We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $\alpha$ of the graph, in a worst-case update time of $O(\log^3 n \log \alpha)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of $O(\log ^2 n \log \alpha)$ for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Since our algorithms have update times with worst-case guarantees, the number of changes to the solution (i.e. the recourse) is naturally limited. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \rho)$ worst-case update time algorithm for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $\rho$. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \alpha)$ worst-case update time algorithm for maintaining a $(1 + \varepsilon) \cdot OPT + 2$ approximation of the optimal out-orientation of a graph with adaptive arboricity $\alpha$. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(\alpha)$ forests.Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety, of problems including maximal matching, $\Delta+1$ coloring, and matrix vector multiplication. All update times are worst-case $O(\alpha+\log^2n \log \alpha)$, where $\alpha$ is the current arboricity of the graph.
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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.
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.
Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the associated reverse-time SDE may be estimated using score-matching. A limitation of this approach is that the forward-time SDE must be run for a sufficiently long time for the final distribution to be approximately Gaussian. In contrast, solving the Schr\"odinger Bridge problem (SB), i.e. an entropy-regularized optimal transport problem on path spaces, yields diffusions which generate samples from the data distribution in finite time. We present Diffusion SB (DSB), an original approximation of the Iterative Proportional Fitting (IPF) procedure to solve the SB problem, and provide theoretical analysis along with generative modeling experiments. The first DSB iteration recovers the methodology proposed by Song et al. (2021), with the flexibility of using shorter time intervals, as subsequent DSB iterations reduce the discrepancy between the final-time marginal of the forward (resp. backward) SDE with respect to the prior (resp. data) distribution. Beyond generative modeling, DSB offers a widely applicable computational optimal transport tool as the continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi, 2013).
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.
Strong spatial mixing (SSM) is an important quantitative notion of correlation decay for Gibbs distributions arising in statistical physics, probability theory, and theoretical computer science. A longstanding conjecture is that the uniform distribution on proper $q$-colorings on a $\Delta$-regular tree exhibits SSM whenever $q \ge \Delta+1$. Moreover, it is widely believed that as long as SSM holds on bounded-degree trees with $q$ colors, one would obtain an efficient sampler for $q$-colorings on all bounded-degree graphs via simple Markov chain algorithms. It is surprising that such a basic question is still open, even on trees, but then again it also highlights how much we still have to learn about random colorings. In this paper, we show the following: (1) For any $\Delta \ge 3$, SSM holds for random $q$-colorings on trees of maximum degree $\Delta$ whenever $q \ge \Delta + 3$. Thus we almost fully resolve the aforementioned conjecture. Our result substantially improves upon the previously best bound which requires $q \ge 1.59\Delta+\gamma^*$ for an absolute constant $\gamma^* > 0$. (2) For any $\Delta\ge 3$ and girth $g = \Omega_\Delta(1)$, we establish optimal mixing of the Glauber dynamics for $q$-colorings on graphs of maximum degree $\Delta$ and girth $g$ whenever $q \ge \Delta+3$. Our approach is based on a new general reduction from spectral independence on large-girth graphs to SSM on trees that is of independent interest. Using the same techniques, we also prove near-optimal bounds on weak spatial mixing (WSM), a closely-related notion to SSM, for the antiferromagnetic Potts model on trees.
We develop a new framework for embedding joint probability distributions in tensor product reproducing kernel Hilbert spaces (RKHS). Our framework accommodates a low-dimensional, normalized and positive model of a Radon-Nikodym derivative, which we estimate from sample sizes of up to several million data points, alleviating the inherent limitations of RKHS modeling. Well-defined normalized and positive conditional distributions are natural by-products to our approach. The embedding is fast to compute and accommodates learning problems ranging from prediction to classification. Our theoretical findings are supplemented by favorable numerical results.
