Demand forecasting is an essential task in retail and manufacturing industries and has been the subject of numerous studies. Conventional popular time-series forecasting methods, such as the ARIMA model, require us to develop a forecasting model for each product. However, when products are frequently replaced and have short sales periods, we do not have enough data to build models individually. This study focuses on zero-shot time-series forecasting methods for demand forecasting with limited data. Zero-shot time-series forecasting is a framework for time-series prediction that does not require fine-tuning with specific time-series data to be predicted. To address the data shortage in practical situations, we propose a zero-shot demand forecasting model that considers exogenous variables. Our experiments with real data demonstrate that our proposed method achieved higher prediction accuracy than existing time-series forecasting methods, especially for products with short sales periods.

**Off-Policy Evaluation (OPE)** aims to assess the effectiveness of counterfactual policies using offline logged data and is frequently utilized to identify the top-$k$ promising policies for deployment in online A/B tests. Existing evaluation metrics for OPE estimators primarily focus on the "accuracy" of OPE or that of downstream policy selection, neglecting risk-return tradeoff and *efficiency* in subsequent online policy deployment. To address this issue, we draw inspiration from portfolio evaluation in finance and develop a new metric, called **SharpeRatio@k**, which measures the risk-return tradeoff and efficiency of policy portfolios formed by an OPE estimator under varying online evaluation budgets ($k$). We first demonstrate, in two example scenarios, that our proposed metric can clearly distinguish between conservative and high-stakes OPE estimators and reliably identify the most *efficient* estimator capable of forming superior portfolios of candidate policies that maximize return with minimal risk during online deployment, while existing evaluation metrics produce only degenerate results. To facilitate a quick, accurate, and consistent evaluation of OPE via SharpeRatio@k, we have also implemented the proposed metric in an open-source software. Using SharpeRatio@k and the software, we conduct a benchmark experiment of various OPE estimators regarding their risk-return tradeoff, presenting several future directions for OPE research.

Abstract We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (Structured semidefinite programs and semialgebraic geometry methods in Robustness and optimization, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a symmetric cone using the usual COP cone. By replacing the usual COP cone appearing in this characterization with the inner- or outer-approximation hierarchy provided by de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248 , 2002) or Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014 , 2012), we obtain an inner- or outer-approximation hierarchy described by semidefinite but not by SOS constraints for the COP matrix cone over the direct product of a nonnegative orthant and a second-order cone. We then compare them with the existing hierarchies provided by Zuluaga et al. (SIAM J Optim 16(4):1076–1091, https://doi.org/10.1137/03060151X , 2006) and Lasserre (Math Program 144:265–276, https://doi.org/10.1007/s10107-013-0632-5 , 2014). Theoretical and numerical examinations imply that we can numerically increase a depth parameter, which determines an approximation accuracy, in the approximation hierarchies derived from de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248 , 2002) and Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014 , 2012), particularly when the nonnegative orthant is small. In such a case, the approximation hierarchy derived from Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014 , 2012) can yield nearly optimal values numerically. Combining the proposed approximation hierarchies with existing ones, we can evaluate the optimal value of COP programming problems more accurately and efficiently.

Uplift modeling aims to optimize treatment policies and is a promising method for causal-based personalization in various domains such as medicine and marketing. However, applying this method to real-world problems faces challenges such as the impossibility of validation and binary treatment limitation. The Contextual Treatment Selection (CTS) algorithm was proposed to overcome the binary treatment limitation and demonstrated state-of-the-art results. However, previous experiments have implied that CTS is cost-ineffective because it requires a large amount of training data. In this paper, we demonstrate that the estimator maximized in CTS is biased against the true metric. We then propose a variance reduced estimator based on the doubly robust estimation technique that provides unbiasedness and desirable variance. We further propose a treatment policy optimization algorithm called VAriance Reduced Treatment Selection (VARTS), which maximizes our estimator. Empirical experiments on synthetic and real-world datasets demonstrated that our method outperforms other existing methods, particularly under realistic conditions such as small sample sizes and high noise levels. These theoretical and empirical results imply that our method can overcome the critical challenges of uplift modeling and should be the first choice for optimizing personalization in various fields.

