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In this critical survey, we analyze typical claims on the relationship between explainable AI (XAI) and fairness to disentangle the multidimensional relationship between these two concepts. Based on a systematic literature review and a subsequent qualitative content analysis, we identify seven archetypal claims from 175 papers on the alleged fairness benefits of XAI. We present crucial caveats with respect to these claims and provide an entry point for future discussions around the potentials and limitations of XAI for specific fairness desiderata. Importantly, we notice that claims are often (i) vague and simplistic, (ii) lacking normative grounding, or (iii) poorly aligned with the actual capabilities of XAI. We encourage to conceive XAI not as an ethical panacea but as one of many tools to approach the multidimensional, sociotechnical challenge of algorithmic fairness. Moreover, when making a claim about XAI and fairness, we emphasize the need to be more specific about what kind of XAI method is used and which fairness desideratum it refers to, how exactly it enables fairness, and who is the stakeholder that benefits from XAI.

The probabilistic formal verification (PFV) of AI systems is in its infancy. So far, approaches have been limited to ad-hoc algorithms for specific classes of models and/or properties. We propose a unifying framework for the PFV of AI systems based onWeighted Model Integration (WMI), which allows to frame the problem in very general terms. Crucially, this reduction enables the verification of many properties of interest, like fairness, robustness or monotonicity, over a wide range of machine learning models, without making strong distributional assumptions. We support the generality of the approach by solving multiple verification tasks with a single, off-the-shelf WMI solver, then discuss the scalability challenges and research directions related to this promising framework.

This study investigates the asymptotic dynamics of alternating minimization applied to optimize a bilinear non-convex function with normally distributed covariates. We employ the replica method from statistical physics in a multi-step approach to precisely trace the algorithm's evolution. Our findings indicate that the dynamics can be described effectively by a two--dimensional discrete stochastic process, where each step depends on all previous time steps, revealing a memory dependency in the procedure. The theoretical framework developed in this work is broadly applicable for the analysis of various iterative algorithms, extending beyond the scope of alternating minimization.

In optimization-based approaches to inverse problems and to statistical estimation, it is common to augment criteria that enforce data fidelity with a regularizer that promotes desired structural properties in the solution. The choice of a suitable regularizer is typically driven by a combination of prior domain information and computational considerations. Convex regularizers are attractive computationally but they are limited in the types of structure they can promote. On the other hand, nonconvex regularizers are more flexible in the forms of structure they can promote and they have showcased strong empirical performance in some applications, but they come with the computational challenge of solving the associated optimization problems. In this paper, we seek a systematic understanding of the power and the limitations of convex regularization by investigating the following questions: Given a distribution, what is the optimal regularizer for data drawn from the distribution? What properties of a data source govern whether the optimal regularizer is convex? We address these questions for the class of regularizers specified by functionals that are continuous, positively homogeneous, and positive away from the origin. We say that a regularizer is optimal for a data distribution if the Gibbs density with energy given by the regularizer maximizes the population likelihood (or equivalently, minimizes cross-entropy loss) over all regularizer-induced Gibbs densities. As the regularizers we consider are in one-to-one correspondence with star bodies, we leverage dual Brunn-Minkowski theory to show that a radial function derived from a data distribution is akin to a ``computational sufficient statistic'' as it is the key quantity for identifying optimal regularizers and for assessing the amenability of a data source to convex regularization.

Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Characteristics of the MLE problem for algebraic statistical models are reflected in the geometry of the \textit{likelihood correspondence}, a variety that ties together data and their maximum likelihood estimators. We construct the ideal of the likelihood correspondence for the large class of toric models and find a Gr\"{o}bner basis in the case of complete and joint independence models arising from multi-way contingency tables. These results provide insight into their properties and offer faster computational strategies for solving the MLE problem.

Image captioning models are typically trained by treating all samples equally, neglecting to account for mismatched or otherwise difficult data points. In contrast, recent work has shown the effectiveness of training models by scheduling the data using curriculum learning strategies. This paper contributes to this direction by actively curating difficult samples in datasets without increasing the total number of samples. We explore the effect of using three data curation methods within the training process: complete removal of an sample, caption replacement, or image replacement via a text-to-image generation model. Experiments on the Flickr30K and COCO datasets with the BLIP and BEiT-3 models demonstrate that these curation methods do indeed yield improved image captioning models, underscoring their efficacy.

The execution of graph algorithms using neural networks has recently attracted significant interest due to promising empirical progress. This motivates further understanding of how neural networks can replicate reasoning steps with relational data. In this work, we study the ability of transformer networks to simulate algorithms on graphs from a theoretical perspective. The architecture that we utilize is a looped transformer with extra attention heads that interact with the graph. We prove by construction that this architecture can simulate algorithms such as Dijkstra's shortest path algorithm, Breadth- and Depth-First Search, and Kosaraju's strongly connected components algorithm. The width of the network does not increase with the size of the input graph, which implies that the network can simulate the above algorithms for any graph. Despite this property, we show that there is a limit to simulation in our solution due to finite precision. Finally, we show a Turing Completeness result with constant width when the extra attention heads are utilized.

Knowledge graphs represent factual knowledge about the world as relationships between concepts and are critical for intelligent decision making in enterprise applications. New knowledge is inferred from the existing facts in the knowledge graphs by encoding the concepts and relations into low-dimensional feature vector representations. The most effective representations for this task, called Knowledge Graph Embeddings (KGE), are learned through neural network architectures. Due to their impressive predictive performance, they are increasingly used in high-impact domains like healthcare, finance and education. However, are the black-box KGE models adversarially robust for use in domains with high stakes? This thesis argues that state-of-the-art KGE models are vulnerable to data poisoning attacks, that is, their predictive performance can be degraded by systematically crafted perturbations to the training knowledge graph. To support this argument, two novel data poisoning attacks are proposed that craft input deletions or additions at training time to subvert the learned model's performance at inference time. These adversarial attacks target the task of predicting the missing facts in knowledge graphs using KGE models, and the evaluation shows that the simpler attacks are competitive with or outperform the computationally expensive ones. The thesis contributions not only highlight and provide an opportunity to fix the security vulnerabilities of KGE models, but also help to understand the black-box predictive behaviour of KGE models.

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.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.