  In this article, we consider a dominant rational self-map $f:X
\dashrightarrow X$ of a normal projective variety defined over a number field.
We study the arithmetic degree $\alpha_k(f)$ for $f$ and $\alpha_k(f,V)$ of a
subvariety $V$, which generalize the classical arithmetic degree
$\alpha_1(f,P)$ of a point $P$. We generalize Yuan's arithmetic version of
Siu's inequality to higher codimensions and utilize it to demonstrate the
existence of the arithmetic degree $\alpha_k(f)$. Furthermore, we establish the
relative degree formula $\alpha_k(f)=\max\{\lambda_k(f),\lambda_{k-1}(f)\}$. In
addition, we prove several basic properties of the arithmetic degree
$\alpha_k(f, V)$ and establish the upper bound $\overline{\alpha}_{k+1}(f,
V)\leq \max\{\lambda_{k+1}(f),\lambda_{k}(f)\}$, which generalizes the
classical result $\overline{\alpha}_f(P)\leq \lambda_1(f)$. Finally, we discuss
a generalized version of the Kawaguchi-Silverman conjecture that was proposed
by Dang et al, and we provide a counterexample to this conjecture.
