** Data used in deep learning is notoriously problematic. For example, data are usually combined from diverse sources, rarely cleaned and vetted thoroughly, and sometimes corrupted on purpose. Intentional corruption that targets the weak spots of algorithms has been studied extensively under the label of "adversarial attacks." In contrast, the arguably much more common case of corruption that reflects the limited quality of data has been studied much less. Such "random" corruptions are due to measurement errors, unreliable sources, convenience sampling, and so forth. These kinds of corruption are common in deep learning, because data are rarely collected according to strict protocols -- in strong contrast to the formalized data collection in some parts of classical statistics. This paper concerns such corruption. We introduce an approach motivated by very recent insights into median-of-means and Le Cam's principle, we show that the approach can be readily implemented, and we demonstrate that it performs very well in practice. In conclusion, we believe that our approach is a very promising alternative to standard parameter training based on least-squares and cross-entropy loss. **

Imputation Maximization Stochastic Approximation with Application to Generalized Linear Mixed Models

** Generalized linear mixed models are useful in studying hierarchical data with possibly non-Gaussian responses. However, the intractability of likelihood functions poses challenges for estimation. We develop a new method suitable for this problem, called imputation maximization stochastic approximation (IMSA). For each iteration, IMSA first imputes latent variables/random effects, then maximizes over the complete data likelihood, and finally moves the estimate towards the new maximizer while preserving a proportion of the previous value. The limiting point of IMSA satisfies a self-consistency property and can be less biased in finite samples than the maximum likelihood estimator solved by score-equation based stochastic approximation (ScoreSA). Numerically, IMSA can also be advantageous over ScoreSA in achieving more stable convergence and respecting the parameter ranges under various transformations such as nonnegative variance components. This is corroborated through our simulation studies where IMSA consistently outperforms ScoreSA. **

Imputation Maximization Stochastic Approximation with Application to Generalized Linear Mixed Models

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