The blaze was started at around 10.45pm on Saturday at an address in Cameron Park, Prestonfield.
A 62-year-old woman had left the home earlier that day, leaving the property empty.
A number of items were stolen before the fire was started.
The man is stocky, white, mid to late 20s, and 5ft 8in tall. He was wearing a white baseball cap, dark clothes and was on a mountain bike.
Detective Sergeant Heather Powell said: “Fortunately no one was within the property when it was set alight and so there were no injuries as a result of this incident.
“Nevertheless, we are treating this extremely seriously and are eager to speak with anyone who was in the area around 10.45pm on Saturday night.”