{"id":"c8abc491-defb-4ccd-907b-fc2d6455994e","arxiv_id":"1908.08264","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"RAWBFST, a least-squares Monte Carlo method with brute-force SVD truncation, provably approximates conditional expectations with derivative weights at any prescribed polynomial convergence rate under sufficient smoothness.","lead":"This paper introduces a Monte Carlo regression method that estimates conditional expectations and derivatives using basis functions tied to both the current and the next time step, with a built-in safeguard against unstable regressions. It proves the estimator can converge at any desired polynomial rate when the problem is smooth, which is relevant for option pricing and for solving nonlinear equations by simulation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.5 proof applies Lemma 5.2 with the wrong event in the probability term; it bounds P(s_i,K^2 ≥ τL), which tends to 1, instead of the truncation probability P(s_i,K^2 < τL). The rate proof is repairable by a sign correction.","rationale":"The reader's weakest_assumption concerned Assumption 2.1 (bounded coefficients and Gaussian-type density) and the heuristic multi-step error propagation in the dynamic programming applications. My read identifies a more immediate issue inside the proof of the central theorem: the application of Lemma 5.2 uses the wrong event in the probability term, and the subsequent display in (29) asserts a bound on the no-truncation probability that is false in the limit. This is load-bearing because it sits at the step where the statistical error is converted into the O(Δ^ρ) rate. However, the surrounding lemmas—especially Lemma 5.5 and the matrix Bernstein bound—provide exactly the bound needed for the complementary event, so the flaw is a localized sign error rather than a fatal gap. The paper also honestly labels the multi-step error propagation as heuristic and the DP sections as exploratory, so those are acknowledged limitations rather than hidden defects. Since the theorem is likely correct after the sign correction, I do not move the verdict away from CONDITIONAL; the reader's verdict already asks for careful independent verification, and this issue reinforces that request. The concrete test above would settle whether the correction suffices.","tokens_in":33745,"tokens_out":37311,"duration_ms":352763,"concrete_test":"Re-derive the application of Lemma 5.2 in the proof of Theorem 2.5 with Γ~_i = {s_i,K^2 ≥ τL}, writing the probability contribution explicitly as max_i P(Γ~_i^c) = max_i P(s_i,K^2 < τL). Then check that Lemma 5.5, via Lemma 4.7 and inequality (19), indeed gives P(s_i,K^2 < τL) ≤ 2K c_{2,paths}^ρ Δ^ρ for Δ ≤ Δ_0. If this bound holds, the theorem's rate proof is valid after the sign correction; if not, the central claim loses its support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 2.5, the set Γ~_i is defined as {s_i,K^2 ≥ τL}, matching Algorithm 2.3's non-truncation event on which the least-squares coefficients are used. Lemma 5.2, however, requires the probability term max_i P(Γ~_i^c), i.e. the probability that the estimator is set to zero, which is P(s_i,K^2 < τL). The proof instead writes P({s_i,K^2 ≥ τL}) in the displayed inequality obtained from Lemma 5.2, and then states in (29) that P({s_i,K^2 ≥ τL}) ≤ C_3 Δ^ρ. This cannot be true in general: as Δ→0, L grows logarithmically, the empirical Gram matrix concentrates around R_Δ, and λ_min(R_Δ)→1 by Lemma 5.3, so s_i,K^2/L → 1 and, for the fixed τ<1 from the theorem, the event s_i,K^2 ≥ τL has probability tending to 1, not O(Δ^ρ). The lemmas actually supply the reverse event: Lemma 5.5, together with Lemma 4.7 and inequality (19), bounds P(s_i,K^2 < L(1−ϵ_Δ)λ_min,Δ) = P(s_i,K^2 < τL) by 2K c_{2,paths}^ρ Δ^ρ. Thus the written proof is internally inconsistent at the point where the statistical error is converted into the final rate. The fix is a sign change in two places: replace P(s_i,K^2 ≥ τL) by P(s_i,K^2 < τL) in the application of Lemma 5.2 and in the statement of (29). With that correction, the argument goes through, because the probability term then matches Lemma 5.5 and the expectation term already uses 1_{s_i,K^2 ≥ τL} as required. This is a genuine proof-presentation flaw in the central theorem, but it is localized and repairable rather than a fundamental counterexample.