{"id":"5f9994e2-ef78-4331-a5d9-210481dbb1e1","arxiv_id":"2501.10051","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A family of Nesterov-type methods with power-law momentum achieves controllable O(1/k^{2α}) convergence for strongly convex objectives at the critical step size, including monotone and proximal variants.","lead":"This paper generalizes the momentum term in Nesterov's accelerated gradient method to a tunable power law, and proves that the resulting algorithms converge at a rate that can be made arbitrarily close to any inverse polynomial at the critical step size. The idea extends to monotone and proximal (FISTA) variants.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's burn-in K must depend on μ (and s,L), not just (α,r); the proof's own asymptotic discriminant condition forces K≳μ^{-2/3} for α≠1, so the stated K(α,r) is false as written.","rationale":"I read the paper in good faith: the Lyapunov construction is original, the phase-space reformulation is coherent, and the claimed O(1/k^{2α}) rate for fixed strongly convex μ is plausible and may well be repairable. The load-bearing weak point is exactly where the reader placed it: the burn-in index K. The theorem states K:=K(α,r), but the proof's asymptotic negativity step has coefficients involving μ and s. In particular, for α≠1 the cross term IV3 in (3.20) has size r√s(α-1)k^{2α-2}, while the strong-convexity diagonal terms SC1 and SC2 have sizes μ s² k^{2α} and μ s k^{2α-1}; forcing the quadratic form to be negative semidefinite requires k³≥r²(α-1)²/(2 μ² s² α)+o(1). Hence K must grow as μ→0. This matters because a literal reading of K(α,r) would survive the limit μ→0 and imply an O(1/k^{2α}) rate for convex functions, contradicting the standard Ω(1/k²) lower bound; the only way out is to make the μ-dependence explicit. The proof also does not actually display the promised 'sum of three non-positive perfect squares'; the discriminant check is the missing step. This is a statement-and-proof defect, not a fatal flaw in the algorithmic idea. The correct fix is to restate K as K(α,r,μ,L,s) and to supply the discriminant/completion-of-squares argument. With that repair, the theorem would deliver what the paper claims: an eventual inverse-polynomial rate of arbitrary degree for each fixed strongly convex problem. I therefore keep the reader's CONDITIONAL verdict; no adjustment is needed, but the conditional should be issued.","tokens_in":19031,"tokens_out":15462,"duration_ms":155125,"concrete_test":"Re-derive the negativity condition for the right-hand side of (3.20) with explicit constants, using the asymptotic expansions (3.22)-(3.24), and solve the discriminant inequality C²≤4D_vD_w for the quadratic form in (v_k, y_k-x*), with D_v,D_w the positive magnitudes of the diagonal coefficients. This yields k³≥r²(α-1)²/(2 μ² s² α)+o(1) for α≠1; hence any valid K in Theorem 3.1 must grow at least as μ^{-2/3}, proving that K(α,r) cannot be independent of μ and requiring the restatement K=K(α,r,μ,L,s).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 3.1: for every f∈S^1_{μ,L} and s∈(0,1/L], if r>2α then there exists K:=K(α,r) such that f(x_k)-f(x*)≤E(K)/(s k^α(k^α+r k^{α-1})) for all k≥K. The proof only establishes that the right-hand side of (3.20) is eventually non-positive, and the required 'eventually' depends on the strong-convexity modulus: to make the quadratic form in (v_k, y_k-x*) negative semidefinite, the SC-terms must dominate the cross term (3.24). Using the paper's own expansions (3.22)-(3.23), the discriminant condition for the 2×2 form is C²≤4D_vD_w, which yields k³≥r²(α-1)²/(2 μ² s² α)+o(1) for α≠1. Thus any valid K must grow at least as μ^{-2/3} as μ→0; the theorem's K(α,r) is therefore false if read literally. If K is allowed to depend on μ (and L,s), the ambiguity disappears and the claimed O(1/k^{2α}) rate is not contradicted by convex lower bounds, but the statement and proof must be amended accordingly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a family of momentum coefficients of the form ((k-1)^α)/(k^α + r k^{α-1}) for forward-backward accelerated methods. For f ∈ S^1_{μ,L} and s ∈ (0,1/L], it claims that when r > 2α there is a positive integer K := K(α,r) such that NAG-α satisfies f(x_k)-f(x*) ≤ E(K)/(s k^α(k^α + r k^{α-1})) for all k ≥ K, i.e. an O(1/k^{2α}) rate. The same rate is claimed for FISTA-α and for the monotonic variants M-NAG-α and M-FISTA-α, and at the critical step size s = 1/L the paper concludes that arbitrarily high inverse-polynomial rates are achievable by choosing α. The proofs are based on a newly constructed Lyapunov function involving mixed and