Officers say they are treating the incident, which happened at 12.20am on Saturday, September 14 in Piccardy Place, as a hate crime.
A 26-year-old had just left the Planet Out venue when he saw that his 27-year-old friend, who he had been drinking with that evening, was lying on the ground. After going to his aid, the 26-year-old was struck in the face by a man, who made a homophobic remark before carrying out the assault.
The man then left the area with a woman, heading towards Leith Walk. He is described as white, aged 20 to 25-years-old, 5ft 8ins tall and of slim build with long ear-length blond hair. He was wearing a dark blue top, light jeans and possibly spoke with a Fife accent.
Detective Sergeant Andy MacMillan said: "This has been a random and unprovoked attack, which left one of the victims with a painful injury to his eye that required medical attention.
"Given the comments heard by the other victim, we are treating this as a homophobic assault and would urge anyone who remembers seeing anything suspicious in Piccardy Place in the early hours of Saturday morning to contact police immediately.
"Similarly, anyone who recognises the description of the suspect, or who has any other information relevant to this investigation, is also asked to come forward."
Those with information can contact Police Scotland on 101 or alternatively, the charity Crimestoppers on 0800 555 111.