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[Improving Deep Neural Networks] week2. Optimization algorithms
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<span class="label label-default">Series</span>
Part 6 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="#mini-batch-gradient-descent" title="Mini-batch gradient descent">Mini-batch gradient descent</a><ul><li><a class="toc-href" href="#batch-vs-mini-batch-gd" title="batch v.s. mini-batch GD">batch v.s. mini-batch GD</a></li></ul></li><li><a class="toc-href" href="#understanding-mini-batch-gradient-descent_1" title="Understanding mini-batch gradient descent">Understanding mini-batch gradient descent</a></li><li><a class="toc-href" href="#exponentially-weighted-moveing-averages" title="Exponentially weighted (moveing) averages">Exponentially weighted (moveing) averages</a></li><li><a class="toc-href" href="#understanding-exponentially-weighted-averages" title="Understanding exponentially weighted averages">Understanding exponentially weighted averages</a></li><li><a class="toc-href" href="#bias-correction-in-exponentially-weighted-averages" title="Bias correction in exponentially weighted averages">Bias correction in exponentially weighted averages</a></li><li><a class="toc-href" href="#gradient-descent-with-momentum" title="Gradient descent with momentum">Gradient descent with momentum</a></li><li><a class="toc-href" href="#rmsprop" title="RMSprop">RMSprop</a></li><li><a class="toc-href" href="#adam-optimization-algorithm" title="Adam optimization algorithm">Adam optimization algorithm</a></li><li><a class="toc-href" href="#learning-rate-decay" title="Learning rate decay">Learning rate decay</a></li><li><a class="toc-href" href="#the-problem-of-local-optima" title="The problem of local optima">The problem of local optima</a></li><li><a class="toc-href" href="#assignment" title="assignment">assignment</a></li></ul></div>
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<p>This week: optimization algos to faster train NN, on large dataset. </p>
<h2 id="mini-batch-gradient-descent">Mini-batch gradient descent</h2>
<h3 id="batch-vs-mini-batch-gd">batch v.s. mini-batch GD</h3>
<p>Compute J on <code>m</code> examples: vectorization, i.e. stacking x(i) y(i) horizontally.<br/>
<code>X = [x(1), ..., x(m)]</code><br/>
<code>Y = [y(1), ..., y(m)]</code><br/>
→ still slow or impossible with large <code>m</code>. </p>
<p>⇒ split all m examples into <em>mini-batches</em>. X^t^, Y^t^<br/>
e.g. mini batch size = 1000. </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image001.png"/> </p>
<p>Minibatch GD:<br/>
each step, run one iteration of GD using X{t}, Y{t} instead of doing with full X, Y.<br/>
one "<strong>epoch</strong>": one pass through all training set<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image003.png"/> </p>
<h2 id="understanding-mini-batch-gradient-descent_1">Understanding mini-batch gradient descent</h2>
<p>with batch-GD: each iteration will decrease cost function.<br/>
in mini-batch: cost J^t^ is computed on different dataset — noisy.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image004.png"/> </p>
<p>How to choose minibatch size </p>
<ul>
<li>
<p>extreme 1: minibatch size = m → batch GD </p>
<p>coverge fastest in each iter (<em>but too long time or impossible per iter</em>) </p>
</li>
<li>
<p>extreme 2: minibatch size = 1 → <em>stochastic GD</em> </p>
<p>noisy, never coverge, (<em>but loose all vectorization speedup)</em><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image006.png"/> </p>
</li>
</ul>
<p><strong>Guidelines on choosing batch size</strong>: </p>
<ul>
<li>small training set (m<2000) → just batch size </li>
<li>
<p>otherwise: </p>
<p>typical minibatch size = 64/128/256/512 (make sure minibatch size fits in CPU/GPU memory) </p>
</li>
</ul>
<h2 id="exponentially-weighted-moveing-averages">Exponentially weighted (moveing) averages</h2>
<p><em>In fact, exp-weighted-avg is an non-parametric estimator/smoother of a series of vlaues.</em> </p>
<p>example: Temperature over the year<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image007.png"/><br/>
→ use a <em>exp-weighted moving average</em> to model this:<br/>
<code>theta[t]</code> = temperature at day t, t = 1,2,3,...<br/>
<code>v[t]</code> = averaged(smoothed) estimate of theta, t = <strong>0</strong>,1,2,3,...<br/>
<strong>exp-weighted average</strong>: <em>recursivly compute v[t].</em><br/>
<code>v[0] = 0, v[t] = 0.9 * v[t-1] + 0.1 * theta[t]</code><br/>
(param: beta = 0.9)<br/>
⇒ v[t] ~= <em>average of theta[t] over the last 1/(1-beta) days</em>. (c.f. next section)<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image009.png"/><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image008.png"/> </p>
