Cheap eats near The Standard Downtown
I don't think there's anything that's going to knock your socks off within short walking distance. You could try Lemonade, Mendocino Farms, and Cabbage Patch. They're all decent lunch options for downtown professionals and easy walks from the Standard. I have no idea if they're open for dinner. I don't think there is any good ethnic restaurant within normal walking distance.
If you're staying at the Standard, I suggest you go to the rooftop and eat there. The food is fine, and IMO the stunning views and fun atmosphere more than make up for anything you're missing out on at the restaurants people have mentioned so far.
214 S Beverly Dr, Beverly Hills, CA
444 S Flower St, Los Angeles, CA 90071
Look at Lemonade for an intriguing array of salads and stews, Sultan for great chicken and decent other Middle Eastern fare, and Wood Spoon is not too far (for inexpensive Brazilian fare). There are also some good places in Grand Central Market.
Grand Central Market
317 S Broadway, Los Angeles, CA 90013
311 W 6th St, Los Angeles, CA 90014
107 W 9th St, Los Angeles, CA 90015
505 S Flower St, Los Angeles, CA 90071
I'm not exactly sure how to work "authentic" into this, but if you're willing to take the DASH bus or Red Line to Little Tokyo I think Suehiro will fit inexpensive and authentic. You might also like Lazy Ox Canteen.
As for innovative but expensive, you have lots of choices nearby.
There are lots of threads on Downtown but I'm not ifeeling linking to them for you at the moment.
If I had to chose one place to eat downtown it would be Rivera.
If you can get to Mo-Chica I think that fits all three but it isn't walking distance.
337 E 1st St, Los Angeles, CA 90012
3655 S Grand Ave, Los Angeles, CA 90007
Lazy Ox Canteen
241 S San Pedro St, Los Angeles, CA 90012
The sashimi lunch special at Sushi Gen is good and an outstanding bargain. Show up early as there's always a wait. Second Street near Central.
IMAO Lazy Ox may be good but doesn't belong on an "inexpensive" list.
Clifton's Cafeteria on Broadway is a dose of freaky old-style Los Angeles. An experience if not a transcendent one.
648 S Broadway, Los Angeles, CA 90014