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Tianyi Lin

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16 papers

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TMLR Journal 2026 Journal Article

Stepwise Guided Policy Optimization: Coloring Your Incorrect Reasoning in GRPO

Peter Chen
Xiaopeng Li
Ziniu Li
Xi Chen
Tianyi Lin

Reinforcement learning (RL) has proven effective in strengthening the reasoning capabilities of large language models (LLMs). A widely adopted method, Group Relative Policy Optimization (GRPO)~\citep{Shao-2024-Deepseekmath}, has shown strong empirical results in training recent reasoning models~\citep{Guo-2025-Deepseek}, but it fails to update the policy when all responses within a group are incorrect (i.e., all-negative-sample groups). This limitation highlights a gap between artificial and human intelligence: unlike humans, who can learn from mistakes, GRPO discards these failure signals. We introduce a simple framework to mitigate the all-negative-sample issue by incorporating response diversity within groups using a \textit{step-wise} judge model, which can be trained directly or adapted from existing LLMs. In a simplified setting, we prove that this diversification accelerates GRPO’s learning dynamics. We then empirically validate Stepwise Guided Policy Optimization (SGPO) across model sizes (7B, 14B, 32B) in both offline and online training on nine reasoning benchmarks (including base and distilled variants). Overall, SGPO improves average performance and is effective in early and mid-training when all-negative groups are prevalent, while improvements are not uniform across every benchmark and depend on the structure and informativeness of negative samples. Finally, SGPO does not require the judge model to generate correct solutions, distinguishing it from knowledge distillation methods.

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NeurIPS Conference 2025 Conference Paper

ComPO: Preference Alignment via Comparison Oracles

Peter Chen
Xi Chen
Wotao Yin
Tianyi Lin

Direct alignment methods are increasingly used for aligning large language models (LLMs) with human preferences. However, these methods suffer from the issues of verbosity and likelihood displacement, which can be driven by the noisy preference pairs that induce similar likelihood for preferred and dispreferred responses. The contributions of this paper are two-fold. First, we propose a new preference alignment method based on zeroth-order, comparison-based optimization via comparison oracles and provide convergence guarantees for its basic scheme. Second, we improve our method using some heuristics and conduct the experiments to demonstrate the flexibility and compatibility of practical scheme in improving the performance of LLMs using noisy preference pairs. Evaluations are conducted across multiple base and instruction-tuned models (Mistral-7B, Llama-3-8B and Gemma-2-9B) with benchmarks (AlpacaEval 2, MT-Bench and Arena-Hard). Experimental results show the effectiveness of our method as an alternative to addressing the limitations of existing direct alignment methods. A highlight of our work is that we evidence the importance of designing specialized methods for preference pairs with distinct likelihood margin, which complements the recent findings in Razin et al (2025).

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JMLR Journal 2025 Journal Article

Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization

Tianyi Lin
Chi Jin
Michael I. Jordan

We provide a unified analysis of two-timescale gradient descent ascent (TTGDA) for solving structured nonconvex minimax optimization problems in the form of $\min_x \max_{y \in Y} f(x, y)$, where the objective function $f(x, y)$ is nonconvex in $x$ and concave in $y$, and the constraint set $Y \subseteq \mathbb{R}^n$ is convex and bounded. In the convex-concave setting, the single-timescale gradient descent ascent (GDA) algorithm is widely used in applications and has been shown to have strong convergence guarantees. In more general settings, however, it can fail to converge. Our contribution is to design TTGDA algorithms that are effective beyond the convex-concave setting, efficiently finding a stationary point of the function $\Phi(\cdot):= \max_{y \in Y} f(\cdot, y)$. We also establish theoretical bounds on the complexity of solving both smooth and nonsmooth nonconvex-concave minimax optimization problems. To the best of our knowledge, this is the first systematic analysis of TTGDA for nonconvex minimax optimization, shedding light on its superior performance in training generative adversarial networks (GANs) and in other real-world application problems. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

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JMLR Journal 2023 Journal Article

