742 problems

Let $c$ be the teacher model's coefficient vector, with $c$ sampled in the paper's model from $\{\pm 1/\sqrt{k}\}^k$. Worst-case fine-tuning tractability conjecture. Fine-tuning sh…

Roots-of-unity cycle conjecture. For any stationary first-order method, if there exists a cyclic trajectory, then there exists a roots-of-unity two-dimensional cycle on the functio…

Let $n\geq6$ be an even integer, and let $P$ be an optimal small $n$-gon. Its skeleton is the graph consisting of the vertices of $P$ together with all line segments joining pairs…

Let $n$ be a positive integer, $A_1,dots, A_n$ be symmetric positive semidefinite matrices, and let $\lVert\,\cdot\,\rVert$ be the spectral norm. For $m\leq n$, compare the average…

Let $S\in\mathcal{M}_d(\mathbb{C})^+$ and let $\mathbf{a}=(a_i)_{i\in\mathbb{I}_k}\in(\mathbb{R}_{>0}^k)^\downarrow$. Define … and, for a strictly convex unitarily invariant norm…

Let $f:\mathbb{R}^m\rightarrow\mathbb{R}$ be a $C^1$ function that is $C^2$ near its critical points, and let $\{z_n\}$ be the sequence generated by the Backtracking GD method from…

Silver-rate optimality conjecture. The asymptotic rate achieved by the silver stepsize schedule is optimal among all non-adaptive stepsize schedules.

Consider the opinion-dynamics system given by equations … , in which the personal conviction values $u_i$ evolve according to … with $a_i\geq 0$ for $i=1,\ldots,N$, and where…

For minimizing convex quadratics with gradient descent using stepsize schedules, the silver convergence rate is $O(1/n^{\log_2 (1+\sqrt{2})})$. Silver-rate optimality conjecture. T…

Scaling-law conjecture. There are more general scaling laws of the critical batch size in relation to $\eta$.

Let $f$ be a quadratic function, and consider the covariance matrix maintained by CMA-ES. Hansen's covariance convergence conjecture. The covariance matrix of CMA-ES should converg…

Ni's conjecture. In the one-dimensional case, the supremum of the total population over all $d>0$ and $m\in\mathcal{M}_1$ is $3$.

Let $p\in(0,(3-\sqrt{5})/2)$ denote the prevalence, and let $N_{opt}^{(S)}$ be the rounded optimal configuration for the Sterrett group-testing scheme. Malinovsky–Albert conjecture…

Let $\bar{L}$ be a continuously differentiable, smooth, and strongly convex function, and consider constant regressors. Let $\{\theta_k\}_{k=0}^{\infty}$ denote the iterates genera…

Normalize the chessboard to the unit square $U=[0,1]^2$. A configuration consists of white and black queen regions, with no queen of one color attacking a queen of the opposite col…

An almost Hadamard matrix in the complex sense is the notion introduced in the cited work, while a complex Hadamard matrix is a unitary matrix whose entries have equal modulus afte…

Tilt-stability conjecture. Under appropriate assumptions, tilt stability of a local minimizer should exclude the existence of critical multipliers associated with that minimizer.

Quadratic worst-case conjecture. The function $f_2({\bm{x}})$ is a worst-case function for the secondary sequence of OGM1$'$ and OGM2$'$, so the displayed lower bound is the tight…

The experiments concern the semi-smooth Newton method applied to positively constrained convex quadratic programming problems whose matrix $Q\in\mathbb R^{n\times n}$ does not sati…

Let $0<\mu<L$, let $p\in\mathbb{N}$, and let $\mathcal{A}$ be a consistent $p$-SCLI optimization algorithm with diagonal inversion matrix and linear coefficient matrices. Let…

An allocation distributes storage across $n$ nodes, and its recovery delay is denoted by $D$; the parameters are the node count $n$, request rate $\lambda$, and threshold $T$. An a…

Three-dimensional covering-density conjecture.

Let $w_{ij}$, $1\leq i<j\leq n$, be arbitrary real coefficients, and let $\xi_1,\ldots,\xi_n$ be independent random signs with…

Nonexistence conjecture for universally optimal 24-point codes. There is no universally optimal spherical code of 24 points in $S^3$.

Caputo–Carlen–Lieb–Loss conjecture. The maximum of the permanent of $A$ is attained either by $I$, with permanent $1$, or by the matrix $n^{-1/p}J$, with permanent $n!/n^{n/p}$. Eq…