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<time class="post-full-meta-date" datetime=" 3 January 2022"> 3 January 2022</time>
<span class="date-divider">/</span>
<a href='/tag/cs231n/'>CS231N</a>
</section>
<h1 class="post-full-title">cs231n - Lecture 10. Recurrent Neural Networks</h1>
</header>
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<div class="kg-card-markdown">
<h2 id="rnn-process-sequences">RNN: Process Sequences</h2>
<p><img src="/assets/images/cs231n_lec10_0.png" alt="png" width="80%", height="80%" /></p>
<ul>
<li>one to one; vanilla neural networks</li>
<li>one to many; e.g. Image Captioning(image to sequence of words)</li>
<li>many to one; e.g. Action Prediction(video sequence to action class)</li>
<li>many to many(1); e.g. Video Captioning(video sequence to caption)</li>
<li>
<p>many to many(2); e.g. Video Classification on frame level</p>
</li>
<li>Why existing convnets are insufficient?:<br />
Variable sequence length inputs and outputs</li>
</ul>
<p><img src="/assets/images/cs231n_lec10_1.png" alt="png" width="70%", height="70%" /></p>
<ul>
<li>
<p>Key idea: RNNs have an “internal state” that is updated as a sequence is processed.</p>
</li>
<li>RNN hidden state update:<br />
\(h_t = f_W(h_{t-1}, x_t)\)<br />
The same function and the same set of parameters are used at every time step.</li>
<li>
<p>RNN output generation: \(y_t = f_{W_hy}(h_t)\)</p>
</li>
<li>Simple(Vanilla) RNN: The state consists of a single hidden vector <em>h</em><br />
$h_t = \mbox{tanh}(W_hh h_{t-1} + W_{xh}x_t)$<br />
$y_t = W_{hy}h_t$</li>
</ul>
<h3 id="sequence-to-sequenceseq2seq-many-to-one--one-to-many">Sequence to Sequence(Seq2Seq): Many-to-One + One-to-Many</h3>
<ul>
<li>
<p>Many-to-One: Encode input sequence in a single vector<br />
One-to-Many: Produce output sequence from single input vector<br />
Encoder produces the last hidden state $h_T$ and decoder uses it as a default $h_0$. Weights($W_1, W_2$) are re-used for each procedure.</p>
</li>
<li>
<p>Example: Character-level Language Model Sampling</p>
</li>
</ul>
<p><img src="/assets/images/cs231n_lec10_2.png" alt="png" width="40%", height="40%" /></p>
<h3 id="backpropagation">Backpropagation</h3>
<ul>
<li>Backpropagation through time: Computationally Expensive<br />
Forward through entire sequence to compute loss, then backward through entire sequence to compute gradient.</li>
<li><strong>Truncated</strong> Backpropagation through time:<br />
Run forward and backward through <strong>chunks of the sequence</strong> instead of whole sequence. Carry hidden states forward in time forever, but only backpropagate for some smaller number of steps.</li>
</ul>
<h3 id="rnn-tradeoffs">RNN tradeoffs</h3>
<ul>
<li>RNN Advantages:
<ul>
<li>Can process any length input</li>
<li>Computation for step t can (in theory) use information from many steps back</li>
<li>Model size doesn’t increase for longer input</li>
<li>Same weights applied on every timestep, so there is symmetry in how inputs are processed.</li>
</ul>
</li>
<li>RNN Disadvantages:
<ul>
<li>Recurrent computation is slow</li>
<li>In practice, difficult to access information from many steps back</li>
</ul>
</li>
</ul>
<h3 id="image-captioning-cnn--rnn">Image Captioning: CNN + RNN</h3>
<ul>
<li>Instead of the final FC layer and the classifier in CNN, use FC output <em>v</em>(say 4096 length vector) to formulate the default hidden state $h_0$ in RNN.
