Symbolic regression (SR) aims to discover explicit mathematical expressions that explain observed data and is widely used in domains where interpretability is essential. Because interpretability requires expressions to reflect meaningful regularities, SR is sensitive to observations that deviate from the dominant relationship. Such irregular observations, or outliers, are common in real-world data and can hinder SR from identifying underlying regularities. Robust regression mitigates this by downweighting observations with large residuals. However, deciding which observations should be treated as outliers is often ambiguous and depends on user interpretation and domain knowledge, a perspective largely overlooked in existing SR studies. This motivates approaches that present multiple candidate expressions, allowing users to examine different residual patterns and choose expressions consistent with their expertise. We propose diversified residual symbolic regression (DRSR), which achieves high predictive accuracy while promoting diversity with respect to residual patterns based on the Quality-Diversity paradigm. DRSR collects multiple expressions that fit the data well but differ in how residuals are distributed, enabling post-search selection aligned with domain knowledge. On a synthetic mixture dataset, DRSR produces more diverse expressions than conventional SR while capturing multiple underlying relationships. On a real-world astronomical dataset, DRSR discovers multiple expressions consistent with known physical relationships.

CMA-ES is a powerful method for continuous black-box optimization, in which search is performed by iteratively adapting a multivariate Gaussian distribution. To further improve its performance, mirrored orthogonal sampling (MOS) has been proposed as an advanced sampling technique. In this study, we show that CMA-ES with MOS can suffer from performance degradation on multimodal problems. We identify cumulative step-size adaptation (CSA), the standard step-size adaptation mechanism of CMA-ES, as a major source of this issue. We also confirm that existing bias-correction methods for CSA are insufficient to fully resolve this problem on multimodal landscapes. To address this limitation, we propose introducing two-point step-size adaptation (TPA) into CMA-ES with MOS. We examine two TPA variants: one based on direct value comparison and another on rank differences. Unlike CSA, TPA does not rely on evolution path statistics and is expected to be robust in MOS settings. Experimental results demonstrate that the value-comparison TPA significantly outperforms the rank-based variant and CSA on multimodal problems. Furthermore, it remains competitive on unimodal problems, indicating that the proposed approach provides a robust alternative to CSA for MOS-based CMA-ES.

Optimization problems in real-world applications across the medical and engineering domains often involve potential risks when evaluating candidate solutions. Safe optimization aims to perform optimization while suppressing unsafe solution evaluations in such situations. For continuous search spaces, there exist safe optimization methods based on evolutionary computation. However, the algorithm development of safe optimization methods for binary search spaces has not been adequately addressed. In this study, we incorporate additional mechanisms for safe optimization into a binary optimization method, the adaptive stochastic natural gradient method (ASNG) with a family of Bernoulli distributions. For safety functions that must be kept non-negative during optimization, the proposed method, safe ASNG, estimates the Lipschitz constants with respect to the Hamming distance by constructing surrogate models of safety functions based on discrete Walsh functions. Then, safe ASNG computes a safe region that consists of safe solutions around the previously evaluated safe solutions. By projecting newly generated solutions to their nearest neighbors within the safe region, safe ASNG suppresses unsafe solution evaluations. Experimental results on benchmark problems on binary domains confirm that, while the comparative methods fail to suppress unsafe solution evaluations, safe ASNG achieves efficient optimization while effectively suppressing unsafe solution evaluations.

The exponential natural evolution strategy (xNES) and the covariance matrix adaptation evolution strategy (CMA-ES) are effective for continuous black-box optimization, but their search performance strongly depends on hyperparameters such as population size and learning rates. Learning rate adaptation CMA-ES (LRA-CMA-ES) improves robustness by adapting the learning rates of the mean vector and covariance matrix; however, it converges slowly on ill-conditioned problems and often fails on multimodal problems. We argue that these issues partly arise because LRA-CMA-ES employs a single learning rate for the covariance matrix. This implicitly couples the updates of the distribution's step-size and shape components, ignoring the established principle in standard CMA-ES that these components require decoupled update dynamics. To address this limitation, we propose step-size and shape separated LRA-xNES (S3-LRA-xNES), which separately adapts learning rates for step-size and shape components. Experimental results demonstrate that S3-LRA-xNES achieves faster convergence on ill-conditioned problems and improved success rates on multimodal problems. Notably, S3-LRA-xNES outperforms LRA-CMA-ES by achieving a median 82.4% reduction in function evaluations on ill-conditioned problems and consistently improving or maintaining 100% success rates across 50 trials on multimodal problems, highlighting the effectiveness of fully separated learning rate adaptation in xNES.

