We consider the $\varepsilon$-Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to $\varepsilon$). This problem was recently shown to be PPA-complete, for $n$ agents and $n$ cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem. For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents.
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We study the stochastic contextual bandit with knapsacks (CBwK) problem, where each action, taken upon a context, not only leads to a random reward but also costs a random resource consumption in a vector form. The challenge is to maximize the total reward without violating the budget for each resource. We study this problem under a general realizability setting where the expected reward and expected cost are functions of contexts and actions in some given general function classes $\mathcal{F}$ and $\mathcal{G}$, respectively. Existing works on CBwK are restricted to the linear function class since they use UCB-type algorithms, which heavily rely on the linear form and thus are difficult to extend to general function classes. Motivated by online regression oracles that have been successfully applied to contextual bandits, we propose the first universal and optimal algorithmic framework for CBwK by reducing it to online regression. We also establish the lower regret bound to show the optimality of our algorithm for a variety of function classes.
This paper models categorical data with two or multiple responses, focusing on the interactions between responses. We propose an efficient iterative procedure based on sufficient dimension reduction. We study the theoretical guarantees of the proposed method under the two- and multiple-response models, demonstrating the uniqueness of the proposed estimator and with the high probability that the proposed method recovers the oracle least squares estimators. For data analysis, we demonstrate that the proposed method is efficient in the multiple-response model and performs better than some existing methods built in the multiple-response models. We apply this modeling and the proposed method to an adult dataset and right heart catheterization dataset and obtain meaningful results.
We study the rank of sub-matrices arising out of kernel functions, $F(\pmb{x},\pmb{y}): \mathbb{R}^d \times \mathbb{R}^d \mapsto \mathbb{R}$, where $\pmb{x},\pmb{y} \in \mathbb{R}^d$ with $F(\pmb{x},\pmb{y})$ is smooth everywhere except along the line $\pmb{x}=\pmb{y}$. Such kernel functions are frequently encountered in a wide range of applications such as $N$ body problems, Green's functions, integral equations, geostatistics, kriging, Gaussian processes, etc. One of the challenges in dealing with these kernel functions is that the corresponding matrix associated with these kernels is large and dense and thereby, the computational cost of matrix operations is high. In this article, we prove new theorems bounding the numerical rank of sub-matrices arising out of these kernel functions. Under reasonably mild assumptions, we prove that the rank of certain sub-matrices is rank-deficient in finite precision. This rank depends on the dimension of the ambient space and also on the type of interaction between the hyper-cubes containing the corresponding set of particles. This rank structure can be leveraged to reduce the computational cost of certain matrix operations such as matrix-vector products, solving linear systems, etc. We also present numerical results on the growth of rank of certain sub-matrices in $1$D, $2$D, $3$D and $4$D, which, not surprisingly, agrees with the theoretical results.
In group testing, the goal is to identify a subset of defective items within a larger set of items based on tests whose outcomes indicate whether at least one defective item is present. This problem is relevant in areas such as medical testing, DNA sequencing, communication protocols, and many more. In this paper, we study (i) a sparsity-constrained version of the problem, in which the testing procedure is subjected to one of the following two constraints: items are finitely divisible and thus may participate in at most $\gamma$ tests; or tests are size-constrained to pool no more than $\rho$ items per test; and (ii) a noisy version of the problem, where each test outcome is independently flipped with some constant probability. Under each of these settings, considering the for-each recovery guarantee with asymptotically vanishing error probability, we introduce a fast splitting algorithm and establish its near-optimality not only in terms of the number of tests, but also in terms of the decoding time. While the most basic formulations of our algorithms require $\Omega(n)$ storage for each algorithm, we also provide low-storage variants based on hashing, with similar recovery guarantees.
In real-world crowdsourcing annotation systems, due to differences in user knowledge and cultural backgrounds, as well as the high cost of acquiring annotation information, the supervision information we obtain might be insufficient and ambiguous. To mitigate the negative impacts, in this paper, we investigate a more general and broadly applicable learning problem, i.e. \emph{semi-supervised partial label learning}, and propose a novel method based on pseudo-labeling and contrastive learning. Following the key inventing principle, our method facilitate the partial label disambiguation process with unlabeled data and at the same time assign reliable pseudo-labels to weakly supervised examples. Specifically, our method learns from the ambiguous labeling information via partial cross-entropy loss. Meanwhile, high-accuracy pseudo-labels are generated for both partial and unlabeled examples through confidence-based thresholding and contrastive learning is performed in a hybrid unsupervised and supervised manner for more discriminative representations, while its supervision increases curriculumly. The two main components systematically work as a whole and reciprocate each other. In experiments, our method consistently outperforms all comparing methods by a significant margin and set up the first state-of-the-art performance for semi-supervised partial label learning on image benchmarks.
