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Gradient descent
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For the analytical method called "steepest descent", see Method of steepest descent.
Gradient descent is a first-order iterative optimization algorithm for finding the minimum of a function. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or approximate gradient) of the function at the current point. If instead one takes steps proportional to the positive of the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.
Gradient descent is also known as steepest descent. However, gradient descent should not be confused with the method of steepest descent for approximating integrals.
Contents
[hide]
1 Description
1.1 Examples
1.2 Limitations
2 Solution of a linear system
3 Solution of a non-linear system
4 Comments
5 Computational examples
5.1 Python
5.2 Scala
5.3 Julia
5.4 C
5.5 Javascript (ES6)
5.6 MATLAB
5.7 R
6 Extensions
6.1 Fast gradient methods
6.2 The momentum method
7 An analogy for understanding gradient descent
8 See also
9 References
10 Sources
11 External links
Description[edit]
Illustration of gradient descent on a series of level sets.
Gradient descent is based on the observation that if the multi-variable function F ( x ) {\displaystyle F(\mathbf {x} )} is defined and differentiable in a neighborhood of a point a {\displaystyle \mathbf {a} } , then F ( x ) {\displaystyle F(\mathbf {x} )} decreases fastest if one goes from a {\displaystyle \mathbf {a} } in the direction of the negative gradient of F {\displaystyle F} at a {\displaystyle \mathbf {a} } , − ∇ F ( a ) {\displaystyle -\nabla F(\mathbf {a} )} . It follows that, if
a n + 1 = a n − γ ∇ F ( a n ) {\displaystyle \mathbf {a} _{n+1}=\mathbf {a} _{n}-\gamma \nabla F(\mathbf {a} _{n})}
for γ {\displaystyle \gamma } small enough, then F ( a n ) ≥ F ( a n + 1 ) {\displaystyle F(\mathbf {a_{n}} )\geq F(\mathbf {a_{n+1}} )} . In other words, the term γ ∇ F ( a ) {\displaystyle \gamma \nabla F(\mathbf {a} )} is subtracted from a {\displaystyle \mathbf {a} } because we want to move against the gradient, toward the minimum. With this observation in mind, one starts with a guess x 0 {\displaystyle \mathbf {x} _{0}} for a local minimum of F {\displaystyle F} , and considers the sequence x 0 , x 1 , x 2 , … {\displaystyle \mathbf {x} _{0},\mathbf {x} _{1},\mathbf {x} _{2},\dots } such that
x n + 1 = x n − γ n ∇ F ( x n ) , n ≥ 0. {\displaystyle \mathbf {x} _{n+1}=\mathbf {x} _{n}-\gamma _{n}\nabla F(\mathbf {x} _{n}),\ n\geq 0.}
We have
F ( x 0 ) ≥ F ( x 1 ) ≥ F ( x 2 ) ≥ ⋯ , {\displaystyle F(\mathbf {x} _{0})\geq F(\mathbf {x} _{1})\geq F(\mathbf {x} _{2})\geq \cdots ,}
so hopefully the sequence ( x n ) {\displaystyle (\mathbf {x} _{n})} converges to the desired local minimum. Note that the value of the step size γ {\displaystyle \gamma } is allowed to change at every iteration. With certain assumptions on the function F {\displaystyle F} (for example, F {\displaystyle F} convex and ∇ F {\displaystyle \nabla F} Lipschitz) and particular choices of γ {\displaystyle \gamma } (e.g., chosen either via a line search that satisfies the Wolfe conditions or the Barzilai-Borwein[citation needed] method shown as following),
γ n = ( x n − x n − 1 ) T [ ∇ F ( x n ) − ∇ F ( x n − 1 ) ] | | ∇ F ( x n ) − ∇ F ( x n − 1 ) | | 2 {\displaystyle \gamma _{n}={\frac {(\mathbf {x} _{n}-\mathbf {x} _{n-1})^{T}[\nabla F(\mathbf {x} _{n})-\nabla F(\mathbf {x} _{n-1})]}{||\nabla F(\mathbf {x} _{n})-\nabla F(\mathbf {x} _{n-1})||^{2}}}}
convergence to a local minimum can be guaranteed. When the function F {\displaystyle F} is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution.
This process is illustrated in the adjacent picture. Here F {\displaystyle F} is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of F {\displaystyle F} is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the function F {\displaystyle F} is minimal.
Examples[edit]
Gradient descent has problems with pathological functions such as the Rosenbrock function shown here.
f ( x 1 , x 2 ) = ( 1 − x 1 ) 2 + 100 ( x 2 − x 1 2 ) 2 . {\displaystyle f(x_{1},x_{2})=(1-x_{1})^{2}+100(x_{2}-{x_{1}}^{2})^{2}.\quad }
The Rosenbrock function has a narrow curved valley which contains the minimum. The bottom of the valley is very flat. Because of the curved flat valley the optimization is zig-zagging slowly with small stepsizes towards the minimum.
The "Zig-Zagging" nature of the method is also evident below, where the gradient descent method is applied to F ( x , y ) = sin  ( 1 2 x 2 − 1 4 y 2 + 3 ) cos  ( 2 x + 1 − e y ) {\displaystyle F(x,y)=\sin \left({\frac {1}{2}}x^{2}-{\frac {1}{4}}y^{2}+3\right)\cos(2x+1-e^{y})} .
Limitations[edit]
For some of the above examples, gradient descent is relatively slow close to the minimum: technically, its asymptotic rate of convergence is inferior to many other methods. For poorly conditioned convex problems, gradient descent increasingly 'zigzags' as the gradients point nearly orthogonally to the shortest direction to a minimum point. For more details, see the comments below.
For non-differentiable functions, gradient methods are ill-defined. For locally Lipschitz problems and especially for convex minimization problems, bundle methods of descent are well-defined. Non-descent methods, like subgradient projection methods, may also be used.[1] These methods are typically slower than gradient descent. Another alternative for non-differentiable functions is to "smooth" the function, or bound the function by a smooth function. In this approach, the smooth problem is solved in the hope that the answer is close to the answer for the non-smooth problem (occasionally, this can be made rigorous).
Solution of a linear system[edit]
Gradient descent can be used to solve a system of linear equations, reformulated as a quadratic minimization problem, e.g., using linear least squares. The solution of
A x − b = 0 {\displaystyle A\mathbf {x} -\mathbf {b} =0}
in the sense of linear least squares is defined as minimizing the function
F ( x ) = ‖ A x − b ‖ 2 . {\displaystyle F(\mathbf {x} )=\|A\mathbf {x} -\mathbf {b} \|^{2}.}
In traditional linear least squares for real A {\displaystyle A} and b {\displaystyle \mathbf {b} } the Euclidean norm is used, in which case
∇ F ( x ) = 2 A T ( A x − b ) . {\displaystyle \nabla F(\mathbf {x} )=2A^{T}(A\mathbf {x} -\mathbf {b} ).}
In this case, the line search minimization, finding the locally optimal step size γ {\displaystyle \gamma } on every iteration, can be performed analytically, and explicit formulas for the locally optimal γ {\displaystyle \gamma } are known.[2]
For solving linear equations, this algorithm is rarely used, with the conjugate gradient method being one of the most popular alternatives. The speed of convergence of gradient descent depends on the ratio of the maximum to minimum eigenvalues of A T A {\displaystyle A^{T}A} , while the speed of convergence of conjugate gradients has a more complex dependence on the eigenvalues, and can benefit from preconditioning. Gradient descent also benefits from preconditioning, but this is not done as commonly.
Solution of a non-linear system[edit]
Gradient descent can also be used to solve a system of nonlinear equations. Below is an example that shows how to use the gradient descent to solve for three unknown variables, x1, x2, and x3. This example shows one iteration of the gradient descent.
Consider the nonlinear system of equations
{ 3 x 1 − cos  ( x 2 x 3 ) − 3 2 = 0 4 x 1 2 − 625 x 2 2 + 2 x 2 − 1 = 0 exp  ( − x 1 x 2 ) + 20 x 3 + 10 π − 3 3 = 0. {\displaystyle {\begin{cases}3x_{1}-\cos(x_{2}x_{3})-{\tfrac {3}{2}}=0\\4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1=0\\\exp(-x_{1}x_{2})+20x_{3}+{\tfrac {10\pi -3}{3}}=0.\end{cases}}}
Let us introduce the associated function
G ( x ) = [ 3 x 1 − cos  ( x 2 x 3 ) − 3 2 4 x 1 2 − 625 x 2 2 + 2 x 2 − 1 exp  ( − x 1 x 2 ) + 20 x 3 + 10 π − 3 3 ] {\displaystyle G(\mathbf {x} )={\begin{bmatrix}3x_{1}-\cos(x_{2}x_{3})-{\tfrac {3}{2}}\\4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1\\\exp(-x_{1}x_{2})+20x_{3}+{\tfrac {10\pi -3}{3}}\\\end{bmatrix}}} ,
where
x = [ x 1 x 2 x 3 ] {\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\end{bmatrix}}} .