First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems

Michael I. Jordan
Tianyi Lin
Manolis Zampetakis

We consider the problem of computing an equilibrium in a class of nonlinear generalized Nash equilibrium problems (NGNEPs) in which the strategy sets for each player are defined by the equality and inequality constraints that may depend on the choices of rival players. While the asymptotic global convergence and local convergence rate of certain algorithms have been extensively investigated, the iteration complexity analysis is still in its infancy. This paper provides two first-order algorithms based on quadratic penalty method (QPM) and augmented Lagrangian method (ALM), respectively, with an accelerated mirror-prox algorithm as the solver in each inner loop. We show the nonasymptotic convergence rate for these algorithms. In particular, we establish the global convergence guarantee for solving monotone and strongly monotone NGNEPs and provide the complexity bounds expressed in terms of the number of gradient evaluations. Experimental results demonstrate the efficiency of our algorithms in practice. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

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JMLR Journal 2022 Journal Article

Accelerating Adaptive Cubic Regularization of Newton's Method via Random Sampling

Xi Chen
Bo Jiang
Tianyi Lin
Shuzhong Zhang

In this paper, we consider an unconstrained optimization model where the objective is a sum of a large number of possibly nonconvex functions, though overall the objective is assumed to be smooth and convex. Our bid to solving such model uses the framework of cubic regularization of Newton's method. As well known, the crux in cubic regularization is its utilization of the Hessian information, which may be computationally expensive for large-scale problems. To tackle this, we resort to approximating the Hessian matrix via sub-sampling. In particular, we propose to compute an approximated Hessian matrix by either uniformly or non-uniformly sub-sampling the components of the objective. Based upon such sampling strategy, we develop accelerated adaptive cubic regularization approaches and provide theoretical guarantees on global iteration complexity of $\O(\epsilon^{-1/3})$ with high probability, which matches that of the original accelerated cubic regularization methods Jiang et al. (2020) using the full Hessian information. Interestingly, we also show that in the worst case scenario our algorithm still achieves an $O(\epsilon^{-5/6}\log(\epsilon^{-1}))$ iteration complexity bound. The proof techniques are new to our knowledge and can be of independent interets. Experimental results on the regularized logistic regression problems demonstrate a clear effect of acceleration on several real data sets. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

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NeurIPS Conference 2022 Conference Paper

First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces

Michael Jordan
Tianyi Lin
Emmanouil-Vasileios Vlatakis-Gkaragkounis

From optimal transport to robust dimensionality reduction, many machine learning applicationscan be cast into the min-max optimization problems over Riemannian manifolds. Though manymin-max algorithms have been analyzed in the Euclidean setting, it has been elusive how theseresults translate to the Riemannian case. Zhang et al. (2022) have recently identified that geodesic convexconcave Riemannian problems admit always Sion’s saddle point solutions. Immediately, an importantquestion that arises is if a performance gap between the Riemannian and the optimal Euclidean spaceconvex concave algorithms is necessary. Our work is the first to answer the question in the negative: We prove that the Riemannian corrected extragradient (RCEG) method achieves last-iterate at alinear convergence rate at the geodesically strongly convex concave case, matching the euclidean one. Our results also extend to the stochastic or non-smooth case where RCEG & Riemanian gradientascent descent (RGDA) achieve respectively near-optimal convergence rates up to factors dependingon curvature of the manifold. Finally, we empirically demonstrate the effectiveness of RCEG insolving robust PCA.

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NeurIPS Conference 2022 Conference Paper

Gradient-Free Methods for Deterministic and Stochastic Nonsmooth Nonconvex Optimization

Tianyi Lin
Zeyu Zheng
Michael Jordan

Nonsmooth nonconvex optimization problems broadly emerge in machine learning and business decision making, whereas two core challenges impede the development of efficient solution methods with finite-time convergence guarantee: the lack of computationally tractable optimality criterion and the lack of computationally powerful oracles. The contributions of this paper are two-fold. First, we establish the relationship between the celebrated Goldstein subdifferential~\citep{Goldstein-1977-Optimization} and uniform smoothing, thereby providing the basis and intuition for the design of gradient-free methods that guarantee the finite-time convergence to a set of Goldstein stationary points. Second, we propose the gradient-free method (GFM) and stochastic GFM for solving a class of nonsmooth nonconvex optimization problems and prove that both of them can return a $(\delta, \epsilon)$-Goldstein stationary point of a Lipschitz function $f$ at an expected convergence rate at $O(d^{3/2}\delta^{-1}\epsilon^{-4})$ where $d$ is the problem dimension. Two-phase versions of GFM and SGFM are also proposed and proven to achieve improved large-deviation results. Finally, we demonstrate the effectiveness of 2-SGFM on training ReLU neural networks with the \textsc{Minst} dataset.

