arXiv subjects

Dmitry Pasechnyuk-Vilensky

Publications and source records attributed to Dmitry Pasechnyuk-Vilensky.

10 recordsLinked to original sources

arXiv2026

Two Answers on Heavy-Ball Dynamics

We answer two questions about the heavy-ball method on smooth strongly convex functions. The first concerns Polyak's tuning. Its dimension-uniform worst-case exponential rate equals the largest growth factor of an interpolable mixture of rotating geometric sequences, and we determine this factor for every condition number. The optimal mixtures form four algebraic families; the classification combines exact polynomial certificates, a rational parametrization, an exact solution in a quintic number field at the critical three-cycle, and verified interval continuation. Two frequencies suffice, one function in dimension five attains the rate at every horizon, the upper and lower bounds differ by constant factors, and the method converges on the whole class if and only if $κ<9+4\sqrt5$, the threshold located by Badithela and Seiler. The second question concerns arbitrary step sizes and momenta, in particular the regions of the parameter plane between the computed Lyapunov and cycle regions. We describe the phase diagram through the angle $Θ(ω)$ between the characteristic polynomials of the two extreme quadratics. The method converges when $Θ(ω)>ω/2$, has a $k$-periodic orbit when $Θ(2πj/k)\leπ/k$, and the remaining regions are rotation-number gaps organized by Farey fractions. A single strict certificate at one growth exponent bounds the rate, so convergence can be certified at the threshold exponent. In the first level of the gaps, fans of Farey fractions accumulating at a resonant endpoint carry periodic orbits with several harmonics, explicit fan-lag inequalities certify convergence, and both behaviors occur in the regions left open before.

math.OC↗

arXiv2026

Chebyshev-Exact Acceleration under Hessian Variation, II: Computer-Assisted Variational Lower Bound

We study two-step recurrences with positive step sizes and nonnegative momentum coefficients whose terminal residual is the Chebyshev minimax polynomial on a prescribed spectral interval $[μ,L]$. These recurrences have the same terminal error on a fixed quadratic, but their first-order responses to stepwise Hessian perturbations can differ. We prove that the sine-Jacobi recurrence of Part I is asymptotically optimal for this response norm within a factor of $1.000004$, uniformly over $0<μ 1.14268353624$, whereas the sine-Jacobi recurrence attains the constant $2\sqrt{c_{\sin}}$ with $c_{\sin}<1.14269218$. The proof reduces the gain to the diagonal energy of a Jacobi Green matrix. A centering lift represents this matrix by an exponential kernel; packing its diagonal into a density on the line preserves the energy and controls the change in the Birman-Schwinger spectrum. A calibrated identity gives a lower bound using the ground parameter and one Fredholm determinant. Two further determinants and a trace observation improve the bound through a pointwise inequality on an explicit compact state region. One rational certificate verifies this inequality by interval arithmetic. A second certificate constructs a density satisfying the selected observations and both normalizations with energy below the sine value. Finally, no $C^1$ function of finitely many observations of the specified kinds gives a locally sharp lower bound at the sine density.

math.OC↗

arXiv2026

Chebyshev-Exact Acceleration under Hessian Variation, I: Sine-Jacobi Method

We study finite-horizon one-gradient realizations with the Chebyshev minimax terminal residual on $[μ,L]$. Under time-dependent Hessian perturbations, the terminal first variation is governed by a time-ordered spectral kernel $K_s(λ,ν)$; its sharp $\ell_2$ gain is $A_N$. For the prefix-exact Chebyshev recurrence, $$A_N^{pref}=\frac4{\sqrt3}\frac{N^{3/2}}{L-μ}ε_N^\star(1+o(1)),$$ and this is sharp in the causal two-term class with Chebyshev exactness at every prefix. For terminal-only exactness, Jacobi coordinates give $P_N=2^{1-N}T_N$: the spectrum is fixed at the midpoint Chebyshev nodes, while the spectral weights parametrize the realizations. The sine weights give a final-exact Jacobi method with the same terminal residual and $$A_N(J_N^{\sin})=2\sqrt{c_{\sin}}\frac{N^{3/2}}{L-μ}ε_N^\star(1+o(1)), 2\sqrt{c_{\sin}}\approx2.1379<4/\sqrt3.$$ Thus the Chebyshev terminal polynomial does not determine the first-order Hessian-drift gain. Momentum is necessary for this order: every Chebyshev-exact stepsize schedule has $A_N\ge2\sqrt{2/3} N^2ε_N^\star/(L-μ)$. Beyond first order, the response to drift $E_s=H_s-H$ of size $δ=\max_s\|E_s\|$ is a finite polynomial expansion, and for both realizations the terminal error is at most $ε_N^\star T_N(1+2δ/(L-μ))$, the Chebyshev bound of the interval widened by $δ$. This bound is attained, holds for objectives with Lipschitz Hessian, and shows that drift reduces the accelerated rate exponent by the factor $1-\sqrt{δ/μ}$. The two realizations thus share the worst-case response and differ in the $\ell_2$-in-time response measured by $A_N$. The experiments show the finite-horizon effect: lower stochastic curvature overhead, larger admissible-block frontiers, accurate time-varying quadratic predictions, and lower restart cost on an endpoint-coupled smooth strongly convex GLM.

