Ya-Ping Hsieh

Panayotis Mertikopoulos, Ya-Ping Hsieh, Volkan Cevher

We develop a flexible stochastic approximation framework for analyzing the long-run behavior of learning in games (both continuous and finite). The proposed analysis template incorporates a wide array of popular learning algorithms, including gradient-based methods, the exponential/multiplicative weights algorithm for learning in finite games, optimistic and bandit variants of the above, etc. In addition to providing an integrated view of these algorithms, our framework further allows us to obtain several new convergence results, both asymptotic and in finite time, in both continuous and finite games. Specifically, we provide a range of criteria for identifying classes of Nash equilibria and sets of action profiles that are attracting with high probability, and we also introduce the notion of coherence, a game-theoretic property that includes strict and sharp equilibria, and which leads to convergence in finite time. Importantly, our analysis applies to both oracle-based and bandit, payoff-based methods - that is, when players only observe their realized payoffs.

Ya-Ping Hsieh, Mohammad Reza Karimi, Andreas Krause et al.

Many modern machine learning applications - from online principal component analysis to covariance matrix identification and dictionary learning - can be formulated as minimization problems on Riemannian manifolds, and are typically solved with a Riemannian stochastic gradient method (or some variant thereof). However, in many cases of interest, the resulting minimization problem is not geodesically convex, so the convergence of the chosen solver to a desirable solution - i.e., a local minimizer - is by no means guaranteed. In this paper, we study precisely this question, that is, whether stochastic Riemannian optimization algorithms are guaranteed to avoid saddle points with probability 1. For generality, we study a family of retraction-based methods which, in addition to having a potentially much lower per-iteration cost relative to Riemannian gradient descent, include other widely used algorithms, such as natural policy gradient methods and mirror descent in ordinary convex spaces. In this general setting, we show that, under mild assumptions for the ambient manifold and the oracle providing gradient information, the policies under study avoid strict saddle points / submanifolds with probability 1, from any initial condition. This result provides an important sanity check for the use of gradient methods on manifolds as it shows that, almost always, the limit state of a stochastic Riemannian algorithm can only be a local minimizer.

Vignesh Ram Somnath, Matteo Pariset, Ya-Ping Hsieh et al.

Diffusion Schrödinger bridges (DSB) have recently emerged as a powerful framework for recovering stochastic dynamics via their marginal observations at different time points. Despite numerous successful applications, existing algorithms for solving DSBs have so far failed to utilize the structure of aligned data, which naturally arises in many biological phenomena. In this paper, we propose a novel algorithmic framework that, for the first time, solves DSBs while respecting the data alignment. Our approach hinges on a combination of two decades-old ideas: The classical Schrödinger bridge theory and Doob's $h$-transform. Compared to prior methods, our approach leads to a simpler training procedure with lower variance, which we further augment with principled regularization schemes. This ultimately leads to sizeable improvements across experiments on synthetic and real data, including the tasks of predicting conformational changes in proteins and temporal evolution of cellular differentiation processes.

Mohammad Reza Karimi, Ya-Ping Hsieh, Andreas Krause

Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain many important challenges: Existing guarantees (1) typically only hold for the averaged iterates rather than the more desirable last iterates, (2) lack convergence metrics that capture the scales of the variables such as Wasserstein distances, and (3) mainly apply to elementary schemes such as stochastic gradient Langevin dynamics. In this paper, we develop a new framework that lifts the above issues by harnessing several tools from the theory of dynamical systems. Our key result is that, for a large class of state-of-the-art sampling schemes, their last-iterate convergence in Wasserstein distances can be reduced to the study of their continuous-time counterparts, which is much better understood. Coupled with standard assumptions of MCMC sampling, our theory immediately yields the last-iterate Wasserstein convergence of many advanced sampling schemes such as proximal, randomized mid-point, and Runge-Kutta integrators. Beyond existing methods, our framework also motivates more efficient schemes that enjoy the same rigorous guarantees.

Matteo Pariset, Ya-Ping Hsieh, Charlotte Bunne et al.

