This paper considers the problem of matrix-variate logistic regression. The fundamental error threshold on estimating coefficient matrices in the logistic regression problem is found by deriving a lower bound on the minimax risk. The focus of this pa
per is on derivation of a minimax risk lower bound for low-rank coefficient matrices. The bound depends explicitly on the dimensions and distribution of the covariates, the rank and energy of the coefficient matrix, and the number of samples. The resulting bound is proportional to the intrinsic degrees of freedom in the problem, which suggests the sample complexity of the low-rank matrix logistic regression problem can be lower than that for vectorized logistic regression. color{red}color{black} The proof techniques utilized in this work also set the stage for development of minimax lower bounds for tensor-variate logistic regression problems.
The presence of correlation is known to make privacy protection more difficult. We investigate the privacy of socially contagious attributes on a network of individuals, where each individual possessing that attribute may influence a number of others
into adopting it. We show that for contagions following the Independent Cascade model there exists a giant connected component of infected nodes, containing a constant fraction of all the nodes who all receive the contagion from the same set of sources. We further show that it is extremely hard to hide the existence of this giant connected component if we want to obtain an estimate of the activated users at an acceptable level. Moreover, an adversary possessing this knowledge can predict the real status (active or inactive) with decent probability for many of the individuals regardless of the privacy (perturbation) mechanism used. As a case study, we show that the Wasserstein mechanism, a state-of-the-art privacy mechanism designed specifically for correlated data, introduces a noise with magnitude of order $Omega(n)$ in the count estimation in our setting. We provide theoretical guarantees for two classes of random networks: Erdos Renyi graphs and Chung-Lu power-law graphs under the Independent Cascade model. Experiments demonstrate that a giant connected component of infected nodes can and does appear in real-world networks and that a simple inference attack can reveal the status of a good fraction of nodes.
We study the best-arm identification problem in multi-armed bandits with stochastic, potentially private rewards, when the goal is to identify the arm with the highest quantile at a fixed, prescribed level. First, we propose a (non-private) successiv
e elimination algorithm for strictly optimal best-arm identification, we show that our algorithm is $delta$-PAC and we characterize its sample complexity. Further, we provide a lower bound on the expected number of pulls, showing that the proposed algorithm is essentially optimal up to logarithmic factors. Both upper and lower complexity bounds depend on a special definition of the associated suboptimality gap, designed in particular for the quantile bandit problem, as we show when the gap approaches zero, best-arm identification is impossible. Second, motivated by applications where the rewards are private, we provide a differentially private successive elimination algorithm whose sample complexity is finite even for distributions with infinite support-size, and we characterize its sample complexity. Our algorithms do not require prior knowledge of either the suboptimality gap or other statistical information related to the bandit problem at hand.
We provide high probability finite sample complexity guarantees for hidden non-parametric structure learning of tree-shaped graphical models, whose hidden and observable nodes are discrete random variables with either finite or countable alphabets. W
e study a fundamental quantity called the (noisy) information threshold, which arises naturally from the error analysis of the Chow-Liu algorithm and, as we discuss, provides explicit necessary and sufficient conditions on sample complexity, by effectively summarizing the difficulty of the tree-structure learning problem. Specifically, we show that the finite sample complexity of the Chow-Liu algorithm for ensuring exact structure recovery from noisy data is inversely proportional to the information threshold squared (provided it is positive), and scales almost logarithmically relative to the number of nodes over a given probability of failure. Conversely, we show that, if the number of samples is less than an absolute constant times the inverse of information threshold squared, then no algorithm can recover the hidden tree structure with probability greater than one half. As a consequence, our upper and lower bounds match with respect to the information threshold, indicating that it is a fundamental quantity for the problem of learning hidden tree-structured models. Further, the Chow-Liu algorithm with noisy data as input achieves the optimal rate with respect to the information threshold. Lastly, as a byproduct of our analysis, we resolve the problem of tree structure learning in the presence of non-identically distributed observation noise, providing conditions for convergence of the Chow-Liu algorithm under this setting, as well.
