Top 24 Local Minima Global Minima The 145 New Answer

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Convex: Local vs Global Minimum

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Local & Global Minima Explained with Examples – Data Analytics

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Local Minima and Global Minima

How to find minima or maxima of a function

Conclusions

Ajitesh Kumar

Local & Global Minima Explained with Examples - Data Analytics — Local & Global Minima Explained with Examples – Data Analytics

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– MATLAB & Simulink A local minimum of a function is a point where the function value is smaller than at nearby points, but … Explains why solvers might not find the smallest minimum.
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Local vs Global Optima

Local & Global Minima Explained with Examples

Optimization problems containing many local minima remains a critical problem in a variety of domains, including operations research, informatics, and material design. Efficient global optimization remains a problem of general research interest, with applications to a range of fields including operations design, network analysis, and bioinformatics. Within the fields of chemical physics and material design, efficient global optimization is particularly important for finding low potential energy configurations of isolated groups of atoms (clusters) and periodic systems (crystals). In case of Machine learning (ML) algorithms, theer is a need for optimising (minimising) the cost or loss function. In order to become very good at finding solutions to optimisation problems (relating to minimising functions) including machine learning based problems, one must get a good understanding of the concepts of Local minima / global minima and local maxima / global maxima. In this post, you will learning about local minima and global minima.

Local Minima and Global Minima

The point at which a function takes the minimum value is called global minima. However, when the goal is to minimize the function and solved using optimization algorithms such as gradient descent, it may so happen that function may appear to have a minimum value at different points. Those several points which appear to be minima but are not the point where the function actually takes the minimum value are called local minima. Machine learning algorithms such as gradient descent algorithms may get stuck in local minima during the training of the models. Gradient descent is able to find local minima most of the time and not global minima because the gradient does not point in the direction of the steepest descent. Current techniques to find global minima either require extremely high iteration counts or a large number of random restarts for good performance. Global optimization problems can also be quite difficult when high loss barriers exist between local minima.

Take a look at the following picture to understand the concept of local minima and global minima.

Fig 1. Local Minima vs Global Minima

When you draw the function in 3-dimensional space, this is how the local minima and global minima will look like:

Fig 2. Local Minima and Global Minima in 3-dimensional space

In order to find whether a point is local minima or global minima, one would need to find all possible minima of the function.

Here is an animation that can help you understand the difference between local and global minima in a better manner.

Fig 3. Animation representing local minima and global minima

Pay attention to some of the following in the above animation:

The gradient at different points is found out.

If the gradient value is positive at a point, it will be required to move to left or reduce the weight.

If the gradient value is negative at a point, it will be required to increment the value of weight.

The above two steps are done until the minima is reached.

The minima could either be local minima or the global minima. There are different techniques which can be used to find local vs global minima. These techniques will be discussed in the future posts.

What are some of the techniques for dealing with local minima problems?

The following are some of the techniques which has been traditionally used to deal with local minima problem:

Careful selection of hand-crafted features

Dependence on learning rate schedules

Using different number of steps

How to find minima or maxima of a function?

In order to find the minima or maxima of the function, one needs to determine the first-order derivative of that function and equate the same to zero (0). This can give points where you may end up finding both, one or more minima and one or more maxima. Mathematically, this can be represented as the following. Solve the below function and you would get the value of x where the function is zero.

$\frac{df(x)}{dx} = 0$

Substitute the value of x in the function and find the value where the function has either minimum values or maximum values. In order to find whether the point is local/global minima or maxima, take the second-order derivative and determine whether the value is positive or negative. If the second-order derivative value is positive, the point is local or global minima and if the second-order derivative value is negative, the point is local or global maxima. The diagram below represents the same.

Fig 3. Local Minima / Global Minima and Local Maxima / Global Maxima

You may want to check out my post on gradient descent to understand the importance of finding the minima of the function in the context of machine learning.

Conclusions

Here is the summary of what you learned about local minima and global minima of the function:

Local vs. Global Optima

Local vs. Global Optima

Why the Solver Does Not Find the Smallest Minimum In general, solvers return a local minimum (or optimum). The result might be a global minimum (or optimum), but this result is not guaranteed. A local minimum of a function is a point where the function value is smaller than at nearby points, but possibly greater than at a distant point.

A global minimum is a point where the function value is smaller than at all other feasible points. Optimization Toolbox™ solvers typically find a local minimum. (This local minimum can be a global minimum.) They find the minimum in the basin of attraction of the starting point. For more information about basins of attraction, see Basins of Attraction. The following are exceptions to this general rule. Linear programming problems and positive definite quadratic programming problems are convex, with convex feasible regions, so they have only one basin of attraction. Depending on the specified options, linprog ignores any user-supplied starting point, and quadprog does not require one, even though you can sometimes speed a minimization by supplying a starting point.

Global Optimization Toolbox functions, such as simulannealbnd , attempt to search more than one basin of attraction.

Searching for a Smaller Minimum If you need a global minimum, you must find an initial value for your solver in the basin of attraction of a global minimum. You can set initial values to search for a global minimum in these ways: Use a regular grid of initial points.

Use random points drawn from a uniform distribution if all of the problem coordinates are bounded. Use points drawn from normal, exponential, or other random distributions if some components are unbounded. The less you know about the location of the global minimum, the more spread out your random distribution should be. For example, normal distributions rarely sample more than three standard deviations away from their means, but a Cauchy distribution (density 1/(π(1 + x 2 ))) makes greatly disparate samples.

Use identical initial points with added random perturbations on each coordinate—bounded, normal, exponential, or other.

If you have a Global Optimization Toolbox license, use the GlobalSearch (Global Optimization Toolbox) or MultiStart (Global Optimization Toolbox) solvers. These solvers automatically generate random start points within bounds. The more you know about possible initial points, the more focused and successful your search will be.

