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  1. Summary

    We present new models and methods for the posterior drift problem where the regression function in the target domain is modelled as a linear adjustment, on an appropriate scale, of that in the source domain, and study the theoretical properties of our proposed estimators in the binary classification problem. The core idea of our model inherits the simplicity and the usefulness of generalized linear models and accelerated failure time models from the classical statistics literature. Our approach is shown to be flexible and applicable in a variety of statistical settings, and can be adopted for transfer learning problems in various domains including epidemiology, genetics and biomedicine. As concrete applications, we illustrate the power of our approach (i) through mortality prediction for British Asians by borrowing strength from similar data from the larger pool of British Caucasians, using the UK Biobank data, and (ii) in overcoming a spurious correlation present in the source domain of the Waterbirds dataset.

     
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  2. We consider the task of meta-analysis in high-dimensional settings in which the data sources are similar but non-identical. To borrow strength across such heterogeneous datasets, we introduce a global parameter that emphasizes interpretability and statistical efficiency in the presence of heterogeneity. We also propose a one-shot estimator of the global parameter that preserves the anonymity of the data sources and converges at a rate that depends on the size of the combined dataset. For high-dimensional linear model settings, we demonstrate the superiority of our identification restrictions in adapting to a previously seen data distribution as well as predicting for a new/unseen data distribution. Finally, we demonstrate the benefits of our approach on a large-scale drug treatment dataset involving several different cancer cell-lines. 
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  3. We consider the problem of estimating the location of a single change point in a network generated by a dynamic stochastic block model mechanism. This model produces community structure in the network that exhibits change at a single time epoch. We propose two methods of estimating the change point, together with the model parameters, before and after its occurrence. The first employs a least-squares criterion function and takes into consideration the full structure of the stochastic block model and is evaluated at each point in time. Hence, as an intermediate step, it requires estimating the community structure based on a clustering algorithm at every time point. The second method comprises the following two steps: in the first one, a least-squares function is used and evaluated at each time point, but ignoring the community structure and only considering a random graph generating mechanism exhibiting a change point. Once the change point is identified, in the second step, all network data before and after it are used together with a clustering algorithm to obtain the corresponding community structures and subsequently estimate the generating stochastic block model parameters. The first method, since it requires knowledge of the community structure and hence clustering at every point in time, is significantly more computationally expensive than the second one. On the other hand, it requires a significantly less stringent identifiability condition for consistent estimation of the change point and the model parameters than the second method; however, it also requires a condition on the misclassification rate of misallocating network nodes to their respective communities that may fail to hold in many realistic settings. Despite the apparent stringency of the identifiability condition for the second method, we show that networks generated by a stochastic block mechanism exhibiting a change in their structure can easily satisfy this condition under a multitude of scenarios, including merging/splitting communities, nodes joining another community, etc. Further, for both methods under their respective identifiability and certain additional regularity conditions, we establish rates of convergence and derive the asymptotic distributions of the change point estimators. The results are illustrated on synthetic data. In summary, this work provides an in-depth investigation of the novel problem of change point analysis for networks generated by stochastic block models, identifies key conditions for the consistent estimation of the change point, and proposes a computationally fast algorithm that solves the problem in many settings that occur in applications. Finally, it discusses challenges posed by employing clustering algorithms in this problem, that require additional investigation for their full resolution. 
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  4. We propose a strategy for computing estimators in some non-standard M-estimation problems, where the data are distributed across different servers and the observations across servers, though independent, can come from heterogeneous sub-populations, thereby violating the identically distributed assumption. Our strategy fixes the super-efficiency phenomenon observed in prior work on distributed computing in (i) the isotonic regression framework, where averaging several isotonic estimates (each computed at a local server) on a central server produces super-efficient estimates that do not replicate the properties of the global isotonic estimator, i.e. the isotonic estimate that would be constructed by transferring all the data to a single server, and (ii) certain types of M-estimation problems involving optimization of discontinuous criterion functions where M-estimates converge at the cube-root rate. The new estimators proposed in this paper work by smoothing the data on each local server, communicating the smoothed summaries to the central server, and then solving a non-linear optimization problem at the central server. They are shown to replicate the asymptotic properties of the corresponding global estimators, and also overcome the super-efficiency phenomenon exhibited by existing estimators. 
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