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Deep Learning For Vision

Back-Propagation and Other Differentiation Algorithms

Back-propagation (backprop) is the central algorithm for computing gradients in deep learning, which enables effective training of neural networks by updating their parameters using gradient-based optimization methods.

1. Introduction

  • Feedforward propagation: Input data flows through the layers of the network, producing predictions.

  • Back-propagation: During training, error information from the loss function is efficiently propagated backward through the network, allowing computation of gradients for all weights and biases.

2. Back-Propagation Algorithm

Steps

  1. Forward Pass:

    • Compute activations for each neuron layer-by-layer from input to output, storing intermediate values (activations, weighted sums).

  2. Loss Calculation:

    • Compute the loss (cost) function J(θ)J(\theta) comparing predicted output to actual target.

  3. Backward Pass (Gradient Calculation):

    • Use the chain rule to efficiently compute the gradient of the loss function with respect to each parameter in the network.

      • Start at the output layer and propagate errors backward to update previous layers.

      • Gradients are calculated layer-by-layer using cached intermediate values.

  4. Parameter Update:

    • Parameters (weights and biases) are updated using a gradient descent optimizer:

      θnew=θold−α∂J∂θ\theta_{\text{new}} = \theta_{\text{old}} - \alpha \frac{\partial J}{\partial \theta}

      where α\alpha is the learning rate.

3. Mathematical Foundation

  • For each layer, partial derivatives of the loss with respect to the layer’s weights are computed using the stored activations and the derivatives of the activation functions (sigmoid, tanh, ReLU etc.).

  • Chain rule of calculus allows efficient computation of these derivatives even for deep networks.

4. Other Differentiation Algorithms

  • Reverse-mode automatic differentiation: Backpropagation is a special case of this algorithm, efficiently calculating derivatives through computational graphs.

  • Numerical differentiation: Approximates gradients by finite differences; less efficient and less accurate than analytical methods used in backpropagation.

  • Forward-mode automatic differentiation: Used for functions with fewer inputs than outputs, less common for deep networks.

5. Applications for Deep Learning in Vision

  • Image classification: Backpropagation enables CNNs to learn hierarchical features from images.

  • Segmentation and detection: Neural networks can learn spatial relationships via error minimization.

  • Generative models: Enables GANs and autoencoders to learn image generation and transformation.

6. Key Points

  • Backpropagation is essential for efficient gradient calculation in deep networks, allowing the rapid training of models with many layers.

  • Automatic differentiation frameworks (PyTorch, TensorFlow) leverage these principles, freeing users from manual gradient computation.

  • Differentiation algorithms underpin the optimization of deep learning models and drive progress in computer vision and AI.