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes RAWBFST, a least-squares Monte Carlo algorithm for approximating conditional expectations of the form E[H_{ι,Δ}(ξ)y(X_2)|X_1], where X_2 is one Euler step and H_{ι,Δ} is a Malliavin-type weight. The algorithm combines a 'regression anytime' basis construction (local Legendre polynomials depending on both current and future states), stratified uniform sampling with importance weighting, and a brute-force SVD truncation of the empirical regression matrix. The central theoretical result, Theorem 2.5, claims that under a Gaussian-type density bound on the law of X_1 and bounded drift/diffusion coefficients, the L2 error can be made to decay as C log(Δ^{-1})^{D/2} Δ^ρ for any prescribed ρ∈N, provided y is smooth enough. Section 4 develops a standalone noiseless interpolation theorem with exponential statistical error via the matrix Bernstein inequality, and Section 5 gives the proof of Theorem 2.5. Numerical experiments cover a second-derivative approximation, an uncertain volatility model, and a five-dimensional BSDE.","tokens_in":34163,"tokens_out":20540,"duration_ms":183149,"significance":"If the main theorem is correct, this is a substantial contribution to least-squares Monte Carlo: it provides a generic estimator whose convergence rate can be tuned to any polynomial order by increasing smoothness, with explicit parameter choices that are prescribed rather than calibrated to the target result. Theorem 4.3, an exponential concentration bound for noiseless SVD-truncated least squares with non-orthonormal bases, is a useful building block in its own right. The numerical experiments are carefully reported and support the theoretical rates. The main limitations are the assumption of bounded coefficients and Gaussian-type densities (which excludes, for example, geometric Brownian motion) and the fact that the backward dynamic programming applications in Sections 3.2 and 3.3 are exploratory: their error propagation through time steps is heuristic rather than proved.","major_comments":[{"comment":"Lemma 5.2 is applied with Γ̃_i = {s_{i,K}^2 ≥ τL}, so its probability term is max_i P(Γ̃_i^c) = max_i P(s_{i,K}^2 < τL). The proof instead writes max_i P(s_{i,K}^2 ≥ τL) and then states in (29) that this quantity is O(Δ^ρ). This is not true: by Lemma 5.3, λ_min(R_Δ)→1, and the empirical Gram matrix concentrates, so s_{i,K}^2/L → 1 in probability as Δ→0; for the fixed τ<1, the event {s_{i,K}^2 ≥ τL} therefore has probability tending to 1, not O(Δ^ρ). Lemma 4.7, inequality (19), and Lemma 5.5 only bound the complementary event {s_{i,K}^2 < τL}. The fix is a sign change in two places: replace ≥ with < in the probability term after Lemma 5.2 and in the first line of (29). The expectation term in (29), which uses 1_{s_{i,K}^2 ≥ τL}, is correct, so with this correction the proof becomes coherent.","section":"Section 5, proof of Theorem 2.5, display after Lemma 5.2 and Eq. (29)"},{"comment":"After bounding inf_α E|y − Σ α_k η_k|^2 by C_4^2 E[|X_2^{Δ,r2,i} − a_i|_2^{2(Q+1)}], the proof states that the term E[|X_2^{Δ,r2,i} − a_i|_2^{2(|ι|_1+ρ+1)}] is of order Δ^{(2Q+2)γ_cube} = Δ^{ρ+|ι|_1}. The exponent must be 2(Q+1), not 2(|ι|_1+ρ+1), to match the Taylor remainder of degree Q; the displayed equality to Δ^{ρ+|ι|_1} is correct only when Q = |ι|_1+ρ. For the general case Q ≥ |ι|_1+ρ allowed by Theorem 2.5, the written exponent gives a different, generally larger bound. This is a localized error, but it appears in the proof of the main theorem and should be corrected.","section":"Section 5, proof of Theorem 2.5, Taylor remainder estimate"}],"minor_comments":[{"comment":"The recursion for m_{q,r} = E[[ξ_1]^q_r] appears to have an incorrect initial value: for the censored normal variable [ξ]_r, m_{0,r} should be 1, not 0. With m_{0,r}=0, the formula gives m_{2,r} negative for small r. Please also check the factor multiplying (1−Φ(r)); a direct integration by parts for the two-sided censored moment gives a factor 2 in the boundary term.","section":"Remark 2.4, moment recursion"},{"comment":"The assertion that (1,...,1) ∈ supp(X^(2)) is not literally true when b ≠ 0, because the support is the cube [-1,1]^D shifted by an O(Δ^{1−γ_cube}) vector. The stated limit is still correct by continuity of the Legendre polynomials