add polynomial regression for tf 2.0

9 files changed, 881 insertions, 15 deletions

diff --git a/Content/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.md b/Content/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.md
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@@ -0,0 +1,241 @@
+---
+date: 2024-03-21 12:46
+description: Predicting n-th degree polynomials using TensorFlow 2.x
+tags: Tutorial, Tensorflow, Colab
+---
+
+# Polynomial Regression Using TensorFlow 2.x
+
+I have a similar post titled [Polynomial Regression Using Tensorflow](/posts/2019-12-16-TensorFlow-Polynomial-Regression.html) that used `tensorflow.compat.v1` (Which still works as of TF 2.16). But, I thought it would be nicer to redo it with newer TF versions. 
+
+I will be skipping all the introductions about polynomial regression and jumping straight to the code. Personally, I prefer using `scikit-learn` for this task.
+
+## Position vs Salary Dataset
+
+Again, we will be using https://drive.google.com/file/d/1tNL4jxZEfpaP4oflfSn6pIHJX7Pachm9/view (Salary vs Position Dataset)
+
+If you are in a Python Notebook environment like Kaggle or Google Colaboratory, you can simply run:
+```Termcap
+!wget --no-check-certificate 'https://docs.google.com/uc?export=download&id=1tNL4jxZEfpaP4oflfSn6pIHJX7Pachm9' -O data.csv
+```
+
+## Code
+
+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
+
+### Imports
+
+```python
+import pandas as pd
+import tensorflow as tf
+import matplotlib.pyplot as plt
+import numpy as np
+```
+
+### Reading the Dataset
+
+```python
+df = pd.read_csv("data.csv")
+```
+
+### Variables and Constants
+
+Here, we initialize the X and Y values as constants, since they are not going to change. The coefficients are defined as variables.
+
+```python
+X = tf.constant(df["Level"], dtype=tf.float32)
+Y = tf.constant(df["Salary"], dtype=tf.float32)
+
+coefficients = [tf.Variable(np.random.randn() * 0.01, dtype=tf.float32) for _ in range(4)]
+```
+
+Here, `X` and `Y` are the values from our dataset. We initialize the coefficients for the equations as small random values.
+
+These coefficients are evaluated by Tensorflow's `tf.math.poyval` 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:
+
+```
+y = (x**3)*coefficients[3] + (x**2)*coefficients[2] + (x**1)*coefficients[1] (x**0)*coefficients[0]
+``` 
+
+Which is equivalent to the general cubic equation:
+
+<script type="text/javascript"
+  src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
+</script>
+
+$$
+y = ax^3 + bx^2 + cx + d
+$$
+
+### Optimizer Selection & Training
+
+```python
+optimizer = tf.keras.optimizers.Adam(learning_rate=0.3)
+num_epochs = 10_000
+
+for epoch in range(num_epochs):
+    with tf.GradientTape() as tape:
+        y_pred = tf.math.polyval(coefficients, X)
+        loss = tf.reduce_mean(tf.square(y - y_pred))
+    grads = tape.gradient(loss, coefficients)
+    optimizer.apply_gradients(zip(grads, coefficients))
+    if (epoch+1) % 1000 == 0:
+        print(f"Epoch: {epoch+1}, Loss: {loss.numpy()}"
+```
+
+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
+
+```python
+final_coefficients = [c.numpy() for c in coefficients]
+print("Final Coefficients:", final_coefficients)
+
+plt.plot(df["Level"], df["Salary"], label="Original Data")
+plt.plot(df["Level"],[tf.math.polyval(final_coefficients, tf.constant(x, dtype=tf.float32)).numpy() for x in df["Level"]])
+plt.ylabel('Salary')
+plt.xlabel('Position')
+plt.title("Salary vs Position")
+plt.show()
+```
+
+
+## Code Snippet for a Polynomial of Degree N
+
+### Using Gradient Tape
+
+This should work regardless of the Keras backend version (2 or 3)
+
+```python
+import tensorflow as tf
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+
+df = pd.read_csv("data.csv")
+
+############################
+## Change Parameters Here ##
+############################
+x_column = "Level"         #
+y_column = "Salary"        #
+degree = 2                 #
+learning_rate = 0.3        #
+num_epochs = 25_000        #
