arXiv subjects

Tianyi Lin

Publications and source records attributed to Tianyi Lin.

At least 37 records · Page 2Linked to original sources

arXiv2025

Positional Bias in Long-Document Ranking: Impact, Assessment, and Mitigation

We tested over 20 Transformer models for ranking long documents (including recent LongP models trained with FlashAttention and RankGPT models "powered" by OpenAI and Anthropic cloud APIs). We compared them with the simple FirstP baseline, which applied the same model to truncated input (up to 512 tokens). On MS MARCO, TREC DL, and Robust04 no long-document model outperformed FirstP by more than 5% (on average). We hypothesized that this lack of improvement is not due to inherent model limitations, but due to benchmark positional bias (most relevant passages tend to occur early in documents), which is known to exist in MS MARCO. To confirm this, we analyzed positional relevance distributions across four long-document corpora (with six query sets) and observed the same early-position bias. Surprisingly, we also found bias in six BEIR collections, which are typically categorized as short-document datasets. We then introduced a new diagnostic dataset, MS MARCO FarRelevant, where relevant spans were deliberately placed beyond the first 512 tokens. On this dataset, many long-context models (including RankGPT) performed at random-baseline level, suggesting overfitting to positional bias. We also experimented with debiasing training data, but with limited success. Our findings (1) highlight the need for careful benchmark design in evaluating long-context models for document ranking, (2) identify model types that are more robust to positional bias, and (3) motivate further work on approaches to debias training data. We release our code and data to support further research.

cs.IR↗

arXiv2025

Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation

Equilibrium computation on Riemannian manifolds provides a unifying framework for numerous problems in machine learning and data analytics. One of the simplest yet most fundamental methods is Riemannian gradient descent (RGD). While its Euclidean counterpart has been extensively studied, it remains unclear how the manifold curvature affects RGD in game-theoretic settings. This paper addresses this gap by establishing new convergence results for \textit{geodesic strongly monotone} games. Our key result shows that RGD attains last-iterate linear convergence in a \textit{geometry-agnostic} fashion, a key property for applications in machine learning. We extend this guarantee to stochastic and adaptive variants -- SRGD and FARGD -- and establish that: (i) the sample complexity of SRGD is geometry-agnostic and optimal with respect to noise; (ii) FARGD matches the convergence rate of its non-adaptive counterpart up to constant factors, while avoiding reliance on the condition number. Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings, underscoring the surprising power of RGD -- despite its simplicity -- in solving a wide spectrum of machine learning problems.

math.OC↗

arXiv2025

ComPO: Preference Alignment via Comparison Oracles

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).

cs.CL↗

arXiv2025

Intern-S1: A Scientific Multimodal Foundation Model

In recent years, a plethora of open-source foundation models have emerged, achieving remarkable progress in some widely attended fields, with performance being quite close to that of closed-source models. However, in high-value but more challenging scientific professional fields, either the fields still rely on expert models, or the progress of general foundation models lags significantly compared to those in popular areas, far from sufficient for transforming scientific research and leaving substantial gap between open-source models and closed-source models in these scientific domains. To mitigate this gap and explore a step further toward Artificial General Intelligence (AGI), we introduce Intern-S1, a specialized generalist equipped with general understanding and reasoning capabilities with expertise to analyze multiple science modal data. Intern-S1 is a multimodal Mixture-of-Experts (MoE) model with 28 billion activated parameters and 241 billion total parameters, continually pre-trained on 5T tokens, including over 2.5T tokens from scientific domains. In the post-training stage, Intern-S1 undergoes offline and then online reinforcement learning (RL) in InternBootCamp, where we propose Mixture-of-Rewards (MoR) to synergize the RL training on more than 1000 tasks simultaneously. Through integrated innovations in algorithms, data, and training systems, Intern-S1 achieved top-tier performance in online RL training. On comprehensive evaluation benchmarks, Intern-S1 demonstrates competitive performance on general reasoning tasks among open-source models and significantly outperforms open-source models in scientific domains, surpassing closed-source state-of-the-art models in professional tasks, such as molecular synthesis planning, reaction condition prediction, predicting thermodynamic stabilities for crystals. Our models are available at https://huggingface.co/internlm/Intern-S1.

