REVIEW 1 major objections 5 minor 137 references
Reinforcement learning can work without absolute state values.
Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →
T0 review · deepseek-v4-flash
2026-08-01 08:26 UTC pith:AGIJK4UI
load-bearing objection Correct theory, honest experiments, but the headline 'competitive' claim rests on baseline numbers that appear to be quoted, not re-run. the 1 major comments →
Relative Value Learning
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper's central discovery is that the additive offset in a value function is unneeded gauge freedom, so the critic's job can be redefined as learning pairwise differences rather than absolute magnitudes. Formally, the pairwise Bellman operator (T^πΔ)(s_i,s_j)=r^π(s_i)-r^π(s_j)+γ E[Δ(s'_i,s'_j)] is a γ-contraction on the space of bounded antisymmetric functions, and its unique fixed point equals Vπ(s_i)−Vπ(s_j). The paper derives bootstrapping targets (1-step, n-step, λ-return) that remain well-posed when successor states are terminal, and it reconstructs generalized advantage estimation from these pairwise differences. When the learned model equals the true difference function, the resul
What carries the argument
The load-bearing object is the pairwise Bellman operator T^π, defined on the Banach space of bounded antisymmetric pairwise functions by (T^πΔ)(s_i,s_j)=r^π(s_i)−r^π(s_j)+γ E[Δ(s'_i,s'_j)]. Because the reward-difference term cancels when two functions are compared, T^π contracts with factor γ and has a unique fixed point equal to the true value differences Vπ(s_i)−Vπ(s_j). Around this operator the paper builds relative TD residuals, 1-step/n-step/λ targets with explicit terminal-state case handling, and the R-GAE estimator; a trajectory-ranking step estimates start-state offsets to reduce the variance introduced by anchoring each rollout's relative values to zero.
Load-bearing premise
The guarantees assume the learned pairwise model Δθ matches the true value differences; since the paper's difference head is a linear function of the encoder and the paper admits this near-linear head may underfit complex value differences, a poor fit would make R-GAE biased and weaken the variance-reduction argument.
What would settle it
On a small tabular MDP with known Vπ, iterate the pairwise Bellman operator numerically and compare its fixed point to Vπ(s_i)−Vπ(s_j) for every pair; then estimate the policy gradient with R-GAE and compare with the true gradient. Any deviation beyond sampling noise would falsify the contraction and unbiasedness claims.
If this is right
- The offset of an absolute critic is a redundant degree of freedom; RV removes it by construction, so reward-shaping shifts and baseline offsets cannot move the learned critic's predictions.
- GAE can be computed without knowing the absolute value of any state, and when the difference model is exact the policy-gradient estimator is unbiased.
- Bootstrapping remains well-defined through terminal states using only observable rewards and nonterminal pairwise differences, so relative value learning is compatible with standard actor-critic training.
- Relative value learning can replace an absolute critic in PPO with no loss of benchmark performance, beating the absolute critic on 30 of 49 Atari games.
- Trajectory ranking provides a data-dependent initialization that reduces the variance inflation from the per-trajectory constant baseline.
Where Pith is reading between the lines
- The pairwise target makes comparison-only feedback natural: in preference-based or human-in-the-loop RL, where the absolute reward scale is unidentifiable but pairwise comparisons remain well-defined, RV's loss is a plausible direct objective. The paper motivates this direction but does not test it.
- The trajectory-ranking offset is a batch-level gauge anchor; one could learn these offsets end-to-end rather than with row-mean-minus-min, which would directly optimize away the residual B_t variance rather than reduce it heuristically.
- The same contraction argument should extend to Q-learning-style relative critics or relative value iteration in off-policy settings, but pairwise state sampling is O(B²), so a structured or learned sampler would be needed to make that practical.
- The linear difference head is the most likely capacity bottleneck; an antisymmetric nonlinear head could recover bias on games where RV trails PPO, a design choice the paper leaves open.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes Relative Value Learning (RV), a critic that directly learns antisymmetric pairwise state-value differences Δ_θ(s_i,s_j) ≈ V^π(s_i) − V^π(s_j) instead of absolute values. It defines a pairwise Bellman operator T^π on bounded antisymmetric functions and proves it is a γ-contraction whose unique fixed point is the true value difference (Thm 3.1). It derives one-step, n-step, and λ-return targets that handle terminal successors using only observable rewards and pairwise differences (Eqs. 19–22), and reconstructs GAE from pairwise differences (R-GAE), showing that, when Δ_θ = Δ^π, the relative advantage differs from standard GAE by a trajectory-constant term that cancels in the policy gradient (Lemma 3.2, Cor. 3.3). To reduce the variance induced by that constant, the paper introduces a trajectory-ranking offset estimator (Sec. 4). Experiments replace the PPO critic with the RV critic on 49 Atari games and report that PPO+RV is competitive with PPO and DAE (Table 1, Fig. 3). The paper also includes ablations of the offset initialization (Fig. 4) and pair sampling (Table 4), hyperparameters, and a limitations section.
