 Dear Fellow Scholars, this is two-minute papers with Karojona Ifehir. Today, we're not going to have the usual visual fireworks that we had with most topics in computer graphics, but I really hope you'll still find this episode enjoyable and stimulating. This episode is also going to be a bit heavy on what optimization is and will talk a little bit at the end about the intuition of the paper itself. We are going to talk about mathematical optimization. This term is not to be confused with the word optimization that we use in our everyday lives for, for instance, improving the efficiency of a computer code or a workflow. This kind of optimization means finding one hopefully optimal solution from a set of possible candidate solutions. An optimization problem is given the following way. One, there's a set of variables we can play with, and two, there's an objective function that we wish to minimize or maximize. Well, this probably sounds great for mathematicians, but for everyone else, maybe this is a bit confusing. Let's build a better understanding of this concept through an example. For instance, let's imagine that we have to cook a meal for our friends from a given set of ingredients. The question is how much salt vegetables and meat goes into the pan. These are variables that we can play with, and the goal is to choose the optimal amount of these ingredients to maximize the tastiness of the meal. Tastiness will be our objective function, and for a moment we shall pretend that tastiness is an objective measure of a meal. This was just one toy example, but the list of applications is endless. In fact, optimization is so incredibly ubiquitous. There is hardly any field of science where some form of it is not used to solve difficult problems. For instance, if we have the plan of a bridge, we can ask it to tell us the minimal amount of building materials we need to build it in a way that it remains stable. We can also optimize the layout of the bridge itself to make sure the inner tension and compression forces line up well. A big part of deep learning is actually also an optimization problem. There are a given set of neurons, and the variables are when they should be activated. And we are fiddling with these variables to minimize the output error, which can be, for instance, our accuracy in guessing whether a picture depicts a muffin or a chiwawa. The question for almost any problem is usually not whether it can be formulated as an optimization problem, but whether it is worth it. And by worth it, I mean the question whether we can solve it quickly and reliably. An optimizer is a technique that is able to solve these optimization problems and offer us a hopefully satisfactory solution to them. There are many algorithms that excel at solving problems of different complexities, but what ties them together is that they are usually handcrafted techniques written by really smart mathematicians. Gradient descent is one of the simplest optimization algorithms where we change each of the variables around a bit and as a result, see if the objective function changes favorably. After finding a direction that leads to the most favorable changes, we shall continue our journey in that direction. What does this mean in practice? Intuitively, in our cooking example, after making several meals, we would ask our guests about the tastiness of these meals. From their responses, we would recognize that adding a bit more salt led to very favorable results and since these people are notorious meat eaters, decreasing the amount of vegetables and increasing the meat content also led to favorable reviews. And we, of course, on the back of this newfound knowledge, will cook more with these variable changes in pursuit of the best possible meal in the history of mankind. This is something that is reasonably close to what gradient descent is in mathematics. A slightly more sophisticated version of gradient descent is also a very popular way of training neural networks. If you have any questions regarding the gradient part, we had an extended two-minute papers episode on what gradients are and how to use them to build an awesome algorithm for light transport. It is available, where? Well, of course, in the video description box, Karoy. Why are you even asking? So what about the paper part? This incredible new work of Google DeepMind shows that an optimization algorithm itself can emerge as a result of learning. An algorithm itself is not considered the same one thing as deciding what an image depicts or how we should grade a student essay. It is an algorithm, a sequence of steps we have to take. If we are talking about output sequences, we'll definitely need to use a recurrent neural network for that. Their proposed learning algorithm can create new optimization techniques that outperform previously existing methods, not everywhere, but on a set of specialized problems. I hope you've enjoyed the journey. We'll talk quite a bit about optimization in the future. You'll love it. Thanks for watching and for your generous support. I'll see you next time.