arXiv subjects

Andre Wibisono

Publications and source records attributed to Andre Wibisono.

At least 19 recordsLinked to original sources

arXiv2026

Accelerated Mixing Time of Randomized Hamiltonian Monte Carlo

We show the Randomized Hamiltonian Monte Carlo (RHMC) algorithm has accelerated mixing time guarantees for sampling from log-concave probability distributions. RHMC proceeds by repeatedly simulating the continuous-time Hamiltonian dynamics for some random integration times, and resetting the velocity to be an independent Gaussian random variable between each simulation. We show that when the target distribution is log-concave and satisfies an $\alpha$-Talagrand inequality (for example, if the target distribution is $\alpha$-strongly log-concave), if we use a random integration time from either the triangular or the exponential distribution with mean $\Theta(\alpha^{-1/2})$, then RHMC converges exponentially fast in KL divergence, and the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(\alpha^{-1/2} \log(\varepsilon^{-1}))$. We also show that when the target distribution is log-concave, if we use a sequence of random integration times from the triangular distribution with exponentially increasing means, then the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(\varepsilon^{-1/2})$. Our analysis relies on a bound on the average KL divergence along Hamiltonian dynamics, which is inspired by an analogous result on accelerated optimization methods based on Hamiltonian dynamics.

stat.ML↗

arXiv2026

Accelerated Convex Optimization via Hamiltonian Dynamics with Deterministic Integration Time

We develop Hamiltonian dynamics-based algorithms for smooth convex optimization that achieve accelerated rates of convergence. By exploiting contraction of averaged Hamiltonian flow trajectories rather than requiring contraction at trajectory endpoints, we show that Hamiltonian dynamics-based optimization methods admit deterministic and accelerated convergence guarantees, extending prior work that is limited to quadratic objectives or holds only in expectation. We analyze an idealized continuous-time algorithm and derive practical discrete-time implementations with optimal first-order complexity, thereby establishing Hamiltonian dynamics as a useful algorithmic primitive for deterministic accelerated convex optimization.

math.OC↗

arXiv2026

Complexity of Non-Log-Concave Sampling in Fisher Information

We study the query complexity of obtaining a relative Fisher information guarantee for sampling from a log-smooth non-log-concave distribution; this is a sampling analog of finding an approximate stationary point in optimization. Our algorithm is based on the proximal sampler, which is an implicit discretization of the Langevin diffusion, and requires an implementation of the backward step known as the restricted Gaussian oracle (RGO). We show that by leveraging the recent results for log-concave sampling with high-accuracy guarantees in R\'enyi divergence, we can obtain an approximate RGO implementation that -- when used with the proximal sampler -- yields a complexity guarantee in relative Fisher information that inherits the same dimension dependence as log-concave sampling, and improves upon prior work for non-log-concave sampling. We also show a converse reduction that any improvement in the dimension dependence in relative Fisher information for non-log-concave sampling will yield an improved dimension dependence for high-accuracy log-concave sampling.

cs.DS↗

arXiv2026

When and Why is Optimistic Multiplicative Weights Slow? The Geometry of Energy Dissipation

This paper studies the convergence of the Optimistic Multiplicative Weights Update algorithm (OMWU) in two player zero-sum games. Recent works have identified instances on which the last-iterate of OMWU can converge arbitrarily slowly, but understanding when and why this slow convergence occurs has remained open. In this work, we develop a new analysis framework that gives sharp, quantitative explanations for this behavior. Our analysis is based on viewing the algorithm's dual iterates as an optimistic skew-gradient descent with respect to an energy function. We prove over the dual iterates that energy is dissipative, and by establishing tight bounds on the magnitude of dissipation, our analysis quantifies the geometric bottlenecks that arise when the corresponding primal iterates are close to the simplex boundary. This further translates into a new linear last-iterate convergence rate in KL divergence on games with a unique and interior Nash equilibrium. Compared to prior work, this new rate contains a much sharper dependence on game-specific constants, and we prove this dependence is optimal. Moreover, these geometric insights further translate into new separations on uniform convergence rates for OMWU. On the one hand, we prove constant lower bounds on the uniform best-iterate convergence rate in KL divergence and total variation distance from Nash. On the other hand, we establish for the $2\times 2$ setting a new ${\widetilde O}(T^{-1/2})$ best-iterate rate in duality gap, improving substantially over prior work. Together, this shows in general that uniform convergence rate guarantees do not transfer across different measures of distance to Nash.

