The stochastic gradient (SG) method can quickly solve a problem with a large number of components in the objective, or a stochastic optimization problem, to a moderate accuracy. The block coordinate descent/update (BCD) method, on the other hand, can quickly solve problems with multiple (blocks of) variables. This paper introduces a method that combines the great features of SG and BCD for problems with many components in the objective and with multiple (blocks of) variables. This paper proposes a block SG (BSG) method for both convex and nonconvex programs. BSG generalizes SG by updating all the blocks of variables in the Gauss–Seidel type (updating the current block depends on the previously updated block), in either a fixed or randomly shuffled order. Although BSG has slightly more work at each iteration, it typically outperforms SG because of BSG’s Gauss–Seidel updates and larger step sizes, the latter of which are determined by the smaller per-block Lipschitz constants. The convergence of BSG is established for both convex and nonconvex cases. In the convex case, BSG has the same order of convergence rate as SG. In the nonconvex case, its convergence is established in terms of the expected violation of a first-order optimality condition. In both cases our analysis is nontrivial since the typical unbiasedness assumption no longer holds. BSG is numerically evaluated on the following problems: stochastic least squares and logistic regression, which are convex, and low-rank tensor recovery and bilinear logistic regression, which are nonconvex. On the convex problems, BSG performed significantly better than SG. On the nonconvex problems, BSG significantly outperformed the deterministic BCD method because the latter tends to stagnate early near local minimizers. Overall, BSG inherits the benefits of both SG approximation and block coordinate updates and is especially useful for solving large-scale nonconvex problems.

We use the signal function from RLWE key exchange to derive an efficient zero knowledge authentication protocol to validate an RLWE key p=as+e with secret s and error e in the Random Oracle Model (ROM). With this protocol, a verifier can validate that a key presented to him by a prover P is of the form p=as+e with s,e small and that the prover knows s. We accompany the description of the protocol with proof to show that it has negligible soundness and completeness error. The soundness of our protocol relies directly on the hardness of the RLWE problem. The protocol is applicable for both LWE and RLWE but we focus on the RLWE based protocol for efficiency and practicality. We also present a variant of the main protocol with a commitment scheme to avoid using the ROM.

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In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups$G$which possess certain “density” properties (see$Moskowitz$, M., On the density theorems of Borel and Furstenberg,$Ark. Mat.$$16$(1978), 11–27, and$Moskowitz$, M., Some results on automorphisms of bounded displacement and bounded cocycles,$Monatsh. Math.$$85$(1978), 323–336). For such groups we show that Γ always has finite index in its normalizer$N$_{$G$}(Γ). We then investigate analogous questions for the automorphism group Aut($G$) proving, under appropriate conditions, that Stab_{Aut($G$)}(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$ , of Aut($G$) has finite index in Stab_{Aut($G$)}(Γ). We test the limits of our results with various examples and counterexamples.