Going for long run in Tokyo?

I am staying in Shinjuku. Would love to go for a long run. Is it rude to just go for a run around the city on the sidewalks? Or should I find a park that's meant for running? This would be early in the morning, so less people out. I also saw that there is a jogging path around the imperial palace. It is says it's only a 6km run from shinjuku. So, around 4 miles. Would it be not a smart idea to run from shinjuku to the imperial palace? I know I could also jump on the subway and get out there. Thanks!

by sqrrl7