+############################
+
+X = tf.constant(df[x_column], dtype=tf.float32)
+Y = tf.constant(df[y_column], dtype=tf.float32)
+
+coefficients = [tf.Variable(np.random.randn() * 0.01, dtype=tf.float32) for _ in range(degree + 1)]
+
+optimizer = tf.keras.optimizers.Adam(learning_rate=learning_rate)
+
+for epoch in range(num_epochs):
+    with tf.GradientTape() as tape:
+        y_pred = tf.math.polyval(coefficients, X)
+        loss = tf.reduce_mean(tf.square(Y - y_pred))
+    grads = tape.gradient(loss, coefficients)
+    optimizer.apply_gradients(zip(grads, coefficients))
+    if (epoch+1) % 1000 == 0:
+        print(f"Epoch: {epoch+1}, Loss: {loss.numpy()}")
+
+final_coefficients = [c.numpy() for c in coefficients]
+print("Final Coefficients:", final_coefficients)
+
+print("Final Equation:", end=" ")
+for i in range(degree+1):
+  print(f"{final_coefficients[i]} * x^{degree-i}", end=" + " if i < degree else "\n")
+
+plt.plot(X, Y, label="Original Data")
+plt.plot(X,[tf.math.polyval(final_coefficients, tf.constant(x, dtype=tf.float32)).numpy() for x in df[x_column]]), label="Our Poynomial"
+plt.ylabel(y_column)
+plt.xlabel(x_column)
+plt.title(f"{x_column} vs {y_column}")
+plt.legend()
+plt.show()
+```
+
+### 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.
+
+```python
+import tensorflow as tf
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+
+df = pd.read_csv("data.csv")
+
+############################
+## Change Parameters Here ##
+############################
+x_column = "Level"         #
+y_column = "Salary"        #
+degree = 2                 #
+learning_rate = 0.3        #
+num_epochs = 25_000        #
+############################
+
+X = tf.constant(df[x_column], dtype=tf.float32)
+Y = tf.constant(df[y_column], dtype=tf.float32)
+
+coefficients = [tf.Variable(np.random.randn() * 0.01, dtype=tf.float32) for _ in range(degree + 1)]
+
+optimizer = tf.keras.optimizers.Adam(learning_rate=learning_rate)
+
+def loss_function():
+  pred_y = tf.math.polyval(coefficients, X)
+  return tf.reduce_mean(tf.square(pred_y - Y))
+
+for epoch in range(num_epochs):
+    optimizer.minimize(loss_function, var_list=coefficients)
+    if (epoch+1) % 1000 == 0:
+        current_loss = loss_function().numpy()
+        print(f"Epoch {epoch+1}: Training Loss: {current_loss}")
+
+final_coefficients = coefficients.numpy()
+print("Final Coefficients:", final_coefficients)
+
+print("Final Equation:", end=" ")
+for i in range(degree+1):
+  print(f"{final_coefficients[i]} * x^{degree-i}", end=" + " if i < degree else "\n")
+
+plt.plot(X, Y, label="Original Data")
+plt.plot(X,[tf.math.polyval(final_coefficients, tf.constant(x, dtype=tf.float32)).numpy() for x in df[x_column]], label="Our Polynomial")
+plt.ylabel(y_column)
+plt.xlabel(x_column)
+plt.legend()
+plt.title(f"{x_column} vs {y_column}")
+plt.show()
+```
+
+
+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:
+
+```python
+bx = tf.pow(coefficients[1], X)
+pred_y = tf.math.multiply(coefficients[0], bx)
+loss = tf.reduce_mean(tf.square(pred_y - Y))
+```
+
+
diff --git a/Resources/images/opengraph/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.png b/Resources/images/opengraph/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.png
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diff --git a/docs/feed.rss b/docs/feed.rss
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+++ b/docs/feed.rss
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 		<title>Navan's Archive</title>
 		<description>Rare Tips, Tricks and Posts</description>
 		<link>https://web.navan.dev/</link><language>en</language>
-		<lastBuildDate>Fri, 15 Mar 2024 15:00:25 -0000</lastBuildDate>
-		<pubDate>Fri, 15 Mar 2024 15:00:25 -0000</pubDate>
+		<lastBuildDate>Thu, 21 Mar 2024 13:54:34 -0000</lastBuildDate>
+		<pubDate>Thu, 21 Mar 2024 13:54:34 -0000</pubDate>
 		<ttl>250</ttl>
 		<atom:link href="https://web.navan.dev/feed.rss" rel="self" type="application/rss+xml"/>
 		
