arXiv subjects

Jungbin Kim

Publications and source records attributed to Jungbin Kim.

6 recordsLinked to original sources

arXiv2026

Track-centric Iterative Learning for Global Trajectory Optimization in Autonomous Racing

This paper presents a global trajectory optimization framework for minimizing lap time in autonomous racing under uncertain vehicle dynamics. Optimizing the trajectory over the full racing horizon is computationally expensive, and tracking such a trajectory in the real world hardly assures global optimality due to uncertain dynamics. Yet, existing work mostly focuses on dynamics learning at the tracking level, without updating the trajectory itself to account for the learned dynamics. To address these challenges, we propose a track-centric approach that directly learns and optimizes the full-horizon trajectory. We first represent trajectories through a track-agnostic parametric space in light of the wavelet transform. This space is then efficiently explored using Bayesian optimization, where the lap time of each candidate is evaluated by running simulations with the learned dynamics. This optimization is embedded in an iterative learning framework, where the optimized trajectory is deployed to collect real-world data for updating the dynamics, progressively refining the trajectory over the iterations. The effectiveness of the proposed framework is validated through simulations and real-world experiments, demonstrating lap time improvement of up to 20.7% over a nominal baseline and consistently outperforming state-of-the-art methods.

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arXiv2025

Horospherically Convex Optimization on Hadamard Manifolds Part I: Analysis and Algorithms

Geodesic convexity (g-convexity) is a natural generalization of convexity to Riemannian manifolds. However, g-convexity lacks many desirable properties satisfied by Euclidean convexity. For instance, the natural notions of half-spaces and affine functions are themselves not g-convex. Moreover, recent studies have shown that the oracle complexity of geodesically convex optimization necessarily depends on the curvature of the manifold (Criscitiello and Boumal, 2022; Criscitiello and Boumal, 2023; Hamilton and Moitra, 2021), a computational bottleneck for several problems, e.g., tensor scaling. Recently, Lewis et al. (2024) addressed this challenge by proving curvature-independent convergence of subgradient descent, assuming horospherical convexity of the objective's sublevel sets. Using a similar idea, we introduce a generalization of convex functions to Hadamard manifolds, utilizing horoballs and Busemann functions as building blocks (as proxies for half-spaces and affine functions). We refer to this new notion as horospherical convexity (h-convexity). We provide algorithms for both nonsmooth and smooth h-convex optimization, which have curvature-independent guarantees exactly matching those from Euclidean space; this includes generalizations of subgradient descent and Nesterov's accelerated method. Motivated by applications, we extend these algorithms and their convergence rates to minimizing a sum of horospherically convex functions, assuming access to a weighted-Fr\'echet-mean oracle.

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arXiv2024

A Proof of the Exact Convergence Rate of Gradient Descent

We prove the exact worst-case convergence rate of gradient descent for smooth strongly convex optimization on $\mathbb{R}^d$. Concretely, assuming that the objective function $f$ is $\mu$-strongly convex and $L$-smooth, we identify the smallest possible value of $\tau$ for which the inequality $f(x_{N})-f_{*}\leq\tau\|x_{0}-x_{*}\|^{2}$ always holds. The result was previously conjectured by Drori and Teboulle for the case $\mu=0$, and by Taylor, Hendrickx, and Glineur for the case $\mu>0$.

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arXiv2024

A Proof of Exact Convergence Rate of Gradient Descent. Part I. Performance Criterion $\Vert \nabla f(x_N)\Vert^2/(f(x_0)-f_*)$

We prove the exact worst-case convergence rate of gradient descent for smooth strongly convex optimization, with respect to the performance criterion $\Vert \nabla f(x_N)\Vert^2/(f(x_0)-f_*)$. The proof differs from the previous one by Rotaru \emph{et al.} [RGP24], and is based on the performance estimation methodology [DT14].

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arXiv2023

Unifying Nesterov's Accelerated Gradient Methods for Convex and Strongly Convex Objective Functions: From Continuous-Time Dynamics to Discrete-Time Algorithms

Although Nesterov's accelerated gradient (NAG) methods have been studied from various perspectives, it remains unclear why the most popular forms of NAG must handle convex and strongly convex objective functions separately. Motivated by this inconsistency, we propose an NAG method that unifies the existing ones for the convex and strongly convex cases. We first design a Lagrangian function that continuously extends the first Bregman Lagrangian to the strongly convex setting. As a specific case of the Euler--Lagrange equation for this Lagrangian, we derive an ordinary differential equation (ODE) model, which we call the unified NAG ODE, that bridges the gap between the ODEs that model NAG for convex and strongly convex objective functions. We then design the unified NAG, a novel momentum method whereby the continuous-time limit corresponds to the unified ODE. The coefficients and the convergence rates of the unified NAG and unified ODE are continuous in the strong convexity parameter $μ$ on $[0, +\infty)$. Unlike the existing popular algorithm and ODE for strongly convex objective functions, the unified NAG and the unified NAG ODE always have superior convergence guarantees compared to the known algorithms and ODEs for non-strongly convex objective functions. This property is beneficial in practical perspective when considering strongly convex objective functions with small $μ$. Furthermore, we extend our unified dynamics and algorithms to the higher-order setting. Last but not least, we propose the unified NAG-G ODE, a novel ODE model for minimizing the gradient norm of strongly convex objective functions. Our unified Lagrangian framework is crucial in the process of constructing this ODE. Fascinatingly, using our novel tool, called the differential kernel, we observe that the unified NAG ODE and the unified NAG-G ODE have an anti-transpose relationship.

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arXiv2022

Accelerated Gradient Methods for Geodesically Convex Optimization: Tractable Algorithms and Convergence Analysis

We propose computationally tractable accelerated first-order methods for Riemannian optimization, extending the Nesterov accelerated gradient (NAG) method. For both geodesically convex and geodesically strongly convex objective functions, our algorithms are shown to have the same iteration complexities as those for the NAG method on Euclidean spaces, under only standard assumptions. To the best of our knowledge, the proposed scheme is the first fully accelerated method for geodesically convex optimization problems. Our convergence analysis makes use of novel metric distortion lemmas as well as carefully designed potential functions. A connection with the continuous-time dynamics for modeling Riemannian acceleration in (Alimisis et al., 2020) is also identified by letting the stepsize tend to zero. We validate our theoretical results through numerical experiments.

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