Vanishing gradient problem

As explained in Deep Learning with R, Third Edition

A vanishing gradient is a training condition in which gradients become extremely small as they propagate through a deep network, slowing or preventing parameter updates in earlier layers.

inputs <- keras_input(shape = c(32, 32, 3))
x <- inputs |> layer_conv_2d(32, 3, activation = "relu")
residual <- x  #1
x <- x |> layer_conv_2d(64, 3, activation = "relu", padding = "same")  #2
residual <- residual |> layer_conv_2d(64, 1) #3
x <- layer_add(c(x, residual))  #4

How Deep Learning with R, Third Edition covers it

How deep it goes

Explains the failure through a backpropagation analogy and shows residual shape matching, without deriving gradient equations.

Examples

Uses a chain of functions, R/Keras convolutional residual blocks, and recurrent layers processing sequences.

What it assumes

Introduces the backpropagation failure with a function-chain analogy before using R/Keras layer operations.

What you can do after

Apply it via R/Keras residual connections, including projections when a block changes feature-map shape.

How it connects

Connects backpropagation, feature hierarchies, residual connections, SimpleRNN, LSTM, and GRU.

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Also explained in

• 

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  Sutskever's ListA vanishing gradient is an optimization failure in which derivatives become too small to support effective parameter updates during neural-network training.
• 

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  Grokking Machine LearningA neural-network training problem in which gradients become extremely small as they are propagated through layers, slowing or preventing learning.
• 

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  Math and Architectures of Deep LearningThe training difficulty in deep networks where repeated multiplication of small derivatives makes early-layer gradients nearly zero.
• 

  liveBookmanning.com
  Deep Learning with Python, Third EditionA training problem in deep networks where gradient signals become too small to effectively update earlier layers.
• 

  liveBookmanning.com
  Inside Deep Learning: Math, Algorithms, ModelsA training instability in which gradients become too small for effective learning updates.
• 

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  Graph Neural Networks in ActionAn optimization problem in which gradients become progressively smaller as they are propagated through many computational steps, hindering learning of long-range dependencies.
• 

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  Evolutionary Deep Learning: Genetic algorithms and neural networksA neural-network training failure in which backpropagated gradients become extremely small across successive layers, limiting effective parameter updates.
• 

  liveBookmanning.com
  Grokking AI Algorithms, Second EditionThe vanishing gradient problem is a training difficulty in which derivatives become extremely small and cause neural-network parameter updates to become ineffective.
• 

  liveBookmanning.com
  Deep Learning with PyTorch, Second EditionThe difficulty in training deep networks when gradients become too small as they propagate backward through many layers.
• 

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  Learn Generative AI with PyTorchA training failure mode in deep networks where gradient signals become too small to effectively update early layers.
• 

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  LLMs in ProductionA vanishing gradient is a training condition in which propagated derivatives become increasingly small and impede parameter updates.
• 

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  Data Without LabelsThe vanishing gradient problem is an optimization difficulty in which gradients become extremely small during learning, hindering the retention of information across many sequence steps.
• 

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  Build a Large Language Model (From Scratch)A training difficulty in deep neural networks where gradients shrink as they propagate backward, hindering learning in early layers.

Compare how 14 books explain vanishing gradient problem →

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 Deep Learning with R, Third Edition. The section links point to the parts of the book it draws on.