### Conference

Currently I am at the SIAM Imaging conference in Hong Kong. It’s a great conference with great people at a great place. I am pretty sure that this will be the only post from here, since the conference is quite intense. I just wanted to report on two ideas that have become clear here, although, they are both pretty easy and probably already widely known, but anyway:

1. Non-convex + convex objective

There are a lot of talks that deal with optimization problems of the form

$\displaystyle \min_u F(u) + G(u).$

Especially, people try to leverage as much structure of the functionals ${F}$ and ${G}$ as possible. Frequently, there arises a need to deal with non-convex parts of the objective, and indeed, there are several approaches around that deal in one way or another with non-convexity of ${F}$ or even ${F+G}$. Usually, in the presence of an ${F}$ that is not convex, it is helpful if ${G}$ has favorable properties, e.g. that still ${F+G}$ is bounded from below, coercive or even convex again. A particularly helpful property is strong convexity of ${G}$ (i.e. ${G}$ stays convex even if you subtract ${\epsilon/2\|\cdot\|^2}$ from it). Here comes the simple idea: If you already allow ${F}$ to be non-convex, but only have a ${G}$ that is merely convex, but not strongly so, you can modify your objective to

$\displaystyle \underbrace{F(u) - \tfrac\epsilon2\|u\|^2}_{\leftarrow F(u)} + \underbrace{G(u) + \tfrac\epsilon2\|u\|^2}_{\leftarrow G(u)}$

for some ${\epsilon>0}$. This will give you strong convexity of ${G}$ and an ${F}$ that is (often) theoretically no worse than it used to be. It appeared to me that this is an idea that Kristian Bredies told me already almost ten years ago and which me made into a paper (together with Peter Maaß) in 2005 which got somehow delayed and published no earlier than 2009.

If your problem has the form

$\displaystyle \min_u F(u) + G(Ku)$

with some linear operator ${K}$ and both ${F}$ and ${G}$ are convex, it has turned out, that it is tremendously helpful for the solution to consider the corresponding saddle point formulation: I.e. using the convex conjugate ${G^*}$ of ${G}$, you write

$\displaystyle \min_u \max_v F(u) + \langle Ku, v\rangle -G^*(v).$

A class of algorithms, that looks like to Arrow-Hurwicz-method at first glance, has been sparked be the method proposed by Chambolle and Pock. This method allows ${F}$ and ${G}$ to be merely convex (no smoothness or strong convexity needed) and only needs the proximal operators for both ${F}$ and ${G^*}$. I also worked on algorithms for slightly more general problems, involving a reformulation of the saddle point problem as a monotone inclusion, with Tom Pock in the paper An accelerated forward-backward algorithm for monotone inclusions and I also should mention this nice approach by Bredies and Sun who consider another reformulation of the monotone inclusion. However, in the spirit of the first point, one should take advantage of all the available structure in the problem, e.g. smoothness of one of the terms. Some algorithm can exploit smoothness of either ${F}$ or ${G^*}$ and only need convexity of the other term. An idea, that has been used for some time already, to tackle the case if ${F}$, say, is a sum of a smooth part and a non-smooth part (and ${G^*}$ is not smooth), is, to dualize the non-smooth part of ${F}$: Say we have ${F = F_1 + F_2}$ with smooth ${F_1}$, then you could write

$\displaystyle \begin{array}{rcl} &\min_u\max_v F_1(u) + F_2(u) + \langle Ku, v\rangle -G^*(v)\\ & \qquad= \max_u \min_{v,w} F_1(u) + \langle u,w\rangle + \langle Ku, v\rangle -G^*(v) - F_2^*(w) \end{array}$

and you are back in business, if your method allows for sums of convex functions in the dual. The trick got the sticky name “dual transportation trick” in a talk by Marc Teboulle here and probably that will help, that I will not forget it from now on…

The mother example of optimization is to solve problems

$\displaystyle \min_{x\in C} f(x)$

for functions ${f:{\mathbb R}^n\rightarrow{\mathbb R}}$ and sets ${C\in{\mathbb R}^n}$. One further classifies problems according to additional properties of ${f}$ and ${C}$: If ${C={\mathbb R}^n}$ one speaks of unconstrained optimization, if ${f}$ is smooth one speaks of smooth optimization, if ${f}$ and ${C}$ are convex one speaks of convex optimization and so on.

