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Topic: Function approximation

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	Backprop Learning Tool
		In (b)-(g), the function is shown as a dotted line, and the neural net approximation (based on the noisy samples shown as circles) is shown as the solid line.
		In each case, the dotted line is the underlying function to be approximated, the solid line is the neural net output, and the open circles indicate the data points used for training.
		Here, the underlying function to be approximated is a sine wave, and is a perturbation term, producing small "blips" or kinks near the peaks of the sine curve.
	neuron.eng.wayne.edu /bpFunctionApprox/bpFunctionApprox.html (1695 words)

	Function approximation - Wikipedia, the free encyclopedia
		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.
		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.
	en.wikipedia.org /wiki/Function_approximation (181 words)

	Function approximation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		The need for function approximations arises in many branches of (The branches of mathematics that are involved in the study of the physical or biological or sociological world) applied mathematics, and (The branch of engineering science that studies (with the aid of computers) computable processes and structures) computer science in particular.
		In general, a function approximation problem asks us to select a (A mathematical relation such that each element of one set is associated with at least one element of another set) function among a well-defined class that closely matches ("approximates") a target function in a task-specific way.
		Depending on the structure of the (Territory over which rule or control is exercised) domain and (Click link for more info and facts about codomain) codomain of g, several techniques for approximating g may be applicable.
	www.absoluteastronomy.com /encyclopedia/f/fu/function_approximation.htm (242 words)

	MT A14303
		You should be able to sketch a graph of the approximation, evaluate the approximation at a point and find the error between the function and the approximation.
		The standard approximating tool for approximating periodic functions or functions that are only defined on a finite interval is a Fourier polynomial, which is really a linear combination of sine and cosine functions.
		Besides the mechanics that you should learn in constructing approximating functions, it is worthwhile to note that the approximations pass the naive test for approximations, that if you take better and better approximations of a function, the approximation eventually becomes indistinguishable from the function we are approximating.
	www.slu.edu /classes/maymk/SeriesGraphs/SeriesNotes.html (810 words)

	Gamasutra - Features - "A Non-Integer Power Function on the Pixel Shader" (08.01.02]
		The normalized error function is also plotted in order to show how the error is distributed to the left and right of the point where both curve crosses.
		In order to optimize the approximation error, one should seek values of A and B for which the maximal error on the left and right side are equal.
		First we need a function that, given approximate values for A and B, is able to compute the maximal error on the left and right side of the crossing point.
	www.gamasutra.com /features/20020801/beaudoin_01.htm (2056 words)

	Artificial Neural Network - Function Approximation
		The functions are approximated by this applet INSIDE the domain of OPEN interval (0, 1), due to the use of hard-limiting functions in the neural network.
		Because piecewise constant functions are used in the approximation, the error in approximation is linearly dependent on the inverse of this parameter.
		A better and more efficient solution would be an approximator which is second order accurate, where the error is inversely proportional to the square of the number of neurons used in the hidden layer.
	www.loyno.edu /~li/home/java/ann (690 words)

	4.1 Linear and Quadratic Approximations (Site not responding. Last check: 2007-10-22)
		A differentiable function f is one that resembles a linear function at close range.
		In one dimension the graph of a function against the independent variable is a curve, and the linear approximation to it is the function whose graph is the tangent line to it at r'.
		The quadratic approximation which we write out in detail in two dimensions, is of great use in determining the nature of a critical point at r', and can be useful in approximating f when the linear approximation is insufficiently accurate.
	www-math.mit.edu /~djk/18_022/chapter04/section01.html (286 words)

	8.1 Value Prediction with Function Approximation
		The flexibility of the function approximator is thus a scarce resource.
		Reaching this goal is sometimes possible for simple function approximators such as linear ones, but is rarely possible for complex function approximators such as artificial neural networks and decision trees.
		In this section we have outlined a framework for combining a wide range of reinforcement learning methods for value prediction with a wide range of function approximation methods, using the backups of the former to generate training examples for the latter.
	www.cs.iitm.ernet.in /~cs670/book/node86.html (1300 words)

	UCES Methods and Analysis Chap. 1.7: Ill-conditioned Algebraic Systems and Function Approximation
		The second application is somewhat related and requires a polynomial approximation to a function that is difficult to compute.
		Many functions are very expensive to evaluate, but often segments of the graphs look like graphs of polynomials.
		Assume the function to be approximated satisfies f(1) = 1, f(1.1) = 2 and f(1.11) = 3.
	www.krellinst.org /UCES/archive/classes/CNA/dir1.7/uces1.7.html (1234 words)

