546 research outputs found
Deep Residual Learning via Large Sample Mean-Field Stochastic Optimization
We study a class of stochastic optimization problems of the mean-field type
arising in the optimal training of a deep residual neural network. We consider
the sampling problem arising from a continuous layer idealization, and
establish the existence of optimal relaxed controls when the training set has
finite size. The core of our paper is to prove the Gamma-convergence of the
sequence of sampled objective functionals, i.e., show that as the size of the
training set grows large, the minimizer of the sampled relaxed problem
converges to that of the limiting optimization problem. We connect the limit of
the large sampled objective functional to the unique solution, in the
trajectory sense, of a nonlinear Fokker-Planck-Kolmogorov (FPK) equation in a
random environment. We construct an example to show that, under mild
assumptions, the optimal network weights can be numerically computed by solving
a second-order differential equation with Neumann boundary conditions in the
sense of distributions
Real-time Information, Uncertainty and Quantum Feedback Control
Feedback is the core concept in cybernetics and its effective use has made
great success in but not limited to the fields of engineering, biology, and
computer science. When feedback is used to quantum systems, two major types of
feedback control protocols including coherent feedback control (CFC) and
measurement-based feedback control (MFC) have been developed. In this paper, we
compare the two types of quantum feedback control protocols by focusing on the
real-time information used in the feedback loop and the capability in dealing
with parameter uncertainty. An equivalent relationship is established between
quantum CFC and non-selective quantum MFC in the form of operator-sum
representation. Using several examples of quantum feedback control, we show
that quantum MFC can theoretically achieve better performance than quantum CFC
in stabilizing a quantum state and dealing with Hamiltonian parameter
uncertainty. The results enrich understanding of the relative advantages
between quantum MFC and quantum CFC, and can provide useful information in
choosing suitable feedback protocols for quantum systems.Comment: 24 page
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