1. Classification, goals and accuracy

Usually, optimization problems do not have a closed form solution. Consequently, optimization is not primarily concerned with calculating solutions to optimization problems, but with algorithms to solve them. However, having a convergent or terminating algorithm is not fully satisfactory without knowing an upper bound on the runtime. There are several concepts one can work with in this respect and one is the iteration complexity. Here, one gives an upper bound on the number of iterations (which are only allowed to use certain operations such as evaluations of the function ${f}$, its gradient ${\nabla f}$, its Hessian, solving linear systems of dimension ${n}$, projecting onto ${C}$, calculating halfspaces which contain ${C}$, or others) to reach a certain accuracy. But also for the notion of accuracy there are several definitions:

• For general problems one can of course desire to be within a certain distance to the optimal point ${x^*}$, i.e. ${\|x-x^*\|\leq \epsilon}$ for the solution ${x^*}$ and a given point ${x}$.
• One could also demand that one wants to be at a point which has a function value close to the optimal one ${f^*}$, i.e, ${f(x) - f^*\leq \epsilon}$. Note that for this and for the first point one could also desire relative accuracy.
• For convex and unconstrained problems, one knowns that the inclusion ${0\in\partial f(x^*)}$ (with the subgradient ${\partial f(x)}$) characterizes the minimizers and hence, accuracy can be defined by desiring that ${\min\{\|\xi\|\ :\ \xi\in\partial f(x)\}\leq \epsilon}$.

It turns out that the first two definitions of accuracy are much to hard to obtain for general problems and even for smooth and unconstrained problems. The main issue is that for general functions one can not decide if a local minimizer is also a solution (i.e. a global minimizer) by only considering local quantities. Hence, one resorts to different notions of accuracy, e.g.

• For a smooth, unconstrained problems aim at stationary points, i.e. find ${x}$ such that ${\|\nabla f(x)\|\leq \epsilon}$.
• For smoothly constrained smooth problems aim at “approximately KKT-points” i.e. a point that satisfies the Karush-Kuhn-Tucker conditions approximately.

(There are adaptions to the nonsmooth case that are in the same spirit.) Hence, it would be more honest not write ${\min_x f(x)}$ in these cases since this is often not really the problem one is interested in. However, people write “solve ${\min_x f(x)}$” all the time even if they only want to find “approximately stationary points”.

2. The gradient method for smooth, unconstrainted optimization

Consider a smooth function ${f:{\mathbb R}^n\rightarrow {\mathbb R}}$ (we’ll say more precisely how smooth in a minute). We make no assumption on convexity and hence, we are only interested in finding stationary points. From calculus in several dimensions it is known that ${-\nabla f(x)}$ is a direction of descent from the point ${x}$, i.e. there is a value ${h>0}$ such that ${f(x - h\nabla f(x))< f(x)}$. Hence, it seems like moving into the direction of the negative gradient is a good idea. We arrive at what is known as gradient method:

$\displaystyle x_{k+1} = x_k - h_k \nabla f(x_k).$

Now let’s be more specific about the smoothness of ${f}$. Of course we need that ${f}$ is differentiable and we also want the gradient to be continuous (to make the evaluation of ${\nabla f}$ stable). It turns out that some more smoothness makes the gradient method more efficient, namely we require that the gradient of ${f}$ is Lipschitz continuous with a known Lipschitz constant ${L}$. The Lipschitz constant can be used to produce efficient stepsizes ${h_k}$, namely, for ${h_k = 1/L}$ one has the estimate

$\displaystyle f(x_k) - f(x_{k+1})\geq \frac{1}{2L}\|\nabla f(x_k)\|^2.$

This inequality is really great because one can use telescoping to arrive at

$\displaystyle \frac{1}{2L}\sum_{k=0}^N \|\nabla f(x_k)\|^2 \leq f(x_0) - f(x_{N+1}) \leq f(x_0) - f^*$

with the optimal value ${f}$ (note that we do not need to know ${f^*}$ for the following). We immediately arrive at

$\displaystyle \min_{0\leq k\leq N} \|\nabla f(x_k)\| \leq \frac{1}{\sqrt{N+1}}\sqrt{2L(f(x_0)-f^*))}.$

That’s already a result on the iteration complexity! Among the first ${N}$ iterates there is one which has a gradient norm of order ${N^{-1/2}}$.

