Ernest K. Ryu

Ernest K. Ryu, Yongxin Chen, Wuchen Li et al.

In many applications such as color image processing, data has more than one piece of information associated with each spatial coordinate, and in such cases the classical optimal mass transport (OMT) must be generalized to handle vector-valued or matrix-valued densities. In this paper, we discuss the vector and matrix optimal mass transport and present three contributions. We first present a rigorous mathematical formulation for these setups and provide analytical results including existence of solutions and strong duality. Next, we present a simple, scalable, and parallelizable methods to solve the vector and matrix-OMT problems. Finally, we implement the proposed methods on a CUDA GPU and present experiments and applications.

TaeHo Yoon, Kibeom Myoung, Keon Lee et al.

Diffusion models have recently shown remarkable success in high-quality image generation. Sometimes, however, a pre-trained diffusion model exhibits partial misalignment in the sense that the model can generate good images, but it sometimes outputs undesirable images. If so, we simply need to prevent the generation of the bad images, and we call this task censoring. In this work, we present censored generation with a pre-trained diffusion model using a reward model trained on minimal human feedback. We show that censoring can be accomplished with extreme human feedback efficiency and that labels generated with a mere few minutes of human feedback are sufficient. Code available at: https://github.com/tetrzim/diffusion-human-feedback.

Sehyun Kwon, Joo Young Choi, Ernest K. Ryu

In many computer vision applications, images are acquired with arbitrary or random rotations and translations, and in such setups, it is desirable to obtain semantic representations disentangled from the image orientation. Examples of such applications include semiconductor wafer defect inspection, plankton microscope images, and inference on single-particle cryo-electron microscopy (cryo-EM) micro-graphs. In this work, we propose Invariant Representation Learning with Implicit Neural Representation (IRL-INR), which uses an implicit neural representation (INR) with a hypernetwork to obtain semantic representations disentangled from the orientation of the image. We show that IRL-INR can effectively learn disentangled semantic representations on more complex images compared to those considered in prior works and show that these semantic representations synergize well with SCAN to produce state-of-the-art unsupervised clustering results.

Sehyun Kwon, Jaeseung Park, Minkyu Kim et al.

Classical clustering methods do not provide users with direct control of the clustering results, and the clustering results may not be consistent with the relevant criterion that a user has in mind. In this work, we present a new methodology for performing image clustering based on user-specified text criteria by leveraging modern vision-language models and large language models. We call our method Image Clustering Conditioned on Text Criteria (IC|TC), and it represents a different paradigm of image clustering. IC|TC requires a minimal and practical degree of human intervention and grants the user significant control over the clustering results in return. Our experiments show that IC|TC can effectively cluster images with various criteria, such as human action, physical location, or the person's mood, while significantly outperforming baselines.

Jongmin Lee, Amin Rakhsha, Ernest K. Ryu et al.

The Value Iteration (VI) algorithm is an iterative procedure to compute the value function of a Markov decision process, and is the basis of many reinforcement learning (RL) algorithms as well. As the error convergence rate of VI as a function of iteration $k$ is $O(γ^k)$, it is slow when the discount factor $γ$ is close to $1$. To accelerate the computation of the value function, we propose Deflated Dynamics Value Iteration (DDVI). DDVI uses matrix splitting and matrix deflation techniques to effectively remove (deflate) the top $s$ dominant eigen-structure of the transition matrix $\mathcal{P}^π$. We prove that this leads to a $\tilde{O}(γ^k |λ_{s+1}|^k)$ convergence rate, where $λ_{s+1}$is $(s+1)$-th largest eigenvalue of the dynamics matrix. We then extend DDVI to the RL setting and present Deflated Dynamics Temporal Difference (DDTD) algorithm. We empirically show the effectiveness of the proposed algorithms.

Zheng Tan, Weizhen Wang, Andrea L. Bertozzi et al.

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.

Seongmin Hong, Suh Yoon Jeon, Kyeonghyun Lee et al.

In latent diffusion models (LDMs), denoising diffusion process efficiently takes place on latent space whose dimension is lower than that of pixel space. Decoder is typically used to transform the representation in latent space to that in pixel space. While a decoder is assumed to have an encoder as an accurate inverse, exact encoder-decoder pair rarely exists in practice even though applications often require precise inversion of decoder. Prior works for decoder inversion in LDMs employed gradient descent inspired by inversions of generative adversarial networks. However, gradient-based methods require larger GPU memory and longer computation time for larger latent space. For example, recent video LDMs can generate more than 16 frames, but GPUs with 24 GB memory can only perform gradient-based decoder inversion for 4 frames. Here, we propose an efficient gradient-free decoder inversion for LDMs, which can be applied to diverse latent models. Theoretical convergence property of our proposed inversion has been investigated not only for the forward step method, but also for the inertial Krasnoselskii-Mann (KM) iterations under mild assumption on cocoercivity that is satisfied by recent LDMs. Our proposed gradient-free method with Adam optimizer and learning rate scheduling significantly reduced computation time and memory usage over prior gradient-based methods and enabled efficient computation in applications such as noise-space watermarking while achieving comparable error levels.

