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We present a novel sampling-based method for estimating probabilities of rare or failure events. Our approach is founded on the Ensemble Kalman filter (EnKF) for inverse problems. Therefore, we reformulate the rare event problem as an inverse problem and apply the EnKF to generate failure samples. To estimate the probability of failure, we use the final EnKF samples to fit a distribution model and apply Importance Sampling with respect to the fitted distribution. This leads to an unbiased estimator if the density of the fitted distribution admits positive values within the whole failure domain. To handle multi-modal failure domains, we localise the covariance matrices in the EnKF update step around each particle and fit a mixture distribution model in the Importance Sampling step. For affine linear limit-state functions, we investigate the continuous-time limit and large time properties of the EnKF update. We prove that the mean of the particles converges to a convex combination of the most likely failure point and the mean of the optimal Importance Sampling density if the EnKF is applied without noise. We provide numerical experiments to compare the performance of the EnKF with Sequential Importance Sampling.

We present the first algorithm for fully dynamic $k$-centers clustering in an arbitrary metric space that maintains an optimal $2+\epsilon$ approximation in $O(k \cdot \operatorname{polylog}(n,\Delta))$ amortized update time. Here, $n$ is an upper bound on the number of active points at any time, and $\Delta$ is the aspect ratio of the data. Previously, the best known amortized update time was $O(k^2\cdot \operatorname{polylog}(n,\Delta))$, and is due to Chan, Gourqin, and Sozio. We demonstrate that the runtime of our algorithm is optimal up to $\operatorname{polylog}(n,\Delta)$ factors, even for insertion-only streams, which closes the complexity of fully dynamic $k$-centers clustering. In particular, we prove that any algorithm for $k$-clustering tasks in arbitrary metric spaces, including $k$-means, $k$-medians, and $k$-centers, must make at least $\Omega(n k)$ distance queries to achieve any non-trivial approximation factor. Despite the lower bound for arbitrary metrics, we demonstrate that an update time sublinear in $k$ is possible for metric spaces which admit locally sensitive hash functions (LSH). Namely, we demonstrate a black-box transformation which takes a locally sensitive hash family for a metric space and produces a faster fully dynamic $k$-centers algorithm for that space. In particular, for a large class of metrics including Euclidean space, $\ell_p$ spaces, the Hamming Metric, and the Jaccard Metric, for any $c > 1$, our results yield a $c(4+\epsilon)$ approximate $k$-centers solution in $O(n^{1/c} \cdot \operatorname{polylog}(n,\Delta))$ amortized update time, simultaneously for all $k \geq 1$. Previously, the only known comparable result was a $O(c \log n)$ approximation for Euclidean space due to Schmidt and Sohler, running in the same amortized update time.

Multi-armed bandits (MAB) are extensively studied in various settings where the objective is to \textit{maximize} the actions' outcomes (i.e., rewards) over time. Since safety is crucial in many real-world problems, safe versions of MAB algorithms have also garnered considerable interest. In this work, we tackle a different critical task through the lens of \textit{linear stochastic bandits}, where the aim is to keep the actions' outcomes close to a target level while respecting a \textit{two-sided} safety constraint, which we call \textit{leveling}. Such a task is prevalent in numerous domains. Many healthcare problems, for instance, require keeping a physiological variable in a range and preferably close to a target level. The radical change in our objective necessitates a new acquisition strategy, which is at the heart of a MAB algorithm. We propose SALE-LTS: Safe Leveling via Linear Thompson Sampling algorithm, with a novel acquisition strategy to accommodate our task and show that it achieves sublinear regret with the same time and dimension dependence as previous works on the classical reward maximization problem absent any safety constraint. We demonstrate and discuss our algorithm's empirical performance in detail via thorough experiments.

