\section{Previous Related Work}
{\it Learning from Label Proportions (LLP).} Early work on LLP by \cite{FreitasK05,HG} applied trained probabilistic models using Monte-Carlo methods, while \cite{Musicant,R10} provided adaptations of standard supervised learning approaches such as SVM, $k$-NN and neural nets, and \cite{ChenLQZ09,StolpeM11} developed clustering based methods for LLP. More specialized techniques were proposed by \cite{QuadriantoSCL09} and later extended by \cite{PatriniNCR14}  to estimate parameters from label proportions for the exponential generative model assuming well-behaved label distributions of bags. An optimization based approach of \cite{YuLKJC13} provided a novel $\propto$-SVM method for LLP. Newer methods involve deep learning~\citep{KDFS15,DZCBV19,LWQTS19,NSJCRR22} and others leverage characteristics of the distribution of bags~\citep{SRR,ZWS22,chen2023learning} while \cite{easy-llp} developed model training techniques on derived \emph{surrogate} labels for instances for random bags. Defining the LLP label proportion regression task in the PAC framework, \cite{YCKJC14}, established bounds on the generalization error bounds for bag-distributions. For the classification setting and specific types of loss functions, bag-to-instance generalization error bounds were shown by \cite{easy-llp, chen2023learning}.

{\it Domain Adaptation.} %
Many of the domain adaptation techniques try to align the source and target distributions by minimizing a distance-measure between domains. 
The work of \cite{long2015learning} generalized deep convolutional neural networks to the domain adaptation scenario, by matching the task-specific hidden representations for the source and target domains in a reproducing kernel Hilbert space. An extension of this work by \cite{long2017deep} proposed a Joint Adaptation Network which aligns the joint distributions of multiple domain-specific hidden layers using a joint maximum mean discrepancy measure. The technique proposed by \cite{ganin2016domain} focuses on learning from features which are indiscriminate with respect to the shift between domains. Recently, \cite{li2023domain} applied domain adaptation to the LLP and proposed a model combining domain-adversarial neural network (DANN) and label regularization, to learn from source-domain bags and predict on instances from a target domain.


