{"id":"f3fb9aec-5e77-448b-a59e-8c0ea301ec30","arxiv_id":"2506.15385","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"S-MEME iteratively fine-tunes a diffusion model using its own score as the exploration reward, provably converging to the maximum-entropy distribution on the learned manifold.","lead":"This paper presents an algorithm that fine-tunes a pre-trained diffusion model so that it generates more varied and unusual samples, using the model's own score as a guide for exploration. It proves that repeated fine-tuning converges to the most diverse distribution over valid designs, and shows promising image generation results.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Entropy objective (Eq. 8) is undefined for measures supported on the low-dimensional manifolds that motivate Ω_pre; Theorem 7.1's convergence target p*_T may not exist.","rationale":"The reader's verdict of CONDITIONAL is appropriate; my read does not change it. The algorithmic idea — using the score as the reward gradient for entropy exploration and sequential mirror-descent fine-tuning — is credible and interesting. The experiments are promising, though the main table lacks error bars and direct entropy measures in the image case. The reason I single out the entropy well-posedness over the undefined \\tilde{Ω} in Assumption 7.1 and the unverified oracle Assumption 7.3 is that those are fixable technical gaps: \\tilde{Ω} can be defined, and Assumption 7.3 is an explicit (if strong) assumption. In contrast, the reference-measure problem affects the statement of the problem being solved: if the support is a low-dimensional manifold, the differential entropy (Eq. 8) is not defined, so the theorem's conclusion is vacuous. Proposition 1's compactness argument does not address this, as compact null sets exist. The paper explicitly advertises the low-dimensional manifold setting, so this is not an edge case but the central regime. A concrete test on the synthetic example can empirically demonstrate whether the intended regime is singular or full-dimensional; if full-dimensional, the concern is mitigated for the experiments but remains for the claimed general manifold setting.","tokens_in":20936,"tokens_out":18059,"duration_ms":173820,"concrete_test":"Run the synthetic experiment of Sec. 8 and estimate the effective dimension of Ω_pre from the 80,000 samples of p_pre_T used for Fig. 3b: compute the sorted eigenvalues of the empirical covariance and the participation ratio (Σλ_i)^2/Σλ_i^2. Independently, recompute the Monte-Carlo entropy estimate of Fig. 3d using Eq. (39) with ODE tolerance tightened by 10x. If the participation ratio is below 1.9 in this 2D example (or, in a higher-d analogue, strictly below d) or the entropy estimate shifts materially under refinement, then p_pre_T is singular (or nearly singular) w.r.t. Lebesgue, and Eq. (8) is not the right objective. This would confirm the concern by demonstrating that the object whose maximizer Theorem 7.1 targets is not well-defined in the paper's intended manifold regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise of Theorem 7.1 is that Problem (7) has a well-defined maximizer p*_T in P(Ω_pre) for H(p) = -∫ p log p dx (Eq. 8; all integrals w.r.t. Lebesgue per Sec. 2). This requires every candidate p to admit a density w.r.t. Lebesgue on R^d. The paper's own motivation is that Ω_pre = supp(p_pre_T) (Eq. 6) approximates a lower-dimensional data manifold with m << d (Fig. 1), and Proposition 1 only proves Ω_pre is compact. Compactness does not imply positive Lebesgue measure. If Ω_pre has Lebesgue measure zero, any µ ∈ P(Ω_pre) is singular w.r.t. Lebesgue; dµ/dx is 0 a.e., so log(dµ/dx) is undefined (or -∞) and H(µ) as defined in Eq. (8) is not a real-valued functional. The maximum-entropy problem is then ill-posed, and the target p*_T of Theorem 7.1 does not exist as a density. The proof of Theorem 7.1 inherits the problem: the mirror flow (MF), p_t = e^{h_t}/∫_Ω e^{h_t}, is undefined when Ω has zero Lebesgue measure. No reference measure on the manifold is ever supplied, so the KL regularization in Eq. (9) also lacks a well-defined dominating measure for the manifold case. Thus the theorem's statement depends on an unstated full-dimensionality assumption that contradicts the paper's manifold motivation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript proposes a framework for exploration with pre-trained diffusion