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1. On The Stability of Approximate Message Passing with Independent Measurement Ensembles

   Nguyen, Dang Qua; Kim, Taejoon. (July 2023, IEEE International Symposium on Information Theory)

   Approximate message passing (AMP) is a scalable, iterative approach to signal recovery. For structured random measurement ensembles, including independent and identically distributed (i.i.d.) Gaussian and rotationally-invariant matrices, the performance of AMP can be characterized by a scalar recursion called state evolution (SE). The pseudo-Lipschitz (polynomial) smoothness is conventionally assumed. In this work, we extend the SE for AMP to a new class of measurement matrices with independent (not necessarily identically distributed) entries. We also extend it to a general class of functions, called controlled functions which are not constrained by the polynomial smoothness; unlike the pseudo-Lipschitz function that has polynomial smoothness, the controlled function grows exponentially. The lack of structure in the assumed measurement ensembles is addressed by leveraging Lindeberg-Feller. The lack of smoothness of the assumed controlled function is addressed by a proposed conditioning technique leveraging the empirical statistics of the AMP instances. The resultants grant the use of the SE to a broader class of measurement ensembles and a new class of functions. 

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2. Nearly Optimal Algorithms for Level Set Estimation

   Mason, Blake; Jain, Lalit; Mukherjee, Subhojyoti; Camilleri, Romain; Jamieson, Kevin; Nowak, Robert (March 2022, International Conference on Artificial Intelligence and Statistics)

   The level set estimation problem seeks to find all points in a domain  where the value of an unknown function 𝑓:→ℝ exceeds a threshold 𝛼 . The estimation is based on noisy function evaluations that may be acquired at sequentially and adaptively chosen locations in  . The threshold value 𝛼 can either be explicit and provided a priori, or implicit and defined relative to the optimal function value, i.e. 𝛼=(1−𝜖)𝑓(𝐱∗) for a given 𝜖>0 where 𝑓(𝐱∗) is the maximal function value and is unknown. In this work we provide a new approach to the level set estimation problem by relating it to recent adaptive experimental design methods for linear bandits in the Reproducing Kernel Hilbert Space (RKHS) setting. We assume that 𝑓 can be approximated by a function in the RKHS up to an unknown misspecification and provide novel algorithms for both the implicit and explicit cases in this setting with strong theoretical guarantees. Moreover, in the linear (kernel) setting, we show that our bounds are nearly optimal, namely, our upper bounds match existing lower bounds for threshold linear bandits. To our knowledge this work provides the first instance-dependent, non-asymptotic upper bounds on sample complexity of level-set estimation that match information theoretic lower bounds. 

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3. A General Framework for Treatment Effect Estimation in Semi-Supervised and High Dimensional Settings

   Chakrabortty, Abhishek; Dai, Guorong (August 2024, arXiv.org)

   In this article, we aim to provide a general and complete understanding of semi-supervised (SS) causal inference for treatment effects. Specifically, we consider two such estimands: (a) the average treatment effect and (b) the quantile treatment effect, as prototype cases, in an SS setting, characterized by two available data sets: (i) a labeled data set of size n, providing observations for a response and a set of high dimensional covariates, as well as a binary treatment indicator; and (ii) an unlabeled data set of size N, much larger than n, but without the response observed. Using these two data sets, we develop a family of SS estimators which are ensured to be: (1) more robust and (2) more efficient than their supervised counterparts based on the labeled data set only. Beyond the 'standard' double robustness results (in terms of consistency) that can be achieved by supervised methods as well, we further establish root-n consistency and asymptotic normality of our SS estimators whenever the propensity score in the model is correctly specified, without requiring specific forms of the nuisance functions involved. Such an improvement of robustness arises from the use of the massive unlabeled data, so it is generally not attainable in a purely supervised setting. In addition, our estimators are shown to be semi-parametrically efficient as long as all the nuisance functions are correctly specified. Moreover, as an illustration of the nuisance estimators, we consider inverse-probability-weighting type kernel smoothing estimators involving unknown covariate transformation mechanisms, and establish in high dimensional scenarios novel results on their uniform convergence rates, which should be of independent interest. Numerical results on both simulated and real data validate the advantage of our methods over their supervised counterparts with respect to both robustness and efficiency. 

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4. Transfer Learning with Kernel Methods

   https://doi.org/10.1038/s41467-023-41215-8  

   Radhakrishnan, Adityanarayanan; Ruiz_Luyten, Max; Prasad, Neha; Uhler, Caroline (September 2023, Nature Communications)

   Abstract Transfer learning refers to the process of adapting a model trained on a source task to a target task. While kernel methods are conceptually and computationally simple models that are competitive on a variety of tasks, it has been unclear how to develop scalable kernel-based transfer learning methods across general source and target tasks with possibly differing label dimensions. In this work, we propose a transfer learning framework for kernel methods by projecting and translating the source model to the target task. We demonstrate the effectiveness of our framework in applications to image classification and virtual drug screening. For both applications, we identify simple scaling laws that characterize the performance of transfer-learned kernels as a function of the number of target examples. We explain this phenomenon in a simplified linear setting, where we are able to derive the exact scaling laws. 

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5. Multi-Agent Reinforcement Learning in Stochastic Networked Systems

   Yiheng Lin, Guannan Qu (January 2021, Advances in Neural Information Processing Systems 34 (NeurIPS 2021))

   We study multi-agent reinforcement learning (MARL) in a stochastic network of agents. The objective is to find localized policies that maximize the (discounted) global reward. In general, scalability is a challenge in this setting because the size of the global state/action space can be exponential in the number of agents. Scalable algorithms are only known in cases where dependencies are static, fixed and local, e.g., between neighbors in a fixed, time-invariant underlying graph. In this work, we propose a Scalable Actor Critic framework that applies in settings where the dependencies can be non-local and stochastic, and provide a finite-time error bound that shows how the convergence rate depends on the speed of information spread in the network. Additionally, as a byproduct of our analysis, we obtain novel finite-time convergence results for a general stochastic approximation scheme and for temporal difference learning with state aggregation, which apply beyond the setting of MARL in networked systems. 

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