potential energies, with the kinetic energy eliminated in the monotonic setting.","tokens_in":19267,"tokens_out":20908,"duration_ms":199684,"significance":"If the main result can be made correct, this is a useful contribution: it gives explicit, non-fitted Lyapunov constructions for a family of accelerated methods at the critical step size, and it extends them to proximal and monotonic variants. The derivation is transparent in structure, and the non-positivity of the Lyapunov difference is intended to follow from algebraic cancellation rather than from numerical tuning. The numerical experiment on a quadratic is suggestive but not decisive. The principal obstacle is that the central theorem is stated with a burn-in K that depends only on (α,r), while the proof forces K to depend on μ and the step-size parameters; this is a load-bearing flaw that can be repaired by restating the theorem and supplying the missing quantitative discriminant check. With that repair, the paper would be a meaningful addition to the literature on parameterized accelerated methods.","major_comments":[{"comment":"","section":"Theorem 3.1 and its proof"},{"comment":"","section":"Proof of Theorem 3.1, after (3.24)"}],"minor_comments":[{"comment":"","section":"Theorem 4.2"},{"comment":"","section":"Section 4.1, around (4.10)"},{"comment":"","section":"Figure 1"},{"comment":"","section":"Section 5"},{"comment":"","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main mathematical issue is the overstatement of K as a function only of (α,r); this is fixable by restating the theorem with K depending on μ, s, and L and by adding the explicit discriminant condition. The reliance on Li et al. [2024b] and Fu and Shi [2024] is a tool usage rather than a circularity concern. I would encourage the editor to request a revision rather than reject, because the Lyapunov construction and the extension scheme are potentially valuable once the burn-in dependence is correctly stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real extension — replacing the linear momentum coefficient with an α-th power and getting a Lyapunov analysis that avoids the kinetic-energy term, then carrying it through to monotone and proximal variants. That is genuinely new relative to Li-Shi-Yuan and Fu-Shi. The O(1/k^{2α}) rate at the critical step size is the sort of claim people will want to pin down.\n\nThe problem is the statement of Theorem 3.1 (and its FISTA/M-FISTA siblings). The theorem says K = K(α,r), with no dependence on μ. But the proof's own asymptotics force K to depend on μ. To make the quadratic form in (v_k, y_k−x*) non-positive, the strong-convexity terms must dominate the cross term, and the discriminant condition gives k³ ≳ r²(α−1)²/(2 μ² s² α) for α≠1. So any valid K grows at least like μ^{-2/3} as μ→0. Read literally, the theorem is false: taking μ→0 would imply an O(1/k^{2α}) rate for convex functions, contradicting the known Ω(1/k²) lower bound. The fix is easy — restate K as K(α,r,μ,L,s) — but as written it overreaches.\n\nThe proof also glosses the key negativity step. It says the SC terms 'reduce the combined sum to three non-positive perfect squares' without displaying them. That is a real gap, but it looks fillable from the expansions (3.22)–(3.24); I do not see a reason the argument cannot be completed with a concrete K. For M-NAG-α the reformulation in (4.2) is plausible, but the proof there is even terser.\n\nCredit where due: the Lyapunov function without kinetic energy is a nice trick, it genuinely simplifies the monotone extension, and the paper is honest in Remark 3.2 and the conclusion about the indispensability of strong convexity. The reliance on Li et al. [2024b] is a tool use, not a circular argument.\n\nVerdict: worth a serious referee, but the authors need to fix the dependence of K and flesh out the quadratic-form argument. If those land, this is a solid theory contribution. I would not cite it as-is.","headline":"Genuinely new α-power momentum family with claimed O(1/k^{2α}) rates at s=1/L, but the main theorem's burn-in K must depend on μ, and the proof hides that dependence.","tokens_in":19828,"tokens_out":3208,"would_cite":false,"duration_ms":30720,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C25","65K05","90C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Replacing NAG's linear momentum by a power-law coefficient yields an O(1/k^{2α}) rate, tunable to any inverse-polynomial degree on strongly convex objectives.","keywords":["accelerated gradient methods","Nesterov acceleration","momentum coefficients","strong convexity","Lyapunov analysis","forward-backward