scholarly journals Encoding-dependent generalization bounds for parametrized quantum circuits

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 582
Author(s):  
Matthias C. Caro ◽  
Elies Gil-Fuster ◽  
Johannes Jakob Meyer ◽  
Jens Eisert ◽  
Ryan Sweke
Keyword(s):  
Large Body ◽  
Quantum Circuits ◽  
Risk Minimization ◽  
Complexity Measures ◽  
Generalization Bounds ◽  
Data Encoding ◽  
Unseen Data ◽  
Out Of Sample ◽  
Rigorous Framework ◽  
Encoding Strategies

A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.

Improving Mean Variance Optimization through Sparse Hedging Restrictions

2015 ◽  
Vol 50 (6) ◽  
pp. 1415-1441 ◽  
Author(s):  
Shingo Goto ◽  
Yan Xu
Keyword(s):  
Covariance Matrix ◽  
Factor Model ◽  
Risk Minimization ◽  
Portfolio Risk ◽  
Finite Samples ◽  
Inverse Covariance Matrix ◽  
Out Of Sample ◽  
Nonnegativity Constraints ◽  
Mean Variance ◽  
Equal Weighting

AbstractIn portfolio risk minimization, the inverse covariance matrix prescribes the hedge trades in which a stock is hedged by all the other stocks in the portfolio. In practice with finite samples, however, multicollinearity makes the hedge trades too unstable and unreliable. By shrinking trade sizes and reducing the number of stocks in each hedge trade, we propose a “sparse” estimator of the inverse covariance matrix. Comparing favorably with other methods (equal weighting, shrunk covariance matrix, industry factor model, nonnegativity constraints), a portfolio formed on the proposed estimator achieves significant out-of-sample risk reduction and improves certainty equivalent returns after transaction costs.

Means to an end: Pro-government militias as a predictive indicator of strategic mass killing

2015 ◽  
Vol 34 (5) ◽  
pp. 461-484 ◽  
Author(s):  
Ore Koren
Keyword(s):  
Political Violence ◽  
Large Body ◽  
Structural Factors ◽  
Mass Killing ◽  
Forecasting Models ◽  
Sample Data ◽  
Out Of Sample ◽  
Structural Indicators ◽  
Predictive Strength ◽  
Predictive Indicator

Forecasting models of state-led mass killing are limited in their use of structural indicators, despite a large body of research that emphasizes the importance of agency and security repertoires in conditioning political violence. I seek to overcome these limitations by developing a theoretical and statistical framework that highlights the advantages of using pro-government militias (PGMs) as a predictive indicator in forecasting models of state-led mass killing. I argue that PGMs can lower the potential costs associated with mass killing for a regime faced with an internal threat, and might hence “tip the balance” in its favor. In estimating a series of statistical models and their receiver–operator characteristic curves to evaluate this hypothesis globally for the years 1981–2007, focusing on 270 internal threat episodes, I find robust support for my expectations: including PGM indicators in state-led mass killing models significantly improves their predictive strength. Moreover, these results hold even when coefficient estimates produced by in-sample data are used to predict state-led mass killing in cross-validation and out-of-sample data for the years 2008–2013. This study hence provides an introductory demonstration of the potential advantages of including security repertoires, in addition to structural factors, in forecasting models.

scholarly journals Generalizable predictive modeling of semantic processing ability from functional brain connectivity

2021 ◽  
Author(s):  
Danting Meng ◽  
Suiping Wang ◽  
Patrick Wong ◽  
Gangyi Feng
Keyword(s):  
Individual Differences ◽  
Predictive Modeling ◽  
Semantic Processing ◽  
Brain Connectivity ◽  
Brain Network ◽  
Conceptual Information ◽  
Human Connectome Project ◽  
Unseen Data ◽  
Out Of Sample ◽  
The Individual

