Polynomial-Time Algorithm for Optimal Stopping with Fixed Accuracy

On the Slow Convergence to Trivial Solutions of Algorithms for Hard Optimization Problems

Hard combinatorial optimization problems, many of which are NP-hard, present fundamental algorithmic challenges. Average-case analysis on random instances has emerged as a powerful framework for understanding typical algorithmic performance beyond worst-case guarantees. A substantial body of work has established negative results: for sufficiently hard instances (often controlled by the underlying graph connectivity/constraints density), no known polynomial-time algorithm can significantly outperform naive heuristics in the double asymptotic limit where both problem size and constraints density tend to infinity. We revisit this picture by studying the finite-size behavior of some optimization algorithms across easy, intermediate, and hard regimes. Through rigorous analysis of large-graph asymptotics combined with numerical experiments on canonical problems (maximum independent set and maximum $K$-SAT), we demonstrate that while algorithms do eventually converge to theoretically predicted bounds, this convergence can be remarkably slow. In the intermediate regime where instances are already highly constrained, local algorithms achieve solutions substantially better than their predicted performance in the high-constraint-density limit. This gap between finite-regime and asymptotic behavior has important practical implications: sophisticated algorithmic design remains crucial even when asymptotic theory predicts inevitable failure.

Anytime Last Iterates in Fixed Dimension

Optimal Krylov on Average

Abstract. We propose an adaptive randomized truncation estimator for Krylov subspace methods that optimizes the trade-off between the solution variance and the computational cost while remaining unbiased. The estimator solves a constrained optimization problem to compute the truncation probabilities on the fly with minimal computational overhead. The problem has a closed-form solution when the improvement of the deterministic algorithm satisfies a diminishing returns property. We prove that obtaining the optimal adaptive truncation distribution is impossible in the general case. Without the diminishing return condition, our estimator provides a suboptimal but still unbiased solution. We present experimental results in Gaussian process (GP) hyperparameter training and competitive physics-informed neural networks problems to demonstrate the effectiveness of our approach. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/RockyL7/AdaptivelySubsampledKrylov.jl and in the supplementary materials ( AdaptivelySubsampledKrylov_jl-master.zip [14.5KB]). [Formula: see text]

Finite-Time Analysis of the Natural Policy Gradient in Finite-Horizon Markov Decision Processes

Natural Policy Gradient (NPG) is a well-established Reinforcement Learning algorithm that underlies widely used methods such as Trust Region Policy Optimization and Proximal Policy Optimization, both of which have demonstrated strong empirical success. In this paper, we study exact NPG in finite-horizon Markov Decision Processes with known dynamics and horizon-dependent transition kernels. We provide the first finite-time convergence guarantees for this algorithm in this setting, for which we consider both constant and increasing step size regimes. With a constant step size $\eta_t=\eta$, we prove that NPG converges sublinearly with a rate of $\mathcal{O}(H^{2}/t)$ after $t$ iterations, where $H$ is the horizon length. We also extend this constant step size analysis to linear MDPs in an exact population-projection oracle under a full support projection distribution, recovering the same sublinear rate as in the tabular setting. Furthermore, with increasing step sizes, we prove that this algorithm achieves a linear convergence rate of $\mathcal{O}\left(\left(1-\frac{1}{\vartheta_\rho}\right)^t\right)$ for a problem-dependent constant $\vartheta_\rho>1$, and the horizon-only robust schedule of the form $\eta_t=\eta_0(H/(H-1))^t$ where $\eta_0>0$ and $H \geq 2$, attains this same geometric rate.

Finite-Time Analysis of Discounted Exponential-Utility Reinforcement Learning

Discounted exponential utility provides a principled criterion for risk-sensitive sequential decision-making, but its nonlinear structure complicates reinforcement learning. A recent work \citep{thoppe2026reinforcement} addressed this difficulty by introducing a Bellman-compatible surrogate and two model-free fixed-point algorithms for optimizing it over stationary policies. However, their main convergence results are asymptotic. In this work, we establish finite-time rates of $\tilde{O} (1/\sqrt{n})$ for the aforementioned two algorithms under asynchronous Markovian sampling, where $n$ is the iteration index and $\tilde{O}$ hides logarithmic expressions. Importantly, we employ parameter-free choices for the stepsize parameter to derive these rate results. For the algorithmically simpler one-timescale method, the main challenge is that its update equation is not directly aligned with the contraction geometry of its underlying power-law operator. We overcome this mismatch by exploiting the boundedness, monotonicity, and homogeneity of the operator to obtain a local pseudo-contraction property for the relative-error dynamics. We then use a Moreau-envelope-based Lyapunov function and Polyak--Ruppert averaging to obtain the stated convergence rate with parameter-free stepsizes. For the two-timescale method, the main challenge is to control a tracking error on the faster timescale. These results provide the first finite-time guarantees for model-free discounted exponential-utility reinforcement learning.

Beyond Optimal Rates in Stochastic Optimization: Trajectory-Adaptive Stopping Rules

Stochastic gradient descent (SGD) is typically analyzed at a deterministic horizon chosen before the algorithm is run, even though practical stopping decisions are made adaptively by inspecting the evolving trajectory. This mismatch creates a fundamental certification problem: fixed-time guarantees do not generally remain valid at data-dependent stopping times, while deterministic horizons derived from worst-case bounds can be highly conservative. We address this problem for strongly convex stochastic optimization by constructing fully observable, trajectory-adaptive upper confidence sequences for the squared distance of the last iterate to the optimizer and the suboptimality of a weighted average. These bounds hold simultaneously over time, attain the optimal $1/t$ decay rate up to iterated-logarithmic factors in the worst case, and adapt to the realized stochastic gradients, allowing SGD to stop as soon as a prescribed accuracy is certified without sacrificing statistical validity. Our approach treats the evolving SGD trajectory as a sequential experiment whose observations provide evidence about the unknown optimization error. To formalize this perspective, we develop new recursive confidence-sequence techniques and a general time-uniform empirical Bernstein inequality for adapted processes with time-varying conditional means and predictable ranges that may grow without bound. We further extend these confidence-sequence constructions to minibatch SGD, with the empirical Bernstein bounds exploiting the realized second-moment structure within each minibatch. Numerical experiments show that the resulting stopping rules can require several orders of magnitude fewer iterations than natural deterministic horizons.