The incident happened at around 4:30am on Nicolson Street near Surgeons’ Hall.
A 30-year old man and a 32-year old woman had been sitting outside the Hall when an unidentified male walked towards them with two other men, who do not appear to have been involved in this incident, from the direction of the city centre.
As he walked past an argument broke out between him and the man who was sitting. They confronted each other and the suspect allegedly punched the man in the head resulting in him falling to the ground. At this point the woman attempted to intervene but was also struck in the head.
The male suspect then made off on foot and was last seen heading south at the junction of Rankeillor Street.
Both parties received medical attention at the scene by paramedics before being transferred to the Royal Infirmary of Edinburgh where they have since been discharged.
Police are keen to hear from anyone who saw the incident or who has any information that can assist with their ongoing enquiries.
The suspect is described as white, aged between 25-30 years old, with short brown hair and has a stocky/athletic build. He wore a light coloured shirt, possibly pinstriped, with dark trousers and a tattoo behind the right ear.
Detective Constable John McShane of Police Scotland, said: “An extensive investigation is underway and officers are now reviewing CCTV footage of this incident as well as carrying out local enquiries.
“Given the description of the attacker and the distinctive tattoo behind his right ear, I am confident someone will know who this man is or may have even spoken to him earlier in the evening.
“Anyone with any relevant information is asked to contact police immediately on 101 or, alternatively information can be passed anonymously through the charity Crimestoppers on 0800 555 111.”