Generic rectangulations are tilings of the unit square by
rectangles such that no 4 rectangles share a corner. Generic
rectangulations are a fundamental object in computer science: not
only appearing in practice (e.g. in VLSI circuit design) and
theory (e.g. being closely related to binary search trees); but
also within many other fundamental combinatorial objects such as
permutations. In this work, we present a versatile algorithmic
framework for exhaustively generating a large variety of different
classes of generic rectangulations. Our algorithms work under very
mild assumptions and apply to a considerable number of interesting
rectangulation classes known from the literature, such as generic
rectangulations, diagonal rectangulations, S-equivalent
rectangulations, 1-sided/area-universal rectangulations, and their
guillotine variants. The resulting generation algorithms are
efficient, in most interesting cases even loopless or constant
amortized time, i.e., each new rectangulation is generated in
constant time in the worst case or on average, respectively.
In this talk we will show:
A simple introduction to rectangulations and some of its
Our algorithmic generation framework
for rectangulation classes and
Correctness proof (sketch) of the framework by relating
rectangulations to permutations.
Fast domain propagation of linear constraints has become a crucial component of today’s best algorithms and solvers for mixed integer programming and pseudo-boolean optimization to achieve peak solving performance. Irregularities in the form of dynamic algorithmic behaviour, dependency structures, and sparsity patterns in the input data make efficient implementations of domain propagation on GPUs and, more generally, on parallel architectures challenging. This is one of the main reasons why domain propagation in state-of-the-art solvers is single thread only. In this paper, we present a new algorithm for domain propagation which (a) avoids these problems and allows for an efficient implementation on GPUs, and is (b) capable of running propagation rounds entirely on the GPU, without any need for synchronization or communication with the CPU. We present extensive computational results which demonstrate the effectiveness of our approach and show that ample speedups are possible on practically relevant problems: on state-of-the- art GPUs, our geometric mean speed-up for reasonably-large instances is around 10x to 20x and can be as high as 195x on favorably-large instances.
Conditional gradients (CG) constitute a class of projection-free first-order algorithms for smooth convex optimization. As such, they are frequently used in solving smooth convex optimization problems over polytopes, for which the computational cost of projections is prohibitive. We will discuss two topics in this talk. First of all, CG algorithms do not enjoy the optimal convergence rates achieved by projection-based first-order accelerated methods; achieving such globally-accelerated rates is information-theoretically impossible. To address this issue, we present Locally Accelerated Conditional Gradients – an algorithmic framework that couples accelerated steps with conditional gradient steps to achieve local acceleration on smooth strongly convex problems. Our approach does not require projections onto the feasible set, but only on (typically low-dimensional) simplices, thus keeping the computational cost of projections at bay. Further, it achieves optimal accelerated local convergence.
The second topic in this talk deals with the setting where first and second-order information about the function is costly to compute, and so we would like to make as much primal progress as possible per oracle call. Constrained second-order convex optimization algorithms are the method of choice when a high accuracy solution to a problem is needed, due to their local quadratic convergence. These algorithms require the solution of a constrained quadratic subproblem at every iteration. In this setting, we present the Second-Order Conditional Gradient Sliding (SOCGS) algorithm, which uses a projection-free algorithm to solve the constrained quadratic subproblems inexactly and uses inexact Hessian oracles (subject to an accuracy requirement). When the feasible region is a polytope, and a lower bound on the primal gap is known, the algorithm converges quadratically in primal gap after a finite number of linearly convergent iterations. Once in the quadratic regime the SOCGS algorithm requires O(log log (1/ε)) first-order and inexact Hessian oracle calls and O(log (1 / ε) ⋅ log log (1/ε)) linear minimization oracle calls to achieve an ε-optimal solution.
We consider a general class of binary packing problems with multiple convex quadratic knapsack constraints. We present three constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation; (2) a greedy strategy; and (3) a randomized rounding method. The practical performance is tested by a computational study of the empirical approximation of these algorithms for problem instances arising in the context of real-world gas transport networks.
