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This paper addresses the problem of determining all optimal integer solutions of a linear integer network flow problem, which we call the all optimal integer flow (AOF) problem. We derive an O(F (m + n) + mn + M ) time algorithm to determine all F many optimal integer flows in a directed network with n nodes and m arcs, where M is the best time needed to find one minimum cost flow. We remark that stopping Hamacher's well-known method for the determination of the K best integer flows at the first sub-optimal flow results in an algorithm with a running time of O(F m(n log n + m) + M ) for solving the AOF problem. Our improvement is essentially made possible by replacing the shortest path sub-problem with a more efficient way to determine a so called proper zero cost cycle using a modified depth-first search technique. As a byproduct, our analysis yields an enhanced algorithm to determine the K best integer flows that runs in O(Kn3 + M ). Besides, we give lower and upper bounds for the number of all optimal integer and feasible integer solutions. Our bounds are based on the fact that any optimal solution can be obtained by an initial optimal tree solution plus a conical combination of incidence vectors of all induced cycles with bounded coefficients.

Generalized linear models are flexible tools for the analysis of diverse datasets, but the classical formulation requires that the parametric component is correctly specified and the data contain no atypical observations. To address these shortcomings we introduce and study a family of nonparametric full rank and lower rank spline estimators that result from the minimization of a penalized power divergence. The proposed class of estimators is easily implementable, offers high protection against outlying observations and can be tuned for arbitrarily high efficiency in the case of clean data. We show that under weak assumptions these estimators converge at a fast rate and illustrate their highly competitive performance on a simulation study and two real-data examples.

In network analysis, how to estimate the number of communities $K$ is a fundamental problem. We consider a broad setting where we allow severe degree heterogeneity and a wide range of sparsity levels, and propose Stepwise Goodness-of-Fit (StGoF) as a new approach. This is a stepwise algorithm, where for $m = 1, 2, \ldots$, we alternately use a community detection step and a goodness-of-fit (GoF) step. We adapt SCORE \cite{SCORE} for community detection, and propose a new GoF metric. We show that at step $m$, the GoF metric diverges to $\infty$ in probability for all $m < K$ and converges to $N(0,1)$ if $m = K$. This gives rise to a consistent estimate for $K$. Also, we discover the right way to define the signal-to-noise ratio (SNR) for our problem and show that consistent estimates for $K$ do not exist if $\mathrm{SNR} \goto 0$, and StGoF is uniformly consistent for $K$ if $\mathrm{SNR} \goto \infty$. Therefore, StGoF achieves the optimal phase transition. Similar stepwise methods (e.g., \cite{wang2017likelihood, ma2018determining}) are known to face analytical challenges. We overcome the challenges by using a different stepwise scheme in StGoF and by deriving sharp results that are not available before. The key to our analysis is to show that SCORE has the {\it Non-Splitting Property (NSP)}. Primarily due to a non-tractable rotation of eigenvectors dictated by the Davis-Kahan $\sin(\theta)$ theorem, the NSP is non-trivial to prove and requires new techniques we develop.

In this paper, we propose a variationally consistent technique for decreasing the maximum eigenfrequencies of structural dynamics related finite element formulations. Our approach is based on adding a symmetric positive-definite term to the mass matrix that follows from the integral of the traction jump across element boundaries. The added term is weighted by a small factor, for which we derive a suitable, and simple, element-local parameter choice. For linear problems, we show that our mass-scaling method produces no adverse effects in terms of spatial accuracy and orders of convergence. We illustrate these properties in one, two and three spatial dimension, for quadrilateral elements and triangular elements, and for up to fourth order polynomials basis functions. To extend the method to non-linear problems, we introduce a linear approximation and show that a sizeable increase in critical time-step size can be achieved while only causing minor (even beneficial) influences on the dynamic response.

We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique solution of the continuous model is at least one. The numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and provides thus a bound on the rate of weak convergence.

We consider the problem of coded distributed computing using polar codes. The average execution time of a coded computing system is related to the error probability for transmission over the binary erasure channel in recent work by Soleymani, Jamali and Mahdavifar, where the performance of binary linear codes is investigated. In this paper, we focus on polar codes and unveil a connection between the average execution time and the scaling exponent $\mu$ of the family of codes. The scaling exponent has emerged as a central object in the finite-length characterization of polar codes, and it captures the speed of convergence to capacity. In particular, we show that (i) the gap between the normalized average execution time of polar codes and that of optimal MDS codes is $O(n^{-1/\mu})$, and (ii) this upper bound can be improved to roughly $O(n^{-1/2})$ by considering polar codes with large kernels. We conjecture that these bounds could be improved to $O(n^{-2/\mu})$ and $O(n^{-1})$, respectively, and provide a heuristic argument as well as numerical evidence supporting this view.

In this paper we consider $L_p$-approximation, $p \in \{2,\infty\}$, of periodic functions from weighted Korobov spaces. In particular, we discuss tractability properties of such problems, which means that we aim to relate the dependence of the information complexity on the error demand $\varepsilon$ and the dimension $d$ to the decay rate of the weight sequence $(\gamma_j)_{j \ge 1}$ assigned to the Korobov space. Some results have been well known since the beginning of this millennium, others have been proven quite recently. We give a survey of these findings and will add some new results on the $L_\infty$-approximation problem. To conclude, we give a concise overview of results and collect a number of interesting open problems.

We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

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.