Estimating the conditional mean function that relates predictive covariates to a response variable of interest is a fundamental task in economics and statistics. In this manuscript, we propose some general nonparametric regression approaches that are widely applicable based on a simple yet significant decomposition of nonparametric functions into a semiparametric model with shape-restricted components. For instance, we observe that every Lipschitz function can be expressed as a sum of a monotone function and a linear function. We implement well-established shape-restricted estimation procedures, such as isotonic regression, to handle the ``nonparametric" components of the true regression function and combine them with a simple sample-splitting procedure to estimate the parametric components. The resulting estimators inherit several favorable properties from the shape-restricted regression estimators. Notably, it is practically tuning parameter free, converges at the minimax optimal rate, and exhibits an adaptive rate when the true regression function is ``simple". We also confirm these theoretical properties and compare the practice performance with existing methods via a series of numerical studies.
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We propose an approach utilizing gamma-distributed random variables, coupled with log-Gaussian modeling, to generate synthetic datasets suitable for training neural networks. This addresses the challenge of limited real observations in various applications. We apply this methodology to both Raman and coherent anti-Stokes Raman scattering (CARS) spectra, using experimental spectra to estimate gamma process parameters. Parameter estimation is performed using Markov chain Monte Carlo methods, yielding a full Bayesian posterior distribution for the model which can be sampled for synthetic data generation. Additionally, we model the additive and multiplicative background functions for Raman and CARS with Gaussian processes. We train two Bayesian neural networks to estimate parameters of the gamma process which can then be used to estimate the underlying Raman spectrum and simultaneously provide uncertainty through the estimation of parameters of a probability distribution. We apply the trained Bayesian neural networks to experimental Raman spectra of phthalocyanine blue, aniline black, naphthol red, and red 264 pigments and also to experimental CARS spectra of adenosine phosphate, fructose, glucose, and sucrose. The results agree with deterministic point estimates for the underlying Raman and CARS spectral signatures.
Existing neural models are demonstrated to struggle with compositional generalization (CG), i.e., the ability to systematically generalize to unseen compositions of seen components. A key reason for failure on CG is that the syntactic and semantic representations of sequences in both the uppermost layer of the encoder and decoder are entangled. However, previous work concentrates on separating the learning of syntax and semantics instead of exploring the reasons behind the representation entanglement (RE) problem to solve it. We explain why it exists by analyzing the representation evolving mechanism from the bottom to the top of the Transformer layers. We find that the ``shallow'' residual connections within each layer fail to fuse previous layers' information effectively, leading to information forgetting between layers and further the RE problems. Inspired by this, we propose LRF, a novel \textbf{L}ayer-wise \textbf{R}epresentation \textbf{F}usion framework for CG, which learns to fuse previous layers' information back into the encoding and decoding process effectively through introducing a \emph{fuse-attention module} at each encoder and decoder layer. LRF achieves promising results on two realistic benchmarks, empirically demonstrating the effectiveness of our proposal.
To address the needs of modeling uncertainty in sensitive machine learning applications, the setup of distributionally robust optimization (DRO) seeks good performance uniformly across a variety of tasks. The recent multi-distribution learning (MDL) framework tackles this objective in a dynamic interaction with the environment, where the learner has sampling access to each target distribution. Drawing inspiration from the field of pure-exploration multi-armed bandits, we provide distribution-dependent guarantees in the MDL regime, that scale with suboptimality gaps and result in superior dependence on the sample size when compared to the existing distribution-independent analyses. We investigate two non-adaptive strategies, uniform and non-uniform exploration, and present non-asymptotic regret bounds using novel tools from empirical process theory. Furthermore, we devise an adaptive optimistic algorithm, LCB-DR, that showcases enhanced dependence on the gaps, mirroring the contrast between uniform and optimistic allocation in the multi-armed bandit literature.
