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<h1>NN Optimization</h1>
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<section class="tex2jax_ignore mathjax_ignore" id="nn-optimization">
<h1>NN Optimization<a class="headerlink" href="#nn-optimization" title="Permalink to this headline">#</a></h1>
<ol class="arabic">
<li><p>(NNO) <span class="math notranslate nohighlight">\(y = \text{ReLU}\left(\sum_{i=0}^{1} v_i d_i\right)\)</span> where <span class="math notranslate nohighlight">\(d_i\)</span> is a dropout layer with a dropout probability of 0.2. If <span class="math notranslate nohighlight">\(v_i\)</span> = [-0.4, 0.8] for <span class="math notranslate nohighlight">\(i\)</span> = [0, 1], the expected value <span class="math notranslate nohighlight">\(E\left[ y \right]\)</span> is ___.</p></li>
<li><p>(NNO) A network is to be trained to detect rose from input-target pairs (x,t). The final layer has one neuron with output (h) and final estimate <span class="math notranslate nohighlight">\(P(\text{Rose} | x; \theta) = \sigma(h)\)</span> To train the network, binary cross entropy loss function is used, i.e.,</p>
<div class="math notranslate nohighlight">
\[\mathcal{L}(\theta) = -t \ln P(\text{Rose} | x; \theta) - (1 - t) \ln P(\text{not Rose} | x; \theta).\]</div>
<p>Then <span class="math notranslate nohighlight">\(\frac{\partial \mathcal{L}(\theta)}{\partial \theta}\)</span> is</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\left(\sigma\left(h\right)-t\right)\frac{\partial h}{\partial\theta}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\left(t-\sigma\left(h\right)\right)\frac{\partial h}{\partial\theta}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\left(\frac{t}{\sigma\left(h\right)}-\frac{1-t}{1-\sigma\left(h\right)}\right)\frac{\partial h}{\partial\theta}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\left(\frac{t}{\sigma\left(h\right)}+\frac{1-t}{1-\sigma\left(h\right)}\right) \frac{\partial h}{\partial\theta}\)</span></p></li>
<li><p>None of the above</p></li>
</ul>
</li>
<li><p>(NNO) For a multi-class classification problem, a neural network is given as follows:
$<span class="math notranslate nohighlight">\(h_{i_1} = \sum_{i_0=1}^{2} w_{i_0i_1} x_{i_0}\)</span><span class="math notranslate nohighlight">\(
\)</span><span class="math notranslate nohighlight">\(v_{i_1} = \sigma(h_{i_1})\)</span><span class="math notranslate nohighlight">\(
\)</span><span class="math notranslate nohighlight">\(h_{i_2}' = \sum_{i_1=1}^{3} w_{i_1i_2}' v_{i_1}\)</span><span class="math notranslate nohighlight">\(
\)</span><span class="math notranslate nohighlight">\(y_{i_2} = \text{softmax}(h_{i_2}') \)</span>$
Use categorical cross-entropy loss to compute the following:</p>
<ol class="arabic simple">
<li><p>Write the loss function (E) in terms of the variables defined above and the targets <span class="math notranslate nohighlight">\(t_{i_2}\)</span>.</p></li>
<li><p><span class="math notranslate nohighlight">\(\frac{\partial E}{\partial y_1}\)</span> (i.e., <span class="math notranslate nohighlight">\(i_2\)</span> = 1).</p></li>
<li><p><span class="math notranslate nohighlight">\(\frac{\partial E}{\partial w_{11}'}\)</span> (i.e., <span class="math notranslate nohighlight">\(i_1\)</span> = <span class="math notranslate nohighlight">\(i_2\)</span> = 1).</p></li>
<li><p><span class="math notranslate nohighlight">\(\frac{\partial E}{\partial w_{11}}\)</span> (i.e., <span class="math notranslate nohighlight">\(i_0\)</span> = <span class="math notranslate nohighlight">\(i_1\)</span> = 1).</p></li>
</ol>
</li>
<li><p>(NNO) The empirical risk over data d is written as <span class="math notranslate nohighlight">\(\mathcal{R}_d\)</span>. In the case of overfitting, the following happens during training</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\mathcal{R}_{train}\)</span> does not decrease further</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathcal{R}_{val}\)</span> does not decrease further and becomes constant</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathcal{R}_{train}\)</span> decreases but <span class="math notranslate nohighlight">\(\mathcal{R}_{val}\)</span> increases</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathcal{R}_{train}\)</span> oscillates</p></li>
<li><p><span class="math notranslate nohighlight">\(\mathcal{R}_{train}\)</span> becomes constant due to parameters being stuck in a local minimum</p></li>
</ul>
</li>
<li><p>(NNO) A recurrent neural network is given as</p>
<div class="math notranslate nohighlight">
\[y_i^{\left(t\right)}=\sigma\left(\sum_{j}{u_{ij}x_j^{\left(t\right)}}+\sum_{j}{w_{ij}y_j^{\left(t-1\right)}}+b_i\right)\]</div>
<p>The gradient <span class="math notranslate nohighlight">\(\frac{\partial y_i^{\left(t\right)}}{\partial b_i}\)</span> is given by</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(y_i^{\left(t\right)}\left(1-y_i^{\left(t\right)}\right)\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(y_i^{\left(t\right)}\left(1-y_i^{\left(t\right)}\right)\left(1+w_{ii}\right)\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\sum_{l=0}^{\infty}{\left(\prod_{k=0}^{l}{y_i^{\left(t-k\right)}\left(1-y_i^{\left(t-k\right)}\right)}\right)w_{ii}^l}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\sum_{l=0}^{\infty}{\left(y_i^{\left(t-l\right)}\left(1-y_i^{\left(t-l\right)}\right)\right)w_{ii}^l}\)</span></p></li>
<li><p>None of the above</p></li>
</ul>
</li>
<li><p>(NNO) For feature selection, a model <span class="math notranslate nohighlight">\(\hat{y}=\sum_{i}{w_ix_i}\)</span> is trained in a way that many <span class="math notranslate nohighlight">\(w_i\)</span> become zero. To achieve this, the regularization term that is added to the loss function is</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\sum_{i}\left|w_i\right|^2\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\sum_{i}\left|w_i\right|\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\({\sum_{i}\left|w_i\right|}^{-1}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\left(y-\hat{y}\right)^2\)</span></p></li>
<li><p>None of the above</p></li>
</ul>
</li>
<li><p>(NNO) <span class="math notranslate nohighlight">\(\tanh{\left(x\right)}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\)</span>.