Schrödinger bridges (SBs) provide an elegant framework for modeling the temporal evolution of populations in physical, chemical, or biological systems. Such natural processes are commonly subject to changes in population size over time due to the emergence of new species or birth and death events. However, existing neural parameterizations of SBs such as diffusion Schrödinger bridges (DSBs) are restricted to settings in which the endpoints of the stochastic process are both probability measures and assume conservation of mass constraints. To address this limitation, we introduce unbalanced DSBs which model the temporal evolution of marginals with arbitrary finite mass. This is achieved by deriving the time reversal of stochastic differential equations with killing and birth terms. We present two novel algorithmic schemes that comprise a scalable objective function for training unbalanced DSBs and provide a theoretical analysis alongside challenging applications on predicting heterogeneous molecular single-cell responses to various cancer drugs and simulating the emergence and spread of new viral variants.

Mohammad Reza Karimi, Ya-Ping Hsieh, Panayotis Mertikopoulos et al.

We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport, but their behavior is much less understood compared to the Euclidean case because of the lack of a global linear structure on the manifold. We overcome this difficulty by introducing a suitable Fermi coordinate frame which allows us to map the asymptotic behavior of the Riemannian Robbins-Monro (RRM) algorithms under study to that of an associated deterministic dynamical system. In so doing, we provide a general template of almost sure convergence results that mirrors and extends the existing theory for Euclidean Robbins-Monro schemes, despite the significant complications that arise due to the curvature and topology of the underlying manifold. We showcase the flexibility of the proposed framework by applying it to a range of retraction-based variants of the popular optimistic / extra-gradient methods for solving minimization problems and games, and we provide a unified treatment for their convergence.

Tatjana Chavdarova, Ya-Ping Hsieh, Michael I. Jordan

Algorithms that solve zero-sum games, multi-objective agent objectives, or, more generally, variational inequality (VI) problems are notoriously unstable on general problems. Owing to the increasing need for solving such problems in machine learning, this instability has been highlighted in recent years as a significant research challenge. In this paper, we provide an overview of recent progress in the use of continuous-time perspectives in the analysis and design of methods targeting the broad VI problem class. Our presentation draws parallels between single-objective problems and multi-objective problems, highlighting the challenges of the latter. We also formulate various desiderata for algorithms that apply to general VIs and we argue that achieving these desiderata may profit from an understanding of the associated continuous-time dynamics.

Mohammad Reza Karimi, Ya-Ping Hsieh, Andreas Krause

Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich mathematical properties. Recently, the Sinkhorn algorithm has been recast within the mirror descent framework, thus benefiting from classical optimization theory insights. Here, we build upon this result by introducing a continuous-time analogue of the Sinkhorn algorithm. This perspective allows us to derive novel variants of Sinkhorn schemes that are robust to noise and bias. Moreover, our continuous-time dynamics not only generalize but also offer a unified perspective on several recently discovered dynamics in machine learning and mathematics, such as the "Wasserstein mirror flow" of (Deb et al. 2023) or the "mean-field Schrödinger equation" of (Claisse et al. 2023).

Adrian Müller, Antoine Gonon, Zebang Shen et al.

Discrete diffusion models are increasingly competitive for language modeling, yet it remains unclear how their denoising objectives organize learning. Although these objectives target the full data distribution, we show that the exact reverse process induces a hierarchy between coarse support information and finer frequency information. For uniform and absorbing (a.k.a. masking) diffusion, we prove that, in the small-noise regime of the final denoising steps, each single-token reverse edit decomposes into a leading scale, determined by whether it moves toward the data support (e.g., grammatically valid sentences), and a finer coefficient, determining relative probabilities within the same scale. Thus, recovering validity structure only requires learning the correct order of magnitude of reverse probabilities, whereas recovering data frequencies requires coefficient-level estimation. The separation is mechanism-dependent: uniform diffusion exhibits a trichotomy into validity-improving, validity-preserving, and validity-worsening edits, while absorbing diffusion places its leading-order mass on validity-improving moves. Experiments on a masked language diffusion model and synthetic regular-language tasks support these predictions: support-localization emerges earlier than within-support frequency ranking, and the contrast between uniform and absorbing diffusion matches the predicted rate separation. Together, our results suggest that discrete diffusion models learn data support before data frequencies.