Semantic processing (SP) is one of the critical abilities of humans for representing and manipulating meaningful and conceptual information. Neuroimaging studies of SP typically collapse data from many subjects, but both its neural organization and behavioral performance vary between individuals. It is not yet understood whether and how the individual variabilities in neural organizations contribute to the individual differences in SP behaviors. Here we aim to identify the neural signatures underlying SP variabilities by analyzing individual functional connectivity (FC) patterns based on a large-sample Human Connectome Project (HCP) dataset and rigorous predictive modeling. We used a two-stage predictive modeling approach to build an internally cross-validated model and to test the model's generalizability with unseen data from different HCP sub-populations and task states as well as other out-of-sample datasets that are independent of the HCP. FC patterns within a putative semantic brain network were significantly predictive of individual SP scores summarized from five semantic tasks. This cross-validated predictive model can be used to predict unseen HCP data. The model generalizability was enhanced with FCs in language tasks than resting state and other task states and was better for females than males. The model constructed from the HCP dataset can be generalized to two independent cohorts that participated in different semantic tasks. FCs connecting to the Perisylvian language network show the most reliable contributions to predictive modeling and the out-of-sample generalization. These findings contribute to our understanding of the neural sources of individual differences in SP, which potentially lay the foundation for personalized education and improve intervention practice for patients with SP and language deficits.

scholarly journals On the learnability of quantum neural networks

2020 ◽  
Author(s):  
Yuxuan Du ◽  
Min-Hsiu Hsieh ◽  
Tongliang Liu ◽  
Shan You ◽  
Dacheng Tao
Keyword(s):  
Convergence Rates ◽  
Learning Model ◽  
Quantum Circuits ◽  
Risk Minimization ◽  
Learning Capability ◽  
Quantum Neural Network ◽  
Empirical Risk ◽  
Gradient Based ◽  
Convergence Performance ◽  
Poor Convergence

Abstract Quantum neural network (QNN), or equivalently, the variational quantum circuits with a gradient-based classical optimizer, has been broadly applied to many experimental proposals for noisy intermediate scale quantum (NISQ) devices. However, the learning capability of QNN remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate noise introduced by NISQ machines. In this study, we theoretically explore the learnability of QNN from the perspective of the trainability and generalization. Particularly, we derive the convergence performance of QNN under the NISQ setting, and identify classes of computationally hard concepts that can be efficiently learned by QNN. Our results demonstrate that large gate noise, few quantum measurements, and deep circuit depth will lead to poor convergence rates of QNN towards the empirical risk minimization. Moreover, we prove that any concept class, which is efficiently learnable by a restricted quantum statistical query (QSQ) learning model, can also be efficiently learned by QNN. Since the restricted QSQ learning model can tackle certain problems such as parity learning with a runtime speedup, our result suggests that QNN established on NISQ devices will retain the quantum advantage. Our work provides the theoretical guidance for developing advanced QNNs and opens up avenues for exploring quantum advantages using NISQ devices.

Convergence and consistency of ERM algorithm with uniformly ergodic Markov chain samples

2016 ◽  
Vol 14 (03) ◽  
pp. 1650013
Author(s):  
Xiaomei Mo ◽  
Jie Xu
Keyword(s):  
Markov Chain ◽  
Convergence Rate ◽  
Fast Convergence ◽  
Empirical Risk Minimization ◽  
Risk Minimization ◽  
Minimization Algorithm ◽  
Ergodic Markov Chain ◽  
Generalization Bounds ◽  
Empirical Risk ◽  
Fast Convergence Rate

This paper studies the convergence rate and consistency of Empirical Risk Minimization algorithm, where the samples need not be independent and identically distributed (i.i.d.) but can come from uniformly ergodic Markov chain (u.e.M.c.). We firstly establish the generalization bounds of Empirical Risk Minimization algorithm with u.e.M.c. samples. Then we deduce that the Empirical Risk Minimization algorithm on the base of u.e.M.c. samples is consistent and owns a fast convergence rate.