models. The maximum-entropy manifold exploration problem (Problem (7)) seeks a policy whose terminal marginal density p^π_T maximizes differential entropy H(μ) = −∫ dμ log(dμ/dx) over the approximate data manifold Ω_pre := supp(p^pre_T) induced by a pre-trained model. The key identity (12), ∇_x δH(p^π_T) = −∇_x log p^π_T ≃ −s^π(·, T), turns entropy-based exploration into score-based fine-tuning, avoiding explicit density estimation. The authors introduce S-MEME (Algorithm 1), a mirror-descent scheme in which each iteration KL-regularizes the previous model by fine-tuning with the negative score as reward gradient, and prove a one-step optimality result under exact score estimation and exact optimization (Theorem 5.2). Under Assumptions 7.1–7.3 (support compatibility, precompactness, and noise/bias conditions on the fine-tuning oracle), Theorem 7.1 claims almost-sure weak convergence of the induced marginal densities to the maximum-entropy distribution on Ω_pre. Experimental results on a synthetic task and on text-to-image generation with Stable Diffusion support the practical claim of increased sample diversity.","tokens_in":21337,"tokens_out":21889,"duration_ms":223407,"significance":"The strength of the paper is conceptual: Eq. (12) is a simple, parameter-free identity that connects the entropy first variation to the score, and the mirror-descent interpretation of sequential fine-tuning gives a principled escalation from the idealized one-step result to a stochastic-approximation analysis. The paper builds on the right external machinery (relative smoothness, mirror flows, and Benaim-style asymptotic pseudotrajectories) and is transparent about the gap between idealized and realistic assumptions, which is commendable. If the theory were fully rigorous, the contribution would be significant: a provable, density-estimation-free method for max-entropy exploration with diffusion models. However, three load-bearing gaps currently prevent that: the well-posedness of the entropy objective on a possibly lower-dimensional support is not established, Assumption 7.1 refers to an undefined set, and Assumption 7.3 is stated at the level of the oracle error and not verified for the proposed solver.","major_comments":[{"comment":"The maximum-entropy objective in Problem (7) is not well defined for the manifold support that motivates it. Eq. (8) defines H(μ) = −∫ dμ log(dμ/dx) with all integrals taken with respect to Lebesgue measure (Section 2), which requires every admissible μ ∈ P(Ω_pre) to have a density with respect to Lebesgue; however, Section 3 and Fig. 1 motivate Ω_pre as an approximation of a low-dimensional data manifold (m ≪ d), and Proposition 1 proves only compactness, which does not imply positive Lebesgue measure. If Ω_pre has Lebesgue measure zero, no μ ∈ P(Ω_pre) admits a density with respect to Lebesgue, H(μ) is undefined (or −∞), the maximizer p*_T asserted in Theorem 7.1 need not exist as a density, and the mirror flow in (MF), p_t = e^{h_t}/∫_Ω e^{h_t}, is undefined because the normalizing Lebesgue integral vanishes; the KL penalty in Eq. (9) similarly lacks a common reference measure on a lower-dimensional Ω_pre. The manuscript needs an explicit regularity assumption (e.g., Ω_pre has nonempty interior and positive Lebesgue measure, and each p^{π_k}_T is absolutely continuous with respect to Lebesgue) or a reference measure on the manifold; the current tension between the manifold motivation and the Lebesgue-density formalism affects Problem (7), Theorem 5.2, and Theorem 7.1.","section":"Section 3, Eqs. (7)-(8); Appendix C, Eq. (MF)"},{"comment":"Assumption 7.1 refers to a set Ω-tilde that is nowhere defined. This is load-bearing rather than cosmetic: the assumptions supp(p^{π_k}_T) ⊂ Ω-tilde for all k and supp(p^{π_k}_j) = Ω-tilde for some j provide the common integration domain on which the dual variables h_k = δH(p^{π_k}_T) = −log p^{π_k}_T are well defined in the proof of Theorem 7.1, and the equality condition implicitly carries the requirement that the relevant densities be absolutely continuous with respect to a common full-dimensional reference measure. As written, the assumption is vacuous; the authors must define Ω-tilde and state its regularity properties, and should justify that the equality supp(p^{π_k}_j) = Ω-tilde is achievable for Algorithm 1.","section":"Section 7.1, Assumption 7.1"},{"comment":"The main convergence guarantee is