{"id":"41e8ae0e-c4aa-44fa-81cf-ab28a9b4c679","arxiv_id":"2507.06528","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"InvestAlign fine-tunes LLMs on datasets generated from closed-form solutions of a simplified optimal investment problem, improving alignment with real investor decisions under herd behavior by 45-61% in MSE.","lead":"The paper proposes InvestAlign, a way to train financial LLM agents without large amounts of real user data, by using the mathematical solution of a simpler investing problem as teaching labels. The approach makes LLM agents match human investment decisions much better than the original models, including on harder problems with herd behavior.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The faster-convergence theorem (Eq. 28) rests on an unproven and likely false inequality (Eq. 27); smoothing a decreasing density with uniform noise can reduce rather than increase overlap with a decreasing model density.","rationale":"The reader's weakest_assumption targeted the empirical validity of the uniform-noise model and support nesting; those are indeed unvalidated. My concern is more specific and more damaging: even if every distributional assumption in Section 4.2 is granted, the derivation of Eq. (27) is a non-sequitur. A discrete counterexample satisfying the stated monotonicity and nesting conditions yields the opposite inequality, so the theoretical result (28) is not established. This is the single most load-bearing concern because it undermines a headline contribution (faster convergence), whereas the empirical alignment results in Section 5 remain plausible and are separately supported by the reported MSE reductions. The reader noted Eq. (27) was 'sketched' and the proof was 'substantially circular,' so we partially agree; however, the flaw is not merely a missing validation step but an invalid inference. The paper can be repaired by removing or weakening the convergence claim, or by adding the missing conditions and a correct proof, so I do not move the verdict from CONDITIONAL to REJECT. The empirical claims, if error bars and a real-user-SFT baseline are added, could support acceptance; the current verdict remains CONDITIONAL.","tokens_in":30519,"tokens_out":16834,"duration_ms":168044,"concrete_test":"Write a script that evaluates (24) and (25) numerically for f̂ = c/x^2 on [a,b], f̃ = f̂ convolved with U(−ε,ε), and a decreasing f_model (e.g., linear or exponential) on P1 ⊃ P̃, scanning a,b,ε; also evaluate the discrete counterexample above. If any instance yields Î ≥ Ĩ, Eq. (27) is refuted and the proof of (28) fails. Alternatively, re-derive the inequality symbolically to identify the missing condition (e.g., log-concavity or a reversed monotonicity) under which the claimed direction would hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.2 claims faster convergence on theory-generated SFT data because ||∇Lhat(w)|| > ||∇Ltilde(w)|| (Eq. 28). The proof reduces to comparing Î = ∫_{P̂} f̂ f_model and Ĩ = ∫_{P̃} f̃ f_model, where f̂ is the Pareto density (7), f̃ is its convolution with U(−ε,ε), and f_model is a decreasing model density. The paper asserts Î < Ĩ < 1 (Eq. 27) 'because f_model is monotonically decreasing.' This implication is not valid: smoothing a decreasing density with uniform noise can either increase or decrease its overlap with another decreasing density. Concrete counterexample: let P̂ have probabilities (0.8, 0.2) on {1,2}, add uniform noise over {0,1} so P̃ has probabilities (0.4, 0.5, 0.1) on {1,2,3}, and let f_model be decreasing with probabilities (0.5, 0.3, 0.2). Then Î = 0.8·0.5 + 0.2·0.3 = 0.46 and Ĩ = 0.4·0.5 + 0.5·0.3 + 0.1·0.2 = 0.37, so Î > Ĩ, giving ||∇Lhat|| < ||∇Ltilde||, the reverse of (28). Even granting the paper's uniform-noise and nesting assumptions, the faster-convergence conclusion does not follow and can fail. This is load-bearing because the abstract and Section 4.2 advertise 'faster parameter convergence' as a key advantage of InvestAlign.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes InvestAlign, a framework for data-scarce supervised fine-tuning (SFT) of LLMs in behavioral finance. Given a complex optimal investment problem, InvestAlign identifies a similar and simple problem (P3: absolute herd behavior with unilateral influence) whose closed-form theoretical solution (Eq. (11), from the authors' prior work) is used to generate synthetic SFT datasets, and fine-tunes base LLMs (GPT-3.5, Qwen-2, Llama-3.1, GLM-4) into 'InvestAgents.' The paper reports that (i) the theoretical P3 solution is statistically consistent with decisions collected from 119 human participants (Section 3.3, Appendix A.6); (ii) InvestAgents reduce overall MSE relative to pre-SFT LLMs by 44–61% on P3 and on the complex problems P1 (relative herd, unilateral influence) and P2 (absolute herd, mutual influence) (Table 1); and (iii) fine-tuning on theoretical data yields faster parameter convergence than fine-tuning on real-user data, supported by a gradient-norm inequality (Eq. (28)) and by experiments in Figures 3–4. The paper also includes ablations, FinGPT baselines, mixed-data robustness checks, and validation of economic hypotheses H1/H2.","tokens_in":30872,"tokens_out":19574,"duration_ms":192047,"significance":"The empirical alignment results, if they hold up, are valuable: they offer a concrete, reproducible route to LLM alignment in a domain where real-user data are costly and privacy-sensitive, and they demonstrate genuine transfer from a solvable proxy problem to harder problems. The paper ships public code, reports consistent MSE reductions across three base models and three problems, and includes several substantiating analyses (hyperparameter ablations in A.9, FinGPT baselines in A.11, mixed-data robustness in A.10, and economic-principle checks in A.12) that go beyond a single headline number. The theoretical faster-convergence claim is the paper's weakest component: the proof of Eqs. (27)–(28) is a one-line assertion that is demonstrably invalid, and the noise model on which it rests is unvalidated. The observed gradient-norm difference in Figures 3–4 may still hold empirically in the tested settings, but it does not establish the advertised theorem.","major_comments":[{"comment":"The claimed proof of the faster-convergence result is invalid. The paper asserts that 'Because f_P1(t)(·) is monotonically decreasing, from (7) and (21), we can prove that Î < Ĩ < 1', but monotonicity of the model density does not determine the comparison of the two overlap integrals Î = ∫_{P̂} f̂ f_model dx and Ĩ = ∫_{P̃} f̃ f_model dx. A discrete counterexample that satisfies all of the paper's assumptions (f̂ decreasing, f̃ = f̂ convolved with symmetric uniform noise, f_model decreasing, supports nested with P̂ ⊂ P̃) is: f̂ on {1,2} with probabilities (0.8, 0.2); noise n ~ U(−1,1) so f̃ on {0,1,2,3} has probabilities (4/15, 1/3, 1/3, 1/15); f_model on {0,1,2,3} = (0.3, 0.3, 0.2, 0.2). Then Î = 0.8·0.3 + 0.2·0.2 = 0.28 > Ĩ = 4/15·0.3 + 1/3·0.3 + 1/3·0.2 + 1/15·0.2 = 0.26, which by the paper's own Eqs. (22)–(23) gives ||∇Lhat(w)|| < ||∇Ltilde(w)||, the reverse of Eq. (28). The faster-convergence theorem is therefore not established even under the paper's stated assumptions; the empirical evidence in Figures 3–4 is a separate matter.","section":"§4.2 / Appendix A.8, Eqs. (27)–(28)"},{"comment":"The convergence analysis rests on assumptions that are neither validated against the collected data nor stated with their evidentiary status: (i) real-user decisions equal the theoretical optimal solution plus i.i.d. U(−ε, ε) noise, with ε never estimated from the 119 participants' data; (ii) the support nesting P̂1(t) ⊂ P̃1(t) ⊂ P1(t); (iii) monotonicity of the LLM output density f_P1(t)(·); and (iv) the constancy of η when α1 and θ1 vary slightly (Appendix A.7). Under assumption (i), the comparison in Eq. (28) is between a clean signal and the same signal corrupted by noise, so the conclusion is substantially entailed by the modelling choice rather than by a property of investor behavior. The statistical validation in Section 3.3 and Appendix A.6 (difference t = −1.075; correlation t = −0.843 against 0.85) tests only whether the theoretical solution is close to real-user data in mean and correlation; it does not test the uniform-noise, independence, support-nesting, or monotonicity assumptions. The paper should either validate these assumptions empirically or re-frame the convergence claim as an empirical observation.","section":"§4.2"},{"comment":"The empirical comparison of gradient norms is confounded by a dataset-size and coverage mismatch. The theoretical SFT dataset contains 10³ samples (10 α1 values × 10 θ1 values × 10 trials) with uniform attribute coverage, whereas the real-user SFT dataset contains at most 119 samples concentrated in a few attribute classes (Section 3.2). The paper's second assumption ('sample size sufficiently large') is therefore false for the real-user dataset. The observed higher gradient norm on theoretical data could be due to the larger, more uniformly covered dataset rather than to the absence of label noise. A matched-sample comparison (e.g., subsampling the theoretical dataset to match the real-user class distribution and size, or upsampling the real-user data) is needed to support the convergence-rate claim.","section":"§4.2, Figures 3–4"},{"comment":"The reported overall MSE for P2 is computed over the union P2 ∪ P3, not over the P2 data alone. Because P3 is exactly the problem on which InvestAgents were fine-tuned, including P3 data in the P2 metric can inflate the apparent cross-task improvement, and the statement that InvestAlign 'reduce[s] the overall MSEs by ... in P2' is not supported by a P2-only measurement. The authors should report the MSE restricted to the 44 P2 participants (or to the attribute classes actually covered by them) and clarify why the union was used.","section":"§5.2, Table 1 (P2 rows)"},{"comment":"The headline claim of 'significantly closer alignment' is supported only by point