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1. Structured Low-Rank Tensors for Generalized Linear Models

   Taki, Batoul A; Sarwate, Anand; Bajwa, Waheed U (August 2023, Transactions on machine learning research)

   Recent works have shown that imposing tensor structures on the coefficient tensor in regression problems can lead to more reliable parameter estimation and lower sample complexity compared to vector-based methods. This work investigates a new low-rank tensor model, called Low Separation Rank (LSR), in Generalized Linear Model (GLM) problems. The LSR model – which generalizes the well-known Tucker and CANDECOMP/PARAFAC (CP) models, and is a special case of the Block Tensor Decomposition (BTD) model – is imposed onto the coefficient tensor in the GLM model. This work proposes a block coordinate descent algorithm for parameter estimation in LSR-structured tensor GLMs. Most importantly, it derives a minimax lower bound on the error threshold on estimating the coefficient tensor in LSR tensor GLM problems. The minimax bound is proportional to the intrinsic degrees of freedom in the LSR tensor GLM problem, suggesting that its sample complexity may be significantly lower than that of vectorized GLMs. This result can also be specialised to lower bound the estimation error in CP and Tucker-structured GLMs. The derived bounds are comparable to tight bounds in the literature for Tucker linear regression, and the tightness of the minimax lower bound is further assessed numerically. Finally, numerical experiments on synthetic datasets demonstrate the efficacy of the proposed LSR tensor model for three regression types (linear, logistic and Poisson). Experiments on a collection of medical imaging datasets demonstrate the usefulness of the LSR model over other tensor models (Tucker and CP) on real, imbalanced data with limited available samples. License: Creative Commons Attribution 4.0 International (CC BY 4.0) 

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2. On Regret with Multiple Best Arms

   Zhu, Yinglun; Nowak, Robert (November 2022, Advances in Neural Information Processing Systems)

   We study a regret minimization problem with the existence of multiple best/near-optimal arms in the multi-armed bandit setting. We consider the case when the number of arms/actions is comparable or much larger than the time horizon, and make no assumptions about the structure of the bandit instance. Our goal is to design algorithms that can automatically adapt to the unknown hardness of the problem, i.e., the number of best arms. Our setting captures many modern applications of bandit algorithms where the action space is enormous and the information about the underlying instance/structure is unavailable. We first propose an adaptive algorithm that is agnostic to the hardness level and theoretically derive its regret bound. We then prove a lower bound for our problem setting, which indicates: (1) no algorithm can be minimax optimal simultaneously over all hardness levels; and (2) our algorithm achieves a rate function that is Pareto optimal. With additional knowledge of the expected reward of the best arm, we propose another adaptive algorithm that is minimax optimal, up to polylog factors, over all hardness levels. Experimental results confirm our theoretical guarantees and show advantages of our algorithms over the previous state-of-the-art. 

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3. SPEED: Experimental Design for Policy Evaluation in Linear Heteroscedastic Bandits

   Mukherjee, Subhojyoti; Xie, Qiaomin; Hanna, Josiah P; Nowak, Robert (April 2024, International Conference on Artificial Intelligence and Statistics)

   In this paper, we study the problem of optimal data collection for policy evaluation in linear bandits. In policy evaluation, we are given a \textit{target} policy and asked to estimate the expected reward it will obtain when executed in a multi-armed bandit environment. Our work is the first work that focuses on such an optimal data collection strategy for policy evaluation involving heteroscedastic reward noise in the linear bandit setting. We first formulate an optimal design for weighted least squares estimates in the heteroscedastic linear bandit setting with the knowledge of noise variances. This design minimizes the mean squared error (MSE) of the estimated value of the target policy and is termed the oracle design. Since the noise variance is typically unknown, we then introduce a novel algorithm, SPEED (\textbf{S}tructured \textbf{P}olicy \textbf{E}valuation \textbf{E}xperimental \textbf{D}esign), that tracks the oracle design and derive its regret with respect to the oracle design. We show that regret scales as 𝑂˜(𝑑3𝑛−3/2) and prove a matching lower bound of Ω(𝑑2𝑛−3/2) . Finally, we evaluate SPEED on a set of policy evaluation tasks and demonstrate that it achieves MSE comparable to an optimal oracle and much lower than simply running the target policy. 

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4. Multitask Bandit Learning Through Heterogeneous Feedback Aggregation

   Wang, Z.; Chaudhuri, K. (January 2021, Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS) 2021)

   In many real-world applications, multiple agents seek to learn how to perform highly related yet slightly different tasks in an online bandit learning protocol. We formulate this problem as the ϵ-multi-player multi-armed bandit problem, in which a set of players concurrently interact with a set of arms, and for each arm, the reward distributions for all players are similar but not necessarily identical. We develop an upper confidence bound-based algorithm, RobustAgg(ϵ), that adaptively aggregates rewards collected by different players. In the setting where an upper bound on the pairwise dissimilarities of reward distributions between players is known, we achieve instance-dependent regret guarantees that depend on the amenability of information sharing across players. We complement these upper bounds with nearly matching lower bounds. In the setting where pairwise dissimilarities are unknown, we provide a lower bound, as well as an algorithm that trades off minimax regret guarantees for adaptivity to unknown similarity structure. 

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5. Private High-Dimensional Hypothesis Testing

   Narayanan, S. (January 2022, Conference on Learning Theory (COLT 2022))

   We provide improved differentially private algorithms for identity testing of high-dimensional distributions. Specifically, for d-dimensional Gaussian distributions with known covariance Σ, we can test whether the distribution comes from N(μ∗,Σ) for some fixed μ∗ or from some N(μ,Σ) with total variation distance at least α from N(μ∗,Σ) with (ε,0)-differential privacy, using only O~(d1/2α2+d1/3α4/3⋅ε2/3+1α⋅ε) samples if the algorithm is allowed to be computationally inefficient, and only O~(d1/2α2+d1/4α⋅ε) samples for a computationally efficient algorithm. We also provide a matching lower bound showing that our computationally inefficient algorithm has optimal sample complexity. We also extend our algorithms to various related problems, including mean testing of Gaussians with bounded but unknown covariance, uniformity testing of product distributions over {−1,1}d, and tolerant testing. Our results improve over the previous best work of Canonne et al.~\cite{CanonneKMUZ20} for both computationally efficient and inefficient algorithms, and even our computationally efficient algorithm matches the optimal \emph{non-private} sample complexity of O(d√α2) in many standard parameter settings. In addition, our results show that, surprisingly, private identity testing of d-dimensional Gaussians can be done with fewer samples than private identity testing of discrete distributions over a domain of size d \cite{AcharyaSZ18}, which refutes a conjectured lower bound of~\cite{CanonneKMUZ20}. 

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