arXiv subjects

Ernest K. Ryu

Publications and source records attributed to Ernest K. Ryu.

At least 19 recordsLinked to original sources

arXiv2026

M-duality in fixed-point problems for general normed spaces

Solving fixed-point problems by reducing residual bounds is one of the central approaches in fixed-point iteration, and, in particular, the Halpern iteration is known to effectively reduce the residual. Recently, a new type of duality theorem has been discovered in the convex optimization and fixed-point problem literature and establish a certain symmetry in the convergence behavior of algorithms that are dual to each other. However, these duality theorems rely on a Lyapunov-style proof structure and mostly focus on Hilbert spaces. In this work, we first establish an M-duality theorem for Mann iterations with Lipschitz operators in general normed spaces, assuming only mild conditions on the coefficients of the iterations. Moreover, leveraging this M-duality, we derive new algorithms that attain a strictly smaller worst-case residual bound than that of the Halpern iteration, which was previously known as the state-of-the-art method in general normed spaces.

math.OC↗

arXiv2026

A Domain-Specific Harness for End-to-End Automation of Optimization Research

We present AutoOPT, a domain-specific harness for end-to-end automation of optimization research. AutoOPT organizes the discovery of optimal first-order methods into four stages: numerical design through the BnB-PEP methodology; symbolic discovery of the analytic description and a convergence proof through frontier large language models (LLMs); formal verification in the Lean 4 proof assistant; and human interpretation and write-up. We demonstrate the framework on two case studies, each of independent interest. The first, lemniscate acceleration, is a new accelerated gradient method for minimizing the gradient norm of a smooth convex function: after $N$ gradient steps it reduces the squared gradient norm at the optimal $O(1/N^{4})$ rate, with a constant governed by the lemniscate constant $\varpi$, a classical elliptic-integral constant. The second is the analytic description of ITEM-f, a method previously known only numerically: for $L$-smooth, $μ$-strongly convex minimization it contracts the function-value gap at an accelerated linear rate with a per-step factor $(1-\sqrt{μ/L})^{2}$. The convergence theorems of both case studies are formalized and machine-checked in Lean 4.

math.OC↗

arXiv2026

H-invariance theory: A complete characterization of minimax optimal fixed-point algorithms

For nonexpansive fixed-point problems, Halpern's method with optimal parameters, its so-called H-dual algorithm, and in fact, an infinite family of algorithms containing them, all exhibit the exactly minimax optimal convergence rates. In this work, we provide a characterization of the complete, exhaustive family of distinct algorithms using predetermined step-sizes, represented as lower triangular H-matrices, which attain the same optimal convergence rate. The characterization is based on polynomials in the entries of the H-matrix that we call H-invariants, whose values stay constant over all optimal H-matrices, together with H-certificates, of which nonnegativity precisely specifies the region of optimality within the common level set of H-invariants. The H-invariance theory we present offers a novel view of optimal acceleration in first-order optimization as a mathematical study of carefully selected invariants, certificates, and structures induced by them.

math.OC↗

arXiv2026

Higher-Order Derivatives Do Not Accelerate the Computation of Fixed Points

The Picard iteration converges to the unique fixed point of a $q$-contractive operator at a linear rate $q^N$, and a lower bound with an affine construction shows that no deterministic method querying only operator values can do better. But what about higher-order methods that query derivatives? A single Jacobian evaluation reveals an affine map entirely, so the affine construction says nothing about higher-order methods. In this work, we show that finite-order derivative information still does not accelerate the worst-case complexity for smooth contractive fixed-point computation. This contrasts with higher-order smooth minimization, where higher-order derivatives do improve worst-case rates for convex and non-convex minimization.

math.OC↗

arXiv2026

Learning Policy from a Single Trajectory in Average-Reward Markov Decision Process

While there is an extensive body of work characterizing the sample complexity of discounted cumulative-reward MDPs, finite sample analyses for average-reward MDPs have been limited, and most existing works rely on restrictive assumptions such as ergodicity or access to a generative model. In this work, we establish the first finite sample complexity guarantees from a single trajectory for weakly communicating average-reward MDPs. To this end, we study the dynamics of a single trajectory in weakly communicating MDPs and based on this analysis, we develop novel model-free methods. Notably, our value-based and policy-based methods provide finite sample complexity guarantees of $\widetilde{O}(1/\varepsilon^2)$ and $\widetilde{O}(1/\varepsilon^4)$ from a single trajectory in weakly communicating MDPs, respectively. Furthermore, we introduce the first model-free method that requires no prior knowledge of problem-dependent quantities for communicating MDPs.

cs.LG↗

arXiv2026

Optimal Acceleration for Proximal Minimization of the Sum of Convex and Strongly Convex Functions

When minimizing the sum of a convex and a strongly convex function, or when finding the zero of the sum of a monotone operator and a strongly monotone operator, Chambolle and Pock (2010) and Davis and Yin (2015) proposed accelerated mechanisms that achieve an $\mathcal{O}(1/N^2)$ convergence rate for the squared distance to the solution, but the optimality of this rate was not established. In this work, we present Fast Douglas--Rachford Splitting (FDR), an accelerated method that improves the constants established in the prior works, and provide a complexity lower bound establishing that both the $\mathcal{O}(1/N^2)$ convergence rate and the leading-order constant of FDR's rate are optimal.

