containing 8 LSTM layers with skip connections. (A skip connection in a neural network is a connection which skips a layer and connects to the next available layer.) The decoder network outputs a probability distribution of words (well, sort of — we’ll talk more about that later), which we sample from to get our [translated] sentence. 🎉

Here’s a scary diagram from the paper:

But there are a few other aspects to the GNMT that are important to note (there’s actually lots of interesting stuff going on in this architecture, so I really recommend you do read the paper ).

Let’s turn our attention to the center of the above diagram. This is a critical part of the GNMT architecture (and GNMT is certainly not the first to use attention) which allows the decoder to focus on certain parts of the encoder’s output as it produces output. Specifically, the GNMT architecture differs from the traditional seq2seq model in that our encoder does not produce a single fixed-width vector (the final hidden state) representing the entire output. Instead, we actually look at the output from each time step, and each time step gives us some latent representation. While decoding, we combine all of these hidden vectors into one context vector using something called soft attention .

$$T\bigl(r, f(z+c)\bigr) = T(r, f) + O\bigl(r^{\rho-1 + \varepsilon}\bigr) + O(\log r). $$

Lemma 4

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])

\(g:(0,+\infty)\rightarrow{R}\) , \(h:(0,+\infty)\rightarrow{R}\) \(g(r)\leq h(r)\) . , \(\alpha>1\) , \(r_{0}>0\) \(g(r)\leq h(\alpha r)\) \(r_{0}\) .