One might now define the objective function
F ( x ) = 1 2 G T ( x ) G ( x ) = 1 2 ( ( 3 x 1 − cos  ( x 2 x 3 ) − 3 2 ) 2 + ( 4 x 1 2 − 625 x 2 2 + 2 x 2 − 1 ) 2 + ( exp  ( − x 1 x 2 ) + 20 x 3 + 1 3 ( 10 π − 3 ) ) 2 ) {\displaystyle F(\mathbf {x} )={\tfrac {1}{2}}G^{\mathrm {T} }(\mathbf {x} )G(\mathbf {x} )={\tfrac {1}{2}}\left(\left(3x_{1}-\cos(x_{2}x_{3})-{\tfrac {3}{2}}\right)^{2}+\left(4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1\right)^{2}+\left(\exp(-x_{1}x_{2})+20x_{3}+{\tfrac {1}{3}}(10\pi -3)\right)^{2}\right)} ,
which we will attempt to minimize. As an initial guess, let us use
x ( 0 ) = [ 0 0 0 ] {\displaystyle \mathbf {x} ^{(0)}={\begin{bmatrix}0\\0\\0\\\end{bmatrix}}} .
We know that
x ( 1 ) = x ( 0 ) − γ 0 ∇ F ( x ( 0 ) ) {\displaystyle \mathbf {x} ^{(1)}=\mathbf {x} ^{(0)}-\gamma _{0}\nabla F(\mathbf {x} ^{(0)})} ,
where
∇ F ( x ( 0 ) ) = J G ( x ( 0 ) ) T G ( x ( 0 ) ) {\displaystyle \nabla F(\mathbf {x} ^{(0)})=J_{G}(\mathbf {x} ^{(0)})^{\mathrm {T} }G(\mathbf {x} ^{(0)})} .
The Jacobian matrix J G {\displaystyle J_{G}} is given by
J G ( x ) = [ 3 sin  ( x 2 x 3 ) x 3 sin  ( x 2 x 3 ) x 2 8 x 1 − 1250 x 2 + 2 0 − x 2 exp  ( − x 1 x 2 ) − x 1 exp  ( − x 1 x 2 ) 20 ] {\displaystyle J_{G}(\mathbf {x} )={\begin{bmatrix}3&\sin(x_{2}x_{3})x_{3}&\sin(x_{2}x_{3})x_{2}\\8x_{1}&-1250x_{2}+2&0\\-x_{2}\exp {(-x_{1}x_{2})}&-x_{1}\exp(-x_{1}x_{2})&20\\\end{bmatrix}}} .
Evaluating J G {\displaystyle J_{G}} and G {\displaystyle G} at x ( 0 ) {\displaystyle \mathbf {x} ^{(0)}} yields
J G ( x ( 0 ) ) = [ 3 0 0 0 2 0 0 0 20 ] , G ( x ( 0 ) ) = [ − 2.5 − 1 10.472 ] {\displaystyle J_{G}\left(\mathbf {x} ^{(0)}\right)={\begin{bmatrix}3&0&0\\0&2&0\\0&0&20\end{bmatrix}},\qquad G(\mathbf {x} ^{(0)})={\begin{bmatrix}-2.5\\-1\\10.472\end{bmatrix}}} .
Thus,
x ( 1 ) = 0 − γ 0 [ − 7.5 − 2 209.44 ] {\displaystyle \mathbf {x} ^{(1)}=0-\gamma _{0}{\begin{bmatrix}-7.5\\-2\\209.44\end{bmatrix}}}
and
F ( x ( 0 ) ) = 0.5 ( ( − 2.5 ) 2 + ( − 1 ) 2 + ( 10.472 ) 2 ) = 58.456 {\displaystyle F\left(\mathbf {x} ^{(0)}\right)=0.5\left((-2.5)^{2}+(-1)^{2}+(10.472)^{2}\right)=58.456} .
An animation showing the first 83 iterations of gradient descent applied to this example. Surfaces are isosurfaces of F ( x ( n ) ) {\displaystyle F(\mathbf {x} ^{(n)})} at current guess x ( n ) {\displaystyle \mathbf {x} ^{(n)}} , and arrows show the direction of descent. Due to a small and constant step size, the convergence is slow.
Now, a suitable γ 0 {\displaystyle \gamma _{0}} must be found such that F ( x ( 1 ) ) ≤ F ( x ( 0 ) ) {\displaystyle F(\mathbf {x} ^{(1)})\leq F(\mathbf {x} ^{(0)})} . This can be done with any of a variety of line search algorithms. One might also simply guess γ 0 = 0.001 {\displaystyle \gamma _{0}=0.001} , which gives
x ( 1 ) = [ 0.0075 0.002 − 0.20944 ] {\displaystyle \mathbf {x} ^{(1)}={\begin{bmatrix}0.0075\\0.002\\-0.20944\\\end{bmatrix}}} .
Evaluating the objective function at this value, yields
F ( x ( 1 ) ) = 0.5 ( ( − 2.48 ) 2 + ( − 1.00 ) 2 + ( 6.28 ) 2 ) = 23.306 {\displaystyle F\left(\mathbf {x} ^{(1)}\right)=0.5\left((-2.48)^{2}+(-1.00)^{2}+(6.28)^{2}\right)=23.306} .
The decrease from F ( x ( 0 ) ) = 58.456 {\displaystyle F(\mathbf {x} ^{(0)})=58.456} to the next step's value of F ( x ( 1 ) ) = 23.306 {\displaystyle F(\mathbf {x} ^{(1)})=23.306} is a sizable decrease in the objective function. Further steps would reduce its value further, until an approximate solution to the system was found.
Comments[edit]
Gradient descent works in spaces of any number of dimensions, even in infinite-dimensional ones. In the latter case the search space is typically a function space, and one calculates the Fréchet derivative of the functional to be minimized to determine the descent direction.[3]
The gradient descent can take many iterations to compute a local minimum with a required accuracy, if the curvature in different directions is very different for the given function. For such functions, preconditioning, which changes the geometry of the space to shape the function level sets like concentric circles, cures the slow convergence. Constructing and applying preconditioning can be computationally expensive, however.
The gradient descent can be combined with a line search, finding the locally optimal step size γ {\displaystyle \gamma } on every iteration. Performing the line search can be time-consuming. Conversely, using a fixed small γ {\displaystyle \gamma } can yield poor convergence.
Methods based on Newton's method and inversion of the Hessian using conjugate gradient techniques can be better alternatives.[4][5] Generally, such methods converge in fewer iterations, but the cost of each iteration is higher. An example is the BFGS method which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated line search algorithm, to find the "best" value of γ . {\displaystyle \gamma .} For extremely large problems, where the computer memory issues dominate, a limited-memory method such as L-BFGS should be used instead of BFGS or the steepest descent.
Gradient descent can be viewed as Euler's method for solving ordinary differential equations x ′ ( t ) = − ∇ f ( x ( t ) ) {\displaystyle x'(t)=-\nabla f(x(t))} of a gradient flow.
Computational examples[edit]
Python[edit]
The gradient descent algorithm is applied to find a local minimum of the function f(x)=x4−3x3+2, with derivative f'(x)=4x3−9x2. Here is an implementation in the Python programming language.
# From calculation, it is expected that the local minimum occurs at x=9/4 cur_x = 6 # The algorithm starts at x=6 gamma = 0.01 # step size multiplier precision = 0.00001 previous_step_size = 1 max_iters = 10000 # maximum number of iterations iters = 0 #iteration counter df = lambda x: 4 * x**3 - 9 * x**2 while (previous_step_size > precision) & (iters < max_iters): prev_x = cur_x cur_x -= gamma * df(prev_x) previous_step_size = abs(cur_x - prev_x) iters+=1 print("The local minimum occurs at", cur_x) #The output for the above will be: ('The local minimum occurs at', 2.2499646074278457)
Scala[edit]
Here is an implementation in Scala programming language
import math._ val curX = 6 val gamma = 0.01 val precision = 0.00001 val previousStepSize = 1/precision def df(x: Double):Double = 4 * pow(x,3) - 9 * pow(x,2) def gredientDescent(precision:Double, previousStepSize : Double, curX: Double ): Double = { if(previousStepSize > precision) { val newX = curX + (-gamma) * df(curX)	println(curX) gredientDescent(precision , abs(newX - curX), newX) } else curX	} val ans = gredientDescent(precision, previousStepSize, curX) println("The local minimum occurs at $ans")
Julia[edit]
Here is the same algorithm implemented in the Julia programming language.
cur_x = 6; gamma = 0.01; precision = 0.00001; previous_step_size = 1; df = (x) -> 4x^3 - 9x^2; while previous_step_size > precision prev_x = cur_x; cur_x -= gamma * df(prev_x); previous_step_size = abs(cur_x - prev_x); end println("The local minimum occurs at $cur_x");
C[edit]
Here is the same algorithm implemented in the C programming language.
