In my experience searching for relatives who emigrated from the former Russian Empire, I've found that documents after 1917 will show a country of residence/birth more specific than "Russia". A town/city may be provided earlier than that on a ship manifest.
If you do happen to find that your Rosenbergs originated in Latvia, please see "LatviaSIG" and "CourlandSIG" in JewishGen (though recently, read that these Special Interest Groups groups soon will be called something else).
In CourlandSIG, regarding Cossacks in Courland, see:
https://www.jewishgen.org/Courland/lipschitz.htm. - "Jewish Communities in Courland: In Memory of My Native Town Windau" , by Dr S. Lipschitz, in "The Jews of Latvia", Association of Latvian and Estonian Jews in Israel", 1971.
Dr. Lipschitz relates a history of the 1905 Revolution in Courland, saying that Cossacks were dispatched to Courland to quell the uprising. My own family history includes stories of Cossack raids in Courland, the increasing frequency of which prompted my family to leave, beginning in 1906.
Researching: Seibart/Seibert/Tzibart; Sandel/Sader;
Allman/Ullman; Shalkovsky/Zalkoff in Latvia