79 problems

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.

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…

Fixed-step-size convergence conjecture. The algorithm should converge with rate

Let $V_{\rho}$ be the invariant set and let $W_{\rho}$ be the local invariant manifold appearing in the construction, with coordinates $(z,v,u,y_1,y_{2:q})$. The $v$-axes are the d…

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…

Silver-schedule optimality conjecture. The silver stepsize schedule is exactly optimal for this performance metric among stepsize schedules in the strongly convex setting.

Global convergence conjecture. The property $\mathcal{E}(g)$ holds when $d_w>1$.

Let a recurrent neural network have loss landscape and associated gradient descent (GD) map. Non-singularity conjecture. The GD map is non-singular for the loss landscape of a recu…

Let GD denote gradient descent, and let the solution manifold be the set of global minimisers of the overparametrised least-squares problem. A tubular neighbourhood is a neighbourh…

Let $\boldsymbol{\theta}^\ast$ be the planted parameter, let $\mathcal{L}_n$ be the loss associated with the Wirtinger-flow objective … and consider the proportional regime $n=Cd$.…

In the setting of quadratically regularized optimal transport, let the support of the optimal coupling be partitioned into sections, and let $d$ denote the underlying dimension and…

Positive growth-rate conjecture. Positive growth-rate conditions on the stepsizes that allow $L_k\alpha_k>1$ should exist.

Let $N$ denote the iteration count, let $L$ be the smoothness constant, and consider inexact gradient descent for minimizing $f(x_N)-f(x_*)$ under $\|x_0-x_*\|\leq R$. Let…

Let $N$ denote the iteration count and let the performance criterion be the minimum squared gradient norm, $\min_{k\in\{1,\ldots,N\}}\|\nabla f(x_k)\|^2$, for minimizing a smooth c…

Let the data lie on the unit sphere, and let $\eta = \gamma/\lambda$ be the stability-threshold step size, where $\gamma$ is the step-size parameter and $\lambda$ is the relevant s…

Consider the gradient descent scheme with convex–concave splitting for the Allen–Cahn equation, and its energy-dissipation estimate, which includes the energy decrease and a neglec…

Geometric lower-semicontinuity conjecture. A geometric condition on $\omega(u_0;r)$ could be used to improve the Hölder exponent in the estimate for the displacement of the iterate…

Intermediate-regime tightness conjecture. The intermediate regime in Theorem 1 is tight. Furthermore, the corresponding worst-case function is bivariate.

Consider the weak signal regime for gradient descent in leaky ReLU two-layer neural networks, and introduce noise to the binary-classification labels. Noise-robustness conjecture.…

Let $sigma_\alpha$ be the activation functions used in the neural network, and let $sigma'_\alpha$ and $sigma”_\alpha$ denote their first and second derivatives. Theorem establishe…

The neural network has layers indexed by $alpha\in[1:L]$, with activation functions $sigma_\alpha$. Theorem establishes the stated gradient descent dynamics under the paper's smoot…

Grimmer's ratio-property conjecture. There is a set of multipliers $\{\lambda_{i,j}\}_{i,j\in I_N^*}$ satisfying this ratio property whenever $i,i'<j-1,j'-1$, and this set proves T…

Let $N$ be the number of gradient descent iterations, let $\bar\mu$ and $L$ denote the function parameters, let $\gamma^*(N,\mu,L)$ be the optimal stepsize, and let $I_N^*$ be the…

Basic g-composable schedule conjecture. For each $n$, every minimax-optimal stepsize schedule solving