  In this paper, we present a first-order projection-free method, namely, the
universal conditional gradient sliding (UCGS) method, for solving
$\varepsilon$-approximate solutions to convex differentiable optimization
problems. For objective functions with H\"older continuous gradients, we show
that UCGS is able to terminate with $\varepsilon$-solutions with at most
$O((M_\nu D_X^{1+\nu}/{\varepsilon})^{2/(1+3\nu)})$ gradient evaluations and
$O((M_\nu D_X^{1+\nu}/{\varepsilon})^{4/(1+3\nu)})$ linear objective
optimizations, where $\nu\in (0,1]$ and $M_\nu>0$ are the exponent and constant
of the H\"older condition. Furthermore, UCGS is able to perform such
computations without requiring any specific knowledge of the smoothness
information $\nu$ and $M_\nu$. In the weakly smooth case when $\nu\in (0,1)$,
both complexity results improve the current state-of-the-art $O((M_\nu
D_X^{1+\nu}/{\varepsilon})^{1/\nu})$ results on first-order projection-free
method achieved by the conditional gradient method. Within the class of
sliding-type algorithms, to the best of our knowledge, this is the first time a
sliding-type algorithm is able to improve not only the gradient complexity but
also the overall complexity for computing an approximate solution. In the
smooth case when $\nu=1$, UCGS matches the state-of-the-art complexity result
but adds more features allowing for practical implementation.
