Gradient descent

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/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 and artificial intelligence 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 Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades.

A simple extension of gradient descent, stochastic gradient descent, serves as the most basic algorithm used for training most deep networks today.

Description

Illustration of gradient descent on a series of level sets

Gradient descent is based on the observation that if the multi-variable function ${\displaystyle f(\mathbf{x})}$ is defined and differentiable in a neighborhood of a point ${\displaystyle \mathbf{a}}$, then ${\displaystyle f(\mathbf{x})}$ decreases fastest if one goes from ${\displaystyle \mathbf{a}}$ in the direction of the negative gradient of ${\displaystyle f}$ at ${\displaystyle \mathbf{a},-\mathrm{\nabla }f(\mathbf{a})}$. It follows that, if

${\displaystyle {\mathbf{a}}_{n+1}={\mathbf{a}}_n-\eta \mathrm{\nabla }f({\mathbf{a}}_n)}$

for a small enough step size or learning rate ${\displaystyle \eta \in {\mathbb{R}}_{+}}$, then ${\displaystyle f({\mathbf{a}}_{\mathbf{n}})\ge f({\mathbf{a}}_{\mathbf{n}\mathbf{+}\mathbf{1}})}$. In other words, the term ${\displaystyle \eta \mathrm{\nabla }f(\mathbf{a})}$ is subtracted from ${\displaystyle \mathbf{a}}$ because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess ${\displaystyle {\mathbf{x}}_0}$ for a local minimum of ${\displaystyle f}$, and considers the sequence ${\displaystyle {\mathbf{x}}_0,{\mathbf{x}}_1,{\mathbf{x}}_2,\dots }$ such that

${\displaystyle {\mathbf{x}}_{n+1}={\mathbf{x}}_n-{\eta }_n\mathrm{\nabla }f({\mathbf{x}}_n),n\ge 0.}$

We have a monotonic sequence

${\displaystyle f({\mathbf{x}}_0)\ge f({\mathbf{x}}_1)\ge f({\mathbf{x}}_2)\ge \cdots ,}$

so the sequence ${\displaystyle ({\mathbf{x}}_n)}$ converges to the desired local minimum. Note that the value of the step size ${\displaystyle \eta }$ is allowed to change at every iteration.

It is possible to guarantee the convergence to a local minimum under certain assumptions on the function ${\displaystyle f}$ (for example, ${\displaystyle f}$ convex and ${\displaystyle \mathrm{\nabla }f}$ Lipschitz) and particular choices of ${\displaystyle \eta }$. Those include the sequence

${\displaystyle {\eta }_n=\frac{\left|{\left({\mathbf{x}}_n-{\mathbf{x}}_{n-1}\right)}^{\mathrm{\top }}\left[\mathrm{\nabla }f({\mathbf{x}}_n)-\mathrm{\nabla }f({\mathbf{x}}_{n-1})\right]\right|}{{\Vert \mathrm{\nabla }f({\mathbf{x}}_n)-\mathrm{\nabla }f({\mathbf{x}}_{n-1})\Vert }^2}}$

as in the Barzilai-Borwein method, or a sequence ${\displaystyle {\eta }_n}$ satisfying the Wolfe conditions (which can be found by using line search). When the function ${\displaystyle f}$ 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, ${\displaystyle f}$ 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 ${\displaystyle f}$ 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 ${\displaystyle f}$ is minimal.

An analogy for understanding gradient descent

Fog in the mountains

The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. People are stuck in the mountains and are trying to get down (i.e., trying to find the global minimum). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible, so they must use local information to find the minimum. They can use the method of gradient descent, which involves looking at the steepness of the hill at their current position, then proceeding in the direction with the steepest descent (i.e., downhill). If they were trying to find the top of the mountain (i.e., the maximum), then they would proceed in the direction of steepest ascent (i.e., uphill). Using this method, they would eventually find their way down the mountain or possibly get stuck in some hole (i.e., local minimum or saddle point), like a mountain lake. 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 people happen to have at that moment. It takes quite some time to measure the steepness of the hill with the instrument. Thus, they should minimize their use of the instrument if they want to get down the mountain before sunset. The difficulty then is choosing the frequency at which they should measure the steepness of the hill so as not to go off track.

In this analogy, the people represent 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 function at that point. The instrument used to measure steepness is differentiation. The direction they choose to travel in aligns with the gradient of the function at that point. The amount of time they travel before taking another measurement is the step size.