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$$\begin{aligned} kn\bigl(\mu r^{k},a,f\bigr)\leq n\bigl(r,a,f\bigl(p(z)\bigr)\bigr) \leq kn\bigl(\lambda r^{k}, a, f\bigr), \\ N\bigl(\mu r^{k},a,f\bigr)+O(\log r)\leq N\bigl(r,a,f\bigl(p(z)\bigr) \bigr)\leq N\bigl(\lambda r^{k}, a, f\bigr)+O(\log r), \\ (1-\varepsilon)T\bigl(\mu r^{k},f\bigr)\leq T\bigl(r,f\bigl(p(z) \bigr)\bigr)\leq (1+\varepsilon)T\bigl(\lambda r^{k},f\bigr), \end{aligned}$$
$$\begin{aligned} \max\{p,q\}T(r,g) \\ \quad = T \Biggl(r,\sum_{\lambda_{1} \in I_{1}, \mu_{1}\in J_{1}}\alpha_{\lambda_{1}, \mu_{1}}(z) \Biggl(\prod_{\nu=1}^{n}f(z+c_{\nu})^{l_{\lambda_{1}, \nu}} \prod_{\nu=1}^{n}g(z+c_{\nu})^{m_{\mu_{1}, \nu}} \Biggr) \Biggr) + S(r,g) \\ \quad \leq \sum_{\nu=1}^{n} \xi_{1,\nu}T\bigl(r, f(z+c_{\nu})\bigr) + \sum _{\nu =1}^{n} \eta_{1,\nu}T\bigl(r, g(z+c_{\nu})\bigr)+S(r,f)+ S(r,g) \\ \quad = \sum_{\nu=1}^{n} \xi_{1,\nu}T \bigl(r, f(z)\bigr) + O\bigl(r^{\rho(f) -1 + \varepsilon}\bigr) + \sum _{\nu=1}^{n} \eta_{1,\nu}T\bigl(r, g(z)\bigr)+O \bigl(r^{\rho(g) -1 + \varepsilon}\bigr) \\ \qquad {} +O(\log r) + S(r,f)+ S(r,g) \\ \quad = \Biggl(\sum_{\nu=1}^{n} \xi_{1,\nu} \Biggr)T\bigl(r, f(z)\bigr) + \Biggl(\sum _{\nu =1}^{n} \eta_{1,\nu} \Biggr)T\bigl(r, g(z) \bigr) \\ \qquad {} +O\bigl(r^{\rho(g) -1 + \varepsilon}\bigr) + O(\log r) + S(r,f)+ S(r,g) \\ \quad = \sigma_{11}T\bigl(r, f(z)\bigr) + \sigma_{12}T \bigl(r, g(z)\bigr)+ O\bigl(r^{\rho(f) -1 + \varepsilon}\bigr)+O\bigl(r^{\rho(g) -1 + \varepsilon}\bigr) \\ \qquad {}+ O(\log r) + S(r,f)+ S(r,g). \end{aligned}$$
(9)
$$\begin{aligned} \bigl(\max\{p,q\}-\sigma_{12} \bigr)T(r,g) \\ \quad \leq \sigma_{11}T\bigl(r, f(z)\bigr)+ O\bigl(r^{\rho(f) -1 + \varepsilon} \bigr)+O\bigl(r^{\rho(g) -1 + \varepsilon}\bigr) \\ \qquad {} + O(\log r) + S(r,f)+ S(r,g). \end{aligned}$$
(10)
$$\begin{aligned} T(r,g) \leq\frac{\sigma_{11}}{\max\{p,q\}-\sigma_{12}}T\bigl(r, f(z)\bigr) + O \bigl(r^{\rho(f) -1 + \varepsilon}\bigr)+O\bigl(r^{\rho(g) -1 + \varepsilon}\bigr) \\ {} + O(\log r) + S(r,f)+ S(r,g). \end{aligned}$$
(11)
$$\begin{aligned} \max\{s,t\}T(r,f) \\ \quad = T \Biggl(r,\sum_{\lambda_{2} \in I_{2}, \mu_{2}\in J_{2}}\beta_{\lambda_{2}, \mu_{2}}(z) \Biggl(\prod_{\nu =1}^{n}f(z+c_{\nu})^{l_{\lambda_{2}, \nu}} \prod_{\nu=1}^{n}g(z+c_{\nu})^{m_{\mu_{2}, \nu}} \Biggr) \Biggr) + S(r,f) \\ \quad \leq \sum_{\nu=1}^{n} \xi_{2,\nu}T\bigl(r, f(z+c_{\nu})\bigr) + \sum _{\nu =1}^{n} \eta_{2,\nu}T\bigl(r, g(z+c_{\nu})\bigr)+S(r,f)+ S(r,g) \\ \quad = \sum_{\nu=1}^{n} \xi_{2,\nu}T \bigl(r, f(z)\bigr) + O\bigl(r^{\rho(f) -1 + \varepsilon}\bigr) + \sum _{\nu=1}^{n} \eta_{2,\nu}T\bigl(r, g(z)\bigr) \\ \qquad {} + O\bigl(r^{\rho(g) -1 + \varepsilon}\bigr) +O(\log r) + S(r,f)+ S(r,g) \\ \quad = \Biggl(\sum_{\nu=1}^{n} \xi_{2,\nu} \Biggr)T\bigl(r, f(z)\bigr) + \Biggl(\sum _{\nu =1}^{n} \eta_{2,\nu} \Biggr)T\bigl(r, g(z) \bigr) \\ \qquad {} + O(\log r) + S(r,f)+ S(r,g) \\ \quad = \sigma_{21}T\bigl(r, f(z)\bigr) + \sigma_{22}T \bigl(r, g(z)\bigr)+ O\bigl(r^{\rho(f) -1 + \varepsilon}\bigr)+O\bigl(r^{\rho(g) -1 + \varepsilon}\bigr) \\ \qquad {}+ O(\log r) + S(r,f)+ S(r,g). \end{aligned}$$
(12)
$$\begin{aligned} \bigl(\max\{s,t\}-\sigma_{21} \bigr)T(r,f) \\ \quad \leq \sigma_{22}T\bigl(r, g(z)\bigr)+ O\bigl(r^{\rho(f) -1 + \varepsilon} \bigr)+O\bigl(r^{\rho(g) -1 + \varepsilon}\bigr) \\ \qquad {}+ O(\log r) + S(r,f)+ S(r,g) \end{aligned}$$
(13)
$$\begin{aligned} T(r,f) \leq \frac{\sigma_{22}}{\max\{s,t\}-\sigma_{21}}T\bigl(r, g(z)\bigr) + O \bigl(r^{\rho(f) -1 + \varepsilon}\bigr)+O\bigl(r^{\rho(g) -1 + \varepsilon}\bigr) \\ {} + O(\log r) + S(r,f)+ S(r,g). \end{aligned}$$
(14)

Using ( 11 ), we can obtain \(\rho(g)\leq\rho(f)\) . Similarly, we can get \(\rho(f)\leq\rho (g)\) from ( 14 ). Therefore, we have \(\rho(f)=\rho(g)\) .

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