{"id":"d1af8197-cd2f-4b1b-a060-a66bd3c5f204","arxiv_id":"2607.18121","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit non-asymptotic residual bounds for Krasnosel'skii–Mann iterations of contractions, with a smooth recovery of the nonexpansive rate as the contraction factor tends to 1.","lead":"This paper gives explicit, non-asymptotic error bounds for the averaged Krasnosel'skii–Mann fixed-point iteration on contracting maps, including the nearly-nonexpansive regime where classic geometric bounds are useless. The bounds come with a principled recipe for choosing the relaxation parameter, which is relevant to optimization algorithms, reinforcement learning, and iterative PDE solvers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the closed-form bound is internally consistent and the κ-dependence is an explicitly stated input, not a correctness gap.","rationale":"The reader's ACCEPT is justified. I checked the core proof chain: recursion validity, Markov-chain interpretation, lattice-path enumeration, hypergeometric transformations, dominated convergence at ℓ=1, and the comparison with the tight bounds S_m. No circularity and no fitted parameters appear. The κ assumption is a genuine practical limitation, but it is explicitly scoped and does not threaten the correctness of the central claim. Hence no change to the verdict.","tokens_in":34882,"tokens_out":31957,"duration_ms":259221,"concrete_test":"Run an independent numerical verification of the central identity: for α ∈ {0.1, 0.5, 0.9}, ℓ ∈ {0.5, 0.9, 0.99}, and m = 0..20, compute R_m by evaluating the exact recursion (12) (or the combinatorial formula (4)) and compare against numerical integration of (7) to 12 significant digits; also confirm the ℓ=1 limit against 2F1(1/2, −m; 2; 4α(1−α)). A mismatch would indicate a mis-derived identity; none is expected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the argument in good faith and cannot identify a load-bearing gap. The central claim is the explicit formula for R_m; the recursion (11) is valid, the cancellation that restricts the double sum to j≥m+1 is secured by ∑_{j=0}^m π^n_j = (1−α)^{n−m}, the lattice-path count matches Theorem 3, and the integral/continuity steps in Theorem 2 check out. The geometric decay follows from ℓ_α < 1, and the ℓ→1 limit is justified by dominated convergence. The only input that can make the bound weak is κ (diameter or a priori bound); the paper states this limitation explicitly and it is a standard applicability condition, not a flaw in the derivation. The omitted Theorem 6 proofs are supporting only. I therefore report no significant objection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives explicit non-asymptotic bounds for the fixed-point residuals of Krasnosel'skii–Mann iterations for ℓ-Lipschitz maps on convex subsets of normed spaces. The main quantity R_m (defined as c_{m,m+1}/α) is shown to satisfy a closed-form expression in terms of (1−ℓ)ℓ_α^m plus an integral/hypergeometric term involving s_α=(1−α+α√ℓ)^2; as ℓ→1 this recovers the Baillon–Bruck nonexpansive bound. The proof strategy combines a recursive estimate (11) with an absorbing Markov chain on Z^2, lattice-path enumeration (Theorem 3), and hypergeometric/integral representations (Propositions 7–8, Theorem 2). The paper also analyses the optimal choice of the relaxation parameter α, gives asymptotic expansions for the minimizer, and extends the bounds to inexact iterations (Theorem 14).","tokens_in":35056,"tokens_out":12794,"duration_ms":126637,"significance":"If correct, this is a substantial contribution to the quantitative analysis of KM iterations. It provides explicit, computable residual bounds that decay geometrically for ℓ<1 and that interpolate smoothly to the nonexpansive Baillon–Bruck bound as ℓ→1. The formulas are derived without any fitted parameters: α is a user-chosen relaxation and ℓ, κ are problem inputs. The derivation is transparent and self-contained, with the lattice-path counting and integral representations carefully justified. The comparison with the minimax-tight transport bounds S_m is honest and non-circular, including a rigorous absolute-gap estimate (Proposition 20) and extensive numerical evidence. The dependence of the bound on the diameter/a-priori constant κ is explicitly stated as an input, which is a standard applicability condition rather than a correctness gap.","major_comments":[],"minor_comments":[{"comment":"Theorem 6 is stated without proof; the text says the proofs 'follow step by step' from [6]. Since this theorem is not used in the derivation of the main results, this is not load-bearing, but the authors should either provide the proofs in an appendix or explicitly state it as a known result with a precise reference.","section":"§2.7, Theorem 6"},{"comment":"The caption says 'Already for ℓ=0.9 the (km) bound improves upon Banach–Picard's coarse bound', but the improvement occurs only for a range of iteration counts, not uniformly in m. Rephrasing to 'can improve over a range of m' would be more accurate.","section":"Figure 1 caption"},{"comment":"There are minor typographical issues, e.g. 