20 problems

Oracle-call optimality conjecture. For the class of methods that require an optimal number of communication rounds, these bounds are also optimal, up to polylogarithmic factors, in…

Extension conjecture. Theorem and Corollary remain valid (in substance) even under local oracles.

Let $d$ denote the data dimension and $n$ the number of training samples in local polynomial interpolation-based gradient descent (LPI-GD). Existing guarantees assume…

Let $p\geq 2$, and consider the class of $p$th-order algorithms for convex-concave minimax optimization defined in the paper. For the constructed hard instance, the paper proves an…

Consider high-order extragradient methods for convex-concave minimax optimization, with complexity measured by the number of high-order oracle calls needed to find an $\epsilon$-so…

Let $f({\bm{x}},{\bm{y}})$ be a convex-concave function with $L_p$-Lipschitz continuous $p$th-order derivatives, and let $\epsilon>0$ be the target accuracy. The authors' upper bou…

Mixed-integer oracle lower-bound conjecture. Under these assumptions, at least

Consider convex-concave minimax problems measured by an accuracy parameter $\epsilon$, with convergence rate expressed in oracle complexity up to polylogarithmic factors. Optimal-r…

Let $\mathcal{J}$ be a finite index set, and for each $j\in\mathcal{J}$ let $f_j$ be a convex $(M_j,v_j)$-Hölder smooth function. Let $x_0$ be the initial point, $x^*$ an optimizer…

The paper studies nonconvex, nonsmooth optimization through oracle complexity. In its smoothing results, the algorithm is assumed to be translation invariant with respect to consta…

Optimal-complexity conjecture. For small values of $\epsilon$, textsc{vrpda} attains optimal oracle complexity for the problem class.

Non-tightness conjecture. For $cmu>0$, the best bound attainable by Theorem for the absolute inaccuracy performance measure is not tight, and a more refined approach is required to…

Optimality conjecture. Under these assumptions, the obtained bound for zeroth-order oracle calculations per node is optimal up to polylogarithmic factors among methods with an opti…

Let $d$ be the dimension, $\varepsilon$ the target accuracy, $K$ the number of queries per parallel round, and $\gamma$ the parameter appearing in the low-dimensional complexity re…

Let $d$ be the dimension, $\varepsilon$ the target accuracy, and $K$ the number of queries made in each parallel round for nonsmooth Lipschitz-continuous optimization over the…

Let $K$ be the convex set in the query model under discussion, and suppose that an optimization oracle is implemented using separation or membership queries when the algorithm know…

Let $f$ be a twice-differentiable, $\text{lambda}$-strongly convex function with $\text{mu}_1$-Lipschitz gradients and $\text{mu}_2$-Lipschitz Hessians, and let an algorithm access…

Consider finite-sum optimization algorithms using second-order information, and let the Newton method provide the reference for iteration cost. Cheap-iteration conjecture. The assu…

Consider finite-sum optimization with $n$ component functions in dimension $d$, and algorithms using adaptive index sampling, so that the sampled indices may depend on the individu…

Let $F(\mathbf{w})=\frac{1}{n}\sum_{i=1}^n f_i(\mathbf{w})$ be a finite-sum optimization objective, and let a randomized sketching method replace its Hessian by a low-rank approxim…