It is well known that this is unlearnable, but we sketch a proof here.

During training, generative large language models (LLMs) are exposed to vast amounts of information, including data relevant to economic modelling, such as geospatial statistics and firm-level financial metrics. If LLMs can effectively retrieve and utilise this knowledge, they could reduce dependence on external data sources that are time-consuming to access, clean, and merge, or that incur financial costs. Moreover, if LLMs accurately represent data, they could support downstream tasks like data imputation and outlier detection. In this study, we evaluate whether and how LLMs can be used for typical economic data processes. Not all knowledge within an LLM may be explicit and retrievable in natural language by prompting the model.

We provide tight upper and lower bounds on the complexity of minimizing the average of m convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient descent (AGD) and an accelerated variant of SVRG are optimal in the deterministic and randomized settings respectively, and that a gradient oracle is sufficient for the optimal rate. For non-smooth functions, having access to prox oracles reduces the complexity and we present optimal methods based on smoothing that improve over methods using just gradient accesses.

It is well known that this is unlearnable, but we sketch a proof here.

Recent studies have increasingly focused on non-asymptotic convergence analyses for actor-critic (AC) algorithms. One such effort introduced a two-timescale critic-actor algorithm for the discounted cost setting using a tabular representation, where the usual roles of the actor and critic are reversed. However, only asymptotic convergence was established there. Subsequently, both asymptotic and non-asymptotic analyses of the critic-actor algorithm with linear function approximation were conducted. In our work, we introduce the first natural critic-actor algorithm with function approximation for the long-run average cost setting and under inequality constraints. We provide the non-asymptotic convergence guarantees for this algorithm. Our analysis establishes optimal learning rates and we also propose a modification to enhance sample complexity. We further show the results of experiments on three different Safety-Gym environments where our algorithm is found to be competitive in comparison with other well known algorithms.

In our notation, the model in Dasgupta et al. [4] would have score function F As presented in Dasgupta et al. Although the model was proposed and analyzed in Dasgupta et al. Remark 2. Note that the mean of The proof is by direct calculation. The following lemma will be helpful in proving the next part. Given a threshold T, temperature hyperparameters,, there exists and a bijection on the set of parameterizations {V!

Multi-step reasoning is a fundamental challenge in artificial intelligence, with applications ranging from mathematical problem-solving to decision-making in dynamic environments. Reinforcement Learning (RL) has shown promise in enabling agents to perform multi-step reasoning by optimizing long-term rewards. However, conventional RL methods struggle with complex reasoning tasks due to issues such as credit assignment, high-dimensional state representations, and stability concerns. Recent advancements in Transformer architectures and hyperbolic geometry have provided novel solutions to these challenges. This paper introduces a new framework that integrates hyperbolic Transformers into RL for multi-step reasoning. The proposed approach leverages hyperbolic embeddings to model hierarchical structures effectively. We present theoretical insights, algorithmic details, and experimental results that include Frontier Math and nonlinear optimal control problems. Compared to RL with vanilla transformer, the hyperbolic RL largely improves accuracy by (32%~44%) on FrontierMath benchmark, (43%~45%) on nonlinear optimal control benchmark, while achieving impressive reduction in computational time by (16%~32%) on FrontierMath benchmark, (16%~17%) on nonlinear optimal control benchmark. Our work demonstrates the potential of hyperbolic Transformers in reinforcement learning, particularly for multi-step reasoning tasks that involve hierarchical structures.

This paper studies clustering algorithms for dynamically evolving graphs $\{G_t\}$, in which new edges (and potential new vertices) are added into a graph, and the underlying cluster structure of the graph can gradually change. The paper proves that, under some mild condition on the cluster-structure, the clusters of the final graph $G_T$ of $n_T$ vertices at time $T$ can be well approximated by a dynamic variant of the spectral clustering algorithm. The algorithm runs in amortised update time $O(1)$ and query time $o(n_T)$. Experimental studies on both synthetic and real-world datasets further confirm the practicality of our designed algorithm.

Bayesian networks (BNs) are a foundational model in machine learning and causal inference. Their graphical structure can handle high-dimensional problems, divide them into a sparse collection of smaller ones, underlies Judea Pearl's causality, and determines their explainability and interpretability. Despite their popularity, there are almost no resources in the literature on how to compute Shannon's entropy and the Kullback-Leibler (KL) divergence for BNs under their most common distributional assumptions. In this paper, we provide computationally efficient algorithms for both by leveraging BNs' graphical structure, and we illustrate them with a complete set of numerical examples. In the process, we show it is possible to reduce the computational complexity of KL from cubic to quadratic for Gaussian BNs.

Network data is prevalent in numerous big data applications including economics and health networks where it is of prime importance to understand the latent structure of network. In this paper, we model the network using the Degree-Corrected Mixed Membership (DCMM) model. In DCMM model, for each node $i$, there exists a membership vector $\boldsymbol{\pi}_ i = (\boldsymbol{\pi}_i(1), \boldsymbol{\pi}_i(2),\ldots, \boldsymbol{\pi}_i(K))$, where $\boldsymbol{\pi}_i(k)$ denotes the weight that node $i$ puts in community $k$. We derive novel finite-sample expansion for the $\boldsymbol{\pi}_i(k)$s which allows us to obtain asymptotic distributions and confidence interval of the membership mixing probabilities and other related population quantities. This fills an important gap on uncertainty quantification on the membership profile. We further develop a ranking scheme of the vertices based on the membership mixing probabilities on certain communities and perform relevant statistical inferences. A multiplier bootstrap method is proposed for ranking inference of individual member's profile with respect to a given community. The validity of our theoretical results is further demonstrated by via numerical experiments in both real and synthetic data examples.