<h2 id="understanding-exponentially-weighted-averages">Understanding exponentially weighted averages</h2>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image010.png"/><br/>
Understanding the math of exp-weighted-average:<br/>
→ unroll the recursive formula:<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image011.png"/> </p>
<p>⇔ v[100] = <em>a convolution of theta[t-100:t] and a exp-decaying function:</em><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image012.png"/> </p>
<p>A small trick on <em>estimating exps</em>: <br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image013.png"/><br/>
That's why in previous section, we say this formula ~= averaging over last 1/(1-beta) (=1/epslon) days' theta.<br/>
e.g. theta[t-10]'s weight is 0.9 ^ 10 ~= 1/e ~= 0.35, i.e. after 10 days, the weight of theta[t-10] is decayed to ~< 1/3. </p>
<p><strong>Advantage</strong> to just computing moving window averages: in implementation, only keep updating a single variable v_theta — very small memory usage. </p>
<h2 id="bias-correction-in-exponentially-weighted-averages">Bias correction in exponentially weighted averages</h2>
<p>To make exp-weighted averages more accurate by <em>bias correction.</em><br/>
Problem with previous implementation in initial phase: </p>
<ul>
<li>v[0] = 0, beta = 0.98 </li>
<li>v[1] = 0.98<em>0 + 0.02 * theta[1] ⇒ </em>v[t] starts lower than theta. *(purple curve VS green curve) </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image014.png"/><br/>
⇒ <strong>Correction</strong>:<br/>
take <code>v[t] / (1 - beta^t)</code> instead of v[t].<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image015.png"/> </p>
<p>In practice: <em>most people don't bother to implement bias correction</em> — just wait for the initial phase to warm up... </p>
<h2 id="gradient-descent-with-momentum">Gradient descent with momentum</h2>
<p>GD with momentum: </p>
<ul>
<li>almost always faster than normal GD </li>
<li>in short: <em>compute exp-weighted-avg of the gradients as gradient to use</em> </li>
</ul>
<p>example: if contour of loss is an ellipse, can't use too large step in GD, and oscillate. → average of steps will be faster. </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image018.png"/> </p>
<p><strong>GD momentum</strong>:<br/>
Use exp-weighted-avg of dW (V_dW) and of db (V_db). — smooth out oscillation steps of normal GD.<br/>
update params with the averaged value V_dW, V_db.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image017.png"/> </p>
<p>Why the name "momentum": <em>rolling down a ball in a bowl</em><br/>
dW/db: ~accelation<br/>
V_dW/B_db: ~velocity at current time (<em>this is why the smoothing average is called 'v'</em>)<br/>
beta: ~friction </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image019.png"/> </p>
<p>in practice: </p>
<ul>
<li>beta=0.9 works well for most cases </li>
<li>no bias correction implemented </li>
<li>can use beta = 0 → ~V_dW is scaled by 1/(1-beta) → use a scaled alpha then. </li>
</ul>
<p>⇒ less intuitive, and couples the tuning of alpha and beta </p>
<h2 id="rmsprop">RMSprop</h2>
<p>"Root-Mean-Square-prop". </p>
<ul>
<li>also for speedup GD </li>
<li>
<p>in short: </p>
<p><em>S =</em> exp-weighted-avg of gradient <em>squared</em> <em>(that's why call param beta2);</em><br/>
when updating params(W,b)<em>, scale dW,db by <strong><em>sqrt of S</em></strong>. </em><code>dW / sqrt(S_dW)</code> </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image022.png"/> </p>
<p>example: ellipse contour, want slow rate in <code>b</code> directrion and fast rate in <code>w</code> direction. </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image023.png"/> </p>
<p>⇒ for each step (dW, db) = gradient of current minibatch: </p>
</li>
<li>
<p>db <em>large</em>, dW <em>small</em> </p>
<p>→ update w by dW/sqrt(S_dW), ⇔ <em>larger</em> step in W direction<br/>
→ db/sqrt(S_db) ⇔ <em>smaller</em> step in b direction → damping out oscillations in b direction </p>
</li>
</ul>
<h2 id="adam-optimization-algorithm">Adam optimization algorithm</h2>
<p>"Adaptive Moment-estimation" </p>
<ul>
<li>A <em>combination</em> of RMSprop and momentum. </li>
<li>Proved to work well for a varity of problems. </li>
</ul>
<p>Algo: </p>
<ul>
<li>maintain both V_dW, V_db (hyper-param=beta1) and S_dW, S_db (hyper-param=beta2) </li>
<li>implement bias correction: V_corrected, S_corrected — divid by (1-beta^t) </li>
<li>param update (W, b): <code>V / sqrt(S)</code><br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image024.png"/> </li>