Zebang Shen, Ya-Ping Hsieh, Niao He

Diffusion models often generate novel samples even when the learned score is only \emph{coarse} -- a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the \emph{manifold hypothesis}, this behavior can instead be explained by coarse scores capturing the \emph{geometry} of the data while discarding the fine-scale distributional structure of the population measure~$μ_{\scriptscriptstyle\mathrm{data}}$. Concretely, whereas estimating the full data distribution $μ_{\scriptscriptstyle\mathrm{data}}$ supported on a $k$-dimensional manifold is known to require the classical minimax rate $\tilde{\mathcal{O}}(N^{-1/k})$, we prove that diffusion models trained with coarse scores can exploit the \emph{regularity of the manifold support} and attain a near-parametric rate toward a \emph{different} target distribution. This target distribution has density uniformly comparable to that of~$μ_{\scriptscriptstyle\mathrm{data}}$ throughout any $\tilde{\mathcal{O}}\bigl(N^{-β/(4k)}\bigr)$-neighborhood of the manifold, where $β$ denotes the manifold regularity. Our guarantees therefore depend only on the smoothness of the underlying support, and are especially favorable when the data density itself is irregular, for instance non-differentiable. In particular, when the manifold is sufficiently smooth, we obtain that \emph{generalization} -- formalized as the ability to generate novel, high-fidelity samples -- occurs at a statistical rate strictly faster than that required to estimate the full population distribution~$μ_{\scriptscriptstyle\mathrm{data}}$.

Riccardo De Santi, Kimon Protopapas, Ya-Ping Hsieh et al.

Flow and diffusion models are typically pre-trained on limited available data (e.g., molecular samples), covering only a fraction of the valid design space (e.g., the full molecular space). As a consequence, they tend to generate samples from only a narrow portion of the feasible domain. This is a fundamental limitation for scientific discovery applications, where one typically aims to sample valid designs beyond the available data distribution. To this end, we address the challenge of leveraging access to a verifier (e.g., an atomic bonds checker), to adapt a pre-trained flow model so that its induced density expands beyond regions of high data availability, while preserving samples validity. We introduce formal notions of strong and weak verifiers and propose algorithmic frameworks for global and local flow expansion via probability-space optimization. Then, we present Flow Expander (FE), a scalable mirror descent scheme that provably tackles both problems by verifier-constrained entropy maximization over the flow process noised state space. Next, we provide a thorough theoretical analysis of the proposed method, and state convergence guarantees under both idealized and general assumptions. Ultimately, we empirically evaluate our method on both illustrative, yet visually interpretable settings, and on a molecular design task showcasing the ability of FE to expand a pre-trained flow model increasing conformer diversity while preserving validity.

Riccardo De Santi, Marin Vlastelica, Ya-Ping Hsieh et al.

Exploration is critical for solving real-world decision-making problems such as scientific discovery, where the objective is to generate truly novel designs rather than mimic existing data distributions. In this work, we address the challenge of leveraging the representational power of generative models for exploration without relying on explicit uncertainty quantification. We introduce a novel framework that casts exploration as entropy maximization over the approximate data manifold implicitly defined by a pre-trained diffusion model. Then, we present a novel principle for exploration based on density estimation, a problem well-known to be challenging in practice. To overcome this issue and render this method truly scalable, we leverage a fundamental connection between the entropy of the density induced by a diffusion model and its score function. Building on this, we develop an algorithm based on mirror descent that solves the exploration problem as sequential fine-tuning of a pre-trained diffusion model. We prove its convergence to the optimal exploratory diffusion model under realistic assumptions by leveraging recent understanding of mirror flows. Finally, we empirically evaluate our approach on both synthetic and high-dimensional text-to-image diffusion, demonstrating promising results.

Riccardo De Santi, Marin Vlastelica, Ya-Ping Hsieh et al.