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JMLR Journal 2022 Journal Article

On the Complexity of Approximating Multimarginal Optimal Transport

Tianyi Lin
Nhat Ho
Marco Cuturi
Michael I. Jordan

We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here between $m$ discrete probability distributions supported each on $n$ support points. First, we show that the standard linear programming (LP) representation of the MOT problem is not a minimum-cost flow problem when $m \geq 3$. This negative result implies that some combinatorial algorithms, e.g., network simplex method, are not suitable for approximating the MOT problem, while the worst-case complexity bound for the deterministic interior-point algorithm remains a quantity of $\tilde{\mathcal{O}}(n^{3m})$. We then propose two simple and deterministic algorithms for approximating the MOT problem. The first algorithm, which we refer to as multimarginal Sinkhorn algorithm, is a provably efficient multimarginal generalization of the Sinkhorn algorithm. We show that it achieves a complexity bound of $\tilde{\mathcal{O}}(m^3n^m\varepsilon^{-2})$ for a tolerance $\varepsilon \in (0, 1)$. This provides a first near-linear time complexity bound guarantee for approximating the MOT problem and matches the best known complexity bound for the Sinkhorn algorithm in the classical OT setting when $m = 2$. The second algorithm, which we refer to as accelerated multimarginal Sinkhorn algorithm, achieves the acceleration by incorporating an estimate sequence and the complexity bound is $\tilde{\mathcal{O}}(m^3n^{m+1/3}\varepsilon^{-4/3})$. This bound is better than that of the first algorithm in terms of $1/\varepsilon$, and accelerated alternating minimization algorithm (Tupitsa et al., 2020) in terms of $n$. Finally, we compare our new algorithms with the commercial LP solver Gurobi. Preliminary results on synthetic data and real images demonstrate the effectiveness and efficiency of our algorithms. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

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JMLR Journal 2022 Journal Article

On the Efficiency of Entropic Regularized Algorithms for Optimal Transport

Tianyi Lin
Nhat Ho
Michael I. Jordan

We present several new complexity results for the entropic regularized algorithms that approximately solve the optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. First, we improve the complexity bound of a greedy variant of Sinkhorn, known as Greenkhorn, from $\tilde{O}(n^2\varepsilon^{-3})$ to $\tilde{O}(n^2\varepsilon^{-2})$. Notably, our result can match the best known complexity bound of Sinkhorn and help clarify why Greenkhorn significantly outperforms Sinkhorn in practice in terms of row/column updates as observed by Altschuler et al. (2017). Second, we propose a new algorithm, which we refer to as APDAMD and which generalizes an adaptive primal-dual accelerated gradient descent (APDAGD) algorithm (Dvurechensky et al., 2018) with a prespecified mirror mapping $\phi$. We prove that APDAMD achieves the complexity bound of $\tilde{O}(n^2\sqrt{\delta}\varepsilon^{-1})$ in which $\delta>0$ stands for the regularity of $\phi$. In addition, we show by a counterexample that the complexity bound of $\tilde{O}(\min\{n^{9/4}\varepsilon^{-1}, n^2\varepsilon^{-2}\})$ proved for APDAGD before is invalid and give a refined complexity bound of $\tilde{O}(n^{5/2}\varepsilon^{-1})$. Further, we develop a deterministic accelerated variant of Sinkhorn via appeal to estimated sequence and prove the complexity bound of $\tilde{O}(n^{7/3}\varepsilon^{-4/3})$. As such, we see that accelerated variant of Sinkhorn outperforms Sinkhorn and Greenkhorn in terms of $1/\varepsilon$ and APDAGD and accelerated alternating minimization (AAM) (Guminov et al., 2021) in terms of $n$. Finally, we conduct the experiments on synthetic and real data and the numerical results show the efficiency of Greenkhorn, APDAMD and accelerated Sinkhorn in practice. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

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ICML Conference 2022 Conference Paper

Online Nonsubmodular Minimization with Delayed Costs: From Full Information to Bandit Feedback

