Despite their appeal as physics-inspired, energy-based and generative nature,
general Boltzmann Machines (BM) are considered intractable to train. This
belief led to simplified models of BMs with restricted intralayer connections
or layer-by-layer training of deep BMs. Recent developments in domain-specific
hardware -- specifically probabilistic computers (p-computer) with
probabilistic bits (p-bit) -- may change established wisdom on the tractability
of deep BMs. In this paper, we show that deep and unrestricted BMs can be
trained using p-computers generating hundreds of billions of Markov Chain Monte
Carlo (MCMC) samples per second, on sparse networks developed originally for
use in D-Wave's annealers. To maximize the efficiency of learning the
p-computer, we introduce two families of Mean-Field Theory assisted learning
algorithms, or xMFTs (x = Naive and Hierarchical). The xMFTs are used to
estimate the averages and correlations during the positive phase of the
contrastive divergence (CD) algorithm and our custom-designed p-computer is
used to estimate the averages and correlations in the negative phase. A custom
Field-Programmable-Gate Array (FPGA) emulation of the p-computer architecture
takes up to 45 billion flips per second, allowing the implementation of CD-$n$
where $n$ can be of the order of millions, unlike RBMs where $n$ is typically 1
or 2. Experiments on the full MNIST dataset with the combined algorithm show
that the positive phase can be efficiently computed by xMFTs without much
degradation when the negative phase is computed by the p-computer. Our
algorithm can be used in other scalable Ising machines and its variants can be
used to train BMs, previously thought to be intractable.

利用概率计算机（p-computer）和意味场理论（Mean-Field Theory）辅助学习算法，该研究提出了用于训练难以处理的波尔兹曼机（Boltzmann Machines）的方法以及在可扩展的伊辛机器（Ising machines）中的应用。

利用概率计算机辅助的均场深度玻尔兹曼学习

Mean-Field Assisted Deep Boltzmann Learning with Probabilistic Computers

Federated learning (FL) enables multiple clients to train a model while
keeping their data private collaboratively. Previous studies have shown that
data heterogeneity between clients leads to drifts across client updates.
However, there are few studies on the relationship between client and global
modes, making it unclear where these updates end up drifting. We perform
empirical and theoretical studies on this relationship by utilizing mode
connectivity, which measures performance change (i.e., connectivity) along
parametric paths between different modes. Empirically, reducing data
heterogeneity makes the connectivity on different paths more similar, forming
more low-error overlaps between client and global modes. We also find that a
barrier to connectivity occurs when linearly connecting two global modes, while
it disappears with considering non-linear mode connectivity. Theoretically, we
establish a quantitative bound on the global-mode connectivity using mean-field
theory or dropout stability. The bound demonstrates that the connectivity
improves when reducing data heterogeneity and widening trained models.
Numerical results further corroborate our analytical findings.

通过模式连接性理论和实证研究，我们发现数据异质性的减少可以提高不同路径上的连接性，形成更多客户端和全局模式之间的低误差重叠，并且我们还发现线性连接两个全局模式时存在连接性的障碍，但考虑非线性模式连接性后，这个障碍消失。

联邦学习的模态连接性与数据异质性

Mode Connectivity and Data Heterogeneity of Federated Learning

We propose the first least-order Galerkin model of an incompressible flow
undergoing two successive supercritical bifurcations of Hopf and pitchfork
type. A key enabler is a mean-field consideration exploiting the symmetry of
the mean flow and the asymmetry of the fluctuation. These symmetries generalize
mean-field theory, e.g. no assumption of slow growth-rate is needed. The
resulting 5-dimensional Galerkin model successfully describes the phenomenogram
of the fluidic pinball, a two-dimensional wake flow around a cluster of three
equidistantly spaced cylinders. The corresponding transition scenario is shown
to undergo two successive supercritical bifurcations, namely a Hopf and a
pitchfork bifurcations on the way to chaos. The generalized mean-field Galerkin
methodology may be employed to describe other transition scenarios.

提出了第一个压缩不可压流的最低阶 Galerkin 模型，它经历了两个连续的 Hopf 超临界分叉和 pitchfork 类型的分叉，并且使用平均场方法来利用平均流动的对称性和扰动的不对称性，其描述了一个三个等间距气缸周围的二维尾流的流球现象，所得到的描述方法可以应用于其他转换方案。

流体弹球连续分岔的低阶模型

Low-order model for successive bifurcations of the fluidic pinball

We express community detection as an inference problem of determining the
most likely arrangement of communities. We then apply belief propagation and
mean-field theory to this problem, and show that this leads to fast, accurate
algorithms for community detection.

本研究将社区检测问题表达为一个推断问题来决定社区最可能的排列方式，并应用信念传播和均场理论解决该问题，证明这些方法可导致快速准确的社区检测算法。