176 research outputs found

    Near-Optimal Pure Exploration in Matrix Games: A Generalization of Stochastic Bandits & Dueling Bandits

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    We study the sample complexity of identifying the pure strategy Nash equilibrium (PSNE) in a two-player zero-sum matrix game with noise. Formally, we are given a stochastic model where any learner can sample an entry (i,j)(i,j) of the input matrix A[1,1]n×mA\in[-1,1]^{n\times m} and observe Ai,j+ηA_{i,j}+\eta where η\eta is a zero-mean 1-sub-Gaussian noise. The aim of the learner is to identify the PSNE of AA, whenever it exists, with high probability while taking as few samples as possible. Zhou et al. (2017) presents an instance-dependent sample complexity lower bound that depends only on the entries in the row and column in which the PSNE lies. We design a near-optimal algorithm whose sample complexity matches the lower bound, up to log factors. The problem of identifying the PSNE also generalizes the problem of pure exploration in stochastic multi-armed bandits and dueling bandits, and our result matches the optimal bounds, up to log factors, in both the settings.Comment: 22 pages, 5 figure

    Query-Efficient Algorithms to Find the Unique Nash Equilibrium in a Two-Player Zero-Sum Matrix Game

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    We study the query complexity of identifying Nash equilibria in two-player zero-sum matrix games. Grigoriadis and Khachiyan (1995) showed that any deterministic algorithm needs to query Ω(n2)\Omega(n^2) entries in worst case from an n×nn\times n input matrix in order to compute an ε\varepsilon-approximate Nash equilibrium, where ε<12\varepsilon<\frac{1}{2}. Moreover, they designed a randomized algorithm that queries O(nlognε2)\mathcal O(\frac{n\log n}{\varepsilon^2}) entries from the input matrix in expectation and returns an ε\varepsilon-approximate Nash equilibrium when the entries of the matrix are bounded between 1-1 and 11. However, these two results do not completely characterize the query complexity of finding an exact Nash equilibrium in two-player zero-sum matrix games. In this work, we characterize the query complexity of finding an exact Nash equilibrium for two-player zero-sum matrix games that have a unique Nash equilibrium (x,y)(x_\star,y_\star). We first show that any randomized algorithm needs to query Ω(nk)\Omega(nk) entries of the input matrix ARn×nA\in\mathbb{R}^{n\times n} in expectation in order to find the unique Nash equilibrium where k=supp(x)k=|\text{supp}(x_\star)|. We complement this lower bound by presenting a simple randomized algorithm that, with probability 1δ1-\delta, returns the unique Nash equilibrium by querying at most O(nk4polylog(nδ))\mathcal O(nk^4\cdot \text{polylog}(\frac{n}{\delta})) entries of the input matrix ARn×nA\in\mathbb{R}^{n\times n}. In the special case when the unique Nash Equilibrium is a pure-strategy Nash equilibrium (PSNE), we design a simple deterministic algorithm that finds the PSNE by querying at most O(n)\mathcal O(n) entries of the input matrix.Comment: 17 page

    The Effect of a Low-Fat, Plant-Based Lifestyle Intervention (CHIP) on Serum HDL Subfraction Levels - A Cohort Study

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    This conference abstract discusses a low-fat plant-based diet and the effect it has on HDL level

    When Does Size Matter? An Empirical Study of Consumer Demographics and Product Package Choices

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    From the Washington University Senior Honors Thesis Abstracts (WUSHTA), 2017. Published by the Office of Undergraduate Research. Joy Zalis Kiefer, Director of Undergraduate Research and Associate Dean in the College of Arts & Sciences; Lindsey Paunovich, Editor; Helen Human, Programs Manager and Assistant Dean in the College of Arts and Sciences Mentor: Tat Cha

    HDL Subfraction Changes with a Low-fat, Plant-based Complete Health Improvement Program (CHIP)

