arXiv subjects

Karen Seidel

Publications and source records attributed to Karen Seidel.

6 recordsLinked to original sources

arXiv2022

What's Wrong with Deep Learning in Tree Search for Combinatorial Optimization

Combinatorial optimization lies at the core of many real-world problems. Especially since the rise of graph neural networks (GNNs), the deep learning community has been developing solvers that derive solutions to NP-hard problems by learning the problem-specific solution structure. However, reproducing the results of these publications proves to be difficult. We make three contributions. First, we present an open-source benchmark suite for the NP-hard Maximum Independent Set problem, in both its weighted and unweighted variants. The suite offers a unified interface to various state-of-the-art traditional and machine learning-based solvers. Second, using our benchmark suite, we conduct an in-depth analysis of the popular guided tree search algorithm by Li et al. [NeurIPS 2018], testing various configurations on small and large synthetic and real-world graphs. By re-implementing their algorithm with a focus on code quality and extensibility, we show that the graph convolution network used in the tree search does not learn a meaningful representation of the solution structure, and can in fact be replaced by random values. Instead, the tree search relies on algorithmic techniques like graph kernelization to find good solutions. Thus, the results from the original publication are not reproducible. Third, we extend the analysis to compare the tree search implementations to other solvers, showing that the classical algorithmic solvers often are faster, while providing solutions of similar quality. Additionally, we analyze a recent solver based on reinforcement learning and observe that for this solver, the GNN is responsible for the competitive solution quality.

cs.LG↗

arXiv2021

Learning from Informants: Relations between Learning Success Criteria

Learning from positive and negative information, so-called \emph{informants}, being one of the models for human and machine learning introduced by E.~M.~Gold, is investigated. Particularly, naturally arising questions about this learning setting, originating in results on learning from solely positive information, are answered. By a carefully arranged argument learners can be assumed to only change their hypothesis in case it is inconsistent with the data (such a learning behavior is called \emph{conservative}). The deduced main theorem states the relations between the most important delayable learning success criteria, being the ones not ruined by a delayed in time hypothesis output. Additionally, our investigations concerning the non-delayable requirement of consistent learning underpin the claim for \emph{delayability} being the right structural property to gain a deeper understanding concerning the nature of learning success criteria. Moreover, we obtain an anomalous \emph{hierarchy} when allowing for an increasing finite number of \emph{anomalies} of the hypothesized language by the learner compared with the language to be learned. In contrast to the vacillatory hierarchy for learning from solely positive information, we observe a \emph{duality} depending on whether infinitely many \emph{vacillations} between different (almost) correct hypotheses are still considered a successful learning behavior.

cs.FL↗

arXiv2021

Learning Languages in the Limit from Positive Information with Finitely Many Memory Changes

We investigate learning collections of languages from texts by an inductive inference machine with access to the current datum and a bounded memory in form of states. Such a bounded memory states (BMS) learner is considered successful in case it eventually settles on a correct hypothesis while exploiting only finitely many different states. We give the complete map of all pairwise relations for an established collection of criteria of successfull learning. Most prominently, we show that non-U-shapedness is not restrictive, while conservativeness and (strong) monotonicity are. Some results carry over from iterative learning by a general lemma showing that, for a wealth of restrictions (the semantic restrictions), iterative and bounded memory states learning are equivalent. We also give an example of a non-semantic restriction (strongly non-U-shapedness) where the two settings differ.

cs.FL↗

arXiv2021

Learning Half-Spaces and other Concept Classes in the Limit with Iterative Learners

In order to model an efficient learning paradigm, iterative learning algorithms access data one by one, updating the current hypothesis without regress to past data. Past research on iterative learning analyzed for example many important additional requirements and their impact on iterative learners. In this paper, our results are twofold. First, we analyze the relative learning power of various settings of iterative learning, including learning from text and from informant, as well as various further restrictions, for example we show that strongly non-U-shaped learning is restrictive for iterative learning from informant. Second, we investigate the learnability of the concept class of half-spaces and provide a constructive iterative algorithm to learn the set of half-spaces from informant.

cs.LG↗

arXiv2019

Random Subgroups of Rationals

This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup $(G,+)$ of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of $(G,+)$; second, what learnability properties can one extract from $G$ and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of $(G,+)$ coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for $G$ with respect to any generating sequence for $G$ is not even semi-decidable, one can build a generating sequence $β$ such that the word problem for $G$ with respect to $β$ is co-recursively enumerable (assuming that the set of generators of $G$ is limit-recursive). In regard to the second question, it is proven that there is a generating sequence $β$ for $G$ such that every non-trivial finitely generated subgroup of $G$ is recursively enumerable and the class of all such subgroups of $G$ is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of $G$ is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of $G$ cannot be syntactically identified in the limit with respect to any generating sequence for $G$. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.

cs.LO↗

arXiv2018

Counting Homomorphisms to Trees Modulo a Prime

Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular, independent sets and colourings. In this article we study the complexity of $\#_p\mathsf{H}$OMS$\mathsf{T}$O$H$, the problem of counting graph homomorphisms from an input graph to a graph $H$ modulo a prime number $p$. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the input graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. However, in subsequent studies on counting modulo $2$ the influence, the structure of $H$ has on the tractability, was shown to persist, yielding similar dichotomies. Our main result shows that for every tree $H$ and every prime $p$ the problem $\#_p\mathsf{H}$OMS$\mathsf{T}$O$H$ is either polynomial time computable or $\#_p\mathsf{P}$-complete. This addresses the conjecture of Faben and Jerrum stating this dichotomy for every graph $H$ when counting modulo 2. In order to prove this result, we study the structural properties of a homomorphism. As an important interim, this study yields a dichotomy for the problem of weighted counting independent sets in a bipartite graph modulo some prime $p$. Our results are the first suggesting that such dichotomies hold not only for the one-bit functions of the modulo~2 case but for the modular counting functions of all primes $p$.

cs.CC↗