In-Context Learning for Attention Scheme: from Single Softmax Regression
  to Multiple Softmax Regression via a Tensor Trick
Yeqi Gao Zhao Song Shenghao Xie
Large language models (LLMs) have brought significant and transformative
changes in human society. These models have demonstrated remarkable
capabilities in natural language understanding and generation, leading to
various advancements and impacts across several domains.
  We consider the in-context learning under two formulation for attention
related regression in this work. Given matrices $A_1 \in \mathbb{R}^{n \times
d}$, and $A_2 \in \mathbb{R}^{n \times d}$ and $B \in \mathbb{R}^{n \times n}$,
the purpose is to solve some certain optimization problems: Normalized version
$\min_{X} \| D(X)^{-1} \exp(A_1 X A_2^\top) - B \|_F^2$ and Rescaled version
$\| \exp(A_1 X A_2^\top) - D(X) \cdot B \|_F^2$. Here $D(X) := \mathrm{diag}(
\exp(A_1 X A_2^\top) {\bf 1}_n )$.
  Our regression problem shares similarities with previous studies on
softmax-related regression. Prior research has extensively investigated
regression techniques related to softmax regression: Normalized version $\|
\langle \exp(Ax) , {\bf 1}_n \rangle^{-1} \exp(Ax) - b \|_2^2$ and Resscaled
version $\| \exp(Ax) - \langle \exp(Ax), {\bf 1}_n \rangle b \|_2^2 $
  In contrast to previous approaches, we adopt a vectorization technique to
address the regression problem in matrix formulation. This approach expands the
dimension from $d$ to $d^2$, resembling the formulation of the regression
problem mentioned earlier.
  Upon completing the lipschitz analysis of our regression function, we have
derived our main result concerning in-context learning.