In this work, we give a statistical characterization of the $\gamma$-regret for arbitrary structured bandit problems, the regret which arises when comparing against a benchmark that is $\gamma$ times the optimal solution. The $\gamma$-regret emerges in structured bandit problems over a function class $\mathcal{F}$ where finding an exact optimum of $f \in \mathcal{F}$ is intractable. Our characterization is given in terms of the $\gamma$-DEC, a statistical complexity parameter for the class $\mathcal{F}$, which is a modification of the constrained Decision-Estimation Coefficient (DEC) of Foster et al., 2023 (and closely related to the original offset DEC of Foster et al., 2021). Our lower bound shows that the $\gamma$-DEC is a fundamental limit for any model class $\mathcal{F}$: for any algorithm, there exists some $f \in \mathcal{F}$ for which the $\gamma$-regret of that algorithm scales (nearly) with the $\gamma$-DEC of $\mathcal{F}$. We provide an upper bound showing that there exists an algorithm attaining a nearly matching $\gamma$-regret. Due to significant challenges in applying the prior results on the DEC to the $\gamma$-regret case, both our lower and upper bounds require novel techniques and a new algorithm.
相關內容
In the Text-to-speech(TTS) task, the latent diffusion model has excellent fidelity and generalization, but its expensive resource consumption and slow inference speed have always been a challenging. This paper proposes Discrete Diffusion Model with Contrastive Learning for Text-to-Speech Generation(DCTTS). The following contributions are made by DCTTS: 1) The TTS diffusion model based on discrete space significantly lowers the computational consumption of the diffusion model and improves sampling speed; 2) The contrastive learning method based on discrete space is used to enhance the alignment connection between speech and text and improve sampling quality; and 3) It uses an efficient text encoder to simplify the model's parameters and increase computational efficiency. The experimental results demonstrate that the approach proposed in this paper has outstanding speech synthesis quality and sampling speed while significantly reducing the resource consumption of diffusion model. The synthesized samples are available at //github.com/lawtherWu/DCTTS.
In this paper, we present distributed fault-tolerant algorithms that approximate the centroid of a set of $n$ data points in $\mathbb{R}^d$. Our work falls into the broader area of approximate multidimensional Byzantine agreement. The standard approach used in existing algorithms is to agree on a vector inside the convex hull of all correct vectors. This strategy dismisses many possibly correct data points. As a result, the algorithm does not necessarily agree on a representative value. To find better convergence strategies for the algorithms, we use the novel concept of defining an approximation of the centroid in the presence of Byzantine adversaries. We show that the standard agreement algorithms do not allow us to compute a better approximation than $2d$ of the centroid in the synchronous case. We investigate the trade-off between the quality of the approximation, the resilience of the algorithm, and the validity of the solution in order to design better approximation algorithms. For the synchronous case, we show that it is possible to achieve an optimal approximation of the centroid with up to $t<n/(d+1)$ Byzantine data points. This approach however does not give any guarantee on the validity of the solution. Therefore, we develop a second approach that reaches a $2\sqrt{d}$-approximation of the centroid, while satisfying the standard validity condition for agreement protocols. We are even able to restrict the validity condition to agreement inside the box of correct data points, while achieving optimal resilience of $t< n/3$. For the asynchronous case, we can adapt all three algorithms to reach the same approximation results (up to a constant factor). Our results suggest that it is reasonable to study the trade-off between validity conditions and the quality of the solution.
The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. The Mapper algorithm requires tuning several parameters in order to generate a "nice" Mapper graph. The paper focuses on selecting the cover parameter. We present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on $G$-means clustering which searches for the optimal number of clusters in $k$-means by conducting iteratively the Anderson-Darling test. Our splitting procedure employs a Gaussian mixture model in order to choose carefully the cover based on the distribution of a given data. Experiments for synthetic and real-world datasets demonstrate that our algorithm generates covers so that the Mapper graphs retain the essence of the datasets.
Quantum computing (QC) introduces a novel mode of computation with the possibility of greater computational power that remains to be exploited $\unicode{x2013}$ presenting exciting opportunities for high performance computing (HPC) applications. However, recent advancements in the field have made clear that QC does not supplant conventional HPC, but can rather be incorporated into current heterogeneous HPC infrastructures as an additional accelerator, thereby enabling the optimal utilization of both paradigms. The desire for such integration significantly affects the development of software for quantum computers, which in turn influences the necessary software infrastructure. To date, previous review papers have investigated various quantum programming tools (QPTs) (such as languages, libraries, frameworks) in their ability to program, compile, and execute quantum circuits. However, the integration effort with classical HPC frameworks or systems has not been addressed. This study aims to characterize existing QPTs from an HPC perspective, investigating if existing QPTs have the potential to be efficiently integrated with classical computing models and determining where work is still required. This work structures a set of criteria into an analysis blueprint that enables HPC scientists to assess whether a QPT is suitable for the quantum-accelerated classical application at hand.
