We consider the problem of dynamically maintaining the convex hull of a set $S$ of points in the plane under the following special sequence of insertions and deletions (called window-sliding updates): insert a point to the right of all points of $S$ and delete the leftmost point of $S$. We propose an $O(|S|)$-space data structure that can handle each update in $O(1)$ amortized time, such that all standard binary-search-based queries on the convex hull of $S$ can be answered in $O(\log |S|)$ time, and the convex hull itself can be output in time linear in the number of its vertices.
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We study the online variant of the Min-Sum Set Cover (MSSC) problem, a generalization of the well-known list update problem. In the MSSC problem, an algorithm has to maintain the time-varying permutation of the list of $n$ elements, and serve a sequence of requests $R_1, R_2, \dots, R_t, \dots$. Each $R_t$ is a subset of elements of cardinality at most $r$. For a requested set $R_t$, an online algorithm has to pay the cost equal to the position of the first element from $R_t$ on its list. Then, it may arbitrarily permute its list, paying the number of swapped adjacent element pairs. We present the first constructive deterministic algorithm for this problem, whose competitive ratio does not depend on $n$. Our algorithm is $O(r^2)$-competitive, which beats both the existential upper bound of $O(r^4)$ by Bienkowski and Mucha [AAAI '23] and the previous constructive bound of $O(r^{3/2} \cdot \sqrt{n})$ by Fotakis et al. [ICALP '20]. Furthermore, we show that our algorithm attains an asymptotically optimal competitive ratio of $O(r)$ when compared to the best fixed permutation of elements.
In this paper, we present a stochastic gradient algorithm for minimizing a smooth objective function that is an expectation over noisy cost samples, and only the latter are observed for any given parameter. Our algorithm employs a gradient estimation scheme with random perturbations, which are formed using the truncated Cauchy distribution from the delta sphere. We analyze the bias and variance of the proposed gradient estimator. Our algorithm is found to be particularly useful in the case when the objective function is non-convex, and the parameter dimension is high. From an asymptotic convergence analysis, we establish that our algorithm converges almost surely to the set of stationary points of the objective function and obtains the asymptotic convergence rate. We also show that our algorithm avoids unstable equilibria, implying convergence to local minima. Further, we perform a non-asymptotic convergence analysis of our algorithm. In particular, we establish here a non-asymptotic bound for finding an epsilon-stationary point of the non-convex objective function. Finally, we demonstrate numerically through simulations that the performance of our algorithm outperforms GSF, SPSA, and RDSA by a significant margin over a few non-convex settings and further validate its performance over convex (noisy) objectives.
We consider the problem of designing deterministic graph algorithms for the model of Massively Parallel Computation (MPC) that improve with the sparsity of the input graph, as measured by the notion of arboricity. For the problems of maximal independent set (MIS), maximal matching (MM), and vertex coloring, we improve the state of the art as follows. Let $\lambda$ denote the arboricity of the $n$-node input graph with maximum degree $\Delta$. MIS and MM: We develop a deterministic low-space MPC algorithm that reduces the maximum degree to $poly(\lambda)$ in $O(\log \log n)$ rounds, improving and simplifying the randomized $O(\log \log n)$-round $poly(\max(\lambda, \log n))$-degree reduction of Ghaffari, Grunau, Jin [DISC'20]. Our approach when combined with the state-of-the-art $O(\log \Delta + \log \log n)$-round algorithm by Czumaj, Davies, Parter [SPAA'20, TALG'21] leads to an improved deterministic round complexity of $O(\log \lambda + \log \log n)$ for MIS and MM in low-space MPC. We also extend above MIS and MM algorithms to work with linear global memory. Specifically, we show that both problems can be solved in deterministic time $O(\min(\log n, \log \lambda \cdot \log \log n))$, and even in $O(\log \log n)$ time for graphs with arboricity at most $\log^{O(1)} \log n$. In this setting, only a $O(\log^2 \log n)$-running time bound for trees was known due to Latypov and Uitto [ArXiv'21]. Vertex Coloring: We present a $O(1)$-round deterministic algorithm for the problem of $O(\lambda)$-coloring in linear-memory MPC with relaxed global memory of $n \cdot poly(\lambda)$ that solves the problem after just one single graph partitioning step. This matches the state-of-the-art randomized round complexity by Ghaffari and Sayyadi [ICALP'19] and improves upon the deterministic $O(\lambda^{\epsilon})$-round algorithm by Barenboim and Khazanov [CSR'18].
The advancement of new digital image sensors has enabled the design of exposure multiplexing schemes where a single image capture can have multiple exposures and conversion gains in an interlaced format, similar to that of a Bayer color filter array. In this paper, we ask the question of how to design such multiplexing schemes for adaptive high-dynamic range (HDR) imaging where the multiplexing scheme can be updated according to the scenes. We present two new findings. (i) We address the problem of design optimality. We show that given a multiplex pattern, the conventional optimality criteria based on the input/output-referred signal-to-noise ratio (SNR) of the independently measured pixels can lead to flawed decisions because it cannot encapsulate the location of the saturated pixels. We overcome the issue by proposing a new concept known as the spatially varying exposure risk (SVE-Risk) which is a pseudo-idealistic quantification of the amount of recoverable pixels. We present an efficient enumeration algorithm to select the optimal multiplex patterns. (ii) We report a design universality observation that the design of the multiplex pattern can be decoupled from the image reconstruction algorithm. This is a significant departure from the recent literature that the multiplex pattern should be jointly optimized with the reconstruction algorithm. Our finding suggests that in the context of exposure multiplexing, an end-to-end training may not be necessary.
