{"id":"8a9aff77-f60f-4f25-8a81-04c0f6a40ad7","arxiv_id":"2412.21149","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"FRM replaces output-space losses with function-space losses, fitting a per-data-point function and approximating the resulting objective with Taylor/Laplace expansions, yielding weighted least squares with a Jacobian-Hessian metric.","lead":"Functional Risk Minimization (FRM) is a new training framework that fits a separate function to each data point, comparing models in function space instead of output space. The authors argue this captures realistic structured noise better than standard ERM and show small gains in linear regression, value estimation, and representation learning experiments.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (2) does not follow from Eq. (1): it maximizes a per-point marginal likelihood under Qθ*, not an estimator of the expected risk, and its normalizer is off by a factor of n.","rationale":"The reader's weakest_assumption identifies the Laplace/Taylor small-perturbation approximation as the load-bearing risk. That is a real concern, and the paper's own footnote concedes the approximation does not justify accuracy of the entire integral. However, the more fundamental problem occurs before the Taylor step: Eq. (2) is not the empirical counterpart of Eq. (1). Eq. (1) is an identity whose argmin is exactly the expected-risk minimizer, provided one could evaluate the expectation under the true P(θ). Eq. (2) replaces the unknown P(θ) with the parametric family Qθ* and then maximizes the resulting marginal likelihood of the labels. This is a different learning criterion. The paper's text says it 'approximates' the cross-entropy on P(y|x) by computing per-point constrained integrals, but no consistency argument connects the maximizer of Eq. (2) to the minimizer of Eq. (1). The normalizer discrepancy (Z^n vs. Z) is a concrete symptom of this: if Eq. (2) were a direct per-datapoint approximation of Eq. (1), the regularization term would scale with n. The reported linear-regression experiment already hints at the problem: under ideal Gaussian bias noise, ERM has lower test loss than FRM, which should not happen if Eq. (2) were an estimator of Eq. (1) in that setting. The reader's rationale does mention the transition from Eq. (1) to Eq. (2) as a heuristic jump, so there is partial agreement, but the formal weakest_assumption field focuses on the later Taylor approximation. I chose the derivation gap as the single most load-bearing concern because it determines whether the proposed objective has any claim to minimize expected risk, independently of how accurately the integral is approximated. The concrete test above would settle whether Eq. (2) actually tracks the risk minimizer in a tractable case; if the test fails, the central claim that FRM is derived from risk minimization is unsupported, and the reader's REJECT verdict is appropriate.","tokens_in":17568,"tokens_out":11214,"duration_ms":129021,"concrete_test":"Use 1-D linear regression with a known true prior P(λ,β) that is not in the Qθ* family, e.g. β|λ ~ N(0,1) and λ ~ 0.5δ_{1} + 0.5δ_{2}. Simulate a large dataset from this prior. Compute exactly: (i) the empirical risk minimizer θ*_ERM and (ii) the maximizer θ*_FRM of Eq. (2) with the corrected -n logZ normalizer, using numerical integration over the constraint lines f_{θ_i}(x_i)=y_i. Increase n and check whether θ*_FRM approaches θ*_ERM. If it does not, Eq. (2) is not an empirical estimator of Eq. (1), and the paper's central derivation claim is falsified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (1) is an identity: with Qθ*(θ) = e^{-E_x L(fθ(x), fθ*(x))}/Z(θ*), the expression H(P,Qθ*) - logZ(θ*) equals E_{θ~P}[E_x L(fθ(x), fθ*(x))], which is exactly the expected risk. Thus the population objective in Eq. (1) has the true risk minimizer as its argmin, independently of the choice of Q. To turn this into a training objective, one must estimate the expectation under the unknown P(θ) from the observed (x_i,y_i) pairs. Eq. (2) instead sums the logs of per-point integrals of e^{-E_x L} over the constraint set f_{θ_i}(x_i)=y_i, minus logZ(θ*). That is the log marginal likelihood of the labels under the parametric model θ_i ~ Qθ*, not an estimator of E_{θ~P} E_x L. Replacing the unknown P(θ) with Qθ* changes the objective, and the paper gives no argument that the maximizer of this misspecified likelihood converges to the minimizer of expected risk. There is also an algebraic slip: the full-data likelihood under Qθ* has normalizer