We consider the statistical analysis of heterogeneous data for prediction in situations where the observations include functions, typically time series. We extend the modeling with Mixtures-of-Experts (ME), as a framework of choice in modeling heterogeneity in data for prediction with vectorial observations, to this functional data analysis context. We first present a new family of ME models, named functional ME (FME) in which the predictors are potentially noisy observations, from entire functions. Furthermore, the data generating process of the predictor and the real response, is governed by a hidden discrete variable representing an unknown partition. Second, by imposing sparsity on derivatives of the underlying functional parameters via Lasso-like regularizations, we provide sparse and interpretable functional representations of the FME models called iFME. We develop dedicated expectation--maximization algorithms for Lasso-like (EM-Lasso) regularized maximum-likelihood parameter estimation strategies to fit the models. The proposed models and algorithms are studied in simulated scenarios and in applications to two real data sets, and the obtained results demonstrate their performance in accurately capturing complex nonlinear relationships and in clustering the heterogeneous regression data.
Loss functions steer the optimization direction of recommendation models and are critical to model performance, but have received relatively little attention in recent recommendation research. Among various losses, we find Softmax loss (SL) stands out for not only achieving remarkable accuracy but also better robustness and fairness. Nevertheless, the current literature lacks a comprehensive explanation for the efficacy of SL. Toward addressing this research gap, we conduct theoretical analyses on SL and uncover three insights: 1) Optimizing SL is equivalent to performing Distributionally Robust Optimization (DRO) on the negative data, thereby learning against perturbations on the negative distribution and yielding robustness to noisy negatives. 2) Comparing with other loss functions, SL implicitly penalizes the prediction variance, resulting in a smaller gap between predicted values and and thus producing fairer results. Building on these insights, we further propose a novel loss function Bilateral SoftMax Loss (BSL) that extends the advantage of SL to both positive and negative sides. BSL augments SL by applying the same Log-Expectation-Exp structure to positive examples as is used for negatives, making the model robust to the noisy positives as well. Remarkably, BSL is simple and easy-to-implement -- requiring just one additional line of code compared to SL. Experiments on four real-world datasets and three representative backbones demonstrate the effectiveness of our proposal. The code is available at //github.com/junkangwu/BSL
Model counting, or counting the satisfying assignments of a Boolean formula, is a fundamental problem with diverse applications. Given #P-hardness of the problem, developing algorithms for approximate counting is an important research area. Building on the practical success of SAT-solvers, the focus has recently shifted from theory to practical implementations of approximate counting algorithms. This has brought to focus new challenges, such as the design of auditable approximate counters that not only provide an approximation of the model count, but also a certificate that a verifier with limited computational power can use to check if the count is indeed within the promised bounds of approximation. Towards generating certificates, we start by examining the best-known deterministic approximate counting algorithm that uses polynomially many calls to a $\Sigma_2^P$ oracle. We show that this can be audited via a $\Sigma_2^P$ oracle with the query constructed over $n^2 \log^2 n$ variables, where the original formula has $n$ variables. Since $n$ is often large, we ask if the count of variables in the certificate can be reduced -- a crucial question for potential implementation. We show that this is indeed possible with a tradeoff in the counting algorithm's complexity. Specifically, we develop new deterministic approximate counting algorithms that invoke a $\Sigma_3^P$ oracle, but can be certified using a $\Sigma_2^P$ oracle using certificates on far fewer variables: our final algorithm uses only $n \log n$ variables. Our study demonstrates that one can simplify auditing significantly if we allow the counting algorithm to access a slightly more powerful oracle. This shows for the first time how audit complexity can be traded for complexity of approximate counting.
