The Complexity of All-switches Strategy Improvement

Strategy improvement is a widely-used and well-studied class of algorithms for solving graph-based infinite games. These algorithms are parameterized by a switching rule, and one of the most natural rules is "all switches" which switches as many edges as possible in each iteration. Continuing a recent line of work, we study all-switches strategy improvement from the perspective of computational complexity. We consider two natural decision problems, both of which have as input a game $G$, a starting strategy $s$, and an edge $e$. The problems are: 1.) The edge switch problem, namely, is the edge $e$ ever switched by all-switches strategy improvement when it is started from $s$ on game $G$? 2.) The optimal strategy problem, namely, is the edge $e$ used in the final strategy that is found by strategy improvement when it is started from $s$ on game $G$? We show $\mathtt{PSPACE}$-completeness of the edge switch problem and optimal strategy problem for the following settings: Parity games with the discrete strategy improvement algorithm of V\"oge and Jurdzi\'nski; mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff games and simple stochastic games with their standard strategy improvement algorithms. We also show $\mathtt{PSPACE}$-completeness of an analogous problem to edge switch for the bottom-antipodal algorithm for finding the sink of an Acyclic Unique Sink Orientation on a cube

Computing Approximate Nash Equilibria in Polymatrix Games

In an $\epsilon$-Nash equilibrium, a player can gain at most $\epsilon$ by unilaterally changing his behaviour. For two-player (bimatrix) games with payoffs in $[0,1]$, the best-known$\epsilon$ achievable in polynomial time is 0.3393. In general, for $n$-player games an $\epsilon$-Nash equilibrium can be computed in polynomial time for an $\epsilon$ that is an increasing function of $n$ but does not depend on the number of strategies of the players. For three-player and four-player games the corresponding values of $\epsilon$ are 0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general $n$-player games where a player's payoff is the sum of payoffs from a number of bimatrix games. There exists a very small but constant $\epsilon$ such that computing an $\epsilon$-Nash equilibrium of a polymatrix game is \PPAD-hard. Our main result is that a $(0.5+\delta)$-Nash equilibrium of an $n$-player polymatrix game can be computed in time polynomial in the input size and $\frac{1}{\delta}$. Inspired by the algorithm of Tsaknakis and Spirakis, our algorithm uses gradient descent on the maximum regret of the players. We also show that this algorithm can be applied to efficiently find a $(0.5+\delta)$-Nash equilibrium in a two-player Bayesian game

A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations

We consider the 3D Cahn-Hilliard equations coupled to, and driven by, the forced, incompressible 3D Navier-Stokes equations. The combination, known as the Cahn-Hilliard-Navier-Stokes (CHNS) equations, is used in statistical mechanics to model the motion of a binary fluid. The potential development of singularities (blow-up) in the contours of the order parameter $\phi$ is an open problem. To address this we have proved a theorem that closely mimics the Beale-Kato-Majda theorem for the $3D$ incompressible Euler equations [Beale et al. Commun. Math. Phys., Commun. Math. Phys., ${\rm 94}$, $61-66 ({\rm 1984})$]. By taking an $L^{\infty}$ norm of the energy of the full binary system, designated as $E_{\infty}$, we have shown that $\int_{0}^{t}E_{\infty}(\tau)\,d\tau$ governs the regularity of solutions of the full 3D system. Our direct numerical simulations (DNSs), of the 3D CHNS equations, for (a) a gravity-driven Rayleigh Taylor instability and (b) a constant-energy-injection forcing, with $128^3$ to $512^3$ collocation points and over the duration of our DNSs, confirm that $E_{\infty}$ remains bounded as far as our computations allow.Comment: 11 pages, 3 figure

Effects of hemodialysis therapy on sit-to-walk characteristics in end stage renal disease patients