Choosing the step size and descent direction

Since using a step size ${\displaystyle \eta }$ that is too small would slow convergence, and a ${\displaystyle \eta }$ too large would lead to overshoot and divergence, finding a good setting of ${\displaystyle \eta }$ is an important practical problem. Philip Wolfe also advocated using "clever choices of the [descent] direction" in practice. While using a direction that deviates from the steepest descent direction may seem counter-intuitive, the idea is that the smaller slope may be compensated for by being sustained over a much longer distance.

To reason about this mathematically, consider a direction ${\displaystyle {\mathbf{p}}_n}$ and step size ${\displaystyle {\eta }_n}$ and consider the more general update:

${\displaystyle {\mathbf{a}}_{n+1}={\mathbf{a}}_n-{\eta }_n{\mathbf{p}}_n}$.

Finding good settings of ${\displaystyle {\mathbf{p}}_n}$ and ${\displaystyle {\eta }_n}$ requires some thought. First of all, we would like the update direction to point downhill. Mathematically, letting ${\displaystyle {\theta }_n}$ denote the angle between ${\displaystyle -\mathrm{\nabla }f({\mathbf{a}}_{\mathbf{n}})}$ and ${\displaystyle {\mathbf{p}}_n}$, this requires that ${\displaystyle \cos {\theta }_n>0.}$ To say more, we need more information about the objective function that we are optimising. Under the fairly weak assumption that ${\displaystyle f}$ is continuously differentiable, we may prove that:

${\displaystyle f({\mathbf{a}}_{n+1})\le f({\mathbf{a}}_n)-{\eta }_n\Vert \mathrm{\nabla }f({\mathbf{a}}_n){\Vert }_2\Vert {\mathbf{p}}_n{\Vert }_2\left(\cos {\theta }_n-\underset{t\in [0,1]}{max}\frac{\Vert \mathrm{\nabla }f({\mathbf{a}}_n-t{\eta }_n{\mathbf{p}}_n)-\mathrm{\nabla }f({\mathbf{a}}_n){\Vert }_2}{\Vert \mathrm{\nabla }f({\mathbf{a}}_n){\Vert }_2}\right)}$

1

This inequality implies that the amount by which we can be sure the function ${\displaystyle f}$ is decreased depends on a trade off between the two terms in square brackets. The first term in square brackets measures the angle between the descent direction and the negative gradient. The second term measures how quickly the gradient changes along the descent direction.

In principle inequality (1) could be optimized over ${\displaystyle {\mathbf{p}}_n}$ and ${\displaystyle {\eta }_n}$ to choose an optimal step size and direction. The problem is that evaluating the second term in square brackets requires evaluating ${\displaystyle \mathrm{\nabla }f({\mathbf{a}}_n-t{\eta }_n{\mathbf{p}}_n)}$, and extra gradient evaluations are generally expensive and undesirable. Some ways around this problem are:

• Forgo the benefits of a clever descent direction by setting ${\displaystyle {\mathbf{p}}_n=\mathrm{\nabla }f({\mathbf{a}}_{\mathbf{n}})}$, and use line search to find a suitable step-size ${\displaystyle {\gamma }_n}$, such as one that satisfies the Wolfe conditions. A more economic way of choosing learning rates is backtracking line search, a method that has both good theoretical guarantees and experimental results. Note that one does not need to choose ${\displaystyle {\mathbf{p}}_n}$ to be the gradient; any direction that has positive inner product with the gradient will result in a reduction of the function value (for a sufficiently small value of ${\displaystyle {\eta }_n}$).
• Assuming that ${\displaystyle f}$ is twice-differentiable, use its Hessian ${\displaystyle {\mathrm{\nabla }}^{2}f}$ to estimate ${\displaystyle \Vert \mathrm{\nabla }f({\mathbf{a}}_n-t{\eta }_n{\mathbf{p}}_n)-\mathrm{\nabla }f({\mathbf{a}}_n){\Vert }_2\approx \Vert t{\eta }_n{\mathrm{\nabla }}^{2}f({\mathbf{a}}_n){\mathbf{p}}_n\Vert .}$Then choose ${\displaystyle {\mathbf{p}}_n}$ and ${\displaystyle {\eta }_n}$ by optimising inequality (1).
• Assuming that ${\displaystyle \mathrm{\nabla }f}$ is Lipschitz, use its Lipschitz constant ${\displaystyle L}$ to bound ${\displaystyle \Vert \mathrm{\nabla }f({\mathbf{a}}_n-t{\eta }_n{\mathbf{p}}_n)-\mathrm{\nabla }f({\mathbf{a}}_n){\Vert }_2\le Lt{\eta }_n\Vert {\mathbf{p}}_n\Vert .}$ Then choose ${\displaystyle {\mathbf{p}}_n}$ and ${\displaystyle {\eta }_n}$ by optimising inequality (1).
• Build a custom model of ${\displaystyle \underset{t\in [0,1]}{max}\frac{\Vert \mathrm{\nabla }f({\mathbf{a}}_n-t{\eta }_n{\mathbf{p}}_n)-\mathrm{\nabla }f({\mathbf{a}}_n){\Vert }_2}{\Vert \mathrm{\nabla }f({\mathbf{a}}_n){\Vert }_2}}$ for ${\displaystyle f}$. Then choose ${\displaystyle {\mathbf{p}}_n}$ and ${\displaystyle {\eta }_n}$ by optimising inequality (1).
• Under stronger assumptions on the function ${\displaystyle f}$ such as convexity, more advanced techniques may be possible.

