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Series: ACO Student Seminar

Information-based complexity is an alternative to Turing complexity that is well-suited for understanding a broad class of convex optimization algorithms. The groundbreaking work of Nemirovski and Yudin provided the basis of the theory, establishing tight lower bounds on the running time of first-order methods in a number of settings. There has been a recent interest on these classical techniques, because of exciting new applications on Machine Learning, Signal Processing, Stochastic Programming, among others. In this talk, we will introduce the rudiments of the theory, some examples, and open problems. Based on joint work with Gábor Braun and Sebastian Pokutta.

Series: ACO Student Seminar

In algorithmic game theory, the price of anarchy framework studies efficiency loss in decentralized environments. In optimization and decision theory, the price of robustness framework explores the tradeoffs between optimality and robustness in the case of single agent decision making under uncertainty. We establish a connection between the two that provides a novel analytic framework for proving tight performance guarantees for distributed systems in uncertain environments.We present applications of this framework to novel variants of atomic congestion games with uncertain costs, for which we provide tight performance bounds under a wide range of risk attitudes. Our results establish that the individual's attitude towards uncertainty has a critical effect on system performance and should therefore be a subject of close and systematic investigation.

Series: ACO Student Seminar

A large variety of Constraint Satisfactoin Problems can be classified as
"computationally hard". In recent years researchers from statistical
mechanics have investigated such problems via non-rigorous methods. The
aim of this talk is to give a brief overview of this work, and of the
extent to which the physics ideas can be turned into rigorous
mathematics. I'm also going to point out various open problems.

Series: ACO Student Seminar

We will begin by formulating the Riemann-Roch theorem for graphs, due to
the speaker and Norine. We will then describe some refinements and
applications.
Refinements include a Riemann-Roch theorem for tropical curves, proved
by Gathmann-Kerber and Mikhalkin-Zharkov, and a Riemann-Roch theorem for
metrized complexes of curves, proved by Amini and the speaker.
Applications include a new proof by Cools-Draisma-Payne-Robeva of the
Brill-Noether theorem in algebraic geometry, a generalization by Amini
and the speaker of the Eisenbud-Harris theory of limit linear series,
and a new Chabauty-Coleman style bound for the number of rational points
on an algebraic curve over the rationals, proved recently by Katz and
Zureick-Brown.

Series: ACO Student Seminar

Understanding how the physiology of organisms arises through the dynamic interaction of the molecular constituents of life is an important goal of molecular systems biology, for which mathematical modeling can be very helpful. Different modeling strategies have been used for this purpose. Dynamic mathematical models can be broadly divided into two classes: continuous, such as systems of differential equations and their stochastic variants and discrete, such as Boolean networks and their generalizations. This talk will focus on the discrete modeling approach. Applications will include the study of stochasticity under this setting. No background in mathematical biology is required, and the talk will be accessible to a broad audience.

Series: ACO Student Seminar

In this talk, we are going to introduce Linearized Proximal Alternating Minimization Algorithm and its variants for total variation based variational model. Since the proposed method does not require any special inner solver (e.g. FFT or DCT), which is quite often required in augmented Lagrangian based approach (ADMM), it shows better performance for large scale problems. In addition, we briefly introduce new regularization method (nonconvex higher order total variation).

Series: ACO Student Seminar

In this talk we consider the problem of finding basic solutions to linear programs where some vertices are excluded. We study the complexity of this and related problems, most of which turn out to be hard. On the other hand, we show that forbidding vertices from 0-1 polytopes can be carried out with a compact extended formulation. A similar result holds for integer programs having a box-integrality property. We discuss some applications of our results.

Series: ACO Student Seminar

The hard-core model has attracted much attention across several
disciplines, representing lattice gases in statistical physics and
independent sets in discrete mathematics and computer science. On
finite graphs, we are given a parameter \lambda, and an independent
set I arises with probability proportional to \lambda^{|I|}. We
are interested in determining the mixing time of local Markov
chains that add or remove a small number of vertices in each step.
On finite regions of Z^2 it is conjectured that there is a phase
transition at some critical point \lambda_c that is approximately
3.79. It is known that local chains are rapidly mixing when
\lambda < 2.3882. We give complementary results showing that
local chains will mix slowly when \lambda > 5.3646 on regions
with periodic (toroidal) boundary conditions and when
\lambda > 7.1031 with non-periodic (free) boundary conditions.
The proofs use a combinatorial characterization of configurations
based on the presence or absence of fault lines and an enumeration
of a new class of self-avoiding walks called taxi walks.
(Joint work with Antonio Blanca, David Galvin and Prasad Tetali)

Series: ACO Student Seminar

Mechanism design for distributed systems is fundamentally concerned with aligning individual incentives with social welfare to avoid socially inefficient outcomes that can arise from agents acting autonomously. One simple and natural approach is to centrally broadcast non-binding advice intended to guide the system to a socially near-optimal state while still harnessing the incentives of individual agents. The analytical challenge is proving fast convergence to near optimal states, and we present the first results that carefully constructed advice vectors yield stronger guarantees. We apply this approach to a broad family of potential games modeling vertex cover and set cover optimization problems in a distributed setting. This class of problems is interesting because finding exact solutions to their optimization problems is NP-hard yet highly inefficient equilibria exist, so a solution in which agents simply locally optimize is not satisfactory. We show that with an arbitrary advice vector, a set cover game quickly converges to an equilibrium with cost of the same order as the square of the social cost of the advice vector. More interestingly, we show how to efficiently construct an advice vector with a particular structure with cost $O(\log n)$ times the optimal social cost, and we prove that the system quickly converges to an equilibrium with social cost of this same order.

Series: ACO Student Seminar

Inpainting, deblurring and denoising images are common tasks required for a number of applications in science and engineering. Since the seminal work of Rudin, Osher and Fatemi, image regularization by total variation (TV) became a standard heuristic for achieving these tasks. In this talk, I will introduce the TV regularization model and some connections with sparse optimization and compressed sensing. Later, I will summarize some of the fastest existing methods for solving TV regularization. Motivated by improving the super-linear (on the dimension) running time of these algorithms, we propose two heuristics for image regularization models: the first one is to replace the TV by the \ell^1 norm of the Laplacian, and the second is a new, to the best of our knowledge, approximation of the TV seminorm, based on a redundant parameterization of the gradient field. We prove that the latter regularizer is an O(log n) approximation of the TV seminorm. This proof is based on basic techniques from Discrete Fourier Analysis and an estimate of the fundamental solutions of the Laplace equation on a grid, due to Mangad. Finally, we present preliminary computational results for the three models, on mid-scale images. This talk will be self-contained. Joint work with Arkadi Nemirovski.