The learning of relations, dynamics, and governing equations from (sampled) data is a prevalent problem in various scientific fields. In the absence of noise, this task can often be solved using the Buchberger-Möller Algorithm, see Möller and Buchberger (1982). In general, however, when we do not have direct access to aforementioned equations, we have access to them indirectly only via noisy sampling. Known methods which extract relations from noisy data are the Approximate Vanishing Ideal Algorithm (AVI) and Vanishing Component Analysis (VCA), see Heldt et al. (2009) and Livni et al. (2013), respectively. These algorithms are often computationally inefficient and only allow for restricted recovery of relations from data. We present a new AVI algorithm and two variants utilizing Conditional Gradient Descent or the Frank-Wolfe Algorithm, see, e.g., Frank and Wolfe (1956) or Levitin and Polyak (1966). Our method is more general, easily regularizable, and can also extract non-homogeneous relations. We then present the efficacy and versatility of our approach for three learning problems: reconstruction of governing equations from data, classification, and regression.
This talk is a basic overview over my PhD projects. I focus on a project on making neural networks interpretable. If a neural network outputs a classification result for a high-dimensional input, we would like to know whether a small set of the input parameters was already decisive for the classification. This problem is formalised by introducing a probabilistic variant of prime implicants. We show that this problem is hard for the class NPPP and remains NP-hard for any non-trivial approximation. We present a heuristic solution strategy for real-world applications that is build on Assumed Density Filtering: A technique to propagate multivariate normal distributions through neural networks. To justify the use of this heuristic we show that in fact no sensible family of distributions can be invariant under ReLu-neural network layers.
In Computer Vision, algorithms are now capable of “intelligent” automatic processing of the images. And these possibilities have been recently recycled in many creative processes by A.I. artists, which outputs sometimes reached hammer prizes in art auctions. This emergence of A.I. art looks like an artification, i.e., a societal process by which a practice becomes commonly acknowledged as an art form. Art history abounds with such examples, e.g., the transitioning from craftmanship to solo painters in the Renaissance.
Hence, rather than assessing the artist’s narratives, this point of view allows understanding A.I. art as a figurehead of the broader ecosystem of the creative industry. The latter indeed highly benefited from recent advances in Computer Vision.
We discuss here the possible intertwinements between A.I. art, the use of Computer Vision in the next generation of creative software, the creative industry, and broader societal consequences.
This is ongoing work with Louis Thiry and Léa Saint-Raymond at Ecole Normale Supérieure in Paris.
During the COGA Retreat 2019 we started research on the Symmetric Rendezvous-on-the-Line-Problem with Unknown Initial Distance: Two robots are placed on the real line with unknown initial distance. They do not know the location of the other robot and do not necessarily share the same sense of “left” and “right” on the line. They move with constant speed over the line and try to achieve rendez-vous using the same (randomized) search strategy in as little time as possible.
In a March 2020 talk (given in the COGA Research Seminar) we presented a family of search strategies that performed well in simulations without being able to determine their exact competitive ratio. During this talk we will present a novel family of search strategies whose precise competitive ratio can be computed using generating functions.
This is WIP by Guillaume, Khai Van, Martin, and Max.
This talk deals with an open problem concerning the expressivity of feedforward neural networks (NNs) with rectified linear unit (ReLU) activations. While it has been shown that every continuous, piecewise linear (CPWL) function can be exactly represented by such an NN with logarithmic depth (with respect to the input dimension), it is still unknown whether there exists any CPWL function that cannot be represented with three layers. Based on a yet unproven theoretical assumption, we show a computer-aided proof of non-existence of a 3-layer ReLU NN that computes the maximum of five scalar input numbers. We achieve this partial result by converting the existence question into a linear programming feasibility problem. This is joint work in progress with Amitabh Basu, Marco Di Summa, and Martin Skutella.
The Frank-Wolfe algorithm has become a popular constrained optimization algorithm for it is simple and projection-free, and it has been successfully applied to a variety of real-world problems. Its main drawback however lies in its convergence rate, which can be excessively slow due to naive descent directions. In this talk, we provide a brief review and propose two methods significantly improving its performance: one for minimizing a convex objective and one for minimizing a (non)convex finite-sum objective. The first method speeds up the Frank-Wolfe algorithm by better aligning the descent direction with that of the negative gradient via a matching pursuit-style subroutine. Although the idea is reasonably natural, its advantage over the state-of-the-art Frank-Wolfe algorithms is quite considerable. The second method blends stochastic Frank-Wolfe algorithms with adaptive gradients, which set entry-wise step-sizes that automatically adjust to the geometry of the problem. Computational experiments on convex and nonconvex objectives demonstrate the advantage of our approach.