add polynomial regression for tf 2.0

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@@ -484,6 +484,273 @@ creating<span class="w"> </span>a<span class="w"> </span>DOS<span class="w"> </s
 		
 		<item>
 			<guid isPermaLink="true">
+				https://web.navan.dev/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html
+			</guid>
+			<title>
+				Polynomial Regression Using TensorFlow 2.x
+			</title>
+			<description>
+				Predicting n-th degree polynomials using TensorFlow 2.x
+			</description>
+			<link>https://web.navan.dev/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html</link>
+			<pubDate>Thu, 21 Mar 2024 12:46:00 -0000</pubDate>
+			<content:encoded><![CDATA[<h1>Polynomial Regression Using TensorFlow 2.x</h1>
+
+<p>I have a similar post titled <a rel="noopener" target="_blank" href="/posts/2019-12-16-TensorFlow-Polynomial-Regression.html">Polynomial Regression Using Tensorflow</a> that used <code>tensorflow.compat.v1</code> (Which still works as of TF 2.16). But, I thought it would be nicer to redo it with newer TF versions. </p>
+
+<p>I will be skipping all the introductions about polynomial regression and jumping straight to the code. Personally, I prefer using <code>scikit-learn</code> for this task.</p>
+
+<h2>Position vs Salary Dataset</h2>
+
+<p>Again, we will be using https://drive.google.com/file/d/1tNL4jxZEfpaP4oflfSn6pIHJX7Pachm9/view (Salary vs Position Dataset)</p>
+
+<p>If you are in a Python Notebook environment like Kaggle or Google Colaboratory, you can simply run:</p>
+
+<div class="codehilite">
+<pre><span></span><code><span class="nt">!wget</span><span class="na"> --no-check-certificate &#39;https</span><span class="p">:</span><span class="nc">//docs.google.com/uc?export</span><span class="o">=</span><span class="l">download&amp;id=1tNL4jxZEfpaP4oflfSn6pIHJX7Pachm9&#39; -O data.csv</span>
+</code></pre>
+</div>
+
+<h2>Code</h2>
+
+<p>If you just want to copy-paste the code, scroll to the bottom for the entire snippet. Here I will try and walk through setting up code for a 3rd-degree (cubic) polynomial</p>
+
+<h3>Imports</h3>
+
+<div class="codehilite">
+<pre><span></span><code><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+</code></pre>
+</div>
+
+<h3>Reading the Dataset</h3>
+
+<div class="codehilite">
+<pre><span></span><code><span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s2">&quot;data.csv&quot;</span><span class="p">)</span>
+</code></pre>
+</div>
+
+<h3>Variables and Constants</h3>
+
+<p>Here, we initialize the X and Y values as constants, since they are not going to change. The coefficients are defined as variables.</p>
+
+<div class="codehilite">
+<pre><span></span><code><span class="n">X</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">&quot;Level&quot;</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">&quot;Salary&quot;</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+
+<span class="n">coefficients</span> <span class="o">=</span> <span class="p">[</span><span class="n">tf</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">()</span> <span class="o">*</span> <span class="mf">0.01</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)]</span>
+</code></pre>
+</div>
+
+<p>Here, <code>X</code> and <code>Y</code> are the values from our dataset. We initialize the coefficients for the equations as small random values.</p>
+
+<p>These coefficients are evaluated by Tensorflow's <code>tf.math.poyval</code> function which returns the n-th order polynomial based on how many coefficients are passed. Since our list of coefficients contains 4 different variables, it will be evaluated as:</p>
+
+<pre><code>y = (x**3)*coefficients[3] + (x**2)*coefficients[2] + (x**1)*coefficients[1] (x**0)*coefficients[0]
+</code></pre>
+
+<p>Which is equivalent to the general cubic equation:</p>
+
+<p><script type="text/javascript"
+  src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></p>
+
+</script>
+
+$$
+y = ax^3 + bx^2 + cx + d
+$$
+
+### Optimizer Selection & Training
+<div class="codehilite">
+
+<pre><span></span><code><span class="n">optimizer</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
+<span class="n">num_epochs</span> <span class="o">=</span> <span class="mi">10_000</span>
+
+<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_epochs</span><span class="p">):</span>
+    <span class="k">with</span> <span class="n">tf</span><span class="o">.</span><span class="n">GradientTape</span><span class="p">()</span> <span class="k">as</span> <span class="n">tape</span><span class="p">:</span>
+        <span class="n">y_pred</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">math</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">coefficients</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">reduce_mean</span><span class="p">(</span><span class="n">tf</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">))</span>
+    <span class="n">grads</span> <span class="o">=</span> <span class="n">tape</span><span class="o">.</span><span class="n">gradient</span><span class="p">(</span><span class="n">loss</span><span class="p">,</span> <span class="n">coefficients</span><span class="p">)</span>
+    <span class="n">optimizer</span><span class="o">.</span><span class="n">apply_gradients</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">grads</span><span class="p">,</span> <span class="n">coefficients</span><span class="p">))</span>
+    <span class="k">if</span> <span class="p">(</span><span class="n">epoch</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Epoch: </span><span class="si">{</span><span class="n">epoch</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s2">, Loss: </span><span class="si">{</span><span class="n">loss</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span><span class="si">}</span><span class="s2">&quot;</span>
+</code></pre>
+
+</div>
+
+
+In TensorFlow 1, we would have been using `tf.Session` instead. 