cs.LG↗

arXiv2025

Deterministic Nonsmooth Nonconvex Optimization

We study the complexity of optimizing nonsmooth nonconvex Lipschitz functions by producing $(δ,ε)$-stationary points. Several recent works have presented randomized algorithms that produce such points using $\tilde O(δ^{-1}ε^{-3})$ first-order oracle calls, independent of the dimension $d$. It has been an open problem as to whether a similar result can be obtained via a deterministic algorithm. We resolve this open problem, showing that randomization is necessary to obtain a dimension-free rate. In particular, we prove a lower bound of $Ω(d)$ for any deterministic algorithm. Moreover, we show that unlike smooth or convex optimization, access to function values is required for any deterministic algorithm to halt within any finite time. On the other hand, we prove that if the function is even slightly smooth, then the dimension-free rate of $\tilde O(δ^{-1}ε^{-3})$ can be obtained by a deterministic algorithm with merely a logarithmic dependence on the smoothness parameter. Motivated by these findings, we turn to study the complexity of deterministically smoothing Lipschitz functions. Though there are efficient black-box randomized smoothings, we start by showing that no such deterministic procedure can smooth functions in a meaningful manner, resolving an open question. We then bypass this impossibility result for the structured case of ReLU neural networks. To that end, in a practical white-box setting in which the optimizer is granted access to the network's architecture, we propose a simple, dimension-free, deterministic smoothing that provably preserves $(δ,ε)$-stationary points. Our method applies to a variety of architectures of arbitrary depth, including ResNets and ConvNets. Combined with our algorithm, this yields the first deterministic dimension-free algorithm for optimizing ReLU networks, circumventing our lower bound.

cs.LG↗

arXiv2025

Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization

We provide a unified analysis of two-timescale gradient descent ascent (TTGDA) for solving structured nonconvex minimax optimization problems in the form of $\min_\textbf{x} \max_{\textbf{y} \in Y} f(\textbf{x}, \textbf{y})$, where the objective function $f(\textbf{x}, \textbf{y})$ is nonconvex in $\textbf{x}$ and concave in $\textbf{y}$, 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 $Φ(\cdot) := \max_{\textbf{y} \in Y} f(\cdot, \textbf{y})$. 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.

cs.LG↗

arXiv2024

A Continuous-Time Perspective on Global Acceleration for Monotone Equation Problems

We propose a new framework to design and analyze accelerated methods that solve general monotone equation (ME) problems $F(x)=0$. Traditional approaches include generalized steepest descent methods and inexact Newton-type methods. If $F$ is uniformly monotone and twice differentiable, these methods achieve local convergence rates while the latter methods are globally convergent thanks to line search and hyperplane projection. However, a global rate is unknown for these methods. The variational inequality methods can be applied to yield a global rate that is expressed in terms of $\|F(x)\|$ but these results are restricted to first-order methods and a Lipschitz continuous operator. It has not been clear how to obtain global acceleration using high-order Lipschitz continuity. This paper takes a continuous-time perspective where accelerated methods are viewed as the discretization of dynamical systems. Our contribution is to propose accelerated rescaled gradient systems and prove that they are equivalent to closed-loop control systems. Based on this connection, we establish the properties of solution trajectories. Moreover, we provide a unified algorithmic framework obtained from discretization of our system, which together with two approximation subroutines yields both existing high-order methods and new first-order methods. We prove that the $p^{th}$-order method achieves a global rate of $O(k^{-p/2})$ in terms of $\|F(x)\|$ if $F$ is $p^{th}$-order Lipschitz continuous and the first-order method achieves the same rate if $F$ is $p^{th}$-order strongly Lipschitz continuous. If $F$ is strongly monotone, the restarted versions achieve local convergence with order $p$ when $p \geq 2$. Our discrete-time analysis is largely motivated by the continuous-time analysis and demonstrates the fundamental role that rescaled gradients play in global acceleration for solving ME problems.

math.OC↗

arXiv2024

On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems

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 $Φ(\cdot) := \max_{\mathbf{y} \in \mathcal{Y}} f(\cdot, \mathbf{y})$ 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.

cs.LG↗

arXiv2024

Doubly Optimal No-Regret Online Learning in Strongly Monotone Games with Bandit Feedback

We consider online no-regret learning in unknown games with bandit feedback, where each player can only observe its reward at each time -- determined by all players' current joint action -- rather than its gradient. We focus on the class of \textit{smooth and strongly monotone} games and study optimal no-regret learning therein. Leveraging self-concordant barrier functions, we first construct a new bandit learning algorithm and show that it achieves the single-agent optimal regret of $\tildeΘ(n\sqrt{T})$ under smooth and strongly concave reward functions ($n \geq 1$ is the problem dimension). We then show that if each player applies this no-regret learning algorithm in strongly monotone games, the joint action converges in the \textit{last iterate} to the unique Nash equilibrium at a rate of $\tildeΘ(nT^{-1/2})$. Prior to our work, the best-known convergence rate in the same class of games is $\tilde{O}(n^{2/3}T^{-1/3})$ (achieved by a different algorithm), thus leaving open the problem of optimal no-regret learning algorithms (since the known lower bound is $Ω(nT^{-1/2})$). Our results thus settle this open problem and contribute to the broad landscape of bandit game-theoretical learning by identifying the first doubly optimal bandit learning algorithm, in that it achieves (up to log factors) both optimal regret in the single-agent learning and optimal last-iterate convergence rate in the multi-agent learning. We also present preliminary numerical results on several application problems to demonstrate the efficacy of our algorithm in terms of iteration count.