algorithms","FISTA","critical step size"],"falsifier":"For a fixed $\\alpha>1$ and $r>2\\alpha$, run NAG-$\\alpha$ on the one-dimensional quadratic $f(x)=\\mu x^2/2$ with $L$ fixed and step $s=1/L$, and compute the exact Lyapunov difference $E(k+1)-E(k)$. Find the smallest $K_\\mu$ such that the difference is non-positive for all $k\\ge K_\\mu$. If $K_\\mu$ remained bounded as $\\mu\\to0$, the theorem would force a better-than-$1/k^2$ rate in the convex limit, contradicting the standard $\\Omega(1/k^2)$ lower bound; so the numerical $K_\\mu$ should grow at least like $1/\\sqrt{\\mu}$, and any observed boundedness would falsify the theorem as stated.","tokens_in":18769,"feed_emoji":"📉","tokens_out":16055,"duration_ms":140054,"temperature":0.7,"pith_summary":"This paper claims that a small modification to Nesterov's accelerated gradient method makes its convergence rate tunable: replacing the linear momentum coefficient $(k-1)/(k+r)$ with the $\\alpha$-power form $(k-1)^\\alpha/(k^\\alpha+r k^{\\alpha-1})$ yields an $O(1/k^{2\\alpha})$ function-error bound on strongly convex objectives, provided $r>2\\alpha$ and the step size does not exceed $1/L$. At the critical step size $s=1/L$, where no rate for forward-backward accelerated methods was previously available, choosing $\\alpha$ large gives an inverse-polynomial rate of any desired degree. The same Lyapunov argument extends the rate to monotone variants and, through the proximal fundamental inequality, to composite problems solved by FISTA-type algorithms. A sympathetic reader would care because the result offers a single parameter family that interpolates between and beyond standard accelerated rates and closes a known gap at the critical step size.","feed_headline":"One parameter gives accelerated methods any inverse-polynomial rate","feed_subtitle":"Generalized momentum gives arbitrarily high inverse-polynomial error decay for strongly convex objectives and FISTA.","key_machinery":"The machine that carries the proof is the $\\alpha$-power momentum coefficient $\\frac{(k-1)^\\alpha}{k^\\alpha+r k^{\\alpha-1}}$ with $r>2\\alpha$, paired with a Lyapunov function $E(k)=s k^\\alpha(k^\\alpha+r k^{\\alpha-1})(f(x_k)-f(x_\\star))+\\frac12\\|\\sqrt{s}(k-1)^\\alpha v_k+r(k-1)^{\\alpha-1}(x_k-x_\\star)\\|^2$. The strong-convexity fundamental inequality (2.3) supplies the terms $SC_1$ and $SC_2$ that cancel the positive cross terms; without them, the Lyapunov difference cannot be forced non-positive. For composite objectives, Lemma 2.3 (the proximal version of the same inequality) plays the identical role. The choice $r>2\\alpha$ makes the leading coefficient of the Lyapunov difference behave like $(2\\alpha-r)k^{2\\alpha-1}$, which is negative, and the remaining positive terms are absorbed once the iteration number passes a threshold $K$.","core_discovery":"At the center of the paper is the claim that for any $\\alpha>0$ and $r>2\\alpha$, the NAG-$\\alpha$ iterates (with $x_0=y_0$) driven by the momentum coefficient $\\frac{(k-1)^\\alpha}{k^\\alpha+r k^{\\alpha-1}}$ converge on $f\\in S^1_{\\mu,L}(\\mathbb{R}^d)$ with step size $s\\in(0,1/L]$ at the rate $$f(x_k)-f(x_\\star)\\le \\frac{E(K)}{s\\,k^\\$\\alpha$(k^\\$\\alpha$+r $k^{{\\alpha-1}}$)},\\qquad k\\ge K,$$ where $K:=K(\\alpha,r)$ is the first index at which the Lyapunov function $E(k)$ becomes non-increasing. The rate is $O(1/k^{2\\alpha})$, so at $s=1/L$ the parameter $\\alpha$ controls the degree of the inverse-polynomial rate. The proof identifies a Lyapunov function tailored to $\\alpha$, shows that with $r>2\\alpha$ its leading difference is negative, and uses the strong-convexity terms to absorb the positive cross terms once $k$ is large. A reformulation of the Lyapunov function in terms of $x_k$ and $y_k$ (without the velocity variable) lets the same argument cover the monotone variants, and the proximal fundamental inequality extends it verbatim to FISTA-$\\alpha$ and M-FISTA-$\\alpha$ for composite objectives.","pith_inferences":["Editorial inference: because $K$ is not stated to depend on $\\mu$, the theorem must be read as allowing $K$ to grow when the condition number grows; if $K$ were $\\mu$-independent, sending $\\mu\\to0$ would imply an $O(1/k^{2\\alpha})$ rate for convex functions with $\\alpha>1$, contradicting the standard $\\Omega(1/k^2)$ lower bound.","Editorial inference: a testable consequence is that the onset iteration should scale roughly like $1/\\sqrt{\\mu}$ on ill-conditioned quadratics; plotting the first $k$ at which the Lyapunov difference becomes non-positive against $1/\\mu$ would reveal whether the threshold is tight.","Editorial inference: the family