Significance. The theoretical core is a genuine contribution: Thm 3.1 is clean and correct; Eq. 20's four terminal cases are exact under the convention that terminal states have zero value; and Lemma 3.2/Cor. 3.3 establish that R-GAE is unbiased in the exact-model limit. The derivations are self-contained and checkable by hand, the code is released, and the Limitations section is candid about the restrictiveness of the linear difference head, the O(B^2) pairing cost, and the uninformative-ranking risk. If the empirical comparison were confirmed under a controlled protocol, the paper would show that a difference-based critic is a viable alternative to absolute critics with no practical penalty, which is of clear interest to the RL community. However, the headline empirical claim currently rests on baseline numbers whose provenance and evaluation protocol are not documented, so the practical significance is not yet established.
major comments (1)
- [§6.1, Table 1; §6 Compute Resources] The headline claim that PPO+RV 'attains competitive performance' and 'exceeds PPO on 30 out of 49 games' is not supported by the evidence as presented. The paper never states whether the PPO and DAE columns were produced in the same codebase with the same environment wrappers, no-op reset behavior, frame skip, evaluation window, and seeds as PPO+RV. The Compute Resources paragraph reports 490 runs (49 games × 10 seeds), which accounts only for PPO+RV; no compute is reported for the baseline columns. If those numbers are quoted from prior papers, differences in preprocessing and evaluation can easily exceed the reported standard deviations, in which case 30/49 wins is close to chance and the bootstrap intervals in Fig. 3 are not a valid comparison. Please re-run the baselines in the same pipeline or clearly state their provenance and protocol, and qualify the competitive claim accordingly
minor comments (5)
- [Cor. 3.3 proof] The proof writes B_t = C(1−(γλ)^{T−t+1})/(1−γλ), omitting the (1−γ) factor present in Lemma 3.2's definition B_t = (1−γ)C Σ_{l=0}^{T−t}(γλ)^l. The conclusion is unaffected because any t-dependent constant vanishes in the score-function identity, but the displayed formula should match Lemma 3.2.
- [Appendix A, Eqs. (32)–(34)] The derivations use '≈' where the equalities are in fact exact under the standard convention V(s_terminal)=0 (e.g., V(s_i)=r_i when s_{i+1} is terminal). Replacing ≈ with = would avoid the impression that the one-step targets are approximate.
- [Fig. 4 / §6.2] The ablation for value initialization is shown only as per-game curves with no aggregate statistic. A table or bootstrap confidence intervals over the 49 games would substantiate the claim that 'relative ranking is needed'.
- [Eq. (28) / Table 3] The critic loss is written as plain MSE, but Table 3 lists 'RV value clipping 0.15'. Please clarify how value clipping enters the critic update and how it interacts with the n-step targets.
- [Abstract / Cor. 3.3] The abstract says R-GAE is 'an unbiased policy-gradient estimator' without the qualifier that this holds when Δ_θ = Δ^π, as assumed in Cor. 3.3. Adding that qualifier would prevent overstatement, especially since the linear difference head in §5.1 may underfit complex value differences (as acknowledged in §7).
Circularity Check
No significant circularity: the theoretical derivations are self-contained and the central claims do not reduce to fitted inputs or self-citations.
full rationale
The paper's load-bearing theory is derived, not assumed. Theorem 3.1 defines the pairwise Bellman operator T^pi from reward differences and the transition kernel, proves gamma-contraction on the sup-norm space of bounded antisymmetric functions, and identifies the unique fixed point with V^pi(s_i)-V^pi(s_j) by subtracting the Bellman equations; uniqueness follows from the Banach fixed-point theorem, so the result is a proof rather than a definitional restatement. Lemma 3.2 and Corollary 3.3 are algebraic telescoping arguments plus the standard score-function identity; the only idealization is the explicitly stated assumption Delta_theta = Delta^pi, which is a consistency condition, not a fitted parameter renamed as a prediction. The 1-step/n-step/lambda targets are TD-style bootstraps with terminal-case corrections expressed in observable rewards and non-terminal pairwise terms, so they do not collapse into the predicted quantity by construction. Section 4.1's trajectory-ranking offset is estimated from the learned model itself, but it is presented as a heuristic initialization, and the Limitations section candidly acknowledges the risks of the near-linear head and uninformative ranking; this affects robustness, not circularity. The empirical comparison with PPO/DAE may be open to methodological criticism because baseline provenance is not stated, but that is an experimental-validity concern, not a circularity concern. There are no load-bearing self-citations, no imported uniqueness claims from the authors' prior work, and no ansatz smuggled in via citation. The derivations are self-contained against the stated assumptions, so the appropriate circularity score is 0.
Axiom & Free-Parameter Ledger
free parameters (5)
- n-step target horizon n =
5
- RV critic loss coefficient c_v =
1.25
- RV value clipping threshold =
0.15
- same-episode pairing probability p =
0.33
- PPO hyperparameter set (γ=0.99, λ=0.95, ϵ=0.1, minibatch 128, rollout T=128, LR 2.5e-4, entropy coef 0.01, epochs 5) =
as listed
axioms (7)
- domain assumption MDP with bounded rewards and γ ∈ [0,1) under a stochastic policy π
- domain assumption Terminal states have value zero (V(terminal) = 0)
- domain assumption Successor states s'_i ~ P^π(·|s_i) and s'_j ~ P^π(·|s_j) are drawn independently
- standard math Banach fixed-point theorem (contraction mapping principle)
- standard math Score-function identity E_{a~π}[∇ log π(a|s)] = 0
- standard math GAE (Schulman et al. 2015b) is the reference estimator being reconstructed
- domain assumption Telescoping Ṽ_θ(s_t) = Σ_{k<t} ∆_θ(s_{k+1}, s_k) equals V^π(s_t) − V^π(s_0) only if ∆_θ = ∆^π
read the original abstract
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 $\Delta(s_i, s_j) = V(s_i) - V(s_j)$. We introduce a pairwise Bellman operator and prove it is a $\gamma$-contraction with a unique fixed point equal to the true value differences, derive well-posed $1$-step, $n$-step and $\lambda$-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.
Figures
Reference graph
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