Mathematical language in scientific communications and educational scenarios is important yet relatively understudied compared to natural languages. Recent works on mathematical language focus either on representing stand-alone mathematical expressions, especially in their natural tree format, or mathematical reasoning in pre-trained natural language models. Existing works on jointly modeling and generating natural and mathematical languages simply treat mathematical expressions as text, without accounting for the rigid structural properties of mathematical expressions. In this paper, we propose a series of modifications to existing language models to jointly represent and generate text and math: representing mathematical expressions as sequences of node tokens in their operator tree format, using math symbol and tree position embeddings to preserve the semantic and structural properties of mathematical expressions, and using a constrained decoding method to generate mathematically valid expressions. We ground our modifications in GPT-2, resulting in a model MathGPT, and demonstrate that it outperforms baselines on mathematical expression generation tasks.
We study the problem of uncertainty quantification via prediction sets, in an online setting where the data distribution may vary arbitrarily over time. Recent work develops online conformal prediction techniques that leverage regret minimization algorithms from the online learning literature to learn prediction sets with approximately valid coverage and small regret. However, standard regret minimization could be insufficient for handling changing environments, where performance guarantees may be desired not only over the full time horizon but also in all (sub-)intervals of time. We develop new online conformal prediction methods that minimize the strongly adaptive regret, which measures the worst-case regret over all intervals of a fixed length. We prove that our methods achieve near-optimal strongly adaptive regret for all interval lengths simultaneously, and approximately valid coverage. Experiments show that our methods consistently obtain better coverage and smaller prediction sets than existing methods on real-world tasks, such as time series forecasting and image classification under distribution shift.
We study the statistical inference of nonlinear stochastic approximation algorithms utilizing a single trajectory of Markovian data. Our methodology has practical applications in various scenarios, such as Stochastic Gradient Descent (SGD) on autoregressive data and asynchronous Q-Learning. By utilizing the standard stochastic approximation (SA) framework to estimate the target parameter, we establish a functional central limit theorem for its partial-sum process, $\boldsymbol{\phi}_T$. To further support this theory, we provide a matching semiparametric efficient lower bound and a non-asymptotic upper bound on its weak convergence, measured in the L\'evy-Prokhorov metric. This functional central limit theorem forms the basis for our inference method. By selecting any continuous scale-invariant functional $f$, the asymptotic pivotal statistic $f(\boldsymbol{\phi}_T)$ becomes accessible, allowing us to construct an asymptotically valid confidence interval. We analyze the rejection probability of a family of functionals $f_m$, indexed by $m \in \mathbb{N}$, through theoretical and numerical means. The simulation results demonstrate the validity and efficiency of our method.
The remarkable positive impact of Deep Neural Networks on many Artificial Intelligence (AI) tasks has led to the development of various high performance algorithms as well as specialized processors and accelerators. In this paper we address this scenario by demonstrating that the principles underlying the modern realization of the general matrix multiplication (GEMM) in conventional processor architectures, are also valid to achieve high performance for the type of operations that arise in deep learning (DL) on an exotic accelerator such as the AI Engine (AIE) tile embedded in Xilinx Versal platforms. In particular, our experimental results with a prototype implementation of the GEMM kernel, on a Xilinx Versal VCK190, delivers performance close to 86.7% of the theoretical peak that can be expected on an AIE tile, for 16-bit integer operands.
We propose a hierarchical local electricity market (LEM) at the primary and secondary feeder levels in a distribution grid, to optimally coordinate and schedule distributed energy resources (DER) and provide valuable grid services like voltage control. At the primary level, we use a current injection-based model that is valid for both radial and meshed, balanced and unbalanced, multi-phase systems. The primary and secondary markets leverage the flexibility offered by DERs to optimize grid operation and maximize social welfare. Numerical simulations on an IEEE-123 bus modified to include DERs, show that the LEM successfully achieves voltage control and reduces overall network costs, while also allowing us to decompose the price and value associated with different grid services so as to accurately compensate DERs.
