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<h1>
[Improving Deep Neural Networks] week1. Practical aspects of Deep Learning
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<span class="label label-default">Series</span>
Part 5 of «Andrew Ng Deep Learning MOOC»
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#setting-up-your-maching-learning-application" title="Setting up your Maching Learning Application">Setting up your Maching Learning Application</a><ul><li><a class="toc-href" href="#train-dev-test-sets" title="Train / Dev / Test sets">Train / Dev / Test sets</a></li><li><a class="toc-href" href="#bias-variance" title="Bias / Variance">Bias / Variance</a></li><li><a class="toc-href" href="#basic-recipe-for-machine-learning" title="Basic Recipe for Machine Learning">Basic Recipe for Machine Learning</a></li></ul></li><li><a class="toc-href" href="#regularizing-your-neural-network_1" title="Regularizing your neural network">Regularizing your neural network</a><ul><li><a class="toc-href" href="#regularization" title="Regularization">Regularization</a><ul><li><a class="toc-href" href="#example-logistic-regression" title="example: logistic regression">example: logistic regression</a></li><li><a class="toc-href" href="#example-nn" title="example: NN">example: NN</a></li></ul></li><li><a class="toc-href" href="#why-regularization-reduces-overfitting_1" title="Why regularization reduces overfitting?">Why regularization reduces overfitting?</a></li><li><a class="toc-href" href="#dropout-regularization" title="Dropout Regularization">Dropout Regularization</a><ul><li><a class="toc-href" href="#dropout-implementation-inverted-dropout" title='dropout implementation: "inverted dropout"'>dropout implementation: "inverted dropout"</a></li></ul></li><li><a class="toc-href" href="#understanding-dropout_1" title="Understanding Dropout">Understanding Dropout</a></li><li><a class="toc-href" href="#other-regularization-methods" title="Other regularization methods">Other regularization methods</a><ul><li><a class="toc-href" href="#data-augmentation" title="data augmentation">data augmentation</a></li><li><a class="toc-href" href="#early-stopping" title="early stopping">early stopping</a></li></ul></li></ul></li><li><a class="toc-href" href="#setting-up-your-optimization-problem_2" title="Setting up your optimization problem">Setting up your optimization problem</a><ul><li><a class="toc-href" href="#normalizing-inputs" title="Normalizing inputs">Normalizing inputs</a><ul><li><a class="toc-href" href="#vanishing-exploding-gradients" title="Vanishing / Exploding gradients">Vanishing / Exploding gradients</a></li></ul></li><li><a class="toc-href" href="#weight-initialization-for-deep-networks_1" title="Weight Initialization for Deep Networks">Weight Initialization for Deep Networks</a></li><li><a class="toc-href" href="#numerical-approximation-of-gradients" title="Numerical approximation of gradients">Numerical approximation of gradients</a></li><li><a class="toc-href" href="#gradient-checking" title="Gradient checking">Gradient checking</a></li><li><a class="toc-href" href="#gradient-checking-implementation-notes" title="Gradient Checking Implementation Notes">Gradient Checking Implementation Notes</a></li></ul></li></ul></div>
</div>
</div>
<h1 id="setting-up-your-maching-learning-application">Setting up your Maching Learning Application</h1>
<h2 id="train-dev-test-sets">Train / Dev / Test sets</h2>
<p>Applied ML: highly iterative process. <em>idea-code-exp loop</em> </p>
<p><strong>splitting data</strong><br/>
splitting data in order to speed up the idea-code-exp loop: <br/>
*training set / dev(hold-out/cross-validataion) set / test set * </p>
<p><strong>split ratio</strong>: </p>
<ul>
<li>with 100~10000 examples: 70/30 or 60/20/20 </li>
<li>with ~1M examples: dev/test set can have much smaller ratio, e.g. 98/1/1 </li>
</ul>
<p><strong>mismatched train/test distribution</strong><br/>
training and test set don't come from the same dist.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image.png"/> </p>
<ul>
<li>rule of thumb: <strong>make sure</strong> <strong>dev and test set come from the same distribution.</strong> </li>
<li>might be OK to only have dev set. — thought in this case no longer have unbiased estimate of performance. </li>
</ul>
<h2 id="bias-variance">Bias / Variance</h2>
<ul>
<li>high variance: <em>overfitting</em> </li>
<li>high bias: <em>underfitting</em> </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image001.png"/><br/>
high base and high variance (worse case): high bias in some region and high variance elsewhere<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image004.png"/> </p>
<p><strong>how to estimate bias&variance</strong><br/>
