We study variants of the secretary problem, where $N$, the number of candidates, is a random variable, and the decision maker wants to maximize the probability of success -- picking the largest number among the $N$ candidates -- using only the relative ranks of the candidates revealed so far. We consider three forms of prior information about $\mathbf p$, the probability distribution of $N$. In the full information setting, we assume $\mathbf p$ to be fully known. In that case, we show that single-threshold type of strategies can achieve $1/e$-approximation to the maximum probability of success among all possible strategies. In the upper bound setting, we assume that $N\leq \overline{n}$ (or $\mathbb E[N]\leq \bar{\mu}$), where $\bar{n}$ (or $\bar{\mu}$) is known. In that case, we show that randomization over single-threshold type of strategies can achieve the optimal worst case probability of success of $\frac{1}{\log(\bar{n})}$ (or $\frac{1}{\log(\bar{\mu})}$) asymptotically. Surprisingly, there is a single-threshold strategy (depending on $\overline{n}$) that can succeed with probability $2/e^2$ for all but an exponentially small fraction of distributions supported on $[\bar{n}]$. In the sampling setting, we assume that we have access to $m$ samples $N^{(1)},\ldots,N^{(m)}\sim_{iid} \mathbf p$. In that case, we show that if $N\leq T$ with probability at least $1-O(\epsilon)$ for some $T\in \mathbb N$, $m\gtrsim \frac{1}{\epsilon^2}\max(\log(\frac{1}{\epsilon}),\epsilon \log(\frac{\log(T)}{\epsilon}))$ is enough to learn a strategy that is at least $\epsilon$-suboptimal, and we provide a lower bound of $\Omega(\frac{1}{\epsilon^2})$, showing that the sampling algorithm is optimal when $\epsilon=O(\frac{1}{\log\log(T)})$.
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From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM.
The paper studies the rewriting problem, that is, the decision problem whether, for a given conjunctive query $Q$ and a set $\mathcal{V}$ of views, there is a conjunctive query $Q'$ over $\mathcal{V}$ that is equivalent to $Q$, for cases where the query, the views, and/or the desired rewriting are acyclic or even more restricted. It shows that, if $Q$ itself is acyclic, an acyclic rewriting exists if there is any rewriting. An analogous statement also holds for free-connex acyclic, hierarchical, and q-hierarchical queries. Regarding the complexity of the rewriting problem, the paper identifies a border between tractable and (presumably) intractable variants of the rewriting problem: for schemas of bounded arity, the acyclic rewriting problem is NP-hard, even if both $Q$ and the views in $\mathcal{V}$ are acyclic or hierarchical. However, it becomes tractable if the views are free-connex acyclic (i.e., in a nutshell, their body is (i) acyclic and (ii) remains acyclic if their head is added as an additional atom).
The ability to derive underlying principles from a handful of observations and then generalize to novel situations -- known as inductive reasoning -- is central to human intelligence. Prior work suggests that language models (LMs) often fall short on inductive reasoning, despite achieving impressive success on research benchmarks. In this work, we conduct a systematic study of the inductive reasoning capabilities of LMs through iterative hypothesis refinement, a technique that more closely mirrors the human inductive process than standard input-output prompting. Iterative hypothesis refinement employs a three-step process: proposing, selecting, and refining hypotheses in the form of textual rules. By examining the intermediate rules, we observe that LMs are phenomenal hypothesis proposers (i.e., generating candidate rules), and when coupled with a (task-specific) symbolic interpreter that is able to systematically filter the proposed set of rules, this hybrid approach achieves strong results across inductive reasoning benchmarks that require inducing causal relations, language-like instructions, and symbolic concepts. However, they also behave as puzzling inductive reasoners, showing notable performance gaps between rule induction (i.e., identifying plausible rules) and rule application (i.e., applying proposed rules to instances), suggesting that LMs are proposing hypotheses without being able to actually apply the rules. Through empirical and human analyses, we further reveal several discrepancies between the inductive reasoning processes of LMs and humans, shedding light on both the potentials and limitations of using LMs in inductive reasoning tasks.
We propose a diarization system, that estimates "who spoke when" based on spatial information, to be used as a front-end of a meeting transcription system running on the signals gathered from an acoustic sensor network (ASN). Although the spatial distribution of the microphones is advantageous, exploiting the spatial diversity for diarization and signal enhancement is challenging, because the microphones' positions are typically unknown, and the recorded signals are initially unsynchronized in general. Here, we approach these issues by first blindly synchronizing the signals and then estimating time differences of arrival (TDOAs). The TDOA information is exploited to estimate the speakers' activity, even in the presence of multiple speakers being simultaneously active. This speaker activity information serves as a guide for a spatial mixture model, on which basis the individual speaker's signals are extracted via beamforming. Finally, the extracted signals are forwarded to a speech recognizer. Additionally, a novel initialization scheme for spatial mixture models based on the TDOA estimates is proposed. Experiments conducted on real recordings from the LibriWASN data set have shown that our proposed system is advantageous compared to a system using a spatial mixture model, which does not make use of external diarization information.