#include <stdio.h> #include <stdlib.h> double dfx(double x) { return 4*(x*x*x) - 9*(x*x); } double abs_val(double x) { return x > 0 ? x : -x; } double gradient_descent(double dx, double error, double gamma, unsigned int max_iters) { double c_error = error + 1; unsigned int iters = 0; double p_error; while(error < c_error && iters < max_iters) { p_error = dx; dx -= dfx(p_error) * gamma; c_error = abs_val(p_error-dx); printf("\nc_error %f\n", c_error); iters++; } return dx; } int main() { double dx = 6; double error = 1e-6; double gamma = 0.01; unsigned int max_iters = 10000; printf("The local minimum is: %f\n", gradient_descent(dx, error, gamma, max_iters)); return 0; }
The above piece of code has to be modified with regard to step size according to the system at hand and convergence can be made faster by using an adaptive step size. In the above case the step size is not adaptive. It stays at 0.01 in all the directions which can sometimes cause the method to fail by diverging from the minimum.
Javascript (ES6)[edit]
Here is an example of gradient descent against randomised values, squareMeter and price, written in Javascript
// adjust training set size const M = 10; // generate random training set const DATA = []; const getRandomIntFromInterval = (min, max) => Math.floor(Math.random() * (max - min + 1) + min); const createRandomPortlandHouse = () => ({ squareMeter: getRandomIntFromInterval(0, 100), price: getRandomIntFromInterval(0, 100), }); for (let i = 0; i < M; i++) { DATA.push(createRandomPortlandHouse()); } const _x = DATA.map(date => date.squareMeter); const _y = DATA.map(date => date.price); // linear regression and gradient descent const LEARNING_RATE = 0.0003; let thetaOne = 0; let thetaZero = 0; const hypothesis = x => thetaZero + thetaOne * x; const learn = (x, y, alpha) => { let thetaZeroSum = 0; let thetaOneSum = 0; for (let i = 0; i < M; i++) { thetaZeroSum += hypothesis(x[i]) - y[i]; thetaOneSum += (hypothesis(x[i]) - y[i]) * x[i]; } thetaZero = thetaZero - (alpha / M) * thetaZeroSum; thetaOne = thetaOne - (alpha / M) * thetaOneSum; } const MAX_ITER = 1500; for (let i = 0; i < MAX_ITER; i++) { learn(_x, _y, LEARNING_RATE); console.log('thetaZero ', thetaZero, 'thetaOne', thetaOne); }
MATLAB[edit]
The following MATLAB code demonstrates a concrete solution for solving the non-linear system of equations presented in the previous section:
{ 3 x 1 − cos  ( x 2 x 3 ) − 3 2 = 0 4 x 1 2 − 625 x 2 2 + 2 x 2 − 1 = 0 exp  ( − x 1 x 2 ) + 20 x 3 + 10 π − 3 3 = 0 {\displaystyle {\begin{cases}3x_{1}-\cos(x_{2}x_{3})-{\tfrac {3}{2}}=0\\4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1=0\\\exp(-x_{1}x_{2})+20x_{3}+{\tfrac {10\pi -3}{3}}=0\end{cases}}}
% Multi-variate vector-valued function G(x) G = @(x) [ 3*x(1) - cos(x(2)*x(3)) - 3/2 ; 4*x(1)^2 - 625*x(2)^2 + 2*x(2) - 1 ; exp(-x(1)*x(2)) + 20*x(3) + (10*pi-3)/3]; % Jacobian of G JG = @(x) [ 3, sin(x(2)*x(3))*x(3), sin(x(2)*x(3))*x(2) ; 8*x(1), -1250*x(2)+2, 0 ; -x(2)*exp(-x(1)*x(2)), -x(1)*exp(-x(1)*x(2)), 20 ]; % Objective function F(x) to minimize in order to solve G(x)=0 F = @(x) 0.5 * sum(G(x).^2); % Gradient of F (partial derivatives) dF = @(x) JG(x).' * G(x); % Parameters GAMMA = 0.001; % step size (learning rate) MAX_ITER = 1000; % maximum number of iterations FUNC_TOL = 0.1; % termination tolerance for F(x) fvals = []; % store F(x) values across iterations progress = @(iter,x) fprintf('iter = %3d: x = %-32s, F(x) = %f\n', ... iter, mat2str(x,6), F(x)); % Iterate iter = 1; % iterations counter x = [0; 0; 0]; % initial guess fvals(iter) = F(x); progress(iter, x); while iter < MAX_ITER && fvals(end) > FUNC_TOL iter = iter + 1; x = x - GAMMA * dF(x); % gradient descent fvals(iter) = F(x); % evaluate objective function progress(iter, x); % show progress end % Plot plot(1:iter, fvals, 'LineWidth',2); grid on; title('Objective Function'); xlabel('Iteration'); ylabel('F(x)'); % Evaluate final solution of system of equations G(x)=0 disp('G(x) = '); disp(G(x)) % Output: % % iter = 1: x = [0;0;0] , F(x) = 58.456136 % iter = 2: x = [0.0075;0.002;-0.20944] , F(x) = 23.306394 % iter = 3: x = [0.015005;0.0015482;-0.335103] , F(x) = 10.617030 % ... % iter = 187: x = [0.683335;0.0388258;-0.52231] , F(x) = 0.101161 % iter = 188: x = [0.684666;0.0389831;-0.522302] , F(x) = 0.099372 % % (converged in 188 iterations after exceeding termination tolerance for F(x))
R[edit]
The following R code is an example of implementing gradient descent algorithm to find the minimum of the function f(x)=x4−3x3+2 in previous section. Note that we are looking for f(x)'s minimum by solving its derivative being equal to zero.
∇ f ( x ) = 4 x 3 − 9 x 2 = 0 {\displaystyle \nabla f(x)=4x^{3}-9x^{2}=0}
And the x can be updated with gradient descent method every iteration in the form of
x ( k + 1 ) = x ( k ) − α ∇ f ( x ( k ) ) {\displaystyle x^{(k+1)}=x^{(k)}-\alpha \nabla f(x^{(k)})}
where k = 1, 2, ..., maximum iteration, and α is the step size.
# set up a stepsize alpha = 0.003 # set up a number of iteration iter = 500 # define the gradient of f(x) = x^4 - 3*x^3 + 2 gradient = function(x) return((4*x^3) - (9*x^2)) # randomly initialize a value to x set.seed(100) x = floor(runif(1)*10) # create a vector to contain all xs for all steps x.All = vector("numeric",iter) # gradient descent method to find the minimum for(i in 1:iter){ x = x - alpha*gradient(x) x.All[i] = x print(x) } # print result and plot all xs for every iteration print(paste("The minimum of f(x) is ", x, sep = "")) plot(x.All, type = "l")
Extensions[edit]
Gradient descent can be extended to handle constraints by including a projection onto the set of constraints. This method is only feasible when the projection is efficiently computable on a computer. Under suitable assumptions, this method converges. This method is a specific case of the forward-backward algorithm for monotone inclusions (which includes convex programming and variational inequalities).[6]
Fast gradient methods[edit]
Another extension of gradient descent is due to Yurii Nesterov from 1983,[7] and has been subsequently generalized. He provides a simple modification of the algorithm that enables faster convergence for convex problems. For unconstrained smooth problems the method is called the Fast Gradient Method (FGM) or the Accelerated Gradient Method (AGM). Specifically, if the differentiable function F {\displaystyle F} is convex and ∇ F {\displaystyle \nabla F} is Lipschitz, and it is not assumed that F {\displaystyle F} is strongly convex, then the error in the objective value generated at each step k {\displaystyle k} by the gradient descent method will be bounded by O ( 1 / k ) {\displaystyle {\mathcal {O}}(1/k)} . Using the Nesterov acceleration technique, the error decreases at O ( 1 / k 2 ) {\displaystyle {\mathcal {O}}(1/k^{2})} .[8] It is known that the rate O ( 1 / k 2 ) {\displaystyle {\mathcal {O}}(1/k^{2})} for the decrease of the cost function is optimal for first-order optimization methods. Nevertheless, there is the opportunity to improve the algorithm by reducing the constant factor. The optimized gradient method (OGM)[9] reduces that constant by a factor of two and is an optimal first-order method for large-scale problems.[10]
For constrained or non-smooth problems, Nesterov's FGM is called the fast proximal gradient method (FPGM), an acceleration of the proximal gradient method.
The momentum method[edit]
Main article: Stochastic gradient descent § Momentum
Yet another extension, that reduces the risk of getting stuck in a local minimum, as well as speeds up the convergence considerably in cases where the process would otherwise zig-zag heavily, is the momentum method, which uses a momentum term in analogy to "the mass of Newtonian particles that move through a viscous medium in a conservative force field".[11] This method is often used as an extension to the backpropagation algorithms used to train artificial neural networks.[12][13]
An analogy for understanding gradient descent[edit]
The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. A person is stuck in the mountains and is trying to get down (i.e. trying to find the minima). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible, so he must use local information to find the minima. He can use the method of gradient descent, which involves looking at the steepness of the hill at his current position, then proceeding in the direction with the steepest descent (i.e. downhill). If he was trying to find the top of the mountain (i.e. the maxima), then he would proceed in the direction steepest ascent (i.e. uphill). Using this method, he would eventually find his way down the mountain. However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires a sophisticated instrument to measure, which the person happens to have at the moment. It takes quite some time to measure the steepness of the hill with the instrument, thus he should minimize his use of the instrument if he wanted to get down the mountain before sunset. The difficulty then is choosing the frequency at which he should measure the steepness of the hill so not to go off track.
