39 problems

Adaptive minimax optimality conjecture. The convergence rate of the SOLIT rule should be adaptively minimax optimal over the whole range of smoothness classes specified by the inde…

Let $n$ be the sample size, let $d$ be the dimension of the covariate, and let $s$ be the smoothness parameter. For mean integrated square error (MISE), consider estimation over an…

Minimax volume conjecture. The minimax volume satisfies, up to a constant factor depending only on $n$,

Minimax-rate conjecture. This sub-Exponential rate of convergence should be minimax optimal up to constants that may depend on the parameters of the mixture model.

Consider the nonparametric density demixing model with $K$ component densities, sample-size parameters $N$ and $n$, and a plug-in estimator of the mixing matrix $\Pi$. Plug-in erro…

Classification-rate-optimality conjecture. The bounds in and remain rate-optimal, including for values of $p$ and $q$ other than $\infty$, analogously to the regression setting.

Combinatorial-term conjecture. The full minimax rate should include this combinatorial term as an additive contribution, in addition to the smoothness term.

Let $\hat{f}$ be a local nonparametric estimator for a regression function, and suppose that it exhibits optimal global sup-norm convergence. For an adversarial perturbation set…

Conjecture. The lower bound in Theorem $$ is not tight.

One-sided estimation lower-bound conjecture. There exist constants $c_0,c,r>0$ such that, for every $s,n\in\mathbb{N}$ satisfying

Let $n$ denote the sample size, and let the convex/concave component be estimated under the $L_{2}(P)$ norm, where $P$ is the relevant covariate distribution. Minimax lower-bound c…

Donoho's conjecture. A different approach is necessary to develop a sequence of non-asymptotic minimax lower bounds that converges to the correct asymptotic limit.

Let $N=(n_1,\dots,n_n)$ be a multinomial vector with parameter $n$ and probabilities $(n^{-1},\dots,n^{-1})$. For each $C>0$, let $c>0$ be a constant independent of $n$ such that t…

Sparsity-dependent lower-bound conjecture. A sharper minimax lower bound should be

Optimality conjecture. The results obtained for general $K^*$, despite being seemingly unsatisfactory, are conjectured to be optimal.

Signal-to-noise gap conjecture. This gap is caused by a loose upper bound, which in turn is caused by insufficiently sharp control of certain cross terms.

Unconditional lower-bound conjecture. Theorem 1 should remain valid without this assumption. The assumption is used to obtain lower bounds for adaptive estimators based on reasonab…

Rate-optimal procedure conjecture. In the general setting, a rate-optimal estimator does not belong to the family

Generalization conjecture. The lower bound can also be generalized to all exponential families.

Let $d$ be the covariate dimension, let $\beta$ and $\bar{\alpha}$ denote the smoothness parameters for the regression function and propensity score, respectively, and let $\gamma$…

Minimax lower-bound conjecture. The lower-bound conclusion stated for the regime $M/\sqrt N\gtrsim 1$ should also hold when $M/\sqrt N\to 0$; in particular, the minimax estimation…

Span-dependent plug-in lower-bound conjecture. There is a Markov reward process for which this bound holds. The claim would show that the corresponding upper bound is sharp up to a…

Let $cM_{cmathsf{var}}(csigvalbd,csigrewbd)$, $cM_{cmathsf{vfun}}(cvalbd,csigrewbd)$, and $cM_{cmathsf{rew}}(crmax,csigrewbd)$ denote the classes of Markov reward processes with, r…

In a periodic independent, identically distributed model, consider quickest change detection under Lorden's minimax criterion, and allow stopping algorithms whose thresholds vary p…

CCK's minimax optimality conjecture. The rate $n^{-1/6}$ is nearly optimal in a minimax sense when $d_n$ is extremely larger than $n$. This conjecture concerns the dependence on…