Value-Function Fitting Task

A Value-Function Fitting Task is a function creation task and that accepts a parametric function family and a training dataset and is required to produce a fitted function according to some fitness function.

• AKA: Function Approximation.
• Context:
  • Input: a Numerically-Labeled Training Dataset; a Function Family; and an Objective Measure.
    • optional: a Function Fitness Measure.
  • output: a Fitted Function.
  • It can range from being a Univariate Function Fitting Task to being a Multivariate Function Fitting Task.
  • It can range from being a Supervised Function Fitting Task to being an Unsupervised Function Fitting Task.
  • It can range from being a Parametric Function Fitting Task to being a Non-Parametric Function Fitting Task.
  • It can range from being a Manual Function Fitting Task to being an Automated Function Fitting Task.
  • It can range from being a Linear Function Fitting Task to being a Non-Linear Function Fitting Task (such as polynomial fitting).
  • It can be solved by a Function Fitting System (that implements a function fitting algorithm).
  • It can support tasks such as: Interpolation, Extrapolation, ...
• Example(s):
  • Least-Squares Function Fitting.
  • Probability Model Fitting, such as conditional-probability function estimation.
  • Regression Model Fitting or a Classification Model Fitting.
  • …
• Counter-Example(s):
  • Statistical Inference Task.
• See: Metric Space Optimization, Numerical Approximation, Taylor Series, Numerical Analysis, Approximation Theory, Smoothing, Mollifier, Model-based Learning.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/function_approximation Retrieved:2015-6-14.
  • The need for 'function approximations arises in many branches of applied mathematics, and computer science in particular. In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way.

    One can distinguish two major classes of function approximation problems: First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).

    Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead. A related problem online time series approximation is to summarize the data in one-pass and construct an approximate representation that can support a variety of timeseries queries with bounds on worst-case error.

    To some extent the different problems (regression, classification, fitness approximation) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.

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