Feature Construction

RBF networks are function approximators whose features are radial basis functions.

Programming

From State to Features

Feature construction asks how an input state is turned into information that a function approximator can use. An RBF network makes a specific choice: its features are radial basis functions. Each feature describes the input in relation to a center, varying according to the input's distance from that center. The RBF network then uses those features to approximate a target function.

How does an input state become a set of radial basis function features, and how do those features vary with distance from each center?

A State Through an RBF Representation

Worked example

A Nearby State and Three Centers

Consider an input state and three radial basis functions, each associated with a different center.

1
Compare distances
The input state is compared with each center. One center may be relatively near the state, while the others may be farther away.
2
Construct feature responses
Each radial basis function produces a feature value based on the input's distance from its center. The feature responses therefore describe the same state from several center-based perspectives.
3
Approximate the target
The RBF network uses the constructed feature representation in a function approximator.

Result

The original state has been converted into a collection of radial basis function features. Changing the state changes its distances from the centers and therefore changes the feature representation.

The important idea is not merely that an RBF network has several features. It is that the features are organized around centers and respond to distance from those centers. A state near one center can receive a different pattern of feature responses from a state that is farther away or closer to another center.

What do you think happens?

Suppose the input state moves closer to one center while the other centers remain where they are. What should happen to the RBF feature representation?

Only the feature associated with the nearest center can change
The feature values can change because the distances to the centers have changed
The representation must remain identical because the centers are fixed

Reveal answer

Answer:

The feature values can change because the distances to the centers have changed.

Fixed centers do not mean fixed feature responses. They mean that the centers remain unchanged while different input states can produce different responses according to their distances from those centers.

Why Fixed Features Are Linear

An RBF network with fixed features is a linear RBF network. The radial basis functions define the representation of the input, but that representation does not change during learning. Learning is therefore applied to the linear function approximator built on top of those fixed features. The network can produce different approximations by changing the parameters of that linear approximator without moving the feature centers or changing their widths.

How can changing only the linear approximator parameters produce different approximations while the RBF centers and widths remain fixed?

The word linear refers to what learning changes in the approximator. It does not mean that the radial basis features are absent or that the input-to-feature relationship is a simple straight line. The defining condition is that the feature representation stays fixed while the linear approximator is learned.

When RBF Learning Becomes Nonlinear

Feature behavior	Linear RBF network	Nonlinear RBF network
RBF feature centers	Fixed during learning	Can be changed during learning
RBF feature widths	Fixed during learning	Can be changed during learning
What learning changes	The linear approximator built on the features	The approximator and the feature representation
Potential fitting ability	Uses the given representation	May fit target functions more precisely
Computational and tuning burden	Lower than when centers and widths are also adapted	Greater computational complexity and often more manual tuning

The distinction depends on whether learning changes only the approximator or also the RBF representation.

What changes during learning in a linear RBF network versus a nonlinear RBF network?

Adapting centers and widths can help the network fit target functions more precisely because the representation itself can be adjusted. That potential precision has a cost: nonlinear RBF methods require more computational complexity and can require more manual tuning before learning becomes robust and efficient.

RBF Methods and Tile Coding

Consideration	RBF network methods	Tile coding
Representation	Uses radial basis functions as features	Uses tile-code regions
Feature evaluation	Requires evaluating radial basis function responses	Uses the tile-code representation
Computational demand	Has additional computational complexity compared with tile coding	Provides the comparison baseline described by the source
Adaptation	Nonlinear versions can change centers and widths	The source does not describe center and width adaptation for tile coding
Tuning	Nonlinear RBF networks can require more manual tuning	The source does not specify an equivalent tuning burden

The source emphasizes the additional computational complexity of RBF network methods compared with tile coding, especially when RBF features are adapted.

RBF networks and tile coding both construct a representation before function approximation, but they do so differently. RBF methods describe an input through radial basis function responses. Tile coding describes it through tile-code regions. According to the source, RBF network methods carry additional computational complexity compared with tile coding, and nonlinear RBF networks add further tuning effort because their centers and widths are learned.

The High-Dimensionality Problem

Higher-dimensional state spaces make both RBF feature construction and tile coding more difficult to make effective and control. As the number of state dimensions increases, there are more dimensions in which centers or tile-code regions must provide useful coverage. The representation can therefore become harder to place and control across the state space.

Why do RBF centers and tile-code regions become harder to place, cover, and control as the number of state dimensions increases?

Best practice

When a state representation has many dimensions, examine not only the approximator but also the feature construction. Ask whether the centers or regions cover the state space in a useful and controllable way, and remember that an adaptable RBF representation may require additional tuning and computation.

Check Your Classification

EASY

A method uses radial basis functions as its features. During learning, the feature centers and widths remain fixed, while the linear approximator is learned. Classify the method and explain why.

Hints

First identify what kind of features the method uses.
Then ask whether learning changes the feature representation.
Use the distinction between fixed features and learned centers or widths.

Your attempt (not saved)

Worked example

Classifying Two RBF Methods

Classify two methods: Method A keeps centers and widths fixed and learns only the linear approximator. Method B learns the linear approximator while also changing centers and widths.

1
Classify Method A
Its RBF features remain fixed, so it is a linear RBF network.
2
Classify Method B
Its feature representation changes during learning because centers and widths are adapted, so it is a nonlinear RBF network.
3
Compare the trade-off
Method B may fit target functions more precisely, but it brings greater computational complexity and often more manual tuning.

Result

Method A is linear because only the approximator is learned. Method B is nonlinear because learning also changes the RBF feature representation.

Common mistake
Calling every RBF network nonlinear.
What it looks like
A network uses radial basis functions but keeps all feature centers and widths fixed.
Why it fails
The source distinguishes linear and nonlinear RBF networks by what learning changes, not simply by the presence of radial basis functions.
Do this instead
Classify the network as linear when the features are fixed and the learned part is the linear approximator.
Common mistake
Assuming fixed centers mean fixed feature values for every input.
What it looks like
The centers stay unchanged, so a learner concludes that the feature representation never changes.
Why it fails
Different input states can have different distances from the fixed centers.
Do this instead
Separate fixed feature parameters from feature responses: fixed centers and widths can still produce different responses for different states.
Common mistake
Describing the extra precision of nonlinear RBF networks without mentioning its cost.
What it looks like
An answer says only that learning centers and widths improves fitting.
Why it fails
The potential fitting improvement is accompanied by greater computational complexity and often more manual tuning.
Do this instead
State both sides of the trade-off: potentially more precise fitting, but more computation and tuning effort.
Common mistake
Ignoring state-space dimension when discussing feature construction.
What it looks like
A representation is evaluated only in a low-dimensional setting and assumed to be equally easy to place and control in a higher-dimensional one.
Why it fails
Higher-dimensional state spaces make centers and tile-code regions harder to place, cover, and control.
Do this instead
Include the dimensionality of the state space when assessing whether a representation will be effective and manageable.