'fix ed' in the abstract and the rendering 'Krasnosel'ski ˘ ı–Mann' in the Disclosure. These do not affect the mathematics.","section":"Abstract and Disclosure"},{"comment":"The proof of (34) uses uniform convergence and strong convexity of Ψ_ℓ. The statement is correct, but it would be helpful to note explicitly that the constants in the O(1/m^{3/2}) term depend on ℓ, as they do in (33).","section":"§4.2.1, Proposition 11"}],"recommendation":"accept","confidential_remarks":"The paper is within the scope of the journal and the central results are rigorously proved. The disclosure of AI assistance in identifying a connection is transparent and does not affect the technical assessment, but the editor may wish to confirm it complies with the journal's authorship and disclosure policies."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Best quick take: this is a genuinely useful paper for anyone who needs finite-time residual bounds for Krasnosel'skii-Mann iterations with contractions near 1. The headline result — the closed form for c_{m,n}, hence R_m, with the smooth ℓ→1 limit recovering Baillon–Bruck — is new and it is actually proved, not postulated. The Markov-chain/lattice-path argument is clean and the counting matches Krattenthaler's formula. Theorem 2's integral representations are the workhorse: they make geometric decay visible for ℓ<1 and give a principled way to pick α, with the asymptotics αℓ(m)≈1−c*_ℓ/m. The inexact-iteration convolution bound in Theorem 14 is a solid bonus. I checked recursion (11), the cancellation restricting the double sum, the dominated convergence at ℓ=1; it holds up.\n\nWhat's not to like: not much in the proofs. The main soft spot is the κ-dependence: every bound scales with κ = diam(C) or an a priori bound on ∥x0−T xm∥. If κ is huge or unavailable, the bound is weak or inapplicable. But the paper says this explicitly and it is a standard condition, not a hidden flaw. Theorem 6's proof is omitted (it is a supporting monotonicity/triangle-inequality result), and the relative-gap conjecture R_m/S_m ≤ sqrt(3/2) is clearly labeled open. The comparison with the optimal-transport bound S_m shows the gap is small numerically; the analytic absolute gap estimate (1−α)^2(m+1)ℓ_α^m/α is conservative but fine.\n\nThe only thing I'd want before citing it in a paper: a quick numerical or symbolic check that formula (4) matches the recursion for a few small m,n, and that the hypergeometric evaluations in Theorem 2 agree with direct summation. They probably do — the structure is coherent — but it costs five minutes. The self-citation pattern is heavy but it is genuine prior work in the same line; not a red flag.\n\nWho should read it: people doing iteration-count estimates for value iteration in discounted MDPs, strong-convexity-adjacent optimization, under-relaxed PDE solvers. It deserves a serious referee; I'd send it out. My own verdict is accept with minor revisions, mostly asking for the Theorem 6 proof sketch and a short discussion of how to certify κ in the MDP/optimization applications.","headline":"Solid, well-proved paper that turns a known recursion into an explicit, usable bound for KM residuals; the κ caveat is real but standard.","tokens_in":35554,"tokens_out":2152,"would_cite":true,"duration_ms":395104,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47J25","47J26","65J15","33C05","33C65","33C90","65K15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives an exact, closed-form expression for the finite-step residual bound of the Krasnosel'skii–Mann iteration, showing geometric decay for contractions and continuous recovery of the nonexpansive bound as the contraction factor","keywords":["fixed point iterations","contractive maps","error bounds","lattice paths","under-relaxed Picard iterations","hypergeometric functions","Krasnosel'skii-Mann iteration","non-asymptotic bounds"],"falsifier":"Take a concrete contractive map on a bounded set, e.g., T x = ℓ·x on [−1,1] with κ=2 and α=1/2, run the KM iteration numerically, and compare the observed residual ∥x_m − T x_m∥ to κ R_m for a range of m and ℓ (say ℓ=0.9, 0.99, m up to 100). Any observed residual exceeding the bound would falsify Theorem 2. Alternatively, verify the equality of the two representations (7) and (4) for R_m at random parameters, since (4) follows from the recursive definition and (7) is claimed as an identity.","tokens_in":34751,"feed_emoji":"🧮","tokens_out":8172,"duration_ms":68542,"temperature":0.7,"pith_summary":"The paper establishes that the Krasnosel'skii–Mann under-relaxed iteration, applied to an ℓ-Lipschitz map on a convex set of diameter at most κ, satisfies the explicit finite-step bound ∥x_m − T x_m∥ ≤ κ R_m, where R_m is given by a closed-form expression with a hypergeometric function and an integral. For strict contractions the bound decays geometrically, and as the contraction factor tends to 1 it continuously recovers the classical nonexpansive bound. The same formulas yield the optimal relaxation parameter and error bounds for inexact iterations. If correct, this converts an asymptotic convergence theorem into a