For instance, the preference of a consumer to choose one product over another constitutes a pairwise comparison between the two products.

In Appendix B, we provide additional experiments supporting our claims. A.1 Proof of Lemma 3.1 We begin with Lemma 3.3 of [2], which gives 1 2 nullx We begin with equation eq. B.1 Condition Number of A is 1 Figure 6: Speed ups with best possible stepsizes vs. batch size (noisy experiments with σ = 0 . Figure 7: Speed up vs. stepsizes for l Figure 8: Speed up vs. stepsizes for logistic regression. Figure 13: Speed ups with best possible stepsizes vs. batch size (noisy experiments with σ = 0 .

Current state-of-the-art document retrieval solutions mainly follow an index-retrieve paradigm, where the index is hard to be directly optimized for the final retrieval target.

We first consider process samples by logistic regression with cluster centers as categorical variables . Intuitively, non-orthonormal centers correlate with each other, which means there is an overlap among categorical variables and makes it hard to identify the decision boundary that leads to a failed classification.

In this section, we describe Nadaraya-Watson (NW) classifier, Local Polynomial (LP) classifier and their convergence rates (Audibert & Tsybakov, 2007). Proof of Corollary 2. Proposition 6 immediately proves the assertion. We basically follow the proof of Chaudhuri & Dasgupta (2014) Theorem 4(b). In Section G.1, we first define symbols In Section G.2, we describe the sketch of the proof and main differences between our proof and that of Section G.3 shows the main body of the Proof, by utilizing several Lemmas listed in A minimum radius whose measure of the ball is larger than t > 0, i.e., r Chaudhuri & Dasgupta (2014) Lemma 21) Then, the assertion is proved. See the following Section G.4 for Lemma 1-7 used in this proof.