Current approaches to empathetic response generation typically encode the entire dialogue history directly and put the output into a decoder to generate friendly feedback. These methods focus on modelling contextual information but neglect capturing the direct intention of the speaker. We argue that the last utterance in the dialogue empirically conveys the intention of the speaker. Consequently, we propose a novel model named InferEM for empathetic response generation. We separately encode the last utterance and fuse it with the entire dialogue through the multi-head attention based intention fusion module to capture the speaker's intention. Besides, we utilize previous utterances to predict the last utterance, which simulates human's psychology to guess what the interlocutor may speak in advance. To balance the optimizing rates of the utterance prediction and response generation, a multi-task learning strategy is designed for InferEM. Experimental results demonstrate the plausibility and validity of InferEM in improving empathetic expression.
This study analyzes the nonasymptotic convergence behavior of the quasi-Monte Carlo (QMC) method with applications to linear elliptic partial differential equations (PDEs) with lognormal coefficients. Building upon the error analysis presented in (Owen, 2006), we derive a nonasymptotic convergence estimate depending on the specific integrands, the input dimensionality, and the finite number of samples used in the QMC quadrature. We discuss the effects of the variance and dimensionality of the input random variable. Then, we apply the QMC method with importance sampling (IS) to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient in bounded domains of $\mathbb{R}^d$, where the random coefficient is modeled as a stationary Gaussian random field parameterized by the trigonometric and wavelet-type basis. We propose two types of IS distributions, analyze their effects on the QMC convergence rate, and observe the improvements.
This paper rigorously shows how over-parameterization changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where $M^* \in \mathbb{R}^{n \times n}$ is a positive semi-definite unknown matrix of rank $r \ll n$, and one uses a symmetric parameterization $XX^\top$ to learn $M^*$. Here $X \in \mathbb{R}^{n \times k}$ with $k > r$ is the factor matrix. We give a novel $\Omega (1/T^2)$ lower bound of randomly initialized GD for the over-parameterized case ($k >r$) where $T$ is the number of iterations. This is in stark contrast to the exact-parameterization scenario ($k=r$) where the convergence rate is $\exp (-\Omega (T))$. Next, we study asymmetric setting where $M^* \in \mathbb{R}^{n_1 \times n_2}$ is the unknown matrix of rank $r \ll \min\{n_1,n_2\}$, and one uses an asymmetric parameterization $FG^\top$ to learn $M^*$ where $F \in \mathbb{R}^{n_1 \times k}$ and $G \in \mathbb{R}^{n_2 \times k}$. Building on prior work, we give a global exact convergence result of randomly initialized GD for the exact-parameterization case ($k=r$) with an $\exp (-\Omega(T))$ rate. Furthermore, we give the first global exact convergence result for the over-parameterization case ($k>r$) with an $\exp(-\Omega(\alpha^2 T))$ rate where $\alpha$ is the initialization scale. This linear convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from $\Omega (1/T^2)$ to linear convergence. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of $\alpha$, recovering the rate in the exact-parameterization case.
In contrast to batch learning where all training data is available at once, continual learning represents a family of methods that accumulate knowledge and learn continuously with data available in sequential order. Similar to the human learning process with the ability of learning, fusing, and accumulating new knowledge coming at different time steps, continual learning is considered to have high practical significance. Hence, continual learning has been studied in various artificial intelligence tasks. In this paper, we present a comprehensive review of the recent progress of continual learning in computer vision. In particular, the works are grouped by their representative techniques, including regularization, knowledge distillation, memory, generative replay, parameter isolation, and a combination of the above techniques. For each category of these techniques, both its characteristics and applications in computer vision are presented. At the end of this overview, several subareas, where continuous knowledge accumulation is potentially helpful while continual learning has not been well studied, are discussed.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
Incompleteness is a common problem for existing knowledge graphs (KGs), and the completion of KG which aims to predict links between entities is challenging. Most existing KG completion methods only consider the direct relation between nodes and ignore the relation paths which contain useful information for link prediction. Recently, a few methods take relation paths into consideration but pay less attention to the order of relations in paths which is important for reasoning. In addition, these path-based models always ignore nonlinear contributions of path features for link prediction. To solve these problems, we propose a novel KG completion method named OPTransE. Instead of embedding both entities of a relation into the same latent space as in previous methods, we project the head entity and the tail entity of each relation into different spaces to guarantee the order of relations in the path. Meanwhile, we adopt a pooling strategy to extract nonlinear and complex features of different paths to further improve the performance of link prediction. Experimental results on two benchmark datasets show that the proposed model OPTransE performs better than state-of-the-art methods.
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