conditional on Assumption 7.3, which is asserted directly on the oracle errors b_k and U_k, but the paper offers no analysis showing that the proposed instantiation of LINEARFINETUNINGSOLVER (Algorithm 2, based on Adjoint Matching) satisfies the bias-decay and summability conditions in terms of its inner-loop parameters (integration steps, gradient steps, batch sizes). Consequently, Theorem 7.1 is close to a restatement of the assumption that the solver tracks the idealized mirror iterates (MD_k): the same assumption also absorbs the unresolved gap between the densities reachable by diffusion policies and the full space P(Ω_pre) over which (MD_k) optimizes. In the proof, the step asserting that the interpolated process is an asymptotic pseudotrajectory of (MF) is delegated to 'standard arguments' without showing how Assumptions 7.1-7.3 control the discretization error for the specific nonlinear map h ↦ −log(e^h/∫ e^h). The theorem would be strengthened by either deriving verifiable bounds on the bias and noise of the concrete solver or restating the result as an explicitly conditional guarantee.","section":"Section 7.1, Assumption 7.3; Appendix C, proof of Theorem 7.1"}],"minor_comments":[{"comment":"The claimed convergence rate of order Õ((log log k)^{−1}) is stated without derivation or a precise reference; either the rate proof should be included or the remark should be removed.","section":"Section 7, Remark after Theorem 7.1"},{"comment":"The proof of Theorem 5.2 uses inconsistent symbols (a free 'μ' in Eq. (27) and in the expression '1 + kμ/(L−μ)', with 'l', 'k', and 'K' conflated), which obscures the argument; the variables should be made consistent throughout the proof.","section":"Appendix B, Eqs. (24)-(27)"},{"comment":"The last row of Table 1 is labeled 'H(p, p^pre_T)' in the header but is described in the text as Gaussian cross-entropy, and the caption of Table 2 says 'FID, CLIP and cross-entropy evaluation' although the table reports Vendi and CLIP scores; the captions and row labels should be corrected.","section":"Section 8, Tables 1 and 2"},{"comment":"The Robbins-Monro conditions are stated for γ_k, but Algorithm 1 exposes only the regularization coefficients α_k = 1/γ_k, and the theoretical schedule γ_k → 0 with summable squares corresponds to α_k → ∞, which does not match the fixed-regularization experimental protocol described in Section 8; the paper should specify the recommended α_k schedule and clarify the relationship between the theoretical step-size conditions and the experiments.","section":"Section 7.1 and Algorithm 1"},{"comment":"The implementation scales the reward gradient by λ (λ = 0.1 in both experiments), while the theory in Sections 5 and 7 uses the unscaled score gradient −s^{k−1}(·, T); the paper should explain how this scaling is absorbed in the analysis (for example, into the step size or the bias/noise terms).","section":"Section 8 and Appendix D"}],"recommendation":"major_revision","confidential_remarks":"This is a borderline paper for me. The algorithmic idea and the score-entropy identity (12) are genuinely nice and likely publishable at a strong venue once the theory is repaired. My main reservations are (i) the unstated full-dimensionality assumption that is in tension with the manifold motivation, (ii) the undefined Ω-tilde in Assumption 7.1, and (iii) the near-tautological dependence of Theorem 7.1 on Assumption 7.3, which is unverified for the Adjoint-Matching instantiation. I also find the related-work claim to be 'the first rigorous theoretical analysis of maximum state entropy exploration over continuous state spaces' premature given these gaps. I recommend major revision rather than rejection, since the issues are addressable within the scope of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a paper worth reading, but the \"provable convergence\" headline oversells what is actually shown.\n\nThe genuinely nice idea is Eq. (12): the gradient of the entropy first variation is the negative score, so you can maximize entropy of a diffusion model's marginal without ever estimating its density. Wrapping that in an iterative mirror-descent fine-tuning scheme (S-MEME) is a clean way to turn a nonlinear distributional objective into a sequence of linear-reward RL fine-tuning problems. The problem formulation — maximum entropy exploration over the implicit support of a pre-trained model — is well motivated, and the synthetic and text-to-image experiments show the intended behavior: higher entropy, more diverse samples, roughly preserved semantic alignment.