 I'm sharing the screen just a second. Is everything all right with the screen? Yes, and we can hear you. OK, great, I'll start. So thank you, Yuri. And I also want to thank the organizers for the invitation. We want to speak about the optimization of full RSB spherical models. Let me start with the definition of the models. These are random functions, which are defined on the unit sphere. Actually, they are even random polynomials, as you see in a second. And for those of you who are familiar with the models, I'm using here a slightly different normalization and working with the unit sphere. The first models are defined as a pure PSP model. So they define you choose a parameter p and use the following formula. Here, x is a vector on the sphere, or in Rm. So x i are the coordinates. And you go over all possible monomials of the free p. And you multiply each one of them by an IID Gaussian. So in some sense, this is the most natural way to define a homogeneous random polynomial. And the mixed PSP models are linear combinations of such models. Here, I'm assuming that the h and p are independent. And to choose a model, you just choose a sequence of parameters gamma p. And we are interested in the large n limit of the study the function as n goes to infinity and gamma p are fixed. And the problem I will talk about is an optimization problem. What we want to do is, given the function, find a point where the function attains roughly the maximum value or maximal energy. So first, let me comment that in these models, if you normalize the maximum by n, then it has a limit, which I denote by e star. This is called the ground state energy. And such configurations or points are called ground state configurations, x star as in the equation. To be precise, what you get as an input are the coefficient j, which defines the random function. And we want an algorithm that produces such a point with minimal time complexity. Of course, if you don't care about that, if you just search for the point by sampling the function, to get within an error of epsilon, it will take you exponential time. And what I'll show you is that in some models, you can optimize in polynomial time. Now, let me define p max as the maximal value of p, such that gamma p is not 0. Because the number of coefficients that you use to define h n is roughly of the order of n to the p max. So the way I stated this problem, I need p max to be finite in order not to have infinite complexity. The theorem I want to show you says that under some condition, which I'll talk about in a minute, assuming this p max is finite, you can optimize within epsilon error in the best complexity you can hope for. Just n to the p max. And this condition is that at zero temperature, the support of the Parisi measure is the whole interval 0, 1. So I'm not going to define exactly what it means, but I will tell you some consequences of this condition that are relevant to the optimization problem. And also, I will tell you about an explicit condition, which is equivalent. So for the spherical models, we know that this condition about the support of the Parisi measure, you can check it by constructing the mixture polynomial. So that's one polynomial in one variable, determined this seed. You just use the gamma p. And you can check whether this function or the second derivative of this function to the power of minus f is concave of 0, 1. That exactly characterizes these models which satisfy this condition about the support of the Parisi measure. Before I move on, let me, I want to mention that in the easy case, namely, when you optimize over the hypercube instead of the hypersphere, a similar result was derived by a Montanari for the SK model and more generally for mixed models by a Lalawi-Montanari insertion. OK, so the following, what I'm going to tell you about in this slide is for any spherical model. So with high probability, there exists a tree whose vertices are points inside the ball and the root is the origin. And the leaves of this tree are points of the sphere of radius 1. It has the following properties. First of all, any two edges on this tree are orthogonal. So this is the maximal orthogonality you can expect from this structure. And secondly, this is relevant to the optimization program because we are interested in points with maximal energy. Any point on the tree, any of the vertices, the energy at any of the vertices is maximal over the sphere of the same radius. OK, and let me remark, on the interior of the ball I define hn by the similar formula we used for the sphere. Lastly, the number of levels in this tree depends on the size of the support of the parallelism. So in this picture, this is the so-called three-step replica symmetry breaking, where you have the depth of the tree is 3. But the case I'm interested in, where the support is the interval 0, 1. In this case, the number of levels goes to infinity, hn goes to infinity, and you'll have vertices already between 0, 1, and t is asymptotically. So we have this huge structure. We are interested in the leaves of this tree. These are points on the sphere of radius 1. We would like to get one of them, for example, that will be a ground state configuration. And of course, we cannot pick a point from the tree. But what I want to explain is how you can use this structure, which you can hope to follow from the origin to the boundary to design an algorithm that optimizes the function. OK, so I want to look at only one path from this tree. So I denote it by xq, and q is the time parameter. And