In turbulence modeling, we are concerned with finding closure models that represent the effect of the subgrid scales on the resolved scales. Recent approaches gravitate towards machine learning techniques to construct such models. However, the stability of machine-learned closure models and their abidance by physical structure (e.g. symmetries, conservation laws) are still open problems. To tackle both issues, we take the `discretize first, filter next' approach. In this approach we apply a spatial averaging filter to existing fine-grid discretizations. The main novelty is that we introduce an additional set of equations which dynamically model the energy of the subgrid scales. Having an estimate of the energy of the subgrid scales, we can use the concept of energy conservation to derive stability. The subgrid energy containing variables are determined via a data-driven technique. The closure model is used to model the interaction between the filtered quantities and the subgrid energy. Therefore the total energy should be conserved. Abiding by this conservation law yields guaranteed stability of the system. In this work, we propose a novel skew-symmetric convolutional neural network architecture that satisfies this law. The result is that stability is guaranteed, independent of the weights and biases of the network. Importantly, as our framework allows for energy exchange between resolved and subgrid scales it can model backscatter. To model dissipative systems (e.g. viscous flows), the framework is extended with a diffusive component. The introduced neural network architecture is constructed such that it also satisfies momentum conservation. We apply the new methodology to both the viscous Burgers' equation and the Korteweg-De Vries equation in 1D. The novel architecture displays superior stability properties when compared to a vanilla convolutional neural network.
In the field of intracity freight transportation, changes in order volume are significantly influenced by temporal and spatial factors. When building subsidy and pricing strategies, predicting the causal effects of these strategies on order volume is crucial. In the process of calculating causal effects, confounding variables can have an impact. Traditional methods to control confounding variables handle data from a holistic perspective, which cannot ensure the precision of causal effects in specific temporal and spatial dimensions. However, temporal and spatial dimensions are extremely critical in the logistics field, and this limitation may directly affect the precision of subsidy and pricing strategies. To address these issues, this study proposes a technique based on flexible temporal-spatial grid partitioning. Furthermore, based on the flexible grid partitioning technique, we further propose a continuous entropy balancing method in the temporal-spatial domain, which named TS-EBCT (Temporal-Spatial Entropy Balancing for Causal Continue Treatments). The method proposed in this paper has been tested on two simulation datasets and two real datasets, all of which have achieved excellent performance. In fact, after applying the TS-EBCT method to the intracity freight transportation field, the prediction accuracy of the causal effect has been significantly improved. It brings good business benefits to the company's subsidy and pricing strategies.
Solving partial differential equations (PDEs) by learning the solution operators has emerged as an attractive alternative to traditional numerical methods. However, implementing such architectures presents two main challenges: flexibility in handling irregular and arbitrary input and output formats and scalability to large discretizations. Most existing architectures are limited by their desired structure or infeasible to scale large inputs and outputs. To address these issues, we introduce an attention-based model called an inducing-point operator transformer (IPOT). Inspired by inducing points methods, IPOT is designed to handle any input function and output query while capturing global interactions in a computationally efficient way. By detaching the inputs/outputs discretizations from the processor with a smaller latent bottleneck, IPOT offers flexibility in processing arbitrary discretizations and scales linearly with the size of inputs/outputs. Our experimental results demonstrate that IPOT achieves strong performances with manageable computational complexity on an extensive range of PDE benchmarks and real-world weather forecasting scenarios, compared to state-of-the-art methods.
Multi-modal fusion is a fundamental task for the perception of an autonomous driving system, which has recently intrigued many researchers. However, achieving a rather good performance is not an easy task due to the noisy raw data, underutilized information, and the misalignment of multi-modal sensors. In this paper, we provide a literature review of the existing multi-modal-based methods for perception tasks in autonomous driving. Generally, we make a detailed analysis including over 50 papers leveraging perception sensors including LiDAR and camera trying to solve object detection and semantic segmentation tasks. Different from traditional fusion methodology for categorizing fusion models, we propose an innovative way that divides them into two major classes, four minor classes by a more reasonable taxonomy in the view of the fusion stage. Moreover, we dive deep into the current fusion methods, focusing on the remaining problems and open-up discussions on the potential research opportunities. In conclusion, what we expect to do in this paper is to present a new taxonomy of multi-modal fusion methods for the autonomous driving perception tasks and provoke thoughts of the fusion-based techniques in the future.
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