In physics, entanglement 'reduces' the entropy of an entity, because the (von Neumann) entropy of, e.g., a composite bipartite entity in a pure entangled state is systematically lower than the entropy of the component sub-entities. We show here that this 'genuinely non-classical reduction of entropy as a result of composition' also holds whenever two concepts combine in human cognition and, more generally, it is valid in human culture. We exploit these results and make a 'new hypothesis' on the nature of entanglement, namely, the production of entanglement in the preparation of a composite entity can be seen as a 'dynamical process of collaboration between its sub-entities to reduce uncertainty', because the composite entity is in a pure state while its sub-entities are in a non-pure, or density, state, as a result of the preparation. We identify within the nature of this entanglement a mechanism of contextual updating and illustrate the mechanism in the example we analyze. Our hypothesis naturally explains the 'non-classical nature' of some quantum logical connectives, as due to Bell-type correlations.
We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite $n$ and $d$, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with $d$ and the bandwidth.
We provide uniform inference and confidence bands for kernel ridge regression (KRR), a widely-used non-parametric regression estimator for general data types including rankings, images, and graphs. Despite the prevalence of these data -- e.g., ranked preference lists in school assignment -- the inferential theory of KRR is not fully known, limiting its role in economics and other scientific domains. We construct sharp, uniform confidence sets for KRR, which shrink at nearly the minimax rate, for general regressors. To conduct inference, we develop an efficient bootstrap procedure that uses symmetrization to cancel bias and limit computational overhead. To justify the procedure, we derive finite-sample, uniform Gaussian and bootstrap couplings for partial sums in a reproducing kernel Hilbert space (RKHS). These imply strong approximation for empirical processes indexed by the RKHS unit ball with logarithmic dependence on the covering number. Simulations verify coverage. We use our procedure to construct a novel test for match effects in school assignment, an important question in education economics with consequences for school choice reforms.
The linear response of a photomultiplier tube (PMT) is a required property for photon counting and reconstruction of the neutrino energy. The linearity valid region and the saturation response of PMT were investigated using a linear-alkyl-benzene (LAB)-based liquid scintillator. A correlation was observed between the two different saturation responses, with pulse-shape distortion and pulse-area decrease. The observed pulse-shape provides useful information for the estimation of the linearity region relative to the pulse-area. This correlation-based diagnosis allows an ${in}$-${situ}$ estimation of the linearity range, which was previously challenging. The measured correlation between the two saturation responses was employed to train an artificial-neural-network (ANN) to predict the decrease in pulse-area from the observed pulse-shape. The ANN-predicted pulse-area decrease enables the prediction of the ideal number of photoelectrons irrelevant to the saturation behavior. This pulse-shape-based machine learning technique offers a novel method for restoring the saturation response of PMTs.
Constraint solving is an elementary way for verification of deep neural networks (DNN). In the domain of AI safety, a DNN might be modified in its structure and parameters for its repair or attack. For such situations, we propose the incremental DNN verification problem, which asks whether a safety property still holds after the DNN is modified. To solve the problem, we present an incremental satisfiability modulo theory (SMT) algorithm based on the Reluplex framework. We simulate the most important features of the configurations that infers the verification result of the searching branches in the old solving procedure (with respect to the original network), and heuristically check whether the proofs are still valid for the modified DNN. We implement our algorithm as an incremental solver called DeepInc, and exerimental results show that DeepInc is more efficient in most cases. For the cases that the property holds both before and after modification, the acceleration can be faster by several orders of magnitude, showing that DeepInc is outstanding in incrementally searching for counterexamples. Moreover, based on the framework, we propose the multi-objective DNN repair problem and give an algorithm based on our incremental SMT solving algorithm. Our repair method preserves more potential safety properties on the repaired DNNs compared with state-of-the-art.
In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. Recently, many flexible machine learning methods have been developed for instrumental variable estimation. However, these methods have at least one of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) only obtaining estimation error rates in terms of pseudometrics (\emph{e.g.,} projected norm) rather than valid metrics (\emph{e.g.,} $L_2$ norm); or (3) imposing the so-called closedness condition that requires a certain conditional expectation operator to be sufficiently smooth. In this paper, we present the first method and analysis that can avoid all three limitations, while still permitting general function approximation. Specifically, we propose a new penalized minimax estimator that can converge to a fixed IV solution even when there are multiple solutions, and we derive a strong $L_2$ error rate for our estimator under lax conditions. Notably, this guarantee only needs a widely-used source condition and realizability assumptions, but not the so-called closedness condition. We argue that the source condition and the closedness condition are inherently conflicting, so relaxing the latter significantly improves upon the existing literature that requires both conditions. Our estimator can achieve this improvement because it builds on a novel formulation of the IV estimation problem as a constrained optimization problem.
