Zihan Zhang

Runlong Zhou, Zihan Zhang, Maryam Fazel et al.

We study horizon-free regret minimization for finite-horizon time-homogeneous tabular Markov decision processes with $S$ states, $A$ actions, horizon $H$, and per-trajectory total reward bounded by $1$. We propose a new algorithm and prove a regret upper bound \[\tilde O(\sqrt{SAK}+S^8A^3)\] with failure probability $δ$, where $K$ is the number of episodes and $\tilde O(\cdot)$ hides $\mathsf{poly}\log(S,A,K,1/δ)$. Thus, the regret is $H$-free and asymptotically optimal, matching the contextual-bandit lower bound $Ω(\sqrt{SAK})$ up to logarithmic factors. This completely removes the $\log H$ dependence from the previous $\tilde O(\sqrt{SAK\log H}+S^2A\log H)$ guarantee of Zhang et al. (2021), and drastically improves the prior best horizon-free regret $\tilde O(\sqrt{S^9A^3K})$ of Zhang et al. (2022) asymptotically. The main technical difficulty is that the optimal value functions $\{V_h^*\}_{h=1}^H$ are time-inhomogeneous even though the transition kernel is time-homogeneous. A direct union bound over all value functions typically incurs an additional $\min\{\log H,S\}$ factor. We avoid this factor by (i) exploiting the monotonicity of $V_h^*$ in $h$ and (ii) non-trivially projecting the value functions onto an $S$-dimensional grid. Our analysis relies on three additional ingredients. First, we introduce a horizon-truncation argument that enables reward-based exploration and removes the cost of a separate reward-free exploration phase. Second, we design a cutting bonus that preserves both optimism and the monotonicity needed for planning. Third, we prove a new bound on total deviation for time-homogeneous MDPs, which controls the clipped variance terms in the cutting bonus with adjustable polynomial dependence on $S$ and without any dependence on $H$. Together, these tools yield an asymptotically optimal horizon-free regret guarantee.

Joshua Brakensiek, Yeyuan Chen, Aaron Putterman et al.

In coding theory, list recoverability is a fundamental concept which robustly captures how ``spread-out'' codewords are in a code. More formally, given a code $C \subseteq Σ^n$ and input lists $S_1, \dots, S_n \subseteq Σ$ of size at most $\ell$, list recoverability requires that there are at most $L$ codewords $c \in C$ such that $c_i \in S_i$ for at least $(1-ρ)n$ choices of $i \in [n]$. List recovery is an important question which has found applications in many areas, including complexity theory, property testing, compressed sensing, streaming algorithms, and cryptography. As our first main result, we establish a tight ``generalized singleton bound''. Formally, we show that for constant $\ell, L,ρ$ and sufficiently large alphabets $Σ$, if we define $R^*=\frac{L+1-\ell}{L}-\frac{L+1}{L}ρ$, it is possible for a $(ρ,\ell,L)$ list-recoverable code to have rate $R^*-ε$ but impossible to have rate $R^*+ε$. One direction of our result already directly generalizes and improves a weaker impossibility result due to Goldberg, Shangguan, and Tamo. For our second main result, we prove that there is a fundamental shortcoming in existing methods that aim to construct explicit, optimal list-recoverable codes. Indeed, recent work has constructed explicit codes achieving list-decoding capacity (along with other related properties) using a framework introduced in the work of Alon--Edmonds--Luby (AEL). We give a meta-analysis of such constructions by presenting an ``AEL framework'' which captures all such recent constructions in the literature. Within this framework, we show that no AEL-based code can break a recently-identified list-recovery barrier for additive and linear codes.