Usually by following one of the recipes above, convergence to a local minimum can be guaranteed. When the function ${\displaystyle f}$ is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution.

Solution of a linear system

The steepest descent algorithm applied to the Wiener filter

Gradient descent can be used to solve a system of linear equations

${\displaystyle \mathbf{A}\mathbf{x}-\mathbf{b}=0}$

reformulated as a quadratic minimization problem. If the system matrix ${\displaystyle \mathbf{A}}$ is real symmetric and positive-definite, an objective function is defined as the quadratic function, with minimization of

${\displaystyle f(\mathbf{x})={\mathbf{x}}^{\mathrm{\top }}\mathbf{A}\mathbf{x}-2{\mathbf{x}}^{\mathrm{\top }}\mathbf{b},}$

so that

${\displaystyle \mathrm{\nabla }f(\mathbf{x})=2(\mathbf{A}\mathbf{x}-\mathbf{b}).}$

For a general real matrix ${\displaystyle \mathbf{A}}$, linear least squares define

${\displaystyle f(\mathbf{x})={\Vert \mathbf{A}\mathbf{x}-\mathbf{b}\Vert }^2.}$

In traditional linear least squares for real ${\displaystyle \mathbf{A}}$ and ${\displaystyle \mathbf{b}}$ the Euclidean norm is used, in which case

${\displaystyle \mathrm{\nabla }f(\mathbf{x})=2{\mathbf{A}}^{\mathrm{\top }}(\mathbf{A}\mathbf{x}-\mathbf{b}).}$

The line search minimization, finding the locally optimal step size ${\displaystyle \eta }$ on every iteration, can be performed analytically for quadratic functions, and explicit formulas for the locally optimal ${\displaystyle \eta }$ are known.

For example, for real symmetric and positive-definite matrix ${\displaystyle \mathbf{A}}$, a simple algorithm can be as follows,

${\displaystyle \begin{array}{rl}& \text{repeat in the loop:}\\ & \mathbf{r}:=\mathbf{b}-\mathbf{A}\mathbf{x}\\ & \eta :={\mathbf{r}}^{\mathrm{\top }}\mathbf{r}/{\mathbf{r}}^{\mathrm{\top }}\mathbf{A}\mathbf{r}\\ & \mathbf{x}:=\mathbf{x}+\eta \mathbf{r}\\ & \text{if }{\mathbf{r}}^{\mathrm{\top }}\mathbf{r}\text{ is sufficiently small, then exit loop}\\ & \text{end repeat loop}\\ & \text{return }\mathbf{x}\text{ as the result}\end{array}}$

To avoid multiplying by ${\displaystyle \mathbf{A}}$ twice per iteration, we note that ${\displaystyle \mathbf{x}:=\mathbf{x}+\eta \mathbf{r}}$ implies ${\displaystyle \mathbf{r}:=\mathbf{r}-\eta \mathbf{A}\mathbf{r}}$, which gives the traditional algorithm,

${\displaystyle \begin{array}{rl}& \mathbf{r}:=\mathbf{b}-\mathbf{A}\mathbf{x}\\ & \text{repeat in the loop:}\\ & \eta :={\mathbf{r}}^{\mathrm{\top }}\mathbf{r}/{\mathbf{r}}^{\mathrm{\top }}\mathbf{A}\mathbf{r}\\ & \mathbf{x}:=\mathbf{x}+\eta \mathbf{r}\\ & \text{if }{\mathbf{r}}^{\mathrm{\top }}\mathbf{r}\text{ is sufficiently small, then exit loop}\\ & \mathbf{r}:=\mathbf{r}-\eta \mathbf{A}\mathbf{r}\\ & \text{end repeat loop}\\ & \text{return }\mathbf{x}\text{ as the result}\end{array}}$

Convergence path of steepest descent method for A = [[2, 2], [2, 3]]

The method is rarely used for solving linear equations, with the conjugate gradient method being one of the most popular alternatives. The number of gradient descent iterations is commonly proportional to the spectral condition number ${\displaystyle \kappa (\mathbf{A})}$ of the system matrix ${\displaystyle \mathbf{A}}$ (the ratio of the maximum to minimum eigenvalues of ${\displaystyle {\mathbf{A}}^{\mathrm{\top }}\mathbf{A}}$), while the convergence of conjugate gradient method is typically determined by a square root of the condition number, i.e., is much faster. Both methods can benefit from preconditioning, where gradient descent may require less assumptions on the preconditioner.