and convergence of the support in Hausdorff distance; please rephrase the argument.","section":"Lemma 5.3, lower bound for m_Δ"},{"comment":"The backward recursion applications assume, without proof, that the one-step error bound of Theorem 2.5 propagates through many time steps at the stated rate. The text already labels these sections exploratory, but it would be helpful to state explicitly in the conclusions that no rigorous error bound for the full backward recursion is proved in this paper.","section":"Sections 3.2 and 3.3, dynamic programming heuristic"},{"comment":"Equation (4) would benefit from parentheses around the exponent of ε; the intended expression is ε^{-D/(Q+1) (1+|ι|_1/ρ)}. The current line breaks make the exponent structure ambiguous.","section":"Throughout, notation"}],"recommendation":"major_revision","confidential_remarks":"The sign error in the proof of Theorem 2.5 is localized and clearly repairable; the surrounding lemmas appear sound and the numerical evidence is consistent with the claimed rates. I recommend asking for a revision rather than rejecting. The paper would also benefit from the authors explicitly acknowledging, in the statement of the main theorem or in the discussion, that the error propagation in the backward dynamic programming applications is not covered by the theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things. First, this is a serious and mostly careful paper: it introduces RAWBFST, proves an exponential statistical error bound for least-squares interpolation with brute-force SVD truncation in the non-orthonormal basis setting (Thm 4.3), and uses it to get an arbitrarily high polynomial rate for conditional expectations with Malliavin weights under smoothness (Thm 2.5). The 'regression anytime' idea is explicitly credited to [6,13,28]; the genuinely new content is the SVD truncation and its concentration analysis.\n\nSecond, there is a real flaw in the proof of Thm 2.5. When the proof applies Lemma 5.2, it writes max_i P(s_i,K^2 ≥ τL), but Lemma 5.2 requires the probability that the estimator is zeroed, i.e. P(s_i,K^2 < τL). The event in the displayed inequality then cannot be bounded by C_3 Δ^ρ: the empirical Gram matrix concentrates, so s_i,K^2/L → λ_min(R_Δ) → 1, and for fixed τ<1 the \"≥ τL\" event has probability tending to 1. Lemma 5.5 bounds the complement. This is a sign error in two places; with the correction the argument goes through. It is localized and repairable, but as printed the proof is internally inconsistent.\n\nWhat the paper does well: the analysis of noiseless regression in Section 4 is sound and self-contained—matrix Bernstein, Gershgorin, and the eigenvalue estimates in Lemma 5.3 are carefully done. The numerical examples support the predicted rates and the parameter choices are transparent. The complexity discussion (Section 2.3) is honest, including the low-dimensional limitation and the log factors.\n\nSoft spots in proportion: besides the sign error, the dynamic programming applications in Sections 3.2 and 3.3 rely on an explicitly heuristic error propagation assumption; the authors say so, but it means those sections are exploratory, not proven. No code or data is shipped, so the numerical tables cannot be independently reproduced. Assumption 2.1 (bounded coefficients, Gaussian upper bound on the density) excludes common financial models like GBM; the authors acknowledge the scope.\n\nWho it's for: anyone working on least-squares Monte Carlo, BSDE schemes, or probabilistic schemes for fully nonlinear PDEs. It deserves a serious referee. I would send it out and require a corrected proof of Theorem 2.5, plus a sentence clarifying the heuristic status of the multi-step error propagation. After that, it's a solid contribution.