scholarly journals Stochastic Online Learning with Probabilistic Graph Feedback

2020 ◽  
Vol 34 (04) ◽  
pp. 4675-4682
Author(s):  
Shuai Li ◽  
Wei Chen ◽  
Zheng Wen ◽  
Kwong-Sak Leung
Keyword(s):  
Online Learning ◽  
Lower Bounds ◽  
Random Graphs ◽  
High Probability ◽  
Upper Bounds ◽  
Directed Edge ◽  
Probabilistic Graph ◽  
One Step ◽  
The One ◽  
Independent Probability

We consider a problem of stochastic online learning with general probabilistic graph feedback, where each directed edge in the feedback graph has probability pij. Two cases are covered. (a) The one-step case, where after playing arm i the learner observes a sample reward feedback of arm j with independent probability pij. (b) The cascade case where after playing arm i the learner observes feedback of all arms j in a probabilistic cascade starting from i – for each (i,j) with probability pij, if arm i is played or observed, then a reward sample of arm j would be observed with independent probability pij. Previous works mainly focus on deterministic graphs which corresponds to one-step case with pij ∈ {0,1}, an adversarial sequence of graphs with certain topology guarantees, or a specific type of random graphs. We analyze the asymptotic lower bounds and design algorithms in both cases. The regret upper bounds of the algorithms match the lower bounds with high probability.

Sharp upper bounds on perfect retrieval in the Hopfield model

1999 ◽  
Vol 36 (03) ◽  
pp. 941-950 ◽  
Author(s):  
Anton Bovier
Keyword(s):  
Fixed Points ◽  
Lower Bounds ◽  
Upper Bound ◽  
Random Variables ◽  
Upper Bounds ◽  
Negative Association ◽  
Hopfield Model ◽  
Gradient Dynamics ◽  
One Step ◽  
The One

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

Sharp upper bounds on perfect retrieval in the Hopfield model

1999 ◽  
Vol 36 (3) ◽  
pp. 941-950 ◽  
Author(s):  
Anton Bovier
Keyword(s):  
Fixed Points ◽  
Lower Bounds ◽  
Upper Bound ◽  
Random Variables ◽  
Upper Bounds ◽  
Negative Association ◽  
Hopfield Model ◽  
Gradient Dynamics ◽  
One Step ◽  
The One

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

RELIABLE INTERNET-BASED MASTER-WORKER COMPUTING IN THE PRESENCE OF MALICIOUS WORKERS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250002 ◽  
Author(s):  
ANTONIO FERNÁNDEZ ANTA ◽  
CHRYSSIS GEORGIOU ◽  
LUIS LÓPEZ ◽  
AGUSTÍN SANTOS
Keyword(s):  
Lower Bounds ◽  
Distributed System ◽  
High Probability ◽  
Upper Bounds ◽  
The Internet ◽  
Decision Strategy ◽  
Single Task ◽  
Asymptotically Optimal ◽  
A Value ◽  
Probability Of Success

We consider a Master-Worker distributed system where a master processor assigns, over the Internet, tasks to a collection of n workers, which are untrusted and might act maliciously. In addition, a worker may not reply to the master, or its reply may not reach the master, due to unavailabilities or failures of the worker or the network. Each task returns a value, and the goal is for the master to accept only correct values with high probability. Furthermore, we assume that the service provided by the workers is not free; for each task that a worker is assigned, the master is charged with a work-unit. Therefore, considering a single task assigned to several workers, our objective is to have the master processor to accept the correct value of the task with high probability, with the smallest possible amount of work (number of workers the master assigns the task). We probabilistically bound the number of faulty processors by assuming a known probability p < 1/2 of any processor to be faulty. Our work demonstrates that it is possible to obtain, with provable analytical guarantees, high probability of correct acceptance with low work. In particular, we first show lower bounds on the minimum amount of (expected) work required, so that any algorithm accepts the correct value with probability of success 1 - ε, where ε ≪ 1 (e.g., 1/n). Then we develop and analyze two algorithms, each using a different decision strategy, and show that both algorithms obtain the same probability of success 1 - ε, and in doing so, they require similar upper bounds on the (expected) work. Furthermore, under certain conditions, these upper bounds are asymptotically optimal with respect to our lower bounds.

scholarly journals Upper Bounds for Online Ramsey Games in Random Graphs

2009 ◽  
Vol 18 (1-2) ◽  
pp. 259-270 ◽  
Author(s):  
MARTIN MARCINISZYN ◽  
RETO SPÖHEL ◽  
ANGELIKA STEGER
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Upper Bounds ◽  
Arbitrary Size ◽  
Threshold Functions ◽  
Current Graph ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove an upper bound on the typical duration of this game ifFis from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

On the Spread of Random Graphs

2014 ◽  
Vol 23 (4) ◽  
pp. 477-504
Author(s):  
LOUIGI ADDARIO-BERRY ◽  
SVANTE JANSON ◽  
COLIN McDIARMID
Keyword(s):  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Complete Graph ◽  
Connected Graph ◽  
High Probability ◽  
Small World ◽  
Regular Graphs ◽  
World Model ◽  
Supercritical Range