<ul>
<li>before: $h = \mbox{tanh}(W_{xh}\ast x+W_{hh}\ast h)$</li>
<li>now: $h=\mbox{tanh}(W_{xh}\ast x + W_{hh}\ast h + W_{ih}\ast v)$</li>
</ul>
</li>
<li>RNN for Image Captioning<br />
Re-sample the previous output $y_{t-1}$ as the next input $x_t$, iterate untill $y_t$ sample takes <code class="language-plaintext highlighter-rouge"><END></code> token.</li>
</ul>
<h3 id="visual-question-answering-rnns-with-attention">Visual Question Answering: RNNs with Attention</h3>
<p><img src="/assets/images/cs231n_lec10_3.png" alt="png" width="80%", height="80%" /></p>
<h3 id="other-tasks">Other tasks</h3>
<ul>
<li>Visual Dialog: Conversations about images</li>
<li>Visual Language Navigation: Go to the living room<br />
Agent encodes instructions in language and uses an RNN to generate a series of movements as the visual input changes after each move.</li>
<li>Visual Question Answering: Dataset Bias<br />
With different types(Image + Question + Answer) of data used, model performances are better.</li>
</ul>
<h3 id="long-short-term-memory-lstm">Long Short Term Memory (LSTM)</h3>
<ul>
<li>Vanilla RNN<br />
\(h_t = \mbox{tanh}(W_{hh}h_{t-1} + W_{xh}x_t) \\
= \mbox{tanh}\left(
(W_{hh} \ W_{hx}) {\begin{pmatrix} h_{t-1} \\ x_t \end{pmatrix}}
\right) \\
= \mbox{tanh}\left(
W {\begin{pmatrix} h_{t-1} \\ x_t \end{pmatrix}}
\right)\)</li>
<li>
<p>\(\frac{\partial h_t}{\partial h_{t-1}} = \mbox{tanh}' (W_{hh}h_{t-1} + W_{xh}x_t)W_{hh}\)<br />
$\frac{\partial L}{\partial W} = \sum_{t=1}^T \frac{\partial L_t}{\partial W}$</p>
\[\begin{align*}
\frac{\partial L_T}{\partial W} &= \frac{\partial L_T}{\partial h_T}
\frac{\partial h_t}{\partial h_{t-1}}\cdots
\frac{\partial h_1}{\partial W} \\
&= \frac{\partial L_T}{\partial h_T}(\prod_{t=2}^T \frac{\partial h_t}{\partial h_{t-1}})\frac{\partial h_1}{\partial W} \\
&= \frac{\partial L_T}{\partial h_T}(\prod_{t=2}^T \mbox{tanh}'(W_{hh}h_{t-1} + W_{xh}x_t))W_{hh}^{T-1} \frac{\partial h_1}{\partial W}
\end{align*}\]
</li>
<li>Problem<br />
As the output of <em>tanh</em> function are in range of <code class="language-plaintext highlighter-rouge">[-1,1]</code> and almost smaller than 1, vanilla RNN has <strong><em>vanishing gradients</em></strong>. If we assume no non-linearity, the gradient will be \(\frac{\partial L_T}{\partial W} = \frac{\partial L_T}{\partial h_T}W_{hh}^{T-1}\frac{\partial h_1}{\partial W}\). In this case, when the largest singular value is greater than 1, we have exploding gradients, while the value is smaller than 1, we have vanishing gradients.</li>
</ul>
<div class="language-python highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="n">H</span> <span class="o">=</span> <span class="mi">5</span> <span class="c1"># dimensionality of hidden state
</span><span class="n">T</span> <span class="o">=</span> <span class="mi">50</span> <span class="c1"># number of time steps
</span><span class="n">Whh</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">H</span><span class="p">)</span>
<span class="c1"># forward pass of an RNN (ignoring inputs x)
</span><span class="n">hs</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">ss</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">hs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
<span class="n">ss</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Whh</span><span class="p">,</span> <span class="n">hs</span><span class="p">[</span><span class="n">t</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="n">hs</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">maximum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ss</span><span class="p">[</span><span class="n">t</span><span class="p">])</span>
<span class="c1"># backward pass of the RNN
</span><span class="n">dhs</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">dss</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">dhs</span><span class="p">[</span><span class="n">T</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">random</span><span class="p">.</span><span class="n">randn</span><span class="p">(</span><span class="n">H</span><span class="p">)</span> <span class="c1">#start off the chain with random gradient
</span><span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">xrange</span><span class="p">(</span><span class="n">T</span><span class="p">)):</span>
<span class="n">dss</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">hs</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">></span> <span class="mi">0</span><span class="p">)</span> <span class="o">*</span> <span class="n">dhs</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="c1"># backprop through the nonlinearity
</span> <span class="n">dhs</span><span class="p">[</span><span class="n">t</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Whh</span><span class="p">.</span><span class="n">T</span><span class="p">,</span> <span class="n">dss</span><span class="p">[</span><span class="n">t</span><span class="p">])</span> <span class="c1"># backprop into previous hidden state
</span> <span class="c1"># "Whh.T" multiplied by "T" times!
