test

Author: akcube

content/blog/basics-of-supervised-learning-linear-regression.md

@@ -1,6 +1,6 @@
 ---
 author: Kishore Kumar
-date: 2025-08-18 06:04:35+0530
+date: 2025-08-18 06:20:02+0530
 doc: 2025-08-18 06:03:53+0530
 title: Basics Of Supervised Learning - Linear Regression
 topics: []
@@ -106,23 +106,71 @@ Finding the best set of parameters $\theta$ now just means finding the best valu
 
 Consider the function $y = (x-3)^2 + 2$. We can find *a* minimum by differentiating it and setting $\frac{dy}{dx} = 0$. This gives us $\frac{dy}{dx} = \frac{d(x^2 - 6x + 9 + 2)}{dx} = 2x - 6 = 0 \implies x = 3$. At $x = 3$, $y = 2$. By double differentiating it, we get $\frac{d^2y}{dx^2} = 2 \gt 0$ which means it's a minimum. Since this curve is concave-up shaped, it has just one minimum and hence it is the *global* minimum. 
 
-```desmos-graph
-y = (x-3)^2 + 2
-A = (3,2)
-```
+<div id="desmos-8a6985bc" style="width: 100%; height: 400px; margin: 20px 0; border: 1px solid #ccc;"></div>
+<script>
+(function() {
+	if (typeof Desmos === 'undefined') {
+		console.error('Desmos API not loaded');
+		return;
+	}
+	var calculator = Desmos.GraphingCalculator(
+		document.getElementById('desmos-8a6985bc'),
+		{
+			expressions: false,
+			settingsMenu: false,
+			zoomFit: true,
+			showGrid: true,
+			showXAxis: true,
+			showYAxis: true
+		}
+	);
+	
+	// Set expressions from parsed content
+		calculator.setExpression({"id": "expr_0", "latex": "y = (x-3)^2 + 2", "color": "#c74440"});
+	calculator.setExpression({"id": "expr_1", "latex": "A = (3,2)", "color": "#2d70b3"});
+	
+	calculator.setMathBounds({
+		left: -10, right: 10,
+		bottom: -10, top: 10
+	});
+})();
+</script>
 
 However, this same approach isn't very feasible for more complicated functions. Sometimes solving for all possible values of $\frac{dy}{dx} = 0$ is difficult (or impossible). Checking the double derivative for complex functions might often be inconclusive and we may need to check higher order functions or use numerical methods. When we're dealing with multiple variables and higher dimensional functions, the computations can get extremely complex and difficult to compute. So instead, people rely on iterative numerical optimization algorithms. 
 
 **Gradient Descent** is one such iterative optimization algorithm used to find the minimum of a function. We start with some random initial values for $\theta$ and repeatedly update them by taking small steps in the direction of the steepest descent of the cost function. Consider this more complex function below:
 
-```desmos-graph
-C(w) = 0.1w^5 - 0.3w^4 - 1.2w^3 + 0.8w^2 + 1.5w
-
-w_0 = 0.5
-(w_0, C(w_0))
-
-T(w) = C(w_0) + (0.5w_0^4 - 1.2w_0^3 - 3.6w_0^2 + 1.6w_0 + 1.5)(w - w_0)
-```
+<div id="desmos-48dbb6d3" style="width: 100%; height: 400px; margin: 20px 0; border: 1px solid #ccc;"></div>
+<script>
+(function() {
+	if (typeof Desmos === 'undefined') {
+		console.error('Desmos API not loaded');
+		return;
+	}
+	var calculator = Desmos.GraphingCalculator(
+		document.getElementById('desmos-48dbb6d3'),
+		{
+			expressions: false,
+			settingsMenu: false,
+			zoomFit: true,
+			showGrid: true,
+			showXAxis: true,
+			showYAxis: true
+		}
+	);
+	
+	// Set expressions from parsed content
+		calculator.setExpression({"id": "expr_0", "latex": "C(w) = 0.1w^5 - 0.3w^4 - 1.2w^3 + 0.8w^2 + 1.5w", "color": "#c74440"});
+	calculator.setExpression({"id": "expr_1", "latex": "w_0 = 0.5", "color": "#2d70b3"});
+	calculator.setExpression({"id": "point_2", "latex": "(w_0, C(w_0))"});
+	calculator.setExpression({"id": "expr_3", "latex": "T(w) = C(w_0) + (0.5w_0^4 - 1.2w_0^3 - 3.6w_0^2 + 1.6w_0 + 1.5)(w - w_0)", "color": "#fa7e19"});
+	
+	calculator.setMathBounds({
+		left: -10, right: 10,
+		bottom: -10, top: 10
+	});
+})();
+</script>
 
 Given any point $w_0$, we can find out the answer to *"Which direction should I move in to reduce the value of the function?"* by computing the derivative (slope) of the function at that point $w_0$. If the slope is positive, we should move left to reduce the value of the function. If it's negative, we should move right. If we do this repeatedly, we'll eventually approach & reach some **local minimum** of the function. The visualization that really helps sell this idea is that of a ball rolling down the 2D hills (curves generated by the function). If we generate *enough* random initial points (or balls) and perform this procedure, we should eventually hit a very good local minimum. 
 

hugo.toml

@@ -32,7 +32,7 @@ defaultMarkdownHandler = "goldmark"
   description = "If we keep holding onto yesterday, what are we going to be tomorrow?"
   keywords = ["kishore", "kumar", "akcube", "iiit", "iiith", "iiit-h", "international institute of information technology", "ICPC", "competitive programming", "optimization", "performance", "algorithms", "finance", "math", "c++", "cpp", "research", "quant", "finance", "ml", "deep learning", "reinforcement learning", "theory"]
   customCSS = []
-  customJS = []
+  customJS = ["https://www.desmos.com/api/v1.11/calculator.js"]
   dateFormat = "2 January 2006"
   emphasisWithDots = true
   since = "2024"