This paper investigates an Internet of Things (IoT) system in which multiple devices are observing some object's physical parameters and then offloading their observations back to the BS in time with opportunistic channel access. Specifically, each device accesses the common channel through contention with a certain probability firstly and then the winner evaluates the instant channel condition and decides to accept the right of channel access or not. We analyze this system through the perspective of Age of Information (AoI), which describes the freshness of observed information. The target is to minimize average AoI by optimizing the probability of device participation in contention and the transmission rate threshold. The problem is hard to solve since the AoI expression in fractional form is complex. We first decompose the original problem into two single-variable optimization sub-problems through Dinkelbach method and Block Coordinate Descent (BCD) method. And then we transform them to Monotonic optimization problems by proving the monotonicity of the objective functions, whose global optimal solution is able to be found through Polyblock algorithm. Numerical results verify the validity of our proposed method.
The growing availability and usage of low precision foating point formats has attracts many interests of developing lower or mixed precision algorithms for scientific computing problems. In this paper we investigate the possibility of exploiting lower precision computing in LSQR for solving discrete linear ill-posed problems. We analyze the choice of proper computing precisions in the two main parts of LSQR, including the construction of Lanczos vectors and updating procedure of iterative solutions. We show that, under some mild conditions, the Lanczos vectors can be computed using single precision without loss of any accuracy of final regularized solutions as long as the noise level is not extremely small. We also show that the most time consuming part for updating iterative solutions can be performed using single precision without sacrificing any accuracy. The results indicate that the most time consuming parts of the algorithm can be implemented using single precision, and thus the performance of LSQR for solving discrete linear ill-posed problems can be significantly enhanced. Numerical experiments are made for testing the single precision variants of LSQR and confirming our results.
The goal of cryptocurrencies is decentralization. In principle, all currencies have equal status. Unlike traditional stock markets, there is no default currency of denomination (fiat), thus the trading pairs can be set freely. However, it is impractical to set up a trading market between every two currencies. In order to control management costs and ensure sufficient liquidity, we must give priority to covering those large-volume trading pairs and ensure that all coins are reachable. We note that this is an optimization problem. Its particularity lies in: 1) the trading volume between most (>99.5%) possible trading pairs cannot be directly observed. 2) It satisfies the connectivity constraint, that is, all currencies are guaranteed to be tradable. To solve this problem, we use a two-stage process: 1) Fill in missing values based on a regularized, truncated eigenvalue decomposition, where the regularization term is used to control what extent missing values should be limited to zero. 2) Search for the optimal trading pairs, based on a branch and bound process, with heuristic search and pruning strategies. The experimental results show that: 1) If the number of denominated coins is not limited, we will get a more decentralized trading pair settings, which advocates the establishment of trading pairs directly between large currency pairs. 2) There is a certain room for optimization in all exchanges. The setting of inappropriate trading pairs is mainly caused by subjectively setting small coins to quote, or failing to track emerging big coins in time. 3) Too few trading pairs will lead to low coverage; too many trading pairs will need to be adjusted with markets frequently. Exchanges should consider striking an appropriate balance between them.
Tie-breaker designs trade off a statistical design objective with short-term gain from preferentially assigning a binary treatment to those with high values of a running variable $x$. The design objective is any continuous function of the expected information matrix in a two-line regression model, and short-term gain is expressed as the covariance between the running variable and the treatment indicator. We investigate how to specify design functions indicating treatment probabilities as a function of $x$ to optimize these competing objectives, under external constraints on the number of subjects receiving treatment. Our results include sharp existence and uniqueness guarantees, while accommodating the ethically appealing requirement that treatment probabilities are non-decreasing in $x$. Under such a constraint, there always exists an optimal design function that is constant below and above a single discontinuity. When the running variable distribution is not symmetric or the fraction of subjects receiving the treatment is not $1/2$, our optimal designs improve upon a $D$-optimality objective without sacrificing short-term gain, compared to the three level tie-breaker designs of Owen and Varian (2020) that fix treatment probabilities at $0$, $1/2$, and $1$. We illustrate our optimal designs with data from Head Start, an early childhood government intervention program.
Deep convolutional neural networks (CNNs) have recently achieved great success in many visual recognition tasks. However, existing deep neural network models are computationally expensive and memory intensive, hindering their deployment in devices with low memory resources or in applications with strict latency requirements. Therefore, a natural thought is to perform model compression and acceleration in deep networks without significantly decreasing the model performance. During the past few years, tremendous progress has been made in this area. In this paper, we survey the recent advanced techniques for compacting and accelerating CNNs model developed. These techniques are roughly categorized into four schemes: parameter pruning and sharing, low-rank factorization, transferred/compact convolutional filters, and knowledge distillation. Methods of parameter pruning and sharing will be described at the beginning, after that the other techniques will be introduced. For each scheme, we provide insightful analysis regarding the performance, related applications, advantages, and drawbacks etc. Then we will go through a few very recent additional successful methods, for example, dynamic capacity networks and stochastic depths networks. After that, we survey the evaluation matrix, the main datasets used for evaluating the model performance and recent benchmarking efforts. Finally, we conclude this paper, discuss remaining challenges and possible directions on this topic.
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