In this analogy, the person represents the algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. The steepness of the hill represents the slope of the error surface at that point. The instrument used to measure steepness is differentiation (the slope of the error surface can be calculated by taking the derivative of the squared error function at that point). The direction he chooses to travel in aligns with the gradient of the error surface at that point. The amount of time he travels before taking another measurement is the learning rate of the algorithm. See the limitation section for a discussion of the limitations of this type of "hill climbing" algorithm.
See also[edit]
Conjugate gradient method
Stochastic gradient descent
Rprop
Delta rule
Wolfe conditions
Preconditioning
BFGS method
Nelder–Mead method
Gauss–Newton algorithm
Hill climbing
References[edit]
Jump up ^ Kiwiel, Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization". Mathematical Programming (Series A). 90 (1). Berlin, Heidelberg: Springer. pp. 1–25. doi:10.1007/PL00011414. ISSN 0025-5610. MR 1819784.
Jump up ^ Yuan, Ya-xiang (1999). "Step-sizes for the gradient method" (PDF). AMS/IP Studies in Advanced Mathematics. Providence, RI: American Mathematical Society. 42 (2): 785.
Jump up ^ G. P. Akilov, L. V. Kantorovich, Functional Analysis, Pergamon Pr; 2 Sub edition, ISBN 0-08-023036-9, 1982
Jump up ^ W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd Ed., Cambridge University Press, New York, 1992
Jump up ^ T. Strutz: Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). 2nd edition, Springer Vieweg, 2016, ISBN 978-3-658-11455-8.
Jump up ^ P. L. Combettes and J.-C. Pesquet, "Proximal splitting methods in signal processing", in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (H. H. Bauschke, R. S. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, Editors), pp. 185-212. Springer, New York, 2011.
Jump up ^ Yu. Nesterov, "Introductory Lectures on Convex Optimization. A Basic Course" (Springer, 2004, ISBN 1-4020-7553-7)
Jump up ^ Fast Gradient Methods, lecture notes by Prof. Lieven Vandenberghe for EE236C at UCLA
Jump up ^ D. Kim, J A Fessler, "Optimized first-order methods for smooth convex minimization" Math. Prog. 151:8-107, Sep. 2016 https://dx.doi.org/10.1007/s10107-015-0949-3
Jump up ^ Yoel Drori, "The exact information-based complexity of smooth convex minimization." https://arxiv.org/abs/1606.01424
Jump up ^ Qian, Ning (January 1999). "On the momentum term in gradient descent learning algorithms" (PDF). Neural Networks. 12 (1): 145–151. Archived from the original (PDF) on 8 May 2014. Retrieved 17 October 2014.
Jump up ^ "Momentum and Learning Rate Adaptation". Willamette University. Retrieved 17 October 2014.
Jump up ^ Geoffrey Hinton; Nitish Srivastava; Kevin Swersky. "6 - 3 - The momentum method". YouTube. Retrieved 18 October 2014. Part of a lecture series for the Coursera online course Neural Networks for Machine Learning.
Sources[edit]
Avriel, Mordecai (2003). Nonlinear Programming: Analysis and Methods. Courier Corporation. ISBN 978-0-486-43227-4.
Snyman, Jan (3 March 2005). Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms. Springer Science & Business Media. ISBN 978-0-387-24348-1.
Homod, Raad Z.; Sahari, Khairul Salleh Mohamed; Almurib, Haider A. F.; Nagi, Farrukh Hafiz (2012-06-01). "Gradient auto-tuned Takagi–Sugeno Fuzzy Forward control of a HVAC system using predicted mean vote index". Energy and Buildings. 49 (6): 254–267. doi:10.1016/j.enbuild.2012.02.013.
Cauchy, Augustin (1847). Méthode générale pour la résolution des systèmes d'équations simultanées. pp. 536–538.
External links[edit]
Using gradient descent in C++, Boost, Ublas for linear regression
Series of Khan Academy videos discusses gradient ascent
Online book teaching gradient descent in deep neural network context
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text:
Grammy Award for Record of the Year
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For the United Kingdom music award, see The Record of the Year.
Grammy Award for Record of the Year
Gilded gramophone trophy presented to Grammy Award winners
Awarded for
quality vocal or instrumental recording tracks
Country
United States
Presented by
National Academy of Recording Arts and Sciences
First awarded
1959
Last awarded
2018
Currently held by
Bruno Mars, "24K Magic" (2018)
Website
www.grammy.com
The Grammy Award for Record of the Year is presented by the National Academy of Recording Arts and Sciences of the United States to "honor artistic achievement, technical proficiency and overall excellence in the recording industry, without regard to sales or chart position."[1][2] The Record of the Year award is one of the four most prestigious categories at the awards (alongside Best New Artist, Song of the Year and Album of the Year) presented annually since the 1st Grammy Awards in 1959. According to the 54th Grammy Awards description guide, the award is presented:
for commercially released singles or tracks of new vocal or instrumental recordings. Tracks from a previous year's album may be entered provided the track was not entered the previous year and provided the album did not win a Grammy. Award to the artist(s), producer(s), recording engineer(s) and/or mixer(s) if other than the artist.[3]
Since the 55th Grammy Awards in 2013, mastering engineers are considered nominees and award recipients in this category.[4]
Record of the Year is related to but is conceptually different from Song of the Year or Album of the Year:
Record of the Year is awarded for a single or for one track from an album. This award goes to the performing artist, the producer, recording engineer and/or mixer for that song. In this sense, "record" means a particular recorded song, not its composition or an album of songs.
Song of the Year is also awarded for a single or individual track, but the recipient of this award is the songwriter who actually wrote the lyrics and/or melodies to the song. "Song" in this context means the song as composed, not its recording.
Album of the Year is awarded for a whole album, and the award is presented to the artist, producer, recording engineer, and mastering engineer for that album. In this context, "album" means a recorded collection of songs (a multi-track LP, CD, or download package), not the individual songs or their compositions.
Contents
[hide]
1 History
2 Process
3 Recipients
4 References
5 External links
History[edit]
The honorees through its history have been:
1959–1965: Artist only.
1966–1998: Artist and producer.
1999–2012: Artist, producer, and recording engineer or mixer.
2013–: Artist, producer, mastering engineer, and recording engineer or mixer.
Tom Coyne holds the record for most wins in this category as a mastering engineer at four times (2015, 2016, 2017 and 2018). Coyne is the only person to win the award four consecutive years (2015–2018). Paul Simon has won three times, more than any other artist ("Mrs. Robinson" in 1969, "Bridge over Troubled Water" in 1971, and "Graceland" in 1988). Tom Elmhirst has won three times as an engineer/mixer (2008, 2012 and 2017).
Roberta Flack was the first artist to win Record of the Year in two consecutive years for the years 1973 ("The First Time Ever I Saw Your Face") and 1974 ("Killing Me Softly with His Song"). This happened again when the group U2 won for the years 2001 ("Beautiful Day") and 2002 ("Walk On"), the only occurrence of an artist winning the award two consecutive years with records from the same album.
Other artists to receive two Grammys for Record of the Year are Henry Mancini ("Moon River", "Days of Wine and Roses"); Art Garfunkel ("Mrs. Robinson", "Bridge over Troubled Water"); The Fifth Dimension ("Up, Up and Away", "Aquarius/Let the Sunshine In"); Eric Clapton ("Tears in Heaven", "Change the World"); Norah Jones ("Don't Know Why", "Here We Go Again"); Mark Ronson ("Rehab", "Uptown Funk"); Adele ("Rolling in the Deep", "Hello") and Bruno Mars ("Uptown Funk", "24K Magic").
Frank Sinatra has the most nominations for Record of the Year for an artist and a male artist with seven nominations; he won the award once in 1967 for "Strangers in the Night". The Beatles have the most Record of the Year nominations for a group; they had four nominations ("I Want to Hold Your Hand", "Yesterday", "Hey Jude", and "Let It Be") but never won the award. Barbra Streisand and Beyoncé have the most Record of the Year nominations amongst female artists with five nods. Streisand has been nominated ("Happy Days Are Here Again", "People", "Evergreen (Love Theme from A Star Is Born)", "You Don't Bring Me Flowers", (with Neil Diamond) and "Woman in Love") but has never received the award.[5] Beyoncé was nominated for "Say My Name" as part of Destiny's Child and four times as a solo act with "Crazy in Love", "Irreplaceable", "Halo", and "Formation".
Christopher Cross is the first artist to receive the Grammy Award for Record of the Year as well as for Song of the Year, Album of the Year, and Best New Artist in one ceremony. Adele is the first female artist to receive the award for Record of the Year, Song of the Year, Best New Artist, and Album of the Year, nonconsecutively. Only six artists took the Record of the Year and Best New Artist awards during the same ceremony: Bobby Darin ("Mack the Knife"), Christopher Cross ("Sailing"), Sheryl Crow ("All I Wanna Do"), Norah Jones ("Don't Know Why"), Amy Winehouse ("Rehab") and Sam Smith ("Stay with Me").
