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Mirror Descent
In mathematics, mirror descent is an iterative optimization algorithm for finding a local minimum of a differentiable function. It generalizes algorithms such as gradient descent and multiplicative weights. History Mirror descent was originally proposed by Nemirovski and Yudin in 1983. Motivation In gradient descent with the sequence of learning rates (\eta_n)_ applied to a differentiable function F, one starts with a guess \mathbf_0 for a local minimum of F, and considers the sequence \mathbf_0, \mathbf_1, \mathbf_2, \ldots such that :\mathbf_=\mathbf_n-\eta_n \nabla F(\mathbf_n),\ n \ge 0. This can be reformulated by noting that :\mathbf_=\arg \min_ \left(F(\mathbf_n) + \nabla F(\mathbf_n)^T (\mathbf - \mathbf_n) + \frac\, \mathbf - \mathbf_n\, ^2\right) In other words, \mathbf_ minimizes the first-order approximation to F at \mathbf_n with added proximity term \, \mathbf - \mathbf_n\, ^2. This squared Euclidean distance term is a particular example of a Bregman di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Algorithm
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called '' convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Minimum
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative'' extrema) or on the entire domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate 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 trajectory that maximizes that function; the procedure is then known as ''gradient ascent''. It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. 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 Has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arkadi Nemirovsky
Arkadi Nemirovski (; born March 14, 1947) is a professor at the H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology. He has been a leader in continuous optimization and is best known for his work on the ellipsoid method, modern interior-point methods and robust optimization. Biography Nemirovski earned a Ph.D. in Mathematics in 1974 from Moscow State University and a Doctor of Sciences in Mathematics degree in 1990 from the Institute of Cybernetics of the Ukrainian Academy of Sciences in Kiev. He has won three prestigious prizes: the Fulkerson Prize, the George B. Dantzig Prize, and the John von Neumann Theory Prize. He was elected a member of the U.S. National Academy of Engineering (NAE) in 2017 "for the development of efficient algorithms for large-scale convex optimization problems", and the U.S National Academy of Sciences (NAS) in 2020. In 2023, Nemirovski and Yurii Nesterov were jointly awarded th2023 WLA Prize in Com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bregman Distance
In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions – notably as either values of the parameter of a parametric model or as a data set of observed values – the resulting distance is a statistical distance. The most basic Bregman divergence is the squared Euclidean distance. Bregman divergences are similar to metrics, but satisfy neither the triangle inequality (ever) nor symmetry (in general). However, they satisfy a generalization of the Pythagorean theorem, and in information geometry the corresponding statistical manifold is interpreted as a (dually) flat manifold. This allows many techniques of optimization theory to be generalized to Bregman divergences, geometrically as generalizations of least squares. Bregman dive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (mathematics), ''epigraph'' (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup \cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap \cap. A twice-differentiable function, differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain of a function, domain. Well-known examples of convex functions of a single variable include a linear function f(x) = cx (where c is a real number), a quadratic function cx^2 (c as a nonnegative real number) and an exponential function ce^x (c as a nonnegative real number). Convex functions pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, 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 minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f(\mathbf) may be defined by: df=\nabla f \cdot d\mathbf where df is the total infinitesimal change in f for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
