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[Neural Networks and Deep Learning] week2. Neural Networks Basics
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<time datetime="2017-09-13T00:00:00+02:00"><i class="fa fa-calendar"></i> Wed, 13 Sep 2017</time>
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<span class="label label-default">Series</span>
Part 2 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="#binary-classification-notation" title="Binary Classification & notation">Binary Classification & notation</a></li><li><a class="toc-href" href="#logistic-regression-as-a-nueral-network_1" title="Logistic Regression as a Nueral Network">Logistic Regression as a Nueral Network</a><ul><li><a class="toc-href" href="#logistic-regression" title="Logistic Regression">Logistic Regression</a></li><li><a class="toc-href" href="#logistic-regression-cost-function" title="Logistic Regression Cost Function">Logistic Regression Cost Function</a></li><li><a class="toc-href" href="#gradient-descent" title="Gradient Descent">Gradient Descent</a></li><li><a class="toc-href" href="#computation-graph" title="Computation Graph">Computation Graph</a></li><li><a class="toc-href" href="#logistic-regression-gradient-descent-computation-graph" title="Logistic Regression Gradient Descent (&computation graph)">Logistic Regression Gradient Descent (&computation; graph)</a></li><li><a class="toc-href" href="#gradient-descent-on-m-examples" title="Gradient Descent on m Examples">Gradient Descent on m Examples</a></li></ul></li><li><a class="toc-href" href="#python-and-vectorization_1" title="Python and Vectorization">Python and Vectorization</a><ul><li><a class="toc-href" href="#vectorization" title="Vectorization">Vectorization</a></li><li><a class="toc-href" href="#vectorizing-logistic-regression" title="Vectorizing Logistic Regression">Vectorizing Logistic Regression</a></li><li><a class="toc-href" href="#broadcasting-in-python" title="Broadcasting in Python">Broadcasting in Python</a></li><li><a class="toc-href" href="#a-note-on-python-numpy-vectors" title="A note on python numpy vectors">A note on python numpy vectors</a></li><li><a class="toc-href" href="#explanation-of-logistic-regression-cost-function-optional" title="Explanation of logistic regression cost function (optional)">Explanation of logistic regression cost function (optional)</a></li></ul></li><li><a class="toc-href" href="#assignments_1" title="Assignments">Assignments</a><ul><li><a class="toc-href" href="#python-numpy-basics" title="python / numpy basics">python / numpy basics</a></li><li><a class="toc-href" href="#logistic-regression-with-a-neural-network-mindset" title="Logistic Regression with a Neural Network mindset">Logistic Regression with a Neural Network mindset</a></li></ul></li></ul></div>
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</div>
<p>This week: <strong>logistic regression</strong>.</p>
<h2 id="binary-classification-notation">Binary Classification & notation</h2>
<p>ex. cat classifier from image
image pixels: 64x64x3
⇒ unroll(flatten) to a feature vector <code>x</code> dim=64x64x3=12288:=<code>n</code> (input dimension)</p>
<p><strong>notation</strong></p>
<ul>
<li>superscript <code>(i)</code> for ith example, e.g. <code>x^(i)</code></li>
<li>superscript <code>[l]</code> for lth layer, e.g. <code>w^[l]</code></li>
<li><code>m</code>: number of data</li>
<li><code>n_x</code>: input dimension, <code>n_y</code>: output dimension.</li>
<li><code>n_h^[l]</code>: number of hidden units for layer l.</li>
<li><code>L</code>: number of layers</li>
<li><code>X</code>: dim=(<code>n_x</code>,<code>m</code>), each <em>column</em> is a training example x^(i).</li>
<li><code>Y</code>: dim=(<code>1</code>,<code>m</code>), one single <code>row</code> matrix.</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image.png"/></p>
<h1 id="logistic-regression-as-a-nueral-network_1">Logistic Regression as a Nueral Network</h1>
<h2 id="logistic-regression">Logistic Regression</h2>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image009.png"/><br/>