Thirty of the winning songs have also won the award for Song of the Year.
Process[edit]
Members of the National Academy of Recording Arts and Sciences nominate their choices for record of the year. A list of the top twenty records is given to the Nominations Review Committee, a specially selected group of anonymous members, who then select the top five records to gain a nomination in the category in a special ballot.[6] The rest of the members then vote a winner from the five nominees.[7]
Recipients[edit]
An asterisk (*) indicates this recording also won Song of the Year.
Year[I]
Winner(s)
Work
Nominees
Ref.
1959 Modugno, DomenicoDomenico Modugno Volare !"Nel Blu Dipinto Di Blu (Volare)" *
"Catch a Falling Star" by Perry Como
"The Chipmunk Song (Christmas Don't Be Late)" by David Seville
"Fever" by Peggy Lee
"Witchcraft" by Frank Sinatra
[8]
1960 Darin, BobbyBobby Darin Mack the Knife !"Mack the Knife"
"A Fool Such as I" by Elvis Presley
"High Hopes" by Frank Sinatra
"Like Young" by André Previn
"The Three Bells" by The Browns
[9]
1961 Faith, PercyPercy Faith Theme from A Summer Place !"Theme from A Summer Place"
"Are You Lonesome Tonight?" by Elvis Presley
"Georgia On My Mind" by Ray Charles
"Mack the Knife" by Ella Fitzgerald
"Nice 'n' Easy" by Frank Sinatra
[10]
1962 Mancini, HenryHenry Mancini Moon River !"Moon River" *
"Big Bad John" by Jimmy Dean
"The Second Time Around" by Frank Sinatra
"Take Five" by The Dave Brubeck Quartet
"(Up a) Lazy River" by Si Zentner
[11]
1963 Bennett, TonyTony Bennett I Left My Heart in San Francisco !"I Left My Heart in San Francisco"
"Desafinado" by Stan Getz & Charlie Byrd
"Fly Me to the Moon Bossa Nova" by Joe Harnell and His Orchestra
"I Can't Stop Loving You" by Ray Charles
"What Kind of Fool Am I?" by Sammy Davis, Jr.
[12]
1964 Mancini, HenryHenry Mancini Days of Wine and Roses !"Days of Wine and Roses" *
"Dominique" by The Singing Nun
"Happy Days Are Here Again" by Barbra Streisand
"I Wanna Be Around" by Tony Bennett
"Wives and Lovers" by Jack Jones
[13]
1965 Gilberto, AstrudAstrud Gilberto and Stan Getz Girl from Ipanema, The !"The Girl from Ipanema"
"Downtown" by Petula Clark
"Hello, Dolly!" by Louis Armstrong
"I Want to Hold Your Hand" by The Beatles
"People" by Barbra Streisand
[14]
1966 Alpert, HerbHerb Alpert and the Tijuana Brass
· produced by Herb Alpert & Jerry Moss A Taste of Honey !"A Taste of Honey"
"Yesterday" by The Beatles
"The 'In' Crowd" by The Ramsey Lewis Trio
"King of the Road" by Roger Miller
"The Shadow of Your Smile" by Tony Bennett
[15]
1967 Sinatra, FrankFrank Sinatra
· produced by Jimmy Bowen Strangers in the Night !"Strangers in the Night"
"Almost Persuaded" by David Houston
"Monday, Monday" by The Mamas & the Papas
"What Now My Love" by Herb Alpert and the Tijuana Brass
"Winchester Cathedral" by The New Vaudeville Band
[16]
1968 Fifth Dimension, The !The 5th Dimension
· produced by Johnny Rivers & Marc Gordon Up, Up and Away !"Up, Up and Away" *
"By the Time I Get to Phoenix" by Glen Campbell
"My Cup Runneth Over" by Ed Ames
"Ode to Billie Joe" by Bobbie Gentry
"Somethin' Stupid" by Nancy Sinatra & Frank Sinatra
[17]
1969 Simon & Garfunkel
· produced by Art Garfunkel, Paul Simon & Roy Halee Mrs. Robinson !"Mrs. Robinson"
"Harper Valley PTA" by Jeannie C. Riley
"Hey Jude" by The Beatles
"Honey" by Bobby Goldsboro
"Wichita Lineman" by Glen Campbell
[18]
1970 Fifth Dimension, The !The 5th Dimension
· produced by Bones Howe Aquarius !"Aquarius/Let the Sunshine In"
"A Boy Named Sue" by Johnny Cash
"Is That All There Is?" by Peggy Lee
"Love Theme from Romeo and Juliet" by Henry Mancini
"Spinning Wheel" by Blood, Sweat & Tears
[19]
1971 Simon & Garfunkel
· produced by Art Garfunkel, Paul Simon & Roy Halee Bridge over Troubled Water !"Bridge over Troubled Water" *
"(They Long to Be) Close to You" by Carpenters
"Everything Is Beautiful" by Ray Stevens
"Fire and Rain" by James Taylor
"Let it Be" by The Beatles
[20]
1972 King, CaroleCarole King
· produced by Lou Adler It's Too Late !"It's Too Late"
"Joy to the World" by Three Dog Night
"My Sweet Lord" by George Harrison
"Theme from Shaft" by Isaac Hayes
"You've Got a Friend" by James Taylor
[21]
1973 Roberta Flack
· produced by Joel Dorn The First Time Ever I Saw Your Face !"The First Time Ever I Saw Your Face" *
"Alone Again (Naturally)" by Gilbert O'Sullivan
"American Pie" by Don McLean
"Song Sung Blue" by Neil Diamond
"Without You" by Nilsson
[22]
1974 Killing Me Softly with His Song !"Killing Me Softly with His Song" *
"Bad, Bad Leroy Brown" by Jim Croce
"Behind Closed Doors" by Charlie Rich
"You Are the Sunshine of My Life" by Stevie Wonder
"You're So Vain" by Carly Simon
[23]
1975 Newton-John, OliviaOlivia Newton-John
· produced by John Farrar I Honestly Love You !"I Honestly Love You"
"Don't Let the Sun Go Down on Me" by Elton John
"Feel Like Makin' Love" by Roberta Flack
"Help Me" by Joni Mitchell
"Midnight at the Oasis" by Maria Muldaur
[24]
1976 Captain & Tennille
· produced by Daryl Dragon Love Will Keep Us Together !"Love Will Keep Us Together"
"At Seventeen" by Janis Ian
"Lyin' Eyes" by Eagles
"Mandy" by Barry Manilow
"Rhinestone Cowboy" by Glen Campbell
[25]
1977 Benson, GeorgeGeorge Benson
· produced by Tommy LiPuma This Masquerade !"This Masquerade"
"Afternoon Delight" by Starland Vocal Band
"50 Ways to Leave Your Lover" by Paul Simon
"I Write the Songs" by Barry Manilow
"If You Leave Me Now" by Chicago
[26]
1978 Eagles !Eagles
· produced by Bill Szymczyk Hotel California !"Hotel California"
"Blue Bayou" by Linda Ronstadt
"Don't It Make My Brown Eyes Blue" by Crystal Gayle
"Evergreen (Love Theme from A Star Is Born)" by Barbra Streisand
"You Light Up My Life" by Debby Boone
[27]
1979 Joel, BillyBilly Joel
· produced by Phil Ramone Just the Way You Are !"Just the Way You Are" *
"Baker Street" by Gerry Rafferty
"Feels So Good" by Chuck Mangione
"Stayin' Alive" by Bee Gees
"You Needed Me" by Anne Murray
[28]
1980 Doobie Brothers, The !The Doobie Brothers
· produced by Ted Templeman What a Fool Believes !"What a Fool Believes" *
"After the Love Has Gone" by Earth, Wind & Fire
"I Will Survive" by Gloria Gaynor
"The Gambler" by Kenny Rogers
"You Don't Bring Me Flowers" by Barbra Streisand & Neil Diamond
[29]
1981 Cross, ChristopherChristopher Cross
· produced by Michael Omartian Sailing !"Sailing" *
"The Rose" by Bette Midler
"Lady" by Kenny Rogers
"Theme from New York, New York" by Frank Sinatra
"Woman in Love" by Barbra Streisand
[29]
1982 Carnes, KimKim Carnes
· produced by Val Garay Bette Davis Eyes !"Bette Davis Eyes" *
"Arthur's Theme (Best That You Can Do)" by Christopher Cross
"(Just Like) Starting Over" by John Lennon
"Endless Love" by Diana Ross and Lionel Richie
"Just the Two of Us" by Grover Washington, Jr. with Bill Withers
[29]
1983 Toto
· produced by Toto Rosanna !"Rosanna"
"Steppin' Out" by Joe Jackson
"Ebony and Ivory" by Paul McCartney and Stevie Wonder
"Always on My Mind" by Willie Nelson
"Chariots of Fire" by Vangelis