Recommender systems widely use implicit feedback such as click data because of its general availability. Although the presence of clicks signals the users' preference to some extent, the lack of such clicks does not necessarily indicate a negative response from the users, as it is possible that the users were not exposed to the items (positive-unlabeled problem). This leads to a difficulty in predicting the users' preferences from implicit feedback. Previous studies addressed the positive-unlabeled problem by uniformly upweighting the loss for the positive feedback data or estimating the confidence of each data having relevance information via the EM-algorithm. However, these methods failed to address the missing-not-at-random problem in which popular or frequently recommended items are more likely to be clicked than other items even if a user does not have a considerable interest in them. To overcome these limitations, we first define an ideal loss function to be optimized to realize recommendations that maximize the relevance and propose an unbiased estimator for the ideal loss. Subsequently, we analyze the variance of the proposed unbiased estimator and further propose a clipped estimator that includes the unbiased estimator as a special case. We demonstrate that the clipped estimator is expected to improve the performance of the recommender system, by considering the bias-variance trade-off. We conduct semi-synthetic and real-world experiments and demonstrate that the proposed method largely outperforms the baselines. In particular, the proposed method works better for less popular items that are less frequently observed in the training data. The findings indicate that the proposed method can better achieve the objective of recommending items with the highest relevance.

Uplift modeling aims to optimize treatment allocation by predicting the net effect of a treatment on each individual (ITE) and is expected to achieve causal-based personalization in medicine, marketing, etc. This approach needs specialized methods to train and evaluate ITE prediction models because the true ITE is unobservable. The conventional uplift modeling requires data to be gathered through randomized controlled trials (RCTs), on the other hand, for non-RCT data, the transformed outcome (TO) is commonly used as an unbiased estimator of ITE. However, it is often impossible to conduct RCTs for ethical and economic reasons, and, in observational data, the unbiasedness of TO is based on the unrealistic assumption that the propensity score of each individual is given. In this paper, we theoretically and quantitatively show TO becomes an unreliable proxy ITE when the propensity score estimator is biased or has a large degree of heterogeneity. We then propose a novel proxy outcome, Switch Doubly Robust, turning on and off the effect of propensity score estimator on the outcome prediction models. We theoretically prove SDR achieves better bias-variance trade-off as a proxy ITE than TO and develop novel prediction (SDRM) and evaluation (SDR-MSE) methods. Furthermore, we experimentally show our methods outperformed existing approaches on synthetic datasets. In addition, we applied them to the Right Heart Catheterization dataset and discovered 20% of patients are actually curable, even though the conventional causal inference methods only showed the average treatment effect is negative. We anticipate our methods to be a standard practice of uplift modeling for observational data and lead to optimized personalization in various fields.

Matrices are often required to be well-conditioned in a wide variety of areas including signal processing. Problems to find the nearest positive definite matrix or the nearest correlation matrix that simultaneously satisfy the condition number constraint and sign constraints are presented in this paper. Both problems can be regarded as those to find a projection to the intersection of the closed convex cone corresponding to the condition number constraint and the convex polyhedron corresponding to the other constraints. Thus, we can apply a successive projection method, which is a classical algorithm for finding the projection to the intersection of multiple convex sets, to these problems. The numerical results demonstrated that the algorithm effectively solved the problems.

A SemiDefinite Programming (SDP) problem is one of the most central problems in mathematical optimization. SDP provides an effective computation framework for many research fields. Some applications, however, require solving a large-scale SDP whose size exceeds the capacity of a single processor both in terms of computation time and available memory. SDPARA (SemiDefinite Programming Algorithm paRAllel package) [Yamashita et al. 2003b] was designed to solve such large-scale SDPs. Its parallel performance is outstanding for general SDPs in most cases. However, the parallel implementation is less successful for some sparse SDPs obtained from applications such as Polynomial Optimization Problems (POPs) or Sensor Network Localization (SNL) problems, since this version of SDPARA cannot directly handle sparse Schur Complement Matrices (SCMs). In this article we improve SDPARA by focusing on the sparsity of the SCM and we propose a new parallel implementation using the formula-cost-based distribution along with a replacement of the dense Cholesky factorization. We verify numerically that these features are key to solving SDPs with sparse SCMs more quickly on parallel computing systems. The performance is further enhanced by multithreading and the new SDPARA attains considerable scalability in general. It also finds solutions for extremely large-scale SDPs arising from POPs which cannot be obtained by other solvers.