Mathematics Behind Backpropagation | Theory and Python Code

Explore a simple neural network with one input, two weights, and a true label. Trace backpropagation and gradient descent toward the target output, using a hidden layer and computation graph.

Perform the forward pass of a neural network before updating weights, calculating y_hat from x, w1, and w2, and use mean squared error to guide updates.

Explore the roadmap to understanding backpropagation by linking derivatives, partial derivatives, gradients, and gradient descent to reduce neural network loss and update weights.

Explore derivatives as measures of a function’s rate of change, illustrated by slopes of tangent lines and numerical examples, laying foundations before partial derivatives and gradients in backpropagation.

Explore a numerical derivative example using f(x)=x^3-3x^2+2x+1 to illustrate the power rule, tangent slope, rate of change, and backpropagation intuition with small delta x.

Learn how partial derivatives isolate each input in multivariable functions, revealing how x, y, or z changes the output, and relate these rates to weight updates in backpropagation.

Learn that a gradient collects all partial derivatives and forms the gradient vector. It points to the steepest increase, guiding gradient descent toward the minimum.

Learn how partial derivatives reveal the rate of change for each input on f(x,y)=x^2+y^2, with gradients six and eight at x=3, y=4.

Explore back propagation by deriving gradients of the loss with respect to w1 and w2 via the chain rule after the forward pass with mean squared error, guiding backpropagation.

Explore the chain rule for composite functions and how multiplying partial derivatives reveals how weights influence the loss in neural networks.

Compute the gradient of the mean squared error with respect to y_hat using the chain rule, yielding dL/dy_hat = y_hat - y to guide backpropagation and adjust W1 and W2.

Visualize the loss function with respect to y_hat, compute its gradient and partial derivative, then use backpropagation and chain rule to adjust weights via gradient descent.

Use the chain rule to compute the gradient of w1 from the partial derivative of y hat, link it to the loss, and prepare backpropagation and gradient descent.

Visualize gradient descent on a 3d loss surface for w1 and w2, tracking how initial weights update iteratively via back propagation to minimize loss and approach the ideal solution.

Learn gradient descent: minimize the loss function like mean squared error by updating weights w1 and w2 opposite the gradient, scaled by the learning rate alpha.

Understand how the learning rate alpha sets the step size in gradient descent, preventing overshoot and slow progress, and how adaptive optimizers like Adagrad, RMSprop, and Adam adjust alpha.

Explore how moving in the opposite direction of the gradient updates weights and predictions via gradient descent, using positive and negative gradients to reduce loss.

Compute the gradients of the loss with respect to w1 and w2 via the chain rule, then update with gradient descent (alpha 0.01) to reduce the mean squared error.

Implement gradient descent and backpropagation from scratch in Google Colab using numpy, performing forward pass and mean squared error for a net with x=2, y=20, w1=2, w2=0.5, learning rate of 0.01.

Implement backpropagation by computing the gradients (partial derivatives) of the loss with respect to w1 and w2 using the chain rule, based on the forward pass and y hat.

Implement a training loop for a simple neural network, initializing weights, performing forward passes, calculating mean squared error, and computing gradients for backpropagation before updating weights in gradient descent.

Implement gradient descent to update neural network weights w1 and w2 using gradients dL/dW1 and dL/dW2, perform forward passes to observe loss reduction across ten training epochs.

Demonstrate back propagation by running a simple neural network, update weights W1 and W2 through gradient descent, and confirm loss decreases toward zero as y hat approaches 20.

Demonstrate backpropagation and gradient descent on a network with two inputs and sigmoid activation. Trace the forward pass, composite functions, and mean squared error to show activation and non-linearity.

Conduct the forward pass to compute h1 and y hat using sigmoid activations with weights w1, w2, and w3. Then assess the mean squared error loss before moving to backpropagation.

Learn to compute backpropagation gradients for weights W1, W2, W3, propagate the error through the sigmoid to the MSE loss, and update with gradient descent at learning rate 0.1.

Derive the sigmoid derivative using the chain rule, showing dyhat/dz2 equals yhat times (1 minus yhat) and explaining how small changes in z2 affect yhat.

Apply the chain rule to relate z2 to the loss by combining the gradient of y hat with respect to z2 and the gradient of the loss with respect to y hat.

Apply the chain rule: dL/dW3 = dL/dZ2 × dZ2/dW3 = -0.11222 × 0.3775 = -0.04239, with h1 = 0.3775, so a negative gradient suggests increasing W3 to minimize the loss.

The lecture demonstrates how to compute the gradient dL/dz1 by chaining partial derivatives: dz2/dh1 = w3 and dh1/dz1 = h1(1−h1). Using dL/dz2 = -0.11222, w3 = 0.5, and h1 = 0.3775, it yields dL/dz1 ≈ -0.01318 and shows how backpropagation proceeds toward W1 and W2.

Demonstrate a forward pass and loss calculation for an advanced neural network in a Colab notebook, implementing sigmoid, its derivative, and gradient descent.

Update w1, w2, and w3 via gradient descent using old values minus learning rate times their loss derivatives. Print epoch, y hat, error, and updated weights.

Run the neural network through ten epochs, compare forward pass predictions y hat with targets, and observe decreasing loss as weights w1, w2, w3 update via backprop and gradient descent.