We study a distributed sampling problem where a set of processors want to output (approximately) independent and identically distributed samples from a joint distribution with the help of a common message from a coordinator. Each processor has access
to a subset of sources from a set of independent sources of shared randomness. We consider two cases -- in the omniscient coordinator setting, the coordinator has access to all these sources of shared randomness, while in the oblivious coordinator setting, it has access to none. All processors and the coordinator may privately randomize. In the omniscient coordinator setting, when the subsets at the processors are disjoint (individually shared randomness model), we characterize the rate of communication required from the coordinator to the processors over a multicast link. For the two-processor case, the optimal rate matches a special case of relaxed Wyners common information proposed by Gastpar and Sula (2019), thereby providing an operational meaning to the latter. We also give an upper bound on the communication rate for the randomness-on-the-forehead model where each processor observes all but one source of randomness and we give an achievable strategy for the general case where the processors have access to arbitrary subsets of sources of randomness. Also, we consider a more general model where the processors observe components of correlated sources (with the coordinator observing all the components), where we characterize the communication rate when all the processors wish to output the same random sequence. In the oblivious coordinator setting, we completely characterize the trade-off region between the communication and shared randomness rates for the general case where the processors have access to arbitrary subsets of sources of randomness.
Many applications of machine learning, such as human health research, involve processing private or sensitive information. Privacy concerns may impose significant hurdles to collaboration in scenarios where there are multiple sites holding data and t
he goal is to estimate properties jointly across all datasets. Differentially private decentralized algorithms can provide strong privacy guarantees. However, the accuracy of the joint estimates may be poor when the datasets at each site are small. This paper proposes a new framework, Correlation Assisted Private Estimation (CAPE), for designing privacy-preserving decentralized algorithms with better accuracy guarantees in an honest-but-curious model. CAPE can be used in conjunction with the functional mechanism for statistical and machine learning optimization problems. A tighter characterization of the functional mechanism is provided that allows CAPE to achieve the same performance as a centralized algorithm in the decentralized setting using all datasets. Empirical results on regression and neural network problems for both synthetic and real datasets show that differentially private methods can be competitive with non-private algorithms in many scenarios of interest.
This work addresses the problem of learning sparse representations of tensor data using structured dictionary learning. It proposes learning a mixture of separable dictionaries to better capture the structure of tensor data by generalizing the separa
ble dictionary learning model. Two different approaches for learning mixture of separable dictionaries are explored and sufficient conditions for local identifiability of the underlying dictionary are derived in each case. Moreover, computational algorithms are developed to solve the problem of learning mixture of separable dictionaries in both batch and online settings. Numerical experiments are used to show the usefulness of the proposed model and the efficacy of the developed algorithms.
Two processors output correlated sequences using the help of a coordinator with whom they individually share independent randomness. For the case of unlimited shared randomness, we characterize the rate of communication required from the coordinator
to the processors over a broadcast link. We also give an achievable trade-off between the communication and shared randomness rates.
In many signal processing and machine learning applications, datasets containing private information are held at different locations, requiring the development of distributed privacy-preserving algorithms. Tensor and matrix factorizations are key com
ponents of many processing pipelines. In the distributed setting, differentially private algorithms suffer because they introduce noise to guarantee privacy. This paper designs new and improved distributed and differentially private algorithms for two popular matrix and tensor factorization methods: principal component analysis (PCA) and orthogonal tensor decomposition (OTD). The new algorithms employ a correlated noise design scheme to alleviate the effects of noise and can achieve the same noise level as the centralized scenario. Experiments on synthetic and real data illustrate the regimes in which the correlated noise allows performance matching with the centralized setting, outperforming previous methods and demonstrating that meaningful utility is possible while guaranteeing differential privacy.
This paper derives sufficient conditions for local recovery of coordinate dictionaries comprising a Kronecker-structured dictionary that is used for representing $K$th-order tensor data. Tensor observations are assumed to be generated from a Kronecke
r-structured dictionary multiplied by sparse coefficient tensors that follow the separable sparsity model. This work provides sufficient conditions on the underlying coordinate dictionaries, coefficient and noise distributions, and number of samples that guarantee recovery of the individual coordinate dictionaries up to a specified error, as a local minimum of the objective function, with high probability. In particular, the sample complexity to recover $K$ coordinate dictionaries with dimensions $m_k times p_k$ up to estimation error $varepsilon_k$ is shown to be $max_{k in [K]}mathcal{O}(m_kp_k^3varepsilon_k^{-2})$.
