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Gradient-based Optimization
In mathematics, gradient descent (also often called steepest descent) is a :First order methods, first-order Iterative algorithm, iterative Mathematical optimization, optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. Description Gradient descent is based on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent In 2D
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\bf) may be defined by: :df=\nabla f \cdot d\bf where ''df'' is the total infinitesimal change in ''f'' for an infinitesimal displacement d\bf, and is seen to be maximal when d\bf is in the direction of the gradie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip Wolfe (mathematician)
Philip Starr "Phil" Wolfe (August 11, 1927 – December 29, 2016) was an American mathematician and one of the founders of convex optimization theory and mathematical programming. Life Wolfe received his bachelor's degree, masters, and Ph.D. degrees from the University of California, Berkeley. He and his wife, Hallie, lived in Ossining, New York. Career In 1954, he was offered an instructorship at Princeton, where he worked on generalizations of linear programming, such as quadratic programming and general non-linear programming, leading to the Frank–Wolfe algorithm in joint work with Marguerite Frank, then a visitor at Princeton. When Maurice Sion was on sabbatical at the Institute for Advanced Study, Sion and Wolfe published in 1957 an example of a zero-sum game without a minimax value. Wolfe joined RAND corporation in 1957, where he worked with George Dantzig, resulting in the now well known Dantzig–Wolfe decomposition method. In 1965, he moved to IBM's Thomas J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiation (mathematics)
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "''y'' = ''mx'' + ''c''". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The ''steepness'', incline, or grade of a line is measured by the absolute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contour Line
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph of the function f(x,y) parallel to the (x,y)-plane. More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value. In cartography, a contour line (often just called a "contour") joins points of equal elevation (height) above a given level, such as mean sea level. A contour map is a map illustrated with contour lines, for example a topographic map, which thus shows valleys and hills, and the steepness or gentleness of slopes. The contour interval of a contour map is the difference in elevation between successive contour lines. The gradient of the function is always perpendicular to the contour lines. When the lines are close together the magnitude of the grad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bowl (vessel)
A bowl is a typically round dish or container generally used for preparing, serving, or consuming food. The interior of a bowl is characteristically shaped like a spherical cap, with the edges and the bottom forming a seamless curve. This makes bowls especially suited for holding liquids and loose food, as the contents of the bowl are naturally concentrated in its center by the force of gravity. The exterior of a bowl is most often round but can be of any shape, including rectangular. The size of bowls varies from small bowls used to hold a single serving of food to large bowls, such as punch bowls or salad bowls, that are often used to hold or store more than one portion of food. There is some overlap between bowls, cups, and plates. Very small bowls, such as the tea bowl, are often called cups, while plates with especially deep wells are often called bowls. In many cultures bowls are the most common kind of vessel used for serving and eating food. Historically small bowls w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent Series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted :S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = \sum_^n a_k. A series is convergent (or converges) if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , 1 produce a convergent series: *: ++++++\cdots = . * Alternating the signs of reciprocals of powers of 2 also produces a convergent series: *: -+-+-+\cdots = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfe Conditions
In the unconstrained minimization problem, the Wolfe conditions are a set of inequalities for performing inexact line search, especially in quasi-Newton methods, first published by Philip Wolfe in 1969. In these methods the idea is to find ::\min_x f(\mathbf) for some smooth f\colon\mathbb R^n\to\mathbb R. Each step often involves approximately solving the subproblem ::\min_ f(\mathbf_k + \alpha \mathbf_k) where \mathbf_k is the current best guess, \mathbf_k \in \mathbb R^n is a search direction, and \alpha \in \mathbb R is the step length. The inexact line searches provide an efficient way of computing an acceptable step length \alpha that reduces the objective function 'sufficiently', rather than minimizing the objective function over \alpha\in\mathbb R^+ exactly. A line search algorithm can use Wolfe conditions as a requirement for any guessed \alpha, before finding a new search direction \mathbf_k. Armijo rule and curvature A step length \alpha_k is said to satisfy the ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
