14,672 research outputs found
On CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations
AbstractFor Toeplitz system of weakly nonlinear equations, by using the separability and strong dominance between the linear and the nonlinear terms and using the circulant and skew-circulant splitting (CSCS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-CSCS and nonlinear CSCS-like iteration methods, respectively. The advantage of these methods is that they do not require accurate computation and storage of Jacobian matrix, and only need to solve linear sub-systems of constant coefficient matrices. Therefore, computational workloads and computer storage may be saved in actual implementations. Theoretical analysis shows that these new iteration methods are local convergent under suitable conditions. Numerical results show that both Picard-CSCS and nonlinear CSCS-like iteration methods are feasible and effective for some cases
Bilinear Graph Neural Network with Neighbor Interactions
Graph Neural Network (GNN) is a powerful model to learn representations and
make predictions on graph data. Existing efforts on GNN have largely defined
the graph convolution as a weighted sum of the features of the connected nodes
to form the representation of the target node. Nevertheless, the operation of
weighted sum assumes the neighbor nodes are independent of each other, and
ignores the possible interactions between them. When such interactions exist,
such as the co-occurrence of two neighbor nodes is a strong signal of the
target node's characteristics, existing GNN models may fail to capture the
signal. In this work, we argue the importance of modeling the interactions
between neighbor nodes in GNN. We propose a new graph convolution operator,
which augments the weighted sum with pairwise interactions of the
representations of neighbor nodes. We term this framework as Bilinear Graph
Neural Network (BGNN), which improves GNN representation ability with bilinear
interactions between neighbor nodes. In particular, we specify two BGNN models
named BGCN and BGAT, based on the well-known GCN and GAT, respectively.
Empirical results on three public benchmarks of semi-supervised node
classification verify the effectiveness of BGNN -- BGCN (BGAT) outperforms GCN
(GAT) by 1.6% (1.5%) in classification accuracy.Codes are available at:
https://github.com/zhuhm1996/bgnn.Comment: Accepted by IJCAI 2020. SOLE copyright holder is IJCAI (International
Joint Conferences on Artificial Intelligence), all rights reserve
Regret Distribution in Stochastic Bandits: Optimal Trade-off between Expectation and Tail Risk
We study the trade-off between expectation and tail risk for regret
distribution in the stochastic multi-armed bandit problem. We fully
characterize the interplay among three desired properties for policy design:
worst-case optimality, instance-dependent consistency, and light-tailed risk.
We show how the order of expected regret exactly affects the decaying rate of
the regret tail probability for both the worst-case and instance-dependent
scenario. A novel policy is proposed to characterize the optimal regret tail
probability for any regret threshold. Concretely, for any given and , our policy achieves a worst-case expected regret
of (we call it -optimal) and an instance-dependent
expected regret of (we call it -consistent), while
enjoys a probability of incurring an regret
( in the worst-case scenario and in the
instance-dependent scenario) that decays exponentially with a polynomial
term. Such decaying rate is proved to be best achievable. Moreover, we discover
an intrinsic gap of the optimal tail rate under the instance-dependent scenario
between whether the time horizon is known a priori or not. Interestingly,
when it comes to the worst-case scenario, this gap disappears. Finally, we
extend our proposed policy design to (1) a stochastic multi-armed bandit
setting with non-stationary baseline rewards, and (2) a stochastic linear
bandit setting. Our results reveal insights on the trade-off between regret
expectation and regret tail risk for both worst-case and instance-dependent
scenarios, indicating that more sub-optimality and inconsistency leave space
for more light-tailed risk of incurring a large regret, and that knowing the
planning horizon in advance can make a difference on alleviating tail risks
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