cs.GT↗

arXiv2026

The Geometry of Efficient Nonconvex Sampling

We present an efficient algorithm for uniformly sampling from an arbitrary compact body $\mathcal{X} \subset \mathbb{R}^n$ from a warm start under isoperimetry and a natural volume growth condition. Our result provides a substantial common generalization of known results for convex bodies and star-shaped bodies. The complexity of the algorithm is polynomial in the dimension, the Poincar\'e constant of the uniform distribution on $\mathcal{X}$ and the volume growth constant of the set $\mathcal{X}$.

cs.DS↗

arXiv2026

Tail-Sensitive KL and R\'enyi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings

Hamiltonian Monte Carlo (HMC) algorithms are among the most widely used sampling methods in high dimensional settings, yet their convergence properties are poorly understood in divergences that quantify relative density mismatch, such as Kullback-Leibler (KL) and R\'enyi divergences. These divergences naturally govern acceptance probabilities and warm-start requirements for Metropolis-adjusted Markov chains. In this work, we develop a framework for upgrading Wasserstein convergence guarantees for unadjusted Hamiltonian Monte Carlo (uHMC) to guarantees in tail-sensitive KL and R\'enyi divergences. Our approach is based on one-shot couplings, which we use to establish a regularization property of the uHMC transition kernel. This regularization allows Wasserstein-2 mixing-time and asymptotic bias bounds to be lifted to KL divergence, and analogous Orlicz-Wasserstein bounds to be lifted to R\'enyi divergence, paralleling earlier work of Bou-Rabee and Eberle (2023) that upgrade Wasserstein-1 bounds to total variation distance via kernel smoothing. As a consequence, our results provide quantitative control of relative density mismatch, clarify the role of discretization bias in strong divergences, and yield principled guarantees relevant both for unadjusted sampling and for generating warm starts for Metropolis-adjusted Markov chains.

stat.ML↗

arXiv2025

Fast and Furious Symmetric Learning in Zero-Sum Games: Gradient Descent as Fictitious Play

This paper investigates the sublinear regret guarantees of two non-no-regret algorithms in zero-sum games: Fictitious Play, and Online Gradient Descent with constant stepsizes. In general adversarial online learning settings, both algorithms may exhibit instability and linear regret due to no regularization (Fictitious Play) or small amounts of regularization (Gradient Descent). However, their ability to obtain tighter regret bounds in two-player zero-sum games is less understood. In this work, we obtain strong new regret guarantees for both algorithms on a class of symmetric zero-sum games that generalize the classic three-strategy Rock-Paper-Scissors to a weighted, n-dimensional regime. Under symmetric initializations of the players' strategies, we prove that Fictitious Play with any tiebreaking rule has $O(\sqrt{T})$ regret, establishing a new class of games for which Karlin's Fictitious Play conjecture holds. Moreover, by leveraging a connection between the geometry of the iterates of Fictitious Play and Gradient Descent in the dual space of payoff vectors, we prove that Gradient Descent, for almost all symmetric initializations, obtains a similar $O(\sqrt{T})$ regret bound when its stepsize is a sufficiently large constant. For Gradient Descent, this establishes the first "fast and furious" behavior (i.e., sublinear regret without time-vanishing stepsizes) for zero-sum games larger than 2x2.

cs.LG↗

arXiv2025

Hamiltonian Descent Algorithms for Optimization: Accelerated Rates via Randomized Integration Time

