Vanishing gradient problem

As explained in Math and Architectures of Deep Learning

The training difficulty in deep networks where repeated multiplication of small derivatives makes early-layer gradients nearly zero.

The vanishing gradient problem makes deep-network training ineffective when gradients propagated toward early layers become nearly zero. With backpropagation, the chain rule multiplies derivatives from later layers backward; repeatedly multiplying small values produces an increasingly smaller gradient. Sigmoid units are especially vulnerable when their preactivation is very high or low, because their derivatives become small. section 11.2.1

ReLU addresses this difficulty with ReLU(x) = max(0, x). Its derivative is 1 for positive inputs and 0 elsewhere, so positive activations do not introduce the small sigmoid derivative that causes vanishing gradients. ReLU therefore supports faster convergence in deep networks and appears throughout VGG-style convolutional blocks. section 11.2.1 section 11.5

Deep convolutional architectures

A deep VGG network stacks convolutional blocks and ReLU layers, avoiding sigmoid-based gradient shrinkage. GoogLeNet addresses the same depth-related risk differently by adding two auxiliary classifiers: intermediate losses provide training signals so the middle of the network does not “die out.” section 11.2.1 section 11.2.2

Figure 11.6 Graph of a 1D sigmoid function (dotted curve) and its derivative (solid curve)

Compare the sigmoid curve with its derivative to see why saturated sigmoid units transmit only small gradients. section 11.2.1

How Math and Architectures of Deep Learning covers it

How deep it goes

Traces gradient shrinkage through the chain rule and derivatives, but defers detailed optimization analysis.

Examples

Uses sigmoid and derivative graphs, the ReLU equation, VGG blocks, and GoogLeNet auxiliary classifiers.

What it assumes

Assumes backpropagation and the chain rule while explaining sigmoid and ReLU behavior in network training.

What you can do after

Evaluate/understand why ReLU and auxiliary losses improve trainability in deep convolutional architectures.

How it connects

Connects backpropagation, chain rule, sigmoid, ReLU, Inception, auxiliary classifiers, and deep convolutional networks.

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Related concepts

• ReLUaddressed by · ReLU, or rectified linear unit, is an activation function that maps an input x to max(0, x).
• residual connectionaddressed by · A residual connection is a neural-network pathway that carries an earlier representation around one or more transformations to a later computation, where it can be combined with the transformed representation.
• LSTMaddressed by · An LSTM is a recurrent neural network architecture with gated memory mechanisms that control information flow and help retain or update information across sequences.
• ResNetaddressed by · ResNet is a family of deep neural-network architectures built from residual blocks with shortcut connections that facilitate information flow and optimization.
• GRUaddressed by · A gated recurrent unit (GRU) is a recurrent neural-network architecture that uses learned gates to regulate information flow and update a hidden state across sequence steps.
• normalizationaddressed by · Normalization is the process of transforming a value, representation, or data structure to satisfy a specified standard, scale, or structural constraint.

This explainer was generated by Manning's knowledge graph from the text of Math and Architectures of Deep Learning. The section links point to the parts of the book it draws on.