Our main results are summarized below.

We conclude that equality in Lemma 3.1 holds if and only if w

Trajectoriestakenby , left : s 7! left, and right : s 7! rightareshownas dashededges, rededges, andgreenedges, respectively. Let ={ left : s 7! left, right : s 7! right}, andletg( )= Es d( )[g(s, )]bethe 0-1 loss.

However, the underlying clean model doesn't always exist for imperfect model Theorem A.1 (Necessary and Sufficient conditions for the existence of the underlying clean model.) . This theorem is a straightforward corollary of Bochner's I. (26) We can also expand the hessian of the log q We can then prove Theorem 2.3. All the experiments conducted in this paper are run on one single NVDIA GTX 3090.