The derivative of <span class="math notranslate nohighlight">\(\tanh(x)\)</span> w.r.t. <span class="math notranslate nohighlight">\(x\)</span> at <span class="math notranslate nohighlight">\(x=\log_e{2}\)</span> is</p>
<ul class="simple">
<li><p>0.6</p></li>
<li><p>0.64</p></li>
<li><p>0.5</p></li>
<li><p>0.75</p></li>
<li><p>None of the above</p></li>
</ul>
</li>
<li><p>(NNO) A non-linear model <span class="math notranslate nohighlight">\(\hat{y}=\sigma(w_0+w_1x)\)</span> is to be optimized using gradient descent algorithm over the loss term <span class="math notranslate nohighlight">\(\mathcal{L}=\frac{1}{2}\left(y-\hat{y}\right)^2\)</span>. Here, <span class="math notranslate nohighlight">\(\sigma\left(\cdot\right)\)</span> is the sigmoid function. The weights are chosen to be <span class="math notranslate nohighlight">\(w_0=\log_e\ \left(2\right),\ w_1=\log_e\left(0.5\right)\)</span>. Learning rate <span class="math notranslate nohighlight">\(\eta=1\)</span>. Find the updated value of <span class="math notranslate nohighlight">\(w_0\)</span> after iterating once over the training data <span class="math notranslate nohighlight">\(\left(x=2,y=2/3\right)\)</span>.</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\log_e\ \left(2\right)-\frac{1}{3}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\log_e\ \left(2\right)-\frac{2}{27}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\log_e\ \left(2\right)+\frac{1}{3}\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(\log_e\ \left(2\right)+\frac{2}{27}\)</span></p></li>
<li><p>None of these</p></li>
</ul>
</li>
<li><p>(NNO) The data for flower (<span class="math notranslate nohighlight">\(R\)</span> or <span class="math notranslate nohighlight">\(L\)</span>) classification is given as</p></li>
</ol>
<div class="math notranslate nohighlight">
\[\begin{split}\mathbf{x}^{\intercal}=\left[\begin{matrix}5&4&5.5&2&3\\2&2.5&3&4.5&4\\\end{matrix}\right], \mathbf{y}^{\intercal}=[R,R,R,L,L].\end{split}\]</div>
<p>What is the first sample, i.e., <span class="math notranslate nohighlight">\(\mathbf{x}[0,:]\)</span>, after normalizing it to zero mean and unit variance (use MLE).
- <span class="math notranslate nohighlight">\([0.86,-1.3]\)</span>
- <span class="math notranslate nohighlight">\([0.27,-1.2]\)</span>
- <span class="math notranslate nohighlight">\([0,0]\)</span>
- <span class="math notranslate nohighlight">\([1.23,-1.32]\)</span>
- None of these</p>
<ol class="arabic simple" start="10">
<li><p>(NNO) For multi-label classification, a neural network has 3 inputs. There is no hidden layer. The output layer has 2 neurons, <span class="math notranslate nohighlight">\(h_j=\sum_{i=1}^3 w_{ij} x_i + b_j\)</span>. If the target is <span class="math notranslate nohighlight">\(t_j\)</span> and the model predicts <span class="math notranslate nohighlight">\(\hat{y}_j\)</span>, the update rule for <span class="math notranslate nohighlight">\(w_{ij}\)</span> is:</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(w_{ij} \leftarrow w_{ij}+\eta (t_j-\hat{y}_j)x_i\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(w_{ij} \leftarrow w_{ij}+\eta (\frac{t_j}{\hat{y}_j}-\frac{1-t_j}{1-\hat{y}_j})x_i\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(w_{ij} \leftarrow w_{ij}+\eta (\frac{t_j}{\hat{y}_j}-\frac{1-t_j}{1-\hat{y}_j})y_j x_i\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(w_{ij} \leftarrow w_{ij}+\eta (t_j\log\hat{y}_j)x_i\)</span></p></li>
<li><p><span class="math notranslate nohighlight">\(w_{ij} \leftarrow w_{ij}+\eta (t_j\log\hat{y}_j + (1-t_j)\log(1-\hat{y}_j))x_i\)</span></p></li>
<li><p>None of these</p></li>
</ul>
</li>
<li><p>(NNO) For hyperparameter tuning, which of the following is found to work best for large search spaces</p>
<ul class="simple">
<li><p>Grid search with narrow grid</p></li>
<li><p>Grid search with wide grid</p></li>
<li><p>Gradient descent with adaptive stepsize</p></li>
<li><p>Pick randomly</p></li>
<li><p>Analytic optimization</p></li>
</ul>
</li>
<li><p>(NNO)</p></li>
</ol>
<ul class="simple">
<li><p>What is Overfitting?</p></li>
<li><p>How would you detect if your model is overfitting?</p></li>
<li><p>What different methods could you use to avoid it?</p></li>
</ul>
</section>
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