In this paper we explore the challenges and strategies for enhancing the robustness of $k$-means clustering algorithms against adversarial manipulations. We evaluate the vulnerability of clustering algorithms to adversarial attacks, emphasising the associated security risks. Our study investigates the impact of incremental attack strength on training, introduces the concept of transferability between supervised and unsupervised models, and highlights the sensitivity of unsupervised models to sample distributions. We additionally introduce and evaluate an adversarial training method that improves testing performance in adversarial scenarios, and we highlight the importance of various parameters in the proposed training method, such as continuous learning, centroid initialisation, and adversarial step-count.
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Database queries are often used to select and rank items as decision support for many applications. As automated decision-making tools become more prevalent, there is a growing recognition of the need to diversify their outcomes. In this paper, we define and study the problem of modifying the selection conditions of an ORDER BY query so that the result of the modified query closely fits some user-defined notion of diversity while simultaneously maintaining the intent of the original query. We show the hardness of this problem and propose a Mixed Integer Linear Programming (MILP) based solution. We further present optimizations designed to enhance the scalability and applicability of the solution in real-life scenarios. We investigate the performance characteristics of our algorithm and show its efficiency and the usefulness of our optimizations.
Quantitative analysis of probabilistic programs aims at deriving tight numerical bounds for probabilistic properties such as expectation and assertion probability, and plays a crucial role in the verification of probabilistic programs. Along this line of research, most existing works consider numerical bounds over the whole state space monolithically and do not consider piecewise bounds. Clearly, monolithic bounds are either conservative, or not expressive and succinct enough in general. To derive more succinct, expressive and precise numerical bounds for probabilistic properties, we propose a novel approach for synthesizing piecewise linear bounds in this work. To this end, we first show how to extract a piecewise feature w.r.t. a given quantitative property from a probabilistic program using latticed $k$-induction that captures a wide and representative class of piecewise bound functions. Second, we develop an algorithmic approach to synthesize piecewise linear upper and lower bounds from the piecewise feature, for which we show that the synthesis of piecewise linear bounds can be reduced to bilinear programming. Third, we implement our approach with the bilinear programming solver Gurobi. The experimental results indicate that our approach is capable of generating tight or even accurate piecewise linear bounds for an extensive set of benchmarks compared with the state of the art.
In this work, we present three linear numerical schemes to model nematic liquid crystals using the Landau-de Gennes $\textbf{Q}$-tensor theory. The first scheme is based on using a truncation procedure of the energy, which allows for an unconditionally energy stable first order accurate decoupled scheme. The second scheme uses a modified second order accurate optimal dissipation algorithm, which gives a second order accurate coupled scheme. Finally, the third scheme uses a new idea to decouple the unknowns from the second scheme which allows us to obtain accurate dynamics while improving computational efficiency. We present several numerical experiments to offer a comparative study of the accuracy, efficiency and the ability of the numerical schemes to represent realistic dynamics.
The present work is devoted to strong approximations of a generalized A\"{i}t-Sahalia model arising from mathematical finance. The numerical study of the considered model faces essential difficulties caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. In this paper, a novel explicit Euler-type scheme is proposed, which is easily implementable and able to preserve positivity of the original model unconditionally, i.e., for any time step-size $h >0$. A mean-square convergence rate of order $0.5$ is also obtained for the proposed scheme in both non-critical and general critical cases. Our work is motivated by the need to justify the multi-level Monte Carlo (MLMC) simulations for the underlying model, where the rate of mean-square convergence is required and the preservation of positivity is desirable particularly for large discretization time steps. Numerical experiments are finally provided to confirm the theoretical findings.