Geometric behavior and residual orthogonality

In steepest descent applied to solving ${\displaystyle \mathbf{A}\mathbf{x}=\mathbf{b}}$, where ${\displaystyle \mathbf{A}}$ is symmetric positive-definite, the residual vectors ${\displaystyle {\mathbf{r}}_k=\mathbf{b}-\mathbf{A}{\mathbf{x}}_k}$ are orthogonal across iterations:

${\displaystyle ⟨{\mathbf{r}}_{k+1},{\mathbf{r}}_k⟩=0.}$

Because each step is taken in the steepest direction, steepest-descent steps alternate between directions aligned with the extreme axes of the elongated level sets. When ${\displaystyle \kappa (\mathbf{A})}$ is large, this produces a characteristic zig–zag path. The poor conditioning of ${\displaystyle \mathbf{A}}$ is the primary cause of the slow convergence, and orthogonality of successive residuals reinforces this alternation.

As shown in the image on the right, steepest descent converges slowly due to the high condition number of ${\displaystyle \mathbf{A}}$, and the orthogonality of residuals forces each new direction to undo the overshoot from the previous step. The result is a path that zigzags toward the solution. This inefficiency is one reason conjugate gradient or preconditioning methods are preferred.

Solution of a non-linear system

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

An animation showing the first 83 iterations of gradient descent applied to this example. Surfaces are isosurfaces of ${\displaystyle f({\mathbf{x}}^{(n)})}$ at current guess ${\displaystyle {\mathbf{x}}^{(n)}}$, and arrows show the direction of descent. Due to a small and constant step size, the convergence is slow.

${\displaystyle \{\begin{array}{l}3x_1-\cos (x_2{x}_3)-\frac{3}{2}=0\\ 4x_1^2-625x_2^2+2x_2-1=0\\ \exp (-x_1{x}_2)+20x_3+\frac{10\pi -3}{3}=0\end{array}}$

Let us introduce the associated function

${\displaystyle G(\mathbf{x})=\left[\begin{array}{c}3x_1-\cos (x_2{x}_3)-\frac{3}{2}\\ 4x_1^2-625x_2^2+2x_2-1\\ \exp (-x_1{x}_2)+20x_3+\frac{10\pi -3}{3}\end{array}\right],}$

where

${\displaystyle \mathbf{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right].}$

One might now define the objective function

${\displaystyle \begin{array}{rl}f(\mathbf{x})& =\frac{1}{2}{G}^{\mathrm{\top }}(\mathbf{x})G(\mathbf{x})\\ & =\frac{1}{2}[{\left(3x_1-\cos (x_2{x}_3)-\frac{3}{2}\right)}^2+{\left(4x_1^2-625x_2^2+2x_2-1\right)}^2+\\ & {\left(\exp (-x_1{x}_2)+20x_3+\frac{10\pi -3}{3}\right)}^2],\end{array}}$

which we will attempt to minimize. As an initial guess, let us use

${\displaystyle {\mathbf{x}}^{(0)}=\mathbf{0}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right].}$

We know that

${\displaystyle {\mathbf{x}}^{(1)}=\mathbf{0}-{\eta }_0\mathrm{\nabla }f(\mathbf{0})=\mathbf{0}-{\eta }_0{J}_G(\mathbf{0}{)}^{\mathrm{\top }}G(\mathbf{0}),}$

where the Jacobian matrix ${\displaystyle J_G}$ is given by

${\displaystyle J_G(\mathbf{x})=\left[\begin{array}{ccc}3& \sin (x_2{x}_3)x_3& \sin (x_2{x}_3)x_2\\ 8x_1& -1250x_2+2& 0\\ -x_2\exp (-x_1{x}_2)& -x_1\exp (-x_1{x}_2)& 20\end{array}\right].}$

We calculate:

${\displaystyle J_G(\mathbf{0})=\left[\begin{array}{ccc}3& 0& 0\\ 0& 2& 0\\ 0& 0& 20\end{array}\right],G(\mathbf{0})=\left[\begin{array}{c}-2.5\\ -1\\ 10.472\end{array}\right].}$