Basins of Attraction If an objective function f(x) is smooth, the vector –∇f(x) points in the direction where f(x) decreases most quickly. The equation of steepest descent, namely d d t x ( t ) = − ∇ f ( x ( t ) ) , yields a path x(t) that goes to a local minimum as t increases. Generally, initial values x(0) that are near each other give steepest descent paths that tend towards the same minimum point. The basin of attraction for steepest descent is the set of initial values that lead to the same local minimum. This figure shows two one-dimensional minima. The figure shows different basins of attraction with different line styles, and indicates the directions of steepest descent with arrows. For this and subsequent figures, black dots represent local minima. Every steepest descent path, starting at a point x(0), goes to the black dot in the basin containing x(0). One-dimensional basins This figure shows how steepest descent paths can be more complicated in more dimensions. One basin of attraction, showing steepest descent paths from various starting points This figure shows even more complicated paths and basins of attraction. Several basins of attraction Constraints can break up one basin of attraction into several pieces. For example, consider minimizing y subject to: y ≥ |x| y ≥ 5 – 4(x–2)2. This figure shows the two basins of attraction with the final points. Code for Generating the Figure myabs = @(x) abs(x); mycurve = @(x) 5 – 4.*(x-2).^2; myline = @(x) max(myabs(x),mycurve(x)); mybrdr = @(x) 1 + heaviside(x-2); myarea = @(x,y) heaviside(y – myline(x)).*mybrdr(x); % myarea = 0 where infeasible, 1 when x < 2, 2 when x > 2 marinv = @(x,y) 1./myarea(x,y); % removes the infeasible region plot(0,0, ‘k.’ , ‘MarkerSize’ ,25, ‘LineWidth’ ,2); hold on grid on plot(11/4,11/4, ‘k.’ , ‘MarkerSize’ ,25, ‘LineWidth’ ,2); fplot(mycurve, ‘k’ ,[.9 3], ‘LineWidth’ ,2) fplot(myline,[-2 5], ‘k’ , ‘LineWidth’ ,2) fsurf(marinv,[-2 5 -1,6], ‘EdgeColor’ , ‘none’ , ‘FaceAlpha’ ,0.5) colormap jet Enter plottools at the command line to add arrows using the arrow drawing tool. The steepest descent paths are straight lines down to the constraint boundaries. From the constraint boundaries, the steepest descent paths travel down along the boundaries. The final point is either (0,0) or (11/4,11/4), depending on whether the initial x-value is above or below 2.

Related Topics

What are Global minima and Local minima in Machine Learning?

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There are a few explanations already, but I figure I would throw some mathematical flavor in the mix.

Let $f : \mathbb{R}^n \to \mathbb{R}$ be a function. A Global Minimum (minima is the plural form) is a point in $\mathbf{x}_0 \in \mathbb{R}^n$ such that for all other $\mathbf{x} \in \mathbb{R}^n$, $f(\mathbf{x}_0) < f(\mathbf{x})$. In English, this means that $\mathbf{x}_0$ is the point where $f$ attains its smallest value. For example, the point $x=1$ is the global minima for the function $x^2 - 2x + 1$. The function $x^3$ has no global minima; I can always find a point which attains a smaller value (mostly by choosing large negative numbers). Let $f : \mathbb{R}^n \to \mathbb{R}$ be a function. A Local Minimum (minima is the plural form) is a point in $\mathbf{x}_0 \in \mathbb{R}^n$ such that $f(\mathbf{x}_0) < f(\mathbf{x}) \iff \lVert\mathbf{x} - \mathbf{x}_0 \rVert< \epsilon $. The "local" here refers to a neighourbood around $\mathbf{x}_0$ (hence the local). It might be the case that some other point $\mathbf{x}_1$ results in smaller output than $\mathbf{x}_0$, which is why local optima can mislead optimization models. So in short: Global minima are the places where the function attains its smallest value. Local minima are places where the function attains its smallest value in a neighbourhood of a point. Global minima are what we want when we optimize a loss function (smallest loss we could ever obtain). Local optima can be good, but by virtue of being local and not global we could do better.

So you have finished reading the local minima global minima topic article, if you find this article useful, please share it. Thank you very much. See more: Local minimum and global minimum, Global minima, how to find global minima, Local minimum neural network, local minima vs local minimum, Local optima, Local minima and maxima, how to avoid local minima in gradient descent

Top 24 Local Minima Global Minima The 145 New Answer

Local & Global Minima Explained with Examples – Data Analytics

Local vs. Global Optima
– MATLAB & Simulink

optimization – What are Global minima and Local minima in Machine Learning? – Cross Validated

What are Local Minima and Global Minima in Gradient Descent? | i2tutorials

Maxima and minima – Wikipedia

Local and Global Maxima and Minima

Local and Global Maxima and Minima

Maxima and Minima of Functions

Local & Global Minima Explained with Examples

Local vs. Global Optima

What are Global minima and Local minima in Machine Learning?

Leave a Comment Cancel reply

Local & Global Minima Explained with Examples – Data Analytics

Local vs. Global Optima – MATLAB & Simulink

optimization – What are Global minima and Local minima in Machine Learning? – Cross Validated

What are Local Minima and Global Minima in Gradient Descent? | i2tutorials

Maxima and minima – Wikipedia

Local and Global Maxima and Minima

Local and Global Maxima and Minima

Maxima and Minima of Functions

Local & Global Minima Explained with Examples

Local vs. Global Optima

What are Global minima and Local minima in Machine Learning?

Leave a Comment Cancel reply

Local vs. Global Optima
– MATLAB & Simulink