Uijeong Jang, Jason D. Lee, Ernest K. Ryu

Low-rank adaptation (LoRA) has become the standard approach for parameter-efficient fine-tuning of large language models (LLM), but our theoretical understanding of LoRA has been limited. In this work, we theoretically analyze LoRA fine-tuning in the neural tangent kernel (NTK) regime with $N$ data points, showing: (i) full fine-tuning (without LoRA) admits a low-rank solution of rank $r\lesssim \sqrt{N}$; (ii) using LoRA with rank $r\gtrsim \sqrt{N}$ eliminates spurious local minima, allowing gradient descent to find the low-rank solutions; (iii) the low-rank solution found using LoRA generalizes well.

Stephen P. Boyd, Tetiana Parshakova, Ernest K. Ryu et al.

We present a novel methodology for convex optimization algorithm design using ideas from electric RLC circuits. Given an optimization problem, the first stage of the methodology is to design an appropriate electric circuit whose continuous-time dynamics converge to the solution of the optimization problem at hand. Then, the second stage is an automated, computer-assisted discretization of the continuous-time dynamics, yielding a provably convergent discrete-time algorithm. Our methodology recovers many classical (distributed) optimization algorithms and enables users to quickly design and explore a wide range of new algorithms with convergence guarantees.

Joo Young Choi, Jaesung R. Park, Inkyu Park et al.

Current state-of-the-art diffusion models employ U-Net architectures containing convolutional and (qkv) self-attention layers. The U-Net processes images while being conditioned on the time embedding input for each sampling step and the class or caption embedding input corresponding to the desired conditional generation. Such conditioning involves scale-and-shift operations to the convolutional layers but does not directly affect the attention layers. While these standard architectural choices are certainly effective, not conditioning the attention layers feels arbitrary and potentially suboptimal. In this work, we show that simply adding LoRA conditioning to the attention layers without changing or tuning the other parts of the U-Net architecture improves the image generation quality. For example, a drop-in addition of LoRA conditioning to EDM diffusion model yields FID scores of 1.91/1.75 for unconditional and class-conditional CIFAR-10 generation, improving upon the baseline of 1.97/1.79.

Junsu Kim, Jaeyeon Kim, Ernest K. Ryu

Low-rank adaptation (LoRA) has become a standard approach for fine-tuning large foundation models. However, our theoretical understanding of LoRA remains limited as prior analyses of LoRA's training dynamics either rely on linearization arguments or consider highly simplified setups. In this work, we analyze the LoRA loss landscape without such restrictive assumptions. We define two regimes: a "special regime", which includes idealized setups where linearization arguments hold, and a "generic regime" representing more realistic setups where linearization arguments do not hold. In the generic regime, we show that LoRA training converges to a global minimizer with low rank and small magnitude, or a qualitatively distinct solution with high rank and large magnitude. Finally, we argue that the zero-initialization and weight decay in LoRA training induce an implicit bias toward the low-rank, small-magnitude region of the parameter space -- where global minima lie -- thus shedding light on why LoRA training usually succeeds in finding global minima.

Jongmin Lee, Ernest K. Ryu

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.

Jongmin Lee, Ernest K. Ryu

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.

Donghwan Rho, Sieun Seo, Hyewon Sung et al.

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.

Jaesung R. Park, Junsu Kim, Gyeongman Kim et al.

Reinforcement learning with verifiable rewards (RLVR) has recently emerged as the leading approach for enhancing the reasoning capabilities of large language models (LLMs). However, RLVR is prone to entropy collapse, where the LLM quickly converges to a near-deterministic form, hindering exploration and progress during prolonged RL training. In this work, we reveal that the clipping mechanism in PPO and GRPO induces biases on entropy. Through theoretical and empirical analyses, we show that clip-low increases entropy, while clip-high decreases it. Further, under standard clipping parameters, the effect of clip-high dominates, resulting in an overall entropy reduction even when purely random rewards are provided to the RL algorithm. Our findings highlight an overlooked confounding factor in RLVR: independent of the reward signal, the clipping mechanism influences entropy, which in turn affects the reasoning behavior. Furthermore, our analysis demonstrates that clipping can be deliberately used to control entropy. Specifically, with a more aggressive clip-low value, one can increase entropy, promote exploration, and ultimately prevent entropy collapse in RLVR training.

Chanwoong Park, Uijeong Jang, Ernest K. Ryu et al.

Sharpness-Aware Minimization (SAM) is a widely used method that steers training toward flatter minimizers, which typically generalize better. In this work, however, we show that SAM can converge to hallucinated minimizers -- points that are not minimizers of the original objective. We theoretically prove the existence of such hallucinated minimizers and establish conditions for local convergence to them. We further provide empirical evidence demonstrating that SAM can indeed converge to these points in practice. Finally, we propose a simple yet effective remedy for avoiding hallucinated minimizers.