dim(x) = n_x
parameters: w (dim=n_x) , b (dim=1)
(alternative notation: adding b to w → add x_0 = 1 to feature x. → will NOT use this notation here
keeping w and b separate make implementation easier )</p>
<p>linear regression: <code>y_hat = w^T*x + b</code>
logistic regssion: <code>y_hat = sigmoid(w^T*x + b)</code>
sigmoid function: S-shaped function
<code>sigmoid(z) = 1 / ( 1 + e^-z)</code>
z large → sigmoid(z) ~= 1
z small → sigmoid(z) ~= 0<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image001.png"/></p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image025.png"/></p>
<h2 id="logistic-regression-cost-function">Logistic Regression Cost Function</h2>
<p>To train model for best parameters (w, b), need to define loss function.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image010.png"/> <br/>
y_hat: between (0,1)
training set: {(x^(i), y^(i)))), i = 1..m}
want: y_hat(i) ~= y(i)</p>
<p><strong>Loss function</strong> <code>L(y_hat, y)</code>: on a <em>single</em> training example (x, y)</p>
<ul>
<li>square error: <code>L(y_hat, y) = (y_hat - y)^2/2</code> <ul>
<li>⇒ <em>not convex</em>, GD not work well, uneasy to optimize</li>
</ul>
</li>
<li>loss function used in logistic regression: </li>
</ul>
<p><code>L(y_hat, y) = -[ylog(y_hat) + (1-y)log(1-y_hat)]</code></p>
<ul>
<li>convex w.r.t. w and b</li>
<li>when y = 1, loss = -log(y_hat) → want y_hat large → y_hat ~=1</li>
<li>when y = 0, loss = -log(1-y_hat) → want y_hat small → y_hat ~=0</li>
</ul>
<p><strong>Cost function</strong> <code>J(w,b)</code>: average on all training sets, only depends on parameters w, b<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image003.png"/> </p>
<h2 id="gradient-descent">Gradient Descent</h2>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image005.png"/><br/>
⇒ minimize <code>J(w,b)</code> wrt. w and b</p>
<ul>
<li><code>J(w,b)</code> is convex ⇒ gradient descent</li>
<li>Initialization: for logistic regression, any init works because of convexity of J, usually init as 0</li>
</ul>
<p>Gradient descent: </p>
<ul>
<li><code>alpha</code> = learning rate</li>
<li>derivative <code>dJ(w)/dw</code> </li>
</ul>
<p>~= slope of function <code>J</code> at point <code>w</code>
~= direction where <code>J</code> <em>grows</em> fastest at point <code>w</code>
<em>denote this as '</em><code>dw</code><em>' in code</em></p>
<ul>
<li>algo: 'take steepest descent'<ul>
<li>from an init value of w_0</li>
<li>repeatedly update w until converge <code>w := w - alpha*dw</code></li>
</ul>
</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image006.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image007.png"/> </p>
<p>In the case of logistic regression, >1 params (<code>w</code> and <code>b</code>) to update:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image008.png"/> </p>
<p>Intuitions about derivatives: <code>f'(a)</code> = slope of function <code>f</code> at <code>a</code> .</p>
<h2 id="computation-graph">Computation Graph</h2>
<p>example: function <code>J(a,b,c) = 3(a+b*c)</code></p>
<p><strong>Forward propagation</strong>: compute J(a,b,c) value:</p>
<ul>
<li>internal u := b*c</li>
<li>internal v := a+u</li>
<li>J = 3 * v</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image011.png"/></p>
<p><strong>Backward propagation</strong>: compute derivatives dJ/da, dJ/db, dJ/dc:</p>
<ul>
<li>J = 3*v → compute dJ/dv</li>
<li>v = a + u → compute dv/da, dv/du</li>
<li>u = bc → compute du/db, du/dc</li>
</ul>
<p>⇒ chain rule<em>: dJ/da is multiplying the derivatives along the path from J back to a</em></p>
<ul>
<li>dJ/da = dJ/dv * dv/da</li>
<li>dJ/db = dJ/dv * dv/du * du/db</li>
<li>
<p>dJ/dc = dJ/dv * dv/du * du/dc</p>
</li>
<li>
<p>In code: <em>denote '</em><code>dvar</code><em>' as d(FinalOutput)/d(var) for simplicity. i.e. da = dJ/da, dv = dJ/dv, etc.</em></p>
</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image014.png"/></p>
<h2 id="logistic-regression-gradient-descent-computation-graph">Logistic Regression Gradient Descent (&computation graph)</h2>
<p>logistic regression loss(on a single training example x,y) L.
