On the Convergence of Decentralized Gradient Descent

Consider the consensus problem of minimizing $f(x)=\sum_{i=1}^n f_i(x)$ where each $f_i$ is only known to one individual agent $i$ out of a connected network of $n$ agents. All the agents shall collaboratively solve this problem and obtain the solution subject to data exchanges restricted to between neighboring agents. Such algorithms avoid the need of a fusion center, offer better network load balance, and improve data privacy. We study the decentralized gradient descent method in which each agent $i$ updates its variable $x_{(i)}$, which is a local approximate to the unknown variable $x$, by combining the average of its neighbors' with the negative gradient step $-\alpha \nabla f_i(x_{(i)})$. The iteration is $$x_{(i)}(k+1) \gets \sum_{\text{neighbor} j \text{of} i} w_{ij} x_{(j)}(k) - \alpha \nabla f_i(x_{(i)}(k)),\quad\text{for each agent} i,$$ where the averaging coefficients form a symmetric doubly stochastic matrix $W=[w_{ij}] \in \mathbb{R}^{n \times n}$. We analyze the convergence of this iteration and derive its converge rate, assuming that each $f_i$ is proper closed convex and lower bounded, $\nabla f_i$ is Lipschitz continuous with constant $L_{f_i}$, and stepsize $\alpha$ is fixed. Provided that $\alpha < O(1/L_h)$ where $L_h=\max_i\{L_{f_i}\}$, the objective error at the averaged solution, $f(\frac{1}{n}\sum_i x_{(i)}(k))-f^*$, reduces at a speed of $O(1/k)$ until it reaches $O(\alpha)$. If $f_i$ are further (restricted) strongly convex, then both $\frac{1}{n}\sum_i x_{(i)}(k)$ and each $x_{(i)}(k)$ converge to the global minimizer $x^*$ at a linear rate until reaching an $O(\alpha)$-neighborhood of $x^*$. We also develop an iteration for decentralized basis pursuit and establish its linear convergence to an $O(\alpha)$-neighborhood of the true unknown sparse signal

The induced interaction in a Fermi gas with a BEC-BCS crossover

We study the effect of the induced interaction on the superfluid transition temperature of a Fermi gas with a BEC-BCS crossover. The Gorkov-Melik-Barkhudarov theory about the induced interaction is extended from the BCS side to the entire crossover, and the pairing fluctuation is treated in the approach by Nozi\`{e}res and Schmitt-Rink. At unitarity, the induced interaction reduces the transition temperature by about twenty percent. In the BCS limit, the transition temperature is reduced by a factor about 2.22, as found by Gorkov and Melik-Barkhudarov. Our result shows that the effect of the induced interaction is important both on the BCS side and in the unitary region.Comment: 11 pages, 3 figures, to be published in PR

Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover

The Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover is derived by the path-integral method. In addition to the standard Ginzburg-Landau equation, a second equation describing the total atom density is obtained. These two coupled equations are necessary to describe both homogeneous and inhomogeneous systems. The Ginzburg-Landau theory is valid near the transition temperature $T_c$ on both sides of the crossover. In the weakly-interacting BEC region, it is also accurate at zero temperature where the Ginzburg-Landau equation can be mapped onto the Gross-Pitaevskii (GP) equation. The applicability of GP equation at finite temperature is discussed. On the BEC side, the fluctuation of the order parameter is studied and the renormalization to the molecule coupling constant is obtained.Comment: 16 pages, 2 figures, to be published in PR

Electron self-energy and effective mass in a single heterostructure

In this paper, we investigate the electron self-energy and effective mass in a single heterostructure using Green-function method. Numerical calculations of the electron self-energy and effective mass for GaAs/AlAs heterostructure are performed. The results show that the self energy (effective mass) of electron, which incorporate the energy of electron coupling to interface-optical phonons and half three-dimension LO phonons, monotonically increase(decrease) from that of interface polaron to that of 3D bulk polaron with the increase of the distance between the position of the electron and interface.Comment: 10 pages, 2 figure

The TRRAP-HAT-Sp1 axis maintains brain homeostasis

The homeostasis of the brain is tightly controlled by the viability and functionality of various cell types, including neurons and glial cells, like oligodendrocytes, astrocytes as well as microglia. Defects of neurogenesis and maintenance of neural cells are associated with multiple neuropathologies, such as Intellectual Disability (ID) and Autism Spectrum Disorders (ASD) among other diseases. HAT and HDAC modulate brain functionality, e.g. memory formation, cognitive function, and neuroprotection, whereas the disturbance of the acetylation profiles has been related to multiple neuropathological diseases. However, how epigenetic regulation participates in the neurodevelopmental, neural differentiation and neurodegenerative processes remains largely unknown. In our studies, we have chosen the HAT adaptor, Trrap, to investigate how the disturbance of acetylation would affect brain functionality. We show that Trrap deletion in post-mitotic neurons results in neurodegeneration. In addition, Trrap deficiency in adult neural stem cells compromises their self-renewal and differentiation. With integrated transcriptomics, epigenomics, and proteomics we identify Sp1 as the master regulator controlled by Trrap-HAT and demonstrate that the Trrap-HAT-Sp1 axis ensures the proper expression of genes involved in microtubule dynamics. We find that Trrap mediates Sp1 binding through the maintenance of the acetylation profile on Sp1 and that acetylation of Sp1 plays an important role, dependent and independent of Trrap, in its transcription activation. Taken together, we demonstrate that Trrap, through its mediated acetylation, is involved in neuroprotection and neural differentiation via the regulation of Sp1 activity. My dissertation provides a novel insight into the role of epigenetic regulation of transcription factors in the maintenance of brain homeostasis and preventing neurodegeneration