AlphaRJM is named for the component that matters less in its own ablation. Appendix E.1 reports that removing the Ornstein-Uhlenbeck return critic and restoring a quantile critic "produces the largest decline in both IC and ICIR." On CSI300, seed 0, IC falls by 0.0118, from 0.0480 to 0.0362. Removing Reward-Jump Memory costs 0.0046, taking IC from 0.0480 to 0.0434. Dropping the energy-distance loss costs 0.0027. On CSI300, seed 0, the critic contributes roughly two and a half times as much as the memory.
The title follows the smaller effect.
The abstract says the ablations "confirm the complementary roles" of all three pieces. They may well be complementary on the loss surface. The naming remains hard to defend. Across the one universe and one seed used for the ablation, the method, title and abstract's opening clause foreground the second-place component.
What AlphaRJM builds
Formulaic alpha discovery in the AlphaGen line frames a symbolic expression as a sequence of token decisions. Payment arrives only after the expression is complete and evaluated. The reward measures the marginal improvement in training-set ensemble IC: r = Q(P_next) minus Q(P_current). The vocabulary contains 62 selectable actions, each expression can reach at most 20 tokens, and the retained pool holds 50 formulas.
AlphaRJM adds two pieces. Reward-Jump Memory is an 8-dimensional latent vector placed beside the encoded partial expression, a 32-d GRU output, and the 32-d permutation-invariant pool encoding. It remains unchanged while tokens are being selected. A terminal event updates it by J = 0.02 times a gated, signed vector. The realized reward and a one-hot outcome drive that jump: accepted, replaced, rejected, or invalid.
The rationale is sensible. Two search paths can reach an identical retained pool after very different histories of rejected and invalid expressions. A policy conditioned only on the pool loses that history.
The second addition is the critic. Every possible next token receives an 88-dimensional conditioning vector built from the expression, pool, memory and action embedding. The critic maps that vector to four scalars: a starting value, long-run level, mean-reversion rate and diffusion scale. Together they parameterise a scalar mean-reverting SDE over an internal coordinate running from 0 to 1.
At the default operating point, the paper integrates the process using 4 Euler-Maruyama steps. It generates 16 particles under the CSI300/CSI500 preset and 32 on CSI800 and CSI1000. An action's score is the particle mean plus an annealed coefficient multiplied by the particle standard deviation. The authors state that this "is used solely as an internal stochastic return critic for symbolic search rather than as a model of asset prices or alpha signals."
The data cover CSI300, CSI500 and CSI800, with CSI1000 reported in an appendix. Daily Qlib data are split into training from 2010 through 2020, validation on 2021, and testing from 2022 through 2024. The target is the 20-day forward close-to-close return, alongside a 42-day variant. Every run is CPU-only on a 16 GB Apple M4 Mac mini.
Closed-form moments, sampled anyway
Appendix F proves that the critic follows a linear OU diffusion. Its conditional law is Gaussian at every positive time, and Equations 58 and 59 give the mean and variance. Corollary F.3 then provides the exact mean and variance for the K-step Euler-Maruyama particles.
The paper therefore derives closed-form expressions for both statistics used in the action score, then estimates them by Monte Carlo with 16 draws at the default operating point. The chosen form has a stated purpose. The authors call it "an inexpensive sampleable return law with separately controlled location, reversion, and stochastic dispersion." Yet they run no comparison with the closed-form moments already derived.
The sensitivity results deepen the puzzle. On CSI300 with seed 0, the authors vary one setting at a time across 8, 16 and 32 particles. Their finding is that "fewer particles generally yield stronger correlation metrics." Better Monte Carlo precision for a known quantity leaves the search worse off.
Particle noise may be serving as exploration jitter. Under that reading, it should be presented and tuned as exploration instead of distributional return modelling. The CSI300 seed-0 sweep also shows relatively little sensitivity to the integration-step count of 2, 4 and 8, which supports the same interpretation.
Do the averages hold up?
At h=20, AlphaRJM records the best mean in 10 of 12 dataset-metric cells. The individual cells are less persuasive.
On CSI300, AlphaRJM reports IC of 0.042±0.007, while AlphaGen reaches 0.041±0.011. Both have RIC of 0.046, although AlphaRJM has the tighter spread, 0.003 vs 0.021. Across four seeds, this is a tie presented as a win, and the paper reports no significance test.
CSI500 shows AlphaRJM at IC 0.046±0.014 against AlphaSAGE at 0.041±0.007. The 0.005 difference remains well within one standard deviation. CSI500 ICIR averages 0.379 with a standard deviation of 0.159. Forty-two percent of the mean. Both methods post RIC of 0.059.
