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CMU Introduction to Deep Learning 11785, Spring 2026: Lecture 5

CMU Introduction to Deep Learning 11785, Spring 2026: Lecture 5

Carnegie Mellon University Deep Learning

1:23:45

Overview

This lecture introduces the fundamental concepts of training neural networks, focusing on gradient descent and backpropagation. It begins with a review of empirical risk minimization and the role of loss functions like cross-entropy. The lecture then delves into notation for network parameters (weights and biases) and explains the gradient descent update rule. A significant portion is dedicated to a calculus refresher, introducing influence diagrams and the chain rule to visualize and compute derivatives for nested functions. The core of the lecture explains the forward and backward passes of backpropagation, detailing how gradients are computed layer by layer. It also touches upon special cases like vector activations, non-differentiable activations (ReLU), and the max activation, and introduces the concept of subgradients. Finally, it discusses the vectorization of these operations for efficiency, particularly on GPUs, and outlines the complete training algorithm.

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Chapters

  • •Review of empirical risk minimization and loss functions (e.g., cross-entropy).
  • •Goal: Minimize loss by adjusting network parameters (weights and biases).
  • •Gradient descent is used to minimize the loss function.
  • •Introduction to notation for network layers, weights, and biases.
  • •Understanding derivatives as multiplicative terms for change.
  • •Influence diagrams represent variable dependencies and their influence.
  • •Arrows in influence diagrams are weighted by derivatives.
  • •Chain rule for nested functions is visualized using influence diagrams.
  • •Illustrative example of a small network with neurons and activation functions.
  • •Notation for affine terms (weighted sum + bias) and neuron outputs.
  • •The forward pass computes all intermediate outputs by passing input through layers.
  • •Intermediate values must be stored for the backward pass.
  • •Backpropagation computes derivatives by working backward from the loss.
  • •It iteratively applies the chain rule to find gradients for weights and biases.
  • •The process starts with the derivative of the loss with respect to the network's output.
  • •Derivatives are computed for activations, affine terms, and finally weights.
  • •Handling vector activations (e.g., Softmax) requires summing over outputs.
  • •Non-differentiable activations (e.g., ReLU) use subgradients at points of non-differentiability.
  • •The max activation function has a derivative of 1 for the maximum input and 0 otherwise.
  • •Representing layers using vector and matrix operations (weights as matrices).
  • •Simplified forward pass equations: Z = WY + B, Y = F(Z).
  • •Vector calculus and Jacobians are used for the backward pass.
  • •Chain rule in vector form: derivatives are right-multiplied as we go backward.
  • •The complete algorithm involves initializing weights, performing forward and backward passes for each instance, and averaging gradients.
  • •Gradient descent uses these averaged gradients to update parameters.
  • •Vectorized operations significantly speed up computation, especially on GPUs.

Key Takeaways

  1. 1Gradient descent is the core optimization algorithm for training neural networks by minimizing a loss function.
  2. 2Backpropagation is an efficient algorithm for computing the gradients of the loss function with respect to all network parameters.
  3. 3Influence diagrams and the chain rule are powerful tools for understanding and deriving gradient computations.
  4. 4The forward pass computes network outputs and stores intermediate values; the backward pass uses these values to compute gradients.
  5. 5Special handling is required for vector activations and non-differentiable activation functions (using subgradients).
  6. 6Vectorizing operations using matrices and vectors significantly simplifies implementation and improves computational efficiency.
  7. 7The derivative of the loss with respect to a weight is the input to the weight multiplied by the derivative of the loss with respect to the output of the neuron.
  8. 8Backpropagation involves iteratively right-multiplying gradients by the derivatives of the current layer's operations as you move backward through the network.
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