→ <em>look at train and dev set error</em> </p>
<ul>
<li>high variance: Err_train << Err_dev — not generalize well </li>
<li>high bias: Err_train ~= Err_dev, and Err_train >> Err_human — not learning well even on training set </li>
<li>high bias <em>and</em> high variance (worse): Err_train >> Err_human, Err_train >> Err_dev </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image003.png"/> </p>
<h2 id="basic-recipe-for-machine-learning">Basic Recipe for Machine Learning</h2>
<p>basic recipe: </p>
<ol>
<li>does algo have high bias ? (look at Err_train) <ul>
<li>if yes → try bigger nn / other architecture </li>
<li>until having low bias (fit well training set) </li>
</ul>
</li>
<li>high variance ? (look at Err_dev) <ul>
<li>if yes → get more data / regularization / other architecture </li>
</ul>
</li>
</ol>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image005.png"/> </p>
<p><strong>bias-variance tradeoff</strong> </p>
<ul>
<li>in pre-DL era, bias and variance are tradeoff (decrease one → increase the other) </li>
<li>in DL era: <em>if getting bigger nn and more data always possible</em>, <em>both can be reduced</em> </li>
</ul>
<p>(when well regularized,) <em>"training a bigger NN almost never hurts."</em> </p>
<h1 id="regularizing-your-neural-network_1">Regularizing your neural network</h1>
<p>2 ways to reduce variance: regularize, or get more data. </p>
<h2 id="regularization">Regularization</h2>
<h3 id="example-logistic-regression">example: logistic regression</h3>
<ul>
<li>params: <code>w</code>, <code>b</code> </li>
<li>cost function <code>J(w,b) = 1/m * L(yhat_i, yi)</code> </li>
</ul>
<p>→ add one more term to cost <code>J</code>: adding L2 norm of <code>w</code>(<em>L2 regularization</em>)<br/>
(lambda: regularization param)<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image006.png"/><br/>
<em>just omit regularizing b</em>: <code>w</code> is high dim, <code>b</code> is single number. </p>
<p>L1 regularization: L1 norm of <code>w</code> → <em>w will be sparse → </em>compressing the model (just a little bit)<br/>
⇒ <em>L2-reg is much often used</em> </p>
<h3 id="example-nn">example: NN</h3>
<ul>
<li>params: <code>w[l]</code>, <code>b[l]</code> for l = 1..N </li>
<li>sum of the norms of each <code>w[l]</code> matrix. </li>
</ul>
<p>⇒ <em>"Frobenius norm"</em> of a matrix: sum (each element squared)<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image007.png"/> </p>
<p><strong>gradient descent</strong>: adding one more term from backprop<br/>
d(1/2m * ||w||) = lambda / m <br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image008.png"/> </p>
<p>L2-reg also called "<strong>weight decay</strong>": <br/>
with L2-reg, looks as if doing the backprop updating, with w being w' = (1-alpha*lambda/m) * w (decayed w)<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image009.png"/> </p>
<h2 id="why-regularization-reduces-overfitting_1">Why regularization reduces overfitting?</h2>
<p><em>why imposing small params prevents overfitting?</em> </p>
<p><strong>intuition 1</strong><br/>
→ heavy regularization <br/>
→ weight ~= 0 <br/>
→ many hidden units' impact are "<em>zeroed-out"</em><br/>
→ simpler NN<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image010.png"/> </p>
<p><strong>intuition 2</strong><br/>
e.g. activation g(z) = tanh(z)<br/>
small z → g(z) ~= linear, <br/>
large z → g(z) flattend<br/>
⇒ large lambda → small w <br/>
→ z small <br/>
→ every layer ~linear<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image011.png"/> </p>
<h2 id="dropout-regularization">Dropout Regularization</h2>
<p>another powerful method of regularization<br/>
<strong>dropout</strong>: <em>For each training example</em>, in each layer, <em>eliminate randomly some of its output values.</em> <br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image012.png"/> <img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image013.png"/> </p>
<h3 id="dropout-implementation-inverted-dropout">dropout implementation: "inverted dropout"</h3>
<p>example: dropout of <em>layer 3</em>, keep_prob = 0.8 (prob of keeping hidden unit)<br/>
→ generate a rand matrix of shape the same shape as activation <code>a[3]</code> </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">d3 = np.random.rand(a3.shape[0], a3.shape[1]) < keep_prob # d3 is bool matrix </span></span>
<span class="code-line"><span class="err">a3 = np.multiply(a3, d3) # element-wise multiply </span></span>
<span class="code-line"><span class="err">a3 /= keep_prob # ****"inverted dropout"****</span></span>
</pre></div>
<p><strong>"inverted dropout": why a3 /= keep_prob (i.e. make a3 larger)?</strong> </p>
<ul>
<li>let's say layer 3 has 50 units, keep_prob = 0.8 </li>
<li>→ ~10 units shut off </li>
<li><code>z[4] = w[4] * a[3] + b[4]</code> </li>
</ul>