Black-box discrete optimization (BB-DO) problems arise in many real-world applications, such as neural architecture search and mathematical model estimation. A key challenge in BB-DO is epistasis among parameters where multiple variables must be modified simultaneously to effectively improve the objective function. Estimation of Distribution Algorithms (EDAs) provide a powerful framework for tackling BB-DO problems. In particular, an EDA leveraging a Variational Autoencoder (VAE) has demonstrated strong performance on relatively low-dimensional problems with epistasis while reducing computational cost. Meanwhile, evolutionary algorithms such as DSMGA-II and P3, which integrate bit-flip-based local search with linkage learning, have shown excellent performance on high-dimensional problems. In this study, we propose a new memetic algorithm that combines VAE-based sampling with local search. The proposed method inherits the strengths of both VAE-based EDAs and local search-based approaches: it effectively handles high-dimensional problems with epistasis among parameters without incurring excessive computational overhead. Experiments on NK landscapes―a challenging benchmark for BB-DO involving epistasis among parameters―demonstrate that our method outperforms state-of-the-art VAE-based EDA methods, as well as leading approaches such as P3 and DSMGA-II.

Multi-objective optimization is crucial in scientific and industrial applications where solutions must balance trade-offs among conflicting objectives. State-of-the-art methods, such as NSGA-III and MOEA/D, can handle many objectives but struggle with coverage issues, particularly in cases involving inverted triangular Pareto fronts or strong nonlinearity. Moreover, NSGA-III often relies on simulated binary crossover, which deteriorates in problems with variable dependencies. In this study, we propose a novel multi-start optimization method that addresses these challenges. Our approach introduces a newly introduced scalarization technique, the Target Point-based Tcheby-cheff Distance (TPTD) method, which significantly improves coverage on problems with inverted triangular Pareto fronts. For efficient multi-start optimization, TPTD leverages a target point defined in the objective space, which plays a critical role in shaping the scalarized function. This is because, if the target points are distributed uniformly, it is expected that single-objective function optimization using TPTD could construct an approximate solution set with good coverage. The positions of the target points are adaptively determined according to the shape of the Pareto front, ensuring improvement in coverage. The proposed method first searches for target points corresponding to objective vectors on the Pareto front boundary using a binary search method and then relocates the target points corresponding to objective vectors within the boundary according to the shape of the boundary. This operation is computationally efficient because the positions of the non-boundary target points are determined in a single relocation. Furthermore, the flexibility of this scalarization allows seamless integration with powerful single-objective optimization methods, such as natural evolution strategies, to efficiently handle variable dependencies. Experimental results on benchmark problems, including those with inverted triangular Pareto fronts, demonstrate that our method outperforms NSGA-II, NSGA-III, and MOEA/D-DE in terms of the Hypervolume indicator. Notably, our approach achieves computational efficiency improvements of up to 474 times over these baselines.

Active learning (AL) is a label-efficient machine learning paradigm that focuses on selectively annotating high-value instances to maximize learning efficiency. Its effectiveness can be further enhanced by incorporating weak supervision, which uses rough yet cost-effective annotations instead of exact (i.e., full) but expensive annotations. We introduce a novel AL framework, Instance-wise Supervision-Level Optimization (ISO), which not only selects the instances to annotate but also determines their optimal annotation level within a fixed annotation budget. Its optimization criterion leverages the value-to-cost ratio (VCR) of each instance while ensuring diversity among the selected instances. In classification experiments, ISO consistently outperforms traditional AL methods and surpasses a state-of-the-art AL approach that combines full and weak supervision, achieving higher accuracy at a lower overall cost. This code is available at https://github.com/matsuo-shinnosuke/ISOAL.

