In what sense is backpropagation a fast algorithm? What is Proof-of-work and How Does it Relate to Bitcoin? to choose a neural network’s hyper-parameters?
Why are deep neural networks hard to train? What’s causing the vanishing gradient problem? Appendix: Is there a simple algorithm for intelligence? If you benefit from the book, please make a small donation. 5, but you can choose the amount. Thanks to all the supporters who made the book possible, with especial thanks to Pavel Dudrenov. Thanks also to all the contributors to the Bugfinder Hall of Fame.
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In the last chapter we saw how neural networks canlearn their weights and biases using the gradient descent algorithm. There was, however, a gap in our explanation: we didn’t discuss how tocompute the gradient of the cost function. Inthis chapter I’ll explain a fast algorithm for computing suchgradients, an algorithm known as backpropagation. This chapter is more mathematically involved than the rest of thebook. If you’re not crazy about mathematics you may be tempted toskip the chapter, and to treat backpropagation as a black box whosedetails you’re willing to ignore. Why take the time to study thosedetails? The reason, of course, is understanding.
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The expression tells us how quicklythe cost changes when we change the weights and biases. And while theexpression is somewhat complex, it also has a beauty to it, with eachelement having a natural, intuitive interpretation. With that said, if you want to skim the chapter, or jump straight tothe next chapter, that’s fine. I’ve written the rest of the book tobe accessible even if you treat backpropagation as a black box. Thereare, of course, points later in the book where I refer back to resultsfrom this chapter. But at those points you should still be able tounderstand the main conclusions, even if you don’t follow all thereasoning. Before discussing backpropagation, let’s warm up with a fastmatrix-based algorithm to compute the output from a neural network.
We actually already briefly saw this algorithmnear the end of the last chapter, but I described it quickly, so it’sworth revisiting in detail. In particular, this is a good way ofgetting comfortable with the notation used in backpropagation, in afamiliar context. Let’s begin with a notation which lets us refer to weights in thenetwork in an unambiguous way. We use a similar notation for the network’s biases and activations.
Think of it as a way of escaping index hell,while remaining precise about what’s going on. The expression is alsouseful in practice, because most matrix libraries provide fast ways ofimplementing matrix multiplication, vector addition, andvectorization. For backpropagation to work we need to make two mainassumptions about the form of the cost function. Before stating thoseassumptions, though, it’s useful to have an example cost function inmind. This assumption will also hold true forall the other cost functions we’ll meet in this book. In particular, it’s notsomething we can modify by changing the weights and biases in any way,i. The backpropagation algorithm is based on common linear algebraicoperations – things like vector addition, multiplying a vector by amatrix, and so on.
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But one of the operations is a little lesscommonly used. Backpropagation is about understanding how changing the weights andbiases in a network changes the cost function. As the input to theneuron comes in, the demon messes with the neuron’s operation. Now, this demon is a good demon, and is trying to help you improve thecost, i.
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In fact, if you dothis things work out quite similarly to the discussion below. But itturns out to make the presentation of backpropagation a little morealgebraically complicated. Plan of attack: Backpropagation is based around fourfundamental equations. I state the four equations below.
Be warned, though: youshouldn’t expect to instantaneously assimilate the equations. Such anexpectation will lead to disappointment. This is a very natural expression. It’s a perfectly good expression, but not the matrix-based form wewant for backpropagation. As you can see, everything in this expression has a nice vector form,and is easily computed using a library such as Numpy.
This equation appears complicated, buteach element has a nice interpretation. In this case, we’ll say the weight learns slowly,meaning that it’s not changing much during gradient descent. Let’s start by looking at the outputlayer. We can obtain similar insights for earlier layers. But I’m speaking of the general tendency.
Summing up, we’ve learnt that a weight will learn slowly if either theinput neuron is low-activation, or if the output neuron has saturated,i. None of these observations is too greatly surprising. Still, theyhelp improve our mental model of what’s going on as a neural networklearns. Furthermore, we can turn this type of reasoning around. This presentation may be disconcerting if you’re unused to the Hadamard product.