Jaesung R. Park, Jaewook J. Suh, Youngjoon Hong et al.

In deep learning, the recently introduced state space models utilize HiPPO (High-order Polynomial Projection Operators) memory units to approximate continuous-time trajectories of input functions using ordinary differential equations (ODEs), and these techniques have shown empirical success in capturing long-range dependencies in long input sequences. However, the mathematical foundations of these ODEs, particularly the singular HiPPO-LegS (Legendre Scaled) ODE, and their corresponding numerical discretizations remain unsettled. In this work, we fill this gap by establishing that HiPPO-LegS ODE is well-posed despite its singularity, albeit without the freedom of arbitrary initial conditions. Further, we establish convergence of the associated numerical discretization schemes for Riemann integrable input functions.

Jongmin Lee, Ernest K. Ryu

Value Iteration (VI) is foundational to the theory and practice of modern reinforcement learning, and it is known to converge at a $\mathcal{O}(γ^k)$-rate, where $γ$ is the discount factor. Surprisingly, however, the optimal rate for the VI setup was not known, and finding a general acceleration mechanism has been an open problem. In this paper, we present the first accelerated VI for both the Bellman consistency and optimality operators. Our method, called Anc-VI, is based on an \emph{anchoring} mechanism (distinct from Nesterov's acceleration), and it reduces the Bellman error faster than standard VI. In particular, Anc-VI exhibits a $\mathcal{O}(1/k)$-rate for $γ\approx 1$ or even $γ=1$, while standard VI has rate $\mathcal{O}(1)$ for $γ\ge 1-1/k$, where $k$ is the iteration count. We also provide a complexity lower bound matching the upper bound up to a constant factor of $4$, thereby establishing optimality of the accelerated rate of Anc-VI. Finally, we show that the anchoring mechanism provides the same benefit in the approximate VI and Gauss--Seidel VI setups as well.

TaeHo Yoon, Youngsuk Park, Ernest K. Ryu et al.

Probabilistic time series forecasting has played critical role in decision-making processes due to its capability to quantify uncertainties. Deep forecasting models, however, could be prone to input perturbations, and the notion of such perturbations, together with that of robustness, has not even been completely established in the regime of probabilistic forecasting. In this work, we propose a framework for robust probabilistic time series forecasting. First, we generalize the concept of adversarial input perturbations, based on which we formulate the concept of robustness in terms of bounded Wasserstein deviation. Then we extend the randomized smoothing technique to attain robust probabilistic forecasters with theoretical robustness certificates against certain classes of adversarial perturbations. Lastly, extensive experiments demonstrate that our methods are empirically effective in enhancing the forecast quality under additive adversarial attacks and forecast consistency under supplement of noisy observations.

Jongmin Lee, Joo Young Choi, Ernest K. Ryu et al.

The tremendous recent progress in analyzing the training dynamics of overparameterized neural networks has primarily focused on wide networks and therefore does not sufficiently address the role of depth in deep learning. In this work, we present the first trainability guarantee of infinitely deep but narrow neural networks. We study the infinite-depth limit of a multilayer perceptron (MLP) with a specific initialization and establish a trainability guarantee using the NTK theory. We then extend the analysis to an infinitely deep convolutional neural network (CNN) and perform brief experiments.

Albert No, TaeHo Yoon, Sehyun Kwon et al.

Generative adversarial networks (GAN) are a widely used class of deep generative models, but their minimax training dynamics are not understood very well. In this work, we show that GANs with a 2-layer infinite-width generator and a 2-layer finite-width discriminator trained with stochastic gradient ascent-descent have no spurious stationary points. We then show that when the width of the generator is finite but wide, there are no spurious stationary points within a ball whose radius becomes arbitrarily large (to cover the entire parameter space) as the width goes to infinity.

Ernest K. Ryu, Kun Yuan, Wotao Yin

Despite remarkable empirical success, the training dynamics of generative adversarial networks (GAN), which involves solving a minimax game using stochastic gradients, is still poorly understood. In this work, we analyze last-iterate convergence of simultaneous gradient descent (simGD) and its variants under the assumption of convex-concavity, guided by a continuous-time analysis with differential equations. First, we show that simGD, as is, converges with stochastic sub-gradients under strict convexity in the primal variable. Second, we generalize optimistic simGD to accommodate an optimism rate separate from the learning rate and show its convergence with full gradients. Finally, we present anchored simGD, a new method, and show convergence with stochastic subgradients.

Ernest K. Ryu, Jialin Liu, Sicheng Wang et al.

Plug-and-play (PnP) is a non-convex framework that integrates modern denoising priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. An advantage of PnP is that one can use pre-trained denoisers when there is not sufficient data for end-to-end training. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this paper, we theoretically establish convergence of PnP-FBS and PnP-ADMM, without using diminishing stepsizes, under a certain Lipschitz condition on the denoisers. We then propose real spectral normalization, a technique for training deep learning-based denoisers to satisfy the proposed Lipschitz condition. Finally, we present experimental results validating the theory.