In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. posed this question for odd cycle transversal and feedback vertex set. We answer it for these two and four further problems, namely connected vertex cover, connected domintaing set, steiner tree, and connected odd cycle transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algorithms than relative to treewidth and prove matching lower bounds. For upper bounds we first extend the idea of Groenland et al.~[STACS~2022] to solve what we call coloring-like problem. Such problems are defined by a symmetric matrix $M$ over $\mathbb{F}_2$ indexed by a set of colors. The goal is to count the number (modulo some prime $p$) of colorings of a graph such that $M$ has a $1$-entry if indexed by the colors of the end-points of any edge. We show that this problem can be solved faster if $M$ has small rank over $\mathbb{F}_p$. We apply this result to get our upper bounds for connected vertex cover and connected dominating set. The upper bounds for odd cycle transversal and feedback vertex set use a subdivision trick to get below the bounds that matrix rank would yield.
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[Alecu et al.: Graph functionality, JCTB2021] define functionality, a graph parameter that generalizes graph degeneracy. They research the relation of functionality to many other graph parameters (tree-width, clique-width, VC-dimension, etc.). Extending their research, we prove logarithmic lower bound for functionality of random graph $G(n,p)$ for large range of $p$. Previously known graphs have functionality logarithmic in number of vertices. We show that for every graph $G$ on $n$ vertices we have $\mathrm{fun}(G) \le O(\sqrt{ n \log n})$ and we give a nearly matching $\Omega(\sqrt{n})$-lower bound provided by projective planes. Further, we study a related graph parameter \emph{symmetric difference}, the minimum of $|N(u) \Delta N(v)|$ over all pairs of vertices of the ``worst possible'' induced subgraph. It was observed by Alecu et al. that $\mathrm{fun}(G) \le \mathrm{sd}(G)+1$ for every graph $G$. We compare $\mathrm{fun}$ and $\mathrm{sd}$ for the class $\mathrm{INT}$ of interval graphs and $\mathrm{CA}$ of circular-arc graphs. We let $\mathrm{INT}_n$ denote the $n$-vertex interval graphs, similarly for $\mathrm{CA}_n$. Alecu et al. ask, whether $\mathrm{fun}(\mathrm{INT})$ is bounded. Dallard et al. answer this positively in a recent preprint. On the other hand, we show that $\Omega(\sqrt[4]{n}) \leq \mathrm{sd}(\mathrm{INT}_n) \leq O(\sqrt[3]{n})$. For the related class $\mathrm{CA}$ we show that $\mathrm{sd}(\mathrm{CA}_n) = \Theta(\sqrt{n})$. We propose a follow-up question: is $\mathrm{fun}(\mathrm{CA})$ bounded?
Branching processes are a class of continuous-time Markov chains (CTMCs) prevalent for modeling stochastic population dynamics in ecology, biology, epidemiology, and many other fields. The transient or finite-time behavior of these systems is fully characterized by their transition probabilities. However, computing them requires marginalizing over all paths between endpoint-conditioned values, which often poses a computational bottleneck. Leveraging recent results that connect generating function methods to a compressed sensing framework, we recast this task from the lens of sparse optimization. We propose a new solution method using variable splitting; in particular, we derive closed form updates in a highly efficient ADMM algorithm. Notably, no matrix products -- let alone inversions -- are required at any step. This reduces computational cost by orders of magnitude over existing methods, and the resulting algorithm is easily parallelizable and fairly insensitive to tuning parameters. A comparison to prior work is carried out in two applications to models of blood cell production and transposon evolution, showing that the proposed method is orders of magnitudes more scalable than existing work.
We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The query points are given by a zero-dimensional parametrization. We design an algorithm which counts the number of connected components of the real curve under study, and decides which query point lie in which connected component, in time log-linear in $N^6$, where $N$ is the maximum of the degrees and coefficient bit-sizes of the polynomials given as input. This matches the currently best-known bound for computing the topology of real plane curves. The main novelty of this algorithm is the avoidance of the computation of the complete topology of the curve.
In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks. The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and neural networks (NNs) act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias / approximation properties of NNs. We derive \textsl{a priori} error estimates in the standard $L^2(\Omega)$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g., spatial mesh size, time step size, and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation.
We study the tractability of the maximum independent set problem from the viewpoint of graph width parameters, with the goal of defining a width parameter that is as general as possible and allows to solve independent set in polynomial-time on graphs where the parameter is bounded. We introduce two new graph width parameters: one-sided maximum induced matching-width (o-mim-width) and neighbor-depth. O-mim-width is a graph parameter that is more general than the known parameters mim-width and tree-independence number, and we show that independent set and feedback vertex set can be solved in polynomial-time given a decomposition with bounded o-mim-width. O-mim-width is the first width parameter that gives a common generalization of chordal graphs and graphs of bounded clique-width in terms of tractability of these problems. The parameter o-mim-width, as well as the related parameters mim-width and sim-width, have the limitation that no algorithms are known to compute bounded-width decompositions in polynomial-time. To partially resolve this limitation, we introduce the parameter neighbor-depth. We show that given a graph of neighbor-depth $k$, independent set can be solved in time $n^{O(k)}$ even without knowing a corresponding decomposition. We also show that neighbor-depth is bounded by a polylogarithmic function on the number of vertices on large classes of graphs, including graphs of bounded o-mim-width, and more generally graphs of bounded sim-width, giving a quasipolynomial-time algorithm for independent set on these graph classes. This resolves an open problem asked by Kang, Kwon, Str{\o}mme, and Telle [TCS 2017].