Thus

${\displaystyle {\mathbf{x}}^{(1)}=\mathbf{0}-{\eta }_0\left[\begin{array}{c}-7.5\\ -2\\ 209.44\end{array}\right],}$

and

${\displaystyle f(\mathbf{0})=0.5\left((-2.5{)}^2+(-1{)}^2+(10.472{)}^2\right)=\mathrm{58.456.}}$

Now, a suitable ${\displaystyle {\eta }_0}$ must be found such that

${\displaystyle f\left({\mathbf{x}}^{(1)}\right)\le f\left({\mathbf{x}}^{(0)}\right)=f(\mathbf{0}).}$

This can be done with any of a variety of line search algorithms. One might also simply guess ${\displaystyle {\eta }_0=0.001,}$ which gives

${\displaystyle {\mathbf{x}}^{(1)}=\left[\begin{array}{c}0.0075\\ 0.002\\ -0.20944\end{array}\right].}$

Evaluating the objective function at this value, yields

${\displaystyle f\left({\mathbf{x}}^{(1)}\right)=0.5\left((-2.48{)}^2+(-1.00{)}^2+(6.28{)}^2\right)=\mathrm{23.306.}}$

The decrease from ${\displaystyle f(\mathbf{0})=58.456}$ to the next step's value of

${\displaystyle f\left({\mathbf{x}}^{(1)}\right)=23.306}$

is a sizable decrease in the objective function. Further steps would reduce its value further until an approximate solution to the system was found.

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.

That gradient descent works in any number of dimensions (finite number at least) can be seen as a consequence of the Cauchy–Schwarz inequality, i.e. the magnitude of the inner (dot) product of two vectors of any dimension is maximized when they are colinear. In the case of gradient descent, that would be when the vector of independent variable adjustments is proportional to the gradient vector of partial derivatives.

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 modified via momentums (Nesterov, Polyak, and Frank–Wolfe) and heavy-ball parameters (exponential moving averages and positive-negative momentum). The main examples of such optimizers are Adam, DiffGrad, Yogi, AdaBelief, etc.

Methods based on Newton's method and inversion of the Hessian using conjugate gradient techniques can be better alternatives. 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 ${\displaystyle \eta .}$ 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.

While it is sometimes possible to substitute gradient descent for a local search algorithm, gradient descent is not in the same family: although it is an iterative method for local optimization, it relies on an objective function's gradient rather than an explicit exploration of a solution space.

Gradient descent can be viewed as applying Euler's method for solving ordinary differential equations ${\displaystyle x^{\prime }(t)=-\mathrm{\nabla }f(x(t))}$ to a gradient flow. In turn, this equation may be derived as an optimal controller for the control system ${\displaystyle x^{\prime }(t)=u(t)}$ with ${\displaystyle u(t)}$ given in feedback form ${\displaystyle u(t)=-\mathrm{\nabla }f(x(t))}$.

Modifications

Gradient descent can converge to a local minimum and slow down in a neighborhood of a saddle point. Even for unconstrained quadratic minimization, gradient descent develops a zig–zag pattern of subsequent iterates as iterations progress, resulting in slow convergence. Multiple modifications of gradient descent have been proposed to address these deficiencies.

Fast gradient methods

Yurii Nesterov has proposed a simple modification that enables faster convergence for convex problems and has been since further generalized. For unconstrained smooth problems, the method is called the fast gradient method (FGM) or the accelerated gradient method (AGM). Specifically, if the differentiable function ${\displaystyle f}$ is convex and ${\displaystyle \mathrm{\nabla }f}$ is Lipschitz, and it is not assumed that ${\displaystyle f}$ is strongly convex, then the error in the objective value generated at each step ${\displaystyle k}$ by the gradient descent method will be bounded by $\mathcal{O}\left(k^{-1}\right)$. Using the Nesterov acceleration technique, the error decreases at $\mathcal{O}\left(k^{-2}\right)$. It is known that the rate ${\displaystyle \mathcal{O}\left(k^{-2}\right)}$ 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) reduces that constant by a factor of two and is an optimal first-order method for large-scale problems.

For constrained or non-smooth problems, Nesterov's FGM is called the fast proximal gradient method (FPGM), an acceleration of the proximal gradient method.

Momentum or heavy ball method

Trying to break the zig-zag pattern of gradient descent, the momentum or heavy ball method uses a momentum term in analogy to a heavy ball sliding on the surface of values of the function being minimized, or to mass movement in Newtonian dynamics through a viscous medium in a conservative force field. Gradient descent with momentum remembers the solution update at each iteration, and determines the next update as a linear combination of the gradient and the previous update. For unconstrained quadratic minimization, a theoretical convergence rate bound of the heavy ball method is asymptotically the same as that for the optimal conjugate gradient method.

