Overparametrization often helps improve the generalization performance. This paper presents a dual view of overparametrization suggesting that downsampling may also help generalize. Focusing on the proportional regime $m\asymp n \asymp p$, where $m$ represents the sketching size, $n$ is the sample size, and $p$ is the feature dimensionality, we investigate two out-of-sample prediction risks of the sketched ridgeless least square estimator. Our findings challenge conventional beliefs by showing that downsampling does not always harm generalization but can actually improve it in certain cases. We identify the optimal sketching size that minimizes out-of-sample prediction risks and demonstrate that the optimally sketched estimator exhibits stabler risk curves, eliminating the peaks of those for the full-sample estimator. To facilitate practical implementation, we propose an empirical procedure to determine the optimal sketching size. Finally, we extend our analysis to cover central limit theorems and misspecified models. Numerical studies strongly support our theory.

High resolution tactile sensing has great potential in autonomous mobile robotics, particularly for legged robots. One particular area where it has significant promise is the traversal of challenging, varied terrain. Depending on whether an environment is slippery, soft, hard or dry, a robot must adapt its method of locomotion accordingly. Currently many multi-legged robots, such as Boston Dynamic's Spot robot, have preset gaits for different surface types, but struggle over terrains where the surface type changes frequently. Being able to automatically detect changes within an environment would allow a robot to autonomously adjust its method of locomotion to better suit conditions, without requiring a human user to manually set the change in surface type. In this paper we report on the first detailed investigation of the properties of a particular bio-inspired tactile sensor, the TacTip, to test its suitability for this kind of automatic detection of surface conditions. We explored different processing techniques and a regression model, using a custom made rig for data collection to determine how a robot could sense directional and general force on the sensor in a variety of conditions. This allowed us to successfully demonstrate how the sensor can be used to distinguish between soft, hard, dry and (wet) slippery surfaces. We further explored a neural model to classify specific surface textures. Pin movement (the movement of optical markers within the sensor) was key to sensing this information, and all models relied on some form of temporal information. Our final trained models could successfully determine the direction the sensor is heading in, the amount of force acting on it, and determine differences in the surface texture such as Lego vs smooth hard surface, or concrete vs smooth hard surface.

We propose a novel surrogate modelling approach to efficiently and accurately approximate the response of complex dynamical systems driven by time-varying exogenous excitations over extended time periods. Our approach, namely manifold nonlinear autoregressive modelling with exogenous input (mNARX), involves constructing a problem-specific exogenous input manifold that is optimal for constructing autoregressive surrogates. The manifold, which forms the core of mNARX, is constructed incrementally by incorporating the physics of the system, as well as prior expert- and domain- knowledge. Because mNARX decomposes the full problem into a series of smaller sub-problems, each with a lower complexity than the original, it scales well with the complexity of the problem, both in terms of training and evaluation costs of the final surrogate. Furthermore, mNARX synergizes well with traditional dimensionality reduction techniques, making it highly suitable for modelling dynamical systems with high-dimensional exogenous inputs, a class of problems that is typically challenging to solve. Since domain knowledge is particularly abundant in physical systems, such as those found in civil and mechanical engineering, mNARX is well suited for these applications. We demonstrate that mNARX outperforms traditional autoregressive surrogates in predicting the response of a classical coupled spring-mass system excited by a one-dimensional random excitation. Additionally, we show that mNARX is well suited for emulating very high-dimensional time- and state-dependent systems, even when affected by active controllers, by surrogating the dynamics of a realistic aero-servo-elastic onshore wind turbine simulator. In general, our results demonstrate that mNARX offers promising prospects for modelling complex dynamical systems, in terms of accuracy and efficiency.

Transformer-based large language models (Vaswani et al., 2017) pretrained with diverse tasks have demonstrated strong ability for in-context learning (ICL), that is, the pretrained models can answer new queries based on a few in-context demonstrations (see, e.g., Brown et al. (2020) and references thereafter). ICL is one of the key abilities contributing to the success of large language models, which allows pretrained models to solve multiple downstream tasks without updating their model parameters. However, the statistical foundation of ICL is still in its infancy. A recent line of research aims to quantify ICL by studying transformers pretrained on the linear regression task with a Gaussian prior (Garg et al., 2022; Akyürek et al., 2022; Li et al., 2023b; Raventós et al., 2023). Specifically, Garg et al. (2022); Akyürek et al. (2022); Li et al. (2023b) study the setting where transformers are pretrained in an online manner using independent linear regression tasks with the same Gaussian prior. They find that such a pretrained transformer can perform ICL on fresh linear regression tasks.

Many causal estimands are only partially identifiable since they depend on the unobservable joint distribution between potential outcomes. Stratification on pretreatment covariates can yield sharper partial identification bounds; however, unless the covariates are discrete with relatively small support, this approach typically requires consistent estimation of the conditional distributions of the potential outcomes given the covariates. Thus, existing approaches may fail under model misspecification or if consistency assumptions are violated. In this study, we propose a unified and model-agnostic inferential approach for a wide class of partially identified estimands, based on duality theory for optimal transport problems. In randomized experiments, our approach can wrap around any estimates of the conditional distributions and provide uniformly valid inference, even if the initial estimates are arbitrarily inaccurate. Also, our approach is doubly robust in observational studies. Notably, this property allows analysts to use the multiplier bootstrap to select covariates and models without sacrificing validity even if the true model is not included. Furthermore, if the conditional distributions are estimated at semiparametric rates, our approach matches the performance of an oracle with perfect knowledge of the outcome model. Finally, we propose an efficient computational framework, enabling implementation on many practical problems in causal inference.