Adapting large-scale foundation flow and diffusion generative models to optimize task-specific objectives while preserving prior information is crucial for real-world applications such as molecular design, protein docking, and creative image generation. Existing principled fine-tuning methods aim to maximize the expected reward of generated samples, while retaining knowledge from the pre-trained model via KL-divergence regularization. In this work, we tackle the significantly more general problem of optimizing general utilities beyond average rewards, including risk-averse and novelty-seeking reward maximization, diversity measures for exploration, and experiment design objectives among others. Likewise, we consider more general ways to preserve prior information beyond KL-divergence, such as optimal transport distances and Renyi divergences. To this end, we introduce Flow Density Control (FDC), a simple algorithm that reduces this complex problem to a specific sequence of simpler fine-tuning tasks, each solvable via scalable established methods. We derive convergence guarantees for the proposed scheme under realistic assumptions by leveraging recent understanding of mirror flows. Finally, we validate our method on illustrative settings, text-to-image, and molecular design tasks, showing that it can steer pre-trained generative models to optimize objectives and solve practically relevant tasks beyond the reach of current fine-tuning schemes.

Xiang Li, Zebang Shen, Ya-Ping Hsieh et al.

Score-based methods, such as diffusion models and Bayesian inverse problems, are often interpreted as learning the data distribution in the low-noise limit ($σ\to 0$). In this work, we propose an alternative perspective: their success arises from implicitly learning the data manifold rather than the full distribution. Our claim is based on a novel analysis of scores in the small-$σ$ regime that reveals a sharp separation of scales: information about the data manifold is $Θ(σ^{-2})$ stronger than information about the distribution. We argue that this insight suggests a paradigm shift from the less practical goal of distributional learning to the more attainable task of geometric learning, which provably tolerates $O(σ^{-2})$ larger errors in score approximation. We illustrate this perspective through three consequences: i) in diffusion models, concentration on data support can be achieved with a score error of $o(σ^{-2})$, whereas recovering the specific data distribution requires a much stricter $o(1)$ error; ii) more surprisingly, learning the uniform distribution on the manifold-an especially structured and useful object-is also $O(σ^{-2})$ easier; and iii) in Bayesian inverse problems, the maximum entropy prior is $O(σ^{-2})$ more robust to score errors than generic priors. Finally, we validate our theoretical findings with preliminary experiments on large-scale models, including Stable Diffusion.

Charlotte Bunne, Ya-Ping Hsieh, Marco Cuturi et al.

The static optimal transport $(\mathrm{OT})$ problem between Gaussians seeks to recover an optimal map, or more generally a coupling, to morph a Gaussian into another. It has been well studied and applied to a wide variety of tasks. Here we focus on the dynamic formulation of OT, also known as the Schrödinger bridge (SB) problem, which has recently seen a surge of interest in machine learning due to its connections with diffusion-based generative models. In contrast to the static setting, much less is known about the dynamic setting, even for Gaussian distributions. In this paper, we provide closed-form expressions for SBs between Gaussian measures. In contrast to the static Gaussian OT problem, which can be simply reduced to studying convex programs, our framework for solving SBs requires significantly more involved tools such as Riemannian geometry and generator theory. Notably, we establish that the solutions of SBs between Gaussian measures are themselves Gaussian processes with explicit mean and covariance kernels, and thus are readily amenable for many downstream applications such as generative modeling or interpolation. To demonstrate the utility, we devise a new method for modeling the evolution of single-cell genomics data and report significantly improved numerical stability compared to existing SB-based approaches.

Maria-Luiza Vladarean, Ahmet Alacaoglu, Ya-Ping Hsieh et al.

We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of combinatorial problems, which involve a number of constraints that is polynomial in the problem dimension. The most important feature of our framework is that only a subset of the constraints is processed at each iteration, thus gaining a computational advantage over prior works that require full passes. Our algorithms rely on variance reduction and smoothing used in conjunction with conditional gradient steps, and are accompanied by rigorous convergence guarantees. Preliminary numerical experiments are provided for illustrating the practical performance of the methods.

