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Maximized objective function of an optimization problem
The value function of an optimization problem gives the value attained by the objective function at a solution, while only depending on the parameters
Value_function
Distance from zero to a number
the absolute value function is idempotent (meaning that the absolute value of any absolute value is itself). The absolute value function of a real number
Absolute_value
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Concept in game theory
coalition (set of players) S {\displaystyle S} , we define the payoff or value function v ( S ) {\displaystyle v(S)} as the total sum of payoffs that the members
Shapley_value
Operation in differential calculus
the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved. For the absolute value function f ( x )
Symmetric_derivative
Topics referred to by the same term
Function value may refer to: In mathematics, the value of a function when applied to an argument. In computer science, a closure. This disambiguation page
Function_value
Branch of mathematics studying functions of a complex variable
real-valued. In other words, a complex function f : C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } may be decomposed into two real-valued functions (
Complex_analysis
Theorem in mathematics
and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating that the average
Mean_value_theorem
Generalized mathematical function
a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for
Multivalued_function
Field of machine learning
two main approaches for achieving this are value function estimation and direct policy search. Value function approaches attempt to find a policy that maximizes
Reinforcement_learning
Association of one output to each input
possible applications of the concept. A function is often denoted by a letter such as f, g or h. The value of a function f at an element x of its domain (that
Function_(mathematics)
Mathematical constants
The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer, half-integer, and
Particular values of the gamma function
Particular_values_of_the_gamma_function
Function returning minus 1, zero or plus 1
In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value −1, +1 or 0 according to whether
Sign_function
Mathematical function with no sudden changes
mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies there
Continuous_function
Mathematical function that outputs real values
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each
Real-valued_function
mathematics, proto-value functions (PVFs) are automatically learned basis functions that are useful in approximating task-specific value functions, providing
Proto-value_function
Instantaneous rate of change (mathematics)
to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists
Derivative
All derivatives have the intermediate value property
theorem states that the derivative of any real-valued function of a real variable has the intermediate value property, that is, that the image of an interval
Darboux's_theorem_(analysis)
Indicator function of positive numbers
function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function named after Oliver Heaviside, the value
Heaviside_step_function
Theory of behavioral economics
It introduces a value function defined over gains and losses rather than final wealth, as well as a probability-weighting function that reflects the
Prospect_theory
Function returning one of only two values
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {−1
Boolean_function
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given point
Probability_density_function
Function valued in a vector space; typically a real or complex one
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional
Vector-valued_function
Function defined by multiple sub-functions
mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned
Piecewise_function
Concept in economics and decision theory
the same utility value. Individual and social utility can be construed as the value of a utility function and a social welfare function, respectively. When
Utility
Function whose values are sets (mathematics)
A set-valued function, also called a correspondence or set-valued relation, is a mathematical function that maps elements from one set, the domain of the
Set-valued_function
Functions in mathematics
convergent sequence of harmonic functions is still harmonic. This is true because every continuous function satisfying the mean value property is harmonic. Consider
Harmonic_function
Variable that represents an argument to a function
arguments are passed to a function. Generally, with call by value, a parameter acts like a new, local variable initialized to the value of the argument. If
Parameter (computer programming)
Parameter_(computer_programming)
Point to which functions converge in analysis
the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears
Limit_of_a_function
Mathematical function whose derivative exists
or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable
Differentiable_function
Notion in mathematics
as values. The value of a function, given the value(s) assigned to its argument(s), is the quantity assumed by the function for these argument values. For
Value_(mathematics)
Mathematical function, denoted exp(x) or e^x
mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted e x
Exponential_function
Average value of a random variable
function given by a function f {\displaystyle f} on the real number line. This means that the probability of X {\displaystyle X} taking on any value in
Expected_value
Model-free reinforcement learning algorithm
K\}} is the smallest value which improves the sample loss and satisfies the sample KL-divergence constraint. Fit value function by regression on mean-squared