This technique is used in stochastic gradient descent and as an extension to the backpropagation algorithms used to train artificial neural networks. In the direction of updating, stochastic gradient descent adds a stochastic property. The weights can be used to calculate the derivatives.

Extensions

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).

Gradient descent is a special case of mirror descent using the squared Euclidean distance as the given Bregman divergence.

Theoretical properties

The properties of gradient descent depend on the properties of the objective function and the variant of gradient descent used (for example, if a line search step is used). The assumptions made affect the convergence rate, and other properties, that can be proven for gradient descent. For example, if the objective is assumed to be strongly convex and lipschitz smooth, then gradient descent converges linearly with a fixed step size. Looser assumptions lead to either weaker convergence guarantees or require a more sophisticated step size selection.

Neuroevolution

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Neuroevolution

Neuroevolution, or neuro-evolution, is a form of artificial intelligence that uses evolutionary algorithms to generate artificial neural networks (ANN), parameters, and rules. It is most commonly applied in artificial life, general game playing and evolutionary robotics. The main benefit is that neuroevolution can be applied more widely than supervised learning algorithms, which require a syllabus of correct input-output pairs. In contrast, neuroevolution requires only a measure of a network's performance at a task. For example, the outcome of a game (i.e., whether one player won or lost) can be easily measured without providing labeled examples of desired strategies. Neuroevolution is commonly used as part of the reinforcement learning paradigm, and it can be contrasted with conventional deep learning techniques that use backpropagation (gradient descent on a neural network) with a fixed topology.

Features

Many neuroevolution algorithms have been defined. One common distinction is between algorithms that evolve only the strength of the connection weights for a fixed network topology (sometimes called conventional neuroevolution), and algorithms that evolve both the topology of the network and its weights (called TWEANNs, for Topology and Weight Evolving Artificial Neural Network algorithms).

A separate distinction can be made between methods that evolve the structure of ANNs in parallel to its parameters (those applying standard evolutionary algorithms) and those that develop them separately (through memetic algorithms).

Comparison with gradient descent

Further information: Gradient descent

Most neural networks use gradient descent rather than neuroevolution. However, around 2017 researchers at Uber stated they had found that simple structural neuroevolution algorithms were competitive with sophisticated modern industry-standard gradient-descent deep learning algorithms, in part because neuroevolution was found to be less likely to get stuck in local minima. In Science, journalist Matthew Hutson speculated that part of the reason neuroevolution is succeeding where it had failed before is due to the increased computational power available in the 2010s.

It can be shown that there is a correspondence between neuroevolution and gradient descent.

Direct and indirect encoding

Evolutionary algorithms operate on a population of genotypes (also referred to as genomes). In neuroevolution, a genotype is mapped to a neural network phenotype that is evaluated on some task to derive its fitness.

In direct encoding schemes the genotype directly maps to the phenotype. That is, every neuron and connection in the neural network is specified directly and explicitly in the genotype. In contrast, in indirect encoding schemes the genotype specifies indirectly how that network should be generated.

Indirect encodings are often used to achieve several aims:

• modularity and other regularities;
• compression of phenotype to a smaller genotype, providing a smaller search space;
• mapping the search space (genome) to the problem domain.

Taxonomy of embryogenic systems for indirect encoding

Traditionally indirect encodings that employ artificial embryogeny (also known as artificial development) have been categorised along the lines of a grammatical approach versus a cell chemistry approach. The former evolves sets of rules in the form of grammatical rewrite systems. The latter attempts to mimic how physical structures emerge in biology through gene expression. Indirect encoding systems often use aspects of both approaches.

Stanley and Miikkulainen propose a taxonomy for embryogenic systems that is intended to reflect their underlying properties. The taxonomy identifies five continuous dimensions, along which any embryogenic system can be placed:

• Cell (neuron) fate: the final characteristics and role of the cell in the mature phenotype. This dimension counts the number of methods used for determining the fate of a cell.
• Targeting: the method by which connections are directed from source cells to target cells. This ranges from specific targeting (source and target are explicitly identified) to relative targeting (e.g., based on locations of cells relative to each other).
• Heterochrony: the timing and ordering of events during embryogeny. Counts the number of mechanisms for changing the timing of events.
• Canalization: how tolerant the genome is to mutations (brittleness). Ranges from requiring precise genotypic instructions to a high tolerance of imprecise mutation.
• Complexification: the ability of the system (including evolutionary algorithm and genotype to phenotype mapping) to allow complexification of the genome (and hence phenotype) over time. Ranges from allowing only fixed-size genomes to allowing highly variable length genomes.

