We propose a novel approach to nonlinear functional regression, called the Mapping-to-Parameter function model, which addresses complex and nonlinear functional regression problems in parameter space by employing any supervised learning technique. Central to this model is the mapping of function data from an infinite-dimensional function space to a finite-dimensional parameter space. This is accomplished by concurrently approximating multiple functions with a common set of B-spline basis functions by any chosen order, with their knot distribution determined by the Iterative Local Placement Algorithm, a newly proposed free knot placement algorithm. In contrast to the conventional equidistant knot placement strategy that uniformly distributes knot locations based on a predefined number of knots, our proposed algorithms determine knot location according to the local complexity of the input or output functions. The performance of our knot placement algorithms is shown to be robust in both single-function approximation and multiple-function approximation contexts. Furthermore, the effectiveness and advantage of the proposed prediction model in handling both function-on-scalar regression and function-on-function regression problems are demonstrated through several real data applications, in comparison with four groups of state-of-the-art methods.
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We propose a nonlinear difference-in-differences method to estimate multivariate counterfactual distributions in classical treatment and control study designs with observational data. Our approach sheds a new light on existing approaches like the changes-in-changes and the classical semiparametric difference-in-differences estimator and generalizes them to settings with multivariate heterogeneity in the outcomes. The main benefit of this extension is that it allows for arbitrary dependence and heterogeneity in the joint outcomes. We demonstrate its utility both on synthetic and real data. In particular, we revisit the classical Card \& Krueger dataset, examining the effect of a minimum wage increase on employment in fast food restaurants; a reanalysis with our method reveals that restaurants tend to substitute full-time with part-time labor after a minimum wage increase at a faster pace. A previous version of this work was entitled "An optimal transport approach to causal inference.
Sequential testing, always-valid $p$-values, and confidence sequences promise flexible statistical inference and on-the-fly decision making. However, unlike fixed-$n$ inference based on asymptotic normality, existing sequential tests either make parametric assumptions and end up under-covering/over-rejecting when these fail or use non-parametric but conservative concentration inequalities and end up over-covering/under-rejecting. To circumvent these issues, we sidestep exact at-least-$\alpha$ coverage and focus on asymptotic calibration and asymptotic optimality. That is, we seek sequential tests whose probability of \emph{ever} rejecting a true hypothesis approaches $\alpha$ and whose expected time to reject a false hypothesis approaches a lower bound on all such asymptotically calibrated tests, both "approaches" occurring under an appropriate limit. We permit observations to be both non-parametric and dependent and focus on testing whether the observations form a martingale difference sequence. We propose the universal sequential probability ratio test (uSPRT), a slight modification to the normal-mixture sequential probability ratio test, where we add a burn-in period and adjust thresholds accordingly. We show that even in this very general setting, the uSPRT is asymptotically optimal under mild generic conditions. We apply the results to stabilized estimating equations to test means, treatment effects, {\etc} Our results also provide corresponding guarantees for the implied confidence sequences. Numerical simulations verify our guarantees and the benefits of the uSPRT over alternatives.
Explaining Pre-Trained Language Models with Attribution Scores: An Analysis in Low-Resource Settings
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.
We consider the differentially private (DP) facility location problem in the so called super-set output setting proposed by Gupta et al. [SODA 2010]. The current best known expected approximation ratio for an $\epsilon$-DP algorithm is $O\left(\frac{\log n}{\sqrt{\epsilon}}\right)$ due to Cohen-Addad et al. [AISTATS 2022] where $n$ denote the size of the metric space, meanwhile the best known lower bound is $\Omega(1/\sqrt{\epsilon})$ [NeurIPS 2019]. In this short note, we give a lower bound of $\tilde{\Omega}\left(\min\left\{\log n, \sqrt{\frac{\log n}{\epsilon}}\right\}\right)$ on the expected approximation ratio of any $\epsilon$-DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.
Subsampling algorithms for various parametric regression models with massive data have been extensively investigated in recent years. However, all existing studies on subsampling heavily rely on clean massive data. In practical applications, the observed covariates may suffer from inaccuracies due to measurement errors. To address the challenge of large datasets with measurement errors, this study explores two subsampling algorithms based on the corrected likelihood approach: the optimal subsampling algorithm utilizing inverse probability weighting and the perturbation subsampling algorithm employing random weighting assuming a perfectly known distribution. Theoretical properties for both algorithms are provided. Numerical simulations and two real-world examples demonstrate the effectiveness of these proposed methods compared to other uncorrected algorithms.