We study the Hamiltonian flow for optimization (HF-opt), which simulates the Hamiltonian dynamics for some integration time and resets the velocity to $0$ to decrease the objective function; this is the optimization analogue of the Hamiltonian Monte Carlo algorithm for sampling. For short integration time, HF-opt has the same convergence rates as gradient descent for minimizing strongly and weakly convex functions. We show that by randomizing the integration time in HF-opt, the resulting randomized Hamiltonian flow (RHF) achieves accelerated convergence rates in continuous time, similar to the rates for the accelerated gradient flow. We study a discrete-time implementation of RHF as the randomized Hamiltonian gradient descent (RHGD) algorithm. We prove that RHGD achieves the same accelerated convergence rates as Nesterov's accelerated gradient descent (AGD) for minimizing smooth strongly and weakly convex functions. We provide numerical experiments to demonstrate that RHGD is competitive with classical accelerated methods such as AGD across all settings and outperforms them in certain regimes.

math.OC↗

arXiv2025

Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality

We study the mixing time guarantee for sampling in relative Fisher information via the Proximal Sampler algorithm, which is an approximate proximal discretization of the Langevin dynamics. We show that when the target probability distribution is strongly log-concave, the relative Fisher information converges exponentially fast along the Proximal Sampler; this matches the exponential convergence rate of the relative Fisher information along the continuous-time Langevin dynamics for strongly log-concave target. When combined with a standard implementation of the Proximal Sampler via rejection sampling, this exponential convergence rate provides a high-accuracy iteration complexity guarantee for the Proximal Sampler in relative Fisher information when the target distribution is strongly log-concave and log-smooth. Our proof proceeds by establishing a strong data processing inequality for relative Fisher information along the Gaussian channel under strong log-concavity, and a data processing inequality along the reverse Gaussian channel for a special distribution. The forward and reverse Gaussian channels compose to form the Proximal Sampler, and these data processing inequalities imply the exponential convergence rate of the relative Fisher information along the Proximal Sampler.

cs.IT↗

arXiv2024

On the Convergence of Min-Max Langevin Dynamics and Algorithm

We study zero-sum games in the space of probability distributions over the Euclidean space $\mathbb{R}^d$ with entropy regularization, in the setting when the interaction function between the players is smooth and strongly convex-strongly concave. We prove an exponential convergence guarantee for the mean-field min-max Langevin dynamics to compute the equilibrium distribution of the zero-sum game. We also study the finite-particle approximation of the mean-field min-max Langevin dynamics, both in continuous and discrete times. We prove biased convergence guarantees for the continuous-time finite-particle min-max Langevin dynamics to the stationary mean-field equilibrium distribution with an explicit bias term which does not scale with the number of particles. We also prove biased convergence guarantees for the discrete-time finite-particle min-max Langevin algorithm to the stationary mean-field equilibrium distribution with an additional bias term which scales with the step size and the number of particles. This provides an explicit iteration complexity for the average particle along the finite-particle algorithm to approximately compute the equilibrium distribution of the zero-sum game.

cs.GT↗

arXiv2024

High-accuracy sampling from constrained spaces with the Metropolis-adjusted Preconditioned Langevin Algorithm

In this work, we propose a first-order sampling method called the Metropolis-adjusted Preconditioned Langevin Algorithm for approximate sampling from a target distribution whose support is a proper convex subset of $\mathbb{R}^{d}$. Our proposed method is the result of applying a Metropolis-Hastings filter to the Markov chain formed by a single step of the preconditioned Langevin algorithm with a metric $\mathscr{G}$, and is motivated by the natural gradient descent algorithm for optimisation. We derive non-asymptotic upper bounds for the mixing time of this method for sampling from target distributions whose potentials are bounded relative to $\mathscr{G}$, and for exponential distributions restricted to the support. Our analysis suggests that if $\mathscr{G}$ satisfies stronger notions of self-concordance introduced in Kook and Vempala (2024), then these mixing time upper bounds have a strictly better dependence on the dimension than when is merely self-concordant. We also provide numerical experiments that demonstrates the practicality of our proposed method. Our method is a high-accuracy sampler due to the polylogarithmic dependence on the error tolerance in our mixing time upper bounds.

stat.CO↗

arXiv2024

Fast Convergence of $\Phi$-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler

We study the mixing time of two popular discrete-time Markov chains in continuous space, the Unadjusted Langevin Algorithm and the Proximal Sampler, which are discretizations of the Langevin dynamics. We extend mixing time analyses for these Markov chains to hold in $\Phi$-divergence. We show that any $\Phi$-divergence arising from a twice-differentiable strictly convex function $\Phi$ converges to $0$ exponentially fast along these Markov chains, under the assumption that their stationary distributions satisfy the corresponding $\Phi$-Sobolev inequality, which holds for example when the target distribution of the Langevin dynamics is strongly log-concave. Our setting includes as special cases popular mixing time regimes, namely the mixing in chi-squared divergence under a Poincar\'e inequality, and the mixing in relative entropy under a log-Sobolev inequality. Our results follow by viewing the sampling algorithms as noisy channels and bounding the contraction coefficients arising in the appropriate strong data processing inequalities.

math.ST↗

arXiv2024

A Symplectic Analysis of Alternating Mirror Descent

Motivated by understanding the behavior of the Alternating Mirror Descent (AMD) algorithm for bilinear zero-sum games, we study the discretization of continuous-time Hamiltonian flow via the symplectic Euler method. We provide a framework for analysis using results from Hamiltonian dynamics, Lie algebra, and symplectic numerical integrators, with an emphasis on the existence and properties of a conserved quantity, the modified Hamiltonian (MH), for the symplectic Euler method. We compute the MH in closed-form when the original Hamiltonian is a quadratic function, and show that it generally differs from the other conserved quantity known previously in that case. We derive new error bounds on the MH when truncated at orders in the stepsize in terms of the number of iterations, $K$, and use these bounds to show an improved $\mathcal{O}(K^{1/5})$ total regret bound and an $\mathcal{O}(K^{-4/5})$ duality gap of the average iterates for AMD. Finally, we propose a conjecture which, if true, would imply that the total regret for AMD scales as $\mathcal{O}\left(K^{\varepsilon}\right)$ and the duality gap of the average iterates as $\mathcal{O}\left(K^{-1+\varepsilon}\right)$ for any $\varepsilon>0$, and we can take $\varepsilon=0$ upon certain convergence conditions for the MH.

math.OC↗

arXiv2024

Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of $\Phi$-Mutual Information

The mixing time of a Markov chain determines how fast the iterates of the Markov chain converge to the stationary distribution; however, it does not control the dependencies between samples along the Markov chain. In this paper, we study the question of how fast the samples become approximately independent along popular Markov chains for continuous-space sampling: the Langevin dynamics in continuous time, and the Unadjusted Langevin Algorithm and the Proximal Sampler in discrete time. We measure the dependence between samples via $\Phi$-mutual information, which is a broad generalization of the standard mutual information, and which is equal to $0$ if and only if the the samples are independent. We show that along these Markov chains, the $\Phi$-mutual information between the first and the $k$-th iterate decreases to $0$ exponentially fast in $k$ when the target distribution is strongly log-concave. Our proof technique is based on showing the Strong Data Processing Inequalities (SDPIs) hold along the Markov chains. To prove fast mixing of the Markov chains, we only need to show the SDPIs hold for the stationary distribution. In contrast, to prove the contraction of $\Phi$-mutual information, we need to show the SDPIs hold along the entire trajectories of the Markov chains; we prove this when the iterates along the Markov chains satisfy the corresponding $\Phi$-Sobolev inequality, which is implied by the strong log-concavity of the target distribution.

math.ST↗

arXiv2024

Optimal score estimation via empirical Bayes smoothing

We study the problem of estimating the score function of an unknown probability distribution $\rho^*$ from $n$ independent and identically distributed observations in $d$ dimensions. Assuming that $\rho^*$ is subgaussian and has a Lipschitz-continuous score function $s^*$, we establish the optimal rate of $\tilde \Theta(n^{-\frac{2}{d+4}})$ for this estimation problem under the loss function $\|\hat s - s^*\|^2_{L^2(\rho^*)}$ that is commonly used in the score matching literature, highlighting the curse of dimensionality where sample complexity for accurate score estimation grows exponentially with the dimension $d$. Leveraging key insights in empirical Bayes theory as well as a new convergence rate of smoothed empirical distribution in Hellinger distance, we show that a regularized score estimator based on a Gaussian kernel attains this rate, shown optimal by a matching minimax lower bound. We also discuss extensions to estimating $\beta$-H\"older continuous scores with $\beta \leq 1$, as well as the implication of our theory on the sample complexity of score-based generative models.