CSI800 is where the claim of "strong and stable gains" parts company with the table. AlphaRJM leads on IC, at 0.036 against the nearest rival, AlphaQCM at 0.029. It also leads on ICIR, with 0.339 against LightGBM at 0.305. AlphaSAGE wins RIC, 0.045 against 0.043, and RICIR, 0.396 against 0.371.
On CSI1000, using two seeds, AlphaRJM has the highest IC at 0.076 and a much larger RIC at 0.102, versus AlphaGen's 0.089. Its ICIR is 0.505, the second-lowest figure in the table. Only the MLP is lower at 0.464, while AlphaQCM reaches 0.563 and LightGBM 0.556. Among the four formula-discovery methods, AlphaRJM has the lowest ICIR, trailing AlphaGen's 0.513 and AlphaSAGE's 0.517. The signal has a larger mean and more variation through time than its direct rivals, evidence that cuts against the stability claim.
The h=42 results use random seed 0. Table 4, which contains the ablation, also uses CSI300 with random seed 0. The sensitivity sweep again uses CSI300 with seed 0. For the long-horizon wealth curve, the authors acknowledge the limitation and write that it "is presented as a complementary illustration rather than as a separate statistical comparison." The h=42 IC and RIC figures and Table 4 warrant the same caveat.
At h=42 on CSI300, a plain MLP ties AlphaRJM at IC 0.046. It beats AlphaGen at 0.021, AlphaQCM at 0.045 and AlphaSAGE at 0.025. A tabular baseline matching the symbolic search machinery at the long horizon receives too little attention.
Training budgets also differ. Each of the three episode-based methods receives 10,000 terminal formula episodes. AlphaGen receives 51,200 executed primitive PPO actions because the authors retain its native training unit instead of rewriting it, and they disclose that choice. The budgets are unmatched. AlphaGen's near-tie on CSI300 carries that asterisk, whichever direction one believes the mismatch cuts.
Gross wealth curves
Figures 1 and 4 provide the paper's only portfolio evidence, and both are gross. Daily contribution is defined as R_d = (1/h) times the mean h-day forward return among the top 20% of valid stocks ranked by predicted score. The calculation uses overlapping windows, rescales them by 1/h, and compounds the result. What emerges is a smoothed view of forecast quality.
No terminal wealth figure or drawdown is printed. Transaction costs are nowhere stated. The portfolio reselects the top 20% of the universe on every trading date, while the paper gives no turnover, cost or capacity estimate. IC of 0.046 with ICIR of 0.379 is a genuine forecast. The paper leaves unanswered whether it survives costs at that reselection frequency, and I would not guess.
Index membership treatment is also unspecified. Current constituents would make the reported IC levels optimistic compared with point-in-time membership; the text does not tell us which was used.
Testing covers 2022 through 2024 in China, a single and largely down regime. There is no sub-period split. Hyperparameters come in two presets, one covering CSI300/CSI500 and another covering CSI800/CSI1000. The sensitivity grid was searched on CSI300 seed 0, so the cross-universe comparison does not use one fixed configuration.
A US test of the mechanism
We are running a version on liquid US-listed equities. This universe differs from every market tested in the paper, which means the figures above do not transfer and our run will not constitute a replication. The paper specifies the splits, the 10,000-episode budget, and the pool insertion and reweighting rules in Appendix B.2. It does not provide the exact grammar or feature-token set. Those are described only as "the same expression grammar family as the controlled formulaic-alpha benchmark," so we would reconstruct them from the released reference implementations.
The mechanism itself can still be tested. Does event-driven memory for accepted, rejected and invalid outcomes improve pool-dependent symbolic search once a pool encoding already exists? Does the OU critic beat a quantile critic under an equal search budget? And can it beat its own analytic mean and variance after the particles are switched off?
Table 4 answers the first question on one universe and one seed, CSI300 with random seed 0. The paper never runs the second comparison. If a particle-free analytic critic matches the sampled version, the SDE machinery reduces to a mean-plus-sigma bonus with a learned reversion parameter. The ablation's largest effect would then cease to support the stochastic return modelling story. A failure to match would identify noise as the operative mechanism, which deserves to be stated plainly.
My verdict would change if CSI500 ICIR dispersion fell from 0.159 around a mean of 0.379 under a paired test with more seeds.
Cheap hardware counts in the paper's favour. A search system that completes on a Mac mini gives a small desk room to iterate, and that matters more than an IC difference of 0.001 across four seeds. The part worth carrying forward is the one identified by the ablation: an action-conditioned uncertainty estimate over discovery returns.