Z(θ*)^n, so the last term in Eq. (2) should be -n logZ(θ*), not -logZ(θ*). As written, the regularizer's strength is underweighted by a factor of n. Since the downstream Taylor/Laplace approximations in Sec. 4.3 inherit this objective, the claimed derivation of FRM from risk minimization is unsupported at its very first step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Functional Risk Minimization (FRM), an alternative to empirical risk minimization in which each training point is assigned its own latent function f_{θ_i} drawn from a learned distribution over functions, and the training objective compares functions rather than outputs. The authors derive a functional-risk objective, provide a variational approximation for language models and a Taylor/Laplace approximation for over-parameterized models, prove a universal distribution theorem for FGMs, argue that common ERM losses are subsumed as special cases, and report experiments in linear regression, offline value estimation, and representation learning for MNIST variants.","tokens_in":17925,"tokens_out":5937,"duration_ms":56697,"significance":"If the central derivation were sound, FRM would be a broad and conceptually interesting alternative to ERM with potential implications for modeling structured noise and for understanding generalization in over-parameterized networks. The paper has real strengths: the FGM formulation is clearly described, the connections to hierarchical Bayes, meta-learning, and the NTK literature are useful, and the universal distribution theorem (Appendix A) and the loss-subsumption lemmas (Appendix B) are nontrivial. The experiments, however, are small-scale and do not compensate for the flaws in the derivation. The paper is clearly written and the approximation strategy is creative, but its load-bearing theoretical claim is not established.","major_comments":[{"comment":"The derivation from expected risk minimization to the FRM objective is not valid. Equation (1) is an identity: substituting Qθ* into H(P(θ), Qθ*(θ)) − log Z(θ*) gives exactly E_{θ∼P}[E_x L(f_θ(x), f_θ*(x))], so its argmin is the expected-risk minimizer regardless of the choice of Q. Equation (2), however, is not an estimator of that expectation; it is the log marginal likelihood of the observed labels under the model θ_i ∼ Qθ* with the per-point hard constraints f_{θ_i}(x_i) = y_i. No argument is given that maximizing this misspecified likelihood converges to the minimizer of expected risk, and the replacement of the unknown P(θ) by Qθ* is an unstated modeling assumption. The claimed derivation therefore fails at its first step.","section":"§4.1, Eqs. (1)–(2)"},{"comment":"The normalizer in Eq. (2) is wrong by a factor of n. If the per-point parameters θ_i are independent draws from Qθ*, the full-data likelihood is ∏_i (1/Z(θ*)) ∫_{f_{θ_i}(x_i)=y_i} e^{-E_x L(f_{θ_i}(x), f_θ*(x))} dθ_i, so the objective should end with −n log Z(θ*) rather than −log Z(θ*). Since this term is described as a label-independent regularizer and is inherited by the approximations in §4.3, the error changes the balance between the data term and the regularizer and can alter the location of the optimum.","section":"§4.1, Eq. (2)"},{"comment":"The tractable Taylor/Laplace approximation requires that the parameter perturbation Δ_i needed to fit each point be small (|Δ_i| << 1) and that f be approximately linear in Δ_i within that range, but the paper never verifies this regime for the models and datasets used in the experiments. The footnote on page 7 concedes that the small-perturbation assumption 'does not justify that this is an accurate approximation of the entire integral.' Because the regression objective (4), the classification log-cdf objective, and the interpretation in §4.4 all inherit this approximation, the empirical claims are not backed by a validated approximation.","section":"§4.3"},{"comment":"The transition to the regression objective is incomplete. Equation (3) still contains a factor Z(θ*) inside the integrand while the outer −log Z(θ*) term is dropped, and the Gaussian density used to obtain Eq. (4) also carries a θ*-dependent normalizer (proportional to |H_{f,L}|^{1/2}) that is omitted. Since the paper explicitly states that both H_{f,L} and J_i depend on θ*, these omissions are not harmless constants and can change the optimum. The regression objective as written is therefore not a faithful consequence of the Laplace approximation.","section":"§4.3, Eqs. (3)–(4)"}],"minor_comments":[{"comment":"The abstract claims 'better performance in