The spread of a connected graph G was introduced by Alon, Boppana and Spencer [1], and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on V(G). We investigate the spread for certain models of sparse random graph, in particular for random regular graphs G(n,d), for Erdős–Rényi random graphs Gn,p in the supercritical range p>1/n, and for a ‘small world’ model. For supercritical Gn,p, we show that if p=c/n with c>1 fixed, then with high probability the spread of the giant component is bounded, and we prove corresponding statements for other models of random graphs, including a model with random edge lengths. We also give lower bounds on the spread for the barely supercritical case when p=(1+o(1))/n. Further, we show that for d large, with high probability the spread of G(n,d) becomes arbitrarily close to that of the complete graph $\mathsf{K}_n$.

scholarly journals The Game Chromatic Number of Dense Random Graphs

10.37236/4391 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Ralph Keusch ◽  
Angelika Steger
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
High Probability ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Game Chromatic Number ◽  
Order Of Magnitude ◽  
Random Structures

Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if at the end, all vertices are colored. The game chromatic number χg(G) is defined as the smallest k for which the first player has a winning strategy.Recently, Bohman, Frieze and Sudakov [Random Structures and Algorithms 2008] analysed the game chromatic number of random graphs and obtained lower and upper bounds of the same order of magnitude. In this paper we improve existing results and show that with high probability, the game chromatic number χg(Gn,p) of dense random graphs with p ≥ e-o(log n) is asymptotically twice as large as the ordinary chromatic number χ(Gn,p).

MINIMIZING CONGESTION OF LAYOUTS FOR ATM NETWORKS WITH FAULTY LINKS

1999 ◽  
Vol 10 (04) ◽  
pp. 503-512 ◽  
Author(s):  
LESZEK GASIENIEC ◽  
EVANGELOS KRANAKIS ◽  
DANNY KRIZANC ◽  
ANDREZEJ PELC
Keyword(s):  
Lower Bounds ◽  
High Probability ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Atm Networks ◽  
Worst Case ◽  
Precise Analysis ◽  
Order Of Magnitude ◽  
Bounded Size ◽  
Fault Distribution

We consider the problem of constructing virtual path layouts for an ATM network consisting of a complete network Kn of n processors in which a certain number of links may fail. Our main goal is to construct layouts which tolerate any configuration of up to f faults and have the least possible congestion. First, we study the minimal congestion of 1-hop f-tolerant layouts in Kn. For any positive integer f we give upper and lower bounds on this minimal congestion and construct f-tolerant layouts with congestion corresponding to the upper bounds. Our results are based on a precise analysis of the diameter of the network Kn[ℱ] which results from Kn by deleting links from a set ℱ of bounded size. Next we study the minimal congestion of h-hop f-tolerant layouts in Kn, for larger values of the number h of hops. We give upper and lower bounds on the order of magnitude of this congestion, based on results for 1-hop layouts. Finally, we consider a random, rather than worst case, fault distribution where links fail independently with constant probability p<1. Our goal now is to construct layouts with low congestion that tolerate the existing faults with high probability. For any p<1, we show the existence of 1-hop layouts in Kn, with congestion O( log n).

scholarly journals Lower Estimate of Clique Size via Edge Coloring

2021 ◽  
Vol 27_NS1 (1) ◽  
pp. 8-15
Author(s):  
Balázs Király ◽  
Sándor Szabó
Keyword(s):  
Lower Bounds ◽  
Numerical Experiments ◽  
Lower Estimate ◽  
Edge Coloring ◽  
Clique Number ◽  
Upper Bounds ◽  
Search Algorithms ◽  
The One ◽  
The Given ◽  
Better Than

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

A Study on the Rearrangement of Dialkyl 1-Aryl-1-hydroxymethylphosphonates to Benzyl Phosphates

2020 ◽  
Vol 24 (4) ◽  
pp. 465-471 ◽  
Author(s):  
Zita Rádai ◽  
Réka Szabó ◽  
Áron Szigetvári ◽  
Nóra Zsuzsa Kiss ◽  
Zoltán Mucsi ◽  
...  
Keyword(s):  
Quantum Chemical ◽  
Room Temperature ◽  
Phase Transfer ◽  
One Pot ◽  
Substrate Dependence ◽  
Triethylbenzylammonium Chloride ◽  
One Step ◽  
The Pudovik Reaction ◽  
The One ◽  
Solid Liquid

The phospha-Brook rearrangement of dialkyl 1-aryl-1-hydroxymethylphosphonates (HPs) to the corresponding benzyl phosphates (BPs) has been elaborated under solid-liquid phase transfer catalytic conditions. The best procedure involved the use of triethylbenzylammonium chloride as the catalyst and Cs2CO3 as the base in acetonitrile as the solvent at room temperature. The substrate dependence of the rearrangement has been studied, and the mechanism of the transformation under discussion was explored by quantum chemical calculations. The key intermediate is an oxaphosphirane. The one-pot version starting with the Pudovik reaction has also been developed. The conditions of this tandem transformation were the same, as those for the one-step HP→BP conversion.

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