This study focuses on mixed-variable black-box optimization (MV-BBO), addressing continuous, integer, and categorical variables. Many real-world MV-BBO problems involve dependencies among these different types of variables, requiring efficient methods to optimize them simultaneously. Recently, stochastic optimization methods leveraging the mechanism of the covariance matrix adaptation evolution strategy have shown promising results in mixed-integer or mixed-category optimization. However, such methods cannot handle the three types of variables simultaneously. In this study, we propose CatCMA with Margin (CatCMAwM), a stochastic optimization method for MV-BBO that jointly optimizes continuous, integer, and categorical variables. CatCMAwM is developed by incorporating a novel integer handling into CatCMA, a mixed-category black-box optimization method employing a joint distribution of multivariate Gaussian and categorical distributions. The proposed integer handling is carefully designed by reviewing existing integer handlings and following the design principles of CatCMA. Even when applied to mixed-integer problems, it stabilizes the marginal probability and improves the convergence performance of continuous variables. Numerical experiments show that CatCMAwM effectively handles the three types of variables, outperforming state-of-the-art Bayesian optimization methods and baselines that simply incorporate existing integer handlings into CatCMA.

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful methods for solving continuous black-box optimization problems. A practically useful aspect of CMA-ES is that it can be used without hyperparameter tuning. However, the hyperparameter settings still have a considerable impact on performance, especially for difficult tasks, such as solving multimodal or noisy problems. This study comprehensively explores the impact of learning rate on CMA-ES performance and demonstrates the necessity of a small learning rate by considering ordinary differential equations. Thereafter, it discusses the setting of an ideal learning rate. Based on these discussions, we develop a novel learning rate adaptation mechanism for CMA-ES that maintains a constant signal-to-noise ratio. Additionally, we investigate the behavior of CMA-ES with the proposed learning rate adaptation mechanism through numerical experiments and compare the results with those obtained for CMA-ES with a fixed learning rate and with population size adaptation. The results show that CMA-ES with the proposed learning rate adaptation works well for multimodal and/or noisy problems without extremely expensive learning rate tuning.

Black-box optimization problems often require simultaneously optimizing different types of variables, such as continuous, integer, and categorical variables. Unlike integer variables, categorical variables do not necessarily have a meaningful order, and the discretization approach of continuous variables does not work well. Although several Bayesian optimization methods can deal with mixed-category black-box optimization (MC-BBO), they suffer from a lack of scalability to high-dimensional problems and internal computational cost. This paper proposes CatCMA, a stochastic optimization method for MC-BBO problems, which employs the joint probability distribution of multivariate Gaussian and categorical distributions as the search distribution. CatCMA updates the parameters of the joint probability distribution in the natural gradient direction. CatCMA also incorporates the acceleration techniques used in the covariance matrix adaptation evolution strategy (CMA-ES) and the stochastic natural gradient method, such as step-size adaptation and learning rate adaptation. In addition, we restrict the ranges of the categorical distribution parameters by margin to prevent premature convergence and analytically derive a promising margin setting. Numerical experiments show that the performance of CatCMA is superior and more robust to problem dimensions compared to state-of-the-art Bayesian optimization algorithms.

In several real-world applications in medical and control engineering, there are unsafe solutions whose evaluations involve inherent risk. This optimization setting is known as safe optimization and formulated as a specialized type of constrained optimization problem with constraints for safety functions. Safe optimization requires performing efficient optimization without evaluating unsafe solutions. A few studies have proposed the optimization methods for safe optimization based on Bayesian optimization and the evolutionary algorithm. However, Bayesian optimization-based methods often struggle to achieve superior solutions, and the evolutionary algorithm-based method fails to effectively reduce unsafe evaluations. This study focuses on CMA-ES as an efficient evolutionary algorithm and proposes an optimization method termed safe CMA-ES. The safe CMA-ES is designed to achieve both safety and efficiency in safe optimization. The safe CMA-ES estimates the Lipschitz constants of safety functions transformed with the distribution parameters using the maximum norm of the gradient in Gaussian process regression. Subsequently, the safe CMA-ES projects the samples to the nearest point in the safe region constructed with the estimated Lipschitz constants. The numerical simulation using the benchmark functions shows that the safe CMA-ES successfully performs optimization, suppressing the unsafe evaluations, while the existing methods struggle to significantly reduce the unsafe evaluations.

