Continuous-Valued Features
RBFs turn the binary idea of coarse coding into a continuous feature response.
Beyond Yes-or-No Regions
A feature can ask whether an input state belongs to a particular region. Coarse coding gives a binary answer: the feature is active or inactive. Radial basis functions, or RBFs, make that response gradual. An RBF feature can take any value from 0 to 1, so its value can express different degrees to which the feature is present.
Following One Feature
Consider one RBF feature organized around a center state. The center state acts as the feature's prototype. For any input state, the feature considers the distance between that input and the center. At the center, the input has no separation from the prototype. As the input moves away, the distance changes, and the feature's response changes with it.
What do you think happens?
What should happen to an RBF feature's response as the input moves from its center state to a more distant state?
Reveal answer
Answer: The response changes as the distance from the center changes. The center has no separation from the prototype, while increasingly distant inputs are interpreted through their greater distance relative to the feature's width.
An RBF feature is organized around a center state and responds according to distance from that center. Its response is therefore based on the relationship between the input and the prototype, not on an arbitrary label attached to the state.
Distance Relative to Width
Distance alone does not fully describe an RBF response. The distance is interpreted relative to the feature's width. Width determines how the feature treats separation from its center: the same input-to-center distance can have a different meaning for a narrow feature than for a wide feature.
Comparing Narrow and Wide Features
Imagine two RBF features with the same center state. One has a narrow width and the other has a wide width. Consider the same input state for both features.
Choose the center: Both features are organized around the same center state, so both compare the input with the same prototype.
Measure separation: Each feature considers the distance between the input state and the center state.
Interpret relative to width: The narrow feature interprets the distance using a smaller width, while the wide feature interprets it using a larger width. Width therefore changes how the same separation is understood.
Compare the responses: The two features can produce different degrees of feature presence even though their center and input are the same, because their widths differ.
A feature's width controls the range of distances over which its response is interpreted as strong or weak. Width is therefore part of the meaning of an RBF feature, not a decorative parameter.
Choosing a Distance Metric
The distance used by an RBF is a modeling choice. The source illustrates the one-dimensional case with Euclidean distance, but the broader pattern is to choose a center, measure distance from that center, and interpret the distance using the feature's width. The most appropriate distance metric depends on the state representation and the task.
Common Interpretation Errors
Treating an RBF feature as only active or inactive.
RBFs extend the binary idea of coarse coding by allowing values from 0 to 1, which express different degrees of feature presence.
Fix:
Interpret the value as a graded response determined by the input's distance from the feature's center.Ignoring the feature's center state.
An RBF feature is organized around a center state, and its response depends on the distance between the input and that center.
Fix:
First identify the center state, then consider the input-to-center distance.Treating width as unrelated to the response.
The response is interpreted relative to the feature's width.
Fix:
Consider both the distance and the width when explaining the feature's response.Assuming one distance metric fits every task.
Distance metrics are modeling choices that should fit the state representation and task.
Fix:
Select or discuss the metric in relation to the states and the task.
Check Your Reasoning
An RBF feature has a center state. An input begins at that center and then moves away. Explain, in words, what information the feature uses at each stage and why its response can represent a degree of feature presence rather than a binary membership decision.
Hints
- Start with the separation between the input and the center at the beginning.
- Identify what changes as the input moves.
- Explain why width matters when interpreting the changing distance.
- Contrast the result with a feature that can only be active or inactive.
- A strong answer should mention the center state, the changing input-to-center distance, interpretation relative to width, and the resulting value range from 0 to 1.
Key Takeaways
- RBFs extend coarse coding from binary feature membership to continuous-valued responses.
- Each RBF feature is organized around a center state and responds according to the distance from that center.
- The response is interpreted relative to the feature's width, so width affects how distances are understood.
- RBF values can range from 0 to 1 and express degrees of feature presence.
- The distance metric is a modeling choice that should fit the state representation and task.