However, from here on it’s getting complicated: We can not say anything about the function values ${f(x_k)}$ and about convergence of the iterates ${x_k}$. And even for convex functions ${f}$ (which allow for more estimates from above and below) one needs some more effort to prove convergence of the functional values to the global minimal one.

But how about convergence of the iterates for the gradient method if convexity is not given? It turns out that this is a hard problem. As illustration, consider the continuous case, i.e. a trajectory of the dynamical system

$\displaystyle \dot x = -\nabla f(x)$

(which is a continuous limit of the gradient method as the stepsize goes to zero). A physical intuition about this dynamical system in ${{\mathbb R}^2}$ is as follows: The function ${f}$ describes a landscape and ${x}$ are the coordinates of an object. Now, if the landscape is slippery the object slides down the landscape and if we omit friction and inertia, the object will always slide in the direction of the negative gradient. Consider now a favorable situation: ${f}$ is smooth, bounded from below and the level sets ${\{f\leq t\}}$ are compact. What can one say about the trajectories of the ${\dot x = -\nabla f(x)}$? Well, it seems clear that one will arrive at a local minimum after some time. But with a little imagination one can see that the trajectory of ${x}$ does not even has to be of finite length! To see this consider a landscape ${f}$ that is a kind of bowl-shaped valley with a path which goes down the hillside in a spiral way such that it winds around the minimum infinitely often. This situation seems somewhat pathological and one usually does not expect situation like this in practice. If you tried to prove convergence of the iterates of gradient or subgradient descent you may have noticed that one sometimes wonders why the proof turns out to be so complicated. The reason lies in the fact that such pathological functions are not excluded. But what functions should be excluded in order to avoid this pathological behavior without restricting to too simple functions?

3. The Kurdyka-Łojasiewicz inequality

Here comes the so-called Kurdyka-Łojasiewicz inequality into play. I do not know its history well, but if you want a pointer, you could start with the paper “On gradients of functions definable in o-minimal structures” by Kurdyka.

The inequality shall be a way to turn a complexity estimate for the gradient of a function into a complexity estimate for the function values. Hence, one would like to control the difference in functional value by the gradient. One way to do so is the following:

Definition 1 Let ${f}$ be a real valued function and assume (without loss of generality) that ${f}$ has a unique minimum at ${0}$ and that ${f(0)=0}$. Then ${f}$ satisfies a Kurdyka-Łojasiewicz inequality if there exists a differentiable function ${\kappa:[0,r]\rightarrow {\mathbb R}}$ on some interval ${[0,r]}$ with ${\kappa'>0}$ and ${\kappa(0)=0}$ such that

$\displaystyle \|\nabla(\kappa\circ f)(x)\|\geq 1$

for all ${x}$ such that ${f(x).

Informally, this definition ensures that that one can “reparameterize the range of the function such that the resulting function has a kink in the minimum and is steep around that minimum”. This definition is due to the above paper by Kurdyka from 1998. In fact it is a slight generalization of the Łowasiewicz inequality (which dates back to a note of Łojasiewicz from 1963) which states that there is some ${C>0}$ and some exponent ${\theta}$ such that in the above situation it holds that

$\displaystyle \|\nabla f(x)\|\geq C|f(x)|^\theta.$

To that that, take ${\kappa(s) = s^{1-\theta}}$ and evaluate the gradient to ${\nabla(\kappa\circ f)(x) = (1-\theta)f(x)^{-\theta}\nabla f(x)}$ to obtain ${1\leq (1-\theta)|f(x)|^{-\theta}\|\nabla f(x)\|}$. This also makes clear that in the case the inequality is fulfilled, the gradient provides control over the function values.