We explore off-policy evaluation and learning (OPE/L) in contextual combinatorial bandits (CCB), where a policy selects a subset in the action space. For example, it might choose a set of furniture pieces (a bed and a drawer) from available items (bed, drawer, chair, etc.) for interior design sales. This setting is widespread in fields such as recommender systems and healthcare, yet OPE/L of CCB remains unexplored in the relevant literature. Typical OPE/L methods such as regression and importance sampling can be applied to the CCB problem, however, they face significant challenges due to high bias or variance, exacerbated by the exponential growth in the number of available subsets. To address these challenges, we introduce a concept of factored action space, which allows us to decompose each subset into binary indicators. This formulation allows us to distinguish between the “main effect” derived from the main actions, and the “residual effect”, originating from the supplemental actions, facilitating more effective OPE. Specifically, our estimator, called OPCB, leverages an importance sampling-based approach to unbiasedly estimate the main effect, while employing regression-based approach to deal with the residual effect with low variance. OPCB achieves substantial variance reduction compared to conventional importance sampling methods and bias reduction relative to regression methods under certain conditions, as illustrated in our theoretical analysis. Experiments demonstrate OPCB’s superior performance over typical methods in both OPE and OPL.

We study off-policy evaluation (OPE) in the problem of slate contextual bandits where a policy selects multi-dimensional actions known as slates. This problem is widespread in recommender systems, search engines, marketing, to medical applications, however, the typical Inverse Propensity Scoring (IPS) estimator suffers from substantial variance due to large action spaces, making effective OPE a significant challenge. The PseudoInverse (PI) estimator has been introduced to mitigate the variance issue by assuming linearity in the reward function, but this can result in significant bias as this assumption is hard-to-verify from observed data and is often substantially violated. To address the limitations of previous estimators, we develop a novel estimator for OPE of slate bandits, called Latent IPS (LIPS), which defines importance weights in a low-dimensional slate abstraction space where we optimize slate abstractions to minimize the bias and variance of LIPS in a data-driven way. By doing so, LIPS can substantially reduce the variance of IPS without imposing restrictive assumptions on the reward function structure like linearity. Through empirical evaluation, we demonstrate that LIPS substantially outperforms existing estimators, particularly in scenarios with non-linear rewards and large slate spaces.

This study targets the mixed-integer black-box optimization (MI-BBO) problem where continuous and integer variables should be optimized simultaneously. The covariance matrix adaptation evolution strategy (CMA-ES), our focus in this study, is a population-based stochastic search method that samples solution candidates from a multivariate Gaussian distribution (MGD), which shows excellent performance in continuous black-box optimization. The parameters of MGD, mean and (co)variance, are updated based on the evaluation value of candidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO with straightforward discretization, however, the variance corresponding to the integer variables becomes much smaller than the granularity of the discretization before reaching the optimal solution, which leads to the stagnation of the optimization. In particular, when binary variables are included in the problem, this stagnation more likely occurs because the granularity of the discretization becomes wider, and the existing integer handling for the CMA-ES does not address this stagnation. To overcome these limitations, we propose a simple integer handling for the CMA-ES based on lower-bounding the marginal probabilities associated with the generation of integer variables in the MGD. The numerical experiments on the MI-BBO benchmark problems demonstrate the efficiency and robustness of the proposed method. Furthermore, to demonstrate the generality of the idea of the proposed method, in addition to the single-objective optimization case, we incorporate it into multi-objective CMA-ES and verify its performance on bi-objective mixed-integer benchmark problems.

The covariance matrix adaptation evolution strategy (CMA-ES) is an efficient continuous black-box optimization method. The CMA-ES possesses many attractive features, including invariance properties and a well-tuned default hyperparameter setting. Moreover, several components to specialize the CMA-ES have been proposed, such as noise handling and constraint handling. To utilize these advantages in mixed-integer optimization problems, the CMA-ES with margin has been proposed. The CMA-ES with margin prevents the premature convergence of discrete variables by the margin correction, in which the distribution parameters are modified to leave the generation probability for changing the discrete variable. The margin correction has been applied to (μ/μw,Λ)-CMA-ES, while this paper introduces the margin correction into (1+1)-CMA-ES, an elitist version of CMA-ES. The (1+1)-CMA-ES is often advantageous for unimodal functions and can be computationally less expensive. To tackle the performance deterioration on mixed-integer optimization, we use the discretized elitist solution as the mean of the sampling distribution and modify the margin correction not to move the elitist solution. The numerical simulation using benchmark functions on mixed-integer, integer, and binary domains shows that (1+1)-CMA-ES with margin outperforms the CMA-ES with margin and is better than or comparable with several specialized methods to a particular search domain.