Patients with end stage renal diseases (ESRD) undergoing hemodialysis (HD) have high morbidity and mortality due to multiple causes; one of which is dramatically higher fall rates than the general population. In spite of the multiple efforts aiming to decrease the high mortality and improve quality of life in ESRD patients, limited success has been achieved. If adequate interventions for fall prevention are to be achieved, the functional and mobility mechanisms consistent with falls in this population must be understood. Human movements such as sit-to-walk (STW) tasks are clinically significant, and analysis of these movements provides a meaningful evaluation of postural and locomotor performance in elderly patients with functional limitations indicative of fall risks. In order to assess the effects of HD therapy on fall risks, 22 sessions of both pre- and post-HD measurements were obtained in six ESRD patients utilizing customized inertial measurement units (IMU). IMU signals were denoised using ensemble empirical mode decomposition and Savistky-Golay filtering methods to detect relevant events for identification of STW phases. The results indicated that patients were slower to get out of the chair (as measured by trunk flexion angular accelerations, time to peak trunk flexion, and overall STW completion time) following the dialysis therapy session. STW is a frequent movement in activities of daily living, and HD therapy may influence the postural and locomotor control of these movements. The analysis of STW movement may assist in not only assessing a patient's physical status, but in identifying HD-related fall risk as well. This preliminary study presents a non-invasive method of kinematic measurement for early detection of increased fall risk in ESRD patients using portable inertial sensors for out-patient monitoring. This can be helpful in understanding the pathogenesis better, and improve awareness in health care providers in targeting interventions to identify individuals at risk for fall

The role of BKM-type theorems in 3D3D Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis

The Beale-Kato-Majda theorem contains a single criterion that controls the behaviour of solutions of the $3D$ incompressible Euler equations. Versions of this theorem are discussed in terms of the regularity issues surrounding the $3D$ incompressible Euler and Navier-Stokes equations together with a phase-field model for the statistical mechanics of binary mixtures called the $3D$ Cahn-Hilliard-Navier-Stokes (CHNS) equations. A theorem of BKM-type is established for the CHNS equations for the full parameter range. Moreover, for this latter set, it is shown that there exists a Reynolds number and a bound on the energy-dissipation rate that, remarkably, reproduces the $Re^{3/4}$ upper bound on the inverse Kolmogorov length normally associated with the Navier-Stokes equations alone. An alternative length-scale is introduced and discussed, together with a set of pseudo-spectral computations on a $128^{3}$ grid.Comment: 3 figures and 3 table

Standard FITS template for simulated astrophysical scenes with the WFIRST coronagraph

The science investigation teams (SITs) for the WFIRST coronagraphic instrument have begun studying the capabilities of the instrument to directly image reflected light off from exoplanets at contrasts down to contrasts of ~10^-9 with respect to the stellar flux. Detection of point sources at these high contrasts requires yield estimates and detailed modeling of the image of the planetary system as it propagates through the telescope optics. While the SITs might generate custom astrophysical scenes, the integrated model, propagated through the internal speckle field, is typically done at JPL. In this white paper, we present a standard file format to ensure a single distribution system between those who produce the raw astrophysical scenes, and JPL modelers who incorporate those scenes into their optical modeling. At its core, our custom file format uses FITS files, and incorporates standards on packaging astrophysical scenes. This includes spectral and astrometric information for planetary and stellar point sources, zodiacal light and extragalactic sources that may appear as contaminants. Adhering to such a uniform data distribution format is necessary, as it ensures seamless work flow between the SITs and modelers at JPL for the goals of understanding limits of the WFIRST coronagraphic instrument.Comment: 8 pages, white pape

Visualization-Driven Time-Series Extraction from Wearable Systems Can Facilitate Differentiation of Passive ADL Characteristics among Stroke and Healthy Older Adults

Wearable technologies allow the measurement of unhindered activities of daily living (ADL) among patients who had a stroke in their natural settings. However, methods to extract meaningful information from large multi-day datasets are limited. This study investigated new visualization-driven time-series extraction methods for distinguishing activities from stroke and healthy adults. Fourteen stroke and fourteen healthy adults wore a wearable sensor at the L5/S1 position for three consecutive days and collected accelerometer data passively in the participant’s naturalistic environment. Data from visualization facilitated selecting information-rich time series, which resulted in classification accuracy of 97.3% using recurrent neural networks (RNNs). Individuals with stroke showed a negative correlation between their body mass index (BMI) and higher-acceleration fraction produced during ADL. We also found individuals with stroke made lower activity amplitudes than healthy counterparts in all three activity bands (low, medium, and high). Our findings show that visualization-driven time series can accurately classify movements among stroke and healthy groups using a deep recurrent neural network. This novel visualization-based time-series extraction from naturalistic data provides a physical basis for analyzing passive ADL monitoring data from real-world environments. This time-series extraction method using unit sphere projections of acceleration can be used by a slew of analysis algorithms to remotely track progress among stroke survivors in their rehabilitation program and their ADL abilities

Parallel approximation of non-interactive zero-sum quantum games

This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players' payoffs. We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that one-turn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space.Comment: 18 page