supervised, unsupervised, and RL experiments,' but the experiments are limited to linear regression, mountain car value estimation, and MNIST-based VAE representations; the RL experiment is offline value estimation rather than policy learning, so the claim is stronger than the evidence.","section":"Abstract and §5"},{"comment":"There are typographical errors, including 'Altough' in §3.1 and an inconsistent citation to 'LeCun (1998)' versus 'LeCun et al. (1998)' in the references and figure captions.","section":"§3.1 and references"},{"comment":"The variational approximation for LLMs is described only at a high level; the distributions P_z and G in the expression for θ are not defined, and the claim that the prompt-based adaptation is 'closely connected to classic VAEs' would benefit from an explicit derivation.","section":"§4.2"},{"comment":"The lemmas showing that MSE, L1, accuracy, and cross-entropy are subcases of functional losses use unnormalized densities (e.g., F ∝ e^{-|β−β_i|^2}); the role of the normalizer, which is central to the FRM objective, is not addressed, so the claimed subsumption is only shown for an informal version of the functional-loss objective.","section":"Appendix B"},{"comment":"Figure 7 lacks details about dataset sizes, the precise noise distributions used, and whether the reported ratios are averages over many trials; without this information the magnitude of the reported gains is difficult to assess.","section":"§5.1"}],"recommendation":"reject","confidential_remarks":"The paper contains an interesting idea and the writing is clear, but the central claim that FRM is derived from expected risk minimization is not supported by the mathematics. The factor-of-n normalizer error is fixable, but the step from Eq. (1) to Eq. (2) is a fundamental change of objective rather than a derivation, and the paper does not supply a consistency argument for the new objective. The small-perturbation approximation is also unverified for the experiments. These are load-bearing issues that cannot be remedied by local corrections; a satisfactory revision would need to reframe the contribution as a heuristic objective with thorough empirical validation, which is beyond the scope of the current manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Fernando—quick take on 2412.21149. The core idea—model each training point with its own latent function, then tie them together in function space—is genuinely new. The FGM framing, the specific FRM objective in Eq. (4), and the subsumption lemmas are real contributions. The universal distribution theorem is a solid proof. The experiments, while small-scale, show the idea has empirical legs, especially the structured-noise examples and the CNN toy.\n\nThe soft spot is the load-bearing claim that FRM is derived from expected risk minimization. It isn't. Eq. (1) is an identity; Eq. (2) is a heuristic substitution of a per-point marginal likelihood for the unknown P(θ). No argument shows the maximizer of Eq. (2) minimizes expected risk. The stress-tester is also right about a concrete algebraic slip: the full-data likelihood has normalizer Z(θ*)^n, so the last term should be -n log Z(θ*), not -log Z(θ*). That's not a minor typo—it changes the regularization strength by a factor of n.\n\nThe Taylor/Laplace step has a related problem: it assumes small per-point parameter perturbations, and the paper's own footnote concedes the approximation isn't justified for the whole integral. For deep networks, nothing verifies that each point can be fit with a small weight change. So the regression objective in Eq. (4) rests on an unverified premise.\n\nI want to be clear about proportion. The framework is interesting enough that I'd rather see it fixed than buried. The derivation gap is fixable: reframe FRM as a variational approximation to a hierarchical Bayes marginal likelihood, not as a risk minimizer, and the objective stands on its own. The empirical work needs error bars and ideally code, but the signal is there.