Reallocation scheduling is one of the most fundamental problems in various areas such as supply chain management, logistics, and transportation science. In this paper, we introduce the reallocation problem that models the scheduling in which products are with fixed cost, non-fungible, and reallocated in parallel, and comprehensively study the complexity of the problem under various settings of the transition time, product size, and capacities. We show that the problem can be solved in polynomial time for a fundamental setting where the product size and transition time are both uniform. We also show that the feasibility of the problem is NP-complete even for little more general settings, which implies that no polynomial-time algorithm constructs a feasible schedule of the problem unless P$=$NP. We then consider the relaxation of the problem, which we call the capacity augmentation, and derive a reallocation schedule feasible with the augmentation such that the completion time is at most the optimal of the original problem. When the warehouse capacity is sufficiently large, we design constant-factor approximation algorithms under all the settings. We also show the relationship between the reallocation problem and the bin packing problem when the warehouse and carry-in capacities are sufficiently large.
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This paper focuses on stochastic saddle point problems with decision-dependent distributions in both the static and time-varying settings. These are problems whose objective is the expected value of a stochastic payoff function, where random variables are drawn from a distribution induced by a distributional map. For general distributional maps, the problem of finding saddle points is in general computationally burdensome, even if the distribution is known. To enable a tractable solution approach, we introduce the notion of equilibrium points -- which are saddle points for the stationary stochastic minimax problem that they induce -- and provide conditions for their existence and uniqueness. We demonstrate that the distance between the two classes of solutions is bounded provided that the objective has a strongly-convex-strongly-concave payoff and Lipschitz continuous distributional map. We develop deterministic and stochastic primal-dual algorithms and demonstrate their convergence to the equilibrium point. In particular, by modeling errors emerging from a stochastic gradient estimator as sub-Weibull random variables, we provide error bounds in expectation and in high probability that hold for each iteration; moreover, we show convergence to a neighborhood in expectation and almost surely. Finally, we investigate a condition on the distributional map -- which we call opposing mixture dominance -- that ensures the objective is strongly-convex-strongly-concave. Under this assumption, we show that primal-dual algorithms converge to the saddle points in a similar fashion.
This work studies how the introduction of the entropic regularization term in unbalanced Optimal Transport (OT) models may alter their homogeneity with respect to the input measures. We observe that in common settings (including balanced OT and unbalanced OT with Kullback-Leibler divergence to the marginals), although the optimal transport cost itself is not homogeneous, optimal transport plans and the so-called Sinkhorn divergences are indeed homogeneous. However, homogeneity does not hold in more general Unbalanced Regularized Optimal Transport (UROT) models, for instance those using the Total Variation as divergence to the marginals. We propose to modify the entropic regularization term to retrieve an UROT model that is homogeneous while preserving most properties of the standard UROT model. We showcase the importance of using our Homogeneous UROT (HUROT) model when it comes to regularize Optimal Transport with Boundary, a transportation model involving a spatially varying divergence to the marginals for which the standard (inhomogeneous) UROT model would yield inappropriate behavior.
Recent advances in quantized compressed sensing and high-dimensional estimation have shown that signal recovery is even feasible under strong non-linear distortions in the observation process. An important characteristic of associated guarantees is uniformity, i.e., recovery succeeds for an entire class of structured signals with a fixed measurement ensemble. However, despite significant results in various special cases, a general understanding of uniform recovery from non-linear observations is still missing. This paper develops a unified approach to this problem under the assumption of i.i.d. sub-Gaussian measurement vectors. Our main result shows that a simple least-squares estimator with any convex constraint can serve as a universal recovery strategy, which is outlier robust and does not require explicit knowledge of the underlying non-linearity. Based on empirical process theory, a key technical novelty is an approximative increment condition that can be implemented for all common types of non-linear models. This flexibility allows us to apply our approach to a variety of problems in non-linear compressed sensing and high-dimensional statistics, leading to several new and improved guarantees. Each of these applications is accompanied by a conceptually simple and systematic proof, which does not rely on any deeper properties of the observation model. On the other hand, known local stability properties can be incorporated into our framework in a plug-and-play manner, thereby implying near-optimal error bounds.
We present a $(1- \varepsilon)$-approximation algorithms for maximum cardinality matchings in disk intersection graphs -- all with near linear running time. We also present estimation algorithm that returns $(1\pm \varepsilon)$-approximation to the size of such matchings -- this algorithms run in linear time for unit disks, and $O(n \log n)$ for general disks (as long as the density is relatively small).
Optimization under uncertainty and risk is indispensable in many practical situations. Our paper addresses stability of optimization problems using composite risk functionals which are subjected to measure perturbations. Our main focus is the asymptotic behavior of data-driven formulations with empirical or smoothing estimators such as kernels or wavelets applied to some or to all functions of the compositions. We analyze the properties of the new estimators and we establish strong law of large numbers, consistency, and bias reduction potential under fairly general assumptions. Our results are germane to risk-averse optimization and to data science in general.
The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between one-dimensional random projections of the two measures. However, the topological, statistical, and computational consequences of this technique have not yet been well-established. In this paper, we aim at bridging this gap and derive various theoretical properties of sliced probability divergences. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of a sliced divergence does not depend on the problem dimension. We finally apply our general results to several base divergences, and illustrate our theory on both synthetic and real data experiments.
We study the class of first-order locally-balanced Metropolis--Hastings algorithms introduced in Livingstone & Zanella (2021). To choose a specific algorithm within the class the user must select a balancing function $g:\mathbb{R} \to \mathbb{R}$ satisfying $g(t) = tg(1/t)$, and a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a universal limiting optimal acceptance rate of 57% and scaling of $n^{-1/3}$ as the dimension $n$ tends to infinity among all members of the class under mild smoothness assumptions on $g$ and when the target distribution for the algorithm is of the product form. In particular we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimise this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, optimal choice of balancing function under a Gaussian noise distribution, and optimal choice of first-order locally-balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings and in particular show that a bi-modal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.
Many-user MAC is an important model for understanding energy efficiency of massive random access in 5G and beyond. Introduced in Polyanskiy'2017 for the AWGN channel, subsequent works have provided improved bounds on the asymptotic minimum energy-per-bit required to achieve a target per-user error at a given user density and payload, going beyond the AWGN setting. The best known rigorous bounds use spatially coupled codes along with the optimal AMP algorithm. But these bounds are infeasible to compute beyond a few (around 10) bits of payload. In this paper, we provide new achievability bounds for the many-user AWGN and quasi-static Rayleigh fading MACs using the spatially coupled codebook design along with a scalar AMP algorithm. The obtained bounds are computable even up to 100 bits and outperform the previous ones at this payload.
We present collaborative similarity embedding (CSE), a unified framework that exploits comprehensive collaborative relations available in a user-item bipartite graph for representation learning and recommendation. In the proposed framework, we differentiate two types of proximity relations: direct proximity and k-th order neighborhood proximity. While learning from the former exploits direct user-item associations observable from the graph, learning from the latter makes use of implicit associations such as user-user similarities and item-item similarities, which can provide valuable information especially when the graph is sparse. Moreover, for improving scalability and flexibility, we propose a sampling technique that is specifically designed to capture the two types of proximity relations. Extensive experiments on eight benchmark datasets show that CSE yields significantly better performance than state-of-the-art recommendation methods.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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