Hedging Strategies with the KTB Futures

2002 ◽  
Vol 10 (2) ◽  
pp. 25-56
Author(s):  
Jae Ha Lee ◽  
Han Deog Hui
Keyword(s):  
Garch Model ◽  
Corporate Bond ◽  
Price Sensitivity ◽  
Risk Minimization ◽  
Hedging Strategies ◽  
Daily Data ◽  
Bond Portfolio ◽  
Hedge Ratios ◽  
Out Of Sample ◽  
The Garch Model

This study explores hedging strategies that use the KTB futures to hedge the price risk of the KTB spot portfolio. The study establishes the price sensitivity, risk-minimization, bivariate GARCH (1,1) models as hedging models, and analyzes their hedging performances. The sample period covers from September 29, 1999 to September 18, 2001. Time-matched prices at 11:00 (11:30) of the KTB futures and spot were used in the analysis. The most important findings may be summarized as follows. First, while the average hedge ration of the price sensitivity model is close to one, both the risk-minimization and GARCH model exhibit hedge ratios that are substantially lower than one. Hedge ratios tend to be greater for daily data than for weekly data. Second, for the daily in-sample data, hedging effectiveness is the highest for the GARCH model with time-varying hedge ratios, but the risk-minimization model with constant hedge ratios is not far behind the GARCH model in its hedging performance. In the case of out-of-sample hedging effectiveness, the GARCH model is the best for the KTB spot portfolio, and the risk-minimization model is the best for the corporate bond portfolio. Third, for daily data, the in-sample hedge shows a better performance than the out-of-sample hedge, except for the risk-minimization hedge against the corporate bond portfolio. Fourth, for the weekly in-sample hedges, the price sensitivity model is the worst and the risk-minimization model is the best in hedging the KTB spot portfolio. While the GARCH model is the best against the KTB +corporate bond portfolio, the risk-minimization model is generally as good as the GARCH model. The risk-minimization model performs the best for the weekly out-of-sample data, and the out-of-sample hedges are better than the in-sample hedges. Fifth, while the hedging performance of the risk-minimization model with daily moving window seems somewhat superior to the traditional risk-minimization model when the trading volume increased one year after the inception of the KTB futures, on the average the traditional model is better than the moving-window model. For weekly data, the traditional model exhibits a better performance. Overall, in the Korean bond markets, investors are encouraged to use the simple risk-minimization model to hedge the price risk of the KTB spot and corporate bond portfolios.

Forecasting is difficult, especially about the future

2013 ◽  
Vol 50 (1) ◽  
pp. 17-31 ◽  
Author(s):  
Kristian Skrede Gleditsch ◽  
Michael D Ward
Keyword(s):  
Conflict Management ◽  
International Conflict ◽  
Large Body ◽  
Strong Support ◽  
Structural Features ◽  
Interstate Conflict ◽  
Baseline Risk ◽  
Conflict Studies ◽  
Important Goal ◽  
Out Of Sample

Prediction is an important goal in the study of international conflict, but a large body of research has found that existing statistical models generally have disappointing predictive abilities. We show that most efforts build on models unlikely to be helpful for prediction. Many models essentially ignore the origins of conflict; studies look either at invariant structural features believed to affect the opportunities of conflict, or at factors that are believed to reduce the baseline risk of conflict, without attempting to identify the potential motivations and contentious issues over which conflicts typically arise. Researchers that have considered how contentious issues may motivate conflict and how these can be managed, using the Issues Correlates of War (ICOW) data, have not considered how these features may inform prediction. We assess the risk of dyadic interstate conflict based on the presence of specific contentious issues and conflict management events that may change the conflict potential of these contentious issues. We evaluate to what extent incorporating contentious issues and conflict management can help improve out-of-sample forecasting, as well as advance our understanding of conflict dynamics. Our results provide strong support for the idea that taking into account contentious issues can inform and improve out-of-sample forecasting.