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    Background and Objectives: Low HDL concentrations are considered an important risk factor for cardiovascular disease. Interventions promoting a low-fat, plant-based eating pattern appear to reduce CVD risk while paradoxically also reducing HDL concentrations. Recent studies show HDL to comprise a range of subfractions, but the role these play in ameliorating the risk of CVD is unclear. The purpose of this study was to characterise changes in HDL subfractions in participants where HDL decreased following the CHIP intervention which promotes a low-fat, plant-based diet, with physical activity. Methods and Study Design: Individuals (n=22; mean age=55.4±16.3 years; 45.5% men, 54.5% women) participating in a CHIP intervention were assessed at baseline and 30 days for changes in BMI, blood pressure, lipid profile, (including large-, intermediate- and small-HDL subfractions) and fasting glucose. Results: HDL significantly decreased (10.6%, pConclusions: This paper discusses specific changes in HDL subfractions when overall-HDL decreases as a response to low fat, whole-food, plant-based eating and exercise. Additional research is required to elucidate the reasons through which behavioural therapies remodel the HDL particle and how this impacts the functional properties of HDL and CVD risk

    The Effect of a Low-Fat, Plant-Based Lifestyle Intervention (CHIP) on Serum HDL Levels and the Implications for Metabolic Syndrome Status - A Cohort Study

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    Background Low levels of high-density lipoproteins (HDL) are considered an important risk factor for cardiovascular disease and constitute one of the criteria for the Metabolic Syndrome (MetS). Lifestyle interventions promoting a low-fat, plant-based eating pattern appear to paradoxically reduce cardiovascular risk but also HDL levels. This study examined the changes in MetS risk factors, in particular HDL, in a large cohort participating in a 30-day lifestyle intervention that promoted a low-fat, plant-based eating pattern. Methods Individuals (n = 5,046; mean age = 57.3 ± 12.9 years; 33.5% men, 66.5% women) participating in a in a Complete Health Improvement Program (CHIP) lifestyle intervention within the United States were assessed at baseline and 30 days for changes in body mass index (BMI), blood pressure (BP), lipid profile and fasting plasma glucose (FPG). Results HDL levels decreased by 8.7% (p Conclusions When people move towards a low-fat, plant-based diet, HDL levels decrease while other indicators of cardiovascular risk improve. This observation raises questions regarding the value of using HDL levels as a predictor of cardiovascular risk in populations who do not consume a typical western diet. As HDL is part of the assemblage of risk factors that constitute MetS, classifying individuals with MetS may not be appropriate in clinical practice or research when applying lifestyle interventions that promote a plant-based eating pattern.[from publisher\u27s website]

    Real world uptake, safety profile and outcomes of docetaxel in newly diagnosed metastatic prostate cancer

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    Objectives: To investigate the uptake, safety and efficacy of docetaxel chemotherapy in hormone-naïve metastatic prostate cancer (mPC) in the first year of use outside of a clinical trial. Subjects/patients and Methods: Patients in the West of Scotland Cancer Network (WoSCAN) with newly diagnosed mPC were identified from the regional multidisciplinary team (MDT) meetings and their treatment details were collected from electronic patient records. The rate of febrile neutropenia, hospitalisations, time to progression and overall survival were compared between those patients who received docetaxel and androgen deprivation therapy (ADT), or ADT alone using survival analysis. Results: Out of 270 eligible patients, 103 received docetaxel (38.1%). 35 patients (34%) were hospitalised and there were 17 episodes of febrile neutropenia (16.5%). Two patients (1.9%) died within 30 days of chemotherapy. Patients who received ADT alone had an increased risk of progression (HR 2.03, 95% CI (1.27, 3.25), log-rank test, p= 0.002) and had an increased risk of death (HR 5.88, 95% CI 2.52, 13.72, log-rank p=0.001) compared to the docetaxel group. The risk of febrile neutropenia was nine times greater if chemotherapy was started within three weeks of ADT initiation (95% CI (1.22,77.72) p= 0.032). Conclusion: Docetaxel chemotherapy in hormone-naïve mPC has significant toxicities, but has a similar effect on time to progression and overall survival as seen in randomised trials. Chemotherapy should be started 3 weeks or more after androgen deprivation
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