cs.LG↗

arXiv2024

Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback

Online gradient descent (OGD) is well known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of $Θ(\log T)$ for strongly convex cost functions; and (2) in the multi-agent setting of strongly monotone games, with each agent employing OGD, we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of $Θ(\frac{1}{T})$. While these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In this paper, we design a fully adaptive OGD algorithm, \textsf{AdaOGD}, that does not require a priori knowledge of these parameters. In the single-agent setting, our algorithm achieves $O(\log^2(T))$ regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs \textsf{AdaOGD} in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of $O(\frac{\log^3 T}{T})$, again optimal up to log factors. We illustrate our algorithms in a learning version of the classical newsvendor problem, where due to lost sales, only (noisy) gradient feedback can be observed. Our results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multi-retailer settings. We also extend our results to the more general setting of exp-concave cost functions and games, using the online Newton step (ONS) algorithm.

cs.GT↗

arXiv2024

Perseus: A Simple and Optimal High-Order Method for Variational Inequalities

This paper settles an open and challenging question pertaining to the design of simple and optimal high-order methods for solving smooth and monotone variational inequalities (VIs). A VI involves finding $x^\star \in \mathcal{X}$ such that $\langle F(x), x - x^\star\rangle \geq 0$ for all $x \in \mathcal{X}$. We consider the setting in which $F$ is smooth with up to $(p-1)^{th}$-order derivatives. For $p = 2$, the cubic regularized Newton method was extended to VIs with a global rate of $O(ε^{-1})$. An improved rate of $O(ε^{-2/3}\log\log(1/ε))$ can be obtained via an alternative second-order method, but this method requires a nontrivial line-search procedure as an inner loop. Similarly, high-order methods based on line-search procedures have been shown to achieve a rate of $O(ε^{-2/(p+1)}\log\log(1/ε))$. As emphasized by Nesterov, however, such procedures do not necessarily imply practical applicability in large-scale applications, and it would be desirable to complement these results with a simple high-order VI method that retains the optimality of the more complex methods. We propose a $p^{th}$-order method that does \textit{not} require any line search procedure and provably converges to a weak solution at a rate of $O(ε^{-2/(p+1)})$. We prove that our $p^{th}$-order method is optimal in the monotone setting by establishing a matching lower bound under a generalized linear span assumption. Our method with restarting attains a linear rate for smooth and uniformly monotone VIs and a local superlinear rate for smooth and strongly monotone VIs. Our method also achieves a global rate of $O(ε^{-2/p})$ for solving smooth and nonmonotone VIs satisfying the Minty condition and when augmented with restarting it attains a global linear and local superlinear rate for smooth and nonmonotone VIs satisfying the uniform/strong Minty condition.

math.OC↗

arXiv2024

A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport

Kernel-based optimal transport (OT) estimators offer an alternative, functional estimation procedure to address OT problems from samples. Recent works suggest that these estimators are more statistically efficient than plug-in (linear programming-based) OT estimators when comparing probability measures in high-dimensions~\citep{Vacher-2021-Dimension}. Unfortunately, that statistical benefit comes at a very steep computational price: because their computation relies on the short-step interior-point method (SSIPM), which comes with a large iteration count in practice, these estimators quickly become intractable w.r.t. sample size $n$. To scale these estimators to larger $n$, we propose a nonsmooth fixed-point model for the kernel-based OT problem, and show that it can be efficiently solved via a specialized semismooth Newton (SSN) method: We show, exploring the problem's structure, that the per-iteration cost of performing one SSN step can be significantly reduced in practice. We prove that our SSN method achieves a global convergence rate of $O(1/\sqrt{k})$, and a local quadratic convergence rate under standard regularity conditions. We show substantial speedups over SSIPM on both synthetic and real datasets.

cs.LG↗

arXiv2023

First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems

We consider the problem of computing an equilibrium in a class of \textit{nonlinear generalized Nash equilibrium problems (NGNEPs)} in which the strategy sets for each player are defined by equality and inequality constraints that may depend on the choices of rival players. While the asymptotic global convergence and local convergence rates of algorithms to solve this problem have been extensively investigated, the analysis of nonasymptotic iteration complexity is still in its infancy. This paper presents two first-order algorithms -- based on the quadratic penalty method (QPM) and augmented Lagrangian method (ALM), respectively -- with an accelerated mirror-prox algorithm as the solver in each inner loop. We establish a global convergence guarantee for solving monotone and strongly monotone NGNEPs and provide nonasymptotic complexity bounds expressed in terms of the number of gradient evaluations. Experimental results demonstrate the efficiency of our algorithms in practice.