estimates of overall MSE reductions (45.59%–61.26%), with no confidence intervals or significance tests. Since the real-user samples are small (119, 80, and 44 participants for P3, P1, and P2) and the LLM responses are averaged over 10 seeds, bootstrap confidence intervals on the MSE reductions or a paired test over attribute classes should be reported to justify the word 'significantly'.","section":"§5.1, Table 1 and Abstract"}],"minor_comments":[{"comment":"The null hypothesis 'mean correlation coefficient does not differ from 0.85' (t-statistic −0.843) uses a threshold (0.85) whose justification is not given; reporting the estimated mean and confidence interval of {ρi} would be more informative.","section":"Appendix A.6"},{"comment":"The text says GPT-3.5 was also evaluated on P2, but Table 1 contains no GPT-3.5 row for P2; the discrepancy should be resolved.","section":"§5.2, Table 1"},{"comment":"The overall MSE averages over attribute classes without weighting by the number of participants in each class, so classes with a single participant count as much as classes with many; a participant-weighted average or a per-class coverage report would be more robust.","section":"§5.1, Eqs. (9)–(10)"},{"comment":"The symbol T denotes both the time horizon and the time set T = [0, T], and in Eqs. (6) and (26) P1(t) is used for both the decision variable and its support range; distinct notation would remove ambiguity.","section":"§3.1"},{"comment":"GLM-4 results are shown in Figure 6 but no GLM-4 entries appear in Table 1; adding them would let readers compare all four models on the same metric.","section":"Appendix A.4"},{"comment":"The derivation of Eq. (21) identifies f_P1(t)(x) with Sigmoid(z) without defining the mapping from the decision value x to the sigmoid input z; this step needs an explicit statement.","section":"§4.2, Eq. (21)"},{"comment":"The section heading 'Comparison of Real-User Data and Pre-SFT LLMs' Investment Decision on P1 and P1' appears to contain a typo; it should likely read 'P1 and P2'.","section":"Appendix A.5"},{"comment":"The claim that 'a larger gradient norm leads to faster convergence' cites a survey of unstable gradients; this is a heuristic rather than a convergence-rate theorem, and even if Eq. (28) were valid, additional assumptions on the loss landscape would be needed to conclude a faster convergence rate.","section":"§4.2, citation (Chen and Xiang, 2018)"}],"recommendation":"major_revision","confidential_remarks":"The paper's empirical core—large, consistent MSE reductions with public code—is credible and worth publishing after revision, but the advertised theoretical result (faster convergence) is not established: the proof of Eq. (27) is invalid and the conclusion is partly built into the unvalidated noise model. I would advise the authors to either supply a correct proof with validated assumptions or soften the convergence claim in the abstract, and to report P2-only metrics and significance tests. Note also that the theoretical solution in Eq. (11) comes from the authors' own prior papers (Wang and Zhao, 2024a,b, 2025); this is legitimate, but the novelty framing in the introduction should make the dependency explicit for readers."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the core empirical result is real: fine-tuning on SFT data generated from the closed-form solution of the simpler P3 improves alignment with real-user decisions on P1 and P2, with 44–61% MSE reductions across models. That holds up on reading. Second, the advertised theoretical result on faster convergence (Eq. 28) has a genuine hole. The stress-test counterexample is valid: Eq. 27 does not follow from monotonicity, and the inequality can reverse under the paper's own assumptions.\n\nWhat is actually new: constructing SFT datasets from the theoretical solution of a similar simple problem and transferring them to harder herd-behavior problems. That specific pipeline is not in the cited literature. The paper also does real work around it: validating the theoretical solution against 119 participants' decisions with t-tests, releasing code, ablating LoRA rank and fine-tuning steps, comparing against a FinGPT baseline, and testing mixed theoretical/real-user datasets. Those are honest, reproducible touches.\n\nThe soft spots are concentrated in Section 4.2. The proof assumes real-user data equal the theoretical solution plus uniform noise, with support nesting and monotone model density. None of that is validated. The stress-test correctly shows the key inequality can flip, so the faster-convergence claim is not established as a theorem. It is an empirical observation with a plausible but incomplete story, and the story is partly circular because real-user data are modeled as a noisy version of the same theoretical solution. Separately, the headline MSE reductions lack confidence intervals, participant pools are small (119, 80, 44), and the main table lacks a real-user-SFT baseline for final alignment; the gradient/loss figures and the mix-SFT appendix only partially fill that gap. These are addressable.