Keisuke Yano

Faris Chaudhry, Anthea Monod, Keisuke Yano

Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can be arbitrarily conservative and fail to capture the geometry actually encountered by optimization trajectories. In this paper, we develop a trajectory-restricted framework for linear convergence based on localized geometric regularity. We introduce restricted variants of the Polyak--Łojasiewicz inequality, error bound, and quadratic growth conditions that are required to hold only on subsets of the domain. We show that classical convergence guarantees extend under these localized conditions, and in key cases, we develop new arguments that yield explicit relationships between the corresponding constants. The resulting rates are governed by geometric quantities associated with the regions traversed by the algorithm. For polyhedral composite problems, we prove that convergence is controlled by restricted Hoffman constants corresponding to the active polyhedral faces visited along the trajectory. Once the iterates enter a well-conditioned face, the effective condition number improves accordingly. Our work provides a geometric quantification for fast local convergence after active-set or manifold identification and more broadly suggests that linear convergence is fundamentally governed by the geometry of the subsets explored by the algorithm, rather than by worst-case global conditioning.

Akifumi Okuno, Keisuke Yano

This paper discusses the estimation of the generalization gap, the difference between generalization performance and training performance, for overparameterized models including neural networks. We first show that a functional variance, a key concept in defining a widely-applicable information criterion, characterizes the generalization gap even in overparameterized settings where a conventional theory cannot be applied. As the computational cost of the functional variance is expensive for the overparameterized models, we propose an efficient approximation of the function variance, the Langevin approximation of the functional variance (Langevin FV). This method leverages only the $1$st-order gradient of the squared loss function, without referencing the $2$nd-order gradient; this ensures that the computation is efficient and the implementation is consistent with gradient-based optimization algorithms. We demonstrate the Langevin FV numerically by estimating the generalization gaps of overparameterized linear regression and non-linear neural network models, containing more than a thousand of parameters therein.

Kento Nakamura, Keisuke Yano, Fumiyasu Komaki

Ranked data appear in many different applications, including voting and consumer surveys. There often exhibits a situation in which data are partially ranked. Partially ranked data is thought of as missing data. This paper addresses parameter estimation for partially ranked data under a (possibly) non-ignorable missing mechanism. We propose estimators for both complete rankings and missing mechanisms together with a simple estimation procedure. Our estimation procedure leverages a graph regularization in conjunction with the Expectation-Maximization algorithm. Our estimation procedure is theoretically guaranteed to have the convergence properties. We reduce a modeling bias by allowing a non-ignorable missing mechanism. In addition, we avoid the inherent complexity within a non-ignorable missing mechanism by introducing a graph regularization. The experimental results demonstrate that the proposed estimators work well under non-ignorable missing mechanisms.