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful methods for solving black-box continuous optimization problems. One practically useful aspect of the CMA-ES is that it can be used without hyperparameter tuning. However, the hyperparameter settings still have a considerable impact, especially for difficult tasks such as solving multimodal or noisy problems. In this study, we investigate whether the CMA-ES with default population size can solve multimodal and noisy problems. To perform this investigation, we develop a novel learning rate adaptation mechanism for the CMA-ES, such that the learning rate is adapted so as to maintain a constant signal-to-noise ratio. We investigate the behavior of the CMA-ES with the proposed learning rate adaptation mechanism through numerical experiments, and compare the results with those obtained for the CMA-ES with a fixed learning rate. The results demonstrate that, when the proposed learning rate adaptation is used, the CMA-ES with default population size works well on multimodal and/or noisy problems, without the need for extremely expensive learning rate tuning.

We study preferential Bayesian optimization (BO) where reliable feedback is limited to pairwise comparison called duels. An important challenge in preferential BO, which uses the preferential Gaussian process (GP) model to represent flexible preference structure, is that the posterior distribution is a computationally intractable skew GP. The most widely used approach for preferential BO is Gaussian approximation, which ignores the skewness of the true posterior. Alternatively, Markov chain Monte Carlo (MCMC) based preferential BO is also proposed. In this work, we first verify the accuracy of Gaussian approximation, from which we reveal the critical problem that the predictive probability of duels can be inaccurate. This observation motivates us to improve the MCMC-based estimation for skew GP, for which we show the practical efficiency of Gibbs sampling and derive the low variance MC estimator. However, the computational time of MCMC can still be a bottleneck in practice. Towards building a more practical preferential BO, we develop a new method that achieves both high computational efficiency and low sample complexity, and then demonstrate its effectiveness through extensive numerical experiments.

In this work, we propose a new variant of natural evolution strategies (NES) for high-dimensional black-box opti-mization problems. The proposed method, CR-FM-NES, extends a recently proposed state-of-the-art NES, Fast Moving Natural Evolution Strategy (FM-NES), in order to be applicable in high-dimensional problems. CR-FM-NES builds on an idea using a restricted representation of a covariance matrix instead of using a full covariance matrix, while inheriting an efficiency of FM-NES. The restricted representation of the covariance matrix enables CR-FM-NES to update parameters of a multivariate normal distribution in linear time and space complexity, which can be applied to high-dimensional problems. Our experimental results reveal that CR-FM-NES does not lose the efficiency of FM-NES, and on the contrary, CR-FM-NES has achieved significant speedup compared to FM-NES on some benchmark problems. Furthermore, our numerical experiments using 200, 600, and 1000-dimensional benchmark problems demonstrate that CR-FM-NES is effective over scalable baseline methods, VD-CMA and Sep-CMA.

This study targets the mixed-integer black-box optimization (MI-BBO) problem where continuous and integer variables should be optimized simultaneously. The CMA-ES, our focus in this study, is a population-based stochastic search method that samples solution candidates from a multivariate Gaussian distribution (MGD), which shows excellent performance in continuous BBO. The parameters of MGD, mean and (co)variance, are updated based on the evaluation value of candidate solutions in the CMA-ES. If the CMA-ES is applied to the MI-BBO with straightforward discretization, however, the variance corresponding to the integer variables becomes much smaller than the granularity of the discretization before reaching the optimal solution, which leads to the stagnation of the optimization. In particular, when binary variables are included in the problem, this stagnation more likely occurs because the granularity of the discretization becomes wider, and the existing modification to the CMA-ES does not address this stagnation. To overcome these limitations, we propose a simple modification of the CMA-ES based on lower-bounding the marginal probabilities associated with the generation of integer variables in the MGD. The numerical experiments on the MI-BBO benchmark problems demonstrate the efficiency and robustness of the proposed method.