</ul>
<p>hyperparameters: </p>
<ul>
<li>alpha: learning rate, <em>needs tuning</em> </li>
<li>beta1: usually 0.9 </li>
<li>beta2: usually 0.999 </li>
<li>epsilon: 10e-8 (not important) </li>
</ul>
<h2 id="learning-rate-decay">Learning rate decay</h2>
<p>slowly reduce learning rate. </p>
<p>In minibatch with fixed learning rate: <em>will never converge</em>.<br/>
⇒ decay learning rate → oscillating around a smaller region of optima.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image026.png"/> </p>
<p>Implementation: </p>
<ul>
<li>1 <em>epoch</em> = 1 pass through whole data. </li>
<li>decay learning rate alpha after each epoch (hyper-param <em>decay-rate</em>):<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image028.png"/> </li>
</ul>
<p>other decay method: </p>
<ul>
<li>exponentially decay alpha: </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image029.png"/> </p>
<ul>
<li>sqrt of epoch_num: </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image030.png"/> </p>
<ul>
<li>discrete staircase: </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image031.png"/> </p>
<ul>
<li>manual decay: when training takes <em>long time</em> </li>
</ul>
<h2 id="the-problem-of-local-optima">The problem of local optima</h2>
<p>low-dimention optimas: <br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image032.png"/> </p>
<p>This gives <em>wrong</em> intuition, in practice (high-dim), most 0-gradient points are <em>saddle points.</em> </p>
<p><img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image034.png"/> </p>
<p><strong>plateau</strong>: region where gradient close to 0 for long time.<br/>
<img alt="" class="img-responsive" src="../images/Ng_DLMooc_c2wk2/pasted_image035.png"/> </p>
<p>take-away: </p>
<ul>
<li><em>unlikely</em> to stuck in bad local optima: D dimentional → ~2^(-D) of chance. </li>
<li><em>plateaus</em> can make learning slow → use momentum/RMSprop/Adam to speedup training. </li>
</ul>
<h2 id="assignment">assignment</h2>
<p><strong>implementing mini-batch GD</strong>:<br/>
shuffle data (note: X and Y's columns are sync-ed!): </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">permutation = list(np.random.permutation(m)) </span></span>
<span class="code-line"><span class="err"> shuffled_X = X[:, permutation] </span></span>
<span class="code-line"><span class="err"> shuffled_Y = Y[:, permutation].reshape((1,m))</span></span>
</pre></div>
<p>→ partition data:<br/>
kth mini batch: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">mini_batch_X = shuffled_X[:, k*mini_batch_size:(k+1)*mini_batch_size] </span></span>
<span class="code-line"><span class="err"> mini_batch_Y = shuffled_Y[:, k*mini_batch_size:(k+1)*mini_batch_size]</span></span>
</pre></div>
<p>detail: when m % minibatch_size != 0: handle last batch (smaller than a regular batch) </p>
</div>
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<div id="toc"><ul><li><a class="toc-href" href="#mini-batch-gradient-descent" title="Mini-batch gradient descent">Mini-batch gradient descent</a><ul><li><a class="toc-href" href="#batch-vs-mini-batch-gd" title="batch v.s. mini-batch GD">batch v.s. mini-batch GD</a></li></ul></li><li><a class="toc-href" href="#understanding-mini-batch-gradient-descent_1" title="Understanding mini-batch gradient descent">Understanding mini-batch gradient descent</a></li><li><a class="toc-href" href="#exponentially-weighted-moveing-averages" title="Exponentially weighted (moveing) averages">Exponentially weighted (moveing) averages</a></li><li><a class="toc-href" href="#understanding-exponentially-weighted-averages" title="Understanding exponentially weighted averages">Understanding exponentially weighted averages</a></li><li><a class="toc-href" href="#bias-correction-in-exponentially-weighted-averages" title="Bias correction in exponentially weighted averages">Bias correction in exponentially weighted averages</a></li><li><a class="toc-href" href="#gradient-descent-with-momentum" title="Gradient descent with momentum">Gradient descent with momentum</a></li><li><a class="toc-href" href="#rmsprop" title="RMSprop">RMSprop</a></li><li><a class="toc-href" href="#adam-optimization-algorithm" title="Adam optimization algorithm">Adam optimization algorithm</a></li><li><a class="toc-href" href="#learning-rate-decay" title="Learning rate decay">Learning rate decay</a></li><li><a class="toc-href" href="#the-problem-of-local-optima" title="The problem of local optima">The problem of local optima</a></li><li><a class="toc-href" href="#assignment" title="assignment">assignment</a></li></ul></div>
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