\n\nWhere I'd push back:\n\n- Theorem 7.1 as stated is not a complete theorem: Assumption 7.1 uses an undefined set Ω̃. Maybe it's a typo for Ω_pre, but as written the reader cannot check the statement. That's fixable, but it should not have gotten to camera-ready in this state.\n\n- Assumption 7.3 effectively assumes the linear fine-tuning oracle has asymptotically vanishing bias and controlled noise. This is a standard stochastic-approximation condition, but the paper never verifies it for Adjoint Matching or any concrete solver. So the result is a reduction: if a solver satisfies those conditions, then S-MEME converges. That is a legitimate contribution, but it is not a convergence guarantee for the implemented algorithm, and the abstract's \"provable convergence under realistic assumptions\" is doing a lot of work.\n\n- The measure-theoretic issue is real. The objective and the mirror flow need densities with respect to Lebesgue. If Ω_pre is a low-dimensional manifold — which is exactly the motivation in Figure 1 — those objects are undefined. The likely resolution is that the noisy diffusion process gives a full-dimensional, thickened support, but the paper never says this. One sentence defining the reference measure would fix it.\n\nMinor: the main image table has no error bars; the appendix Vendi table does, so this is easy to repair.\n\nBottom line: a serious referee should engage. The algorithmic idea is good and the experiments are directionally convincing. The theory needs a corrected statement, a clarification of the measure assumption, and an honest positioning of Assumption 7.3. I'd accept for review and ask for major revision, not reject.","headline":"A genuinely useful exploration idea built on the score-as-entropy-gradient identity, but the convergence theorem is shakier than the abstract suggests and needs a corrected statement.","tokens_in":21802,"tokens_out":4232,"would_cite":true,"duration_ms":46737,"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":"Exploration of a learned design space can be posed as entropy maximization over a diffusion model's implicit manifold; sequential fine-tuning with the model's own score provably converges to the maximum-entropy distribution.","keywords":["maximum entropy exploration","diffusion models","score function","mirror descent","sequential fine-tuning","manifold hypothesis","stochastic optimal control","entropy maximization"],"falsifier":"Pre-train a diffusion model on samples lying exactly on a known lower-dimensional set, e.g. a circle in $\\mathbb{R}^2$ or a 2D surface in $\\mathbb{R}^{10}$, and run S-MEME. If the framework is sound, the fine-tuned model's samples should converge weakly to the uniform distribution on that set; if differential entropy is undefined on zero-measure supports, the iterates should fail to converge or spread unevenly along the intrinsic geometry. A complementary check is to measure, on the paper's own 2D example, the oracle bias $b_k = \\mathbb{E}[\\delta H(p^{\\pi_k}_T) - \\delta H(p^k_\\sharp)\\mid\\mathcal{G}_k]$ as the linear solver's inner iterations grow, since Assumption 7.3 requires this bias to decay to zero.","tokens_in":20684,"feed_emoji":"🧭","tokens_out":13859,"duration_ms":120977,"temperature":0.7,"pith_summary":"The paper attempts to turn exploration—generating genuinely novel designs rather than near-copies of the training data—into a well-defined optimization problem. Its claim is that a pre-trained diffusion model can be fine-tuned so that its samples spread out uniformly over the design manifold the model has implicitly learned, and that this is exactly the solution of a maximum-entropy problem over that manifold. The key move is an identity: $\\nabla_x\\delta H(p^\\pi_T) = -\\nabla_x\\log p^\\pi_T$, so the gradient of the entropy's first variation is the negative score function, a quantity the diffusion model already predicts, and the algorithm never has to estimate densities. The paper proves that with an exact score and an exact optimization oracle a single fine-tuning step is optimal, and that under realistic noise and bias conditions the sequential S-MEME algorithm converges weakly almost surely to the maximum-entropy distribution. This matters because it offers a principled, scalable route to novel valid