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We prove that there are continuum-many axiomatic extensions of the full Lambek calculus with exchange that have the deductive interpolation property. Further, we extend this result to both classical and intuitionistic linear logic as well as their multiplicative-additive fragments. None of the logics we exhibit have the Craig interpolation property, but we show that they all enjoy a guarded form of Craig interpolation.

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.

In many industrial applications, obtaining labeled observations is not straightforward as it often requires the intervention of human experts or the use of expensive testing equipment. In these circumstances, active learning can be highly beneficial in suggesting the most informative data points to be used when fitting a model. Reducing the number of observations needed for model development alleviates both the computational burden required for training and the operational expenses related to labeling. Online active learning, in particular, is useful in high-volume production processes where the decision about the acquisition of the label for a data point needs to be taken within an extremely short time frame. However, despite the recent efforts to develop online active learning strategies, the behavior of these methods in the presence of outliers has not been thoroughly examined. In this work, we investigate the performance of online active linear regression in contaminated data streams. Our study shows that the currently available query strategies are prone to sample outliers, whose inclusion in the training set eventually degrades the predictive performance of the models. To address this issue, we propose a solution that bounds the search area of a conditional D-optimal algorithm and uses a robust estimator. Our approach strikes a balance between exploring unseen regions of the input space and protecting against outliers. Through numerical simulations, we show that the proposed method is effective in improving the performance of online active learning in the presence of outliers, thus expanding the potential applications of this powerful tool.

In a multi-agent system, agents can cooperatively learn a model from data by exchanging their estimated model parameters, without the need to exchange the locally available data used by the agents. This strategy, often called federated learning, is mainly employed for two reasons: (i) improving resource-efficiency by avoiding to share potentially large datasets and (ii) guaranteeing privacy of local agents' data. Efficiency can be further increased by adopting a beyond-5G communication strategy that goes under the name of Over-the-Air Computation. This strategy exploits the interference property of the wireless channel. Standard communication schemes prevent interference by enabling transmissions of signals from different agents at distinct time or frequency slots, which is not required with Over-the-Air Computation, thus saving resources. In this case, the received signal is a weighted sum of transmitted signals, with unknown weights (fading channel coefficients). State of the art papers in the field aim at reconstructing those unknown coefficients. In contrast, the approach presented here does not require reconstructing channel coefficients by complex encoding-decoding schemes. This improves both efficiency and privacy.

Let a polytope $P$ be defined by a system $A x \leq b$. We consider the problem of counting the number of integer points inside $P$, assuming that $P$ is $\Delta$-modular, where the polytope $P$ is called $\Delta$-modular if all the rank sub-determinants of $A$ are bounded by $\Delta$ in the absolute value. We present a new FPT-algorithm, parameterized by $\Delta$ and by the maximal number of vertices in $P$, where the maximum is taken by all r.h.s. vectors $b$. We show that our algorithm is more efficient for $\Delta$-modular problems than the approach of A. Barvinok et al. To this end, we do not directly compute the short rational generating function for $P \cap Z^n$, which is commonly used for the considered problem. Instead, we use the dynamic programming principle to compute its particular representation in the form of exponential series that depends on a single variable. We completely do not rely to the Barvinok's unimodular sign decomposition technique. Using our new complexity bound, we consider different special cases that may be of independent interest. For example, we give FPT-algorithms for counting the integer points number in $\Delta$-modular simplices and similar polytopes that have $n + O(1)$ facets. As a special case, for any fixed $m$, we give an FPT-algorithm to count solutions of the unbounded $m$-dimensional $\Delta$-modular subset-sum problem.

Performative prediction, as introduced by Perdomo et al. (2020), is a framework for studying social prediction in which the data distribution itself changes in response to the deployment of a model. Existing work on optimizing accuracy in this setting hinges on two assumptions that are easily violated in practice: that the performative risk is convex over the deployed model, and that the mapping from the model to the data distribution is known to the model designer in advance. In this paper, we initiate the study of tractable performative prediction problems that do not require these assumptions. To tackle this more challenging setting, we develop a two-level zeroth-order optimization algorithm, where one level aims to compute the distribution map, and the other level reparameterizes the performative prediction objective as a function of the induced data distribution. Under mild conditions, this reparameterization allows us to transform the non-convex objective into a convex one and achieve provable regret guarantees. In particular, we provide a regret bound that is sublinear in the total number of performative samples taken and only polynomial in the dimension of the model parameter.

Gun violence is a major problem in contemporary American society, with tens of thousands injured each year. However, relatively little is known about the effects on family members and how effects vary across subpopulations. To study these questions and, more generally, to address a gap in the causal inference literature, we present a framework for the study of effect modification or heterogeneous treatment effects in difference-in-differences designs. We implement a new matching technique, which combines profile matching and risk set matching, to (i) preserve the time alignment of covariates, exposure, and outcomes, avoiding pitfalls of other common approaches for difference-in-differences, and (ii) explicitly control biases due to imbalances in observed covariates in subgroups discovered from the data. Our case study shows significant and persistent effects of nonfatal firearm injuries on several health outcomes for those injured and on the mental health of their family members. Sensitivity analyses reveal that these results are moderately robust to unmeasured confounding bias. Finally, while the effects for those injured are modified largely by the severity of the injury and its documented intent, for families, effects are strongest for those whose relative's injury is documented as resulting from an assault, self-harm, or law enforcement intervention.

Most existing studies on linear bandits focus on the one-dimensional characterization of the overall system. While being representative, this formulation may fail to model applications with high-dimensional but favorable structures, such as the low-rank tensor representation for recommender systems. To address this limitation, this work studies a general tensor bandits model, where actions and system parameters are represented by tensors as opposed to vectors, and we particularly focus on the case that the unknown system tensor is low-rank. A novel bandit algorithm, coined TOFU (Tensor Optimism in the Face of Uncertainty), is developed. TOFU first leverages flexible tensor regression techniques to estimate low-dimensional subspaces associated with the system tensor. These estimates are then utilized to convert the original problem to a new one with norm constraints on its system parameters. Lastly, a norm-constrained bandit subroutine is adopted by TOFU, which utilizes these constraints to avoid exploring the entire high-dimensional parameter space. Theoretical analyses show that TOFU improves the best-known regret upper bound by a multiplicative factor that grows exponentially in the system order. A novel performance lower bound is also established, which further corroborates the efficiency of TOFU.

Training a neural network (NN) typically relies on some type of curve-following method, such as gradient descent (GD) (and stochastic gradient descent (SGD)), ADADELTA, ADAM or limited memory algorithms. Convergence for these algorithms usually relies on having access to a large quantity of observations in order to achieve a high level of accuracy and, with certain classes of functions, these algorithms could take multiple epochs of data points to catch on. Herein, a different technique with the potential of achieving dramatically better speeds of convergence, especially for shallow networks, is explored: it does not curve-follow but rather relies on 'decoupling' hidden layers and on updating their weighted connections through bootstrapping, resampling and linear regression. By utilizing resampled observations, the convergence of this process is empirically shown to be remarkably fast and to require a lower amount of data points: in particular, our experiments show that one needs a fraction of the observations that are required with traditional neural network training methods to approximate various classes of functions.