math.OC↗

arXiv2026

Nesterov Acceleration with Operator Decomposition

We propose Nesterov acceleration with Operator Decomposition (NOD), which extends Nesterov's accelerated gradient descent (NAG) from smooth strongly convex optimization to the broader setting of strongly monotone, Lipschitz operators. The key insight is to decompose the operator into cyclically monotone and monotone components, with the Asplund decomposition providing the tightest such representation, and to have the algorithm utilize the decomposed oracles. NOD and its analysis subsume the classical theory of Nesterov acceleration and yield an iteration complexity for finding an $ε$-accurate solution of \[ Θ\left(\sqrt{\frac{L_ϕ}μ + \frac{L_{\mathbb{S}}^2}{μ^2}} \,\log \frac{1}ε\right), \] where $μ$ is the strong monotonicity parameter, $L_ϕ$ is the Lipschitz constant of the cyclically monotone component, and $L_{\mathbb{S}}$ is the Lipschitz constant of the (possibly acyclic) monotone remainder.

math.OC↗

arXiv2026

Nesterov Flow May Travel Infinitely Long to Converge to a Minimizer

Recent work has established that the trajectory of the Nesterov ODE, a the continuous-time model of Nesterov's accelerated gradient method, exhibits point convergence towards a minimizer of a convex potential. A natural next question is whether this point convergence can be upgraded to rectifiability, namely whether the convergent orbit has finite path length. This work provides the answer in the negative by constructing a differentiable convex potential in $\mathbb{R}^2$ for which the flow converges to its minimizer but still accumulates infinite path length. All proofs of this work are due entirely to an internal model at OpenAI.

math.OC↗

arXiv2026

Policy Gradient Algorithms in Average-Reward Multichain MDPs

While there is an extensive body of research analyzing policy gradient methods for discounted cumulative-reward MDPs, prior work on policy gradient methods for average-reward MDPs has been limited, with most existing results restricted to ergodic or unichain settings. In this work, we first establish a policy gradient theorem for average-reward multichain MDPs based on the invariance of the classification of recurrent and transient states. Building on this foundation, we develop refined analyses and obtain a collection of convergence and sample-complexity results that advance the understanding of this setting. In particular, we show that the proposed $α$-clipped policy mirror ascent algorithm attains an $ε$-optimal policy with respect to positive policies.

math.OC↗

arXiv2026

Why Policy Gradient Algorithms Work for Undiscounted Total-Reward MDPs

The classical policy gradient method is the theoretical and conceptual foundation of modern policy-based reinforcement learning (RL) algorithms. Most rigorous analyses of such methods, particularly those establishing convergence guarantees, assume a discount factor $γ< 1$. In contrast, however, a recent line of work on policy-based RL for large language models uses the undiscounted total-reward setting with $γ= 1$, rendering much of the existing theory inapplicable. In this paper, we provide analyses of the policy gradient method for undiscounted expected total-reward infinite-horizon MDPs based on two key insights: (i) the classification of the MDP states into recurrent and transient states is invariant over the set of policies that assign strictly positive probability to every action (as is typical in deep RL models employing a softmax output layer) and (ii) the classical state visitation measure (which may be ill-defined when $γ= 1$) can be replaced with a new object that we call the transient visitation measure.

cs.LG↗

arXiv2026

Optimal Non-Asymptotic Rates of Value Iteration for Average-Reward Markov Decision Processes

While there is an extensive body of research on the analysis of Value Iteration (VI) for discounted cumulative-reward MDPs, prior work on analyzing VI for (undiscounted) average-reward MDPs has been limited, and most prior results focus on asymptotic rates in terms of Bellman error. In this work, we conduct refined non-asymptotic analyses of average-reward MDPs, obtaining a collection of convergence results that advance our understanding of the setup. Among our new results, most notable are the $\mathcal{O}(1/k)$-rates of Anchored Value Iteration on the Bellman error under the multichain setup and the span-based complexity lower bound that matches the $\mathcal{O}(1/k)$ upper bound up to a constant factor of $8$ in the weakly communicating and unichain setups

math.OC↗

arXiv2026

Sharpness-Aware Minimization Can Hallucinate Minimizers

Sharpness-Aware Minimization (SAM) is widely used to seek flatter minima -- often linked to better generalization. In its standard implementation, SAM updates the current iterate using the loss gradient evaluated at a point perturbed by distance $ρ$ along the normalized gradient direction. We show that, for some choices of $ρ$, SAM can stall at points where this shifted (perturbed-point) gradient vanishes despite a nonzero original gradient, and therefore, they are not stationary points of the original loss. We call these points hallucinated minimizers, prove their existence under simple nonconvex landscape conditions (e.g., the presence of a local minimizer and a local maximizer), and establish sufficient conditions for local convergence of the SAM iterates to them. We corroborate this failure mode in neural network training and observe that it aligns with SAM's performance degradation often seen at large $ρ$. Finally, as a practical safeguard, we find that a short initial SGD warm-start before enabling SAM mitigates this failure mode and reduces sensitivity to the choice of $ρ$.

