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This paper investigates the achievable region and precoder design for multiple access wiretap (MAC-WT) channels, where each user transmits both secret and open (i.e., non-confidential) messages. All these messages are intended for the legitimate receiver (or Bob for brevity) and the eavesdropper (Eve) is interested only in the secret messages of all users. By allowing users with zero secret message rate to act as conventional MAC channel users with no wiretapping, we show that the achievable region of the discrete memoryless (DM) MAC-WT channel given in [1] can be enlarged. In [1], the achievability was proven by considering the two-user case, making it possible to prove a key auxiliary lemma by directly using the Fourier-Motzkin elimination procedure. However, this approach does not generalize to the case with any number of users. In this paper, we provide a new region that generally enlarges that in [1] and provide general achievability proof. Furthermore, we consider the Gaussian vector (GV) MAC-WT channel and maximize the sum secrecy rate by precoder design. Although this non-convex problem can be solved by the majorization minimization (MM) technique, this suffers from an extremely high computational complexity. Instead, we propose a simultaneous diagonalization based low-complexity (SDLC) method to maximize the secrecy rate of a simple single-user wiretap channel, and then use this method to iteratively optimize the covariance matrix of each user in the GV MAC-WT channel at hand. Simulation results show that, compared with existing approaches, the SDLC scheme achieves similar secrecy performance but requires much lower complexity. It is also shown that the system spectral efficiency can be significantly increased by simultaneously transmitting secret and open messages.

We analyze a general class of bilevel problems, in which the upper-level problem consists in the minimization of a smooth objective function and the lower-level problem is to find the fixed point of a smooth contraction map. This type of problems include instances of meta-learning, equilibrium models, hyperparameter optimization and data poisoning adversarial attacks. Several recent works have proposed algorithms which warm-start the lower level problem, i.e. they use the previous lower-level approximate solution as a staring point for the lower-level solver. This warm-start procedure allows one to improve the sample complexity in both the stochastic and deterministic settings, achieving in some cases the order-wise optimal sample complexity. However, there are situations, e.g., meta learning and equilibrium models, in which the warm-start procedure is not well-suited or ineffective. In this work we show that without warm-start, it is still possible to achieve order-wise optimal or near-optimal sample complexity. In particular, we propose a simple method which uses stochastic fixed point iterations at the lower-level and projected inexact gradient descent at the upper-level, that reaches an $\epsilon$-stationary point using $O(\epsilon^{-2})$ and $\tilde{O}(\epsilon^{-1})$ samples for the stochastic and the deterministic setting, respectively. Finally, compared to methods using warm-start, our approach yields a simpler analysis that does not need to study the coupled interactions between the upper-level and lower-level iterates

We study the feature-based newsvendor problem, in which a decision-maker has access to historical data consisting of demand observations and exogenous features. In this setting, we investigate feature selection, aiming to derive sparse, explainable models with improved out-of-sample performance. Up to now, state-of-the-art methods utilize regularization, which penalizes the number of selected features or the norm of the solution vector. As an alternative, we introduce a novel bilevel programming formulation. The upper-level problem selects a subset of features that minimizes an estimate of the out-of-sample cost of ordering decisions based on a held-out validation set. The lower-level problem learns the optimal coefficients of the decision function on a training set, using only the features selected by the upper-level. We present a mixed integer linear program reformulation for the bilevel program, which can be solved to optimality with standard optimization solvers. Our computational experiments show that the method accurately recovers ground-truth features already for instances with a sample size of a few hundred observations. In contrast, regularization-based techniques often fail at feature recovery or require thousands of observations to obtain similar accuracy. Regarding out-of-sample generalization, we achieve improved or comparable cost performance.

Most of the security services in the connected world of cyber-physical systems necessitate authenticating a large number of nodes privately. In this paper, the private authentication problem is considered which consists of a certificate authority, a verifier (or some verifiers), many legitimate users (provers), and an arbitrary number of attackers. Each legitimate user wants to be authenticated (using his personal key) by the verifier(s), while simultaneously staying completely anonymous (even to the verifier). On the other hand, an attacker must fail to be authenticated. We analyze this problem from an information-theoretical perspective and propose a general interactive information-theoretic model for the problem. As a metric to measure the reliability, we consider the normalized total key rate whose maximization has a trade-off with establishing privacy. The problem is considered in two different scenarios: single-server scenario (only one verifier is considered, which all the provers are connected to) and multi-server scenario ($N$ verifiers are assumed, where each verifier is connected to a subset of users). For both scenarios, two regimes are considered: finite size regime (i.e., the variables are elements of a finite field) and asymptotic regime (i.e., the variables are considered to have large enough length). We propose achievable schemes that satisfy the completeness, soundness, and privacy properties in both single-server and multi-server scenarios in all cases. In the finite size regime, the main idea is to generate the authentication keys according to a secret sharing scheme. We show that the proposed scheme in the special case of multi-server authentication in the finite size regime is optimal. In the asymptotic regime, we use a random binning based scheme that relies on the joint typicality to generate the authentication keys.

We introduce a novel Quadratic Unconstrained Binary Optimization (QUBO) formulation method for spanning tree problems. Instead of encoding the presence of edges in the tree individually, we opt to encode spanning trees as a permutation problem. We apply our method to four NP-hard spanning tree variants, namely the k-minimum spanning tree, degree-constrained minimum spanning tree, minimum leaf spanning tree, and maximum leaf spanning tree. Our main result is a formulation with $\mathcal{O}(|V|k)$ variables for the k-minimum spanning tree problem, beating related strategies that need $\mathcal{O}(|V|^{2})$ variables.