math.OC↗

arXiv2026

Information-Theoretic Upper Bounds for Deterministic Noise in Zeroth-Order Convex Optimization

We study zeroth-order convex optimization on Euclidean balls with function values corrupted by a fixed, uniformly bounded deterministic perturbation. For a query budget $T$, the maximum admissible level of noise (MALN) is the supremum of noise levels for which an $\eps$-accurate solution can be found with probability $1-β$. We construct an explicit hard family: the support function of a spherical belt combined with an affine branch along a hidden random direction. It yields finite-budget upper bounds on the MALN for Lipschitz, strongly convex, uniformly convex, smooth, Hölder, and smooth strongly convex classes. For Lipschitz objectives the bound has order $\eps^2\sqrt\ell/(\sqrt nMR)+\eps\ell/n$, where $\ell=\log(2(T+1)/(1-β))$ and $n-1\ge32\ell$; this is the scale of Li and Risteski with explicit logarithmic factors. A tangential-gradient estimate for the Moreau envelope yields the smooth scale $\eps^{3/2}/(\sqrt n\sqrt L\,R)$ and, for $L-μ\ge5μ$, the smooth strongly convex scale $\eps\sqrt{μ/((L-μ)n)}$. These bounds are tight: on explicit parameter ranges and subclasses, polynomial-query two-point methods tolerate noise of the order of the terms of order $n^{-1/2}$, so that the MALN is determined up to $O(\sqrt\ell)$ and absolute constants whenever these terms dominate. At $L=μ$, simplex interpolation with $n+1$ queries tolerates noise $R\sqrt{2μ\eps/n}$, which is optimal up to $O(\sqrt\ell)$ for $\eps\leμR^2/3$, while with at most $n$ exact queries no randomized algorithm guarantees success probability greater than $1/2$ uniformly over the class. We also quantify noise tolerance near this endpoint and illustrate the hiding mechanism and the breakdown of a two-point method numerically.

math.OC↗

arXiv2026

Learning Latent Graph Geometry via Fixed-Point Schrödinger-Type Activation: A Theoretical Study

We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schrödinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable implicit graph layer. To learn the graph itself, we optimize over the stratified moduli space of weighted graphs and equip each stratum with a non-degenerate Kähler-Hessian metric that keeps natural-gradient descent and face crossing well posed. We then show that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, and that it admits a penalized global relaxation whose stationary states converge to the exact one as the penalty parameter tends to infinity. Reverse-mode differentiation is recovered as the adjoint of the exact global system, and the penalized adjoint converges to it in the same limit. Finally, under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.

cs.LG↗

arXiv2026

Order-Sensitive Sequential Interventions on Ideal Lattices

We study sequential interventions under prerequisite constraints. In this setting, admissible intervention sequences are paths in the ideal lattice of a finite prerequisite poset rather than unconstrained action strings. We give an exact local-to-global theory of order sensitivity on this state space. First, we prove that any two admissible paths with the same endpoints differ by a finite sequence of elementary diamond swaps. Second, for edge-additive path valuations, we show that path-independence is equivalent to vanishing diamond curvature, yielding an endpoint potential with a canonical Möbius parameterization on the ideal lattice. Third, we prove that a local diamond field is induced by an edge-based path model if and only if it satisfies cube consistency, with uniqueness after fixing a reference-tree gauge. Under reduced-state longitudinal assumptions, supported reference paths identify reference-path scores, whereas local order effects require two-sided support of both orders on each diamond. These results yield exact planning consequences, including an order-insensitivity bound and dynamic programming on the truncated ideal lattice.

