How did you know the loss is “continue” ?
Even if you don’t load the states of optimizer, note that model parameters are loaded, the loss won’t seem like training from scratch.
Back to Adam, assume you stop training at step=t-1, at resuming, if you load the optimizer states, m_t and v_t will be where you stop training (m_t-1 → m_t).
However, if not, they will be m_0 and v_0 (m_t-1 → m_0). The states just differ.