+
+Here we are using `GradientTape()` instead, to keep track of the loss evaluation and coefficients. This is crucial, as our optimizer needs these gradients to be able to optimize our coefficients.
+
+Our loss function is Mean Squared Error (MSE)
+
+$$
+= \frac{1}{n}\sum_{i=1}^{n} (Y_i - \^{Y_i})
+$$
+
+Where $\^{Y_i}$ is the predicted value and $Y_i$ is the actual value
+
+### Plotting Final Coefficients
+<div class="codehilite">
+
+<pre><span></span><code><span class="n">final_coefficients</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coefficients</span><span class="p">]</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Final Coefficients:&quot;</span><span class="p">,</span> <span class="n">final_coefficients</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">&quot;Level&quot;</span><span class="p">],</span> <span class="n">df</span><span class="p">[</span><span class="s2">&quot;Salary&quot;</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Original Data&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="s2">&quot;Level&quot;</span><span class="p">],[</span><span class="n">tf</span><span class="o">.</span><span class="n">math</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">final_coefficients</span><span class="p">,</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">df</span><span class="p">[</span><span class="s2">&quot;Level&quot;</span><span class="p">]])</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Salary&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Position&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Salary vs Position&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</code></pre>
+
+</div>
+
+
+
+## Code Snippet for a Polynomial of Degree N
+
+### Using Gradient Tape
+
+This should work regardless of the Keras backend version (2 or 3)
+<div class="codehilite">
+
+<pre><span></span><code><span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s2">&quot;data.csv&quot;</span><span class="p">)</span>
+
+<span class="c1">############################</span>
+<span class="c1">## Change Parameters Here ##</span>
+<span class="c1">############################</span>
+<span class="n">x_column</span> <span class="o">=</span> <span class="s2">&quot;Level&quot;</span>         <span class="c1">#</span>
+<span class="n">y_column</span> <span class="o">=</span> <span class="s2">&quot;Salary&quot;</span>        <span class="c1">#</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>                 <span class="c1">#</span>
+<span class="n">learning_rate</span> <span class="o">=</span> <span class="mf">0.3</span>        <span class="c1">#</span>
+<span class="n">num_epochs</span> <span class="o">=</span> <span class="mi">25_000</span>        <span class="c1">#</span>
+<span class="c1">############################</span>
+
+<span class="n">X</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">x_column</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">y_column</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+
+<span class="n">coefficients</span> <span class="o">=</span> <span class="p">[</span><span class="n">tf</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">()</span> <span class="o">*</span> <span class="mf">0.01</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">degree</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
+
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">learning_rate</span><span class="o">=</span><span class="n">learning_rate</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_epochs</span><span class="p">):</span>
+    <span class="k">with</span> <span class="n">tf</span><span class="o">.</span><span class="n">GradientTape</span><span class="p">()</span> <span class="k">as</span> <span class="n">tape</span><span class="p">:</span>
+        <span class="n">y_pred</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">math</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">coefficients</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+        <span class="n">loss</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">reduce_mean</span><span class="p">(</span><span class="n">tf</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">Y</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">))</span>
+    <span class="n">grads</span> <span class="o">=</span> <span class="n">tape</span><span class="o">.</span><span class="n">gradient</span><span class="p">(</span><span class="n">loss</span><span class="p">,</span> <span class="n">coefficients</span><span class="p">)</span>
+    <span class="n">optimizer</span><span class="o">.</span><span class="n">apply_gradients</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">grads</span><span class="p">,</span> <span class="n">coefficients</span><span class="p">))</span>
+    <span class="k">if</span> <span class="p">(</span><span class="n">epoch</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Epoch: </span><span class="si">{</span><span class="n">epoch</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s2">, Loss: </span><span class="si">{</span><span class="n">loss</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+
+<span class="n">final_coefficients</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coefficients</span><span class="p">]</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Final Coefficients:&quot;</span><span class="p">,</span> <span class="n">final_coefficients</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Final Equation:&quot;</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s2">&quot; &quot;</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+  <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">final_coefficients</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="si">}</span><span class="s2"> * x^</span><span class="si">{</span><span class="n">degree</span><span class="o">-</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s2">&quot; + &quot;</span> <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">degree</span> <span class="k">else</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Original Data&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,[</span><span class="n">tf</span><span class="o">.</span><span class="n">math</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">final_coefficients</span><span class="p">,</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">df</span><span class="p">[</span><span class="n">x_column</span><span class="p">]]),</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Our Poynomial&quot;</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="n">y_column</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="n">x_column</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">x_column</span><span class="si">}</span><span class="s2"> vs </span><span class="si">{</span><span class="n">y_column</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</code></pre>
+
+</div>
+
+
+### Without Gradient Tape
+
+This relies on the Optimizer's `minimize` function and uses the `var_list` parameter to update the variables.