math.OC↗

arXiv2023

Projection Robust Wasserstein Distance and Riemannian Optimization

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.

cs.LG↗

arXiv2022

Monotone Inclusions, Acceleration and Closed-Loop Control

We propose and analyze a new dynamical system with a closed-loop control law in a Hilbert space $\mathcal{H}$, aiming to shed light on the acceleration phenomenon for \textit{monotone inclusion} problems, which unifies a broad class of optimization, saddle point and variational inequality (VI) problems under a single framework. Given $A: \mathcal{H} \rightrightarrows \mathcal{H}$ that is maximal monotone, we propose a closed-loop control system that is governed by the operator $I - (I + λ(t)A)^{-1}$, where a feedback law $λ(\cdot)$ is tuned by the resolution of the algebraic equation $λ(t)\|(I + λ(t)A)^{-1}x(t) - x(t)\|^{p-1} = θ$ for some $θ> 0$. Our first contribution is to prove the existence and uniqueness of a global solution via the Cauchy-Lipschitz theorem. We present a simple Lyapunov function for establishing the weak convergence of trajectories via the Opial lemma and strong convergence results under additional conditions. We then prove a global ergodic convergence rate of $O(t^{-(p+1)/2})$ in terms of a gap function and a global pointwise convergence rate of $O(t^{-p/2})$ in terms of a residue function. Local linear convergence is established in terms of a distance function under an error bound condition. Further, we provide an algorithmic framework based on the implicit discretization of our system in a Euclidean setting, generalizing the large-step HPE framework. Although the discrete-time analysis is a simplification and generalization of existing analyses for a bounded domain, it is largely motivated by the above continuous-time analysis, illustrating the fundamental role that the closed-loop control plays in acceleration in monotone inclusion. A highlight of our analysis is a new result concerning $p^{th}$-order tensor algorithms for monotone inclusion problems, complementing the recent analysis for saddle point and VI problems.

math.OC↗

arXiv2022

On the Complexity of Deterministic Nonsmooth and Nonconvex Optimization

In this paper, we present several new results on minimizing a nonsmooth and nonconvex function under a Lipschitz condition. Recent work shows that while the classical notion of Clarke stationarity is computationally intractable up to some sufficiently small constant tolerance, the randomized first-order algorithms find a $(δ, ε)$-Goldstein stationary point with the complexity bound of $\tilde{O}(δ^{-1}ε^{-3})$, which is independent of dimension $d \geq 1$~\citep{Zhang-2020-Complexity, Davis-2022-Gradient, Tian-2022-Finite}. However, the deterministic algorithms have not been fully explored, leaving open several problems in nonsmooth nonconvex optimization. Our first contribution is to demonstrate that the randomization is \textit{necessary} to obtain a dimension-independent guarantee, by proving a lower bound of $Ω(d)$ for any deterministic algorithm that has access to both $1^{st}$ and $0^{th}$ oracles. Furthermore, we show that the $0^{th}$ oracle is \textit{essential} to obtain a finite-time convergence guarantee, by showing that any deterministic algorithm with only the $1^{st}$ oracle is not able to find an approximate Goldstein stationary point within a finite number of iterations up to sufficiently small constant parameter and tolerance. Finally, we propose a deterministic smoothing approach under the \textit{arithmetic circuit} model where the resulting smoothness parameter is exponential in a certain parameter $M > 0$ (e.g., the number of nodes in the representation of the function), and design a new deterministic first-order algorithm that achieves a dimension-independent complexity bound of $\tilde{O}(Mδ^{-1}ε^{-3})$.

math.OC↗

arXiv2022

Gradient-Free Methods for Deterministic and Stochastic Nonsmooth Nonconvex Optimization

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 $(δ,ε)$-Goldstein stationary point of a Lipschitz function $f$ at an expected convergence rate at $O(d^{3/2}δ^{-1}ε^{-4})$ 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.

math.OC↗

arXiv2022

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

From optimal transport to robust dimensionality reduction, a plethora of machine learning applications can be cast into the min-max optimization problems over Riemannian manifolds. Though many min-max algorithms have been analyzed in the Euclidean setting, it has proved elusive to translate these results to the Riemannian case. Zhang et al. [2022] have recently shown that geodesic convex concave Riemannian problems always admit saddle-point solutions. Inspired by this result, we study whether a performance gap between Riemannian and optimal Euclidean space convex-concave algorithms is necessary. We answer this question in the negative-we prove that the Riemannian corrected extragradient (RCEG) method achieves last-iterate convergence at a linear rate in the geodesically strongly-convex-concave case, matching the Euclidean result. Our results also extend to the stochastic or non-smooth case where RCEG and Riemanian gradient ascent descent (RGDA) achieve near-optimal convergence rates up to factors depending on curvature of the manifold.

math.OC↗