patzer2: So, where does white try to improve here. After two masters have looked at the position and came to no certain conclusion, and Kasparov himself has played the first eight moves as black, it is indeed a difficult task.
However, I'm willing to play the role of Don Quixote and charge in with a little analysis:
First, I recommend 9. h3 Anand's preferred move as in DeFirmian vs E Perelshteyn, 2002 or Naiditsch vs Leko, 2003 or Anand vs Kasparov, 2003 It seems to me that this move gives white gets much more counterplay and flexibility than with the previously more popular 9. f3.
Second, white should consider 10. h4 as in Ljubojevic vs Miles, 1986 or Shirov vs A David, 1996 I think 10. Qd2 Nxd4!gives black too easy an initiative, and the opening explorer seems to validate this as white only wins 16.4% while black wins 36.4% after this sequence (10. Qd2 Nxd4!). Yet with 10. h4, transposing to the Yugoslav attack in the Sicilian Dragon, white wins 53% of the time, while black only wins 29%, albeit in a small sample of only 17 games.
Third, white should consider 12. 0-0 as opposed to castling long into the better developing black attack. The opening explorer shows black wins 66.7% when white plays 12. 0-0-0, but only 6.7% after 12. 0-0 (though it draws 73.3%).
Fourth, white might consider 17. Bxc7. However, black will get a strong attack on the king for the exchange after 17. ..Qxc7 18. Nd5 Nxd5 19. Bxd5 Bf5 20. Kb1 Rc8 22. Be4 Bxe4 23. fxe4 b4. And this is the sort of positions that dragon players love. So I'm not too keen on it, even though it may be better than the lost position occurring in the game.