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Despite their sophisticated capabilities, large language models (LLMs) encounter a major hurdle in effective assessment. This paper first revisits the prevalent evaluation method-multiple choice question answering (MCQA), which allows for straightforward accuracy measurement. Through a comprehensive evaluation of 24 models across 11 benchmarks, we highlight several potential drawbacks of MCQA, for instance, the inconsistency between the MCQA evaluation and the generation of open-ended responses in practical scenarios. In response, we introduce an RWQ-Elo rating system, engaging 24 LLMs such as GPT-4, GPT-3.5, Google-Gemini-Pro and LLaMA-1/-2, in a two-player competitive format, with GPT-4 serving as the judge. Each LLM receives an Elo rating thereafter. This system is designed to mirror real-world usage, and for this purpose, we have compiled a new benchmark called ``Real-world questions'' (RWQ), comprising 20,772 authentic user inquiries. Additionally, we thoroughly analyze the characteristics of our system and compare it with prior leaderboards like AlpacaEval and MT-Bench. Our analysis reveals the stability of our RWQ-Elo system, the feasibility of registering new models, and its potential to reshape LLM leaderboards.

Geometric quantum mechanics, through its differential-geometric underpinning, provides additional tools of analysis and interpretation that bring quantum mechanics closer to classical mechanics: state spaces in both are equipped with symplectic geometry. This opens the door to revisiting foundational questions and issues, such as the nature of quantum entropy, from a geometric perspective. Central to this is the concept of geometric quantum state -- the probability measure on a system's space of pure states. This space's continuity leads us to introduce two analysis tools, inspired by Renyi's information theory, to characterize and quantify fundamental properties of geometric quantum states: the quantum information dimension that is the rate of geometric quantum state compression and the dimensional geometric entropy that monitors information stored in quantum states. We recount their classical definitions, information-theoretic meanings, and physical interpretations, and adapt them to quantum systems via the geometric approach. We then explicitly compute them in various examples and classes of quantum system. We conclude commenting on future directions for information in geometric quantum mechanics.

Prior work has studied the computational complexity of computing optimal strategies to commit to in Stackelberg or leadership games, where a leader commits to a strategy which is observed by one or more followers. We extend this setting to one where the leader can additionally commit to outcome-conditional utility transfers. We characterize the computational complexity of finding optimal strategies in normal-form and Bayesian games, giving a mix of efficient algorithms and NP-hardness results. Finally, we allow the leader to also commit to a signaling scheme which induces a correlated equilibrium. In this setting, optimal commitments can be found in polynomial time for arbitrarily many players.

We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any $n \times n$ matrix pencil $(A,B)$. The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel, and Dumitriu [Technical Report 2010]. We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni, and Srivastava [Foundations of Computational Mathematics 2022]. In particular, we show that perturbing and scaling $(A,B)$ regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of $(A,B)$ in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible $S,T$ and diagonal $D$ such that $||A - SDT^{-1}||_2 \leq \varepsilon$ and $||B - ST^{-1}||_2 \leq \varepsilon$ in at most $O \left(\log^2 \left( \frac{n}{\varepsilon} \right) T_{\text{MM}}(n) \right)$ operations, where $T_{\text{MM}}(n)$ is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact arithmetic matrix pencil diagonalization.

We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time. In this work we study the case where $G$ is the complete graph on $n$ vertices and the number of particles is $M=n/2+\alpha n^{1/2} + o(n^{1/2})$, $\alpha\in \mathbb{R}$. This choice of $M$ corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by $n^{-1/2}$, converges in $p$-th mean, as $n\rightarrow \infty$ and for any $p \in \mathbb{R}$, to a continuous and almost surely positive random variable $T_\alpha$. We find that $T_\alpha$ is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that $\mathbb{E}[T_0] = \pi^{3/2}/\sqrt{7}$, and furthermore we formulate explicit asymptotics when $|\alpha|$ gets large that quantify the transition into and out of the critical window. We also study the random variable counting the total number of jumps that are performed by the particles until the dispersion time is reached and prove that, if rescaled by $n\ln n$, it converges to $2/7$ in probability.

Attribution scores indicate the importance of different input parts and can, thus, explain model behaviour. Currently, prompt-based models are gaining popularity, i.a., due to their easier adaptability in low-resource settings. However, the quality of attribution scores extracted from prompt-based models has not been investigated yet. In this work, we address this topic by analyzing attribution scores extracted from prompt-based models w.r.t. plausibility and faithfulness and comparing them with attribution scores extracted from fine-tuned models and large language models. In contrast to previous work, we introduce training size as another dimension into the analysis. We find that using the prompting paradigm (with either encoder-based or decoder-based models) yields more plausible explanations than fine-tuning the models in low-resource settings and Shapley Value Sampling consistently outperforms attention and Integrated Gradients in terms of leading to more plausible and faithful explanations.

Bayesian optimization (BO) is a powerful approach for optimizing complex and expensive-to-evaluate black-box functions. Its importance is underscored in many applications, notably including hyperparameter tuning, but its efficacy depends on efficiently balancing exploration and exploitation. While there has been substantial progress in BO methods, striking this balance remains a delicate process. In this light, we present LLAMBO, a novel approach that integrates the capabilities of Large Language Models (LLM) within BO. At a high level, we frame the BO problem in natural language, enabling LLMs to iteratively propose and evaluate promising solutions conditioned on historical evaluations. More specifically, we explore how combining contextual understanding, few-shot learning proficiency, and domain knowledge of LLMs can improve model-based BO. Our findings illustrate that LLAMBO is effective at zero-shot warmstarting, and enhances surrogate modeling and candidate sampling, especially in the early stages of search when observations are sparse. Our approach is performed in context and does not require LLM finetuning. Additionally, it is modular by design, allowing individual components to be integrated into existing BO frameworks, or function cohesively as an end-to-end method. We empirically validate LLAMBO's efficacy on the problem of hyperparameter tuning, highlighting strong empirical performance across a range of diverse benchmarks, proprietary, and synthetic tasks.

We revisit the problem of spurious modes that are sometimes encountered in partial differential equations discretizations. It is generally suspected that one of the causes for spurious modes is due to how boundary conditions are treated, and we use this as the starting point of our investigations. By regarding boundary conditions as algebraic constraints on a differential equation, we point out that any differential equation with homogeneous boundary conditions also admits a typically infinite number of hidden or implicit boundary conditions. In most discretization schemes, these additional implicit boundary conditions are violated, and we argue that this is what leads to the emergence of spurious modes. These observations motivate two definitions of the quality of computed eigenvalues based on violations of derivatives of boundary conditions on the one hand, and on the Grassmann distance between subspaces associated with computed eigenspaces on the other. Both of these tests are based on a standardized treatment of boundary conditions and do not require a priori knowledge of eigenvalue locations. The effectiveness of these tests is demonstrated on several examples known to have spurious modes. In addition, these quality tests show that in most problems, about half the computed spectrum of a differential operator is of low quality. The tests also specifically identify the low accuracy modes, which can then be projected out as a type of model reduction scheme.

It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.