\n\nRecommendation: accept peer review with the expectation of a minor revision fixing the event.","headline":"A strong, genuinely useful LSMC paper whose central theorem is currently marred by a localized sign error in the written proof — repairable, but the theorem as printed is not proven.","tokens_in":34726,"tokens_out":2907,"would_cite":true,"duration_ms":27449,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C05","60H35","62G08","90C39"],"pacs":[],"model":"deepseek-v4-flash","headline":"A least-squares Monte Carlo algorithm that, under sufficient smoothness, estimates conditional expectations with Malliavin weights at any chosen polynomial convergence rate.","keywords":["least-squares Monte Carlo","regression anytime","Malliavin weights","SVD truncation","conditional expectation","backward stochastic differential equations","dynamic programming","convergence rates"],"falsifier":"Take the one-dimensional example of Section 3.1 but with unbounded coefficients, e.g. $X_2=X_1+\\mu X_1\\Delta+\\sigma X_1\\sqrt{\\Delta}\\xi$ (geometric Brownian motion), keep the same smooth $y$, and measure $\\mathbb{E}\\int|\\hat z-z|^2\\mu_1(dx)$ for $\\rho=2$; if the observed rate is slower than $\\Delta^2$ (up to log factors), the boundedness assumption is essential. Alternatively, in a bounded-coefficient example, compute the empirical probability that the smallest singular value falls below $\\tau L$ as $L$ grows; if that probability does not decay exponentially, the matrix Bernstein bound behind Theorem 4.3 would need revision.","tokens_in":33496,"feed_emoji":"📈","tokens_out":6093,"duration_ms":55520,"temperature":0.7,"pith_summary":"This paper introduces a least-squares Monte Carlo algorithm, 'Regression Anytime with Brute-Force SVD Truncation' (RAWBFST), for approximating conditional expectations that carry stochastic Malliavin derivative weights—random weights that turn a conditional expectation into an approximate derivative. Such objects appear in dynamic programming equations for nonlinear option pricing and in probabilistic discretization schemes for fully nonlinear parabolic PDEs. The central result is that, when the target function is sufficiently smooth, the mean squared error of the estimator can be made to decay at any prescribed polynomial rate in the time step $\\Delta$, rather than at the usual square-root Monte Carlo rate. The design achieves this by using localized Legendre polynomials in the 'later' state, computing their conditional expectations in closed form, and applying a brute-force SVD truncation that kills ill-conditioned regressions. If the claim holds, users can trade smoothness for speed, making the method a practical building block for backward stochastic differential equations and related recursion problems.","feed_headline":"Monte Carlo estimator hits any polynomial rate if target is smooth","feed_subtitle":"A regression-anytime scheme with SVD truncation removes variance blow-up and speeds up nonlinear option pricing and BSDE solvers.","key_machinery":"The machinery has three parts. First, 'regression anytime' basis functions: products of an indicator of a cube in the $X_1$-space and Legendre polynomials of total degree at most $Q$ in the $X_2$-variable, rescaled to the cube. Because the $X_1$-factor is pulled out of the conditional expectation, the expectation of the basis times the Malliavin weight is available in closed form through moments of a truncated normal, so the estimator includes automatic differentiation. Second, importance sampling: the law of $X_1$ is reweighted to a stratified uniform distribution on a cube, so the expected regression matrix is nearly orthonormal and its extremal eigenvalues are bounded away from zero. Third, brute-force SVD truncation: for each empirical regression matrix, either the ordinary least-squares solution is kept when the smallest singular value is above the threshold $\\tau L$, or all coefficients are set to zero, preventing variance explosions. The key estimate is Theorem 4.3, whose proof uses the matrix Bernstein inequality to show that the bad-event probability decays exponentially in the number of samples $L$, making the statistical error negligible relative to the polynomial approximation error.","core_discovery":"The central claim is Theorem 2.5: fix a requested rate $\\rho\\in\\mathbb{N}$; if $y\\in C_b^{Q+1}(\\mathbb{R}^D)$ with $Q\\ge|\\iota|_1+\\rho$, then Algorithm 2.3 with the stated parameter choices produces an estimator $\\hat z$ satisfying $\\mathbb{E}\\int |\\mathbb{E}[H_{\\iota,\\Delta}(\\xi)y(X_2)|X_1=x]-\\hat z(x)|^2\\mu_1(dx)\\le C\\log(\\Delta^{-1})^{D/2}\\Delta^\\rho$ for all sufficiently small $\\Delta$. Here $H_{\\iota,\\Delta}$ is a scaled multivariate Hermite weight, and $X_2$ is one Euler step starting from $X_1$. The engine behind the rate is Theorem 4.3: for a fixed basis, noiseless least-squares regression with