How do I convert an NSDictionary to a Swift Dictionary<String, NSObject>?

Decode your dictionary as Dictionary<String, NSObject> instead of NSDictionary.

let payload = aDecoder.decodeObjectOfClass(NSDictionary.self, forKey: "payload") as! Dictionary<String, NSObject>

Since you are using Cocoa Touch to serialize and deserialize, they are serialized as NSDictionary, but you can cast it to a Swift Dictionary<,>, as per WWDC 2014 Intermediary Swift and Advanced Swift videos.

You can also decode as NSDictionary and then explicitly cast the object like so:

self.init(actionName: actionName, payload: payload as! Dictionary<String, NSObject>, timestamp: timestamp)

Basically, you are playing here with covariance/contravariance.

Are you sure that that's the correct declaration for init? The argument order is different than the one you're calling in your Objective-C code.

So, if you want a precise reimplementation, the last call should probably be self.init(actionName: actionName, timestamp: timestamp, payload: payload). Calling the initializer with the wrong order of arguments would result in exactly the error message you're getting.