Last time we built a neural network in python that made really bad predictions of your score on a test Based on how many hours you slept and how many hours you studied the night before? This time we’ll focus on the theory of making those predictions better we can initialize the network we built last time and pass in our normalized Data x Using our forward method and have a look at our estimate of y y hat Right now our predictions are pretty inaccurate to improve our model we first need to quantify exactly how wrong our predictions are well do this with a cost function a Cost function allows us to Express exactly how wrong or costly our model is given our examples One way to compute an overall cost is to take each error value square it and add these values together Multiplying by one half will make things simpler down the road now that we have a cost our job is to minimize it and someone says they’re training a network what they really mean is that they’re minimizing a cost function Our cost is a function of two things Our examples, and the weights on our synapses We don’t have much control over our data. So will minimize our cost by changing the weights Conceptually this is a pretty simple concept, we have a collection of nine individual weights. And we’re saying that there is some combination of w’s that will make our cost, j, as small as possible. When I first saw this problem in Machine learning I thought I’ll just try all the weights until I find the best one after all I have a computer enter the curse of dimensionality Here’s the problem let’s pretend for a second that we only have one weight instead of nine To find the ideal value for our weight that will minimize our cost We need to try a bunch of values for w let’s say we test a thousand values That doesn’t seem so bad; After all, my computer is pretty fast It takes about point zero four seconds to check a thousand different weight values for our neural Network Since we’ve competed the cost for a wide range of values of w we can just pick the one with the smallest cost Let that be our weight, and now we’ve trained our network So you may be thinking that point zero four seconds the trainer network is not so bad And we haven’t even optimized anything yet. Plus, there are other way faster languages than python out there Before we optimize though, let’s consider the full complexity of the problem Remember the point zero four seconds required is only for one weight, and we have nine total Let’s next consider two weights for a moment, to maintain the same precision We now need to check 1,000 times a thousand or a million values. This is a lot of work even for a fast computer. After our million evaluations we found our solution But it took an agonizing 40 seconds the real curse of dimensionality kicks in as we continue to add dimensions Searching through three weights would take a billion evaluations or 11 hours searching through all nine weights We need for our simple Neural Network would take one quadrillion 268 Trillion 391 billion 679 million three hundred and fifty thousand five hundred and eighty three and a half years For that reason that just try everything or brute Force optimization method is clearly not going to work Let’s return to the one-dimensional case and see if we can be more clever Let’s evaluate our cost function for a specific value of w if w is 1.1 for example We can run our cost function and see that J is 2.8 We haven’t learned much yet, but let’s try to add a little more information to what we [already] know What if we could figure out which way was downhill if we could we would know whether to make w smaller or larger to? Decrease the cost We could test the cost function immediately to the left and to the right of our test point and see which is smaller This is called numerical estimation and is sometimes a good approach but for us there is a better way Let’s look at our equation so far we have five equations, but we could really think of them as one big approach and since we have one big equation that uniquely Determines our cost J from x y w1 and W2 We can [use] our good friend calculus to find exactly what looking for We want to know which way is downhill that is what is the rate of change of J with respect to w? Also known as the derivative and in this case since we’re just considering one weight at a time. This is a partial derivative We can derive an expression for DJ 2w That will give us the rate of change of J with respect to w for any value of w if DJ Dw is positive then the cost function is going uphill if Dj. Dw is negative and the cost function is going Downhill now We can really speed things up since we [know] in which direction the cost decreases We can save all the time that we would have spent searching in the wrong direction We can save you even more computational time by iteratively taking steps downhill and stopping when the cost stopped getting smaller This method is known as gradient descent and although it may not seem so impressive in one dimension it is capable of incredible speed ups in higher dimensions in Fact in our final video will show that what would have taken 10 to the 27th function evaluations with our brute Force method Will take less than a hundred evaluations with gradient descent? Gradient descent allows us to find needles in very very very large haystacks Now before we celebrate too much here. There is a restriction What if our cost function doesn’t always go in the same direction? What if it goes up and then back down? The mathematical name for this is non convex And it could really throw off our gradient descent algorithm by getting it stuck in a local minimum instead of our ideal Global Minima One of the reasons we chose our cost function to be the sum of squared errors was to exploit [the] convex Nature of quadratic equations We know that the graph of y equals x squared is a nice convex parabola and it turns out that higher dimensional versions are to another piece of the puzzle here is that depending on how we use our data it might not matter if our function is convex or not if We use our examples one at a time instead of all at once Sometimes it won’t matter our function is convex. We will still find a good solution this is called stochastic gradient descent So maybe we shouldn’t be afraid of non convex loss functions as neural Network Wizard Iyanla Kuhn says in his excellent Talk who is afraid of non convex Loss functions? The details of gradient descent are a deep topic for another day for now We’re going to do our gradient descent batch style Where we use all our examples at once and the way we’ve set up our cost function will keep things nice and convex Next time we’ll compute and code up our gradients