designs in domains such as molecule and material discovery, where mimicking existing data is not enough.","feed_headline":"Retraining a diffusion model provably maximizes exploration entropy","feed_subtitle":"S-MEME fine-tunes with only the model's own score and converges to the most uniform sampling of its learned manifold.","key_machinery":"The argument is carried by three linked objects. First, the score-entropy identity $\\nabla_x\\delta H(p^\\pi_T) = -\\nabla_x\\log p^\\pi_T \\simeq -s^\\pi(\\cdot,T)$: it converts the first variation of entropy, which naively requires the log-density, into the score function that the diffusion model is trained to output, removing density estimation from the loop. Second, the mirror-descent reading of fine-tuning, in which $D_{KL}(p^\\pi_T, p^{k-1}_T)$ is the Bregman divergence of the entropic mirror map $Q = -H$, and the objective is 1-smooth and 1-strongly convex relative to itself, which makes an idealized single step optimal and grounds the iterative scheme. Third, the mirror-flow analysis: the dual variables $h_k = \\delta H(p^k_T) = -\\log p^k_T$ interpolate to an asymptotic pseudotrajectory of $\\dot{h}_t = \\delta H(p_t)$, $p_t = \\delta(-H)^*(h_t)$, and the entropy $H$ acts as a strict Lyapunov function for this flow, so the unique internally chain-transitive set is the uniform distribution on $\\Omega_{pre}$, to which the iterates converge almost surely.","core_discovery":"The paper's central claim is that maximum-entropy exploration over the support of a pre-trained diffusion model can be solved as mirror descent over probability measures, with the KL divergence as the Bregman distance and the model's own score as the gradient direction. Linearizing the entropy functional $H$ at the current iterate gives $\\delta H(p^\\pi_T) = -\\log p^\\pi_T$, and since $\\nabla_x\\delta H(p^\\pi_T) = -\\nabla_x\\log p^\\pi_T \\simeq -s^\\pi(\\cdot,T)$, the reward gradient for fine-tuning is the negative score the diffusion model already computes. Each step of the proposed S-MEME algorithm (Score-based Maximum Entropy Manifold Exploration) solves $\\arg\\max_\\pi \\langle\\delta H(p^{k-1}_T), p^\\pi_T\\rangle - \\alpha_k D_{KL}(p^\\pi_T, p^{k-1}_T)$, which is a mirror descent step with entropic mirror map $Q = -H$; because $F = -H$ is 1-smooth and 1-strongly convex relative to $Q$, a single such step is optimal when scores and optimization are exact. In the realistic case the discrete iterates form an asymptotic pseudotrajectory of the continuous-time mirror flow $\\dot{h}_t = \\delta H(p_t)$, $p_t = \\delta(-H)^*(h_t)$, whose only internally chain-transitive set is the uniform measure on $\\Omega_{pre}$. Theorem 7.1 concludes that the induced marginals converge weakly almost surely to $p^*_T \\in \\arg\\max_{p_T\\in\\mathcal{P}(\\Omega_{pre})} H(p_T)$, the maximum-entropy distribution on the pre-trained manifold.","pith_inferences":["Beyond Shannon entropy, the same sequential fine-tuning loop should work for any distributional objective whose first variation is available through the score—for instance Rényi entropies or submodular exploration measures—since only the first variation enters the linearized step.","In the strictly low-dimensional regime, the natural reformulation would maximize entropy with respect to the manifold's intrinsic volume rather than Lebesgue measure; a testable consequence is that naive S-MEME would spread samples unevenly along a curve or surface.","A practical diagnostic suggested by the analysis is the gap between the linearized iterate and the exact mirror descent step; if that gap does not shrink when the inner solver is run longer, the solver's bias, not the score error, is the binding constraint.","The image results hint at a general-purpose recipe—moderate $\\alpha$, a few outer iterations, and an external validity checker—for turning any pre-trained generative model into a data-augmentation engine for design domains beyond images."],"forward_implications":["Exploration needs no uncertainty quantification and no density estimator: the already-trained score network supplies the reward gradient, which is what makes the method scalable to high-dimensional text-to-image models.","With an exact score and an exact linear solver, a single fine-tuning step already produces the maximum-entropy model (Theorem 5.2); the sequential loop exists to correct the bias and noise of realistic solvers.","The regularization