A pervasive methodological error is the post-hoc interpretation of $p$-values. A $p$-value $p$ is not the level at which we reject the null, it is the level at which we would have rejected the null had we chosen level $p$. We introduce post-hoc $p$-values, that do admit this interpretation. We show that $p$ is a post-hoc $p$-value if and only if $1/p$ is an $e$-value. This implies that the product of independent post-hoc $p$-values is a post-hoc $p$-value, making them easy to combine. If we permit external randomization, we find any non-randomized post-hoc $p$-value can be trivially improved. However, we find only (essentially) non-randomized post-hoc $p$-values can be arbitrarily merged through multiplication. Our results extend to post-hoc anytime validity in a sequential setting. Moreover, we introduce two-way post-hoc $p$-values, whose reciprocal is also post-hoc under the alternative. Likelihood ratios are two-way post-hoc $p$-values, which supports their 'direct' interpretation often purported in the context of Bayes factors and links their interpretation to post-hoc $p$-values. Finally, we extend to geometric post-hoc validity and show that GRO $e$-values are the reciprocal of post-hoc $p$-values that minimize the geometric post-hoc error under the alternative.
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Model selection for a given target task can be costly, as it may entail extensive annotation of the quality of outputs of different models. We introduce DiffUse, an efficient method to make an informed decision between candidate text generation models. DiffUse reduces the required amount of preference annotations, thus saving valuable time and resources in performing evaluation. DiffUse intelligently selects instances by clustering embeddings that represent the semantic differences between model outputs. Thus, it is able to identify a subset of examples that are more informative for preference decisions. Our method is model-agnostic, and can be applied to any text generation model. Moreover, we propose a practical iterative approach for dynamically determining how many instances to annotate. In a series of experiments over hundreds of model pairs, we demonstrate that DiffUse can dramatically reduce the required number of annotations -- by up to 75% -- while maintaining high evaluation reliability.
Consider a multi-class labelling problem, where the labels can take values in $[k]$, and a predictor predicts a distribution over the labels. In this work, we study the following foundational question: Are there notions of multi-class calibration that give strong guarantees of meaningful predictions and can be achieved in time and sample complexities polynomial in $k$? Prior notions of calibration exhibit a tradeoff between computational efficiency and expressivity: they either suffer from having sample complexity exponential in $k$, or needing to solve computationally intractable problems, or give rather weak guarantees. Our main contribution is a notion of calibration that achieves all these desiderata: we formulate a robust notion of projected smooth calibration for multi-class predictions, and give new recalibration algorithms for efficiently calibrating predictors under this definition with complexity polynomial in $k$. Projected smooth calibration gives strong guarantees for all downstream decision makers who want to use the predictor for binary classification problems of the form: does the label belong to a subset $T \subseteq [k]$: e.g. is this an image of an animal? It ensures that the probabilities predicted by summing the probabilities assigned to labels in $T$ are close to some perfectly calibrated binary predictor for that task. We also show that natural strengthenings of our definition are computationally hard to achieve: they run into information theoretic barriers or computational intractability. Underlying both our upper and lower bounds is a tight connection that we prove between multi-class calibration and the well-studied problem of agnostic learning in the (standard) binary prediction setting.
Large-scale machine learning problems make the cost of hyperparameter tuning ever more prohibitive. This creates a need for algorithms that can tune themselves on-the-fly. We formalize the notion of "tuning-free" algorithms that can match the performance of optimally-tuned optimization algorithms up to polylogarithmic factors given only loose hints on the relevant problem parameters. We consider in particular algorithms that can match optimally-tuned Stochastic Gradient Descent (SGD). When the domain of optimization is bounded, we show tuning-free matching of SGD is possible and achieved by several existing algorithms. We prove that for the task of minimizing a convex and smooth or Lipschitz function over an unbounded domain, tuning-free optimization is impossible. We discuss conditions under which tuning-free optimization is possible even over unbounded domains. In particular, we show that the recently proposed DoG and DoWG algorithms are tuning-free when the noise distribution is sufficiently well-behaved. For the task of finding a stationary point of a smooth and potentially nonconvex function, we give a variant of SGD that matches the best-known high-probability convergence rate for tuned SGD at only an additional polylogarithmic cost. However, we also give an impossibility result that shows no algorithm can hope to match the optimal expected convergence rate for tuned SGD with high probability.
With the development of large language models, multiple AIs are now made available for code generation (such as ChatGPT and StarCoder) and are adopted widely. It is often desirable to know whether a piece of code is generated by AI, and furthermore, which AI is the author. For instance, if a certain version of AI is known to generate vulnerable code, it is particularly important to know the creator. Existing approaches are not satisfactory as watermarking codes are challenging compared with watermarking text data, as codes can be altered with relative ease via widely-used code refactoring methods. In this work, we propose ACW (AI Code Watermarking), a novel method for watermarking AI-generated codes. ACW is efficient as it requires no training or fine-tuning and works in a black-box manner. It is resilient as the watermark cannot be easily removed or tampered through common code refactoring methods. The key idea of ACW is to selectively apply a set of carefully-designed semantic-preserving, idempotent code transformations, whose presence (or absence) allows us to determine the existence of the watermark. Our experimental results show that ACW is effective (i.e., achieving high accuracy, true positive rates and false positive rates), resilient and efficient, significantly outperforming existing approaches.
