We study security functions which can serve to establish semantic security for the two central problems of information-theoretic security: the wiretap channel, and privacy amplification for secret key generation. The security functions are functional forms of mosaics of combinatorial designs, more precisely, of group divisible designs and balanced incomplete block designs. Every member of a mosaic is associated with a unique color, and each color corresponds to a unique message or key value. Every block index of the mosaic corresponds to a public seed shared between the two trusted communicating parties. The seed set should be as small as possible. We give explicit examples which have an optimal or nearly optimal trade-off of seed length versus color (i.e., message or key) rate. We also derive bounds for the security performance of security functions given by functional forms of mosaics of designs.
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We consider optimization problems in which the goal is find a $k$-dimensional subspace of $\reals^n$, $k<<n$, which minimizes a convex and smooth loss. Such problemsgeneralize the fundamental task of principal component analysis (PCA) to include robust and sparse counterparts, and logistic PCA for binary data, among others. While this problem is not convex it admits natural algorithms with very efficient iterations and memory requirements, which is highly desired in high-dimensional regimes however, arguing about their fast convergence to a global optimal solution is difficult. On the other hand, there exists a simple convex relaxation for which convergence to the global optimum is straightforward, however corresponding algorithms are not efficient when the dimension is very large. In this work we present a natural deterministic sufficient condition so that the optimal solution to the convex relaxation is unique and is also the optimal solution to the original nonconvex problem. Mainly, we prove that under this condition, a natural highly-efficient nonconvex gradient method, which we refer to as \textit{gradient orthogonal iteration}, when initialized with a "warm-start", converges linearly for the nonconvex problem. We also establish similar results for the nonconvex projected gradient method, and the Frank-Wolfe method when applied to the convex relaxation. We conclude with empirical evidence on synthetic data which demonstrate the appeal of our approach.
We extend a previous framework for designing differentially private (DP) mechanisms via randomized graph colorings that was restricted to binary functions, corresponding to colorings in a graph, to multi-valued functions. As before, datasets are nodes in the graph and any two neighboring datasets are connected by an edge. In our setting, we assume each dataset has a preferential ordering for the possible outputs of the mechanism, which we refer to as a rainbow. Different rainbows partition the graph of datasets into different regions. We show that when the DP mechanism is pre-specified at the boundary of such regions, at most one optimal mechanism can exist. Moreover, if the mechanism is to behave identically for all same-rainbow boundary datasets, the problem can be greatly simplified and solved by means of a morphism to a line graph. We then show closed form expressions for the line graph in the case of ternary functions. Treatment of ternary queries in this paper displays enough richness to be extended to higher-dimensional query spaces with preferential query ordering, but the optimality proof does not seem to follow directly from the ternary proof.
We explore a family of information measures that stems from R\'enyi's $\alpha$-Divergences with $\alpha<0$. In particular, we extend the definition of Sibson's $\alpha$-Mutual Information to negative values of $\alpha$ and show several properties of these objects. Moreover, we highlight how this family of information measures is related to functional inequalities that can be employed in a variety of fields, including lower-bounds on the Risk in Bayesian Estimation Procedures.
In this paper, we propose Nesterov Accelerated Shuffling Gradient (NASG), a new algorithm for the convex finite-sum minimization problems. Our method integrates the traditional Nesterov's acceleration momentum with different shuffling sampling schemes. We show that our algorithm has an improved rate of $\mathcal{O}(1/T)$ using unified shuffling schemes, where $T$ is the number of epochs. This rate is better than that of any other shuffling gradient methods in convex regime. Our convergence analysis does not require an assumption on bounded domain or a bounded gradient condition. For randomized shuffling schemes, we improve the convergence bound further. When employing some initial condition, we show that our method converges faster near the small neighborhood of the solution. Numerical simulations demonstrate the efficiency of our algorithm.