Over-parameterized models like deep nets and random forests have become very popular in machine learning. However, the natural goals of continuity and differentiability, common in regression models, are now often ignored in modern overparametrized, locally-adaptive models. We propose a general framework to construct a global continuous and differentiable model based on a weighted average of locally learned models in corresponding local regions. This model is competitive in dealing with data with different densities or scales of function values in different local regions. We demonstrate that when we mix kernel ridge and polynomial regression terms in the local models, and stitch them together continuously, we achieve faster statistical convergence in theory and improved performance in various practical settings.

Motivated by a recent literature on the double-descent phenomenon in machine learning, we consider highly over-parameterized models in causal inference, including synthetic control with many control units. In such models, there may be so many free parameters that the model fits the training data perfectly. We first investigate high-dimensional linear regression for imputing wage data and estimating average treatment effects, where we find that models with many more covariates than sample size can outperform simple ones. We then document the performance of high-dimensional synthetic control estimators with many control units. We find that adding control units can help improve imputation performance even beyond the point where the pre-treatment fit is perfect. We provide a unified theoretical perspective on the performance of these high-dimensional models. Specifically, we show that more complex models can be interpreted as model-averaging estimators over simpler ones, which we link to an improvement in average performance. This perspective yields concrete insights into the use of synthetic control when control units are many relative to the number of pre-treatment periods.

In this paper, we revisit the problem of sparse linear regression in the local differential privacy (LDP) model. Existing research in the non-interactive and sequentially local models has focused on obtaining the lower bounds for the case where the underlying parameter is $1$-sparse, and extending such bounds to the more general $k$-sparse case has proven to be challenging. Moreover, it is unclear whether efficient non-interactive LDP (NLDP) algorithms exist. To address these issues, we first consider the problem in the $\epsilon$ non-interactive LDP model and provide a lower bound of $\Omega(\frac{\sqrt{dk\log d}}{\sqrt{n}\epsilon})$ on the $\ell_2$-norm estimation error for sub-Gaussian data, where $n$ is the sample size and $d$ is the dimension of the space. We propose an innovative NLDP algorithm, the very first of its kind for the problem. As a remarkable outcome, this algorithm also yields a novel and highly efficient estimator as a valuable by-product. Our algorithm achieves an upper bound of $\tilde{O}({\frac{d\sqrt{k}}{\sqrt{n}\epsilon}})$ for the estimation error when the data is sub-Gaussian, which can be further improved by a factor of $O(\sqrt{d})$ if the server has additional public but unlabeled data. For the sequentially interactive LDP model, we show a similar lower bound of $\Omega({\frac{\sqrt{dk}}{\sqrt{n}\epsilon}})$. As for the upper bound, we rectify a previous method and show that it is possible to achieve a bound of $\tilde{O}(\frac{k\sqrt{d}}{\sqrt{n}\epsilon})$. Our findings reveal fundamental differences between the non-private case, central DP model, and local DP model in the sparse linear regression problem.

As machine learning models are increasingly being employed to make consequential decisions in real-world settings, it becomes critical to ensure that individuals who are adversely impacted (e.g., loan denied) by the predictions of these models are provided with a means for recourse. While several approaches have been proposed to construct recourses for affected individuals, the recourses output by these methods either achieve low costs (i.e., ease-of-implementation) or robustness to small perturbations (i.e., noisy implementations of recourses), but not both due to the inherent trade-offs between the recourse costs and robustness. Furthermore, prior approaches do not provide end users with any agency over navigating the aforementioned trade-offs. In this work, we address the above challenges by proposing the first algorithmic framework which enables users to effectively manage the recourse cost vs. More specifically, our framework Probabilistically ROBust rEcourse (PROBE) lets users choose the probability with which a recourse could get invalidated (recourse invalidation rate) if small changes are made to the recourse i.e., the recourse is implemented somewhat noisily. To this end, we propose a novel objective function which simultaneously minimizes the gap between the achieved (resulting) and desired recourse invalidation rates, minimizes recourse costs, and also ensures that the resulting recourse achieves a positive model prediction. We develop novel theoretical results to characterize the recourse invalidation rates corresponding to any given instance w.r.t. Experimental evaluation with multiple real world datasets demonstrates the efficacy of the proposed framework. Machine learning (ML) models are increasingly being deployed to make a variety of consequential decisions in domains such as finance, healthcare, and policy. Consequently, there is a growing emphasis on designing tools and techniques which can provide recourse to individuals who have been adversely impacted by the predictions of these models (Voigt & Von dem Bussche, 2017).

We consider the problem of model selection in a high-dimensional sparse linear regression model under the differential privacy framework. In particular, we consider the problem of differentially private best subset selection and study its utility guarantee. We adopt the well-known exponential mechanism for selecting the best model, and under a certain margin condition, we establish its strong model recovery property. However, the exponential search space of the exponential mechanism poses a serious computational bottleneck. To overcome this challenge, we propose a Metropolis-Hastings algorithm for the sampling step and establish its polynomial mixing time to its stationary distribution in the problem parameters $n,p$, and $s$. Furthermore, we also establish approximate differential privacy for the final estimates of the Metropolis-Hastings random walk using its mixing property. Finally, we also perform some illustrative simulations that echo the theoretical findings of our main results.