\n\nVerdict: send to peer review. A serious referee can untangle the derivation, check the algebra, and push for a proper empirical study. But don't accept it in this form. The overreach in 'derived' and the missing n are exactly what peer review should catch.","headline":"FRM is a genuinely new objective with a real derivation gap at its heart; worth refereeing, not accepting as-is.","tokens_in":18510,"tokens_out":2783,"would_cite":false,"duration_ms":27085,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Training a network should compare functions, not just outputs, and this paper derives an objective that does exactly that.","keywords":["functional risk minimization","empirical risk minimization","function space","functional generative models","over-parameterization","Laplace approximation","generalization","hierarchical Bayes"],"falsifier":"Measure the actual per-point parameter perturbations required to fit a deep network to a real dataset (e.g., by solving $\\min_{\\theta_i: f_{\\theta_i}(x_i)=y_i}\\|\\theta_i-\\theta^*\\|$ for each point) and compare their size with the curvature scale of the FRM Hessian. If typical perturbations are not small, the Gaussian/linearized objective (3) will diverge from the exact FGM integral (2), and the claimed equivalence and the reported gains should not hold on such data.","tokens_in":17348,"feed_emoji":"🎯","tokens_out":6254,"duration_ms":51336,"temperature":0.7,"pith_summary":"This paper proposes Functional Risk Minimization (FRM), a replacement for Empirical Risk Minimization (ERM) in which the training loss compares functions rather than single outputs. The central idea is that each training point $(x_i, y_i)$ is generated by its own latent function $f_{\\theta_i}$ that fits it exactly, with parameters drawn from a distribution centered at the learned model. From this Functional Generative Model the authors derive an objective, approximate it with a Laplace/Taylor expansion valid for over-parameterized networks, and show it subsumes ERM for common losses while improving results in linear regression, a CNN tagging task, a VAE representation benchmark, and a reinforcement-learning value function problem. If the framework holds, standard deep learning training could be replaced or supplemented by an objective that explicitly models structured per-point variability, offering a route to understanding generalization in the over-parameterized regime.","feed_headline":"New training rule beats ERM by comparing functions, not outputs","feed_subtitle":"Each data point gets its own latent function, yielding gains in supervised, unsupervised, and RL benchmarks.","key_machinery":"The Functional Generative Model (FGM) is the central object: each datapoint $(x_i,y_i)$ is assigned its own function $f_{\\theta_i}$ with $y_i=f_{\\theta_i}(x_i)$ and $\\theta_i \\sim P(\\cdot|\\theta^*)$. The argument is carried by the derived FRM objective, which is then made tractable by a Laplace approximation in parameter space and a first-order Taylor expansion of the network output, $f_{\\theta^*+\\Delta}(x) \\approx f_{\\theta^*}(x) + J_{\\theta^*}(x)^T \\Delta$, reducing the per-point integral to a Gaussian integral over a hyperplane (regression) or half-space (classification). In the small-perturbation limit the objective becomes a weighted least-squares loss with covariance $J_i^T H^{-1}_{f,L} J_i$, which is what the experiments optimize.","core_discovery":"The paper claims that the risk-minimization problem can be reframed in function space: instead of minimizing $L(y, f_{\\theta^*}(x))$ over outputs, one minimizes a cross-entropy between the unknown data-dependent distribution over functions $P(\\theta)$ and a loss-induced family $Q_{\\theta^*}(\\theta) \\propto e^{-\\mathbb{E}_x[L(f_\\theta(x), f_{\\theta^*}(x))]}$, penalized by the normalizer $-\\log Z(\\theta^*)$. The resulting FRM objective is $\\arg\\max_{\\theta^*}\\sum_i \\log \\int_{\\theta_i: f_{\\theta_i}(x_i)=y_i} e^{-\\mathbb{E}_x[L(f_{\\theta_i}(x), f_{\\theta^*}(x))]} d\\theta_i - \\log Z(\\theta^*)$, and the paper argues that, for common losses, ERM is a special case in which noise is restricted to output parameters. Under a small-perturbation Laplace approximation this objective becomes a weighted regression or a Gaussian-log-cdf classification loss, and the paper reports that this new objective yields better test performance than ERM in its experiments.","pith_inferences":["A direct extension the authors do not pursue: the Hessian in the FRM objective could be estimated online during training, turning FRM into a practical regularizer that does not require a second pass or full Hessian inversion.","The framework suggests a testable criterion for when to prefer FRM over ERM: estimate whether per-point function variability (e.g., style, pose, color) dominates output noise; if it does, FRM's objective should give larger gains.","Because FRM models each point by its own function, it may naturally handle non-stationary or out-of-distribution data where a single function cannot fit all points, connecting to test-time adaptation without an explicit fine-tuning