Autonomous agents operating in real-world scenarios frequently encounter uncertainty and make decisions based on incomplete information.

In this Appendix, we will derive the fixed-point equations for the order parameters presented in the main text, following and generalising the analysis in Ref. [ Saddle-point equations The saddle-point equations are derived straightforwardly from the obtained free energy functionally extremising with respect to all parameters. The zero-regularisation limit of the logistic loss can help us study the separability transition. N 5 + \ 1 p 0, 1 d 5. (66) As a result, given that \ 2( 0, 1 ], the smaller value for which E is finite is U This result has been generalised immediately afterwards by Pesce et al. Ref. [ 59 ] for the Gaussian case, we can obtain the following fixed-point equations, 8 > > > > > >< > > > > > >: E = Mean universality Following Ref. [ In our case, this condition is simpler than in Ref. [ We see that mean-independence in this setting is indeed verified. Numerical experiments Numerical experiments regarding the quadratic loss with ridge regularisation were performed by computing the Moore-Penrose pseudoinverse solution.

In the following subsections, we provide theoretical derivations. In this subsection, we provide a formal description of the consistency property of score matching. Assumption A.4. (Compactness) The parameter space is compact. Assumption A.5. (Identifiability) There exists a set of parameters A.3 are the conditions that ensure A.7 lead to the uniform convergence property [ In the following Lemma A.9 and Proposition A.10, we examine the sufficient condition for We show that the sufficient conditions stated in Lemma A.9 can be satisfied using the Figure A1: An illustration of the relationship between the variables discussed in Proposition 4.1, Lemma A.12, and Lemma A.13. The properties of KL divergence and Fisher divergence presented in the last two rows are derived in Lemmas A.12 In this section, we provide formal derivations for Proposition 4.1, Lemma A.12, and Lemma A.13. Based on Remark A.14, the following holds: D In this section, we elaborate on the experimental setups and provide the detailed configurations for the experiments presented in Section 5 of the main manuscript.

Kernel density estimation (KDE) is integral to a range of generative and discriminative tasks in machine learning.

We thank all the anonymous reviewers for their constructive feedback. We address each comment as follows. R1-Q2:Just using the predicted mask to concat. R1-Q3:Refine the predicted mask with CRF . SEAM show that CRF ( vs CONT A) is only effective in the first round, i .

We thank the reviewers for thoughtful feedback. "researchers working on pairwise comparisons and preference learning should find this paper to be interesting and Furthermore, we note that we also plan to make our code available as soon as the review period concludes. In our derivation, we pose the problem in a noiseless environment only for simplicity. For similar reasons, we also did not compare our method against algorithms utilizing different models of preference. As with any recommender system, practical considerations are important.

We thank all the reviewers for their time and valuable feedback. We didn't tune the kernel bandwidth but simply apply the widely used median trick [2]. The thinning factor is common in MCMC and has been used for decades. We will add sensitive analysis on that. The reference in L.100 should be referred to equation between L.80 and L.81 and we will fix it as well as the other Appendix A.4 should be Appendix C and Appendix A.5 should be Appendix C.1.