193 research outputs found
Second Time Scale of the Metastability of Reversible Inclusion Processes
We investigate the second time scale of the metastable behavior of the
reversible inclusion process in an extension of the study by [Bianchi, Dommers,
and Giardin\`a, Electronic Journal of Probability, 22: 1-34, 2017], which
presented the first time scale of the same model and conjectured the scheme of
multiple time scales. We show that is indeed the correct second
time scale for the most general class of reversible inclusion processes, and
thus prove the first conjecture of the foresaid study. Here, denotes the
number of particles, and denotes the small scale of randomness of the
system. The main obstacles of this research arise in calculating the sharp
asymptotics for the capacities, and in the fact that the methods employed in
the former study are not directly applicable due to the complex geometry of
particle configurations. To overcome these problems, we first thoroughly
examine the landscape of the transition rates to obtain a proper test function
of the equilibrium potential, which provides the upper bound for the
capacities. Then, we modify the induced test flow and precisely estimate the
equilibrium potential near the metastable valleys to obtain the correct lower
bound for the capacities.Comment: 51 pages, 3 figure
Spectral gap of the symmetric inclusion process
We consider the symmetric inclusion process on a general finite graph. Our
main result establishes universal upper and lower bounds for the spectral gap
of this interacting particle system in terms of the spectral gap of the random
walk on the same graph. In the regime in which the gamma-like reversible
measures of the particle systems are log-concave, our bounds match, yielding a
version for the symmetric inclusion process of the celebrated Aldous' spectral
gap conjecture originally formulated for the interchange process. Finally, by
means of duality techniques, we draw analogous conclusions for an interacting
diffusion-like unbounded conservative spin system known as Brownian energy
process.Comment: 16 page
Recommended from our members
Cloud Scheduling with Deep RL: Learning from Demonstration and Competition
As Cloud's adoption surges across industries, the limitations of its default scheduler, particularly on large scales or for jobs outside of its initial design scope, have become increasingly prominent. With the expansion of cloud usage, the industry is facing increased demands for integrating diverse cloud architectures. However, the default schedulers in various cloud services were primarily engineered with a focus on predictable tasks that exhibit minimal variance. Despite this need, clear and adaptable strategies to navigate these complex scenarios remain elusive, largely due to inherent design challenges.
To address these issues, this thesis presents Dejavu. Dejavu combines reinforcement learning with neural networks to learn and resolve scheduling problems more effectively. To tackle the extended training time associated with reinforcement learning, we have applied pretraining using demonstrations from existing scheduling heuristics, thereby improving training efficiency in our cloud scheduling solution. This process prepares the neural network for subsequent reinforcement learning. A robust reward function is devised to push Dejavu to compete with, and eventually outperform, the exploited heuristics and other existing algorithms.
The experimental results demonstrate the efficacy of Dejavu, showing remarkable improvements in key metrics. Specifically, compared to Kubernetes' default scheduler, it boosts resource utilization by 6% and shortens scheduling time by 3% during the scheduling period. Thus, it represents a significant leap forward in cloud scheduling, offering improved efficiency and versatility.</p
Metastability of the three-state Potts model with general interactions
We consider the Potts model on a two-dimensional periodic rectangular lattice
with general coupling constants , where are the
possible spin values (or colors). The resulting energy landscape is thus
significantly more complex than in the original Ising or Potts models. The
system evolves according to a Glauber-type spin-flipping dynamics. We focus on
a region of the parameter space where there are two symmetric metastable states
and a stable state, and the height of a direct path between the metastable
states is equal to the height of a direct path between any metastable state and
the stable state. We study the metastable transition time in probability and in
expectation, the mixing time of the dynamics and the spectral gap of the system
when the inverse temperature tends to infinity. Then, we identify all
the critical configurations that are visited with high probability during the
metastable transition.Comment: 35 pages, 8 figure
Transformers meet Stochastic Block Models: Attention with Data-Adaptive Sparsity and Cost
To overcome the quadratic cost of self-attention, recent works have proposed
various sparse attention modules, most of which fall under one of two groups:
1) sparse attention under a hand-crafted patterns and 2) full attention
followed by a sparse variant of softmax such as -entmax. Unfortunately,
the first group lacks adaptability to data while the second still requires
quadratic cost in training. In this work, we propose SBM-Transformer, a model
that resolves both problems by endowing each attention head with a
mixed-membership Stochastic Block Model (SBM). Then, each attention head
data-adaptively samples a bipartite graph, the adjacency of which is used as an
attention mask for each input. During backpropagation, a straight-through
estimator is used to flow gradients beyond the discrete sampling step and
adjust the probabilities of sampled edges based on the predictive loss. The
forward and backward cost are thus linear to the number of edges, which each
attention head can also choose flexibly based on the input. By assessing the
distribution of graphs, we theoretically show that SBM-Transformer is a
universal approximator for arbitrary sequence-to-sequence functions in
expectation. Empirical evaluations under the LRA and GLUE benchmarks
demonstrate that our model outperforms previous efficient variants as well as
the original Transformer with full attention. Our implementation can be found
in https://github.com/sc782/SBM-Transformer .Comment: 19 pages, 8 figure
- …