In this paper, self-shielded quad-mode substrate integrated waveguide (SIW) bandpass filters (BPFs) are proposed by using a novel method of mixed-mode cavity, which is achieved by adding a metal patch to the middle substrate layer in a SIW cavity. The mixed-mode cavity can support two pairs of degenerate dual modes, i.e., SIW-cavity modes and strip-patch modes. Characteristics of the proposed mixed-mode SIW cavity are analyzed in detail for filters’ design. For the demonstration, an unbalanced quad-mode filter structure is proposed and analyzed, which can realize single-and dual-band filtering responses. Then, a balanced quad-mode filter structure is proposed and analyzed on the basis of unbalanced quad-mode filter and its odd symmetry of the four resonant modes. The balanced filter can also realize single-and dual-band filtering responses with common-mode inherently at passbands under differential-mode operation. Finally, two unbalanced and two balanced quad-mode filters with single-and dual-band responses are designed, fabricated and measured. The measured results agree well with the designed ones, which verifies the proposed method. The proposed mixed-mode SIW filters have the merits of high selectivity, compact size and high shielding from nearby circuits.

Federated bilevel learning has received increasing attention in various emerging machine learning and communication applications. Recently, several Hessian-vector-based algorithms have been proposed to solve the federated bilevel optimization problem. However, several important properties in federated learning such as the partial client participation and the linear speedup for convergence (i.e., the convergence rate and complexity are improved linearly with respect to the number of sampled clients) in the presence of non-i.i.d. datasets, still remain open. In this paper, we fill these gaps by proposing a new federated bilevel algorithm named FedMBO with a novel client sampling scheme in the federated hypergradient estimation. We show that FedMBO achieves a convergence rate of $\mathcal{O}\big(\frac{1}{\sqrt{nK}}+\frac{1}{K}+\frac{\sqrt{n}}{K^{3/2}}\big)$ on non-i.i.d. datasets, where $n$ is the number of participating clients in each round, and $K$ is the total number of iteration. This is the first theoretical linear speedup result for non-i.i.d. federated bilevel optimization. Extensive experiments validate our theoretical results and demonstrate the effectiveness of our proposed method.

In many statistical learning problems, it is desired that the optimal solution conform to an a priori known sparsity structure represented by a directed acyclic graph. Inducing such structures by means of convex regularizers requires nonsmooth penalty functions that exploit group overlapping. Our study focuses on evaluating the proximal operator of the latent overlapping group lasso developed by Jacob et al. in 2009 . We implemented an alternating direction method of multiplier with a sharing scheme to solve large-scale instances of the underlying optimization problem efficiently. In the absence of strong convexity, global linear convergence of the algorithm is established using the error bound theory. More specifically, the paper contributes to establishing primal and dual error bounds when the nonsmooth component in the objective function does not have a polyhedral epigraph. We also investigate the effect of the graph structure on the speed of convergence of the algorithm. Detailed numerical simulation studies over different graph structures supporting the proposed algorithm and two applications in learning are provided. Summary of Contribution: The paper proposes a computationally efficient optimization algorithm to evaluate the proximal operator of a nonsmooth hierarchical sparsity-inducing regularizer and establishes its convergence properties. The computationally intensive subproblem of the proposed algorithm can be fully parallelized, which allows solving large-scale instances of the underlying problem. Comprehensive numerical simulation studies benchmarking the proposed algorithm against five other methods on the speed of convergence to optimality are provided. Furthermore, performance of the algorithm is demonstrated on two statistical learning applications related to topic modeling and breast cancer classification. The code along with the simulation studies and benchmarks are available on the corresponding author’s GitHub website for evaluation and future use.