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We study two classic variants of block-structured integer programming. Two-stage stochastic programs are integer programs of the form $\{A_i \mathbf{x} + D_i \mathbf{y}_i = \mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$, where $A_i$ and $D_i$ are bounded-size matrices. On the other hand, $n$-fold programs are integer programs of the form $\{{\sum_{i=1}^n C_i\mathbf{y}_i=\mathbf{a}} \textrm{ and } D_i\mathbf{y}_i=\mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$, where again $C_i$ and $D_i$ are bounded-size matrices. It is known that solving these kind of programs is fixed-parameter tractable when parameterized by the maximum dimension among the relevant matrices $A_i,C_i,D_i$ and the maximum absolute value of any entry appearing in the constraint matrix. We show that the parameterized tractability results for two-stage stochastic and $n$-fold programs persist even when one allows large entries in the global part of the program. More precisely, we prove that: - The feasibility problem for two-stage stochastic programs is fixed-parameter tractable when parameterized by the dimensions of matrices $A_i,D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we allow matrices $A_i$ to have arbitrarily large entries. - The linear optimization problem for $n$-fold integer programs that are uniform -- all matrices $C_i$ are equal -- is fixed-parameter tractable when parameterized by the dimensions of matrices $C_i$ and $D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we require that $C_i=C$ for all $i=1,\ldots,n$, but we allow $C$ to have arbitrarily large entries. In the second result, the uniformity assumption is necessary; otherwise the problem is $\mathsf{NP}$-hard already when the parameters take constant values. Both our algorithms are weakly polynomial: the running time is measured in the total bitsize of the input.

We study stability properties of the expected utility function in Bayesian optimal experimental design. We provide a framework for this problem in a non-parametric setting and prove a convergence rate of the expected utility with respect to a likelihood perturbation. This rate is uniform over the design space and its sharpness in the general setting is demonstrated by proving a lower bound in a special case. To make the problem more concrete we proceed by considering non-linear Bayesian inverse problems with Gaussian likelihood and prove that the assumptions set out for the general case are satisfied and regain the stability of the expected utility with respect to perturbations to the observation map. Theoretical convergence rates are demonstrated numerically in three different examples.

The Causal Roadmap outlines a systematic approach to our research endeavors: define quantity of interest, evaluate needed assumptions, conduct statistical estimation, and carefully interpret of results. At the estimation step, it is essential that the estimation algorithm be chosen thoughtfully for its theoretical properties and expected performance. Simulations can help researchers gain a better understanding of an estimator's statistical performance under conditions unique to the real-data application. This in turn can inform the rigorous pre-specification of a Statistical Analysis Plan (SAP), not only stating the estimand (e.g., G-computation formula), the estimator (e.g., targeted minimum loss-based estimation [TMLE]), and adjustment variables, but also the implementation of the estimator -- including nuisance parameter estimation and approach for variance estimation. Doing so helps ensure valid inference (e.g., 95% confidence intervals with appropriate coverage). Failing to pre-specify estimation can lead to data dredging and inflated Type-I error rates.

Ising solvers offer a promising physics-based approach to tackle the challenging class of combinatorial optimization problems. However, typical solvers operate in a quadratic energy space, having only pair-wise coupling elements which already dominate area and energy. We show that such quadratization can cause severe problems: increased dimensionality, a rugged search landscape, and misalignment with the original objective function. Here, we design and quantify a higher-order Hopfield optimization solver, with 28nm CMOS technology and memristive couplings for lower area and energy computations. We combine algorithmic and circuit analysis to show quantitative advantages over quadratic Ising Machines (IM)s, yielding 48x and 72x reduction in time-to-solution (TTS) and energy-to-solution (ETS) respectively for Boolean satisfiability problems of 150 variables, with favorable scaling.

During the development of large language models (LLMs), the scale and quality of the pre-training data play a crucial role in shaping LLMs' capabilities. To accelerate the research of LLMs, several large-scale datasets, such as C4 [1], Pile [2], RefinedWeb [3] and WanJuan [4], have been released to the public. However, most of the released corpus focus mainly on English, and there is still lack of complete tool-chain for extracting clean texts from web data. Furthermore, fine-grained information of the corpus, e.g. the quality of each text, is missing. To address these challenges, we propose in this paper a new complete tool-chain EvalWeb to extract Chinese clean texts from noisy web data. First, similar to previous work, manually crafted rules are employed to discard explicit noisy texts from the raw crawled web contents. Second, a well-designed evaluation model is leveraged to assess the remaining relatively clean data, and each text is assigned a specific quality score. Finally, we can easily utilize an appropriate threshold to select the high-quality pre-training data for Chinese. Using our proposed approach, we release the largest and latest large-scale high-quality Chinese web text ChineseWebText, which consists of 1.42 TB and each text is associated with a quality score, facilitating the LLM researchers to choose the data according to the desired quality thresholds. We also release a much cleaner subset of 600 GB Chinese data with the quality exceeding 90%.

Pre-trained language models can be surprisingly adept at tasks they were not explicitly trained on, but how they implement these capabilities is poorly understood. In this paper, we investigate the basic mathematical abilities often acquired by pre-trained language models. Concretely, we use mechanistic interpretability techniques to explain the (limited) mathematical abilities of GPT-2 small. As a case study, we examine its ability to take in sentences such as "The war lasted from the year 1732 to the year 17", and predict valid two-digit end years (years > 32). We first identify a circuit, a small subset of GPT-2 small's computational graph that computes this task's output. Then, we explain the role of each circuit component, showing that GPT-2 small's final multi-layer perceptrons boost the probability of end years greater than the start year. Finally, we find related tasks that activate our circuit. Our results suggest that GPT-2 small computes greater-than using a complex but general mechanism that activates across diverse contexts.

Cloud computing and the evolution of management methodologies such as Lean Management or Agile entail a profound transformation in both system construction and maintenance approaches. These practices are encompassed within the term "DevOps." This descriptive approach to an information system or application, alongside the configuration of its constituent components, has necessitated the development of descriptive languages paired with specialized engines for automating systems administration tasks. Among these, the tandem of Ansible (engine) and YAML (descriptive language) stands out as the two most prevalent tools in the market, facing notable competition mainly from Terraform. The current document presents an inquiry into a solution for generating and managing Ansible YAML roles and playbooks, utilizing Generative LLMs (Language Models) to translate human descriptions into code. Our efforts are focused on identifying plausible directions and outlining the potential industrial applications. Note: For the purpose of this experiment, we have opted against the use of Ansible Lightspeed. This is due to its reliance on an IBM Watson model, for which we have not found any publicly available references. Comprehensive information regarding this remarkable technology can be found [1] directly on our partner's website, RedHat.

We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the geometry of the domain.

This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard inner product. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring-Wirth non-expansive condition under non-standard metrics while preserving isometry. Building upon the novel vector transport operators, we extend the modified Polak-Ribi$\acute{e}$re-Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis.

Novel categories are commonly defined as those unobserved during training but present during testing. However, partially labelled training datasets can contain unlabelled training samples that belong to novel categories, meaning these can be present in training and testing. This research is the first to generalise between what we call observed-novel and unobserved-novel categories within a new learning policy called open-set learning with augmented category by exploiting unlabelled data or Open-LACU. After surveying existing learning policies, we introduce Open-LACU as a unified policy of positive and unlabelled learning, semi-supervised learning and open-set recognition. Subsequently, we develop the first Open-LACU model using an algorithmic training process of the relevant research fields. The proposed Open-LACU classifier achieves state-of-the-art and first-of-its-kind results.