In a widely-studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. Then, the goal is to provide an optimal solution set, i.e., a set containing an optimal solution for each non-parametric problem obtained by fixing a parameter vector. For many multi-parametric optimization problems, however, an optimal solution set of minimum cardinality can contain super-polynomially many solutions. Consequently, no polynomial-time exact algorithms can exist for these problems even if $\textsf{P}=\textsf{NP}$. We propose an approximation method that is applicable to a general class of multi-parametric optimization problems and outputs a set of solutions with cardinality polynomial in the instance size and the inverse of the approximation guarantee. This method lifts approximation algorithms for non-parametric optimization problems to their parametric version and provides an approximation guarantee that is arbitrarily close to the approximation guarantee of the approximation algorithm for the non-parametric problem. If the non-parametric problem can be solved exactly in polynomial time or if an FPTAS is available, our algorithm is an FPTAS. Further, we show that, for any given approximation guarantee, the minimum cardinality of an approximation set is, in general, not $\ell$-approximable for any natural number $\ell$ less or equal to the number of parameters, and we discuss applications of our results to classical multi-parametric combinatorial optimizations problems. In particular, we obtain an FPTAS for the multi-parametric minimum $s$-$t$-cut problem, an FPTAS for the multi-parametric knapsack problem, as well as an approximation algorithm for the multi-parametric maximization of independence systems problem.

Temporal graphs have been recently introduced to model changes to a given network that occur throughout a fixed period of time. We introduce and investigate the Temporal $\Delta$ Independent Set problem, a temporal variant of the well known Independent Set problem. This problem is e.g. motivated in the context of finding conflict-free schedules for maximum subsets of tasks, that have certain (changing) constraints on each day they need to be performed. We are specifically interested in the case where each task needs to be performed in a certain time-interval on each day and two tasks are in conflict on a day if their time-intervals overlap on that day. This leads us to considering Temporal $\Delta$ Independent Set on the restricted class of temporal unit interval graphs, i.e. temporal graphs where each layer is unit interval. We present several hardness results for this problem, as well as two algorithms: The first is an constant-factor approximation algorithm for instances where $\tau$, the total number of time steps (layers) of the temporal graph, and $\Delta$, a parameter that allows us to model some tolerance in the conflicts, are constants. For the second result we use the notion of order preservation for temporal unit interval graphs that, informally, requires the intervals of every layer to obey a common ordering. We provide an FPT algorithm parameterized by the size of minimum vertex deletion set to order preservation.

Real-world data often comes in compressed form. Analyzing compressed data directly (without decompressing it) can save space and time by orders of magnitude. In this work, we focus on fundamental sequence comparison problems and try to quantify the gain in time complexity when the underlying data is highly compressible. We consider grammar compression, which unifies many practically relevant compression schemes. For two strings of total length $N$ and total compressed size $n$, it is known that the edit distance and a longest common subsequence (LCS) can be computed exactly in time $\tilde{O}(nN)$, as opposed to $O(N^2)$ for the uncompressed setting. Many applications need to align multiple sequences simultaneously, and the fastest known exact algorithms for median edit distance and LCS of $k$ strings run in $O(N^k)$ time. This naturally raises the question of whether compression can help to reduce the running time significantly for $k \geq 3$, perhaps to $O(N^{k/2}n^{k/2})$ or $O(Nn^{k-1})$. Unfortunately, we show lower bounds that rule out any improvement beyond $\Omega(N^{k-1}n)$ time for any of these problems assuming the Strong Exponential Time Hypothesis. At the same time, we show that approximation and compression together can be surprisingly effective. We develop an $\tilde{O}(N^{k/2}n^{k/2})$-time FPTAS for the median edit distance of $k$ sequences. In comparison, no $O(N^{k-\Omega(1)})$-time PTAS is known for the median edit distance problem in the uncompressed setting. For two strings, we get an $\tilde{O}(N^{2/3}n^{4/3})$-time FPTAS for both edit distance and LCS. In contrast, for uncompressed strings, there is not even a subquadratic algorithm for LCS that has less than a polynomial gap in the approximation factor. Building on the insight from our approximation algorithms, we also obtain results for many distance measures including the edit, Hamming, and shift distances.