Transformer architectures have led to remarkable progress in many state-of-art applications. However, despite their successes, modern transformers rely on the self-attention mechanism, whose time- and space-complexity is quadratic in the length of the input. Several approaches have been proposed to speed up self-attention mechanisms to achieve sub-quadratic running time; however, the large majority of these works are not accompanied by rigorous error guarantees. In this work, we establish lower bounds on the computational complexity of self-attention in a number of scenarios. We prove that the time complexity of self-attention is necessarily quadratic in the input length, unless the Strong Exponential Time Hypothesis (SETH) is false. This argument holds even if the attention computation is performed only approximately, and for a variety of attention mechanisms. As a complement to our lower bounds, we show that it is indeed possible to approximate dot-product self-attention using finite Taylor series in linear-time, at the cost of having an exponential dependence on the polynomial order.

The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is frequently pointed out as one of the bottlenecks in the computations. The second bottleneck being the large and numerous linear systems to be solved arising from the use of Newton's method to solve nonlinear systems of equations. In this paper, we will address the first issue. We will see how under mild assumptions the assemblage procedure may be rewritten using a completely loop-free algorithm. Our approach leads to a small matrix-matrix multiplication for which we may count on highly optimized algorithms.

The solution of multistage stochastic linear problems (MSLP) represents a challenge for many applications. Long-term hydrothermal dispatch planning (LHDP) materializes this challenge in a real-world problem that affects electricity markets, economies, and natural resources worldwide. No closed-form solutions are available for MSLP and the definition of non-anticipative policies with high-quality out-of-sample performance of is crucial. Linear decision rules (LDR) provide an interesting simulation-based framework for finding high-quality policies to MSLP through two-stage stochastic models. In practical applications, however, the number of parameters to be estimated when using an LDR may be close or higher than the number of scenarios of the sample average approximation problem, thereby generating an in-sample overfit and poor performances in out-of-sample simulations. In this paper, we propose a novel regularized LDR to solve MSLP based on the AdaLASSO (adaptive least absolute shrinkage and selection operator). The goal is to use the parsimony principle as largely studied in high-dimensional linear regression models to obtain better out-of-sample performance for a LDR applied to MSLP. Computational experiments show that the overfit threat is non-negligible when using the classical non-regularized LDR to solve the LHDP, one of the most studied MSLP with relevant applications in industry. Our analysis highlights the following benefits of the proposed framework in comparison to the non-regularized benchmark: 1) significant reductions in the number of non-zero coefficients (model parsimony), 2) substantial cost reductions in out-of-sample evaluations, and 3) improved spot-price profiles.

We study exact active learning of binary and multiclass classifiers with margin. Given an $n$-point set $X \subset \mathbb{R}^m$, we want to learn any unknown classifier on $X$ whose classes have finite strong convex hull margin, a new notion extending the SVM margin. In the standard active learning setting, where only label queries are allowed, learning a classifier with strong convex hull margin $\gamma$ requires in the worst case $\Omega\big(1+\frac{1}{\gamma}\big)^{(m-1)/2}$ queries. On the other hand, using the more powerful seed queries (a variant of equivalence queries), the target classifier could be learned in $O(m \log n)$ queries via Littlestone's Halving algorithm; however, Halving is computationally inefficient. In this work we show that, by carefully combining the two types of queries, a binary classifier can be learned in time $\operatorname{poly}(n+m)$ using only $O(m^2 \log n)$ label queries and $O\big(m \log \frac{m}{\gamma}\big)$ seed queries; the result extends to $k$-class classifiers at the price of a $k!k^2$ multiplicative overhead. Similar results hold when the input points have bounded bit complexity, or when only one class has strong convex hull margin against the rest. We complement the upper bounds by showing that in the worst case any algorithm needs $\Omega\big(k m \log \frac{1}{\gamma}\big)$ seed and label queries to learn a $k$-class classifier with strong convex hull margin $\gamma$.

Current deep learning research is dominated by benchmark evaluation. A method is regarded as favorable if it empirically performs well on the dedicated test set. This mentality is seamlessly reflected in the resurfacing area of continual learning, where consecutively arriving sets of benchmark data are investigated. The core challenge is framed as protecting previously acquired representations from being catastrophically forgotten due to the iterative parameter updates. However, comparison of individual methods is nevertheless treated in isolation from real world application and typically judged by monitoring accumulated test set performance. The closed world assumption remains predominant. It is assumed that during deployment a model is guaranteed to encounter data that stems from the same distribution as used for training. This poses a massive challenge as neural networks are well known to provide overconfident false predictions on unknown instances and break down in the face of corrupted data. In this work we argue that notable lessons from open set recognition, the identification of statistically deviating data outside of the observed dataset, and the adjacent field of active learning, where data is incrementally queried such that the expected performance gain is maximized, are frequently overlooked in the deep learning era. Based on these forgotten lessons, we propose a consolidated view to bridge continual learning, active learning and open set recognition in deep neural networks. Our results show that this not only benefits each individual paradigm, but highlights the natural synergies in a common framework. We empirically demonstrate improvements when alleviating catastrophic forgetting, querying data in active learning, selecting task orders, while exhibiting robust open world application where previously proposed methods fail.