	Rational Approximation Experiments
		We use the arcsin function because it is monotonic and infinitely differentible, except at arcsin(1).
		As I understand it, a Pade approximation is a formal transformation from a series like a Taylor or Chebyshev approximation to a ratio.
		Given the difficulty in approximating a function over an interval, I tried to simplify the problem by selecting a sequence of points on the function, and finding a Thiele rational interpolation thru them.
	www.ecse.rpi.edu /Homepages/wrf/Research/Short_Notes/arcsin (991 words)

	RLFA: Reinforcement Learning and Function Approximation Reading Group
		Function approximation (FA) techniques should come at rescue, but our understanding of the interplay between function approximation and RL methods is limited in a number of important ways.
		The limit function is shown to satisfy a fixed point equation of the Bellman type, where the fixed point operator depends on the stationary distribution of the exploration policy and approximation properties of the function approximation method.
		Function approximation is essential for problems with a large state space, as it facilitates compact representation and enables generalization.
	rlai.cs.ualberta.ca /RLAI/RLFA.html (4526 words)

	VFA workshop abstracts
		The key computational idea underlying reinforcement learning is the iterative approximation of the value function---the mapping from states (or state-action pairs) to an estimate of the long-term future reward obtainable from that state.
		In value iteration and related TD algorithms for learning value functions, values all across the state space must be repeatedly and recursively re-estimated, which can cause instability in the context of function approximator error.
		I used a local function approximator known as a CMAC---essentially a linear learner using a sparse, coarse-coded representation of the state.
	www.cs.cmu.edu /People/reinf/ml95/abstracts.html (2166 words)

	Publications on Function Approximation (RL)
		This thesis is a study of practical methods to estimate value functions with feedforward neural netw...
		Local linear function approximators are often preferred to feedforward neural networks to estimate v...
		Approximate value iteration in the reinforcement learning context.
	www-anw.cs.umass.edu /rlr/fa.html (1356 words)

	8.4 Control with Function Approximation
		But with function approximation there is no single trace corresponding to a state, just a trace for each component of
		One approach that seems to work well for linear, gradient-descent function approximation methods with binary features is to treat the features as if they were states for the purposes of replacing traces.
		This idea can also be extended to the case of linear function approximation with binary features.
	www.cs.iitm.ernet.in /~cs670/book/node89.html (1140 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations (Site not responding. Last check: 2007-10-22)
		A method for the approximate construction of the nucleon Green's function is presented.
		The development is such that the approximate Green's function automatically has the same analytical properties as the exact one.
		The approximation results in a linear integral equation for the Green's function which is completely renormalized.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4232789 (154 words)

	8.1 Value Prediction with Function Approximation
		The flexibility of the function approximator is thus a scarce resource.
		Reaching this goal is sometimes possible for simple function approximators such as linear ones, but is rarely possible for complex function approximators such as artificial neural networks and decision trees.
		In this section we have outlined a framework for combining a wide range of reinforcement learning methods for value prediction with a wide range of function approximation methods, using the backups of the former to generate training examples for the latter.
	www.cs.ualberta.ca /~sutton/book/ebook/node86.html (1300 words)

	OhioLINK ETD: Skelly, Margaret
		This dissertation investigates the incorporation of function approximation and hierarchy into reinforcement learning for use in an adaptive control setting through empirical studies.
		The function approximation method chosen provides local generalization, which provides for a controlled diffusion of information throughout the task space.
		As a consequence, the experiments conducted with function approximation demonstrate how greater levels of abstraction, as determined by the amount of information diffusion, can accelerate learning in tasks where similar states call for similar actions.
	rave.ohiolink.edu /etdc/view?acc_num=case1081357818 (401 words)

	Learning and Value Function Approximation in Complex Decision Processes (Site not responding. Last check: 2007-10-22)
		Unfortunately, exact computation of the value function typically requires time and storage that grow proportionately with the number of states, and consequently, the enormous state spaces that arise in practical applications render the algorithms intractable.
		Given a selection of basis functions, the algorithm updates weights during simulation of the system such that the weighted combination of basis functions ultimately approximates a value function.
		Each scenario is used to construct a basis function that maps states to future rewards contingent on the future realization of the scenario.
	web.cps.msu.edu /rlr/pub/VanRoy1.html (407 words)

	Function approximation in reinforcement learning (Site not responding. Last check: 2007-10-22)
		Most of the theoretical convergence results for RL algorithms assume a tabular representation of the value function, in which the value of each state is stored in a separate memory location.
		The first approach is to represent the value function of the RL agent using function approximation methods developed in the context of supervised learning, such as neural networks or linear function approximators.
		An alternative approach is to use a function approximator to represent a parameterized policy.
	rl.cs.mcgill.ca /Projects/fa.html (352 words)