as computation graph:</p>
<ul>
<li>z = wx + b</li>
<li>a := sigmoid(z) (=y_hat, 'logit'?)</li>
<li>loss function L(a,y) = - [y(loga) + (1-y)log(1-a)]</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image015.png"/></p>
<h2 id="gradient-descent-on-m-examples">Gradient Descent on m Examples</h2>
<p><em>cost function</em>, i.e. on all training sets.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image016.png"/><br/>
J(w,b) = avg{L(x,y), for all m examples}
→ by linearity of derivative: dJ/dw = avg(dL/dw), just average dw^(i) over all indices i.</p>
<p>In implementation: use <strong>vectorization</strong> as much as possible, get rid of for loops.</p>
<h1 id="python-and-vectorization_1">Python and Vectorization</h1>
<h2 id="vectorization">Vectorization</h2>
<p><em>avoid explicit for-loops whenever possible</em>
e.g. z = w^T * x + b
in numpy:
<code>z = np.dot(w, x) + b</code>
<em>~300 times faster than explicit for loop</em></p>
<p>more examples:
u = A*v matrix multiplication
→ <code>u =</code> <code>np.dot(A, v)</code>
note: <code>A * v</code> would element-wise multiply
u = exp(v) element-wise operation: exponential/log/abs/...
→ <code>u = np.exp(v)</code> <code>/ np.log(v) / np.abs(v) / v**2 / 1/v</code></p>
<h2 id="vectorizing-logistic-regression">Vectorizing Logistic Regression</h2>
<p>implementation before: two for-loops( 1 for each training set, 1 for each feature vector).</p>
<ul>
<li>training input <code>X = [x(1), ... , x(m)]</code>, X.dim = (n_x, m)</li>
<li>weight <code>w^T = [w_1, ... , w_nx]</code>, w.dim = (n_x, 1)</li>
</ul>
<p><strong>Fwd propagation</strong><br/>
z(i) = w^T * x(i) + b, i = 1..m,
→ Z := [z(1)...z(m)] = w^T * X + [b...b], Z.dim = (1, m), stack horizentally
→ <code>Z = np.dot(w.T, X) + b</code> (scalar b <em>auto broadcasted</em> to a row vector)
a(i) = sigmoid( z(i) ) = y_hat(i)
→ A := [a(1)...a(m)] = sigmoid(Z), sigmoid is vectorized</p>
<p><strong>Bkwd propagation: gradient computation</strong><br/>
<code>dz(i) = a(i) - y(i)</code>
→ stack horizentally:
Y = [y(1)...y(m)]
dZ := [dz(1)...dz(m)] = A - Y
graidents:
<code>dw = sum( x(i) * dz(i) ) / m</code>, dw.dim = (nx, 1)
<code>db = sum( dz(i) ) / m</code>
→
db = 1/m * np.sum(dZ)
dw = 1/m * X*dz^T<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image017.png"/></p>
<p>efficient back-prop implementation:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image018.png"/></p>
<h2 id="broadcasting-in-python">Broadcasting in Python</h2>
<p>example: calculate percentage of calories from carb/protein/fat for each food — without fooloop<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image019.png"/><br/>
two lines of numpy code:
A = np.array([[...]..]) # A.dim = (3,4)
cal = A.sum(axis=0) # total calories
percentage = 100 * A / cal.reshape(1,4) # percentage.dim = (1,4)</p>
<ul>
<li><code>axis=0</code>→ sum <em>vertically, </em><code>axis=1</code><em> → sum horizentally</em></li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image020.png"/></p>
<ul>
<li><code>reshape(a,b)</code> → redundant here, just to make sure shape correct, <em>reshape call is cheap</em>. </li>
<li><code>A / cal</code> → (3<em>4 matrix) / (1</em>4 matrix) → <strong>broadcasting</strong></li>
</ul>
<p>more broadcasting examples:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image021.png"/><br/>
General principle: computing (m,n) matrix with (1,n) matrix
⇒ the (1,n) matrix is <em>auto expanded to a (m,n) matrix</em> by copying the row m times, to match the shape, calculate element-wise<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image022.png"/></p>
<h2 id="a-note-on-python-numpy-vectors">A note on python numpy vectors</h2>
<p>flexibility of broadcasting: both advantage and <em>weakness</em>.