everything is parameterized so that at time q, the point is at radius square root of q. And I linearly interpolate between the points because the tree is discrete. And this path will have two properties inherited from the tree. The first one is a geometric property. Each segment is orthogonal to the current position because the current position is sum of segments on the tree, and we said that all of them are orthogonal. And the second property is, again, just what we had for the tree, but now for this path in the full RSV case, the energy at any point is maximal over the sphere of same radius. And now what I want to do is just think of how I can construct such a path with these two properties by myself without having any access to the ultrometric tree. So all the information I take is that there exists such a path, and I take it as only an inspiration for the algorithm. I don't need to use anything for the proofs from the tree. OK, so the geometric property we can just impose by building this path by concatenating many small intervals which are always orthogonal to the current position. So VI are just a direction which I will have to choose, and xi will be just the sum of the VI up to this point, normalize correctly. Now what is left to do is to choose the direction so that we maximize the energy at each step. And if you think about it, maybe the first guess is to use a gradient, but assuming that everything went well up to the I step, that would mean that the energy at the point we have reached is maximal. Therefore, the gradient should be roughly 0. And so there is no point of using the gradient for this. And what you can do as the second best guess is to use the Hessian. Yes, I'm going to look at the Hessian on the orthogonal plane to the point where I'm at after I steps. And you can choose essentially any direction which is in the span of the eigenvectors corresponding to the largest eigenpallet. Actually, this algorithm works. This algorithm will lead you to a point with the right energy. So let me describe the algorithm precisely. Just pick some large number k, which would be the number of steps we take from the origin to the sphere of radius 1. And some small number of delta, which would be some arrow. Finish realization is not very important. You can choose any point you want with radius 1 over square k. And then at each step, I'm going to do the following. I will choose the direction p. So it has length 1. And it's orthogonal to the current position. I also want to require that the projection onto the gradient is going to be positive. This is just I'm going to use a Taylor expansion. So that condition only means that the gradient would not ruin anything. And note also that if I have a direction v, which does not satisfy this inequality, I can just flip it. I can take minus v and get this property. The important thing is that you're going to choose v so that the second order term in a Taylor approximation would be the maximum you can get up to an arrow of delta. This is the same delta that we start with. And all you're going to do is update the step. You're going to take a small step in this direction. You continue to start from the origin and continue until you reach the sphere of radius 1. What you can guarantee is that for the models for which the support of the Parisi measure is 0, 1, is the interval 0, 1, you get the following lower bound on the energy. It's the maximal energy you can get up to an arrow of epsilon. And this epsilon can be as small as you wish, provided that k is large enough and delta is smaller. And as for time complexity, each iteration, each run of this algorithm takes essentially n to the degree of nu, the mixture. And we have constant that depends on delta and we need to multiply this by k. So that would be the constant that appeared in the original theorem. Now let me sketch the proof. It's going to be just one slide. So we have this path that we construct. And we choose the direction according to the Hessian. And what we want to do is control the energy on this path. So a priori, we need to control the Hessian on a random path because this path is chosen randomly. But you can do even better. You can control the Hessian over all points inside the ball of radius 1. And roughly speaking, what you can show is that if you look at the Hessian on the orthogonal space, then its edge is going to be approximately 2 times nu double prime of q to the power of minus half. So now to control the energy, at each step, I'm just going to Taylor expand up to second order. The first order term I just removed because in the algorithm, I've chosen the direction to have positive projection onto the gradient. So I'm just going to neglect this. And this third term is just the error that you get from that expansion. And of course, the orange part here corresponds to the approximation of the edge here. What happens is that if you concatenate many such intervals, we'll approximate the bar you get on the energy. We'll approximate the corresponding integral. So that would be just the integral from 0 to q of this quantity, nu double prime of q to the minus half. And for the models for which the support of the Parisian measure is the interval 0, 1, we know what we can analyze the Parisian formula explicitly. And we can show that the maximum, it is well known, the maximum is given by the same integral. So you just control the energy on the path. And you know by some other argument that this is really the maximal energy. And therefore, you're done. And that's all.