If a tenant approaches you with evidence that there are cockroaches on the property and you don't act to fix it within a reasonable period of time, you could reasonably terminate your lease or withhold your rent. Tenants should contact the local housing authority if the landlord refuses to act. It's never a good idea to withhold rent without permission to pay for a pest control service. That approach could cause you to be evicted or even sued.
It is in the best interest of both parties to resolve the issue amicably and as soon as possible. Try to keep a cool head and reach a mutual agreement in the first instance. In Georgia landlord and tenant laws, it is not recommended that tenants withhold rent when the landlord fails to complete repairs. If a tenant withholds rent, the landlord may consider that they have not paid the rent and apply for eviction.
Generally, for a court to approve a request for a rental security deposit, problems with rental property must be serious enough to involve a potential health risk or make the property uninhabitable. If pest outbreaks are severe enough to limit the habitability of a rental unit, the landlord may be held responsible for pest control to make the property habitable again.