cs.LG↗

arXiv2026

Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof

The Nesterov accelerated gradient method, introduced in 1983, has been a cornerstone of optimization theory and practice. Yet the question of its point convergence had remained open. In this work, we resolve this longstanding open problem in the affirmative. The discovery of the proof was heavily assisted by ChatGPT, a proprietary large language model, and we describe the process through which its assistance was elicited.

math.OC↗

arXiv2025

Computer-Assisted Design of Accelerated Composite Optimization Methods: OptISTA

The accelerated composite optimization method FISTA (Beck, Teboulle 2009) is suboptimal by a constant factor, and we present a new method OptISTA that improves FISTA by a constant factor of 2. The performance estimation problem (PEP) has recently been introduced as a new computer-assisted paradigm for designing optimal first-order methods. In this work, we present a double-function stepsize-optimization PEP methodology that poses the optimization over fixed-step first-order methods for composite optimization as a finite-dimensional nonconvex QCQP, which can be practically solved through spatial branch-and-bound algorithms, and use it to design the exact optimal method OptISTA for the composite optimization setup. We then establish the exact optimality of OptISTA under the large-scale assumption with a lower-bound construction that extends the semi-interpolated zero-chain construction (Drori, Taylor 2022) to the double-function setup of composite optimization. By establishing exact optimality, our work concludes the search for the fastest first-order methods, with respect to the performance measure of worst-case function value suboptimality, for the proximal, projected-gradient, and proximal-gradient setups involving a smooth convex function and a closed proper convex function.

math.OC↗

arXiv2025

Uniqueness of DRS as the 2 Operator Resolvent-Splitting and Impossibility of 3 Operator Resolvent-Splitting

Given the success of Douglas--Rachford splitting (DRS), it is natural to ask whether DRS can be generalized. Are there other 2 operator resolvent-splittings sharing the favorable properties of DRS? Can DRS be generalized to 3 operators? This work presents the answers: no and no. In a certain sense, DRS is the unique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is impossible without lifting, where lifting roughly corresponds to enlarging the problem size. The impossibility result further raises a question. How much lifting is necessary to generalize DRS to 3 operators? This work presents the answer by providing a novel 3 operator resolvent-splitting with provably minimal lifting that directly generalizes DRS.

math.OC↗

arXiv2025

Finite-Time Bounds for Average-Reward Fitted Q-Iteration

Although there is an extensive body of work characterizing the sample complexity of discounted-return offline RL with function approximations, prior work on the average-reward setting has received significantly less attention, and existing approaches rely on restrictive assumptions, such as ergodicity or linearity of the MDP. In this work, we establish the first sample complexity results for average-reward offline RL with function approximation for weakly communicating MDPs, a much milder assumption. To this end, we introduce Anchored Fitted Q-Iteration, which combines the standard Fitted Q-Iteration with an anchor mechanism. We show that the anchor, which can be interpreted as a form of weight decay, is crucial for enabling finite-time analysis in the average-reward setting. We also extend our finite-time analysis to the setup where the dataset is generated from a single-trajectory rather than IID transitions, again leveraging the anchor mechanism.

cs.LG↗

arXiv2025

Traveling Salesman-Based Token Ordering Improves Stability in Homomorphically Encrypted Language Models

As users increasingly interact with large language models (LLMs) using private information, secure and encrypted communication becomes essential. Homomorphic encryption (HE) provides a principled solution by enabling computation directly on encrypted data. Although prior work has explored aspects of running LLMs under HE, the challenge of text generation, particularly next-token prediction, has received limited attention and remains a key obstacle to practical encrypted interaction. In this work, we propose a TSP-based token reordering strategy to address the difficulties of encrypted text generation, together with a post-processing step that further reduces approximation error. Theoretical analysis and experimental results demonstrate that our method prevents collapse, improves coherence in generated text, and preserves data privacy throughout. Overall, our contributions advance the feasibility of practical and privacy-preserving LLM inference.

cs.LG↗

arXiv2025

STORK: Faster Diffusion And Flow Matching Sampling By Resolving Both Stiffness And Structure-Dependence

Diffusion models (DMs) and flow-matching models have demonstrated remarkable performance in image and video generation. However, such models require a significant number of function evaluations (NFEs) during sampling, leading to costly inference. Consequently, quality-preserving fast sampling methods that require fewer NFEs have been an active area of research. However, prior training-free sampling methods fail to simultaneously address two key challenges: the stiffness of the ODE (i.e., the non-straightness of the velocity field) and dependence on the semi-linear structure of the DM ODE (which limits their direct applicability to flow-matching models). In this work, we introduce the Stabilized Taylor Orthogonal Runge--Kutta (STORK) method, addressing both design concerns. We demonstrate that STORK consistently improves the quality of diffusion and flow-matching sampling for image and video generation. Code is available at https://github.com/ZT220501/STORK.

cs.CV↗