Are you currently searching the HACLA Section 8 listings? If not, go there and see what is listed. However, it depends on how many Landlords are listed with section 8 in the Pico/Robertson area.
Also, even if you do find a 1 bedroom that is not listed as section 8, contact the Landlord and tell them you're voucher holder. A landlord can apply to be on a sectoin 8 list. Sometimes, Landlords are also happy to have a voucher holder because it gaurantees a check at the begining of each month.
Yeprem P. Davoodian