Denoising diffusion probabilistic models (DDPMs) have pioneered new state-of-the-art results in disciplines such as computer vision and computational biology for diverse tasks ranging from text-guided image generation to structure-guided protein design. Along this latter line of research, methods have recently been proposed for generating 3D molecules using equivariant graph neural networks (GNNs) within a DDPM framework. However, such methods are unable to learn important geometric properties of 3D molecules, as they adopt molecule-agnostic and non-geometric GNNs as their 3D graph denoising networks, which notably hinders their ability to generate valid large 3D molecules. In this work, we address these gaps by introducing the Geometry-Complete Diffusion Model (GCDM) for 3D molecule generation, which outperforms existing 3D molecular diffusion models by significant margins across conditional and unconditional settings for the QM9 dataset and the larger GEOM-Drugs dataset, respectively, and generates more novel and unique unconditional 3D molecules for the QM9 dataset compared to previous methods. Importantly, we demonstrate that the geometry-complete denoising process of GCDM learned for 3D molecule generation enables the model to generate a significant proportion of valid and energetically-stable large molecules at the scale of GEOM-Drugs, whereas previous methods fail to do so with the features they learn. Additionally, we show that extensions of GCDM can not only effectively design 3D molecules for specific protein pockets but also that GCDM's geometric features can be repurposed to consistently optimize the geometry and chemical composition of existing 3D molecules for molecular stability and property specificity, demonstrating new versatility of molecular diffusion models. Our source code and data are freely available at https://github.com/BioinfoMachineLearning/Bio-Diffusion.
Learning from non-stationary data streams is a research direction that gains increasing interest as more data in form of streams becomes available, for example from social media, smartphones, or industrial process monitoring. Most approaches assume that the ground truth of the samples becomes available (possibly with some delay) and perform supervised online learning in the test-then-train scheme. While this assumption might be valid in some scenarios, it does not apply to all settings. In this work, we focus on scarcely labeled data streams and explore the potential of self-labeling in gradually drifting data streams. We formalize this setup and propose a novel online $k$-nn classifier that combines self-labeling and demand-based active learning.
In settings with few treated units, Difference-in-Differences (DID) estimators are not consistent, and are not generally asymptotically normal. This poses relevant challenges for inference. While there are inference methods that are valid in these settings, some of these alternatives are not readily available when there is variation in treatment timing and heterogeneous treatment effects; or for deriving uniform confidence bands for event-study plots. We present alternatives in settings with few treated units that are valid with variation in treatment timing and/or that allow for uniform confidence bands.
We present a unified framework for deriving PAC-Bayesian generalization bounds. Unlike most previous literature on this topic, our bounds are anytime-valid (i.e., time-uniform), meaning that they hold at all stopping times, not only for a fixed sample size. Our approach combines four tools in the following order: (a) nonnegative supermartingales or reverse submartingales, (b) the method of mixtures, (c) the Donsker-Varadhan formula (or other convex duality principles), and (d) Ville's inequality. Our main result is a PAC-Bayes theorem which holds for a wide class of discrete stochastic processes. We show how this result implies time-uniform versions of well-known classical PAC-Bayes bounds, such as those of Seeger, McAllester, Maurer, and Catoni, in addition to many recent bounds. We also present several novel bounds. Our framework also enables us to relax traditional assumptions; in particular, we consider nonstationary loss functions and non-i.i.d. data. In sum, we unify the derivation of past bounds and ease the search for future bounds: one may simply check if our supermartingale or submartingale conditions are met and, if so, be guaranteed a (time-uniform) PAC-Bayes bound.
This paper introduces a class of investment project's profitability metrics that includes the net present value (NPV) criterion (which labels a project as weakly profitable if its NPV is nonnegative), internal rate of return (IRR), profitability index (PI), payback period (PP), and discounted payback period (DPP) as special cases. We develop an axiomatic characterization of this class, as well as of the mentioned conventional metrics within the class. The proposed approach offers several key contributions. First, it provides a unified interpretation of profitability metrics as indicators of a project's financial stability across various economic scenarios. Second, it reveals that, except for the NPV criterion, a profitability metric is inherently undefined for some projects. In particular, this implies that any extension of IRR to the space of all projects does not meet a set of reasonable conditions. A similar conclusion is valid for the other mentioned conventional metrics. For each of these metrics, we offer a characterization of the pairs of comparable projects and identify the largest set of projects to which the metric can be unequivocally extended. Third, our study identifies conditions under which the application of one metric is superior to others, helping to guide decision-makers in selecting the most appropriate metric for specific investment contexts.