computable, parameter-explicit guarantee useful for optimization, Markov decision processes, and under-relaxed numerical solvers.","feed_headline":"Explicit formula gives geometric error bounds for KM iterations","feed_subtitle":"Residuals shrink geometrically for contractions and match the nonexpansive bound as ℓ→1.","key_machinery":"The paper's engine is the family of coefficients c_{m,n} bounding distances between iterates by κ c_{m,n}. These coefficients satisfy a two-term recursion, and the paper interprets them as absorption probabilities in a Markov chain on Z^2 with two absorbing states. A bijection between winning paths and simple lattice paths under the diagonal allows exact counting via binomial differences, giving closed-form sums for c_{m,n}. From there, probabilistic representations in terms of binomial counting processes, followed by the residue theorem, produce the integral and hypergeometric formulas for the residual bound R_m. The same chain perspective, with rewards added at transient states, yields the","core_discovery":"The central claim is an exact identity for the residual bound. For every m≥0, R_m equals (1−ℓ)ℓ_α^m plus a positive integral times s_α^m, with s_α=(1−α+α√ℓ)^2; for ℓ<1 this can be written using an Euler integral as R_m = (1−ℓ)ℓ_α^m + (ℓ/(1−√ℓ)^2) F_1(3/2;−m,1;3;η,−ξ) s_α^m, and at ℓ=1 it reduces to 2F1(1/2,−m;2;4α(1−α)). The derivation shows that the recursive distance coefficients c_{m,n} are absorption probabilities of a Markov chain on Z^2 whose winning paths are counted by lattice-path enumeration, yielding finite sums first and then the integral form. The identity smoothly connects the contractive and nonexpansive regimes, answering the motivating question.","pith_inferences":["The explicit rate s_α^m implies a crossover iteration count m* where geometric decay overtakes the nonexpansive O(1/√m) behavior; estimating m* from the formula could guide adaptive algorithms that switch from KM to the unrelaxed iteration.","The bound suggests a quantitative 'near-nonexpansive' regime: for ℓ=1−ε the initial iterations track the nonexpansive envelope for roughly O(1/ε) steps before contraction dominates.","The same Markov-chain/lattice-path technique may extend to variable relaxation parameters α_m or to stochastic variants of KM, where closed-form bounds are largely missing.","If the conjectured relative gap R_m/S_m ≤ √(3/2) holds, then the closed-form bound is within 23% of the minimax-tight bound, making it a safe drop-in for rigorous stopping criteria."],"forward_implications":["Every ℓ-Lipschitz KM iterate on a set of diameter ≤κ satisfies ∥x_m − T x_m∥ ≤ κ R_m, with R_m computable in closed form for any m, α, ℓ.","For ℓ<1 the bound decays geometrically, so long-horizon problems such as discounted Markov decision processes with discount factor near 1 get explicit non-asymptotic guarantees.","The formula reproduces the classical nonexpansive bound as ℓ→1, so the contractive and nonexpansive regimes are bridged by a single continuous estimate.","The uniqueness and asymptotics of the optimal relaxation parameter give a principled selection rule: for ℓ≤1/2 use α=1, and for ℓ>1/2 use α≈1−c*_ℓ/m.","For inexact iterations, the error bound takes the form of a discrete convolution of the residual bound with the noise sequence, so the effect of computation errors can be propagated explicitly."],"fun_headline_variants":["KM iteration errors nailed by lattice-path counting","Combinatorics unlocks exact KM iteration bounds","New proof: KM residuals decay at known rate","KM fixed-point distance bound from Markov chain","Explicit KM error formula via lattice path sums"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The bound requires a finite, known a priori diameter κ of the domain (or a uniform bound on ∥x_0 − T x_m∥); if κ is unknown or extremely large, the explicit residual bound is either unavailable or too loose to be useful.","fun_headline_variants_meta":{"raw":{"variants":["KM iteration errors nailed by lattice-path counting","Combinatorics unlocks exact KM iteration bounds","New proof: KM residuals decay at known rate","KM fixed-point distance bound from Markov chain","Explicit KM error formula via lattice path sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000108,"raw_usage":{"total_tokens":880,"prompt_tokens":737,"completion_tokens":143,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":74}},"tokens_in":481,"tokens_out":143,"duration_ms":2804,"temperature":1.0,"reasoning_tokens":74,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T15:55:58.906774+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete contractive map on a bounded set, e.g., T x = ℓ·x on [−1,1] with κ=2 and α=1/2, run the KM iteration numerically, and compare the observed residual ∥x_m − T x_m∥ to κ R_m for a range of m and ℓ (say ℓ=0.9, 0.99, m up to 100). Any observed residual exceeding the bound would falsify Theorem 2. Alternatively, verify the equality of the two representations (7) and (4) for R_m at random parameters, since (4) follows from the recursive definition and (7) is claimed as an identity.","supporting_citations":[],"review_version":1}