There’s an alternative approach, based on conventional matrix multiplication, which some readers may find enlightening. All four are consequences of thechain rule from multivariable calculus. If you’re comfortable withthe chain rule, then I strongly encourage you to attempt thederivation yourself before reading on. These also follow from the chain rule, in a mannersimilar to the proofs of the two equations above.
I leave them to youas an exercise. That completes the proof of the four fundamental equations ofbackpropagation. But it’s reallyjust the outcome of carefully applying the chain rule. A little lesssuccinctly, we can think of backpropagation as a way of computing thegradient of the cost function by systematically applying the chainrule from multi-variable calculus.
That’s all there really is tobackpropagation – the rest is details. The backpropagation equations provide us with a way of computing thegradient of the cost function. Examining the algorithm you can see why it’s calledbackpropagation. It may seem peculiar thatwe’re going through the network backward. But if you think about theproof of backpropagation, the backward movement is a consequence ofthe fact that the cost is a function of outputs from the network. How should we modify the backpropagation algorithm in this case? Rewrite the backpropagation algorithm for this case.
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In practice, it’s common to combine backpropagation with alearning algorithm such as stochastic gradient descent, in which wecompute the gradient for many training examples. Of course, to implement stochastic gradient descent in practice youalso need an outer loop generating mini-batches of training examples,and an outer loop stepping through multiple epochs of training. Having understood backpropagation in the abstract, we can nowunderstand the code used in the last chapter to implementbackpropagation. Recall fromthat chapter that the code was contained in the update_mini_batchand backprop methods of the Network class. The code forthese methods is a direct translation of the algorithm describedabove.
The backprop method follows the algorithm in thelast section closely. There is one small change – we use a slightlydifferent approach to indexing the layers. Python can use negative indices in lists. It’s possible to modify the backpropagation algorithm so that it computes the gradients for all training examples in a mini-batch simultaneously. To answer thisquestion, let’s consider another approach to computing the gradient.
Imagine it’s the early days of neural networks research. Maybe it’sthe 1950s or 1960s, and you’re the first person in the world to thinkof using gradient descent to learn! But to make the idea work youneed a way of computing the gradient of the cost function. It’s simple conceptually, andextremely easy to implement, using just a few lines of code. Certainly, it looks much more promising than the idea of using thechain rule to compute the gradient!
Unfortunately, while this approach appears promising, when youimplement the code it turns out to be extremely slow. To understandwhy, imagine we have a million weights in our network. This should be plausible, but it requires some analysis to make a careful statement. And so the total cost ofbackpropagation is roughly the same as making just two forward passesthrough the network. This speedup was first fully appreciated in 1986, and it greatlyexpanded the range of problems that neural networks could solve.
That, in turn, caused a rush of people using neural networks. Ofcourse, backpropagation is not a panacea. Even in the late 1980speople ran up against limits, especially when attempting to usebackpropagation to train deep neural networks, i. As I’ve explained it, backpropagation presents two mysteries. But can we go any deeper,and build up more intuition about what is going on when we do allthese matrix and vector multiplications?
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The second mystery is howsomeone could ever have discovered backpropagation in the first place? Let’s try to carry this out. However,it has a nice intuitive interpretation. What theequation tells us is that every edge between two neurons in thenetwork is associated with a rate factor which is just the partialderivative of one neuron’s activation with respect to the otherneuron’s activation. What I’ve been providing up to now is a heuristic argument, a way ofthinking about what’s going on when you perturb a weight in a network.
Let me sketch out a line of thinking you could use to further developthis argument. That’s easy to do with a bit ofcalculus. Having done that, you could then try to figure out how towrite all the sums over indices as matrix multiplications. This turnsout to be tedious, and requires some persistence, but notextraordinary insight. After doing all this, and then simplifying asmuch as possible, what you discover is that you end up with exactlythe backpropagation algorithm! Now, I’m not going to work through all this here.
It’s messy andrequires considerable care to work through all the details. If you’reup for a challenge, you may enjoy attempting it. And even if not, Ihope this line of thinking gives you some insight into whatbackpropagation is accomplishing. What about the other mystery – how backpropagation could have beendiscovered in the first place?
