Our general setup leads to a number of interesting findings. We outline precise conditions that decide the sign of the optimal setting $\lambda_{\opt}$ for the ridge parameter $\lambda$ and confirm the implicit $\ell_2$ regularization effect of overparameterization, which theoretically justifies the surprising empirical observation that $\lambda_{\opt}$ can be \textit{negative} in the overparameterized regime. We also characterize the double descent phenomenon for principal component regression (PCR) when $\vX$ and $\vbeta_{\star}$ are both anisotropic. Finally, we determine the optimal weighting matrix $\vSigma_w$ for both the ridgeless ($\lambda\to 0$) and optimally regularized ($\lambda = \lambda_{\opt}$) case, and demonstrate the advantage of the weighted objective over standard ridge regression and PCR.

We thank all reviewers for their helpful feedback. Below we address the questions and comments individually. We will correct typos in the main text and bibliography, and refer to Figure 1 in the introduction. We apologize for the confusion. The V AMP framework does not capture our "aligned" or "misalgined" cases.

The design of training objective is central to training time-series forecasting models. Existing training objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which we found leading to the following two issues: (1) overlook the label autocorrelation effect among future steps, leading to biased training objective; (2) fail to set heterogeneous task weights for different forecasting tasks corresponding to varying future steps, limiting the forecasting performance. To fill this gap, we propose a novel quadratic-form weighted training objective, addressing both of the issues simultaneously. Specifically, the off-diagonal elements of the weighting matrix account for the label autocorrelation effect, whereas the non-uniform diagonals are expected to match the most preferable weights of the forecasting tasks with varying future steps. To achieve this, we propose a Quadratic Direct Forecast (QDF) learning algorithm, which trains the forecast model using the adaptively updated quadratic-form weighting matrix. Experiments show that our QDF effectively improves performance of various forecast models, achieving state-of-the-art results. Code is available at https://anonymous.4open.science/r/QDF-8937.

These works have shown better performance for data generation.

Factor graph optimization serves as a fundamental framework for robotic perception, enabling applications such as pose estimation, simultaneous localization and mapping (SLAM), structure-from-motion (SfM), and situational awareness. Traditionally, these methods solve unconstrained least squares problems using algorithms such as Gauss-Newton and Levenberg-Marquardt. However, extending factor graphs with native support for equality constraints can improve solution accuracy and broaden their applicability, particularly in optimal control. In this paper, we propose a novel extension of factor graphs that seamlessly incorporates equality constraints without requiring additional optimization algorithms. Our approach maintains the efficiency and flexibility of existing second-order optimization techniques while ensuring constraint feasibility. To validate our method, we apply it to an optimal control problem for velocity tracking in autonomous vehicles and benchmark our results against state-of-the-art constraint handling techniques. Additionally, we introduce ecg2o, a header-only C++ library that extends the widely used g2o factor graph library by adding full support for equality-constrained optimization. This library, along with demonstrative examples and the optimal control problem, is available as open source at https://github.com/snt-arg/ecg2o

Learning feature interactions is the key to success for the large-scale CTR prediction in Ads ranking and recommender systems. In industry, deep neural network-based models are widely adopted for modeling such problems. Researchers proposed various neural network architectures for searching and modeling the feature interactions in an end-to-end fashion. However, most methods only learn static feature interactions and have not fully leveraged deep CTR models' representation capacity. In this paper, we propose a new model: DynInt. By extending Polynomial-Interaction-Network (PIN), which learns higher-order interactions recursively to be dynamic and data-dependent, DynInt further derived two modes for modeling dynamic higher-order interactions: dynamic activation and dynamic parameter. In dynamic activation mode, we adaptively adjust the strength of learned interactions by instance-aware activation gating networks. In dynamic parameter mode, we re-parameterize the parameters by different formulations and dynamically generate the parameters by instance-aware parameter generation networks. Through instance-aware gating mechanism and dynamic parameter generation, we enable the PIN to model dynamic interaction for potential industry applications. We implement the proposed model and evaluate the model performance on real-world datasets. Extensive experiment results demonstrate the efficiency and effectiveness of DynInt over state-of-the-art models.

Graph Neural Networks (GNNs) have received much attention recent years and have achieved state-of-the-art performances in many fields. The deeper GNNs can theoretically capture deeper neighborhood information. However, they often suffer from problems of over-fitting and over-smoothing. In order to incorporate deeper information while preserving considerable complexity and generalization ability, we propose Adaptive Graph Diffusion Networks with Hop-wise Attention (AGDNs-HA). We stack multi-hop neighborhood aggregations of different orders into single layer. Then we integrate them with the help of hop-wise attention, which is learnable and adaptive for each node. Experimental results on the standard dataset with semi-supervised node classification task show that our proposed methods achieve significant improvements.

Spatial econometric research typically relies on the assumption that the spatial dependence structure is known in advance and is represented by a deterministic spatial weights matrix. Contrary to classical approaches, we investigate the estimation of sparse spatial dependence structures for regular lattice data. In particular, an adaptive least absolute shrinkage and selection operator (lasso) is used to select and estimate the individual connections of the spatial weights matrix. To recover the spatial dependence structure, we propose cross-sectional resampling, assuming that the random process is exchangeable. The estimation procedure is based on a two-step approach to circumvent simultaneity issues that typically arise from endogenous spatial autoregressive dependencies. The two-step adaptive lasso approach with cross-sectional resampling is verified using Monte Carlo simulations. Eventually, we apply the procedure to model nitrogen dioxide ($\mathrm{NO_2}$) concentrations and show that estimating the spatial dependence structure contrary to using prespecified weights matrices improves the prediction accuracy considerably.