brute-force SVD truncation has statistical error that decays exponentially in the number of samples, because the matrix Bernstein inequality controls the probability that the smallest singular value of the empirical regression matrix falls below its threshold. That exponential control is what lets the number of basis functions and the number of samples grow at the same rate, converting a smoothness assumption into a tunable polynomial convergence rate.","pith_inferences":["A natural testable extension is to weaken Assumption 2.1: for processes with unbounded coefficients such as geometric Brownian motion, the Gaussian-density and boundedness conditions fail; one could check numerically whether a growing truncation radius still yields a rate close to the stated polynomial rate or whether the rate degrades to a Monte Carlo rate.","The paper's backward error propagation in the UVM and BSDE examples is heuristic rather than proven; if the one-step rate of Theorem 2.5 does not propagate through many nested conditional expectations, the practical speed-up in long-horizon problems would be smaller than the one-step analysis suggests.","The brute-force SVD rule is binary: keep all coefficients or set all to zero. A natural alternative is a soft or partial truncation that keeps the largest singular values; the exponential tail bound suggests such a variant might retain the rate while reducing bias, but this is not explored in the paper.","The noiseless regression analysis treats the design points as random samples; replacing them with a deterministic quasi-Monte Carlo sequence could remove the log factors and give exact polynomial rates, though the eigenvalue control would need to be re-established for that setting."],"forward_implications":["For a fixed smoothness class, the user can choose $\\rho$ and obtain mean squared error of order $\\Delta^\\rho$; in terms of computational complexity, achieving accuracy $\\epsilon$ costs roughly $\\epsilon^{-D/(Q+1)(1+|\\iota|_1/\\rho)}$, which is better than the $\\epsilon^{-2}$ Monte Carlo baseline when the dimension-to-smoothness ratio is small.","The estimator simultaneously approximates Malliavin derivatives of the regression function, so one run of the algorithm supplies derivative terms needed in backward recursions for nonlinear option pricing and BSDE solvers.","In dynamic programming applications, each time step only requires one-step Euler samples; no full path simulation or path storage is needed, and per-cube regressions are independent, keeping memory use proportional to the number of coefficients rather than the number of trajectories.","The SVD-truncation lemma (Theorem 4.3) applies to any bounded basis with controlled eigenvalues, so the convergence-rate mechanism is not tied to Legendre polynomials and can be reused with other basis families.","The noiseless interpolation result matches, up to log factors, the optimal rates for interpolation with random design, giving a theoretical benchmark for regression-later-type algorithms beyond the specific RAWBFST setting."],"supporting_citations":[{"why":"Introduces the 'regression later' idea that RAWBFST generalizes, motivating basis functions depending on future state variables.","marker":"[19]"},{"why":"Shows Malliavin derivatives can be handled within regression-later, supplying the variance-reduction perspective that RAWBFST exploits.","marker":"[8]"},{"why":"Provides partial convergence results for regression-later estimates that Theorem 2.5 extends and strengthens.","marker":"[10]"},{"why":"Gives stability and accuracy bounds for least-squares with orthonormal bases, which Theorem 4.3 generalizes to non-orthonormal bases.","marker":"[14]"},{"why":"Introduces conditioned least-squares approximation, an abstract predecessor of the brute-force SVD truncation.","marker":"[15]"},{"why":"Supplies the matrix Bernstein inequality used to bound the probability that SVD truncation is triggered.","marker":"[44]"},{"why":"Provides the stratified regression Monte Carlo scheme for BSDEs that serves as the main algorithmic and numerical comparison baseline.","marker":"[21]"},{"why":"Establishes optimal rates for interpolation with random design that Theorem 4.9 matches up to a log factor.","marker":"[30]"},{"why":"Supplies the Aronson-type density estimate used to justify the Gaussian