We show that the principal types of the closed terms of the affine fragment of $\lambda$-calculus, with respect to a simple type discipline, are structurally isomorphic to their interpretations, as partial involutions, in a natural Geometry of Interaction model \`a la Abramsky. This permits to explain in elementary terms the somewhat awkward notion of linear application arising in Geometry of Interaction, simply as the resolution between principal types using an alternate unification algorithm. As a consequence, we provide an answer, for the purely affine fragment, to the open problem raised by Abramsky of characterising those partial involutions which are denotations of combinatory terms.
Community detection is a crucial task in network analysis that can be significantly improved by incorporating subject-level information, i.e. covariates. However, current methods often struggle with selecting tuning parameters and analyzing low-degree nodes. In this paper, we introduce a novel method that addresses these challenges by constructing network-adjusted covariates, which leverage the network connections and covariates with a unique weight to each node based on the node's degree. Spectral clustering on network-adjusted covariates yields an exact recovery of community labels under certain conditions, which is tuning-free and computationally efficient. We present novel theoretical results about the strong consistency of our method under degree-corrected stochastic blockmodels with covariates, even in the presence of mis-specification and sparse communities with bounded degrees. Additionally, we establish a general lower bound for the community detection problem when both network and covariates are present, and it shows our method is optimal up to a constant factor. Our method outperforms existing approaches in simulations and a LastFM app user network, and provides interpretable community structures in a statistics publication citation network where $30\%$ of nodes are isolated.
We prove the completeness of a first-order analogue of the Fischer Servi logic $\mathsf{FS}$ with respect to its expected birelational semantics. To this end we introduce the notion of the $\textit{trace model}$ and, much like in a canonical model argument, prove a truth lemma. We conclude by examining a number of other first-order Fischer Servi logics, including the first-order analogue of $\mathsf{FSS4}$, whose completeness can be similarly proved.
Multidimensional scaling (MDS) is the act of embedding proximity information about a set of $n$ objects in $d$-dimensional Euclidean space. As originally conceived by the psychometric community, MDS was concerned with embedding a fixed set of proximities associated with a fixed set of objects. Modern concerns, e.g., that arise in developing asymptotic theories for statistical inference on random graphs, more typically involve studying the limiting behavior of a sequence of proximities associated with an increasing set of objects. Standard results from the theory of point-to-set maps imply that, if $n$ is fixed and a sequence of proximities converges, then the limit of the embedded structures is the embedded structure of the limiting proximities. But what if $n$ increases? It then becomes necessary to reformulate MDS so that the entire sequence of embedding problems can be viewed as a sequence of optimization problems in a fixed space. We present such a reformulation and derive some consequences.
Joint image-text embedding is the bedrock for most Vision-and-Language (V+L) tasks, where multimodality inputs are jointly processed for visual and textual understanding. In this paper, we introduce UNITER, a UNiversal Image-TExt Representation, learned through large-scale pre-training over four image-text datasets (COCO, Visual Genome, Conceptual Captions, and SBU Captions), which can power heterogeneous downstream V+L tasks with joint multimodal embeddings. We design three pre-training tasks: Masked Language Modeling (MLM), Image-Text Matching (ITM), and Masked Region Modeling (MRM, with three variants). Different from concurrent work on multimodal pre-training that apply joint random masking to both modalities, we use conditioned masking on pre-training tasks (i.e., masked language/region modeling is conditioned on full observation of image/text). Comprehensive analysis shows that conditioned masking yields better performance than unconditioned masking. We also conduct a thorough ablation study to find an optimal setting for the combination of pre-training tasks. Extensive experiments show that UNITER achieves new state of the art across six V+L tasks (over nine datasets), including Visual Question Answering, Image-Text Retrieval, Referring Expression Comprehension, Visual Commonsense Reasoning, Visual Entailment, and NLVR2.
To provide more accurate, diverse, and explainable recommendation, it is compulsory to go beyond modeling user-item interactions and take side information into account. Traditional methods like factorization machine (FM) cast it as a supervised learning problem, which assumes each interaction as an independent instance with side information encoded. Due to the overlook of the relations among instances or items (e.g., the director of a movie is also an actor of another movie), these methods are insufficient to distill the collaborative signal from the collective behaviors of users. In this work, we investigate the utility of knowledge graph (KG), which breaks down the independent interaction assumption by linking items with their attributes. We argue that in such a hybrid structure of KG and user-item graph, high-order relations --- which connect two items with one or multiple linked attributes --- are an essential factor for successful recommendation. We propose a new method named Knowledge Graph Attention Network (KGAT) which explicitly models the high-order connectivities in KG in an end-to-end fashion. It recursively propagates the embeddings from a node's neighbors (which can be users, items, or attributes) to refine the node's embedding, and employs an attention mechanism to discriminate the importance of the neighbors. Our KGAT is conceptually advantageous to existing KG-based recommendation methods, which either exploit high-order relations by extracting paths or implicitly modeling them with regularization. Empirical results on three public benchmarks show that KGAT significantly outperforms state-of-the-art methods like Neural FM and RippleNet. Further studies verify the efficacy of embedding propagation for high-order relation modeling and the interpretability benefits brought by the attention mechanism.
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