467 research outputs found
Moduli Space of Paired Punctures, Cyclohedra and Particle Pairs on a Circle
In this paper, we study a new moduli space ,
which is obtained from by identifying pairs of
punctures. We find that this space is tiled by cyclohedra, and
construct the canonical form for each chamber. We also find the corresponding
Koba-Nielsen factor can be viewed as the potential of the system of
pairs of particles on a circle, which is similar to the original case of
where the system is particles on a line. We
investigate the intersection numbers of chambers equipped with Koba-Nielsen
factors. Then we construct cyclohedra in kinematic space and show that the
scattering equations serve as a map between the interior of worldsheet
cyclohedron and kinematic cyclohedron. Finally, we briefly discuss string-like
integrals over such moduli space.Comment: 23 pages, 7 figure
Convex Optimization for Linear Query Processing under Approximate Differential Privacy
Differential privacy enables organizations to collect accurate aggregates
over sensitive data with strong, rigorous guarantees on individuals' privacy.
Previous work has found that under differential privacy, computing multiple
correlated aggregates as a batch, using an appropriate \emph{strategy}, may
yield higher accuracy than computing each of them independently. However,
finding the best strategy that maximizes result accuracy is non-trivial, as it
involves solving a complex constrained optimization program that appears to be
non-linear and non-convex. Hence, in the past much effort has been devoted in
solving this non-convex optimization program. Existing approaches include
various sophisticated heuristics and expensive numerical solutions. None of
them, however, guarantees to find the optimal solution of this optimization
problem.
This paper points out that under (, )-differential privacy,
the optimal solution of the above constrained optimization problem in search of
a suitable strategy can be found, rather surprisingly, by solving a simple and
elegant convex optimization program. Then, we propose an efficient algorithm
based on Newton's method, which we prove to always converge to the optimal
solution with linear global convergence rate and quadratic local convergence
rate. Empirical evaluations demonstrate the accuracy and efficiency of the
proposed solution.Comment: to appear in ACM SIGKDD 201
Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation
This paper provides a large deviation principle for Non-Markovian, Brownian
motion driven stochastic differential equations with random coefficients.
Similar to Gao and Liu \cite{GL}, this extends the corresponding results
collected in Freidlin and Wentzell \cite{FreidlinWentzell}. However, we use a
different line of argument, adapting the PDE method of Fleming \cite{Fleming}
and Evans and Ishii \cite{EvansIshii} to the path-dependent case, by using
backward stochastic differential techniques. Similar to the Markovian case, we
obtain a characterization of the action function as the unique bounded solution
of a path-dependent version of the Eikonal equation. Finally, we provide an
application to the short maturity asymptotics of the implied volatility surface
in financial mathematics
HARL: Hierarchical Adaptive Reinforcement Learning Based Auto Scheduler for Neural Networks
To efficiently perform inference with neural networks, the underlying tensor
programs require sufficient tuning efforts before being deployed into
production environments. Usually, enormous tensor program candidates need to be
sufficiently explored to find the one with the best performance. This is
necessary to make the neural network products meet the high demand of
real-world applications such as natural language processing, auto-driving, etc.
Auto-schedulers are being developed to avoid the need for human intervention.
However, due to the gigantic search space and lack of intelligent search
guidance, current auto-schedulers require hours to days of tuning time to find
the best-performing tensor program for the entire neural network.
In this paper, we propose HARL, a reinforcement learning (RL) based
auto-scheduler specifically designed for efficient tensor program exploration.
HARL uses a hierarchical RL architecture in which learning-based decisions are
made at all different levels of search granularity. It also automatically
adjusts exploration configurations in real-time for faster performance
convergence. As a result, HARL improves the tensor operator performance by 22%
and the search speed by 4.3x compared to the state-of-the-art auto-scheduler.
Inference performance and search speed are also significantly improved on
end-to-end neural networks
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