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We consider the problem of ranking a set of objects based on their performance when the measurement of said performance is subject to noise. In this scenario, the performance is measured repeatedly, resulting in a range of measurements for each object. If the ranges of two objects do not overlap, then we consider one object as 'better' than the other, and we expect it to receive a higher rank; if, however, the ranges overlap, then the objects are incomparable, and we wish them to be assigned the same rank. Unfortunately, the incomparability relation of ranges is in general not transitive; as a consequence, in general the two requirements cannot be satisfied simultaneously, i.e., it is not possible to guarantee both distinct ranks for objects with separated ranges, and same rank for objects with overlapping ranges. This conflict leads to more than one reasonable way to rank a set of objects. In this paper, we explore the ambiguities that arise when ranking with ties, and define a set of reasonable rankings, which we call partial rankings. We develop and analyse three different methodologies to compute a partial ranking. Finally, we show how performance differences among objects can be investigated with the help of partial ranking.

Several decades ago the Proximal Point Algorithm (PPA) started to gain a long-lasting attraction for both abstract operator theory and numerical optimization communities. Even in modern applications, researchers still use proximal minimization theory to design scalable algorithms that overcome nonsmoothness. Remarkable works as \cite{Fer:91,Ber:82constrained,Ber:89parallel,Tom:11} established tight relations between the convergence behaviour of PPA and the regularity of the objective function. In this manuscript we derive nonasymptotic iteration complexity of exact and inexact PPA to minimize convex functions under $\gamma-$Holderian growth: $\BigO{\log(1/\epsilon)}$ (for $\gamma \in [1,2]$) and $\BigO{1/\epsilon^{\gamma - 2}}$ (for $\gamma > 2$). In particular, we recover well-known results on PPA: finite convergence for sharp minima and linear convergence for quadratic growth, even under presence of deterministic noise. Moreover, when a simple Proximal Subgradient Method is recurrently called as an inner routine for computing each IPPA iterate, novel computational complexity bounds are obtained for Restarting Inexact PPA. Our numerical tests show improvements over existing restarting versions of the Subgradient Method.

Deep Neural Networks (DNNs) are known to be vulnerable to backdoor attacks, posing concerning threats to their reliable deployment. Recent research reveals that backdoors can be erased from infected DNNs by pruning a specific group of neurons, while how to effectively identify and remove these backdoor-associated neurons remains an open challenge. In this paper, we investigate the correlation between backdoor behavior and neuron magnitude, and find that backdoor neurons deviate from the magnitude-saliency correlation of the model. The deviation inspires us to propose a Magnitude-based Neuron Pruning (MNP) method to detect and prune backdoor neurons. Specifically, MNP uses three magnitude-guided objective functions to manipulate the magnitude-saliency correlation of backdoor neurons, thus achieving the purpose of exposing backdoor behavior, eliminating backdoor neurons and preserving clean neurons, respectively. Experiments show our pruning strategy achieves state-of-the-art backdoor defense performance against a variety of backdoor attacks with a limited amount of clean data, demonstrating the crucial role of magnitude for guiding backdoor defenses.

Reasoning capabilities are crucial for Large Language Models (LLMs), yet a notable gap exists between English and non-English languages. To bridge this disparity, some works fine-tune LLMs to relearn reasoning capabilities in non-English languages, while others replace non-English inputs with an external model's outputs such as English translation text to circumvent the challenge of LLM understanding non-English. Unfortunately, these methods often underutilize the built-in skilled reasoning and useful language understanding capabilities of LLMs. In order to better utilize the minds of reasoning and language understanding in LLMs, we propose a new method, namely MindMerger, which merges LLMs with the external language understanding capabilities from multilingual models to boost the multilingual reasoning performance. Furthermore, a two-step training scheme is introduced to first train to embeded the external capabilities into LLMs and then train the collaborative utilization of the external capabilities and the built-in capabilities in LLMs. Experiments on three multilingual reasoning datasets and a language understanding dataset demonstrate that MindMerger consistently outperforms all baselines, especially in low-resource languages. Without updating the parameters of LLMs, the average accuracy improved by 6.7% and 8.0% across all languages and low-resource languages on the MGSM dataset, respectively.

