V-SHARE: Provable Convergence Guarantees for Advantage-based Algorithms

We establish convergence bounds for deep $V$-learning with horizon $H$. The algorithm fits a scalar value function to targets from executed transitions and selects actions using a predictive model and the value function. For current observed-successor targets with fresh true-kernel outcomes, the conditional mean is $\mathcal{T}^βV$, which averages over behavior-policy actions. The Bellman optimality update is $\mathcal{T} V$. We decompose the update error into six residuals: fitting, transition reuse, target construction, replay, action selection, and exploration. Under $L^s$ concentrability, their $L^p$ norms ($p=s/(s-1)$) control expected $L^1$ policy loss. The bound explicitly weights residuals from only the last $H-1$ update blocks, plus an initialization term for shorter runs. We quantify the cost of a shared sampling distribution across horizon levels. For statistical error bounds of order $n^{-ν}$, we derive optimal continuous allocations and an integer allocation whose objective is within a factor $2^ν$ of the constrained optimum. A margin condition with exponent $α$ gives action error of order $Λ^{1+α/p}$, where $Λ$ combines network drift and score error; a one-step construction proves the exponent sharp. Bounds on the distance between frozen and optimal scores transfer an optimal-gap condition to frozen-iterate gap bounds while retaining the mass of optimal ties. Survival probabilities and coverage conditions at deployment yield bounds for policies selected with approximate scores. Separate spatial ReLU networks per horizon level give a conditional neural regression rate, and the finite-state case gives a log-free expected fit rate. These results give expected policy-loss consistency for the fixed-horizon generative-reset approximate-ERM procedure with exact action scores and provide an explicit residual-decay criterion for FIFO/interleaved SGD.

Action values are foundational to many control algorithms such as Q-learning. Therefore, efficient action-value learning is central to reinforcement learning (RL). However, learning them can be slow, requiring many updates to move values from their initialization, typically near zero, to their true values, which may be far from zero. Moreover, action-value learning algorithms typically update each state-action pair independently, without learning a value that is common to all actions within a state. In this paper, we address these inefficiencies by introducing the mean-expansion layer, which accelerates action-value learning by sharing values across actions within a state and by changing the problem from directly learning potentially large action-values to learning a lower-norm representation of them. In deep RL, this layer can be applied as a parameter-free addition to Q-network architectures without altering the underlying algorithm. Applied to deep Q-networks and implicit quantile networks, it improves aggregate performance across 57 Atari 2600 games while increasing action gaps and dramatically reducing value overestimation.

In reinforcement learning, critics typically estimate absolute state values $V(s)$, estimating how good a particular situation is in isolation. However, it turns out that only differences in value are relevant for control. Motivated by this, we propose Relative Value Learning (RV), a framework that learns value differences directly via an antisymmetric function $Δ(s_i, s_j) = V(s_i) - V(s_j)$. We introduce a pairwise Bellman operator and prove it is a $γ$-contraction with a unique fixed point equal to the true value differences, derive well-posed $1$-step, $n$-step and $λ$-return targets and reconstruct generalized advantage estimation from pairwise differences to obtain an unbiased policy-gradient estimator (R-GAE). Beyond theoretical results, we integrate RV with PPO and achieve competitive performance on the Atari benchmark (49 ALE games) compared to standard PPO, indicating that relative value estimation is an effective alternative to absolute critics.