Tianyi Lin
Aldo Pacchiano
Yaodong Yu
Michael I. Jordan

Motivated by applications to online learning in sparse estimation and Bayesian optimization, we consider the problem of online unconstrained nonsubmodular minimization with delayed costs in both full information and bandit feedback settings. In contrast to previous works on online unconstrained submodular minimization, we focus on a class of nonsubmodular functions with special structure, and prove regret guarantees for several variants of the online and approximate online bandit gradient descent algorithms in static and delayed scenarios. We derive bounds for the agent’s regret in the full information and bandit feedback setting, even if the delay between choosing a decision and receiving the incurred cost is unbounded. Key to our approach is the notion of $(\alpha, \beta)$-regret and the extension of the generic convex relaxation model from \citet{El-2020-Optimal}, the analysis of which is of independent interest. We conduct and showcase several simulation studies to demonstrate the efficacy of our algorithms.

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ICML Conference 2020 Conference Paper

Finite-Time Last-Iterate Convergence for Multi-Agent Learning in Games

Tianyi Lin
Zhengyuan Zhou
Panayotis Mertikopoulos
Michael I. Jordan

In this paper, we consider multi-agent learning via online gradient descent in a class of games called $\lambda$-cocoercive games, a fairly broad class of games that admits many Nash equilibria and that properly includes unconstrained strongly monotone games. We characterize the finite-time last-iterate convergence rate for joint OGD learning on $\lambda$-cocoercive games; further, building on this result, we develop a fully adaptive OGD learning algorithm that does not require any knowledge of problem parameter (e. g. cocoercive constant $\lambda$) and show, via a novel double-stopping time technique, that this adaptive algorithm achieves same finite-time last-iterate convergence rate as non-adaptive counterpart. Subsequently, we extend OGD learning to the noisy gradient feedback case and establish last-iterate convergence results–first qualitative almost sure convergence, then quantitative finite-time convergence rates– all under non-decreasing step-sizes. To our knowledge, we provide the first set of results that fill in several gaps of the existing multi-agent online learning literature, where three aspects–finite-time convergence rates, non-decreasing step-sizes, and fully adaptive algorithms have been unexplored before.

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NeurIPS Conference 2020 Conference Paper

Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm

Tianyi Lin
Nhat Ho
Xi Chen
Marco Cuturi
Michael Jordan

We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of $m$ discrete probability measures supported on a finite metric space of size $n$. We show first that the constraint matrix arising from the standard linear programming (LP) representation of the FS-WBP is \textit{not totally unimodular} when $m \geq 3$ and $n \geq 3$. This result resolves an open question pertaining to the relationship between the FS-WBP and the minimum-cost flow (MCF) problem since it proves that the FS-WBP in the standard LP form is not an MCF problem when $m \geq 3$ and $n \geq 3$. We also develop a provably fast \textit{deterministic} variant of the celebrated iterative Bregman projection (IBP) algorithm, named \textsc{FastIBP}, with a complexity bound of $\tilde{O}(mn^{7/3}\varepsilon^{-4/3})$, where $\varepsilon \in (0, 1)$ is the desired tolerance. This complexity bound is better than the best known complexity bound of $\tilde{O}(mn^2\varepsilon^{-2})$ for the IBP algorithm in terms of $\varepsilon$, and that of $\tilde{O}(mn^{5/2}\varepsilon^{-1})$ from accelerated alternating minimization algorithm or accelerated primal-dual adaptive gradient algorithm in terms of $n$. Finally, we conduct extensive experiments with both synthetic data and real images and demonstrate the favorable performance of the \textsc{FastIBP} algorithm in practice.

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ICML Conference 2020 Conference Paper

On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems

Tianyi Lin
Chi Jin 0001
Michael I. Jordan

We consider nonconvex-concave minimax problems, $\min_{\mathbf{x}} \max_{\mathbf{y} \in \mathcal{Y}} f(\mathbf{x}, \mathbf{y})$, where $f$ is nonconvex in $\mathbf{x}$ but concave in $\mathbf{y}$ and $\mathcal{Y}$ is a convex and bounded set. One of the most popular algorithms for solving this problem is the celebrated gradient descent ascent (GDA) algorithm, which has been widely used in machine learning, control theory and economics. Despite the extensive convergence results for the convex-concave setting, GDA with equal stepsize can converge to limit cycles or even diverge in a general setting. In this paper, we present the complexity results on two-time-scale GDA for solving nonconvex-concave minimax problems, showing that the algorithm can find a stationary point of the function $\Phi(\cdot): = \max_{\mathbf{y} \in \mathcal{Y}} f(\cdot, \mathbf{y})$ efficiently. To the best our knowledge, this is the first nonasymptotic analysis for two-time-scale GDA in this setting, shedding light on its superior practical performance in training generative adversarial networks (GANs) and other real applications.