\n\nThe paper is for people building financial agents or aligning LLMs with behavioral-economic decision processes. They will get a useful recipe and a cautionary example of why convergence claims need tighter derivations.\n\nRecommendation: send it to peer review. The empirical alignment result is worth publishing. Ask the authors to either prove the convergence inequality properly, or remove it from the abstract and Section 4.2 and present the faster-convergence evidence as empirical. Also request confidence intervals and a real-user-SFT final alignment baseline.","headline":"A practical empirical alignment result that leans on a shaky theoretical convergence proof; the 44–61% MSE reductions are real, but Eq. 28 needs either a proper proof or a demotion to observation.","tokens_in":31393,"tokens_out":2073,"would_cite":true,"duration_ms":26433,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that synthetic data from a solvable investment model can replace scarce real-user data when aligning language models to human investors.","keywords":["LLM alignment","supervised fine-tuning","herd behavior","optimal investment","behavioral finance","synthetic training data","data scarcity","convergence rate"],"falsifier":"Using the paper's collected P3 data, compute the residuals $\\tilde{P}_1^i(t) - \\hat{P}_1^i(t)$ for each participant and time step; if they are visibly non-uniform, serially correlated, or shifted away from zero, the assumed noise model in Section 4.2 is violated, and one can directly test whether the theoretical-data gradient norm still exceeds the real-data gradient norm when the empirical residual distribution is used instead of $U(-\\varepsilon, \\varepsilon)$.","tokens_in":30230,"feed_emoji":"📈","tokens_out":7889,"duration_ms":77406,"temperature":0.7,"pith_summary":"The paper proposes that the shortage of real-user data for aligning language models with investor behavior can be sidestepped by generating supervised fine-tuning (SFT) data from the theoretical solution of a simpler, similar optimal-investment problem. It studies a two-agent portfolio-choice setting with herd behavior, in which one agent's decisions influence another's; for the simplest case (absolute herd behavior, unilateral influence) a closed-form optimal decision rule exists. The authors argue that fine-tuning on this theoretically generated data converges faster than fine-tuning on collected user decisions, and that the resulting InvestAgent tracks real-user decisions more closely than the base model, both in the simple problem and in two harder variants where the theoretical solution is computationally heavy. If correct, this offers a data-efficient route to behavioral alignment in finance without collecting large amounts of personal investment data.","feed_headline":"Synthetic theory data aligns AI investor agents with real choices","feed_subtitle":"Fine-tuning on a closed-form investment solution cuts error to real-user decisions by up to 61 percent.","key_machinery":"The engine is P3's closed-form optimal decision rule, Eq. (11), which expresses A1's optimal investment $\\hat{P}_1(t)$ as a function of risk-aversion coefficients $\\alpha_1, \\alpha_2$, influence coefficient $\\theta_1$, market parameters $r, v, \\sigma$, and a numerically computed constant $\\eta$. The paper uses this rule to generate $10^3$ training samples over varied investment attributes. The convergence argument compares cross-entropy gradient norms for theoretical versus real-user data; assuming real-user decisions equal the theoretical solution plus i.i.d. uniform noise, the theoretical-data loss has a larger gradient norm, which in a locally convex loss landscape implies faster parameter convergence.","core_discovery":"The central claim is that supervised fine-tuning on SFT datasets built from the closed-form solution of the simple problem P3 (absolute herd behavior with unilateral influence) produces InvestAgents whose investment decisions align with real-user decisions better than models before fine-tuning, in P3 and also in the complex problems P1 (relative herd behavior, unilateral influence) and P2 (absolute herd behavior, mutual influence). Across the tested models, the paper reports overall mean squared error reductions of 45.59–61.26% for P3, 44.52–56.69% for P1, and 45.77–60.92% for P2 relative to the unfine-tuned baselines. The faster-convergence claim is stated as the gradient-norm inequality $\\|\\nabla \\hat{L}(w)\\| > \\|\\nabla \\tilde{L}(w)\\|$ in Eq. (28), and is supported by measured gradient norms and training-loss curves.","pith_inferences":["(Editorial inference) The faster-convergence proof relies on the idealization that human noise is uniform and independent; a natural next test is whether inequality (28) still holds when residuals are skewed, heteroskedastic, or autocorrelated.","(Editorial inference) The method's usefulness will be governed by how far a solvable toy problem can deviate from the