In order to develop machine learning and deep learning models that take into account the guidelines and principles of trustworthy AI, a novel information theoretic trustworthy AI framework is introduced. A unified approach to "privacy-preserving interpretable and transferable learning" is considered for studying and optimizing the tradeoffs between privacy, interpretability, and transferability aspects. A variational membership-mapping Bayesian model is used for the analytical approximations of the defined information theoretic measures for privacy-leakage, interpretability, and transferability. The approach consists of approximating the information theoretic measures via maximizing a lower-bound using variational optimization. The study presents a unified information theoretic approach to study different aspects of trustworthy AI in a rigorous analytical manner. The approach is demonstrated through numerous experiments on benchmark datasets and a real-world biomedical application concerned with the detection of mental stress on individuals using heart rate variability analysis.
Missing data is frequently encountered in practice. Propensity score estimation is a popular tool for handling such missingness. The propensity score is often developed using a model for the response probability, which can be subject to model misspecification. In this paper, we consider an alternative approach of estimating the inverse of the propensity scores using the density ratio function. The smoothed density ratio function is obtained by the solution to the information projection onto the space satisfying the moment conditions on the balancing scores. By including the covariates for the outcome regression models only into the density ratio model, we can achieve efficient propensity score estimation. Penalized regression is used to identify important covariates. We further extend the proposed approach to the multivariate missing case. Some limited simulation studies are presented to compare with the existing methods.
Understanding the source of the superior generalization ability of NNs remains one of the most important problems in ML research. There have been a series of theoretical works trying to derive non-vacuous bounds for NNs. Recently, the compression of information stored in weights (IIW) is proved to play a key role in NNs generalization based on the PAC-Bayes theorem. However, no solution of IIW has ever been provided, which builds a barrier for further investigation of the IIW's property and its potential in practical deep learning. In this paper, we propose an algorithm for the efficient approximation of IIW. Then, we build an IIW-based information bottleneck on the trade-off between accuracy and information complexity of NNs, namely PIB. From PIB, we can empirically identify the fitting to compressing phase transition during NNs' training and the concrete connection between the IIW compression and the generalization. Besides, we verify that IIW is able to explain NNs in broad cases, e.g., varying batch sizes, over-parameterization, and noisy labels. Moreover, we propose an MCMC-based algorithm to sample from the optimal weight posterior characterized by PIB, which fulfills the potential of IIW in enhancing NNs in practice.
The design of experiments involves an inescapable compromise between covariate balance and robustness. This paper provides a formalization of this trade-off and introduces an experimental design that allows experimenters to navigate it. The design is specified by a robustness parameter that bounds the worst-case mean squared error of an estimator of the average treatment effect. Subject to the experimenter's desired level of robustness, the design aims to simultaneously balance all linear functions of potentially many covariates. The achieved level of balance is better than previously known possible and considerably better than what a fully random assignment would produce. We show that the mean squared error of the estimator is bounded by the minimum of the loss function of an implicit ridge regression of the potential outcomes on the covariates. The estimator does not itself conduct covariate adjustment, so one can interpret the approach as regression adjustment by design. Finally, we provide non-asymptotic tail bounds for the estimator, which facilitate the construction of conservative confidence intervals.
In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations.
Graph Convolutional Networks (GCNs) have proved to be a most powerful architecture in aggregating local neighborhood information for individual graph nodes. Low-rank proximities and node features are successfully leveraged in existing GCNs, however, attributes that graph links may carry are commonly ignored, as almost all of these models simplify graph links into binary or scalar values describing node connectedness. In our paper instead, links are reverted to hypostatic relationships between entities with descriptional attributes. We propose GCN-LASE (GCN with Link Attributes and Sampling Estimation), a novel GCN model taking both node and link attributes as inputs. To adequately captures the interactions between link and node attributes, their tensor product is used as neighbor features, based on which we define several graph kernels and further develop according architectures for LASE. Besides, to accelerate the training process, the sum of features in entire neighborhoods are estimated through Monte Carlo method, with novel sampling strategies designed for LASE to minimize the estimation variance. Our experiments show that LASE outperforms strong baselines over various graph datasets, and further experiments corroborate the informativeness of link attributes and our model's ability of adequately leveraging them.
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