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(\Delta + \log^* n)$ communication rounds; here $n$ is the number of nodes and $\Delta$ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on $n$ is optimal: these problems cannot be solved in $o(\log^* n)$ rounds even if $\Delta = 2$. However, the dependency on $\Delta$ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least $1-1/n$ requires $\Omega(\min\{\Delta,\log \log n / \log \log \log n\})$ rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires $\Omega(\min\{\Delta, \log n / \log \log n\})$ rounds; this is an improvement over prior lower bounds also as a function of $n$.

We study the problem of how to construct a set of policies that can be composed together to solve a collection of reinforcement learning tasks. Each task is a different reward function defined as a linear combination of known features. We consider a specific class of policy compositions which we call set improving policies (SIPs): given a set of policies and a set of tasks, a SIP is any composition of the former whose performance is at least as good as that of its constituents across all the tasks. We focus on the most conservative instantiation of SIPs, set-max policies (SMPs), so our analysis extends to any SIP. This includes known policy-composition operators like generalized policy improvement. Our main contribution is a policy iteration algorithm that builds a set of policies in order to maximize the worst-case performance of the resulting SMP on the set of tasks. The algorithm works by successively adding new policies to the set. We show that the worst-case performance of the resulting SMP strictly improves at each iteration, and the algorithm only stops when there does not exist a policy that leads to improved performance. We empirically evaluate our algorithm on a grid world and also on a set of domains from the DeepMind control suite. We confirm our theoretical results regarding the monotonically improving performance of our algorithm. Interestingly, we also show empirically that the sets of policies computed by the algorithm are diverse, leading to different trajectories in the grid world and very distinct locomotion skills in the control suite.

We address the problem of model selection for the finite horizon episodic Reinforcement Learning (RL) problem where the transition kernel $P^*$ belongs to a family of models $\mathcal{P}^*$ with finite metric entropy. In the model selection framework, instead of $\mathcal{P}^*$, we are given $M$ nested families of transition kernels $\cP_1 \subset \cP_2 \subset \ldots \subset \cP_M$. We propose and analyze a novel algorithm, namely \emph{Adaptive Reinforcement Learning (General)} (\texttt{ARL-GEN}) that adapts to the smallest such family where the true transition kernel $P^*$ lies. \texttt{ARL-GEN} uses the Upper Confidence Reinforcement Learning (\texttt{UCRL}) algorithm with value targeted regression as a blackbox and puts a model selection module at the beginning of each epoch. Under a mild separability assumption on the model classes, we show that \texttt{ARL-GEN} obtains a regret of $\Tilde{\mathcal{O}}(d_{\mathcal{E}}^*H^2+\sqrt{d_{\mathcal{E}}^* \mathbb{M}^* H^2 T})$, with high probability, where $H$ is the horizon length, $T$ is the total number of steps, $d_{\mathcal{E}}^*$ is the Eluder dimension and $\mathbb{M}^*$ is the metric entropy corresponding to $\mathcal{P}^*$. Note that this regret scaling matches that of an oracle that knows $\mathcal{P}^*$ in advance. We show that the cost of model selection for \texttt{ARL-GEN} is an additive term in the regret having a weak dependence on $T$. Subsequently, we remove the separability assumption and consider the setup of linear mixture MDPs, where the transition kernel $P^*$ has a linear function approximation. With this low rank structure, we propose novel adaptive algorithms for model selection, and obtain (order-wise) regret identical to that of an oracle with knowledge of the true model class.

Graph Convolutional Networks (GCNs) have recently become the primary choice for learning from graph-structured data, superseding hash fingerprints in representing chemical compounds. However, GCNs lack the ability to take into account the ordering of node neighbors, even when there is a geometric interpretation of the graph vertices that provides an order based on their spatial positions. To remedy this issue, we propose Geometric Graph Convolutional Network (geo-GCN) which uses spatial features to efficiently learn from graphs that can be naturally located in space. Our contribution is threefold: we propose a GCN-inspired architecture which (i) leverages node positions, (ii) is a proper generalisation of both GCNs and Convolutional Neural Networks (CNNs), (iii) benefits from augmentation which further improves the performance and assures invariance with respect to the desired properties. Empirically, geo-GCN outperforms state-of-the-art graph-based methods on image classification and chemical tasks.