In a large pot of boiling, salted water, place the potatoes one variety at a time, the purple potatoes last, and cook until fork tender. Remove with a slotted spoon into an ice bath. Drain all potatoes. Cut the new potatoes and purple potatoes into quarters. Slice the fingerling potatoes lengthwise.
Place the beets into the pot of boiling water and cook until fork tender, about 15 to 20 minutes. Rinse under cold water and remove the skin. Wear gloves to avoid turning your skin purple. Slice beets into wedges.
In a small bowl, whisk together the rice wine vinegar, olive oil, honey and Dijon until an emulsion has formed. Season with salt and pepper.
In a large bowl, toss all of the vegetables except the lettuce and the olives with approximately 2 tablespoons of vinaigrette. Marinate for 15 minutes.
In a separate large bowl, toss a handful of greens per person with 2 tablespoons of vinaigrette per person. Plate the lettuce and then place a mound of mixed vegetables on top of the lettuce.
Top each salad with a 1 to 2 ounce portion of the albacore tuna. Garnish with the olives. Serve immediately.