[29]
1984 Jackson, MichaelMichael Jackson
· produced by Michael Jackson & Quincy Jones Beat It !"Beat It"
"Flashdance... What a Feeling" by Irene Cara
"Every Breath You Take" by The Police
"All Night Long (All Night)" by Lionel Richie
"Maniac" by Michael Sembello
[29]
1985 Turner, TinaTina Turner
· produced by Terry Britten What's Love Got to Do with It !"What's Love Got to Do with It" *
"Hard Habit to Break" by Chicago
"Girls Just Want to Have Fun" by Cyndi Lauper
"The Heart of Rock & Roll" by Huey Lewis and the News
"Dancing in the Dark" by Bruce Springsteen
[29]
1986 USA for Africa
· produced by Quincy Jones We Are the World !"We Are the World" *
"Money for Nothing" by Dire Straits
"The Boys of Summer" by Don Henley
"The Power of Love" by Huey Lewis and the News
"Born in the U.S.A." by Bruce Springsteen
[29]
1987 Winwood, SteveSteve Winwood
· produced by Russ Titelman & Steve Winwood Higher Love !"Higher Love"
"Sledgehammer" by Peter Gabriel
"Greatest Love of All" by Whitney Houston
"Addicted to Love" by Robert Palmer
"That's What Friends Are For" by Dionne Warwick & Friends
[29]
1988 Simon, PaulPaul Simon
· produced by Paul Simon Graceland !"Graceland"
"La Bamba" by Los Lobos
"I Still Haven't Found What I'm Looking For" by U2
"Luka" by Suzanne Vega
"Back in the High Life Again" by Steve Winwood
[29]
1989 McFerrin, BobbyBobby McFerrin
· produced by Linda Goldstein Don't Worry, Be Happy !"Don't Worry, Be Happy" *
"Giving You the Best That I Got" by Anita Baker
"Fast Car" by Tracy Chapman
"Man in the Mirror" by Michael Jackson
"Roll With It" by Steve Winwood
[29]
1990 Midler, BetteBette Midler
· produced by Arif Mardin Wind Beneath My Wings !"Wind Beneath My Wings" *
"The End of the Innocence" by Don Henley
"She Drives Me Crazy" by Fine Young Cannibals
"We Didn't Start the Fire" by Billy Joel
"The Living Years" by Mike + The Mechanics
[29]
1991 Collins, PhilPhil Collins
· produced by Hugh Padgham & Phil Collins Another Day in Paradise !"Another Day in Paradise"
"Vision of Love" by Mariah Carey
"U Can't Touch This" by MC Hammer
"From a Distance" by Bette Midler
"Nothing Compares 2 U" by Sinéad O'Connor
[29]
1992 Cole, NatalieNatalie Cole with Nat King Cole
· produced by David Foster Unforgettable !"Unforgettable" *
"(Everything I Do) I Do It for You" by Bryan Adams
"Baby Baby" by Amy Grant
"Something to Talk About" by Bonnie Raitt
"Losing My Religion" by R.E.M.
[29]
1993 Clapton, EricEric Clapton
· produced by Russ Titelman Tears in Heaven !"Tears in Heaven" *
"Achy Breaky Heart" by Billy Ray Cyrus
"Beauty and the Beast" by Celine Dion & Peabo Bryson
"Constant Craving" by k.d. lang
"Save the Best for Last" by Vanessa Williams
[29]
1994 Houston, WhitneyWhitney Houston
· produced by David Foster I Will Always Love You !"I Will Always Love You"
"A Whole New World" by Peabo Bryson & Regina Belle
"The River of Dreams" by Billy Joel
"If I Ever Lose My Faith in You" by Sting
"Harvest Moon" by Neil Young
[29]
1995 Crow, SherylSheryl Crow
· produced by Bill Bottrell All I Wanna Do !"All I Wanna Do"
"I'll Make Love to You" by Boyz II Men
"He Thinks He'll Keep Her" by Mary Chapin Carpenter
"Love Sneakin' Up On You" by Bonnie Raitt
"Streets of Philadelphia" by Bruce Springsteen
[30]
1996 Seal
· produced by Trevor Horn Kiss from a Rose !"Kiss from a Rose" *
"One Sweet Day" by Mariah Carey & Boyz II Men
"Gangsta's Paradise" by Coolio
"One of Us" by Joan Osborne
"Waterfalls" by TLC
[31]
1997 Clapton, EricEric Clapton
· produced by Babyface Change the World !"Change the World" *
"Give Me One Reason" by Tracy Chapman
"Because You Loved Me" by Celine Dion
"Ironic" by Alanis Morissette
"1979" by The Smashing Pumpkins
[32]
1998 Colvin, ShawnShawn Colvin
· produced by John Leventhal Sunny Came Home !"Sunny Came Home" *
"Where Have All the Cowboys Gone?" by Paula Cole
"Everyday Is a Winding Road" by Sheryl Crow
"MMMBop" by Hanson
"I Believe I Can Fly" by R. Kelly
[33]
1999 Dion, CelineCeline Dion
· engineered/mixed by David Gleeson, Humberto Gatica & Simon Franglen;
· produced by James Horner, Simon Franglen & Walter Afanasieff My Heart Will Go On !"My Heart Will Go On" *
"The Boy Is Mine" by Brandy" & Monica
"Iris" by Goo Goo Dolls
"Ray of Light" by Madonna
"You're Still the One" by Shania Twain
[34]
2000 Santana featuring Rob Thomas
· engineered/mixed by David Thoener,
· produced by Matt Serletic Smooth !"Smooth" *
"I Want It That Way" by Backstreet Boys
"Believe" by Cher
"Livin' la Vida Loca" by Ricky Martin
"No Scrubs" by TLC
[35]
2001 U2
· engineered/mixed by Richard Rainey & Steve Lillywhite;
· produced by Brian Eno & Daniel Lanois Beautiful Day !"Beautiful Day" *
"Say My Name" by Destiny's Child
"I Try" by Macy Gray
"Music" by Madonna
"Bye Bye Bye" by *NSYNC
[36]
2002 Walk On !"Walk On"
"Video" by India.Arie
"Fallin'" by Alicia Keys
"Ms. Jackson" by OutKast
"Drops of Jupiter (Tell Me)" by Train
[37]
2003 Jones, NorahNorah Jones
· engineered/mixed by Jay Newland;
· produced by Arif Mardin, Jay Newland & Norah Jones Don't Know Why !"Don't Know Why" *
"A Thousand Miles" by Vanessa Carlton
"Without Me" by Eminem
"Dilemma" by Nelly & Kelly Rowland
"How You Remind Me" by Nickelback
[38]
2004 Coldplay
· engineered/mixed by Coldplay, Ken Nelson & Mark Phythian;
· produced by Coldplay & Ken Nelson Clocks !"Clocks"
"Crazy in Love" by Beyoncé & Jay Z
"Where Is the Love?" by The Black Eyed Peas & Justin Timberlake
"Lose Yourself" by Eminem
"Hey Ya!" by OutKast
[39]
2005 Charles, RayRay Charles featuring Norah Jones
· engineered/mixed by Al Schmitt, Mark Fleming, & Terry Howard;
· produced by John R. Burk "Here We Go Again"
"Let's Get It Started" by The Black Eyed Peas
"American Idiot" by Green Day
"Heaven" by Los Lonely Boys
"Yeah" by Usher featuring Lil Jon & Ludacris
[40]
2006 Green Day
· engineered/mixed by Chris Lord-Alge & Doug McKean,
· produced by Green Day & Rob Cavallo Boulevard of Broken Dreams !"Boulevard of Broken Dreams"
"We Belong Together" by Mariah Carey
"Feel Good Inc." by Gorillaz
"Hollaback Girl" by Gwen Stefani
"Gold Digger" by Kanye West
[41]
2007 Dixie Chicks
· engineered/mixed by Chris Testa, Jim Scott & Richard Dodd;
· produced by Rick Rubin Not Ready to Make Nice !"Not Ready to Make Nice" *
"Be Without You" by Mary J. Blige
"You're Beautiful" by James Blunt
"Crazy" by Gnarls Barkley
"Put Your Records On" by Corinne Bailey Rae
[42]
2008 Winehouse, AmyAmy Winehouse
· engineered/mixed by Tom Elmhirst, Vaughan Merrick, Dom Morley, Mark Ronson & Gabriel Roth;
· produced by Mark Ronson Rehab !"Rehab" *
"Irreplaceable" by Beyoncé
"The Pretender" by the Foo Fighters
"Umbrella" by Rihanna featuring Jay Z
"What Goes Around.../...Comes Around" by Justin Timberlake
[43]
2009 Krauss, AlisonAlison Krauss and Robert Plant
· engineered/mixed by Mike Piersante;
· produced by T-Bone Burnett Please Read the Letter !"Please Read the Letter"
"Chasing Pavements" by Adele
"Viva la Vida" by Coldplay
"Bleeding Love" by Leona Lewis
"Paper Planes" by M.I.A.