Deep networks are well-known to be fragile to adversarial attacks, and adversarial training is one of the most popular methods used to train a robust model. To take advantage of unlabeled data, recent works have applied adversarial training to contrastive learning (Adversarial Contrastive Learning; ACL for short) and obtain promising robust performance. However, the theory of ACL is not well understood. To fill this gap, we leverage the Rademacher complexity to analyze the generalization performance of ACL, with a particular focus on linear models and multi-layer neural networks under $\ell_p$ attack ($p \ge 1$). Our theory shows that the average adversarial risk of the downstream tasks can be upper bounded by the adversarial unsupervised risk of the upstream task. The experimental results validate our theory.
Kernel methods are learning algorithms that enjoy solid theoretical foundations while suffering from important computational limitations. Sketching, which consists in looking for solutions among a subspace of reduced dimension, is a well studied approach to alleviate these computational burdens. However, statistically-accurate sketches, such as the Gaussian one, usually contain few null entries, such that their application to kernel methods and their non-sparse Gram matrices remains slow in practice. In this paper, we show that sparsified Gaussian (and Rademacher) sketches still produce theoretically-valid approximations while allowing for important time and space savings thanks to an efficient \emph{decomposition trick}. To support our method, we derive excess risk bounds for both single and multiple output kernel problems, with generic Lipschitz losses, hereby providing new guarantees for a wide range of applications, from robust regression to multiple quantile regression. Our theoretical results are complemented with experiments showing the empirical superiority of our approach over SOTA sketching methods.
Despite their omnipresence in modern NLP, characterizing the computational power of transformer neural nets remains an interesting open question. We prove that transformers whose arithmetic precision is logarithmic in the number of input tokens (and whose feedforward nets are computable using space linear in their input) can be simulated by constant-depth logspace-uniform threshold circuits. This provides insight on the power of transformers using known results in complexity theory. For example, if $\mathsf L \neq \mathsf P$ (i.e., not all poly-time problems can be solved using logarithmic space), then transformers cannot even accurately solve linear equalities or check membership in an arbitrary context-free grammar with empty productions. Our result intuitively emerges from the transformer architecture's high parallelizability. We thus speculatively introduce the idea of a fundamental parallelism tradeoff: any model architecture as parallelizable as the transformer will obey limitations similar to it. Since parallelism is key to training models at massive scale, this suggests a potential inherent weakness of the scaling paradigm.
Recently, several studies (Zhou et al., 2021a; Zhang et al., 2021b; Kim et al., 2021; Zhou and Gu, 2022) have provided variance-dependent regret bounds for linear contextual bandits, which interpolates the regret for the worst-case regime and the deterministic reward regime. However, these algorithms are either computationally intractable or unable to handle unknown variance of the noise. In this paper, we present a novel solution to this open problem by proposing the first computationally efficient algorithm for linear bandits with heteroscedastic noise. Our algorithm is adaptive to the unknown variance of noise and achieves an $\tilde{O}(d \sqrt{\sum_{k = 1}^K \sigma_k^2} + d)$ regret, where $\sigma_k^2$ is the variance of the noise at the round $k$, $d$ is the dimension of the contexts and $K$ is the total number of rounds. Our results are based on an adaptive variance-aware confidence set enabled by a new Freedman-type concentration inequality for self-normalized martingales and a multi-layer structure to stratify the context vectors into different layers with different uniform upper bounds on the uncertainty. Furthermore, our approach can be extended to linear mixture Markov decision processes (MDPs) in reinforcement learning. We propose a variance-adaptive algorithm for linear mixture MDPs, which achieves a problem-dependent horizon-free regret bound that can gracefully reduce to a nearly constant regret for deterministic MDPs. Unlike existing nearly minimax optimal algorithms for linear mixture MDPs, our algorithm does not require explicit variance estimation of the transitional probabilities or the use of high-order moment estimators to attain horizon-free regret. We believe the techniques developed in this paper can have independent value for general online decision making problems.
This work considers the question of how convenient access to copious data impacts our ability to learn causal effects and relations. In what ways is learning causality in the era of big data different from -- or the same as -- the traditional one? To answer this question, this survey provides a comprehensive and structured review of both traditional and frontier methods in learning causality and relations along with the connections between causality and machine learning. This work points out on a case-by-case basis how big data facilitates, complicates, or motivates each approach.
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