upper bound in Assumption 2.1.","marker":"[32]"}],"fun_headline_variants":["SVD-truncated regression achieves any polynomial MC rate","Brute-force SVD truncation gives any polynomial MC rate","MC regression hits any polynomial rate via SVD truncation","Regression anytime: SVD truncation yields any polynomial rate","SVD-truncated least-squares MC: arbitrary polynomial convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rate proof assumes the current state $X_1$ has a density dominated by a Gaussian density and that the drift and diffusion coefficients are bounded, so the state space can be safely truncated to a cube whose radius grows like $\\sqrt{\\log\\Delta^{-1}}$; if that assumption fails, the truncation bias and the conditioning of the local regressions are no longer controlled.","fun_headline_variants_meta":{"raw":{"variants":["SVD-truncated regression achieves any polynomial MC rate","Brute-force SVD truncation gives any polynomial MC rate","MC regression hits any polynomial rate via SVD truncation","Regression anytime: SVD truncation yields any polynomial rate","SVD-truncated least-squares MC: arbitrary polynomial convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000838,"raw_usage":{"total_tokens":3634,"prompt_tokens":909,"completion_tokens":2725,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":525,"completion_tokens_details":{"reasoning_tokens":2641}},"tokens_in":525,"tokens_out":2725,"duration_ms":17382,"temperature":1.0,"reasoning_tokens":2641,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:45:07.725603+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the one-dimensional example of Section 3.1 but with unbounded coefficients, e.g. $X_2=X_1+\\mu X_1\\Delta+\\sigma X_1\\sqrt{\\Delta}\\xi$ (geometric Brownian motion), keep the same smooth $y$, and measure $\\mathbb{E}\\int|\\hat z-z|^2\\mu_1(dx)$ for $\\rho=2$; if the observed rate is slower than $\\Delta^2$ (up to log factors), the boundedness assumption is essential. Alternatively, in a bounded-coefficient example, compute the empirical probability that the smallest singular value falls below $\\tau L$ as $L$ grows; if that probability does not decay exponentially, the matrix Bernstein bound behind Theorem 4.3 would need revision.","supporting_citations":[{"cited_title":"(2004) Simulation for American options: Regression now or regression later? In: Niederreiter, H","cited_arxiv_id":null,"evidence_quote":"Introduces the 'regression later' idea that RAWBFST generalizes, motivating basis functions depending on future state variables."},{"cited_title":"(2012) Least squares Monte Carlo for BSDEs","cited_arxiv_id":null,"evidence_quote":"Shows Malliavin derivatives can be handled within regression-later, supplying the variance-reduction perspective that RAWBFST exploits."},{"cited_title":"Fast Convergence of Regress-Later Estimates in Least Squares Monte Carlo","cited_arxiv_id":"1309.5274","evidence_quote":"Provides partial convergence results for regression-later estimates that Theorem 2.5 extends and strengthens."},{"cited_title":"A., Leviatan, D","cited_arxiv_id":null,"evidence_quote":"Gives stability and accuracy bounds for least-squares with orthonormal bases, which Theorem 4.3 generalizes to non-orthonormal bases."},{"cited_title":"(2017) Optimal weighted least-squares methods","cited_arxiv_id":null,"evidence_quote":"Introduces conditioned least-squares approximation, an abstract predecessor of the brute-force SVD truncation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the matrix Bernstein inequality used to bound the probability that SVD truncation is triggered."},{"cited_title":"G., Turkedjiev, P., V´ azquez, C","cited_arxiv_id":null,"evidence_quote":"Provides the stratified regression Monte Carlo scheme for BSDEs that serves as the main algorithmic and numerical comparison baseline."},{"cited_title":"(2013) Optimal global rates of convergence for interpolation problems with random design","cited_arxiv_id":null,"evidence_quote":"Establishes optimal rates for interpolation with random design that Theorem 4.9 matches up to a log factor."},{"cited_title":"(2010) On some non asymptotic bounds for the Euler scheme","cited_arxiv_id":null,"evidence_quote":"Supplies the Aronson-type density estimate used to justify the Gaussian upper bound in Assumption 2.1."}],"review_version":1}