+
+This will not work with Keras 3 backend in TF 2.16.0 and above unless you switch to the legacy backend.
+<div class="codehilite">
+
+<pre><span></span><code><span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s2">&quot;data.csv&quot;</span><span class="p">)</span>
+
+<span class="c1">############################</span>
+<span class="c1">## Change Parameters Here ##</span>
+<span class="c1">############################</span>
+<span class="n">x_column</span> <span class="o">=</span> <span class="s2">&quot;Level&quot;</span>         <span class="c1">#</span>
+<span class="n">y_column</span> <span class="o">=</span> <span class="s2">&quot;Salary&quot;</span>        <span class="c1">#</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>                 <span class="c1">#</span>
+<span class="n">learning_rate</span> <span class="o">=</span> <span class="mf">0.3</span>        <span class="c1">#</span>
+<span class="n">num_epochs</span> <span class="o">=</span> <span class="mi">25_000</span>        <span class="c1">#</span>
+<span class="c1">############################</span>
+
+<span class="n">X</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">x_column</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+<span class="n">Y</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">df</span><span class="p">[</span><span class="n">y_column</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span>
+
+<span class="n">coefficients</span> <span class="o">=</span> <span class="p">[</span><span class="n">tf</span><span class="o">.</span><span class="n">Variable</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">()</span> <span class="o">*</span> <span class="mf">0.01</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">degree</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
+
+<span class="n">optimizer</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="n">learning_rate</span><span class="o">=</span><span class="n">learning_rate</span><span class="p">)</span>
+
+<span class="k">def</span> <span class="nf">loss_function</span><span class="p">():</span>
+  <span class="n">pred_y</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">math</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">coefficients</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span>
+  <span class="k">return</span> <span class="n">tf</span><span class="o">.</span><span class="n">reduce_mean</span><span class="p">(</span><span class="n">tf</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">pred_y</span> <span class="o">-</span> <span class="n">Y</span><span class="p">))</span>
+
+<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_epochs</span><span class="p">):</span>
+    <span class="n">optimizer</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">loss_function</span><span class="p">,</span> <span class="n">var_list</span><span class="o">=</span><span class="n">coefficients</span><span class="p">)</span>
+    <span class="k">if</span> <span class="p">(</span><span class="n">epoch</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+        <span class="n">current_loss</span> <span class="o">=</span> <span class="n">loss_function</span><span class="p">()</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+        <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Epoch </span><span class="si">{</span><span class="n">epoch</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s2">: Training Loss: </span><span class="si">{</span><span class="n">current_loss</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+
+<span class="n">final_coefficients</span> <span class="o">=</span> <span class="n">coefficients</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Final Coefficients:&quot;</span><span class="p">,</span> <span class="n">final_coefficients</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Final Equation:&quot;</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s2">&quot; &quot;</span><span class="p">)</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+  <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">final_coefficients</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="si">}</span><span class="s2"> * x^</span><span class="si">{</span><span class="n">degree</span><span class="o">-</span><span class="n">i</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s2">&quot; + &quot;</span> <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">degree</span> <span class="k">else</span> <span class="s2">&quot;</span><span class="se">\n</span><span class="s2">&quot;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Original Data&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,[</span><span class="n">tf</span><span class="o">.</span><span class="n">math</span><span class="o">.</span><span class="n">polyval</span><span class="p">(</span><span class="n">final_coefficients</span><span class="p">,</span> <span class="n">tf</span><span class="o">.</span><span class="n">constant</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span><span class="o">.</span><span class="n">numpy</span><span class="p">()</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">df</span><span class="p">[</span><span class="n">x_column</span><span class="p">]],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Our Polynomial&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="n">y_column</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="n">x_column</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">x_column</span><span class="si">}</span><span class="s2"> vs </span><span class="si">{</span><span class="n">y_column</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</code></pre>
+
+</div>
+
+
+
+As always, remember to tweak the parameters and choose the correct model for the job. A polynomial regression model might not even be the best model for this particular dataset.
+
+## Further Programming
+
+How would you modify this code to use another type of nonlinear regression? Say, $ y = ab^x $
+
+Hint: Your loss calculation would be similar to:
+<div class="codehilite">
+
+<pre><span></span><code><span class="n">bx</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">coefficients</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">X</span><span class="p">)</span>
+<span class="n">pred_y</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">math</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">coefficients</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">bx</span><span class="p">)</span>
+<span class="n">loss</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">reduce_mean</span><span class="p">(</span><span class="n">tf</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">pred_y</span> <span class="o">-</span> <span class="n">Y</span><span class="p">))</span>
+</code></pre>
+
+<p></div></p>
+]]></content:encoded>
+		</item>
+		
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