coefficient $\\alpha$ trades validity against exploration: large $\\alpha$ stays near the pre-trained model, small $\\alpha$ reaches low-density regions, enabling guided data augmentation when a validity checker is available.","When the noise-and-bias conditions of Assumption 7.3 hold with the Robbins-Monro step rule, convergence is almost sure and in the weak topology, at a rate of order $\\tilde{O}((\\log\\log k)^{-1})$."],"supporting_citations":[{"why":"Supplies the mirror-flow framework and the entropy-Fenchel-dual system (MF) used for the continuous-time analysis.","marker":"(Hsieh et al., 2019)"},{"why":"Provides the stochastic-approximation theorem that asymptotic pseudotrajectories converge to internally chain-transitive sets.","marker":"(Benaïm, 2006)"},{"why":"Defines asymptotic pseudotrajectories and chain-transitive sets, the bridge between the discrete iterates and the flow.","marker":"(Benaïm & Hirsch, 1996)"},{"why":"Contributes relative smoothness and strong convexity and the three-point Bregman property behind the one-step guarantee.","marker":"(Lu et al., 2018)"},{"why":"Adjoint Matching is the first-order solver used as LINEARFINETUNINGSOLVER in Algorithm 1.","marker":"(Domingo-Enrich et al., 2024)"},{"why":"Frames pre-trained diffusion scores as continuous-time RL actions, the policy view underlying Problem (7).","marker":"(Zhao et al., 2024)"},{"why":"Defines the probability-flow ODE and score-based sampling used by Proposition 1 and the implementation.","marker":"(Song et al., 2020)"},{"why":"Supplies the mirror-flow asymptotic-pseudotrajectory argument and the quoted convergence rate.","marker":"(Karimi et al., 2024)"}],"fun_headline_variants":["Diffusion score fine-tuning provably maximizes entropy","S-MEME: provable max-entropy exploration via scores","Mirror descent on scores yields provable entropy gain","Fine-tune diffusion with own score for max entropy","Provable exploration by entropy maximization on manifold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire problem is only well-defined if every distribution $p^{\\pi_k}_T$ has an ordinary density with respect to Lebesgue measure on $\\mathbb{R}^d$, because differential entropy and the KL regularization in the objective are defined only for such densities; if the learned manifold is truly lower-dimensional, the maximum-entropy problem and its convergence guarantee rest on a reference measure that the paper never supplies.","fun_headline_variants_meta":{"raw":{"variants":["Diffusion score fine-tuning provably maximizes entropy","S-MEME: provable max-entropy exploration via scores","Mirror descent on scores yields provable entropy gain","Fine-tune diffusion with own score for max entropy","Provable exploration by entropy maximization on manifold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000179,"raw_usage":{"total_tokens":1373,"prompt_tokens":1089,"completion_tokens":284,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":705,"completion_tokens_details":{"reasoning_tokens":207}},"tokens_in":705,"tokens_out":284,"duration_ms":3616,"temperature":1.0,"reasoning_tokens":207,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:36:04.590770+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pre-train a diffusion model on samples lying exactly on a known lower-dimensional set, e.g. a circle in $\\mathbb{R}^2$ or a 2D surface in $\\mathbb{R}^{10}$, and run S-MEME. If the framework is sound, the fine-tuned model's samples should converge weakly to the uniform distribution on that set; if differential entropy is undefined on zero-measure supports, the iterates should fail to converge or spread unevenly along the intrinsic geometry. A complementary check is to measure, on the paper's own 2D example, the oracle bias $b_k = \\mathbb{E}[\\delta H(p^{\\pi_k}_T) - \\delta H(p^k_\\sharp)\\mid\\mathcal{G}_k]$ as the linear solver's inner iterations grow, since Assumption 7.3 requires this bias to decay to zero.","supporting_citations":[{"cited_title":"Finding mixed nash equilibria of generative adversarial networks","cited_arxiv_id":null,"evidence_quote":"Supplies the mirror-flow framework and the entropy-Fenchel-dual system (MF) used for the continuous-time analysis."},{"cited_title":"R., Hsieh, Y.-P., and Krause, A","cited_arxiv_id":null,"evidence_quote":"Supplies the mirror-flow asymptotic-pseudotrajectory argument and the quoted convergence rate."}],"review_version":2}