Tradeoffs in Expressivity
To understand the tradeoffs in expressivity within neural networks, you need to be clear about the concepts of network depth and network width. The depth of a neural network refers to the number of layers through which data passes from input to output, excluding the input layer itself. Each layer can be seen as a stage in a sequence of function compositions, where the output of one layer becomes the input to the next. The width of a network is the number of neurons in a given layer, typically measured by the largest layer in the network. Both depth and width play crucial roles in a network's ability to approximate complex functions, but they do so in fundamentally different ways. Width allows a network to process more features or patterns in parallel, while depth enables the network to build hierarchical representations by composing simpler functions into more complex ones.
For some functions, if you restrict a neural network to have only a small number of layers (limited depth), you may need an exponentially larger number of neurons per layer (width) to represent those functions accurately. Specifically, there exist functions that a deep network can represent with a modest number of parameters, but any shallow network would require an exponential increase in width to achieve the same expressive power.
This result highlights why deep networks can be far more efficient than wide, shallow networks for certain tasks. Because deep networks use hierarchical composition, as discussed in earlier chapters, they can build up complex features layer by layer. Each layer extracts and combines features from the previous layer, allowing the network to represent intricate patterns with relatively few neurons at each stage. In contrast, a shallow network must capture all interactions in a single step, which often requires a dramatic increase in width and, consequently, the total number of parameters. This efficiency of depth is not just a theoretical curiosity — it is a practical reason why deep learning has become so successful in modeling high-dimensional, structured data such as images, speech, and text.
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To understand the tradeoffs in expressivity within neural networks, you need to be clear about the concepts of network depth and network width. The depth of a neural network refers to the number of layers through which data passes from input to output, excluding the input layer itself. Each layer can be seen as a stage in a sequence of function compositions, where the output of one layer becomes the input to the next. The width of a network is the number of neurons in a given layer, typically measured by the largest layer in the network. Both depth and width play crucial roles in a network's ability to approximate complex functions, but they do so in fundamentally different ways. Width allows a network to process more features or patterns in parallel, while depth enables the network to build hierarchical representations by composing simpler functions into more complex ones.
For some functions, if you restrict a neural network to have only a small number of layers (limited depth), you may need an exponentially larger number of neurons per layer (width) to represent those functions accurately. Specifically, there exist functions that a deep network can represent with a modest number of parameters, but any shallow network would require an exponential increase in width to achieve the same expressive power.
This result highlights why deep networks can be far more efficient than wide, shallow networks for certain tasks. Because deep networks use hierarchical composition, as discussed in earlier chapters, they can build up complex features layer by layer. Each layer extracts and combines features from the previous layer, allowing the network to represent intricate patterns with relatively few neurons at each stage. In contrast, a shallow network must capture all interactions in a single step, which often requires a dramatic increase in width and, consequently, the total number of parameters. This efficiency of depth is not just a theoretical curiosity — it is a practical reason why deep learning has become so successful in modeling high-dimensional, structured data such as images, speech, and text.
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