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Various model-based diagnosis scenarios require the computation of most preferred fault explanations. Existing algorithms that are sound (i.e., output only actual fault explanations) and complete (i.e., can return all explanations), however, require exponential space to achieve this task. As a remedy, we propose two novel diagnostic search algorithms, called RBF-HS (Recursive Best-First Hitting Set Search) and HBF-HS (Hybrid Best-First Hitting Set Search), which build upon tried and tested techniques from the heuristic search domain. RBF-HS can enumerate an arbitrary predefined finite number of fault explanations in best-first order within linear space bounds, without sacrificing the desirable soundness or completeness properties. The idea of HBF-HS is to find a trade-off between runtime optimization and a restricted space consumption that does not exceed the available memory. In extensive experiments on real-world diagnosis cases we compared our approaches to Reiter's HS-Tree, a state-of-the-art method that gives the same theoretical guarantees and is as general(ly applicable) as the suggested algorithms. For the computation of minimum-cardinality fault explanations, we find that (1) RBF-HS reduces memory requirements substantially in most cases by up to several orders of magnitude, (2) in more than a third of the cases, both memory savings and runtime savings are achieved, and (3) given the runtime overhead is significant, using HBF-HS instead of RBF-HS reduces the runtime to values comparable with HS-Tree while keeping the used memory reasonably bounded. When computing most probable fault explanations, we observe that RBF-HS tends to trade memory savings more or less one-to-one for runtime overheads. Again, HBF-HS proves to be a reasonable remedy to cut down the runtime while complying with practicable memory bounds.

Collective intelligence is a fundamental trait shared by several species of living organisms. It has allowed them to thrive in the diverse environmental conditions that exist on our planet. From simple organisations in an ant colony to complex systems in human groups, collective intelligence is vital for solving complex survival tasks. As is commonly observed, such natural systems are flexible to changes in their structure. Specifically, they exhibit a high degree of generalization when the abilities or the total number of agents changes within a system. We term this phenomenon as Combinatorial Generalization (CG). CG is a highly desirable trait for autonomous systems as it can increase their utility and deployability across a wide range of applications. While recent works addressing specific aspects of CG have shown impressive results on complex domains, they provide no performance guarantees when generalizing towards novel situations. In this work, we shed light on the theoretical underpinnings of CG for cooperative multi-agent systems (MAS). Specifically, we study generalization bounds under a linear dependence of the underlying dynamics on the agent capabilities, which can be seen as a generalization of Successor Features to MAS. We then extend the results first for Lipschitz and then arbitrary dependence of rewards on team capabilities. Finally, empirical analysis on various domains using the framework of multi-agent reinforcement learning highlights important desiderata for multi-agent algorithms towards ensuring CG.

Inspired by a mathematical riddle involving fuses, we define the "fusible numbers" as follows: $0$ is fusible, and whenever $x,y$ are fusible with $|y-x|<1$, the number $(x+y+1)/2$ is also fusible. We prove that the set of fusible numbers, ordered by the usual order on $\mathbb R$, is well-ordered, with order type $\varepsilon_0$. Furthermore, we prove that the density of the fusible numbers along the real line grows at an incredibly fast rate: Letting $g(n)$ be the largest gap between consecutive fusible numbers in the interval $[n,\infty)$, we have $g(n)^{-1} \ge F_{\varepsilon_0}(n-c)$ for some constant $c$, where $F_\alpha$ denotes the fast-growing hierarchy. Finally, we derive some true statements that can be formulated but not proven in Peano Arithmetic, of a different flavor than previously known such statements: PA cannot prove the true statement "For every natural number $n$ there exists a smallest fusible number larger than $n$." Also, consider the algorithm "$M(x)$: if $x<0$ return $-x$, else return $M(x-M(x-1))/2$." Then $M$ terminates on real inputs, although PA cannot prove the statement "$M$ terminates on all natural inputs."

In this paper, we propose a direct parallel-in-time (PinT) algorithm for time-dependent problems with first- or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix $V$ grows as $n$ is increased, where $n$ is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing $V$ and $V^{-1}$, by which we prove that $\mathrm{Cond}_2(V)=\mathcal{O}(n^{2})$. This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as $n$ grows and thus, compared to other direct PinT algorithms, a much larger $n$ can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.

This paper characterizes the inherent power of evolutionary algorithms. This power depends on the computational properties of the genetic encoding. With some encodings, two parents recombined with a simple crossover operator can sample from an arbitrary distribution of child phenotypes. Such encodings are termed \emph{expressive encodings} in this paper. Universal function approximators, including popular evolutionary substrates of genetic programming and neural networks, can be used to construct expressive encodings. Remarkably, this approach need not be applied only to domains where the phenotype is a function: Expressivity can be achieved even when optimizing static structures, such as binary vectors. Such simpler settings make it possible to characterize expressive encodings theoretically: Across a variety of test problems, expressive encodings are shown to achieve up to super-exponential convergence speed-ups over the standard direct encoding. The conclusion is that, across evolutionary computation areas as diverse as genetic programming, neuroevolution, genetic algorithms, and theory, expressive encodings can be a key to understanding and realizing the full power of evolution.