Variably scaled kernels and mapped bases constructed via the so-called fake nodes approach are two different strategies to provide adaptive bases for function interpolation. In this paper, we focus on kernel-based interpolation and we present what we call mapped variably scaled kernels, which take advantage of both strategies. We present some theoretical analysis and then we show their efficacy via numerical experiments. Moreover, we test such a new basis for image reconstruction tasks in the framework of hard X-ray astronomical imaging.
The stochastic approximation (SA) algorithm is a widely used probabilistic method for finding a zero or a fixed point of a vector-valued funtion, when only noisy measurements of the function are available. In the literature to date, one makes a distinction between ``synchronous'' updating, whereby every component of the current guess is updated at each time, and ``asynchronous'' updating, whereby only one component is updated. In this paper, we study an intermediate situation that we call ``batch asynchronous stochastic approximation'' (BASA), in which, at each time instant, \textit{some but not all} components of the current estimated solution are updated. BASA allows the user to trade off memory requirements against time complexity. We develop a general methodology for proving that such algorithms converge to the fixed point of the map under study. These convergence proofs make use of weaker hypotheses than existing results. Specifically, existing convergence proofs require that the measurement noise is a zero-mean i.i.d\ sequence or a martingale difference sequence. In the present paper, we permit biased measurements, that is, measurement noises that have nonzero conditional mean. Also, all convergence results to date assume that the stochastic step sizes satisfy a probabilistic analog of the well-known Robbins-Monro conditions. We replace this assumption by a purely deterministic condition on the irreducibility of the underlying Markov processes. As specific applications to Reinforcement Learning, we analyze the temporal difference algorithm $TD(\lambda)$ for value iteration, and the $Q$-learning algorithm for finding the optimal action-value function. In both cases, we establish the convergence of these algorithms, under milder conditions than in the existing literature.
As an intrinsic and fundamental property of big data, data heterogeneity exists in a variety of real-world applications, such as precision medicine, autonomous driving, financial applications, etc. For machine learning algorithms, the ignorance of data heterogeneity will greatly hurt the generalization performance and the algorithmic fairness, since the prediction mechanisms among different sub-populations are likely to differ from each other. In this work, we focus on the data heterogeneity that affects the prediction of machine learning models, and firstly propose the \emph{usable predictive heterogeneity}, which takes into account the model capacity and computational constraints. We prove that it can be reliably estimated from finite data with probably approximately correct (PAC) bounds. Additionally, we design a bi-level optimization algorithm to explore the usable predictive heterogeneity from data. Empirically, the explored heterogeneity provides insights for sub-population divisions in income prediction, crop yield prediction and image classification tasks, and leveraging such heterogeneity benefits the out-of-distribution generalization performance.
Recently, contrastive learning (CL) has emerged as a successful method for unsupervised graph representation learning. Most graph CL methods first perform stochastic augmentation on the input graph to obtain two graph views and maximize the agreement of representations in the two views. Despite the prosperous development of graph CL methods, the design of graph augmentation schemes -- a crucial component in CL -- remains rarely explored. We argue that the data augmentation schemes should preserve intrinsic structures and attributes of graphs, which will force the model to learn representations that are insensitive to perturbation on unimportant nodes and edges. However, most existing methods adopt uniform data augmentation schemes, like uniformly dropping edges and uniformly shuffling features, leading to suboptimal performance. In this paper, we propose a novel graph contrastive representation learning method with adaptive augmentation that incorporates various priors for topological and semantic aspects of the graph. Specifically, on the topology level, we design augmentation schemes based on node centrality measures to highlight important connective structures. On the node attribute level, we corrupt node features by adding more noise to unimportant node features, to enforce the model to recognize underlying semantic information. We perform extensive experiments of node classification on a variety of real-world datasets. Experimental results demonstrate that our proposed method consistently outperforms existing state-of-the-art baselines and even surpasses some supervised counterparts, which validates the effectiveness of the proposed contrastive framework with adaptive augmentation.
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