Molecular geometry

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Molecular_geometry

Geometry of the water molecule with values for O-H bond length and for H-O-H bond angle between two bonds

Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that determine the position of each atom.

Molecular geometry influences several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism and biological activity. The angles between bonds that an atom forms depend only weakly on the rest of a molecule, i.e. they can be understood as approximately local and hence transferable properties.

Determination

The molecular geometry can be determined by various spectroscopic methods and diffraction methods. IR, microwave and Raman spectroscopy can give information about the molecule geometry from the details of the vibrational and rotational absorbance detected by these techniques. X-ray crystallography, neutron diffraction and electron diffraction can give molecular structure for crystalline solids based on the distance between nuclei and concentration of electron density. Gas electron diffraction can be used for small molecules in the gas phase. NMR and FRET methods can be used to determine complementary information including relative distances, dihedral angles, angles, and connectivity. Molecular geometries are best determined at low temperature because at higher temperatures the molecular structure is averaged over more accessible geometries (see next section). Larger molecules often exist in multiple stable geometries (conformational isomerism) that are close in energy on the potential energy surface. Geometries can also be computed by ab initio quantum chemistry methods to high accuracy. The molecular geometry can be different as a solid, in solution, and as a gas.

The position of each atom is determined by the nature of the chemical bonds by which it is connected to its neighboring atoms. The molecular geometry can be described by the positions of these atoms in space, evoking bond lengths of two joined atoms, bond angles of three connected atoms, and torsion angles (dihedral angles) of three consecutive bonds.

Influence of thermal excitation

Since the motions of the atoms in a molecule are determined by quantum mechanics, "motion" must be defined in a quantum mechanical way. The overall (external) quantum mechanical motions translation and rotation hardly change the geometry of the molecule. (To some extent rotation influences the geometry via Coriolis forces and centrifugal distortion, but this is negligible for the present discussion.) In addition to translation and rotation, a third type of motion is molecular vibration, which corresponds to internal motions of the atoms such as bond stretching and bond angle variation. The molecular vibrations are harmonic (at least to good approximation), and the atoms oscillate about their equilibrium positions, even at the absolute zero of temperature. At absolute zero all atoms are in their vibrational ground state and show zero point quantum mechanical motion, so that the wavefunction of a single vibrational mode is not a sharp peak, but approximately a Gaussian function (the wavefunction for n = 0 depicted in the article on the quantum harmonic oscillator). At higher temperatures the vibrational modes may be thermally excited (in a classical interpretation one expresses this by stating that "the molecules will vibrate faster"), but they oscillate still around the recognizable geometry of the molecule.

To get a feeling for the probability that the vibration of molecule may be thermally excited, we inspect the Boltzmann factor β ≡ exp(−⁠ΔE/kT⁠), where ΔE is the excitation energy of the vibrational mode, k the Boltzmann constant and T the absolute temperature. At 298 K (25 °C), typical values for the Boltzmann factor β are:

• β = 0.089 for ΔE = 500 cm⁻¹
• β = 0.008 for ΔE = 1000 cm⁻¹
• β = 0.0007 for ΔE = 1500 cm⁻¹.

(The reciprocal centimeter is an energy unit that is commonly used in infrared spectroscopy; 1 cm⁻¹ corresponds to 1.23984×10⁻⁴ eV). When an excitation energy is 500 cm⁻¹, then about 8.9 percent of the molecules are thermally excited at room temperature. To put this in perspective: the lowest excitation vibrational energy in water is the bending mode (about 1600 cm⁻¹). Thus, at room temperature less than 0.07 percent of all the molecules of a given amount of water will vibrate faster than at absolute zero.

As stated above, rotation hardly influences the molecular geometry. But, as a quantum mechanical motion, it is thermally excited at relatively (as compared to vibration) low temperatures. From a classical point of view it can be stated that at higher temperatures more molecules will rotate faster, which implies that they have higher angular velocity and angular momentum. In quantum mechanical language: more eigenstates of higher angular momentum become thermally populated with rising temperatures. Typical rotational excitation energies are on the order of a few cm⁻¹. The results of many spectroscopic experiments are broadened because they involve an averaging over rotational states. It is often difficult to extract geometries from spectra at high temperatures, because the number of rotational states probed in the experimental averaging increases with increasing temperature. Thus, many spectroscopic observations can only be expected to yield reliable molecular geometries at temperatures close to absolute zero, because at higher temperatures too many higher rotational states are thermally populated.

Bonding

Molecules, by definition, are most often held together with covalent bonds involving single, double, and/or triple bonds, where a "bond" is a shared pair of electrons (the other method of bonding between atoms is called ionic bonding and involves a positive cation and a negative anion).