	Adaptive Encoding Strongly Improves Function Approximation with CMAC -- Eldracher et al. 9 (2): 403 -- Neural ...
		as a good function approximator with local generalization abilities.
		Depending on the smoothness of the function to be approximated, the
		function approximation for smooth functions are obtained in shorter
	neco.mitpress.org /cgi/content/abstract/9/2/403 (187 words)

	[No title]
		The capabilities of the MLP can be viewed from three different perspectives: the first has to do with its ability to implement logic functions [24], the second with its ability to partition the pattern space for classification problems [13], and the third with its ability to implement nonlinear transformations for function approximation problems [12].
		In [7], a neuron-adaptive activation function built as a piecewise approximation with suitable cubic splines was found to have arbitrary shape and allowed the overall size of the neural network to be reduced, thus trading connection complexity with activation function complexity.
		Such functions exist due to the fact that the solutions to some differential equations that emerge from practical problems are incapable of being expressed by elementary functions.
	www.pcs.cnu.edu /~mzhang/CPSC450_550/BestPaperPresentation/Rainfall_NAN_ICAI04Format.doc (1658 words)

	8 Generalization and Function Approximation
		We have so far assumed that our estimates of value functions are represented as a table with one entry for each state or for each state-action pair.
		The kind of generalization we require is often called function approximation because it takes examples from a desired function (e.g., a value function) and attempts to generalize from them to construct an approximation of the entire function.
		Function approximation is an instance of supervised learning, the primary topic studied in machine learning, artificial neural networks, pattern recognition, and statistical curve fitting.
	web.cs.ualberta.ca /~sutton/book/8/node1.html (322 words)

	Efficient Value Function Approximation Using Regression Trees - Wang, Dietterich (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		Many applications of reinforcement learning have relied on neural network function approximators, which are very slow to train and require substantial parameter tweaking to obtain good performance.
		Other reinforcement learning studies have applied nearest neighbor and CMAC function approximators, but these cannot scale to problems with many features, especially if some features are irrelevant.
		with function approximation, that is, we estimate and represent the value function as a function of state features and actions (e.g.
	citeseer.ist.psu.edu /wang99efficient.html (573 words)

	3 Function Approximation
		Smoothness classes which are particularly well suited to the study of approximation properties of wavelet bases are given by the scale of Besov spaces
		The display contains two plots: the upper shows a jump function, the lower the mother wavelet coefficients corresponding to their position in scale and time.
		The large coefficients are caused by the discontinuity (center) and by boundary effects since we use a periodic wavelet transform for a nonperiodic function.
	sunsite.univie.ac.at /XploRe/tutorials/wavenode4.html (262 words)

	Function Approximation Using Visual Basic - R. Berman (Site not responding. Last check: 2007-10-22)
		Example 0: Compute Exp(x) and the approximation 1+x, as well as the error estimate x^2/2.
		Problem 0: Compute Log(1+x) and the approximation x, as well as the error estimate x^2/2.
		Compute Log(1+x) and the approximation x, as well as the error estimate x^2/2.
	www.math.wayne.edu /~rberman/syllabi/ex_0.htm (252 words)

	Robotics Institute: Stable Function Approximation in Dynamic Programming
		We provide a proof of convergence for a wide class of temporal difference methods involving function approximators such as k-nearest-neighbor, and show experimentally that these methods can be useful.
		The proof is based on a view of function approximators as expansion or contraction mappings.
		In addition, we present a novel view of approximate value iteration: an approximate algorithm for one environment turns out to be an exact algorithm for a different environment.
	www.ri.cmu.edu /pubs/pub_3258.html (269 words)

	Experiment 3: Function Approximation / Inductive Transfer
		This experimental case study will demonstrate that IS can successfully learn in a changing environment where the tasks to be solved become more and more difficult over time (inductive transfer).
		Our system is exposed to a sequence of more and more complex function approximation problems.
		The corresponding ``program'' and the (very high) probabilities of its instructions and parameters are shown in Table 4.
	www.idsia.ch /~juergen/mljssalevin/node11.html (623 words)

	Multi-Player Residual Advantage Learning With General Function Approximation
		For a given value function V, and a given state x, the Bellman residual is defined to be the difference between the two sides of the Bellman equation.
		is a function of the rate of change in the gradient of the error surface, and is therefore different for different regions of state space.
		The action performed by the plane is a function of the state and the action of the missile.
	www.leemon.com /papers/sim_tech (4917 words)

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