The works of Łojasiewicz and Kurdyka show that a large class of functions ${f}$ fulfill the respective inequalities, e.g. piecewise analytic function and even a larger class (termed o-minimal structures) which I haven’t fully understood yet. Since the Kurdyka-Łojasiewicz inequality allows to turn estimates from ${\|\nabla f(x_k)\|}$ into estimates of ${|f(x_k)|}$ it plays a key role in the analysis of descent methods. It somehow explains, that one really never sees pathological behavior such as infinite minimization paths in practice. Lately there have been several works on further generalization of the Kurdyka-Łojasiewicz inequality to the non-smooth case, see e.g. Characterizations of Lojasiewicz inequalities: subgradient flows, talweg, convexity by Bolte, Daniilidis, Ley and Mazet Convergence of non-smooth descent methods using the Kurdyka-Łojasiewicz inequality by Noll (however, I do not try to give an overview over the latest developments here). Especially, here at the French-German-Polish Conference on Optimization which takes place these days in Krakow, the Kurdyka-Łojasiewicz inequality has popped up several times.

The second day of SSVM started with an invited lecture of Tony Lindeberg, who has written one very influential and very early book about scale space theory. His talk was both a tour through scale space and a recap of the recent developments in the field. Especially he show how the time aspect could be incorporated into scale space analysis by a close inspection of how receptive fiels are working. There were more talks but I only took notes from the talk of Jan Lellmann who talked about the problem of generating an elevation map from a few given level lines. One application of this could be to observe coastlines at different tides and then trying the reconstruct the full height map at the coast. One specific feature here is that the surface one looks for may have ridges which stem from kinks in the level lines and these ridges are important features of the surface. He argued that a pure convex regularization will not work and proposed to use more input namely a vector field which is derived from the contour lines such that the vector somehow “follows the ridges”, i.e. it connects to level lines in a correct way.

Finally another observation I had today: Well, this is not a trend, but a notion which I heard for the first time here but which sounds very natural is the informal classification of data terms in variational models as “weak” or “strong”. For example, a denoising data term ${\|u-u^0\|^2_{L^2(\Omega)}}$ is a strong data term because it gives tight information on the whole set ${\Omega}$. On the other hand, an inpainting data term ${u|_{\Omega\setminus\Omega'} = u^0|_{\Omega\setminus\Omega'}}$ is a weak data term because it basically tell nothing within the region ${\Omega'}$.

For afternoon the whole conference has been on tour to three amazing places:

• the Riegersburg, which is not only an impressive castle but also features interesting exhibitions about old arm and witches,
• the Zotter chocolate factory where they make amazing chocolate in mind-boggling varieties,
• and to Schloss Kronberg for the conference dinner (although it was pretty tough to start eating the dinner after visiting Zotter…).

I am at the SSVM 13 and the first day is over. The conference is in a very cozy place in Austria, namely the Schloss Seggau which is located on a small hill near the small town Leibnitz in Styria (which is in Austria but totally unaffected by the floods). The conference is also not very large but there is a good crowd of interesting people here and the program reads interesting.

The time schedule is tight and I don’t know if I can make it to post something everyday. Especially, I will not be able to blog about every talk.

From the first day I have the impression that two things have appeared frequently at different places:

• Differential geometry: Tools from differential geometry like the notion of geodesics or Riemannian metrics have not only been used by Martin Rumpf to model different types of spaces of shapes but also to interpolate between textures.
• Primal dual methods: Variational models should be convex these days and if they are not you should make them convex somehow. Then you can use primal dual methods. Somehow the magic of primal dual methods is that they are incredibly flexible. Also they work robustly and reasonably fast.

Probably, there are more trends to see in the next days.