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NeurIPS Conference 2020 Conference Paper

Projection Robust Wasserstein Distance and Riemannian Optimization

Tianyi Lin
Chenyou Fan
Nhat Ho
Marco Cuturi
Michael Jordan

Projection robust Wasserstein (PRW) distance, or Wasserstein projection pursuit (WPP), is a robust variant of the Wasserstein distance. Recent work suggests that this quantity is more robust than the standard Wasserstein distance, in particular when comparing probability measures in high-dimensions. However, it is ruled out for practical application because the optimization model is essentially non-convex and non-smooth which makes the computation intractable. Our contribution in this paper is to revisit the original motivation behind WPP/PRW, but take the hard route of showing that, despite its non-convexity and lack of nonsmoothness, and even despite some hardness results proved by~\citet{Niles-2019-Estimation} in a minimax sense, the original formulation for PRW/WPP \textit{can} be efficiently computed in practice using Riemannian optimization, yielding in relevant cases better behavior than its convex relaxation. More specifically, we provide three simple algorithms with solid theoretical guarantee on their complexity bound (one in the appendix), and demonstrate their effectiveness and efficiency by conducing extensive experiments on synthetic and real data. This paper provides a first step into a computational theory of the PRW distance and provides the links between optimal transport and Riemannian optimization.

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ICML Conference 2019 Conference Paper

On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms

Tianyi Lin
Nhat Ho
Michael I. Jordan

We provide theoretical analyses for two algorithms that solve the regularized optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. We show that a greedy variant of the classical Sinkhorn algorithm, known as the Greenkhorn algorithm, can be improved to $\bigOtil\left(n^2/\varepsilon^2\right)$, improving on the best known complexity bound of $\bigOtil\left(n^2/\varepsilon^3\right)$. This matches the best known complexity bound for the Sinkhorn algorithm and helps explain why the Greenkhorn algorithm outperforms the Sinkhorn algorithm in practice. Our proof technique is based on a primal-dual formulation and provide a tight upper bound for the dual solution, leading to a class of adaptive primal-dual accelerated mirror descent (APDAMD) algorithms. We prove that the complexity of these algorithms is $\bigOtil\left(n^2\sqrt{\gamma}/\varepsilon\right)$ in which $\gamma \in (0, n]$ refers to some constants in the Bregman divergence. Experimental results on synthetic and real datasets demonstrate the favorable performance of the Greenkhorn and APDAMD algorithms in practice.

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ECAI Conference 2016 Conference Paper

On Stochastic Primal-Dual Hybrid Gradient Approach for Compositely Regularized Minimization

Linbo Qiao
Tianyi Lin
Yu-Gang Jiang 0001
Fan Yang
Wei Liu 0005
Xicheng Lu

We consider a wide spectrum of regularized stochastic minimization problems, where the regularization term is composite with a linear function. Examples of this formulation include graph-guided regularized minimization, generalized Lasso and a class of &ell; 1 regularized problems. The computational challenge is that the closed-form solution of the proximal mapping associated with the regularization term is not available due to the imposed linear composition. Fortunately, the structure of the regularization term allows us to reformulate it as a new convex-concave saddle point problem which can be solved using the Primal-Dual Hybrid Gradient (PDHG) approach. However, this approach may be inefficient in realistic applications as computing the full gradient of the expected objective function could be very expensive when the number of input data samples is considerably large. To address this issue, we propose a Stochastic PDHG (SPDHG) algorithm with either uniformly or non-uniformly averaged iterates. Through uniformly averaged iterates, the SPDHG algorithm converges in expectation withrate for general convex objectives and O( log  ( t) /t) rate for strongly convex objectives, respectively. While with non-uniformly averaged iterates, the SPDHG algorithm is expected to converge with O( 1/t) rate for strongly convex objectives. Numerical experiments on different genres of datasets demonstrate that our proposed algorithm outperforms other competing algorithms.

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