  While large language models based on the transformer architecture have
demonstrated remarkable in-context learning (ICL) capabilities, understandings
of such capabilities are still in an early stage, where existing theory and
mechanistic understanding focus mostly on simple scenarios such as learning
simple function classes. This paper takes initial steps on understanding ICL in
more complex scenarios, by studying learning with representations. Concretely,
we construct synthetic in-context learning problems with a compositional
structure, where the label depends on the input through a possibly complex but
fixed representation function, composed with a linear function that differs in
each instance. By construction, the optimal ICL algorithm first transforms the
inputs by the representation function, and then performs linear ICL on top of
the transformed dataset. We show theoretically the existence of transformers
that approximately implement such algorithms with mild depth and size.
Empirically, we find trained transformers consistently achieve near-optimal ICL
performance in this setting, and exhibit the desired dissection where lower
layers transforms the dataset and upper layers perform linear ICL. Through
extensive probing and a new pasting experiment, we further reveal several
mechanisms within the trained transformers, such as concrete copying behaviors
on both the inputs and the representations, linear ICL capability of the upper
layers alone, and a post-ICL representation selection mechanism in a harder
mixture setting. These observed mechanisms align well with our theory and may
shed light on how transformers perform ICL in more realistic scenarios.