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $X\xrightarrow{f} Y \xrightarrow{g} Z$ such that $M\cong \ker{g}/\mathrm{im}{f}$. It runs in time $O(|X|^3+|Y|^3+|Z|^3)$ and requires $O(|X|^2+|Y|^2+|Z|^2)$ memory, where $|\cdot |$ denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gr\"obner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

We study the problem of selling information to a data-buyer who faces a decision problem under uncertainty. We consider the classic Bayesian decision-theoretic model pioneered by [Blackwell, 1951, 1953]. Initially, the data buyer has only partial information about the payoff-relevant state of the world. A data seller offers additional information about the state of the world. The information is revealed through signaling schemes, also referred to as experiments. In the single-agent setting, any mechanism can be represented as a menu of experiments. [Bergemann et al., 2018] present a complete characterization of the revenue-optimal mechanism in a binary state and binary action environment. By contrast, no characterization is known for the case with more actions. In this paper, we consider more general environments and study arguably the simplest mechanism, which only sells the fully informative experiment. In the environment with binary state and $m\geq 3$ actions, we provide an $O(m)$-approximation to the optimal revenue by selling only the fully informative experiment and show that the approximation ratio is tight up to an absolute constant factor. An important corollary of our lower bound is that the size of the optimal menu must grow at least linearly in the number of available actions, so no universal upper bound exists for the size of the optimal menu in the general single-dimensional setting. For multi-dimensional environments, we prove that even in arguably the simplest matching utility environment with 3 states and 3 actions, the ratio between the optimal revenue and the revenue by selling only the fully informative experiment can grow immediately to a polynomial of the number of agent types. Nonetheless, if the distribution is uniform, we show that selling only the fully informative experiment is indeed the optimal mechanism.

Given a graph function, defined on an arbitrary set of edge weights and node features, does there exist a Graph Neural Network (GNN) whose output is identical to the graph function? In this paper, we fully answer this question and characterize the class of graph problems that can be represented by GNNs. We identify an algebraic condition, in terms of the permutation of edge weights and node features, which proves to be necessary and sufficient for a graph problem to lie within the reach of GNNs. Moreover, we show that this condition can be efficiently verified by checking quadratically many constraints. Note that our refined characterization on the expressive power of GNNs are orthogonal to those theoretical results showing equivalence between GNNs and Weisfeiler-Lehman graph isomorphism heuristic. For instance, our characterization implies that many natural graph problems, such as min-cut value, max-flow value, and max-clique size, can be represented by a GNN. In contrast, and rather surprisingly, there exist very simple graphs for which no GNN can correctly find the length of the shortest paths between all nodes. Note that finding shortest paths is one of the most classical problems in Dynamic Programming (DP). Thus, the aforementioned negative example highlights the misalignment between DP and GNN, even though (conceptually) they follow very similar iterative procedures. Finally, we support our theoretical results by experimental simulations.

We introduce the $\texttt{$k$-experts}$ problem - a generalization of the classic Prediction with Expert's Advice framework. Unlike the classic version, where the learner selects exactly one expert from a pool of $N$ experts at each round, in this problem, the learner can select a subset of $k$ experts at each round $(1\leq k\leq N)$. The reward obtained by the learner at each round is assumed to be a function of the $k$ selected experts. The primary objective is to design an online learning policy with a small regret. In this pursuit, we propose $\texttt{SAGE}$ ($\textbf{Sa}$mpled Hed$\textbf{ge}$) - a framework for designing efficient online learning policies by leveraging statistical sampling techniques. For a wide class of reward functions, we show that $\texttt{SAGE}$ either achieves the first sublinear regret guarantee or improves upon the existing ones. Furthermore, going beyond the notion of regret, we fully characterize the mistake bounds achievable by online learning policies for stable loss functions. We conclude the paper by establishing a tight regret lower bound for a variant of the $\texttt{$k$-experts}$ problem and carrying out experiments with standard datasets.

Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}(\log n/\epsilon^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/\epsilon^{2})$ or $\tilde{O}((\log n)^{6}/\epsilon^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/\epsilon$. For the stochastic setting, our algorithm outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}((\log n)^{2}/\epsilon^{4})$ iterations. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients, which can find negative curvature near saddle points and achieve the polynomial speedup in $\log n$ compared to the perturbed gradient descent methods. Finally, we also perform numerical experiments that support our results.