example: adding column vec and a row vec → get a matrix instead of throwing exceptions.
>>> a
array([1, 2, 3])
>>> b
array([[1],
[2]])
>>> a + b
array([[2, 3, 4],
[3, 4, 5]])</p>
<p><strong>Tips and trick to eliminate bugs</strong></p>
<p><strong>avoid</strong> <strong>rank-1 array</strong>: <br/>
<code>a.shape = (x,)</code>
this is <em>neither row nor column vector</em>, have non-intuitive effects.
>>> a = np.array([1,2,3])
>>> a.shape
(3,) # NOT (3,1)
>>> a.T
array([1, 2, 3])
>>> np.dot(a, a.T) # Mathematically would expact a matrix, if a is column vec
14
>>> a.T.shape
(3,)</p>
<p>⇒ <em>do <strong><em>not</em></strong> use rank-1 arraies, use column/row vectors</em>
>>> a2 = a.reshape((-1, 1)) # A column vector -- (5,1) matrix.
>>> a2
array([[1],
[2],
[3]])
>>> a2.T
array([[1, 2, 3]]) # Note: two brackets!</p>
<p><strong>add assertions</strong><br/>
<code>assert(a.shape == (3,1))</code></p>
<h2 id="explanation-of-logistic-regression-cost-function-optional">Explanation of logistic regression cost function (optional)</h2>
<p>Justisfy why we use this form of cost function:
y_hat ~= chance of y==1 given x
want to express P(y|x) using y_hat and y
P(y|x) as func(y, y_hat) at different values of y:</p>
<ul>
<li>if y = 1: P(y|x) = P(y=1|x) = y_hat</li>
<li>if y = 0: P(y|x) = P(y=0|x) = 1 - y_hat</li>
</ul>
<p>⇒ wrap the two cases <em>in one single formula</em>: using exponent of y and (1-y)<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image023.png"/><br/>
⇒ take log of P(y|x) ⇒ loss function (for a single training example)<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image024.png"/><br/>
⇒ aggregate over all training examples i = 1..m:
(assume: data are iid)
P(labels in training set) = multiply( P(y(i)|x(i) )
take log → log(P(labels in training set)) = sum( log P(y(i)|x(i) ) = - J
<strong>maximizing likelihood = minimizing cost function</strong></p>
<h1 id="assignments_1">Assignments</h1>
<h2 id="python-numpy-basics">python / numpy basics</h2>
<ul>
<li>np.reshape() / np.shape</li>
<li>calculate norm: <code>np.linalg.norm()</code></li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image026.png"/></p>
<ul>
<li><code>keepdims=True</code>: </li>
</ul>
<p>axes that are reduced will be <em>kept</em> (with size=1)</p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err"> >>> a</span></span>
<span class="code-line"><span class="err"> array([[ 0.01014617, 0.08222027, -0.59608242],</span></span>
<span class="code-line"><span class="err"> [-0.18495204, -1.50409531, -1.03853663],</span></span>
<span class="code-line"><span class="err"> [ 0.03995499, -0.67679544, 0.11513247]])</span></span>
<span class="code-line"><span class="err"> >>> a.sum(keepdims=1)</span></span>
<span class="code-line"><span class="err"> array([[-3.75300795]])</span></span>
<span class="code-line"><span class="err"> >>> a.sum()</span></span>
<span class="code-line"><span class="err"> -3.7530079538833663</span></span>
</pre></div>
<ul>
<li><a href="https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html">broadcasting</a></li>
<li>softmax: </li>
</ul>
<p>softmax for row vec:
x.shape = (1,n), x = [x1,...xn]
y = softmax(x), y.shape = (1,n), <code>yi = exp(xi) / sum( exp(xi) )</code>
softmax for matrix
X.shape = (m,n)
Y = softmax(X) = [softmax(row-i of X)], Y.shape = (m, 1)</p>
<h2 id="logistic-regression-with-a-neural-network-mindset">Logistic Regression with a Neural Network mindset</h2>
<ul>
<li>input preprocessing</li>
</ul>
<p>input dataset shape = (m, num_px, num_px, 3)
→ <em>reshape</em> to one column per example, shape = (num_px<em>num_px</em>3, ~~m~~)
→ <em>center & standardize</em> data: <code>x' = (xi - x_mean) / std(x)</code>,
but <em>for images:</em> just divide by 255.0 (max pixel value), convenient and works almost as well.</p>
<ul>
<li>params initialization</li>
</ul>
<p>For logistic regression (cost function convex), just init to zeros is OK.