@@ -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>
+		
+		<item>
+			<guid isPermaLink="true">
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+	<li><a href="/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html">Polynomial Regression Using TensorFlow 2.x</a></li>
+	<ul>
+		<li>Predicting n-th degree polynomials using TensorFlow 2.x</li>
+		<li>Published On: 2024-03-21 12:46</li>
+		<li>Tags: 
+			
+            <a href='/tags/Tutorial.html'>Tutorial</a>, 
+			
+            <a href='/tags/Tensorflow.html'>Tensorflow</a>, 
+			
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+			
+	</ul>
+
+
 	<li><a href="/posts/2024-03-15-setting-up-macos-for-8088-dos-dev.html">Cross-Compiling Hello World for DOS on macOS</a></li>
 	<ul>
 		<li>This goes through compiling Open Watcom 2 and creating simple hello-world exampls</li>
@@ -104,19 +119,6 @@
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+
+	<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>
+
+	<blockquote>If you have scrolled this far, consider subscribing to my mailing list <a href="https://listmonk.navan.dev/subscription/form">here.</a> You can subscribe to either a specific type of post you are interested in, or subscribe to everything with the "Everything" list.</blockquote>
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\ No newline at end of file
diff --git a/docs/posts/index.html b/docs/posts/index.html
index 2d9d613..d886b19 100644
--- a/docs/posts/index.html
+++ b/docs/posts/index.html
@@ -52,6 +52,21 @@
 
 <ul>
 
+	<li><a href="/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html">Polynomial Regression Using TensorFlow 2.x</a></li>
+	<ul>
+		<li>Predicting n-th degree polynomials using TensorFlow 2.x</li>
+		<li>Published On: 2024-03-21 12:46</li>
+		<li>Tags: 
+			
+            <a href='/tags/Tutorial.html'>Tutorial</a>, 
+			
+            <a href='/tags/Tensorflow.html'>Tensorflow</a>, 
+			
+            <a href='/tags/Colab.html'>Colab</a>
+			
+	</ul>
+
+
 	<li><a href="/posts/2024-03-15-setting-up-macos-for-8088-dos-dev.html">Cross-Compiling Hello World for DOS on macOS</a></li>
 	<ul>
 		<li>This goes through compiling Open Watcom 2 and creating simple hello-world exampls</li>
diff --git a/docs/tags/Colab.html b/docs/tags/Colab.html
index a3721a7..fd8ef08 100644
--- a/docs/tags/Colab.html
+++ b/docs/tags/Colab.html
@@ -49,6 +49,21 @@
 
 <ul>
 
+	<li><a href="/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html">Polynomial Regression Using TensorFlow 2.x</a></li>
+	<ul>
+		<li>Predicting n-th degree polynomials using TensorFlow 2.x</li>
+		<li>Published On: 2024-03-21 12:46</li>
+		<li>Tags: 
+			
+            <a href='/tags/Tutorial.html'>Tutorial</a>, 
+			
+            <a href='/tags/Tensorflow.html'>Tensorflow</a>, 
+			
+            <a href='/tags/Colab.html'>Colab</a>
+			
+	</ul>
+
+
 	<li><a href="/posts/2020-07-01-Install-rdkit-colab.html">Installing RDKit on Google Colab</a></li>
 	<ul>
 		<li>Install RDKit on Google Colab with one code snippet.</li>
diff --git a/docs/tags/Tensorflow.html b/docs/tags/Tensorflow.html
index 3bcb911..04006bb 100644
--- a/docs/tags/Tensorflow.html
+++ b/docs/tags/Tensorflow.html
@@ -49,6 +49,21 @@
 
 <ul>
 
+	<li><a href="/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html">Polynomial Regression Using TensorFlow 2.x</a></li>
+	<ul>
+		<li>Predicting n-th degree polynomials using TensorFlow 2.x</li>
+		<li>Published On: 2024-03-21 12:46</li>
+		<li>Tags: 
+			
+            <a href='/tags/Tutorial.html'>Tutorial</a>, 
+			
+            <a href='/tags/Tensorflow.html'>Tensorflow</a>, 
+			
+            <a href='/tags/Colab.html'>Colab</a>
+			
+	</ul>
+
+
 	<li><a href="/posts/2019-12-16-TensorFlow-Polynomial-Regression.html">Polynomial Regression Using TensorFlow</a></li>
 	<ul>
 		<li>Polynomial regression using TensorFlow</li>