Large Language Models (LLMs) have become central to advancing automation and decision-making across various sectors, raising significant ethical questions. This study proposes a comprehensive comparative analysis of the most advanced LLMs to assess their moral profiles. We subjected several state-of-the-art models to a selection of ethical dilemmas and found that all the proprietary ones are mostly utilitarian and all of the open-weights ones align mostly with values-based ethics. Furthermore, when using the Moral Foundations Questionnaire, all models we probed - except for Llama 2- displayed a strong liberal bias. Lastly, in order to causally intervene in one of the studied models, we propose a novel similarity-specific activation steering technique. Using this method, we were able to reliably steer the model's moral compass to different ethical schools. All of these results showcase that there is an ethical dimension in already deployed LLMs, an aspect that is generally overlooked.

Most popular benchmarks for comparing LLMs rely on a limited set of prompt templates, which may not fully capture the LLMs' abilities and can affect the reproducibility of results on leaderboards. Many recent works empirically verify prompt sensitivity and advocate for changes in LLM evaluation. In this paper, we consider the problem of estimating the performance distribution across many prompt variants instead of finding a single prompt to evaluate with. We introduce PromptEval, a method for estimating performance across a large set of prompts borrowing strength across prompts and examples to produce accurate estimates under practical evaluation budgets. The resulting distribution can be used to obtain performance quantiles to construct various robust performance metrics (e.g., top 95% quantile or median). We prove that PromptEval consistently estimates the performance distribution and demonstrate its efficacy empirically on three prominent LLM benchmarks: MMLU, BIG-bench Hard, and LMentry. For example, PromptEval can accurately estimate performance quantiles across 100 prompt templates on MMLU with a budget equivalent to two single-prompt evaluations. Our code and data can be found at //github.com/felipemaiapolo/prompt-eval.

Accounting for uncertainty in Data quality is important for accurate statistical inference. We aim to an optimal conservative allocation for a large universe of assets in mean-variance portfolio (MVP), which is the worst choice within uncertainty in data distribution. Unlike the low dimensional MVP studied in Blanchet et al. (2022, Management Science), the large number of assets raises a challenging problem in quantifying the uncertainty, due to the big deviation of the sample covariance matrix from the population version. To overcome this difficulty, we propose a data-adaptive method to quantify the uncertainty with the help of a factor structure. Monte-Carlo Simulation is conducted to show the superiority of our method in high-dimensional cases, that, avoiding the over-conservative results in Blanchet et al. (2022), our allocation is closer to the oracle version in terms of risk minimization and expected portfolio return controlling.

Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over Wasserstein distance. In this paper we prove that the infinite dimensionality of the space of probabilities drastically deteriorates its sample complexity, which is slower than any polynomial rate in the sample size. We propose a new distance that preserves many desirable properties of the former while achieving a parametric rate of convergence. In particular, our distance 1) metrizes weak convergence; 2) can be estimated numerically through samples with low complexity; 3) can be bounded analytically from above and below. The main ingredient are integral probability metrics, which lead to the name hierarchical IPM.

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior - an approximation of the steady solution - which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

Recommender systems are widely used in big information-based companies such as Google, Twitter, LinkedIn, and Netflix. A recommender system deals with the problem of information overload by filtering important information fragments according to users' preferences. In light of the increasing success of deep learning, recent studies have proved the benefits of using deep learning in various recommendation tasks. However, most proposed techniques only aim to target individuals, which cannot be efficiently applied in group recommendation. In this paper, we propose a deep learning architecture to solve the group recommendation problem. On the one hand, as different individual preferences in a group necessitate preference trade-offs in making group recommendations, it is essential that the recommendation model can discover substitutes among user behaviors. On the other hand, it has been observed that a user as an individual and as a group member behaves differently. To tackle such problems, we propose using an attention mechanism to capture the impact of each user in a group. Specifically, our model automatically learns the influence weight of each user in a group and recommends items to the group based on its members' weighted preferences. We conduct extensive experiments on four datasets. Our model significantly outperforms baseline methods and shows promising results in applying deep learning to the group recommendation problem.