[44]
2010 Kings of Leon
· engineered/mixed by Jacquire King;
· produced by Jacquire King & Angelo Petraglia Use Somebody !"Use Somebody"
"Halo" by Beyoncé
"I Gotta Feeling" by The Black Eyed Peas
"Poker Face" by Lady Gaga
"You Belong with Me" by Taylor Swift
[45]
2011 Lady Antebellum
· engineered/mixed by Clarke Schleicher;
· produced by Lady Antebellum & Paul Worley Need You Now !"Need You Now" *
"Nothin' on You" by B.o.B featuring Bruno Mars
"Love the Way You Lie" by Eminem featuring Rihanna
"Fuck You" by CeeLo Green
"Empire State of Mind" by Jay Z featuring Alicia Keys
[46]
2012 Adele
· engineered/mixed by Tom Elmhirst & Mark Rankin
· produced by Paul Epworth Rolling in the Deep !"Rolling in the Deep" *
"Holocene" by Bon Iver
"Grenade" by Bruno Mars
"The Cave" by Mumford & Sons
"Firework" by Katy Perry
[47]
2013 Gotye featuring Kimbra
· engineered/mixed by Wally De Backer, François Tétaz & William Bowden
· master engineered by William Bowden
· produced by Wally De Backer Somebody That I Used to Know !"Somebody That I Used to Know"
"Lonely Boy" by The Black Keys
"Stronger (What Doesn't Kill You)" by Kelly Clarkson
"We Are Young" by fun. featuring Janelle Monáe
"Thinkin Bout You" by Frank Ocean
"We Are Never Ever Getting Back Together" by Taylor Swift
[48]
2014 Daft Punk featuring Pharrell Williams & Nile Rodgers
· engineered/mixed by Peter Franco, Mick Guzauski, Florian Lagatta & Daniel Lerner
· master engineered by Antoine "Chab" Chabert & Bob Ludwig
· produced by Thomas Bangalter & Guy-Manuel de Homem-Christo "Get Lucky"
"Radioactive" by Imagine Dragons
"Royals" by Lorde
"Locked Out of Heaven" by Bruno Mars
"Blurred Lines" by Robin Thicke featuring T.I. & Pharrell Williams
[49]
2015 Sam Smith
· engineered/mixed by Steve Fitzmaurice, Jimmy Napes & Steve Price
· master engineered by Tom Coyne
· produced by Steve Fitzmaurice, Rodney Jerkins & Jimmy Napes "Stay with Me" (Darkchild version) *
"Fancy" by Iggy Azalea featuring Charli XCX
"Chandelier" by Sia
"Shake It Off" by Taylor Swift
"All About That Bass" by Meghan Trainor
[50]
2016 Mark Ronson featuring Bruno Mars
· engineered/mixed by Josh Blair, Riccardo Damian, Serban Ghenea, Wayne Gordon, John Hanes, Inaam Haq, Boo Mitchell, Charles Moniz & Mark Ronson
· master engineered by Tom Coyne
· produced by Jeff Bhasker, Bruno Mars & Mark Ronson "Uptown Funk"
"Really Love" by D'Angelo & The Vanguard
"Thinking Out Loud" by Ed Sheeran
"Blank Space" by Taylor Swift
"Can't Feel My Face" by The Weeknd
[51]
2017 Adele
· engineered/mixed by Julian Burg, Tom Elmhirst, Emile Haynie, Greg Kurstin, Liam Nolan, Alex Pasco & Joe Visciano
· master engineered by Tom Coyne & Randy Merrill
· produced by Greg Kurstin "Hello" *
"Formation" by Beyoncé
"7 Years" by Lukas Graham
"Work" by Rihanna featuring Drake
"Stressed Out" by Twenty One Pilots
[52]
2018 Bruno Mars
· engineered/mixed by Serban Ghenea, John Hanes & Charles Moniz
· master engineered by Tom Coyne
· produced by Shampoo Press & Curl “24K Magic”
"Redbone" by Childish Gambino
"Despacito (Remix)" by Luis Fonsi and Daddy Yankee featuring Justin Bieber
"The Story of O.J." by Jay-Z
"Humble" by Kendrick Lamar
[53]
^[I] Each year is linked to the article about the Grammy Awards held that year.
References[edit]
General
"Past Winners Search". National Academy of Recording Arts and Sciences. Retrieved March 4, 2011. Note: User must select the "General" category as the genre under the search feature.
Specific
Jump up ^ "Overview". National Academy of Recording Arts and Sciences. Retrieved March 16, 2011.
Jump up ^ "Grammy Awards at a Glance". Los Angeles Times. Tribune Company. Retrieved February 4, 2011.
Jump up ^ "Category Mapper". National Academy of Recording Arts and Sciences. Archived from the original on June 4, 2012. Retrieved November 25, 2011.
Jump up ^ "The Recording Academy Announces Board Of Trustees Meeting Results". National Academy of Recording Arts and Sciences. June 8, 2012. Retrieved June 11, 2012.
Jump up ^ Grein, Paul (2009-12-07). "Boost Your Grammy IQ | Stop The Presses! - Yahoo Music". New.music.yahoo.com. Retrieved 2014-05-23.
Jump up ^ Behind Grammy's Closed Door
Jump up ^ Wyman, Bill (2011-02-11). "The Grammys: the secret committee that alters the membership's nominations". Slate.com. Retrieved 2014-05-23.
Jump up ^ "Grammy Awards 1959 (May)". IndiaServer. Retrieved July 20, 2011.
Jump up ^ "Grammy Awards 1959". IndiaServer.
Jump up ^ "Grammy Awards 1961". IndiaServer.
Jump up ^ "Grammy Awards 1962". Awards & Shows.
Jump up ^ "Grammy Awards 1963". IndiaServer.
Jump up ^ "Grammy Awards 1964". IndiaServer.
Jump up ^ "Grammy Awards 1965". IndiaServer.
Jump up ^ "Grammy Awards Nominees 1966 - Grammy Award Winners 1966". www.awardsandshows.com. Retrieved April 26, 2017.
Jump up ^ "Grammy Awards 1967". Awards & Shows.
Jump up ^ "Grammy Awards 1968". IndiaServer.
Jump up ^ "Grammy Awards Nominees 1969 - Grammy Award Winners 1969". www.awardsandshows.com. Retrieved April 26, 2017.
Jump up ^ "Grammy Awards 1970". IndiaServer.
Jump up ^ "Grammy Awards 1971". IndiaServer.
Jump up ^ "Grammy Awards 1972". IndiaServer.
Jump up ^ "Grammy Awards 1973". IndiaServer.
Jump up ^ "Grammy Awards 1974". IndiaServer.
Jump up ^ "Grammy Awards Nominee 1975 - Grammy Award Winners 1975". www.awardsandshows.com. Retrieved April 26, 2017.
Jump up ^ "Grammy Awards 1976". IndiaServer.
Jump up ^ "Grammy Awards 1977". IndiaServer.
Jump up ^ "Grammy Awards 1978". IndiaServer.
Jump up ^ "Grammy Award Nominees 1979 - Grammy Award Winners 1979". www.awardsandshows.com. Retrieved April 26, 2017.
^ Jump up to: a b c d e f g h i j k l m n o "Grammy Awards: Record of the Year". Rock on the Net. Retrieved December 10, 2011.
Jump up ^ "The 37th Grammy Nominations". Los Angeles Times. Tribune Company. January 6, 1995. p. 1. Retrieved July 12, 2010.
Jump up ^ "List of Grammy nominees". CNN. January 4, 1996. Retrieved July 12, 2010.
Jump up ^ Campbell, Mary (January 8, 1997). "Babyface is up for 12 Grammy awards". Milwaukee Journal Sentinel. Retrieved July 12, 2010.
Jump up ^ "40th Annual Grammy Award Nominations". Digital Hit. Retrieved December 9, 2011.
Jump up ^ "1999 Grammy Nominees". NME. IPC Media. November 27, 1998. Archived from the original on October 11, 2012. Retrieved July 12, 2010.
Jump up ^ "42nd Annual Grammy Awards nominations". CNN. January 4, 2000. Archived from the original on July 22, 2012. Retrieved July 12, 2010.
Jump up ^ "43rd Grammy Awards". CNN. February 21, 2001. Archived from the original on November 6, 2008. Retrieved July 12, 2010.
Jump up ^ "Complete List Of Grammy Nominees". CBS News. January 4, 2002. Retrieved July 12, 2010.
Jump up ^ "45 Grammy Nom List" (PDF). Archived from the original (PDF) on April 26, 2012.
Jump up ^ "They're All Contenders". The New York Times. December 5, 2003. Retrieved July 12, 2010.
Jump up ^ "Grammy Award nominees in top categories". USA Today. Gannett Company. February 7, 2005. Retrieved July 12, 2010.
Jump up ^ "The Complete List of Grammy Nominations". The New York Times. The New York Times Company. December 8, 2005. p. 1. Retrieved July 12, 2010.
Jump up ^ "49th Annual Grammy Awards Winners List". National Academy of Recording Arts and Sciences. Archived from the original on July 10, 2009. Retrieved July 12, 2010.
Jump up ^ "Grammy 2008 Winners List". MTV. February 10, 2008. Retrieved July 12, 2010.
Jump up ^ "Grammy 2009 Winners List". MTV. February 8, 2009. Retrieved July 12, 2010.