</span></code></pre></div></div>
<ul>
<li>
<p>For exploding gradients: control with gradient clipping.<br />
For vanishing gradients: change the architecture, LSTM introduced.</p>
</li>
<li>
<p>LSTM:<br />
\(\begin{pmatrix} i \\ f \\ o \\ g \end{pmatrix} =
\begin{pmatrix} \sigma \\ \sigma \\ \sigma \\ \mbox{tanh}\end{pmatrix} W \begin{pmatrix} h_{t-1} \\ x_t \end{pmatrix}\)<br />
\(c_t = f \odot c_{t-1} + i \odot g\), <em>memory cell update</em><br />
\(h_t = o \odot \mbox{tanh}(c_t)\), <em>hidden state update</em><br />
where <em>W</em> is a stack of $W_h$ and $W_x$</p>
</li>
</ul>
<p><img src="/assets/images/cs231n_lec10_4.png" alt="png" width="60%", height="60%" /><br />
i: Input gate, whether to write to cell<br />
f: Forget gate, Whether to erase cell<br />
o: Output gate, How much to reveal cell<br />
g: Gate gate, How much to write to cell</p>
<p><img src="/assets/images/cs231n_lec10_5.png" alt="png" width="60%", height="60%" /></p>
<ul>
<li>
<p>Backpropagation from $c_t$ to $c_{t-1}$ only elementwise multiplication by <em>f</em>, no matrix multiply by <em>W</em>. Notice that the gradient contains the <em>f</em> gate’s vector of activations; it allows better control of gradients values, using suitable parameter updates of the forget gate. Also notice that are added through the <em>f, i, g,</em> and <em>o</em> gates, we can have better balancing of gradient values.</p>
</li>
<li>
<p>Recall: “PlainNets” vs. ResNets<br />
ResNet is to PlainNet what LSTM is to RNN, kind of.<br />
<em>Additive skip connections</em></p>
</li>
<li>
<p>Do LSTMs solve the vanishing gradient problem?:<br />
The LSTM architecture makes it easier for the RNN to preserve information over many timesteps. e.g. If $f=1$ and $i=0$, then the information of that cell is preserved indefinitely. By contrast, it’s harder for vanilla RNN to learn a recurrent weight matrix $W_h$ that preserves information in hidden state.<br />
LSTM doesn’t guarantee that there is no vanishing/exploding gradient, but it does provide an easier way for the model to learn long-distance dependencies.</p>
</li>
<li>
<p>in between: Highway Networks, <em>Srivastava et al, 2015, [arXiv:1505.00387v2]</em><br />
A new architecture designed to ease gradient-based training of very deep networks. To regulate the flow of information and enlarge the possibility of studying extremely deep and efficient architectures.<br />
$g = T(x, W_T)$, $y = g \odot H(x, W_H) + (1-g)\odot x$</p>
</li>
</ul>
<h3 id="other-rnn-variants">Other RNN Variants</h3>
<ul>
<li>Neural Architecture Search(NAS) with Reinforcement Learning, <em>Zoph et Le, 2017</em>
<ul>
<li>RNN to design model; idea that we can represent the model architecture with a variable-length string.</li>
<li>Apply reinforcement learning on a neural network to maximize the accuracy(as a reward) on validation set, find a good architecture.</li>
</ul>
</li>
<li>GRU; smaller LSTM, <em>“Learning phrase representations using rnn encoder-decoder for statistical machine translation”, Cho et al., 2014</em></li>
<li><em>“An Empirical Exploration of Recurrent Network Architectures”, Jozefowicz et al., 2015</em></li>
<li><em>LSTM: A Search Space Odyssey, Greff et al., 2015</em></li>
</ul>
<h3 id="recurrence-for-vision">Recurrence for Vision</h3>
<ul>
<li>LSTM wer a good default choice until this year</li>
<li>Use variants like GRU if you want faster compute and less parameters</li>
<li>Use transformers (next lecture) as they are dominating NLP models</li>
<li>almost everyday there is a new vision transformer model</li>
</ul>
<h3 id="summary">Summary</h3>
<ul>
<li>RNNs allow a lot of flexibility in architecture design</li>
<li>Vanilla RNNs are simple but don’t work very well</li>
<li>Common to use LSTM or GRU: their additive interactions improve gradient flow</li>
<li>Backward flow of gradients in RNN can explode or vanish. Exploding is controlled with gradient clipping. Vanishing is controlled with additive interactions (LSTM)</li>
<li>Better/simpler architectures are a hot topic of current research, as well as new paradigms for reasoning over sequences</li>
<li>Better understanding (both theoretical and empirical) is needed.</li>
</ul>
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