Jian Ding

Guanyi Chen, Jian Ding, Shuyang Gong et al.

Detection of correlation in a pair of random graphs is a fundamental statistical and computational problem that has been extensively studied in recent years. In this work, we consider a pair of correlated (sparse) stochastic block models $\mathcal{S}(n,\tfracλ{n};k,ε;s)$ that are subsampled from a common parent stochastic block model $\mathcal S(n,\tfracλ{n};k,ε)$ with $k=O(1)$ symmetric communities, average degree $λ=O(1)$, divergence parameter $ε$, and subsampling probability $s$. For the detection problem of distinguishing this model from a pair of independent Erdős-Rényi graphs with the same edge density $\mathcal{G}(n,\tfrac{λs}{n})$, we focus on tests based on \emph{low-degree polynomials} of the entries of the adjacency matrices, and we determine the threshold that separates the easy and hard regimes. More precisely, we show that this class of tests can distinguish these two models if and only if $s> \min \{ \sqrtα, \frac{1}{λε^2} \}$, where $α\approx 0.338$ is the Otter's constant and $\frac{1}{λε^2}$ is the Kesten-Stigum threshold. Combining a reduction argument in \cite{Li25+}, our hardness result also implies low-degree hardness for partial recovery and detection (to independent block models) when $s< \min \{ \sqrtα, \frac{1}{λε^2} \}$. Finally, our proof of low-degree hardness is based on a conditional variant of the low-degree likelihood calculation.

Ofer Dekel, Jian Ding, Tomer Koren et al.

We study a new class of online learning problems where each of the online algorithm's actions is assigned an adversarial value, and the loss of the algorithm at each step is a known and deterministic function of the values assigned to its recent actions. This class includes problems where the algorithm's loss is the minimum over the recent adversarial values, the maximum over the recent values, or a linear combination of the recent values. We analyze the minimax regret of this class of problems when the algorithm receives bandit feedback, and prove that when the minimum or maximum functions are used, the minimax regret is $\tilde Ω(T^{2/3})$ (so called hard online learning problems), and when a linear function is used, the minimax regret is $\tilde O(\sqrt{T})$ (so called easy learning problems). Previously, the only online learning problem that was known to be provably hard was the multi-armed bandit with switching costs.

Ofer Dekel, Jian Ding, Tomer Koren et al.

We study the adversarial multi-armed bandit problem in a setting where the player incurs a unit cost each time he switches actions. We prove that the player's $T$-round minimax regret in this setting is $\widetildeΘ(T^{2/3})$, thereby closing a fundamental gap in our understanding of learning with bandit feedback. In the corresponding full-information version of the problem, the minimax regret is known to grow at a much slower rate of $Θ(\sqrt{T})$. The difference between these two rates provides the \emph{first} indication that learning with bandit feedback can be significantly harder than learning with full-information feedback (previous results only showed a different dependence on the number of actions, but not on $T$.) In addition to characterizing the inherent difficulty of the multi-armed bandit problem with switching costs, our results also resolve several other open problems in online learning. One direct implication is that learning with bandit feedback against bounded-memory adaptive adversaries has a minimax regret of $\widetildeΘ(T^{2/3})$. Another implication is that the minimax regret of online learning in adversarial Markov decision processes (MDPs) is $\widetildeΘ(T^{2/3})$. The key to all of our results is a new randomized construction of a multi-scale random walk, which is of independent interest and likely to prove useful in additional settings.