w = np.zeros((dim,1))
b = 0.0</p>
<ul>
<li>Fwd prop: compute cost function</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image027.png"/><br/>
input <code>X</code> (shape = nx*m, one column per example)→ logits <code>Z</code> → activations <code>A=sigmoid(Z)</code>→ cost <code>J</code></p>
<ul>
<li>Bkwd prop</li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c1wk2//pasted_image028.png"/></p>
<ul>
<li>Optimization</li>
</ul>
<p>gradient descent: w := w - alpha*dw</p>
<ul>
<li>Predict: using learned params</li>
</ul>
<p>Yhat = A = sigmoid(wT * X + b)</p>
</div>
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目录</h4>
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<div id="toc"><ul><li><a class="toc-href" href="#binary-classification-notation" title="Binary Classification & notation">Binary Classification & notation</a></li><li><a class="toc-href" href="#logistic-regression-as-a-nueral-network_1" title="Logistic Regression as a Nueral Network">Logistic Regression as a Nueral Network</a><ul><li><a class="toc-href" href="#logistic-regression" title="Logistic Regression">Logistic Regression</a></li><li><a class="toc-href" href="#logistic-regression-cost-function" title="Logistic Regression Cost Function">Logistic Regression Cost Function</a></li><li><a class="toc-href" href="#gradient-descent" title="Gradient Descent">Gradient Descent</a></li><li><a class="toc-href" href="#computation-graph" title="Computation Graph">Computation Graph</a></li><li><a class="toc-href" href="#logistic-regression-gradient-descent-computation-graph" title="Logistic Regression Gradient Descent (&computation graph)">Logistic Regression Gradient Descent (&computation; graph)</a></li><li><a class="toc-href" href="#gradient-descent-on-m-examples" title="Gradient Descent on m Examples">Gradient Descent on m Examples</a></li></ul></li><li><a class="toc-href" href="#python-and-vectorization_1" title="Python and Vectorization">Python and Vectorization</a><ul><li><a class="toc-href" href="#vectorization" title="Vectorization">Vectorization</a></li><li><a class="toc-href" href="#vectorizing-logistic-regression" title="Vectorizing Logistic Regression">Vectorizing Logistic Regression</a></li><li><a class="toc-href" href="#broadcasting-in-python" title="Broadcasting in Python">Broadcasting in Python</a></li><li><a class="toc-href" href="#a-note-on-python-numpy-vectors" title="A note on python numpy vectors">A note on python numpy vectors</a></li><li><a class="toc-href" href="#explanation-of-logistic-regression-cost-function-optional" title="Explanation of logistic regression cost function (optional)">Explanation of logistic regression cost function (optional)</a></li></ul></li><li><a class="toc-href" href="#assignments_1" title="Assignments">Assignments</a><ul><li><a class="toc-href" href="#python-numpy-basics" title="python / numpy basics">python / numpy basics</a></li><li><a class="toc-href" href="#logistic-regression-with-a-neural-network-mindset" title="Logistic Regression with a Neural Network mindset">Logistic Regression with a Neural Network mindset</a></li></ul></li></ul></div>
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