<p>⇒ a[3] have random 20% units shut off <br/>
→ <em>w[4]</em>a[3] will be reduced by 20% in expection* </p>
<ul>
<li>inverted dropout: a3 /= keep_prob, to <em>keep expected value a3 remains unchanged</em>. </li>
<li>(No dropout at test time) → inverted dropout <em>avoids scaling problem at test time</em> </li>
</ul>
<p><strong>making predictions at test time</strong><br/>
NOT use dropout at test time ⇒ don't want output to be random at test time... </p>
<h2 id="understanding-dropout_1">Understanding Dropout</h2>
<p><em>why randomly shut units prevents overfitting ?</em> </p>
<p><strong>Intuition: can't rely on any one input feature → have to spread out weight</strong><br/>
spread weights ~→ smaller L2 norm (shrink weights)<br/>
Can be formally proven: dropout is equal to <em>adaptive</em> L2-reg, with penalty of different weight being different. </p>
<p>For one hidden unit: any of it input features (from prev layer) can go out at random<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image014.png"/> </p>
<p><strong>Implementation details</strong> </p>
<ul>
<li>vary keep_prob for different layer </li>
</ul>
<p>→ <em>smaller keep_prob for larger layer</em> </p>
<ul>
<li>usually no dropout (or very small dropout) for input layer... </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image016.png"/> </p>
<p><strong>Downside of dropout</strong><br/>
cost function J <em>no longer well-defined </em>(because output yhat is random)<br/>
→ can no longer plot cost-iter curve<br/>
→ turn off dropout before plotting the curve </p>
<h2 id="other-regularization-methods">Other regularization methods</h2>
<h3 id="data-augmentation">data augmentation</h3>
<p>adding more training example is expensive <br/>
→ vary existing training data (e.g. flipping/rand-distortions of training image for cats) </p>
<h3 id="early-stopping">early stopping</h3>
<p>plot Err or J to #iterations <em>for both train and dev set.</em><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image017.png"/> </p>
<p><strong>Downside of early-stopping</strong>: <br/>
<em>optimization cost J</em> and <em>not overfitting</em> should be separated task ("Orthogonalization")<br/>
→ early-stopping couples the two jobs. </p>
<p>upside of early stopping: no need to try different values of regularization param (lambda) → finds "mid-size w" at once. </p>
<h1 id="setting-up-your-optimization-problem_2">Setting up your optimization problem</h1>
<p>How to speed up training (i.e. optimize J) </p>
<h2 id="normalizing-inputs">Normalizing inputs</h2>
<p>normalize input: </p>
<ol>
<li>substract mean </li>
<li>normalize variance </li>
</ol>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image018.png"/><br/>
detail: in data splitting, <em>use the same meu/sigma to normalize test set !</em> </p>
<p><strong>why normalizing input ?</strong><br/>
if features x1 x2 are on different scales → w1 and w2 not same scale<br/>
J is more symmetric, easier to optimize<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image019.png"/> </p>
<h3 id="vanishing-exploding-gradients">Vanishing / Exploding gradients</h3>
<p>One problem in training very deep NN: vanishing/exploding gradients. </p>
<p>example: a very deep NN, each layer 2 units, linear activation g(z)=z, ignore bias b[l] = 0.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image021.png"/><br/>
linear activations → y is just a linear transformation of x<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image022.png"/> </p>
<ul>
<li>assuming each w[l] = 1.5 * Identity_matrix ⇒ activations increase exponentially </li>
<li>assuming each w[l] = 0.5 * Id ⇒ activations decrease exponentially </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image023.png"/><br/>
yhat too large or too small → hard to train </p>
<h2 id="weight-initialization-for-deep-networks_1">Weight Initialization for Deep Networks</h2>
<p>A partial solution of vanishing/exploding gradient problem: <em>carefully initialize weights</em>. </p>
<p><strong>single neuron example:</strong><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image024.png"/> </p>
<ul>
<li>y = g(w*x), g = relu </li>
<li>n = # inputs for </li>
</ul>
<p>z = w1<em>x1 + ... + wn</em>xn, <br/>
if wi are initzed randomly<br/>
→ large ns ⇒ z will be large ! <br/>
⇒ <strong>set var(wi) = 1/n</strong> (2/n in practice) to keep z in similar scale for diffent #inputs<br/>
initialization code: <br/>
<code>w[l] = np.random.randn(shape[l]) * np.sqrt( 2 / n[l-1] ) # n[l-1] = #inputs for layer-l</code> </p>
<p><strong>other variants</strong><br/>
when activation function g = tanh <br/>
⇒ use var(wi) = 1/n ("<strong>Xavier initialization</strong>")<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image025.png"/> </p>
<h2 id="numerical-approximation-of-gradients">Numerical approximation of gradients</h2>