We study offline recommender learning from explicit rating feedback in the presence of selection bias. A current promising solution for dealing with the bias is the inverse propensity score (IPS) estimation. However, the existing propensity-based methods can suffer significantly from the propensity estimation bias. In fact, most of the previous IPS-based methods require some amount of missing-completely-at-random (MCAR) data to accurately estimate the propensity. This leads to a critical self-contradiction; IPS is ineffective without MCAR data, even though it originally aims to learn recommenders from only missing-not-at-random feedback. To resolve this propensity contradiction, we derive a propensity-independent generalization error bound and propose a novel algorithm to minimize the theoretical bound via adversarial learning. Our theory and algorithm do not require a propensity estimation procedure, thereby leading to a well-performing rating predictor without the true propensity information. Extensive experiments demonstrate that the proposed algorithm is superior to a range of existing methods both in rating prediction and ranking metrics in practical settings without MCAR data. Full version of the paper (including the appendix) is available at: https://arxiv.org/abs/1910.07295.

Practitioners often encounter challenging real-world problems that involve a simultaneous optimization of multiple objectives in a complex search space. To address these problems, we propose a practical multiobjective Bayesian optimization algorithm. It is an extension of the widely used Tree-structured Parzen Estimator (TPE) algorithm, called Multiobjective Tree-structured Parzen Estimator (MOTPE). We demonstrate that MOTPE approximates the Pareto fronts of a variety of benchmark problems and a convolutional neural network design problem better than existing methods through the numerical results. We also investigate how the configuration of MOTPE affects the behavior and the performance of the method and the effectiveness of asynchronous parallelization of the method based on the empirical results.

Hyperparameter optimization (HPO), formulated as black-box optimization (BBO), is recognized as essential for automation and high performance of machine learning approaches. The CMA-ES is a promising BBO approach with a high degree of parallelism, and has been applied to HPO tasks, often under parallel implementation, and shown superior performance to other approaches including Bayesian optimization (BO). However, if the budget of hyperparameter evaluations is severely limited, which is often the case for end users who do not deserve parallel computing, the CMA-ES exhausts the budget without improving the performance due to its long adaptation phase, resulting in being outperformed by BO approaches. To address this issue, we propose to transfer prior knowledge on similar HPO tasks through the initialization of the CMA-ES, leading to significantly shortening the adaptation time. The knowledge transfer is designed based on the novel definition of task similarity, with which the correlation of the performance of the proposed approach is confirmed on synthetic problems. The proposed warm starting CMA-ES, called WS-CMA-ES, is applied to different HPO tasks where some prior knowledge is available, showing its superior performance over the original CMA-ES as well as BO approaches with or without using the prior knowledge.

This paper presents a natural evolution strategy for implicitly constrained black-box function optimization. The black-box function optimization is challenging because explicit representations of objective functions are not given, and only evaluation values of solutions can be used. In implicitly constrained black-box function problems, constraints are not explicitly given, and only the feasibility of a solution is obtained when the objective function is evaluated. In other words, the amount of constraint violation cannot be obtained, which makes the optimization difficult. Natural Evolution Strategies (NES) is one of the promising frameworks for black-box function optimization. DX-NES is an improved version of xNES which is a promising NES using a multivariate normal distribution as the probability distribution. DX-NES has been reported to show good performance on unconstrained black-box function optimization problems. However, DX-NES has a serious problem in that its performance degrades when applied to implicitly constrained problems. In order to address the problem, we propose DX-NES taking account of Implicit Constraint (DX-NES-IC). In experiments using benchmark problems and a lens system design problem, DX-NES-IC showed better performance than DX-NES, xNES, CMA-ES, and those with the resampling technique in terms of the number of successful trials and that of evaluations, where the resampling technique is a constraint handling method which can be used for implicitly constrained problems.