Proximal_policy_optimization
Point where function's value is zero
sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle x} of the domain
Zero_of_a_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly
Sigmoid_function
Mathematical relation assigning a probability event to a cost
decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables
Loss_function
Continuous function on an interval takes on every value between its values at the ends
mathematical analysis, the intermediate value theorem states that if f {\displaystyle f} is a continuous function whose domain contains the interval [a
Intermediate_value_theorem
Constants of the mathematical zeta function
partial sums would grow indefinitely large. The zeta function values listed below include function values at the negative even numbers ( s = − 2 , − 4 {\displaystyle
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Sequence of program instructions invokable by other software
as COBOL and BASIC, make a distinction between functions that return a value (typically called "functions") and those that do not (typically called "subprogram"
Function (computer programming)
Function_(computer_programming)
Largest and smallest value taken by a function at a given point
analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extrema, they
Maximum_and_minimum
Real function with secant line between points above the graph itself
mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the
Convex_function
Matrix of second derivatives
second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix
Hessian_matrix
Necessary condition for optimality associated with dynamic programming
relationship between the value function in one period and the value function in the next period. The relationship between these two value functions is called the
Bellman_equation
Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood
Holomorphic_function
Reinforcement learning algorithms
function π ( a | s ) {\displaystyle \pi (a|s)} , while the critic estimates either the value function V ( s ) {\displaystyle V(s)} , the action-value
Actor-critic_algorithm
Type of function in mathematics
becomes analytic as a function of t {\displaystyle t} . Typical examples of functions that are not analytic are The absolute value function x ↦ | x | {\displaystyle
Analytic_function
Class of reinforcement learning algorithms
methods. Unlike value-based methods which learn a value function to derive a policy, policy optimization methods directly learn a policy function π {\displaystyle
Policy_gradient_method
Degree of differentiability of a function or map
function has class C k {\displaystyle C^{k}} . For example, the absolute value function f ( x ) = | x | {\displaystyle f(x)=|x|} has class C 0 {\displaystyle
Smoothness
Nearest integers from a number
the floor function for negative numbers. For an integer n, ⌊n⌋ = ⌈n⌉ = n. Although floor(x + 1) and ceil(x) are equal for non-integer values of x, and
Floor_and_ceiling_functions
Assignment of numbers to points in space
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. The region U may be a set in some Euclidean space
Scalar_field
Function that outputs either true or false
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B
Boolean-valued_function
Logarithm to the base of the mathematical constant e
logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to
Natural_logarithm
Function that is continuous everywhere but differentiable nowhere
mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere
Weierstrass_function
Extension of the factorial function
the second kind. (Euler's integral of the first kind is the beta function.) The value Γ ( 1 ) {\displaystyle \Gamma (1)} can be calculated as Γ ( 1 ) =
Gamma_function
Point where a mathematical object behaves irregularly
reciprocal function f ( x ) = 1 / x {\displaystyle f(x)=1/x} has a singularity at x = 0 {\displaystyle x=0} , where the value of the function is not defined
Singularity_(mathematics)
Special mathematical function defined as sin(x)/x
both cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic
Sinc_function
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
Mathematical concept
example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by
Inverse_function
Method of mathematical integration
the values of a function freely, while preserving the value of the integral. This process of rearrangement can convert a very pathological function into
Lebesgue_integral
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Mathematical functions
{\displaystyle \sinh ^{-1}} ). For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding hyperbolic angle
Inverse_hyperbolic_functions
Mathematical function with multiple real-number arguments
function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function"
Function of several real variables
Function_of_several_real_variables
Loss function used in robust regression
&{\text{otherwise.}}\end{cases}}} This function is quadratic for small values of a, and linear for large values, with equal values and slopes of the different sections
Huber_loss
Matrix of partial derivatives of a vector-valued function
calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member
Integer-valued_function
Uniform restraint of the change in functions
positive real number δ {\displaystyle \delta } such that function values over any function domain interval of the size δ {\displaystyle \delta } are
Uniform_continuity
Function with a multiplicative scaling behaviour
the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the
Homogeneous_function
Theorem in mathematics
complex-valued functions of a complex variable. It generalizes to functions from n-tuples (of real or complex numbers) to n-tuples, and to functions between
Inverse_function_theorem