<p><strong>checking the derivative computation</strong><br/>
example: f(x) = x ^ 3<br/>
→ <em>vary x by epsilon</em> to approximate f'(x), <em>use 2-sided difference</em><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image026.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image027.png"/> </p>
<p>error order = O(epsilon^2) for 2-sided difference<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image028.png"/> </p>
<h2 id="gradient-checking">Gradient checking</h2>
<p><strong>Verify</strong> that your implementation is correct. — help finding out bugs in implementation early. </p>
<ul>
<li>concat all params into a big vector <code>theta</code> </li>
<li>concat all dW[l] db[l] into big vector <code>d_theta</code> </li>
<li>to <strong>check if d_theta is correct</strong>: construct a <code>d_theta_approx</code> vector </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image029.png"/><br/>
⇒ <br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image030.png"/> </p>
<p>How to check "approximate":<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk1//pasted_image031.png"/> </p>
<h2 id="gradient-checking-implementation-notes">Gradient Checking Implementation Notes</h2>
<ul>
<li>Dont' use checking in training: constructing d_theta_approx is slow </li>
<li>When check fails: look at components to try to find bug </li>
<li>Remember regularization: J contains reg term as well </li>
<li>Doesn't work with dropout: J not well defined (random variable), turn dropout off before checking. </li>
<li>Run check at random initialization (w,b~=0), then again after some training(w,b~>0) </li>
</ul>
</div>
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<div id="toc"><ul><li><a class="toc-href" href="#setting-up-your-maching-learning-application" title="Setting up your Maching Learning Application">Setting up your Maching Learning Application</a><ul><li><a class="toc-href" href="#train-dev-test-sets" title="Train / Dev / Test sets">Train / Dev / Test sets</a></li><li><a class="toc-href" href="#bias-variance" title="Bias / Variance">Bias / Variance</a></li><li><a class="toc-href" href="#basic-recipe-for-machine-learning" title="Basic Recipe for Machine Learning">Basic Recipe for Machine Learning</a></li></ul></li><li><a class="toc-href" href="#regularizing-your-neural-network_1" title="Regularizing your neural network">Regularizing your neural network</a><ul><li><a class="toc-href" href="#regularization" title="Regularization">Regularization</a><ul><li><a class="toc-href" href="#example-logistic-regression" title="example: logistic regression">example: logistic regression</a></li><li><a class="toc-href" href="#example-nn" title="example: NN">example: NN</a></li></ul></li><li><a class="toc-href" href="#why-regularization-reduces-overfitting_1" title="Why regularization reduces overfitting?">Why regularization reduces overfitting?</a></li><li><a class="toc-href" href="#dropout-regularization" title="Dropout Regularization">Dropout Regularization</a><ul><li><a class="toc-href" href="#dropout-implementation-inverted-dropout" title='dropout implementation: "inverted dropout"'>dropout implementation: "inverted dropout"</a></li></ul></li><li><a class="toc-href" href="#understanding-dropout_1" title="Understanding Dropout">Understanding Dropout</a></li><li><a class="toc-href" href="#other-regularization-methods" title="Other regularization methods">Other regularization methods</a><ul><li><a class="toc-href" href="#data-augmentation" title="data augmentation">data augmentation</a></li><li><a class="toc-href" href="#early-stopping" title="early stopping">early stopping</a></li></ul></li></ul></li><li><a class="toc-href" href="#setting-up-your-optimization-problem_2" title="Setting up your optimization problem">Setting up your optimization problem</a><ul><li><a class="toc-href" href="#normalizing-inputs" title="Normalizing inputs">Normalizing inputs</a><ul><li><a class="toc-href" href="#vanishing-exploding-gradients" title="Vanishing / Exploding gradients">Vanishing / Exploding gradients</a></li></ul></li><li><a class="toc-href" href="#weight-initialization-for-deep-networks_1" title="Weight Initialization for Deep Networks">Weight Initialization for Deep Networks</a></li><li><a class="toc-href" href="#numerical-approximation-of-gradients" title="Numerical approximation of gradients">Numerical approximation of gradients</a></li><li><a class="toc-href" href="#gradient-checking" title="Gradient checking">Gradient checking</a></li><li><a class="toc-href" href="#gradient-checking-implementation-notes" title="Gradient Checking Implementation Notes">Gradient Checking Implementation Notes</a></li></ul></li></ul></div>
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