Improving Minimum-Variance Portfolios by Alleviating Overdispersion of Eigenvalues

2019 ◽  
Vol 55 (8) ◽  
pp. 2700-2731
Author(s):  
Fangquan Shi ◽  
Lianjie Shu ◽  
Aijun Yang ◽  
Fangyi He
Keyword(s):  
Covariance Matrix ◽  
General Framework ◽  
Minimum Variance ◽  
Risk Minimization ◽  
Portfolio Risk ◽  
Computationally Efficient ◽  
Estimation Errors ◽  
Inverse Covariance Matrix ◽  
Sample Covariance ◽  
Out Of Sample

In portfolio risk minimization, the inverse covariance matrix of returns is often unknown and has to be estimated in practice. Yet the eigenvalues of the sample covariance matrix are often overdispersed, leading to severe estimation errors in the inverse covariance matrix. To deal with this problem, we propose a general framework by shrinking the sample eigenvalues based on the Schatten norm. The proposed framework has the advantage of being computationally efficient as well as structure-free. The comparative studies show that our approach behaves reasonably well in terms of reducing out-of-sample portfolio risk and turnover.

EVOLUTION OF INFLUENZA A NUCLEOTIDE SEGMENTS THROUGH THE LENS OF DIFFERENT COMPLEXITY MEASURES

2018 ◽  
Vol 21 (05) ◽  
pp. 1850009
Author(s):  
IGOR BALAZ ◽  
TAICHI HARUNA
Keyword(s):  
Shannon Entropy ◽  
Influenza A ◽  
Evolutionary Dynamics ◽  
Large Body ◽  
Epidemiological Data ◽  
Influenza Viruses ◽  
Complexity Measures ◽  
Genome Wide ◽  
Complex Process ◽  
Entropy Measures

Evolution of influenza viruses is a highly complex process that is still poorly understood. Multiyear persistence of similar variants and accumulating evidences of existence of multigenic traits indicates that influenza viruses operate as integrated units and not only as sets of distinct genes. However, there is still no consensus on whether it is the case, and to what extent. One of the main problems is the lack of framework for analyzing and interpreting large body of available high dimensional genomic, clinical and epidemiological data. By reducing dimensionality of data we intend to show whether in addition to gene-centric selective pressure, the evolution of influenza RNA segments is also shaped by their mutual interactions. Therefore, we will analyze how different complexity/entropy measures (Shannon entropy, topological entropy and Lempel–Ziv complexity) can be used to study evolution of nucleotide segments of different influenza subtypes, while reducing data dimensionality. We show that, at the nucleotide level, multiyear clusters of genome-wide entropy/complexity correlations emerged during the H1N1 pandemic in 2009. Our data are the first empirical results that indirectly support the suggestion that a component of influenza evolutionary dynamics involves correlation between RNA segments. Of all used complexity/entropy measures, Shannon entropy shows the best correlation with epidemiological data.

scholarly journals Graph-based simulation of quantum computation in the density matrix representation

2005 ◽  
Vol 5 (2) ◽  
pp. 113-130
Author(s):  
G. F. Viamontes ◽  
I. L. Markov ◽  
J. P. Hayes
Keyword(s):  
Density Matrix ◽  
Matrix Representation ◽  
Major Complication ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Simulation Performance ◽  
Data Encoding ◽  
Simulation Techniques ◽  
A New Technique ◽  
Memory Resources

Quantum-mechanical phenomena are playing an increasing role in information processing, as transistor sizes approach the nanometer level, and quantum circuits and data encoding methods appear in the securest forms of communication. Simulating such phenomena efficiently is exceedingly difficult because of the vast size of the quantum state space involved. A major complication is caused by errors (noise) due to unwanted interactions between the quantum states and the environment. Consequently, simulating quantum circuits and their associated errors using the density matrix representation is potentially significant in many applications, but is well beyond the computational abilities of most classical simulation techniques in both time and memory resources. The size of a density matrix grows exponentially with the number of qubits simulated, rendering array-based simulation techniques that explicitly store the density matrix intractable. In this work, we propose a new technique aimed at efficiently simulating quantum circuits that are subject to errors. In particular, we describe new graph-based algorithms implemented in the simulator QuIDDPro/D. While previously reported graph-based simulators operate in terms of the state-vector representation, these new algorithms use the density matrix representation. To gauge the improvements offered by QuIDDPro/D, we compare its simulation performance with an optimized array-based simulator called QCSim. Empirical results, generated by both simulators on a set of quantum circuit benchmarks involving error correction, reversible logic, communication, and quantum search, show that the graph-based approach far outperforms the array-based approach for these circuits.

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