recurrence solving

Existing iterative solvers for large-scale linear systems suffer from strong dependence on the global condition number and coarse-grained complexity analyses. Method: We introduce the *spectral tail condition number* $kappa_ell$, a new fine-grained spectral measure, and develop a refined time-complexity framework. Our approach formally defines $kappa_ell$, integrates it with the Sketch-and-Project paradigm, Nesterov acceleration, determinant point process sampling, and universality theory for Gaussian matrices, thereby exposing an intrinsic connection between iteration complexity and the matrix multiplication exponent $omega$. Contribution/Results: Our analysis achieves a sharper separation between deterministic and randomized algorithms, yielding an $ ilde{O}(kappa_ell n^2 log(1/varepsilon))$ bound for computing an $varepsilon$-accurate solution—valid for $ell$ up to $O(n^{0.729})$. This significantly improves the fine-grained analysis of the conjugate gradient method and establishes a novel theoretical benchmark for iterative algorithm design.

This paper addresses the problem of selecting one state from each of multiple state types to construct an *m*-state Markov chain that minimizes the average reward (i.e., “gain”) under its stationary distribution—a problem arising from optimal binary AIFV-*m* lossless coding design. Conventional approaches require exponential-time enumeration. We introduce the novel concept of the “Markov chain polytope,” modeling state selection as searching for extreme points of a convex envelope lying beneath a specific hyperplane in convex geometry. Leveraging a separation oracle and the ellipsoid method, we devise the first polynomial-time algorithm for exact optimization. Our framework unifies the construction of optimal Markov chains across several lossless compression paradigms, reducing computational complexity from exponential to polynomial time. This yields the first efficient, exact solver for AIFV-*m* code design and related problems.

This work addresses the unbiased estimation of solutions to Poisson equations and asymptotic variances of ergodic averages for Markov chains. We propose a novel unbiased Monte Carlo method based on exact coupling—i.e., coupling that meets almost surely in a finite random number of steps. Our approach is the first to systematically extend exact coupling techniques to both Poisson equation solving and asymptotic variance estimation, ensuring strict unbiasedness, finiteness of all moments, and finite expected computation time—without truncation or bias correction. Theoretically, under standard ergodicity assumptions, we establish unbiasedness, moment existence, and convergence of the estimator. Empirically, we validate its effectiveness and robustness across synthetic and real-world models.

Randomized algorithms instruction often suffers from a disconnect between theoretical foundations and practical implementation, alongside insufficient coverage of modern developments. Method: This project develops a systematic lecture-note framework for advanced undergraduate and graduate students, integrating core probabilistic tools—including expectation, Chernoff and Hoeffding bounds, martingales, Markov chains, and the Lovász Local Lemma—with classical and cutting-edge algorithmic analyses (e.g., randomized quicksort, hashing, MCMC, approximate counting, and derandomization). Notably, it is the first to incorporate foundational quantum computing concepts and distributed randomized algorithms into a course at this level. Contribution/Results: The resulting resource is logically coherent, self-contained, and immediately deployable in teaching. It has served as the primary textbook for Yale University’s CPSC 4690/5690 course for multiple years and is widely adopted as a key reference in randomized algorithms courses across numerous global institutions.

This work addresses parameterized approximation algorithms for Vertex Cover and 3-Hitting Set. Methodologically, it introduces a novel randomized branching paradigm grounded in an equivalence between the algorithm’s recursive structure and a binary stochastic process. Leveraging a type-theoretic adaptation of Sanov’s theorem, the framework performs large-deviation analysis on bivariate recurrence relations, yielding an analytically tractable master theorem for asymptotic running time. Contribution-wise, this is the first unified theoretical framework providing rigorous approximation-ratio–dependent guarantees across multiple approximation factors. It substantially improves worst-case time complexity over prior deterministic branching approaches and overcomes fundamental analytical limitations inherent in traditional branching analysis. The framework establishes a general methodology for characterizing the asymptotic performance of parameterized approximation algorithms, bridging stochastic analysis and combinatorial optimization.

This study addresses the runtime distribution of probabilistic programs with counters and discrete states (GCP). Building upon probabilistic pushdown automata, we formalize an operational semantics for such programs and rigorously establish, for the first time, that the (sub-)probability generating function of their runtime is algebraic. By characterizing this algebraic function through the roots of its kernel polynomial and leveraging singularity analysis together with the theory of formal power series, we devise a complete algorithm to compute the dominant singularities and radius of convergence. This enables the derivation of precise asymptotic expansions and exponential upper bounds for the runtime distribution. In the single-state subclass, our method achieves theoretical completeness.

This work addresses the finite-time convergence of stochastic iterative algorithms for fixed-point equations accessible only through a noisy oracle. The authors propose a norm-independent, unified Lyapunov function framework constructed via a generalized Moreau envelope, which integrates Lyapunov stability theory with stochastic approximation analysis. This framework accommodates complex settings such as Markovian noise, seminorm contractive operators, and dissipative operators, yielding sharp non-asymptotic convergence bounds in both high-probability and mean-square senses. As a result, it provides a unified and refined finite-time convergence guarantee for a broad class of algorithms, including stochastic gradient descent, linear stochastic approximation, Q-learning, and temporal difference learning.

To address the fundamental challenge of limited parallelizability in sequential models, this paper introduces a unified theoretical framework based on linear dynamical systems (LDS). It formally models classical fixed-point algorithms—including Newton’s method, Picard iteration, and the Jacobi method—as approximate linearizations of nonlinear recurrences. From a dynamical systems perspective, the framework characterizes convergence conditions, applicability domains, and inherent parallelization potential, thereby unifying diverse iterative schemes. Methodologically, it integrates LDS modeling, nonlinear recurrence analysis, and approximate linearization techniques to relax the conventional sequential dependency assumption. The results advance the theoretical understanding of existing parallel sequence modeling approaches and provide principled design guidelines for scalable, efficient parallel algorithms. This work establishes a cohesive theoretical foundation for accelerating sequence modeling computations through parallelization.

This study addresses the problem of efficiently identifying the initial state in partially observable Markov chains, particularly under passive observation and limited-efficiency constraints. To this end, the authors introduce and systematically analyze, for the first time, a novel model termed “Markov chains with support for backtracking,” which permits algorithms to strategically revert the process to prior states to accelerate learning or decision-making. The key contributions include establishing the equivalence between non-adaptive and adaptive backtracking strategies in terms of state distinguishability, and constructing a non-adaptive strategy whose query complexity exceeds that of the optimal adaptive strategy by only a polynomial factor—a gap proven to be unavoidable. The theoretical analysis integrates probability theory, information theory, and computational complexity, thereby establishing a new analytical framework for reasoning about backtracking mechanisms.

Non-asymptotic mixing-time analysis of finite-state ergodic Markov chains—both reversible and irreversible—remains challenging, particularly due to the lack of a unified convergence characterization for irreversible chains. Method: We develop an operator-theoretic framework based on orthogonal projections of the transition operator in the ℓ²(π) space, using matrix norms to quantify convergence rates. Contribution/Results: We establish, for the first time, submultiplicativity of pointwise χ²-divergence for irreversible chains, yielding explicit, computable bounds dependent on spectral structure, algebraic–geometric multiplicity gaps, and condition numbers of similarity transformations. We apply this framework to momentum-based samplers, revealing their diffusive behavior via hypercontractivity and regression analysis. For irreversible triangular random walks on graphs, we derive the tightest known mixing-time bound to date and demonstrate near-optimal O(n¹·⁹⁷) convergence even under V-shaped target distributions.