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It is well known that the traditional Jensen inequality is proved by lower bounding the given convex function, $f(x)$, by the tangential affine function that passes through the point $(E\{X\},f(E\{X\}))$, where $E\{X\}$ is the expectation of the random variable $X$. While this tangential affine function yields the tightest lower bound among all lower bounds induced by affine functions that are tangential to $f$, it turns out that when the function $f$ is just part of a more complicated expression whose expectation is to be bounded, the tightest lower bound might belong to a tangential affine function that passes through a point different than $(E\{X\},f(E\{X\}))$. In this paper, we take advantage of this observation, by optimizing the point of tangency with regard to the specific given expression, in a variety of cases, and thereby derive several families of inequalities, henceforth referred to as ``Jensen-like'' inequalities, which are new to the best knowledge of the author. The degree of tightness and the potential usefulness of these inequalities is demonstrated in several application examples related to information theory.

In this work, we derive sharp non-asymptotic deviation bounds for weighted sums of Dirichlet random variables. These bounds are based on a novel integral representation of the density of a weighted Dirichlet sum. This representation allows us to obtain a Gaussian-like approximation for the sum distribution using geometry and complex analysis methods. Our results generalize similar bounds for the Beta distribution obtained in the seminal paper Alfers and Dinges [1984]. Additionally, our results can be considered a sharp non-asymptotic version of the inverse of Sanov's theorem studied by Ganesh and O'Connell [1999] in the Bayesian setting. Based on these results, we derive new deviation bounds for the Dirichlet process posterior means with application to Bayesian bootstrap. Finally, we apply our estimates to the analysis of the Multinomial Thompson Sampling (TS) algorithm in multi-armed bandits and significantly sharpen the existing regret bounds by making them independent of the size of the arms distribution support.

We study the balanced $k$-way hypergraph partitioning problem, with a special focus on its practical applications to manycore scheduling. Given a hypergraph on $n$ nodes, our goal is to partition the node set into $k$ parts of size at most $(1+\epsilon)\cdot \frac{n}{k}$ each, while minimizing the cost of the partitioning, defined as the number of cut hyperedges, possibly also weighted by the number of partitions they intersect. We show that this problem cannot be approximated to within a $n^{1/\text{poly} \log\log n}$ factor of the optimal solution in polynomial time if the Exponential Time Hypothesis holds, even for hypergraphs of maximal degree 2. We also study the hardness of the partitioning problem from a parameterized complexity perspective, and in the more general case when we have multiple balance constraints. Furthermore, we consider two extensions of the partitioning problem that are motivated from practical considerations. Firstly, we introduce the concept of hyperDAGs to model precedence-constrained computations as hypergraphs, and we analyze the adaptation of the balanced partitioning problem to this case. Secondly, we study the hierarchical partitioning problem to model hierarchical NUMA (non-uniform memory access) effects in modern computer architectures, and we show that ignoring this hierarchical aspect of the communication cost can yield significantly weaker solutions.

We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=\omega(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=\Omega(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+\epsilon)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

This paper investigates in which cases continuous optimization for directed acyclic graph (DAG) structure learning can and cannot perform well and why this happens, and suggests possible directions to make the search procedure more reliable. Reisach et al. (2021) suggested that the remarkable performance of several continuous structure learning approaches is primarily driven by a high agreement between the order of increasing marginal variances and the topological order, and demonstrated that these approaches do not perform well after data standardization. We analyze this phenomenon for continuous approaches assuming equal and non-equal noise variances, and show that the statement may not hold in either case by providing counterexamples, justifications, and possible alternative explanations. We further demonstrate that nonconvexity may be a main concern especially for the non-equal noise variances formulation, while recent advances in continuous structure learning fail to achieve improvement in this case. Our findings suggest that future works should take into account the non-equal noise variances formulation to handle more general settings and for a more comprehensive empirical evaluation. Lastly, we provide insights into other aspects of the search procedure, including thresholding and sparsity, and show that they play an important role in the final solutions.

We analyze the bit complexity of efficient algorithms for fundamental optimization problems, such as linear regression, $p$-norm regression, and linear programming (LP). State-of-the-art algorithms are iterative, and in terms of the number of arithmetic operations, they match the current time complexity of multiplying two $n$-by-$n$ matrices (up to polylogarithmic factors). However, previous work has typically assumed infinite precision arithmetic, and due to complicated inverse maintenance techniques, the actual running times of these algorithms are unknown. To settle the running time and bit complexity of these algorithms, we demonstrate that a core common subroutine, known as \emph{inverse maintenance}, is backward-stable. Additionally, we show that iterative approaches for solving constrained weighted regression problems can be accomplished with bounded-error pre-conditioners. Specifically, we prove that linear programs can be solved approximately in matrix multiplication time multiplied by polylog factors that depend on the condition number $\kappa$ of the matrix and the inner and outer radius of the LP problem. $p$-norm regression can be solved approximately in matrix multiplication time multiplied by polylog factors in $\kappa$. Lastly, linear regression can be solved approximately in input-sparsity time multiplied by polylog factors in $\kappa$. Furthermore, we present results for achieving lower than matrix multiplication time for $p$-norm regression by utilizing faster solvers for sparse linear systems.

If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,v\}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families.

Training machine learning models in a meaningful order, from the easy samples to the hard ones, using curriculum learning can provide performance improvements over the standard training approach based on random data shuffling, without any additional computational costs. Curriculum learning strategies have been successfully employed in all areas of machine learning, in a wide range of tasks. However, the necessity of finding a way to rank the samples from easy to hard, as well as the right pacing function for introducing more difficult data can limit the usage of the curriculum approaches. In this survey, we show how these limits have been tackled in the literature, and we present different curriculum learning instantiations for various tasks in machine learning. We construct a multi-perspective taxonomy of curriculum learning approaches by hand, considering various classification criteria. We further build a hierarchical tree of curriculum learning methods using an agglomerative clustering algorithm, linking the discovered clusters with our taxonomy. At the end, we provide some interesting directions for future work.

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.

Lots of learning tasks require dealing with graph data which contains rich relation information among elements. Modeling physics system, learning molecular fingerprints, predicting protein interface, and classifying diseases require that a model to learn from graph inputs. In other domains such as learning from non-structural data like texts and images, reasoning on extracted structures, like the dependency tree of sentences and the scene graph of images, is an important research topic which also needs graph reasoning models. Graph neural networks (GNNs) are connectionist models that capture the dependence of graphs via message passing between the nodes of graphs. Unlike standard neural networks, graph neural networks retain a state that can represent information from its neighborhood with an arbitrary depth. Although the primitive graph neural networks have been found difficult to train for a fixed point, recent advances in network architectures, optimization techniques, and parallel computation have enabled successful learning with them. In recent years, systems based on graph convolutional network (GCN) and gated graph neural network (GGNN) have demonstrated ground-breaking performance on many tasks mentioned above. In this survey, we provide a detailed review over existing graph neural network models, systematically categorize the applications, and propose four open problems for future research.