In the context of linear regression, we construct a data-driven convex loss function with respect to which empirical risk minimisation yields optimal asymptotic variance in the downstream estimation of the regression coefficients. Our semiparametric approach targets the best decreasing approximation of the derivative of the log-density of the noise distribution. At the population level, this fitting process is a nonparametric extension of score matching, corresponding to a log-concave projection of the noise distribution with respect to the Fisher divergence. The procedure is computationally efficient, and we prove that our procedure attains the minimal asymptotic covariance among all convex $M$-estimators. As an example of a non-log-concave setting, for Cauchy errors, the optimal convex loss function is Huber-like, and our procedure yields an asymptotic efficiency greater than 0.87 relative to the oracle maximum likelihood estimator of the regression coefficients that uses knowledge of this error distribution; in this sense, we obtain robustness without sacrificing much efficiency. Numerical experiments confirm the practical merits of our proposal.
The pathfinding problem, which aims to identify a collision-free path between two points, is crucial for many applications, such as robot navigation and autonomous driving. Classic methods, such as A$^*$ search, perform well on small-scale maps but face difficulties scaling up. Conversely, data-driven approaches can improve pathfinding efficiency but require extensive data labeling and lack theoretical guarantees, making it challenging for practical applications. To combine the strengths of the two methods, we utilize the imperative learning (IL) strategy and propose a novel self-supervised pathfinding framework, termed imperative learning-based A$^*$ (iA$^*$). Specifically, iA$^*$ is a bilevel optimization process where the lower-level optimization is dedicated to finding the optimal path by a differentiable A$^*$ search module, and the upper-level optimization narrows down the search space to improve efficiency via setting suitable initial values from a data-driven model. Besides, the model within the upper-level optimization is a fully convolutional network, trained by the calculated loss in the lower-level optimization. Thus, the framework avoids extensive data labeling and can be applied in diverse environments. Our comprehensive experiments demonstrate that iA$^*$ surpasses both classical and data-driven methods in pathfinding efficiency and shows superior robustness among different tasks, validated with public datasets and simulation environments.
We show a generalization of the crossing lemma for multi-graphs drawn on orientable surfaces in which pairs of edges are assumed to be drawn by non-homotopic simple arcs which pairwise cross at most $k$ times.
This paper proposes to develop a new variant of the two-time-scale stochastic approximation to find the roots of two coupled nonlinear operators, assuming only noisy samples of these operators can be observed. Our key idea is to leverage the classic Ruppert-Polyak averaging technique to dynamically estimate the operators through their samples. The estimated values of these averaging steps will then be used in the two-time-scale stochastic approximation updates to find the desired solution. Our main theoretical result is to show that under the strongly monotone condition of the underlying nonlinear operators the mean-squared errors of the iterates generated by the proposed method converge to zero at an optimal rate $O(1/k)$, where $k$ is the number of iterations. Our result significantly improves the existing result of two-time-scale stochastic approximation, where the best known finite-time convergence rate is $O(1/k^{2/3})$. We illustrate this result by applying the proposed method to develop new reinforcement learning algorithms with improved performance.
We prove that the single-site Glauber dynamics for sampling proper $q$-colorings mixes in $O_\Delta(n\log n)$ time on line graphs with $n$ vertices and maximum degree $\Delta$ when $q>(1+o(1))\Delta$. The main tool in our proof is the matrix trickle-down theorem developed by Abdolazimi, Liu and Oveis Gharan (FOCS, 2021).
We consider the problem of finding weights and biases for a two-layer fully connected neural network to fit a given set of data points as well as possible, also known as EmpiricalRiskMinimization. Our main result is that the associated decision problem is $\exists\mathbb{R}$-complete, that is, polynomial-time equivalent to determining whether a multivariate polynomial with integer coefficients has any real roots. Furthermore, we prove that algebraic numbers of arbitrarily large degree are required as weights to be able to train some instances to optimality, even if all data points are rational. Our result already applies to fully connected instances with two inputs, two outputs, and one hidden layer of ReLU neurons. Thereby, we strengthen a result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021]. A consequence of this is that a combinatorial search algorithm like the one by Arora, Basu, Mianjy and Mukherjee [ICLR 2018] is impossible for networks with more than one output dimension, unless $\mathsf{NP}=\exists\mathbb{R}$.
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