Data-Dependent Stability of Stochastic Gradient Descent

We establish a data-dependent notion of algorithmic stability for Stochastic Gradient Descent (SGD), and employ it to develop novel generalization bounds. This is in contrast to previous distribution-free algorithmic stability results for SGD which depend on the worst-case constants. By virtue of the data-dependent argument, our bounds provide new insights into learning with SGD on convex and non-convex problems. In the convex case, we show that the bound on the generalization error depends on the risk at the initialization point. In the non-convex case, we prove that the expected curvature of the objective function around the initialization point has crucial influence on the generalization error. In both cases, our results suggest a simple data-driven strategy to stabilize SGD by pre-screening its initialization. As a corollary, our results allow us to show optimistic generalization bounds that exhibit fast convergence rates for SGD subject to a vanishing empirical risk and low noise of stochastic gradient

Quantum dynamics of a two-level emitter with modulated transition frequency

The resonant quantum dynamics of an excited two-level emitter is investigated via classical modulation of its transition frequency while simultaneously the radiator interacts with a broadband electromagnetic field reservoir. The frequency of modulation is selected to be of the order of the bare-state spontaneous decay rate. In this way, one can induce quantum interference effects and, consequently, quantum coherences among multiple decaying transition pathways. Depending on the modulation depth and its absolute phase, both the spontaneous emission and the frequency shift may be conveniently modified and controlled.Comment: 8 pages, 6 figure

Accelerating the Fourier split operator method via graphics processing units

Current generations of graphics processing units have turned into highly parallel devices with general computing capabilities. Thus, graphics processing units may be utilized, for example, to solve time dependent partial differential equations by the Fourier split operator method. In this contribution, we demonstrate that graphics processing units are capable to calculate fast Fourier transforms much more efficiently than traditional central processing units. Thus, graphics processing units render efficient implementations of the Fourier split operator method possible. Performance gains of more than an order of magnitude as compared to implementations for traditional central processing units are reached in the solution of the time dependent Schr\"odinger equation and the time dependent Dirac equation

Diamond Integrated Optomechanical Circuits

Diamond offers unique material advantages for the realization of micro- and nanomechanical resonators due to its high Young's modulus, compatibility with harsh environments and superior thermal properties. At the same time, the wide electronic bandgap of 5.45eV makes diamond a suitable material for integrated optics because of broadband transparency and the absence of free-carrier absorption commonly encountered in silicon photonics. Here we take advantage of both to engineer full-scale optomechanical circuits in diamond thin films. We show that polycrystalline diamond films fabricated by chemical vapour deposition provide a convenient waferscale substrate for the realization of high quality nanophotonic devices. Using free-standing nanomechanical resonators embedded in on-chip Mach-Zehnder interferometers, we demonstrate efficient optomechanical transduction via gradient optical forces. Fabricated diamond resonators reproducibly show high mechanical quality factors up to 11,200. Our low cost, wideband, carrier-free photonic circuits hold promise for all-optical sensing and optomechanical signal processing at ultra-high frequencies