naturally raises the question of whether $\\alpha$ could be allowed to grow with $k$ to recover a linear rate at $s=1/L$; the paper's fixed-degree polynomial analysis does not address that regime.","Editorial inference: extensions to inexact or stochastic oracles are not covered, because those settings break the exact cancellation that makes the Lyapunov difference tractable; any such extension would require a new argument."],"forward_implications":["At step size $s=1/L$, Theorem 3.1 and Corollary 3.3 give $f(x_k)-f(x_\\star)\\le L E(K)/(k^\\alpha(k^\\alpha+r k^{\\alpha-1}))$ for all $k\\ge K$, so choosing $\\alpha$ large with $r>2\\alpha$ yields an inverse-polynomial rate of any desired degree on strongly convex objectives.","For composite objectives $\\Phi=f+g$ with $f\\in S^1_{\\mu,L}$ and convex $g$, the same $O(1/k^{2\\alpha})$ bound applies to FISTA-$\\alpha$ and M-FISTA-$\\alpha$, using the $s$-proximal subgradient and the proximal fundamental inequality.","The monotone variants M-NAG-$\\alpha$ and M-FISTA-$\\alpha$ never increase the objective value and retain the controllable $O(1/k^{2\\alpha})$ rate, so monitoring the objective during iterations does not sacrifice the acceleration guarantee.","The rate is non-asymptotic only after the threshold $K=K(\\alpha,r)$; before that threshold the Lyapunov difference is uncontrolled, so the theorem predicts a parameter-dependent transient phase whose length is not quantified in the paper.","If the strong convexity assumption is dropped, the terms $SC_1$ and $SC_2$ vanish and the Lyapunov difference can no longer be forced non-positive; the paper's mechanism does not extend to merely convex functions."],"supporting_citations":[{"why":"Poses the open problem about forward-backward accelerated methods on strongly convex functions that motivates the critical-step-size analysis.","marker":"Chambolle and Pock [2016]"},{"why":"Supplies the proximal fundamental inequality (Lemma 4) that carries the FISTA-α and M-FISTA-α extensions.","marker":"Li et al. [2024b]"},{"why":"Provides the monotone update construction and Lyapunov strategy that M-NAG-α and M-FISTA-α inherit.","marker":"Fu and Shi [2024]"},{"why":"Introduces FISTA and the composite/proximal inequality on which the generalization to Φ=f+g builds.","marker":"Beck and Teboulle [2009]"},{"why":"Gives the phase-space (implicit-velocity) reformulation used to derive the mixed-energy term of the Lyapunov function.","marker":"Shi et al. [2022]"},{"why":"Defines the baseline NAG whose linear momentum coefficient the α-power family generalizes.","marker":"Nesterov [1983]"}],"fun_headline_variants":["Set α to get any inverse-polynomial rate in accelerated methods","Controllable momentum: arbitrary polynomial speedup at critical step size","New momentum family gives adjustable O(1/k^{2α}) convergence","α-power momentum tunes NAG and FISTA to any polynomial rate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof only controls the Lyapunov function after an iteration threshold $K$, and the theorem states $K:=K(\\alpha,r)$ without making the dependence on the strong-convexity parameter $\\mu$ explicit; the argument needs $K$ to grow when $\\mu$ shrinks, so the promised rate may start arbitrarily late for nearly convex problems.","fun_headline_variants_meta":{"raw":{"variants":["Set α to get any inverse-polynomial rate in accelerated methods","Controllable momentum: arbitrary polynomial speedup at critical step size","New momentum family gives adjustable O(1/k^{2α}) convergence","α-power momentum tunes NAG and FISTA to any polynomial rate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1668,"prompt_tokens":1220,"completion_tokens":448,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":836,"completion_tokens_details":{"reasoning_tokens":372}},"tokens_in":836,"tokens_out":448,"duration_ms":5226,"temperature":1.0,"reasoning_tokens":372,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:25:19.721648+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed $\\alpha>1$ and $r>2\\alpha$, run NAG-$\\alpha$ on the one-dimensional quadratic $f(x)=\\mu x^2/2$ with $L$ fixed and step $s=1/L$, and compute the exact Lyapunov difference $E(k+1)-E(k)$. Find the smallest $K_\\mu$ such that the difference is non-positive for all $k\\ge K_\\mu$. If $K_\\mu$ remained bounded as $\\mu\\to0$, the theorem would force a better-than-$1/k^2$ rate in the convex limit, contradicting the standard $\\Omega(1/k^2)$ lower bound; so the numerical $K_\\mu$ should grow at least like $1/\\sqrt{\\mu}$, and any observed boundedness would falsify the theorem as stated.","supporting_citations":[],"review_version":1}