A typical assumption in supervised machine learning is that the train (source) and test (target) datasets follow completely the same distribution. This assumption is, however, often violated in uncertain real-world applications, which motivates the study of learning under covariate shift. In this setting, the naive use of adaptive hyperparameter optimization methods such as Bayesian optimization does not work as desired since it does not address the distributional shift among different datasets. In this work, we consider a novel hyperparameter optimization problem under the i>multi-source covariate shift whose goal is to find the optimal hyperparameters for a target task of interest using only unlabeled data in a target task and labeled data inmultiple source tasks. To conduct efficient hyperparameter optimization for the target task, it is essential to estimate the target objective using only the available information. To this end, we construct the variance reduced estimator that unbiasedly approximates the target objective with a desirable variance property. Building on the proposed estimator, we provide a general and tractable hyperparameter optimization procedure, which works preferably in our setting with a no-regret guarantee. The experiments demonstrate that the proposed framework broadens the applications of automated hyperparameter optimization.

In deep learning, hyperparameters can severely affect the learning model performance. The Nelder-Mead (NM) method is known for showing a superior performance for hyperparameter optimization in deep learning. An initial simplex, one of the initial NM method's values, is usually determined randomly while the search performance strongly depends on the shape of the initial simplex. Therefore, it is necessary to determine a proper initial simplex as previous researchers have proposed methods to construct an initial simplex from one starting point in the bounded search space. In this study, we verified how these methods for constructing an initial simplex contribute to improving the result of hyperparameter optimization in deep learning, by using a simple model and a complicated model. A smaller initial simplex may fail to optimization by bad local minima because there are some bad local minima in both learning models. We concluded that the starting point is not necessarily located close to the origin, and that a larger initial simplex contributes to improving the result of hyperparameter optimization in deep learning.

To achieve the most energy-efficient operation, this brief presents a circuit design technique for separating the power supply voltage (VDD) of flip-flops (FFs) from that of combinational circuits, called the higher voltage FF (HVFF). Although VDDscaling can reduce the energy, the minimum operating voltage (VDDmin) of FFs prevents the operation at the optimum supply voltage that minimizes the energy, because the VDDminof FFs is higher than the optimum supply voltage. In HVFF, the VDDof combinational logic gates is reduced below the VDDminof FFs while keeping the VDDof FFs at their VDDmin. This makes it possible to minimize the energy without power and delay penalties at the nominal supply voltage (1.2 V) as well as without FF topological difications. A 16-bit integer unit with HVFF is fabricated in a 65-nm CMOS process, and measurement results show that HVFF reduces the minimum energy by 13% compared with the conventional operation, which is 1/10 times smaller than the energy at the nominal supply voltage.

In this paper, a closed-form expression for estimating a minimum operating voltage (VDDmin) of CMOS logic gates is proposed. VDDmin is defined as the minimum supply voltage at which circuits can operate correctly. VDDmin of combinational circuits can be written as a linear function of the square-root of logarithm of the number of logic gates and its slope is proportional to the standard deviation of the within-die variation in the threshold voltage difference between PMOS and NMOS transistors. The proposed expression is verified with Monte Carlo simulations using various gate chains. The verification reveals that VDDmin of inverter chains can be estimated within 11% error. The expression is also verified with silicon measurements in a 65nm CMOS process.

Many of us in the field of ultra-low-Vddprocessors experience difficulty in assessing the sub/near threshold circuit techniques proposed by earlier papers. This paper investigates five major pitfalls which are often not appreciated by researchers when claiming that their circuits outperform others by working at a lower Vddwith a higher energy-efficiency. These pitfalls include: i) overlook the impacts of different technologies and different Vthdefinitions, ii) only emphasize energy reduction but ignore severe throughput degradation, or expect impractical pipelining depth and parallelism degree to compensate this throughput degradation, iii) unrealistically assume that memory's Vddand energy could scale as well as standard cells, iv) use the highest temperature as the worst timing corner as in the super-threshold, but in fact negative temperature becomes much more detrimental in the sub/near threshold regime, v) pursue just-in-need Vddto compensate effects of PVT, but without considering the high energy loss on DC-DC converters. Therefore, the actual energy benefit from using a sub/near threshold Vddcan be greatly overestimated. This work provides some design guidelines and silicon evidence to ultra-low-Vddsystems. The outlined pitfalls also shed light on future directions in this field.