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A \emph{general branch-and-bound tree} is a branch-and-bound tree which is allowed to use general disjunctions of the form $\pi^{\top} x \leq \pi_0 \,\vee\, \pi^{\top}x \geq \pi_0 + 1$, where $\pi$ is an integer vector and $\pi_0$ is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists when we add Gaussian noise to the coefficients of the cross polytope, thus showing that polynomial-size "smoothed analysis" upper bound is not possible. The results in this paper can be viewed as the branch-and-bound analog of the seminal paper by Chv\'atal et al. \cite{chvatal1989cutting}, who proved lower bounds for the Chv\'atal-Gomory rank.

This paper presents a tractable sufficient condition for the consistency of maximum likelihood estimators (MLEs) in partially observed diffusion models, stated in terms of stationary distributions of the associated test processes, under the assumption that the set of unknown parameter values is finite. We illustrate the tractability of this sufficient condition by verifying it in the context of a latent price model of market microstructure. Finally, we describe an algorithm for computing MLEs in partially observed diffusion models and test it on historical data to estimate the parameters of the latent price model.

We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove method's stability and convergence with spectral accuracy in space. The temporal order depends on solution's regularity in time. Further, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result we find asymptotic exact values of error constants along with their remainders for discretizations of Caputo derivative and fractional integrals. These constants are the smallest possible which improves the previously established results from the literature.

We determine the exact minimax rate of a Gaussian sequence model under bounded convex constraints, purely in terms of the local geometry of the given constraint set $K$. Our main result shows that the minimax risk (up to constant factors) under the squared $L_2$ loss is given by $\epsilon^{*2} \wedge \operatorname{diam}(K)^2$ with \begin{align*} \epsilon^* = \sup \bigg\{\epsilon : \frac{\epsilon^2}{\sigma^2} \leq \log M^{\operatorname{loc}}(\epsilon)\bigg\}, \end{align*} where $\log M^{\operatorname{loc}}(\epsilon)$ denotes the local entropy of the set $K$, and $\sigma^2$ is the variance of the noise. We utilize our abstract result to re-derive known minimax rates for some special sets $K$ such as hyperrectangles, ellipses, and more generally quadratically convex orthosymmetric sets. Finally, we extend our results to the unbounded case with known $\sigma^2$ to show that the minimax rate in that case is $\epsilon^{*2}$.

ML models are typically trained using large datasets of high quality. However, training datasets often contain inconsistent or incomplete data. To tackle this issue, one solution is to develop algorithms that can check whether a prediction of a model is certifiably robust. Given a learning algorithm that produces a classifier and given an example at test time, a classification outcome is certifiably robust if it is predicted by every model trained across all possible worlds (repairs) of the uncertain (inconsistent) dataset. This notion of robustness falls naturally under the framework of certain answers. In this paper, we study the complexity of certifying robustness for a simple but widely deployed classification algorithm, $k$-Nearest Neighbors ($k$-NN). Our main focus is on inconsistent datasets when the integrity constraints are functional dependencies (FDs). For this setting, we establish a dichotomy in the complexity of certifying robustness w.r.t. the set of FDs: the problem either admits a polynomial time algorithm, or it is coNP-hard. Additionally, we exhibit a similar dichotomy for the counting version of the problem, where the goal is to count the number of possible worlds that predict a certain label. As a byproduct of our study, we also establish the complexity of a problem related to finding an optimal subset repair that may be of independent interest.

We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for $d=1$ and $d=2$. We consider a class of jump diffusion processes whose invariant density belongs to some H\"older space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate $\frac{1}{T}$, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for $d=1$ and is equal to $\frac{\log T}{T}$ for $d=2$. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates $\{\frac{1}{T},\frac{\log T}{T}\}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.

A prediction interval covers a future observation from a random process in repeated sampling, and is typically constructed by identifying a pivotal quantity that is also an ancillary statistic. Analogously, a tolerance interval covers a population percentile in repeated sampling and is often based on a pivotal quantity. One approach we consider in non-normal models leverages a link function resulting in a pivotal quantity that is approximately normally distributed. In settings where this normal approximation does not hold we consider a second approach for tolerance and prediction based on a confidence interval for the mean. These methods are intuitive, simple to implement, have proper operating characteristics, and are computationally efficient compared to Bayesian, re-sampling, and machine learning methods. This is demonstrated in the context of multi-site clinical trial recruitment with staggered site initiation, real-world time on treatment, and end-of-study success for a clinical endpoint.

We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically. For discretization, we employ quadtree-like grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions.

Communication in Millimeter wave (mmWave) band relies on narrow beams due to directionality, high path loss, and shadowing. One can use beam alignment (BA) techniques to find and adjust the direction of these narrow beams. In this paper, BA at the base station (BS) is considered, where the BS sends a set of BA packets to scan different angular regions while the user listens to the channel and sends feedback to the BS for each received packet. It is assumed that the packets and feedback received at the user and BS, respectively, can be correctly decoded. Motivated by practical constraints such as propagation delay, a feedback delay for each BA packet is considered. At the end of the BA, the BS allocates a narrow beam to the user including its angle of departure for data transmission and the objective is to maximize the resulting expected beamforming gain. A general framework for studying this problem is proposed based on which a lower bound on the optimal performance as well as an optimality achieving scheme are obtained. Simulation results reveal significant performance improvements over the state-of-the-art BA methods in the presence of feedback delay.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.