step.","The Laplace approximation restricts the framework to small perturbations; an empirical study measuring actual per-point perturbation sizes in trained deep networks would delimit the regime where the derived objective is a faithful surrogate for the true FGM likelihood."],"forward_implications":["If FRM is correct, training objectives for over-parameterized networks should include a term measuring how easily each training point can be fit by a small parameter perturbation, not just the output error.","For common losses (MSE, L1, accuracy, cross-entropy), ERM becomes a restricted case of FRM in which only output-side parameters vary, so FRM is a strict generalization of standard practice.","In the approximate form, FRM can be interpreted as explicitly finding the simplest hyper-model $\\{\\theta_1,\\dots,\\theta_n\\}$ fitting the data, offering a concrete explanation for why over-parameterized ERM generalizes: it implicitly performs the same kind of minimization.","The variational approximation for LLMs suggests functional noise can be modeled in adaptation space (prompts, low-rank updates), which could make FRM-style objectives scalable to large language models.","The reported experiments imply that the benefits are largest when data variability is structured (translation, color, slope noise) rather than i.i.d. output noise, so FRM should be preferred on diverse real-world datasets."],"supporting_citations":[{"why":"Defines ERM, the baseline framework that FRM seeks to replace.","marker":"Vapnik & Chervonenkis, 1969"},{"why":"Documents that over-parameterized networks memorize random labels, motivating the generalization account in section 4.4.","marker":"Zhang et al., 2017"},{"why":"Supplies the neural tangent kernel linearization assumption that justifies the small-perturbation Taylor expansion in section 4.3.","marker":"Jacot et al., 2018"},{"why":"Supplies the variational method used to approximate the FRM integral in section 4.2.","marker":"Kingma & Welling, 2013"},{"why":"With Maddison et al., 2016, provides the Gumbel-softmax construction used to show cross-entropy is an FGM subcase.","marker":"Jang et al., 2016"},{"why":"Provides the concrete distribution used alongside Jang et al. to prove the cross-entropy equivalence.","marker":"Maddison et al., 2016"},{"why":"Supports the claim that CNNs capture natural structured variability, used in figure 1 and the VAE experiments.","marker":"Ulyanov et al., 2018"},{"why":"Provides implicit differentiation tools that scale the Taylor approximation to neural networks.","marker":"Blondel et al., 2021"}],"fun_headline_variants":["FRM: Each data point gets its own function, beats ERM","Forget outputs: compare functions to minimize risk","Functional risk minimization: per-point functions improve on ERM","Better than ERM? Compare functions, not outputs","FRM gives each point a function and better handles noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything in the tractable FRM objective rests on the assumption that each training point can be fit by a very small perturbation of the network's weights, so that a first-order Taylor expansion of the function and a Laplace approximation of the integral are accurate; the paper itself notes this approximation does not justify accuracy over the entire integral.","fun_headline_variants_meta":{"raw":{"variants":["FRM: Each data point gets its own function, beats ERM","Forget outputs: compare functions to minimize risk","Functional risk minimization: per-point functions improve on ERM","Better than ERM? Compare functions, not outputs","FRM gives each point a function and better handles noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000463,"raw_usage":{"total_tokens":2300,"prompt_tokens":914,"completion_tokens":1386,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":1305}},"tokens_in":530,"tokens_out":1386,"duration_ms":12568,"temperature":1.0,"reasoning_tokens":1305,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:01:46.674494+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the actual per-point parameter perturbations required to fit a deep network to a real dataset (e.g., by solving $\\min_{\\theta_i: f_{\\theta_i}(x_i)=y_i}\\|\\theta_i-\\theta^*\\|$ for each point) and compare their size with the curvature scale of the FRM Hessian. If typical perturbations are not small, the Gaussian/linearized objective (3) will diverge from the exact FGM integral (2), and the claimed equivalence and the reported gains should not hold on such data.","supporting_citations":[],"review_version":1}