The framework of Integral Quadratic Constraints (IQC) introduced by (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov's accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions.
Finally, @cite_7 gives a possible geometric interpretation of why NAM accelerates convergence. For their NAM-type method, the result @math is obtained, basically re-deriving .
@cite_7
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Protocol design requires understanding state distributed across many nodes, complex message exchanges, and with competing traAEc. Traditional analysis tools (such as packet traces) too often hide protocol dynamics in a mass of extraneous detail. This paper presents nam , a network animator that provides packet-level animation and protocol-speci c graphs to aid the design and debugging of new network protocols. Taking data from network simulators (such as ns) or live networks, nam was one of the rst tools to provide general purpose, packet-level, network animation. Nam now integrates traditional time-event plots of protocol actions and scenario editing capabilities. We describe how nam visualizes protocol and network dynamics. In this paper, we propose a new deep learning approach, called neural association model (NAM), for probabilistic reasoning in artificial intelligence. We propose to use neural networks to model association between any two events in a domain. Neural networks take one event as input and compute a conditional probability of the other event to model how likely these two events are to be associated. The actual meaning of the conditional probabilities varies between applications and depends on how the models are trained. In this work, as two case studies, we have investigated two NAM structures, namely deep neural networks (DNN) and relation-modulated neural nets (RMNN), on several probabilistic reasoning tasks in AI, including recognizing textual entailment, triple classification in multi-relational knowledge bases and commonsense reasoning. Experimental results on several popular datasets derived from WordNet, FreeBase and ConceptNet have all demonstrated that both DNNs and RMNNs perform equally well and they can significantly outperform the conventional methods available for these reasoning tasks. Moreover, compared with DNNs, RMNNs are superior in knowledge transfer, where a pre-trained model can be quickly extended to an unseen relation after observing only a few training samples. To further prove the effectiveness of the proposed models, in this work, we have applied NAMs to solving challenging Winograd Schema (WS) problems. Experiments conducted on a set of WS problems prove that the proposed models have the potential for commonsense reasoning. Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of @math functions over a convex constraint set @math where both @math and @math are large. In such problems, sub-sampling as a way to reduce @math can offer great amount of computational efficiency. Within the context of second order methods, we first give quantitative local convergence results for variants of Newton's method where the Hessian is uniformly sub-sampled. Using random matrix concentration inequalities, one can sub-sample in a way that the curvature information is preserved. Using such sub-sampling strategy, we establish locally Q-linear and Q-superlinear convergence rates. We also give additional convergence results for when the sub-sampled Hessian is regularized by modifying its spectrum or Levenberg-type regularization. Finally, in addition to Hessian sub-sampling, we consider sub-sampling the gradient as way to further reduce the computational complexity per iteration. We use approximate matrix multiplication results from randomized numerical linear algebra (RandNLA) to obtain the proper sampling strategy and we establish locally R-linear convergence rates. In such a setting, we also show that a very aggressive sample size increase results in a R-superlinearly convergent algorithm. While the sample size depends on the condition number of the problem, our convergence rates are problem-independent, i.e., they do not depend on the quantities related to the problem. Hence, our analysis here can be used to complement the results of our basic framework from the companion paper, [38], by exploring algorithmic trade-offs that are important in practice. Abstract Let the linear system Ax = b be rectangular but solvable. If A is a large sparse matrix, then one possibility to solve the system is to use the iterative method of Kaczmarz. Even in the case when the system is unsolvable, this method is applicable if the relaxation parameters are kept fixed in a certain interval. To get a meaningful result, however, one has to apply strong underrelaxation. We give an interpretation of the limit in terms of a generalized inverse of A and derive bounds with respect to the least-squares solution. Using a result of Buoni and Varga, we estimate the speed of convergence. It is further shown that the eigenvalues of the iteration matrix are nearly real numbers, and we propose to use Chebyshev acceleration. This results in a significant speedup.