target problem before the learned behavior stops transferring; comparing InvestAlign against a model fine-tuned on a size-matched random sample of real-user data would sharpen this.","(Editorial inference) Extending the approach to other behavioral biases such as loss aversion or overconfidence would require closed-form solutions of the same style for those biases, or a hybrid dataset blending theory with a thin layer of real data.","(Editorial inference) Because the paper measures alignment as closeness to the average human decision, a complementary test would be whether InvestAgent reproduces the full distribution of human choices, including the spread and individual heterogeneity, not just the mean."],"forward_implications":["If correct, LLM-based investment agents can be aligned to herd-affected human decision-making without collecting large proprietary datasets of individual investment choices.","The same recipe—solve a simple solvable analog, generate SFT samples, fine-tune—could in principle be applied to other decision problems in behavioral finance where stylized closed-form solutions exist.","Theoretical-data fine-tuning should be preferred to real-user data when the theoretical model is a good proxy, because it converges faster; adding a smaller amount of real-user data can further reduce error.","The trained InvestAgents reproduce two economic hypotheses: investment decisions converge as herd influence increases, and stronger mutual influence lowers expected total terminal wealth.","Fine-tuning on a general financial-domain dataset performs worse on this task than fine-tuning on the theoretically generated herd-behavior dataset."],"supporting_citations":[{"why":"Supplies the baseline continuous-time optimal investment model that all three problems build on.","marker":"(Merton, 1969)"},{"why":"Provides the closed-form solution to P3 that generates the SFT dataset.","marker":"(Wang and Zhao, 2024b)"},{"why":"Defines relative herd behavior and the P1 problem used as one complex target.","marker":"(Wang and Zhao, 2024a)"},{"why":"Defines mutual influence and the P2 problem used as the other complex target.","marker":"(Wang and Zhao, 2025)"},{"why":"Justifies the exponential utility form used in the agents' objectives.","marker":"(Rogers, 2013)"},{"why":"Supplies the convolution formula used to derive the density functions in the gradient-norm comparison.","marker":"(Rényi, 2007)"},{"why":"Provides the local convexity premise used to equate larger gradient norm with faster convergence.","marker":"(Boyd and Vandenberghe, 2004)"},{"why":"Used to support the claim that larger gradient norms drive faster descent in early training.","marker":"(Chen and Xiang, 2018)"},{"why":"Provides the LoRA fine-tuning method used in the convergence and alignment experiments.","marker":"(Hu et al., 2021)"},{"why":"Provides the general financial-domain baseline datasets used as a comparison in the alignment experiments.","marker":"(Yang et al., 2023a)"}],"fun_headline_variants":["Synthetic theory data cuts AI investor alignment error by 61%","Theory-based SFT data aligns LLM investor agents with real choices","InvestAlign: theory-generated training data improves investor alignment","Synthetic solutions from theory align LLMs to investor decisions up to 61% better"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that real people's investment decisions are the mathematically optimal solution plus independent random noise spread evenly within fixed bounds, and that the range of possible human choices fits inside the range the model can produce; if human decisions contain systematic biases that the optimal-investment-plus-herding model does not capture, the faster-convergence inequality and the alignment benefit do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Synthetic theory data cuts AI investor alignment error by 61%","Theory-based SFT data aligns LLM investor agents with real choices","InvestAlign: theory-generated training data improves investor alignment","Synthetic solutions from theory align LLMs to investor decisions up to 61% better"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001077,"raw_usage":{"total_tokens":4508,"prompt_tokens":950,"completion_tokens":3558,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":3482}},"tokens_in":566,"tokens_out":3558,"duration_ms":26080,"temperature":1.0,"reasoning_tokens":3482,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:01:29.984474+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Using the paper's collected P3 data, compute the residuals $\\tilde{P}_1^i(t) - \\hat{P}_1^i(t)$ for each participant and time step; if they are visibly non-uniform, serially correlated, or shifted away from zero, the assumed noise model in Section 4.2 is violated, and one can directly test whether the theoretical-data gradient norm still exceeds the real-data gradient norm when the empirical residual distribution is used instead of $U(-\\varepsilon, \\varepsilon)$.","supporting_citations":[],"review_version":1}