Jump up ^ "The 52nd Annual Grammy Awards Nominees List". National Academy of Recording Arts and Sciences. Archived from the original on June 18, 2010. Retrieved July 12, 2010.
Jump up ^ "53rd Annual Grammy Awards nominees list". Los Angeles Times. Tribune Company. Retrieved December 2, 2010.
Jump up ^ "2011 – 54th Annual GRAMMY Awards Nominees And Winners: General Field". The Recording Academy. November 30, 2011.
Jump up ^ "Dan Auerbach, Fun., Jay Z, Mumford & Sons, Frank Ocean, Kanye West Lead 55th GRAMMY Nominations". Retrieved 30 December 2012.
Jump up ^ "Jay Z Tops 56th GRAMMY Nominations With Nine". Retrieved 6 December 2013.
Jump up ^ "Grammy Awards 2015: The Complete Winners List". Rolling Stone. Retrieved April 26, 2017.
Jump up ^ Billboard.com, 7 December 2015
Jump up ^ Billboard.com, 6 December 2016
Jump up ^ Field Grammy.com, 28 November 2017
External links[edit]
Official website
[show]
v
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e
Grammy Award for Record of the Year
1959−1980
"Nel Blu Dipinto Di Blu (Volare)" by Domenico Modugno (1959)
"Mack the Knife" by Bobby Darin (1960)
"Theme from A Summer Place" by Percy Faith (1961)
"Moon River" by Henry Mancini (1962)
"I Left My Heart in San Francisco" by Tony Bennett (1963)
"Days of Wine and Roses" by Henry Mancini (1964)
"The Girl from Ipanema" by Astrud Gilberto & Stan Getz (1965)
"A Taste of Honey" by Herb Alpert and the Tijuana Brass (1966)
"Strangers in the Night" by Frank Sinatra (1967)
"Up, Up and Away" by The 5th Dimension (Billy Davis Jr., Florence LaRue, Marilyn McCoo, Lamont McLemore, Ron Townson) (1968)
"Mrs. Robinson" by Simon & Garfunkel (Art Garfunkel, Paul Simon) (1969)
"Aquarius/Let the Sunshine In" by The 5th Dimension (Billy Davis Jr., Florence LaRue, Marilyn McCoo, Lamont McLemore, Ron Townson) (1970)
"Bridge over Troubled Water" by Simon & Garfunkel (Art Garfunkel, Paul Simon) (1971)
"It's Too Late" by Carole King (1972)
"The First Time Ever I Saw Your Face" by Roberta Flack (1973)
"Killing Me Softly with His Song" by Roberta Flack (1974)
"I Honestly Love You" by Olivia Newton-John (1975)
"Love Will Keep Us Together" by Captain & Tennille (Daryl Dragon, Toni Tennille) (1976)
"This Masquerade" by George Benson (1977)
"Hotel California" by Eagles (Don Felder, Glenn Frey, Don Henley, Randy Meisner, Joe Walsh) (1978)
"Just the Way You Are" by Billy Joel (1979)
"What a Fool Believes" by The Doobie Brothers (Jeffrey Baxter, John Hartman, Keith Knudsen, Michael McDonald, Tiran Porter, Patrick Simmons) (1980)
1981−2000
"Sailing" by Christopher Cross (1981)
"Bette Davis Eyes" by Kim Carnes (1982)
"Rosanna" by Toto (Bobby Kimball, Steve Lukather, David Paich, Jeff Porcaro, David Hungate, Steve Porcaro) (1983)
"Beat It" by Michael Jackson (1984)
"What's Love Got to Do with It" by Tina Turner (1985)
"We Are the World" by USA for Africa (1986)
"Higher Love" by Steve Winwood (1987)
"Graceland" by Paul Simon (1988)
"Don't Worry, Be Happy" by Bobby McFerrin (1989)
"Wind Beneath My Wings" by Bette Midler (1990)
"Another Day in Paradise" by Phil Collins (1991)
"Unforgettable" by Natalie Cole with Nat King Cole (1992)
"Tears in Heaven" by Eric Clapton (1993)
"I Will Always Love You" by Whitney Houston (1994)
"All I Wanna Do" by Sheryl Crow (1995)
"Kiss from a Rose" by Seal (1996)
"Change the World" by Eric Clapton (1997)
"Sunny Came Home" by Shawn Colvin (1998)
"My Heart Will Go On" by Celine Dion (1999)
"Smooth" by Santana (Rodney Holmes, Tony Lindsay, Karl Perazzo, Raul Rekow, Benny Rietveld, Carlos Santana, Chester Thompson) featuring Rob Thomas (2000)
2001−present
"Beautiful Day" by U2 (Bono, Adam Clayton, The Edge, Larry Mullen Jr.) (2001)
"Walk On" by U2 (Bono, Adam Clayton, The Edge, Larry Mullen Jr.) (2002)
"Don't Know Why" by Norah Jones (2003)
"Clocks" by Coldplay (Guy Berryman, Jon Buckland, Will Champion, Phil Harvey, Chris Martin) (2004)
"Here We Go Again" by Ray Charles and Norah Jones (2005)
"Boulevard of Broken Dreams" by Green Day (Billie Joe Armstrong, Mike Dirnt, Frank Edwin Wright III) (2006)
"Not Ready to Make Nice" by Dixie Chicks (Martie Maguire, Natalie Maines, Emily Robison) (2007)
"Rehab" by Amy Winehouse (2008)
"Please Read the Letter" by Alison Krauss and Robert Plant (2009)
"Use Somebody" by Kings of Leon (Caleb Followill, Jared Followill, Matthew Followill, Nathan Followill) (2010)
"Need You Now" by Lady Antebellum (Hillary Scott, Charles Kelley, Dave Haywood) (2011)
"Rolling in the Deep" by Adele (2012)
"Somebody That I Used to Know" by Gotye featuring Kimbra (2013)
"Get Lucky" by Daft Punk featuring Pharrell Williams & Nile Rodgers (2014)
"Stay with Me" (Darkchild version) by Sam Smith (2015)
"Uptown Funk" by Mark Ronson featuring Bruno Mars (2016)
"Hello" by Adele (2017)
"24K Magic" by Bruno Mars (2018)
[show]
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Ceremony year
1959
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1965
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[show]
v
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Grammy Award categories
General
Record of the Year
Album of the Year
Song of the Year
Best New Artist
Pop
Best Pop Solo Performance
Best Pop Duo/Group Performance
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Best Traditional Pop Vocal Album
Dance/Electronic
Best Dance Recording
Best Dance/Electronic Album
Rock
Best Rock Performance
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Best Rock Song
Best Rock Album
Alternative
Best Alternative Music Album
R&B
Best R&B Performance
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Best R&B Song
Best R&B Album
Best Urban Contemporary Album
Rap/Hip-Hop
Best Rap Performance
Best Rap/Sung Performance
Best Rap Song
Best Rap Album
Country
Best Country Solo Performance
Best Country Duo/Group Performance
Best Country Song
Best Country Album
Jazz
Best Improvised Jazz Solo
Best Jazz Vocal Album
Best Jazz Instrumental Album
Best Large Jazz Ensemble Album
Best Latin Jazz Album
Gospel/
Contemporary Christian Music
Best Gospel Performance/Song
Best Gospel Album
Best Contemporary Christian Music Performance/Song
Best Contemporary Christian Music Album
Latin
Best Latin Pop Album
Best Latin Rock, Urban or Alternative Album
Best Regional Mexican Music Album (including Tejano)
Best Tropical Latin Album
American Roots
Best Americana Album
Best Bluegrass Album
Best Traditional Blues Album
Best Contemporary Blues Album
Best Folk Album
Best Regional Roots Music Album
Best American Roots Performance
Best American Roots Song
Music for Visual Media
Best Compilation Soundtrack for Visual Media
Best Score Soundtrack for Visual Media
Best Song Written for Visual Media
Composing/Arranging
Best Instrumental Composition
Best Arrangement, Instrumental or A Cappella
Best Arrangement, Instrumental and Vocals
Crafts
Best Historical Album
Best Recording Package
Best Boxed or Special Limited Edition Package
Best Album Notes
Production/Engineering
Best Engineered Album, Non-Classical
Best Engineered Album, Classical
Best Remixed Recording
Best Surround Sound Album
Producer of the Year, Non-Classical
Producer of the Year, Classical
Classical
Best Orchestral Performance
Best Opera Recording
Best Choral Performance
Best Chamber Music/Small Ensemble Performance
Best Classical Instrumental Solo
Best Classical Vocal Solo
Best Classical Compendium
Best Classical Contemporary Composition
Music Video
Best Music Video
Best Music Film
Others
Contemporary Instrumental
Best Contemporary Instrumental Album
New Age
Best New Age Album
Reggae
Best Reggae Album
World Music
Best World Music Album
Spoken Word
Best Spoken Word Album
Children's
Best Children's Album
Comedy
Best Comedy Album
Musical Theatre
Best Musical Theater Album
Special Awards
Lifetime Achievement Award
Trustees Award
Legend Award
Grammy Hall of Fame
MusiCares Person of the Year
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