  Predicting simple function classes has been widely used as a testbed for
developing theory and understanding of the trained Transformer's in-context
learning (ICL) ability. In this paper, we revisit the training of Transformers
on linear regression tasks, and different from all the existing literature, we
consider a bi-objective prediction task of predicting both the conditional
expectation $\mathbb{E}[Y|X]$ and the conditional variance Var$(Y|X)$. This
additional uncertainty quantification objective provides a handle to (i) better
design out-of-distribution experiments to distinguish ICL from in-weight
learning (IWL) and (ii) make a better separation between the algorithms with
and without using the prior information of the training distribution.
Theoretically, we show that the trained Transformer reaches near Bayes-optimum,
suggesting the usage of the information of the training distribution. Our
method can be extended to other cases. Specifically, with the Transformer's
context window $S$, we prove a generalization bound of
$\tilde{\mathcal{O}}(\sqrt{\min\{S, T\}/(n T)})$ on $n$ tasks with sequences of
length $T$, providing sharper analysis compared to previous results of
$\tilde{\mathcal{O}}(\sqrt{1/n})$. Empirically, we illustrate that while the
trained Transformer behaves as the Bayes-optimal solution as a natural
consequence of supervised training in distribution, it does not necessarily
perform a Bayesian inference when facing task shifts, in contrast to the
\textit{equivalence} between these two proposed in many existing literature. We
also demonstrate the trained Transformer's ICL ability over covariates shift
and prompt-length shift and interpret them as a generalization over a meta
distribution.
