Recommender systems aim to increase user actions such as clicks and purchases. Typical evaluations of recommenders regard the purchase of a recommended item as a success. However, the item may have been purchased even without the recommendation. An uplift is defned as an increase in user actions caused by recommendations. Situations with and without a recommendation cannot both be observed for a specifc user-item pair at a given time instance, making uplift-based evaluation and optimization challenging. This paper proposes new evaluation metrics and optimization methods for the uplift in a recommender system. We apply a causal inference framework to estimate the average uplift for the ofine evaluation of recommenders. Our evaluation protocol leverages both purchase and recommendation logs under a currently deployed recommender system, to simulate the cases both with and without recommendations. This enables the ofine evaluation of the uplift for newly generated recommendation lists. For optimization, we need to defne positive and negative samples that are specifc to an uplift-based approach. For this purpose, we deduce four classes of items by observing purchase and recommendation logs. We derive the relative priorities among these four classes in terms of the uplift and use them to construct both pointwise and pairwise sampling methods for uplift optimization. Through dedicated experiments with three public datasets, we demonstrate the efectiveness of our optimization methods in improving the uplift. CCS CONCEPTS • Information systems → Recommender systems; • Computing methodologies → Learning from implicit feedback.

Increasing users' positive interactions, such as purchases or clicks, is an important objective of recommender systems. Recommenders typically aim to select items that users will interact with. If the recommended items are purchased, an increase in sales is expected. However, the items could have been purchased even without recommendation. Thus, we want to recommend items that results in purchases caused by recommendation. This can be formulated as a ranking problem in terms of the causal effect. Despite its importance, this problem has not been well explored in the related research. It is challenging because the ground truth of causal effect is unobservable, and estimating the causal effect is prone to the bias arising from currently deployed recommenders. This paper proposes an unbiased learning framework for the causal effect of recommendation. Based on the inverse propensity scoring technique, the proposed framework first constructs unbiased estimators for ranking metrics. Then, it conducts empirical risk minimization on the estimators with propensity capping, which reduces variance under finite training samples. Based on the framework, we develop an unbiased learning method for the causal effect extension of a ranking metric. We theoretically analyze the unbiasedness of the proposed method and empirically demonstrate that the proposed method outperforms other biased learning methods in various settings. CCS Concepts: • Information systems → Recommender systems; • Computing methodologies → Learning from implicit feedback.

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We correct the proof of the theorem in the previous paper presented by the first named author, which concerns Sturm bounds for Siegel modular forms of degree 2 and of even weights modulo a prime number dividing 2 · 3. We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers.2. As mentioned in Introduction, in the case where p | 2 · 3 and k is even, the first named author stated the same property in [7]. However, the proof has some gaps and its method can give only more larger bounds. We give a new proof in subsection 5.1.3. We note that M k (Γ 2 ) = {0} if k is odd and k < 35.4. Other type bounds also were given in [8].By the result of [1] and a similar argument to them, we can prove the following.Corollary 2.3. Let Γ ⊂ Γ 2 be a congruence subgroup with level N, k ∈ Z ≥0 and f ∈ M k (Γ) Op . We put i = [Γ 2 : Γ]. For ν ∈ Z ≥1 , assume that a f (m, r, n) ≡ 0 mod p ν for all m, r, n ∈ 1 N Z with 0 ≤ m, n ≤ b ki . and 4mn − r 2 ≥ 0, then we have f ≡ 0 mod p ν .