math.ST↗

arXiv2023

Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm

We propose a new method called the Metropolis-adjusted Mirror Langevin algorithm for approximate sampling from distributions whose support is a compact and convex set. This algorithm adds an accept-reject filter to the Markov chain induced by a single step of the Mirror Langevin algorithm (Zhang et al., 2020), which is a basic discretisation of the Mirror Langevin dynamics. Due to the inclusion of this filter, our method is unbiased relative to the target, while known discretisations of the Mirror Langevin dynamics including the Mirror Langevin algorithm have an asymptotic bias. For this algorithm, we also give upper bounds for the number of iterations taken to mix to a constrained distribution whose potential is relatively smooth, convex, and Lipschitz continuous with respect to a self-concordant mirror function. As a consequence of the reversibility of the Markov chain induced by the inclusion of the Metropolis-Hastings filter, we obtain an exponentially better dependence on the error tolerance for approximate constrained sampling. We also present numerical experiments that corroborate our theoretical findings.

stat.CO↗

arXiv2023

Extragradient Type Methods for Riemannian Variational Inequality Problems

Riemannian convex optimization and minimax optimization have recently drawn considerable attention. Their appeal lies in their capacity to adeptly manage the non-convexity of the objective function as well as constraints inherent in the feasible set in the Euclidean sense. In this work, we delve into monotone Riemannian Variational Inequality Problems (RVIPs), which encompass both Riemannian convex optimization and minimax optimization as particular cases. In the context of Euclidean space, it is established that the last-iterates of both the extragradient (EG) and past extragradient (PEG) methods converge to the solution of monotone variational inequality problems at a rate of $O\left(\frac{1}{\sqrt{T}}\right)$ (Cai et al., 2022). However, analogous behavior on Riemannian manifolds remains an open question. To bridge this gap, we introduce the Riemannian extragradient (REG) and Riemannian past extragradient (RPEG) methods. We demonstrate that both exhibit $O\left(\frac{1}{\sqrt{T}}\right)$ last-iterate convergence. Additionally, we show that the average-iterate convergence of both REG and RPEG is $O\left(\frac{1}{{T}}\right)$, aligning with observations in the Euclidean case (Mokhtari et al., 2020). These results are enabled by judiciously addressing the holonomy effect so that additional complications in Riemannian cases can be reduced and the Euclidean proof inspired by the performance estimation problem (PEP) technique or the sum-of-squares (SOS) technique can be applied again.

math.OC↗

arXiv2023

Provable Acceleration of Heavy Ball beyond Quadratics for a Class of Polyak-Łojasiewicz Functions when the Non-Convexity is Averaged-Out

Heavy Ball (HB) nowadays is one of the most popular momentum methods in non-convex optimization. It has been widely observed that incorporating the Heavy Ball dynamic in gradient-based methods accelerates the training process of modern machine learning models. However, the progress on establishing its theoretical foundation of acceleration is apparently far behind its empirical success. Existing provable acceleration results are of the quadratic or close-to-quadratic functions, as the current techniques of showing HB's acceleration are limited to the case when the Hessian is fixed. In this work, we develop some new techniques that help show acceleration beyond quadratics, which is achieved by analyzing how the change of the Hessian at two consecutive time points affects the convergence speed. Based on our technical results, a class of Polyak-Łojasiewicz (PL) optimization problems for which provable acceleration can be achieved via HB is identified. Moreover, our analysis demonstrates a benefit of adaptively setting the momentum parameter. (Update: 08/29/2023) Erratum is added in Appendix J. This is an updated version that fixes an issue in the previous version. An additional condition needs to be satisfied for the acceleration result of HB beyond quadratics in this work, which naturally holds when the dimension is one or, more broadly, when the Hessian is diagonal. We elaborate on the issue in Appendix J.

math.OC↗