Ya-Ping Hsieh, Panayotis Mertikopoulos, Volkan Cevher

Compared to ordinary function minimization problems, min-max optimization algorithms encounter far greater challenges because of the existence of periodic cycles and similar phenomena. Even though some of these behaviors can be overcome in the convex-concave regime, the general case is considerably more difficult. On that account, we take an in-depth look at a comprehensive class of state-of-the art algorithms and prevalent heuristics in non-convex / non-concave problems, and we establish the following general results: a) generically, the algorithms' limit points are contained in the ICT sets of a common, mean-field system; b) the attractors of this system also attract the algorithms in question with arbitrarily high probability; and c) all algorithms avoid the system's unstable sets with probability 1. On the surface, this provides a highly optimistic outlook for min-max algorithms; however, we show that there exist spurious attractors that do not contain any stationary points of the problem under study. In this regard, our work suggests that existing min-max algorithms may be subject to inescapable convergence failures. We complement our theoretical analysis by illustrating such attractors in simple, two-dimensional, almost bilinear problems.

Parameswaran Kamalaruban, Yu-Ting Huang, Ya-Ping Hsieh et al.

We introduce a sampling perspective to tackle the challenging task of training robust Reinforcement Learning (RL) agents. Leveraging the powerful Stochastic Gradient Langevin Dynamics, we present a novel, scalable two-player RL algorithm, which is a sampling variant of the two-player policy gradient method. Our algorithm consistently outperforms existing baselines, in terms of generalization across different training and testing conditions, on several MuJoCo environments. Our experiments also show that, even for objective functions that entirely ignore potential environmental shifts, our sampling approach remains highly robust in comparison to standard RL algorithms.

Ya-Ping Hsieh, Chen Liu, Volkan Cevher

We reconsider the training objective of Generative Adversarial Networks (GANs) from the mixed Nash Equilibria (NE) perspective. Inspired by the classical prox methods, we develop a novel algorithmic framework for GANs via an infinite-dimensional two-player game and prove rigorous convergence rates to the mixed NE, resolving the longstanding problem that no provably convergent algorithm exists for general GANs. We then propose a principled procedure to reduce our novel prox methods to simple sampling routines, leading to practically efficient algorithms. Finally, we provide experimental evidence that our approach outperforms methods that seek pure strategy equilibria, such as SGD, Adam, and RMSProp, both in speed and quality.

Ya-Ping Hsieh, Ali Kavis, Paul Rolland et al.

We consider the problem of sampling from constrained distributions, which has posed significant challenges to both non-asymptotic analysis and algorithmic design. We propose a unified framework, which is inspired by the classical mirror descent, to derive novel first-order sampling schemes. We prove that, for a general target distribution with strongly convex potential, our framework implies the existence of a first-order algorithm achieving $\tilde{O}(ε^{-2}d)$ convergence, suggesting that the state-of-the-art $\tilde{O}(ε^{-6}d^5)$ can be vastly improved. With the important Latent Dirichlet Allocation (LDA) application in mind, we specialize our algorithm to sample from Dirichlet posteriors, and derive the first non-asymptotic $\tilde{O}(ε^{-2}d^2)$ rate for first-order sampling. We further extend our framework to the mini-batch setting and prove convergence rates when only stochastic gradients are available. Finally, we report promising experimental results for LDA on real datasets.

Ya-Ping Hsieh, Volkan Cevher

Information concentration of probability measures have important implications in learning theory. Recently, it is discovered that the information content of a log-concave distribution concentrates around their differential entropy, albeit with an unpleasant dependence on the ambient dimension. In this work, we prove that if the potentials of the log-concave distribution are exp-concave, which is a central notion for fast rates in online and statistical learning, then the concentration of information can be further improved to depend only on the exp-concavity parameter, and hence, it can be dimension independent. Central to our proof is a novel yet simple application of the variance Brascamp-Lieb inequality. In the context of learning theory, our concentration-of-information result immediately implies high-probability results to many of the previous bounds that only hold in expectation.