I case anybody wondered: I did not make it to produce a single post from Oberwolfach last week. For me it was an unusual dense week. Not only that there were many interesting talks on a tight schedule. The “talk-free” parts of the days I had a lot of interesting and stimulating discussions and the night life did not permit the writing posts either. Sorry – but you may probably read about the outcome of some of the discussions in future blog post…

Guess where this is:

Well, I spoiled the solution in the title anyway (and of course, some of you recognized the sculpture in the last picture): I am in Oberwolfach this week at the workshop “Computational Inverse Problems”. Maybe I’ll find some time to blog about some talks but I am not sure if I can make it. Usually the Oberwolfach workshops are quite intense and hence, blogging will be too much…

Anyway, the autumn looks great here these days:

ISMP is over now and I’m already home. I do not have many things to report on from the last day. This is not due the lower quality of the talks but due to the fact that I was a little bit exhausted, as usual at the end of a five-day conference. However, I collect a few things for the record:

• In the morning I visited the semi-planary by Xiaojun Chenon non-convex and non-smooth minimization with smoothing methods. Not surprisingly, she treated the problem

$\displaystyle \min_x f(x) + \|x\|_p^p$

with convex and smooth ${f:{\mathbb R}^n\rightarrow{\mathbb R}}$ and ${0. She proposed and analyzed smoothing methods, that is, to smooth the problem a bit to obtain a Lipschitz-continuous objective function ${\phi_\epsilon}$, minimizing this and then gradually decreasing ${\epsilon}$. This works, as she showed. If I remember correctly, she also treated “iteratively reweighted least squares” as I described in my previous post. Unfortunately, she did not include the generalized forward-backward methods based on ${\text{prox}}$-functions for non-convex functions. Kristian and I pursued this approach in our paper Minimization of non-smooth, non-convex functionals by iterative thresholding and some special features of our analysis include:

• A condition which excludes some (but not all) local minimizers from being global.
• An algorithm which avoids this non-global minimizers by carefully adjusting the steplength of the method.
• A result that the number of local minimizers is still finite, even if the problem is posed in ${\ell^2({\mathbb N})}$ and not in ${{\mathbb R}^n}$.

Most of our results hold true, if the ${p}$-quasi-norm is replaced by functions of the form

$\displaystyle \sum_n \phi_n(|x_n|)$

with special non-convex ${\phi}$, namely fulfilling a list of assumptions like

• ${\phi'(x) \rightarrow \infty}$ for ${x\rightarrow 0}$ (infinite slope at ${0}$) and ${\phi(x)\rightarrow\infty}$ for ${x\rightarrow\infty}$ (mild coercivity),
• ${\phi'}$ strictly convex on ${]0,\infty[}$ and ${\phi'(x)/x\rightarrow 0}$ for ${x\rightarrow\infty}$,
• for each ${b>0}$ there is ${a>0}$ such that for ${x it holds that ${\phi(x)>ax^2}$, and
• local integrability of some section of ${\partial\phi'(x) x}$.

As one easily sees, ${p}$-quasi-norms fulfill the assumptions and some other interesting functions as well (e.g. some with very steep slope at ${0}$ like ${x\mapsto \log(x^{1/3}+1)}$).

• Jorge Nocedalgave a talk on second-order methods for non-smooth problems and his main example was a functional like

$\displaystyle \min_x f(x) + \|x\|_1$

with a convex and smooth ${f}$, but different from Xiaojun Chen, he only considered the ${1}$-norm. His talked is among the best planary talks I have ever attended and it was a great pleasure to listen to him. He carefully explained things and put them in perspective. In the case he skipped slides, he made me feel that I either did not miss an important thing, or understood them even though he didn’t show them He argued that it is not necessarily more expensive to use second order information in contrast to first order methods. Indeed, the ${1}$-norm can be used to reduce the number of degrees of freedom for a second order step. What was pretty interesting is, that he advocated semismooth Newton methods for this problem. Roland and I pursued this approach some time ago in our paper A Semismooth Newton Method for Tikhonov Functionals with Sparsity Constraints and, if I remember correctly (my notes are not complete at this point), his family of methods included our ssn-method. The method Roland and I proposed worked amazingly well in the cases in which it converged but the method suffered from non-global convergence. We had some preliminary ideas for globalization, which we could not tune enough to retain the speed of the method, and abandoned the topic. Now, that the topic will most probably be revived by the community, I am looking forward to fresh ideas here.

Next Page »