S-shaped curve
{\displaystyle L} is the carrying capacity, the supremum of the values of the function; k {\displaystyle k} is the logistic growth rate, the steepness
Logistic_function
Function related to statistics and probability theory
function of θ {\textstyle \theta } , a possible value of the deterministic but unknown parameter Θ {\textstyle \Theta } , is the likelihood function,
Likelihood_function
Generalized function whose value is zero everywhere except at zero
continuous function f. The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point
Dirac_delta_function
Mathematical function with convex lower level sets
In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the
Quasiconvex_function
Special function defined by an integral
approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value x. The logarithmic integral
Logarithmic_integral_function
Specific values of a multivalued function
branch that takes a real value for small positive values of the variable. A principal value is the value at a point of the function defined by the principal
Principal_value
Class of reinforcement learning algorithm
this framework, each policy is first evaluated by its corresponding value function. Then, based on the evaluation result, greedy search is completed to
Model-free (reinforcement learning)
Model-free_(reinforcement_learning)
Method by which value is transferred between parties
Silvio Gesell believed that the function of money as a store of value is fundamentally incompatible with its function as a medium of exchange for maximum
Medium_of_exchange
Program function without side effects
In computer programming, a pure function is a function that has the following properties: the function return values are identical for identical arguments
Pure_function
Subfield of number theory
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula
Special_values_of_L-functions
Functions such that f(–x) equals f(x) or –f(x)
odd, the absolute value of that function is an even function. The sum of two even functions is even. The sum of two odd functions is odd. The difference
Even_and_odd_functions
Strong form of uniform continuity
cosine function, is bounded above by 1 in absolute value. Lipschitz continuous functions that are not everywhere differentiable The function f ( x )
Lipschitz_continuity
Optimality condition in optimal control theory
conditions for optimality of a control with respect to a loss function. Its solution is the value function of the optimal control problem which, once known, can
Hamilton–Jacobi–Bellman equation
Hamilton–Jacobi–Bellman_equation
Type of mathematical function
mathematics, a constant function is a function whose (output) value is the same for every input value. As a real-valued function of a real-valued argument, a constant
Constant_function
Generalization of a Markov decision process
expected reward value for each belief state, compactly represented by the optimal value function V ∗ {\displaystyle V^{*}} . This value function is solution
Partially observable Markov decision process
Partially_observable_Markov_decision_process
Angle of complex number about real axis
real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex numbers. The principal value of this function is
Argument_(complex_analysis)
Study of rates of change
the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real
Differential_calculus
Theorem in mathematics and economics
theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of
Envelope_theorem
function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values
Subadditive_set_function
Theorem in real analysis
Rolle's theorem (or lemma) states that a real-valued differentiable function which attains equal values at two distinct points must have a stationary
Rolle's_theorem
Discrete-variable probability distribution
exactly equal to some value. Sometimes it is also known as the discrete probability density function. The probability mass function is often the primary
Probability_mass_function
Mathematical function, inverse of an exponential function
tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete
Logarithm
Generalization of derivatives to real-valued functions
interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value function f ( x ) = | x | {\displaystyle
Subderivative
Differentiable function whose derivative is not Riemann integrable
In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination
Volterra's_function
Logarithm of a complex number
equally spaced along a vertical line in the complex plane. A complex-valued function log : U → C {\displaystyle \log \colon U\to \mathbb {C} } , defined
Complex_logarithm
Mathematics of real numbers and real functions
differentiable function satisfies the intermediate value property; weaker than continuity, but still a restricted class of functions. In more advanced
Real_analysis
Function that is holomorphic on the whole complex plane
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane
Entire_function
Continuous real function on a closed interval has a maximum and a minimum
In real analysis, the extreme value theorem states that if a real-valued function f {\displaystyle f} is continuous on the closed and bounded interval
Extreme_value_theorem
Function computable with bounded loops
\end{cases}}\end{aligned}}} Interpretation: The function f {\displaystyle f} acts as a for-loop from 0 {\displaystyle 0} up to the value of its first argument. The rest
Primitive_recursive_function
Integral expressing the amount of overlap of one function as it is shifted over another
The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the
Convolution
Value approached by a mathematical object
a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to
Limit_(mathematics)
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