There was, however, a gap in our explanation: That's quite a gap! In this chapter I'll explain a fast algorithm for computing such gradients, an algorithm known as backpropagation.

Only gradually do they develop other shots, learning to chip, draw and fade the ball, building on and modifying their basic swing. In a similar way, up to now we've focused on understanding the backpropagation algorithm. It's our "basic swing", the foundation for learning in most work on neural networks.

In this chapter I explain a suite of techniques which can be used to improve on our vanilla implementation of backpropagation, and so improve the way our networks learn. The techniques we'll develop in this chapter include: I'll also overview several other techniques in less depth. The discussions are largely independent of one another, and so you may jump ahead if you wish.

We'll also implement many of the techniques in running code, and use them to improve the results obtained on the handwriting classification problem studied in Chapter 1. Of course, we're only covering a few of the many, many techniques which have been developed for use in neural nets.

The philosophy is that the best entree to the plethora of available techniques is in-depth study of a few of the most important. Mastering those important techniques is not just useful in its own right, but will also deepen your understanding of what problems can arise when you use neural networks.

That will leave you well prepared to quickly pick up other techniques, as you need them. The cross-entropy cost function Most of us find it unpleasant to be wrong. Soon after beginning to learn the piano I gave my first performance before an audience. I was nervous, and began playing the piece an octave too low.

I got confused, and couldn't continue until someone pointed out my error. I was very embarrassed. Yet while unpleasant, we also learn quickly when we're decisively wrong.

You can bet that the next time I played before an audience I played in the correct octave! By contrast, we learn more slowly when our errors are less well-defined. Ideally, we hope and expect that our neural networks will learn fast from their errors.

Is this what happens in practice? To answer this question, let's look at a toy example.

The example involves a neuron with just one input: We'll train this neuron to do something ridiculously easy: Of course, this is such a trivial task that we could easily figure out an appropriate weight and bias by hand, without using a learning algorithm.A Time-line for the History of Mathematics (Many of the early dates are approximates) This work is under constant revision, so come back later.

Please report any errors to me at [email protected] Let us assume that the axis of symmetry for the parabola is vertical.

The standard equation of a parabola with a vertical axis is #y = ax^2 + bx + c#, where a, b, and c are real numbers -- with #a ne 0#.. Using the coordinates of the above points, we know that. You can put this solution on YOUR website! If you can't don't want to use the matrix method, you also use the elimination method: write a quadratic function in standard form that goes through the given points (1,3) (2,1) (-2,).

(We will discuss projectile motion using parametric equations here in the Parametric Equations section.). Note that the independent variable represents time, not distance; sometimes parabolas represent the distance on the \(x\)-axis and the height on the \(y\)-axis, and the shapes are pfmlures.com versus distance would be the path or trajectory .

Worksheet A, Quadratic functions MATH , (SOLUTIONS) pfmlures.com the quadratic function with the given vertex and point. Put your answer in standard form. In the last chapter we saw how neural networks can learn their weights and biases using the gradient descent algorithm.

There was, however, a gap in our explanation: we didn't discuss how to compute the gradient of the cost function. That's quite a gap!

In this chapter I'll explain a fast algorithm for computing such gradients, an algorithm known as backpropagation.

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Standards for Mathematical Practice | Common Core State Standards Initiative