Developing computational models for automatic affect prediction requires valid self-reports about individuals' emotional interpretations of stimuli. In this article, we highlight the important influence of the temporal distance between a stimulus event and the moment when its experience is reported on the provided information's validity. This influence stems from the time-dependent and time-demanding nature of the involved cognitive processes. As such, reports can be collected too late: forgetting is a widely acknowledged challenge for accurate descriptions of past experience. For this reason, methods striving for assessment as early as possible have become increasingly popular. However, here we argue that collection may also occur too early: descriptions about very recent stimuli might be collected before emotional processing has fully converged. Based on these notions, we champion the existence of a temporal distance for each type of stimulus that maximizes the validity of self-reports -- a "right" time. Consequently, we recommend future research to (1) consciously consider the potential influence of temporal distance on affective self-reports when planning data collection, (2) document the temporal distance of affective self-reports wherever possible as part of corpora for computational modelling, and finally (3) and explore the effect of temporal distance on self-reports across different types of stimuli.
Deep Learning (DL) is prevalently used in various industries to improve decision-making and automate processes, driven by the ever-evolving DL libraries and compilers. The correctness of DL systems is crucial for trust in DL applications. As such, the recent wave of research has been studying the automated synthesis of test-cases (i.e., DNN models and their inputs) for fuzzing DL systems. However, existing model generators only subsume a limited number of operators, lacking the ability to pervasively model operator constraints. To address this challenge, we propose NeuRI, a fully automated approach for generating valid and diverse DL models composed of hundreds of types of operators. NeuRI adopts a three-step process: (i) collecting valid and invalid API traces from various sources; (ii) applying inductive program synthesis over the traces to infer the constraints for constructing valid models; and (iii) using hybrid model generation which incorporates both symbolic and concrete operators. Our evaluation shows that NeuRI improves branch coverage of TensorFlow and PyTorch by 24% and 15% over the state-of-the-art model-level fuzzers. NeuRI finds 100 new bugs for PyTorch and TensorFlow in four months, with 81 already fixed or confirmed. Of these, 9 bugs are labelled as high priority or security vulnerability, constituting 10% of all high-priority bugs of the period. Open-source developers regard error-inducing tests reported by us as "high-quality" and "common in practice".
We present a method of explainable artificial intelligence (XAI), "What I Know (WIK)", to provide additional information to verify the reliability of a deep learning model by showing an example of an instance in a training dataset that is similar to the input data to be inferred and demonstrate it in a remote sensing image classification task. One of the expected roles of XAI methods is verifying whether inferences of a trained machine learning model are valid for an application, and it is an important factor that what datasets are used for training the model as well as the model architecture. Our data-centric approach can help determine whether the training dataset is sufficient for each inference by checking the selected example data. If the selected example looks similar to the input data, we can confirm that the model was not trained on a dataset with a feature distribution far from the feature of the input data. With this method, the criteria for selecting an example are not merely data similarity with the input data but also data similarity in the context of the model task. Using a remote sensing image dataset from the Sentinel-2 satellite, the concept was successfully demonstrated with reasonably selected examples. This method can be applied to various machine-learning tasks, including classification and regression.
Designing molecular structures with desired chemical properties is an essential task in drug discovery and material design. However, finding molecules with the optimized desired properties is still a challenging task due to combinatorial explosion of candidate space of molecules. Here we propose a novel \emph{decomposition-and-reassembling} based approach, which does not include any optimization in hidden space and our generation process is highly interpretable. Our method is a two-step procedure: In the first decomposition step, we apply frequent subgraph mining to a molecular database to collect smaller size of subgraphs as building blocks of molecules. In the second reassembling step, we search desirable building blocks guided via reinforcement learning and combine them to generate new molecules. Our experiments show that not only can our method find better molecules in terms of two standard criteria, the penalized $\log P$ and drug-likeness, but also generate drug molecules with showing the valid intermediate molecules.