18 problems

Rate-optimality conjecture. The $\alpha$-dependent price in the convergence rate, namely $\mathcal{O}(\varepsilon^{-(4+\alpha)})$ for $\alpha\in(0,1)$ and…

Consider the variance of linear statistics with respect to $mu_n$ for the three methods listed in the source, including MCMC and Gibbs-measure backgrounds. Gibbs-measure improvemen…

Let $\prescript{\mathrm{Ext}}{}{\Phi}^{\ell}$ and $\prescript{\mathrm{Ext}}{}{\Psi}^{\ell}$ denote the extrapolated fine- and coarse-level quantities at level $\ell$. Assume that…

Let $\mathcal{A}(M,P,N)$ be the single-level estimator, and let $\bm X_N^P$ denote the particle-system realization. For $M$ samples and $P$ particles, variance-decay conjecture. Th…

Variance-reduction conjecture. The repulsion operator may produce variance reduction for smooth functions for a wide range of point processes.

The stochastic deep unrolling methods use minibatches of operators to estimate gradients, and variance-reduction tricks can be incorporated into these networks. Variance-reduction…

Consider the finite-sum saddle-point problem discussed above, with $m_G$ component functions having smoothness parameter $L_G$, and strong-convexity/strong-concavity parameters…

Consider the distributed empirical risk minimization (ERM) setting, in which a fixed finite collection of objectives is optimized, and let FedAc denote the federated accelerated st…

Large-step-size conjecture. All these minibatch proximal stochastic gradient methods admit step-size choices of order

SVRPDA-I acceleration conjecture. The dependence of SVRPDA-I's complexity on the condition number $\kappa$ can be further improved using a similar acceleration technique. The conje…

Sensitivity-ordering conjecture. If the components of $\mathbf{X}$ are arranged in decreasing order of their sensitivity in estimating the function $h$, then this arrangement gives…

The expected smoothness constant governs the convergence of stochastic variance-reduced methods, and the total complexity of SAGA decreases linearly with the mini-batch size up to…

Schmidt et al.'s conjecture. There exist nonuniform sampling probabilities for which the iteration complexity of SAGA improves to

The sparse variant is an algorithmic variant of the variance-reduced three-operator splitting method discussed above; its iterates have a convergence rate measured as a function of…

The discussion concerns mini-batch SAGA applied with a mini-batch size $tau$ to stochastic optimization problems, where the total complexity is the product of $tau$ and the iterati…

Consider distributed without-replacement SVRG, in which the data are randomly split among machines and each machine processes batches of its local data without replacement. Let the…

Let $F$ be the objective function, let $\boldsymbol{w}^*$ be a minimizer of $F$, and let $\boldsymbol{w}_t$ denote the iterates of without-replacement SVRG. The analysis currently…

The stochastic reaction network is simulated using a multilevel adaptive reaction-splitting method with a deterministic time-change control variate based on the mean field. Uniquen…