In fact, if you follow the approach Ijust sketched you will discover a proof of backpropagation. Unfortunately, the proof is quite a bit longer and more complicatedthan the one I described earlier in this chapter. I am, of course, asking you to trust me on this, butthere really is no great mystery to the origin of the earlier proof. It’s just a lot of hard work simplifying the proof I’ve sketched inthis section.
In academic work, please cite this book as: Michael A. This means you’re free to copy, share, and build on this book, but not to sell it. If you’re interested in commercial use, please contact me. Access to this page has been denied because we believe you are using automation tools to browse the website. PC Games news and reviews from PCGamesN. April Fool’s Day has somehow become gaming’s big holiday, as the industry comes together this time every year to celebrate the fun that can be had in poking fun at itself, its fans, and of course the humble games press. Here are some of the best April Fools goofs from an alternate universe’s games industry.
Far Cry 5’s adorable companion Boomer is no exception. But do we do when our time with virtual dogs impedes the time we spend with our real furry friends? Luckily, Ubisoft have the answer in the form of the Squeaky AR-C, which will let you roleplay your adventures with Boomer alongside the real dog of your choice – or at least keep them occupied while you hunt down some new weapons. CD Projekt need a DOGE As a Designer of Game Environments working in CD PROJEKT RED you’ll be responsible for digging up bugs, fetching missing code and sniffing up plot holes.
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Remedy’s mysterious P7 will sadly never see the light of day, as the studio behind Max Payne, Alan Wake, and Quantum Break have decided to quit making videogames in order to focus on their true passion. Remedy will transfer their expertise in digital storytelling to making dark roasts for your drinking pleasure. Though it may be difficult, since Sam Lake has apparently never even tasted the stuff before. The best April Fool’s gags are always those that provide us with something tangible and fun – even if only temporarily. Taking us back to the days when definitively not-age-appropriate entertainment kept getting kid’s cartoons, Overkill have launched a trailer for Payday: The Animated series, complete with an appropriately rockin’ 80s theme song. A bonafide holiday like April 1st is the perfect opportunity to announce more holidays for your game, and Blizzard’s World of Warcraft team certainly understand the synergy. They’ve got a new series of micro-holidays on the way, ranging from level one Battlegrounds to repeatable repeating daily quests.
There’s even ten-second holiday in there for bonus celebrations. While the English-language Blizzard team did alright for AFD, it’s the Korean division that got the true star with World of Warcraft: Dance Battle. Yep, that’s Alliance versus Horde on the arcade dance floor. Even if you don’t speak Korean, you should definitely check out the images on the official site. Rocket League has hats, as all modern videogames do. They got very large on April Fool’s Day.
It’s like NBA Jam’s big head mode all over again. We’re just about due to see the next Halo, and judging by Microsoft’s increasing cross-platform plans it’s almost certainly going to be on PC. Halo: Battle Royale on Reddit, complete with a monthly subscription called Spartan Pass for boosted experience gains. The developers who didn’t announce a new, totally real battle royale game instead decided to detail dating sims based on popular franchises. Mega Man: Date My Robot Master recasts the the series’ robot masters as high school students, giving you the opportunity to date Wood Man, Gemini Man, and Splash Woman. The boss orders should be very interesting here.
There are few games as heavily anticipated as CD Projekt’s Cyberpunk 2077, and to everyone’s great shock it actually came out yesterday. Er, rather, some game called Cyberpunk 2O77 did. Aside from noted rebel Gym Sterling, of course. The team behind Black Mesa have done an excellent job of modernizing the original Half-Life, but no game can be truly modern without loot boxes. Luckily for us, Loot Mesa is on the way, offering cosmetics including weapon and character skins to really spice up Gordon Freeman’s first adventure. Oh, and they’re doing a Last Freeman Standing battle royale mode, because of course they are.
Gaijin Games began their annual April Fool’s fun at 00:01 Kamchatka time and are running it through 23:59 Hawaii time – the better to get in as much tomfoolery as possible. Silent Thunder, which you can play through the War Thunder client. PC gamers are used to upgrading their machines, and Razer’s April Fool’s offering is a nanobot-infused energy drink that lets us upgrade our own cells. Chug a vial of Razer’s Project Venom v2 and you’re ready to go head to head with the pros, with better response times and your own RGB lighting scheme.