We investigate a more generalized form of submodular maximization, referred to as $k$-submodular maximization, with applications across social networks and machine learning domains. In this work, we propose the multilinear extension of $k$-submodular functions and unified Frank-Wolfe-type frameworks based on that. Our frameworks accomodate 1) monotone or non-monotone functions, and 2) various constraint types including matroid constraints, knapsack constraints, and their combinations. Notably, we attain an asymptotically optimal $1/2$-approximation for monotone $k$-submodular maximization problems with knapsack constraints, surpassing the previous $1/3$-approximation. The foundation for our analysis stems from new insights into specific linear and monotone properties pertaining to the multilinear extension.
This article presents MAPS$^2$ : a distributed algorithm that allows multi-robot systems to deliver coupled tasks expressed as Signal Temporal Logic (STL) constraints. Classical control theoretical tools addressing STL constraints either adopt a limited fragment of the STL formula or require approximations of min/max operators, whereas works maximising robustness through optimisation-based methods often suffer from local minima, relaxing any completeness arguments due to the NP-hard nature of the problem. Endowed with probabilistic guarantees, MAPS$^2$ provides an anytime algorithm that iteratively improves the robots' trajectories. The algorithm selectively imposes spatial constraints by taking advantage of the temporal properties of the STL. The algorithm is distributed, in the sense that each robot calculates its trajectory by communicating only with its immediate neighbours as defined via a communication graph. We illustrate the efficiency of MAPS$^2$ by conducting extensive simulation and experimental studies, verifying the generation of STL satisfying trajectories.
We study the complexity of producing $(\delta,\epsilon)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $\Omega(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(d\delta^{-1}\epsilon^{-3})$, which is optimal (up to numerical constants) with respect to $d$ and also optimal with respect to the accuracy parameters $\delta,\epsilon$, thus solving an open question due to Lin et al. (NeurIPS'22). Moreover, the convergence rate achieved by our algorithm is also optimal for smooth objectives, proving that in the nonconvex stochastic zero-order setting, nonsmooth optimization is as easy as smooth optimization. We provide algorithms that achieve the aforementioned convergence rate in expectation as well as with high probability. Our analysis is based on a simple yet powerful geometric lemma regarding the Goldstein-subdifferential set, which allows utilizing recent advancements in first-order nonsmooth nonconvex optimization.
We provide estimates on the fat-shattering dimension of aggregation rules of real-valued function classes. The latter consists of all ways of choosing $k$ functions, one from each of the $k$ classes, and computing a pointwise function of them, such as the median, mean, and maximum. The bound is stated in terms of the fat-shattering dimensions of the component classes. For linear and affine function classes, we provide a considerably sharper upper bound and a matching lower bound, achieving, in particular, an optimal dependence on $k$. Along the way, we improve several known results in addition to pointing out and correcting a number of erroneous claims in the literature.
We study the complexity of the problem of verifying differential privacy for while-like programs working over boolean values and making probabilistic choices. Programs in this class can be interpreted into finite-state discrete-time Markov Chains (DTMC). We show that the problem of deciding whether a program is differentially private for specific values of the privacy parameters is PSPACE-complete. To show that this problem is in PSPACE, we adapt classical results about computing hitting probabilities for DTMC. To show PSPACE-hardness we use a reduction from the problem of checking whether a program almost surely terminates or not. We also show that the problem of approximating the privacy parameters that a program provides is PSPACE-hard. Moreover, we investigate the complexity of similar problems also for several relaxations of differential privacy: R\'enyi differential privacy, concentrated differential privacy, and truncated concentrated differential privacy. For these notions, we consider gap-versions of the problem of deciding whether a program is private or not and we show that all of them are PSPACE-complete.
In this paper we consider a mathematical model which describes the equilibrium of two elastic rods attached to a nonlinear spring. We derive the variational formulation of the model which is in the form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the problem, then we state and prove some convergence results, for which we provide the corresponding mechanical interpretation. Next, we turn to the numerical approximation of the problem based on a finite element scheme. We use a relaxation method to solve the discrete problems that we implement on the computer. Using this method, we provide numerical simulations which validate our convergence results.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191