Molecular geometries can be specified in terms of 'bond lengths', 'bond angles' and 'torsional angles'. The bond length is defined to be the average distance between the nuclei of two atoms bonded together in any given molecule. A bond angle is the angle formed between three atoms across at least two bonds. For four atoms bonded together in a chain, the torsional angle is the angle between the plane formed by the first three atoms and the plane formed by the last three atoms.

There exists a mathematical relationship among the bond angles for one central atom and four peripheral atoms (labeled 1 through 4) expressed by the following determinant. This constraint removes one degree of freedom from the choices of (originally) six free bond angles to leave only five choices of bond angles. (The angles θ₁₁, θ₂₂, θ₃₃, and θ₄₄ are always zero and that this relationship can be modified for a different number of peripheral atoms by expanding/contracting the square matrix.)

$${\displaystyle 0=\left|\begin{array}{cccc}\cos {\theta }_{11}& \cos {\theta }_{12}& \cos {\theta }_{13}& \cos {\theta }_{14}\\ \cos {\theta }_{21}& \cos {\theta }_{22}& \cos {\theta }_{23}& \cos {\theta }_{24}\\ \cos {\theta }_{31}& \cos {\theta }_{32}& \cos {\theta }_{33}& \cos {\theta }_{34}\\ \cos {\theta }_{41}& \cos {\theta }_{42}& \cos {\theta }_{43}& \cos {\theta }_{44}\end{array}\right|}$$

Molecular geometry is determined by the quantum mechanical behavior of the electrons. Using the valence bond approximation this can be understood by the type of bonds between the atoms that make up the molecule. When atoms interact to form a chemical bond, the atomic orbitals of each atom are said to combine in a process called orbital hybridisation. The two most common types of bonds are sigma bonds (usually formed by hybrid orbitals) and pi bonds (formed by unhybridized p orbitals for atoms of main group elements). The geometry can also be understood by molecular orbital theory where the electrons are delocalised.

An understanding of the wavelike behavior of electrons in atoms and molecules is the subject of quantum chemistry.

Isomers

Isomers are types of molecules that share a chemical formula but have difference geometries, resulting in different properties:

• A pure substance is composed of only one type of isomer of a molecule (all have the same geometrical structure).
• Structural isomers have the same chemical formula but different physical arrangements, often forming alternate molecular geometries with very different properties. The atoms are not bonded (connected) together in the same orders.
  • Functional isomers are special kinds of structural isomers, where certain groups of atoms exhibit a special kind of behavior, such as an ether or an alcohol.
• Stereoisomers may have many similar physicochemical properties (melting point, boiling point) and at the same time very different biochemical activities. This is because they exhibit a handedness that is commonly found in living systems. One manifestation of this chirality or handedness is that they have the ability to rotate polarized light in different directions.
• Protein folding concerns the complex geometries and different isomers that proteins can take.

Types of molecular structure

A bond angle is the geometric angle between two adjacent bonds. Some common shapes of simple molecules include:

• Linear: In a linear model, atoms are connected in a straight line. The bond angles are set at 180°. For example, carbon dioxide and nitric oxide have a linear molecular shape.
• Trigonal planar: Molecules with the trigonal planar shape are somewhat triangular and in one plane (flat). Consequently, the bond angles are set at 120°. For example, boron trifluoride.
• Angular: Angular molecules (also called bent or V-shaped) have a non-linear shape. For example, water (H₂O), which has an angle of about 105°. A water molecule has two pairs of bonded electrons and two unshared lone pairs.
• Tetrahedral: Tetra- signifies four, and -hedral relates to a face of a solid, so "tetrahedral" literally means "having four faces". This shape is found when there are four bonds all on one central atom, with no extra unshared electron pairs. In accordance with the VSEPR (valence-shell electron pair repulsion theory), the bond angles between the electron bonds are arccos(−⁠1/3⁠) = 109.47°. For example, methane (CH₄) is a tetrahedral molecule.
• Octahedral: Octa- signifies eight, and -hedral relates to a face of a solid, so "octahedral" means "having eight faces". The bond angle is 90 degrees. For example, sulfur hexafluoride (SF₆) is an octahedral molecule.
• Trigonal pyramidal: A trigonal pyramidal molecule has a pyramid-like shape with a triangular base. Unlike the linear and trigonal planar shapes but similar to the tetrahedral orientation, pyramidal shapes require three dimensions in order to fully separate the electrons. Here, there are only three pairs of bonded electrons, leaving one unshared lone pair. Lone pair – bond pair repulsions change the bond angle from the tetrahedral angle to a slightly lower value. For example, ammonia (NH₃).