The core numbers of vertices in a graph are one of the most well-studied cohesive subgraph models because of the linear running time. In practice, many data graphs are dynamic graphs that are continuously changing by inserting or removing edges. The core numbers are updated in dynamic graphs with edge insertions and deletions, which is called core maintenance. When a burst of a large number of inserted or removed edges come in, we have to handle these edges on time to keep up with the data stream. There are two main sequential algorithms for core maintenance, \textsc{Traversal} and \textsc{Order}. It is proved that the \textsc{Order} algorithm significantly outperforms the \alg{Traversal} algorithm over all tested graphs with up to 2,083 times speedups. To the best of our knowledge, all existing parallel approaches are based on the \alg{Traversal} algorithm; also, their parallelism exists only for affected vertices with different core numbers, which will reduce to sequential when all vertices have the same core numbers. In this paper, we propose a new parallel core maintenance algorithm based on the \alg{Order} algorithm. Importantly, our new approach always has parallelism, even for the graphs where all vertices have the same core numbers. Extensive experiments are conducted over real-world, temporal, and synthetic graphs on a 64-core machine. The results show that for inserting and removing 100,000 edges using 16-worker, our method achieves up to 289x and 10x times speedups compared with the most efficient existing method, respectively.
Consider a mechanism that cannot observe how many players there are directly, but instead must rely on their self-reports to know how many are participating. Suppose the players can create new identities to report to the auctioneer at some cost $c$. The usual mechanism design paradigm is equivalent to implicitly assuming that $c$ is infinity for all players, while the usual Sybil attacks literature is that it is zero or finite for one player (the attacker) and infinity for everyone else (the 'honest' players). The false-name proof literature largely assumes the cost to be 0. We consider a model with variable costs that unifies these disparate streams. A paradigmatic normal form game can be extended into a Sybil game by having the action space by the product of the feasible set of identities to create action where each player chooses how many players to present as in the game and their actions in the original normal form game. A mechanism is (dominant) false-name proof if it is (dominant) incentive-compatible for all the players to self-report as at most one identity. We study mechanisms proposed in the literature motivated by settings where anonymity and self-identification are the norms, and show conditions under which they are not Sybil-proof. We characterize a class of dominant Sybil-proof mechanisms for reward sharing and show that they achieve the efficiency upper bound. We consider the extension when agents can credibly commit to the strategy of their sybils and show how this can break mechanisms that would otherwise be false-name proof.
The majority of machine learning methods can be regarded as the minimization of an unavailable risk function. To optimize the latter, given samples provided in a streaming fashion, we define a general stochastic Newton algorithm and its weighted average version. In several use cases, both implementations will be shown not to require the inversion of a Hessian estimate at each iteration, but a direct update of the estimate of the inverse Hessian instead will be favored. This generalizes a trick introduced in [2] for the specific case of logistic regression, by directly updating the estimate of the inverse Hessian. Under mild assumptions such as local strong convexity at the optimum, we establish almost sure convergences and rates of convergence of the algorithms, as well as central limit theorems for the constructed parameter estimates. The unified framework considered in this paper covers the case of linear, logistic or softmax regressions to name a few. Numerical experiments on simulated data give the empirical evidence of the pertinence of the proposed methods, which outperform popular competitors particularly in case of bad initializa-tions.
Data in Knowledge Graphs often represents part of the current state of the real world. Thus, to stay up-to-date the graph data needs to be updated frequently. To utilize information from Knowledge Graphs, many state-of-the-art machine learning approaches use embedding techniques. These techniques typically compute an embedding, i.e., vector representations of the nodes as input for the main machine learning algorithm. If a graph update occurs later on -- specifically when nodes are added or removed -- the training has to be done all over again. This is undesirable, because of the time it takes and also because downstream models which were trained with these embeddings have to be retrained if they change significantly. In this paper, we investigate embedding updates that do not require full retraining and evaluate them in combination with various embedding models on real dynamic Knowledge Graphs covering multiple use cases. We study approaches that place newly appearing nodes optimally according to local information, but notice that this does not work well. However, we find that if we continue the training of the old embedding, interleaved with epochs during which we only optimize for the added and removed parts, we obtain good results in terms of typical metrics used in link prediction. This performance is obtained much faster than with a complete retraining and hence makes it possible to maintain embeddings for dynamic Knowledge Graphs.
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.
In order to answer natural language questions over knowledge graphs, most processing pipelines involve entity and relation linking. Traditionally, entity linking and relation linking has been performed either as dependent sequential tasks or independent parallel tasks. In this paper, we propose a framework called "EARL", which performs entity linking and relation linking as a joint single task. EARL uses a graph connection based solution to the problem. We model the linking task as an instance of the Generalised Travelling Salesman Problem (GTSP) and use GTSP approximate algorithm solutions. We later develop EARL which uses a pair-wise graph-distance based solution to the problem.The system determines the best semantic connection between all keywords of the question by referring to a knowledge graph. This is achieved by exploiting the "connection density" between entity candidates and relation candidates. The "connection density" based solution performs at par with the approximate GTSP solution.We have empirically evaluated the framework on a dataset with 5000 questions. Our system surpasses state-of-the-art scores for entity linking task by reporting an accuracy of 0.65 to 0.40 from the next best entity linker.
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