Appendix for “ On the Interaction E ↵ ects Between Prediction and Clustering ”

Then a is monotonically decreasing with respect to ↵ if and only if a(✓i) a(✓j). Case 1: ✓i = ✓j . Then a(✓i) = a(✓j), b(✓i) = b(✓j) and the statements holds. Case 2: ✓i 6= ✓j . Then both the following conditions must be true. b(✓j) b(✓i) + i (✓j) i (✓i) > 0 (8) b(✓j) b(✓i) + j (✓j) j (✓i) < 0 (9) If Eq. (8) did not hold, then ✓j would have been optimal at ↵ = i, i.e. ✓j 2 argmin✓2⇥ b(✓) + i (✓). Likewise, if Eq. (9) did not hold, then ✓i 2 argmin✓2⇥ b(✓) + j (✓). Together, they imply b(✓j) b(✓i) + i (✓j) i (✓i) > b(✓j) b(✓i) + j (✓j) j (✓i) i (✓j) i (✓i) > j (✓j) j (✓i) (i j)( (✓j) (✓i)) > 0 (✓j) (✓i) < 0 since i j < 0. Plugging this into Eq. (8), b(✓j) b(✓i) + i (✓j) i (✓i) > 0 b(✓j) b(✓i) > i( (✓i) (✓j)) b(✓j) b(✓i) > 0 (10) which proves the second statement. Finally, plugging Eq. (10) into Eq. (9) concludes the proof. (1 j)(b(✓j) b(✓i)) + j(a(✓j) a(✓i)) < 0 a(✓j) a(✓i) < 0