Expressivity and Function Classes
When you study neural networks, one of the most important concepts to understand is expressivity. In this context, expressivity refers to the range of functions that a neural network can approximate, given its architecture and parameters. Expressivity is not just about whether a network can theoretically approximate a function, but also about how efficiently it can do so in terms of size and complexity.
A function class is a set of functions that share certain properties, such as smoothness or the number of variables. In neural networks, the function class is determined by the architecture: the number of layers (depth), the number of units per layer (width), and the types of activation functions used. The choice of architecture restricts or expands the function class the network can represent.
The architecture of a neural network — specifically, its width and depth — directly shapes its expressivity. Increasing the width of a network, by adding more neurons to a layer, enables the network to represent more complex functions in a single step. However, merely making a network wider does not always allow it to represent every possible function efficiently. In fact, there are certain functions that a shallow but very wide network can approximate, but only with an impractically large number of neurons.
On the other hand, increasing the depth of a network, by stacking more layers, allows the network to build hierarchical representations. Deeper networks can express certain functions with far fewer parameters compared to shallow networks, as they compose simple transformations into more complex ones. This means that while shallow networks can, in theory, approximate any continuous function (as described by the Universal Approximation Theorem), they may require exponentially more units than a deeper network to achieve the same result.
Understanding the interplay between width and depth is crucial for designing neural networks that are both expressive and efficient. The limitations of shallow networks highlight why depth is often preferred in practice, especially when modeling functions with intricate structure.
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When you study neural networks, one of the most important concepts to understand is expressivity. In this context, expressivity refers to the range of functions that a neural network can approximate, given its architecture and parameters. Expressivity is not just about whether a network can theoretically approximate a function, but also about how efficiently it can do so in terms of size and complexity.
A function class is a set of functions that share certain properties, such as smoothness or the number of variables. In neural networks, the function class is determined by the architecture: the number of layers (depth), the number of units per layer (width), and the types of activation functions used. The choice of architecture restricts or expands the function class the network can represent.
The architecture of a neural network — specifically, its width and depth — directly shapes its expressivity. Increasing the width of a network, by adding more neurons to a layer, enables the network to represent more complex functions in a single step. However, merely making a network wider does not always allow it to represent every possible function efficiently. In fact, there are certain functions that a shallow but very wide network can approximate, but only with an impractically large number of neurons.
On the other hand, increasing the depth of a network, by stacking more layers, allows the network to build hierarchical representations. Deeper networks can express certain functions with far fewer parameters compared to shallow networks, as they compose simple transformations into more complex ones. This means that while shallow networks can, in theory, approximate any continuous function (as described by the Universal Approximation Theorem), they may require exponentially more units than a deeper network to achieve the same result.
Understanding the interplay between width and depth is crucial for designing neural networks that are both expressive and efficient. The limitations of shallow networks highlight why depth is often preferred in practice, especially when modeling functions with intricate structure.
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