math.CO↗

arXiv2026

Heterogeneous-Horizon Exact-Weight Local SGD

We study adaptive aggregation for heterogeneous local SGD in convex finite-sum optimization, allowing heterogeneous local horizons, minibatch sizes, gradient noise, and participation. We introduce HEW-Local SGD, a corrected local-SGD method that chooses nodewise server weights by minimizing an explicit one-round upper bound on the next objective value. This yields an exact local-control formulation with a threshold simplex update, separable amplitude updates, and a one-step guarantee under arbitrary predictable participation. We also introduce two post-local variants: a corrected heterogeneous method and a simpler homogeneous specialization. We establish one-step guarantees and global benchmark-style convergence results. In the regimes where comparison is appropriate, the theory matches the qualitative communication-efficient picture of recent LocalSGD/SCAFFOLD analyses, while also giving explicit guarantees for unequal local horizons.

math.OC↗

arXiv2026

Cubic Regularized Newton Method with Variance Reduction for Finite-sum Non-convex Problems

We study finite-sum non-convex optimization $\min_{x\in\mathbb{R}^d} F(x) \;=\; \frac{1}{n}\sum_{i=1}^n f_i(x)$ and analyze a variance-reduced cubic Newton method based on EMA-smoothed SARAH estimators for both gradient and Hessian information. The method combines a coarse stochastic backbone with a terminal homotopy refinement: once the iterates enter a certified small-step regime, the algorithm decreases the regularization level geometrically, shortens the stage length, and increases the mini-batch size at the reciprocal rate while restarting exact finite-sum snapshots at each stage. We work under average squared gradient smoothness and average mean-cubed Hessian smoothness, thereby avoiding the trajectory-wise Hessian boundedness assumption that is often used in related analyses. Under these assumptions and a standard inexact cubic-subproblem certificate, we establish that the method returns an $(\varepsilon,\sqrt{L_2\varepsilon})$-second-order stationary point with total finite-sum oracle complexity $n+\widetilde{\mathcal O}\!\left(n^{1/2}\varepsilon^{-3/2}\right)$. The analysis separates into a coarse progress phase, which yields the $n^{1/2}$-scaled stochastic backbone, and a terminal local bootstrap, which supplies the pointwise accuracy needed to turn the model step certificate into a true second-order certificate.

math.OC↗

arXiv2025

Ethic Duality: A Homological Framework for Primal-Dual Problems

We develop a homological duality framework based on a contravariant functor $D=\operatorname{Hom}_E(-,R)$ with dualizing object $R$. A morphism is called ethic when it satisfies the canonical double-dual compatibility $D^2(f)η=ηf$. In the derived setting, the functor $\mathrm{RHom}_E(-,R)$ produces a graded family of Ext-groups that measure all failures of this compatibility. The first layer $\operatorname{Ext}^1$ identifies primal-dual gaps, while higher $\operatorname{Ext}^k$ provide a systematic hierarchy of derived obstructions to exactness. This formulation specializes uniformly across several classical domains. In linear and conic optimization, Farkas- and Slater-type exactness criteria correspond to the vanishing of $\operatorname{Ext}^1$, and integer duality gaps coincide with torsion Ext-classes. In graph theory, Kirchhoff- and Baker-Norine-type dualities arise as instances of ethic exactness. In dynamical systems, the higher derived layers encode nonvanishing persistence phenomena. Additional examples include social-choice configurations, categorical factorization in scattering formalisms, coding-theoretic duality, and Bellman-type recurrences, all appearing as concrete instances of Ext-controlled exactness. All resulting invariants are stable under derived Morita equivalence and depend only on the dualizing pair $(E,R)$. The framework therefore provides a substrate-independent criterion for primal-dual exactness and a uniform homological description of its obstructions.

math.CT↗

arXiv2025

Moduli space of optimization algorithms

We introduce a geometric and operator-theoretic formalism viewing optimization algorithms as discrete connections on a space of update operators. Each iterative method is encoded by two coupled channels-drift and diffusion-whose algebraic curvature measures the deviation from ideal reversibility and determines the attainable order of accuracy. Flat connections correspond to methods whose updates commute up to higher order and thus achieve minimal numerical dissipation while preserving stability. The formalism recovers classical gradient, proximal, and momentum schemes as first-order flat cases and extends naturally to stochastic, preconditioned, and adaptive algorithms through perturbations controlled by curvature order. Explicit gauge corrections yield higher-order variants with guaranteed energy monotonicity and noise stability. The associated determinantal and isomonodromic formulations yield exact nonasymptotic bounds and describe acceleration effects via Painlevé-type invariants and Stokes corrections.

math.OC↗