Diffusion models, which convert noise into new data instances by learning to reverse a Markov diffusion process, have become a cornerstone in contemporary generative modeling. While their practical power has now been widely recognized, the theoretical underpinnings remain far from mature. In this work, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein) score functions. For a popular deterministic sampler (based on the probability flow ODE), we establish a convergence rate proportional to $1/T$ (with $T$ the total number of steps), improving upon past results; for another mainstream stochastic sampler (i.e., a type of the denoising diffusion probabilistic model), we derive a convergence rate proportional to $1/\sqrt{T}$, matching the state-of-the-art theory. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results characterize how $\ell_2$ score estimation errors affect the quality of the data generation processes. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without resorting to toolboxes for SDEs and ODEs. Further, we design two accelerated variants, improving the convergence to $1/T^2$ for the ODE-based sampler and $1/T$ for the DDPM-type sampler, which might be of independent theoretical and empirical interest.
We propose an autonomous exploration algorithm designed for decentralized multi-robot teams, which takes into account map and localization uncertainties of range-sensing mobile robots. Virtual landmarks are used to quantify the combined impact of process noise and sensor noise on map uncertainty. Additionally, we employ an iterative expectation-maximization inspired algorithm to assess the potential outcomes of both a local robot's and its neighbors' next-step actions. To evaluate the effectiveness of our framework, we conduct a comparative analysis with state-of-the-art algorithms. The results of our experiments show the proposed algorithm's capacity to strike a balance between curbing map uncertainty and achieving efficient task allocation among robots.
Deep reinforcement learning algorithms are usually impeded by sampling inefficiency, heavily depending on multiple interactions with the environment to acquire accurate decision-making capabilities. In contrast, humans rely on their hippocampus to retrieve relevant information from past experiences of relevant tasks, which guides their decision-making when learning a new task, rather than exclusively depending on environmental interactions. Nevertheless, designing a hippocampus-like module for an agent to incorporate past experiences into established reinforcement learning algorithms presents two challenges. The first challenge involves selecting the most relevant past experiences for the current task, and the second challenge is integrating such experiences into the decision network. To address these challenges, we propose a novel method that utilizes a retrieval network based on task-conditioned hypernetwork, which adapts the retrieval network's parameters depending on the task. At the same time, a dynamic modification mechanism enhances the collaborative efforts between the retrieval and decision networks. We evaluate the proposed method across various tasks within a multitask scenario in the Minigrid environment. The experimental results demonstrate that our proposed method significantly outperforms strong baselines.
Few-shot learning, a challenging task in machine learning, aims to learn a classifier adaptable to recognize new, unseen classes with limited labeled examples. Meta-learning has emerged as a prominent framework for few-shot learning. Its training framework is originally a task-level learning method, such as Model-Agnostic Meta-Learning (MAML) and Prototypical Networks. And a recently proposed training paradigm called Meta-Baseline, which consists of sequential pre-training and meta-training stages, gains state-of-the-art performance. However, as a non-end-to-end training method, indicating the meta-training stage can only begin after the completion of pre-training, Meta-Baseline suffers from higher training cost and suboptimal performance due to the inherent conflicts of the two training stages. To address these limitations, we propose an end-to-end training paradigm consisting of two alternative loops. In the outer loop, we calculate cross entropy loss on the entire training set while updating only the final linear layer. In the inner loop, we employ the original meta-learning training mode to calculate the loss and incorporate gradients from the outer loss to guide the parameter updates. This training paradigm not only converges quickly but also outperforms existing baselines, indicating that information from the overall training set and the meta-learning training paradigm could mutually reinforce one another. Moreover, being model-agnostic, our framework achieves significant performance gains, surpassing the baseline systems by approximate 1%.
We introduce a generic framework that reduces the computational cost of object detection while retaining accuracy for scenarios where objects with varied sizes appear in high resolution images. Detection progresses in a coarse-to-fine manner, first on a down-sampled version of the image and then on a sequence of higher resolution regions identified as likely to improve the detection accuracy. Built upon reinforcement learning, our approach consists of a model (R-net) that uses coarse detection results to predict the potential accuracy gain for analyzing a region at a higher resolution and another model (Q-net) that sequentially selects regions to zoom in. Experiments on the Caltech Pedestrians dataset show that our approach reduces the number of processed pixels by over 50% without a drop in detection accuracy. The merits of our approach become more significant on a high resolution test set collected from YFCC100M dataset, where our approach maintains high detection performance while reducing the number of processed pixels by about 70% and the detection time by over 50%.
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