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Anthony Man-Cho So

Publications and source records attributed to Anthony Man-Cho So.

At least 55 records · Page 3Linked to original sources
arXiv2023

A Stability Analysis of Fine-Tuning a Pre-Trained Model

Fine-tuning a pre-trained model (such as BERT, ALBERT, RoBERTa, T5, GPT, etc.) has proven to be one of the most promising paradigms in recent NLP research. However, numerous recent works indicate that fine-tuning suffers from the instability problem, i.e., tuning the same model under the same setting results in significantly different performance. Many recent works have proposed different methods to solve this problem, but there is no theoretical understanding of why and how these methods work. In this paper, we propose a novel theoretical stability analysis of fine-tuning that focuses on two commonly used settings, namely, full fine-tuning and head tuning. We define the stability under each setting and prove the corresponding stability bounds. The theoretical bounds explain why and how several existing methods can stabilize the fine-tuning procedure. In addition to being able to explain most of the observed empirical discoveries, our proposed theoretical analysis framework can also help in the design of effective and provable methods. Based on our theory, we propose three novel strategies to stabilize the fine-tuning procedure, namely, Maximal Margin Regularizer (MMR), Multi-Head Loss (MHLoss), and Self Unsupervised Re-Training (SURT). We extensively evaluate our proposed approaches on 11 widely used real-world benchmark datasets, as well as hundreds of synthetic classification datasets. The experiment results show that our proposed methods significantly stabilize the fine-tuning procedure and also corroborate our theoretical analysis.

cs.LG↗
arXiv2023

ReSync: Riemannian Subgradient-based Robust Rotation Synchronization

This work presents ReSync, a Riemannian subgradient-based algorithm for solving the robust rotation synchronization problem, which arises in various engineering applications. ReSync solves a least-unsquared minimization formulation over the rotation group, which is nonsmooth and nonconvex, and aims at recovering the underlying rotations directly. We provide strong theoretical guarantees for ReSync under the random corruption setting. Specifically, we first show that the initialization procedure of ReSync yields a proper initial point that lies in a local region around the ground-truth rotations. We next establish the weak sharpness property of the aforementioned formulation and then utilize this property to derive the local linear convergence of ReSync to the ground-truth rotations. By combining these guarantees, we conclude that ReSync converges linearly to the ground-truth rotations under appropriate conditions. Experiment results demonstrate the effectiveness of ReSync.

math.OC↗
arXiv2023

On the Estimation Performance of Generalized Power Method for Heteroscedastic Probabilistic PCA

The heteroscedastic probabilistic principal component analysis (PCA) technique, a variant of the classic PCA that considers data heterogeneity, is receiving more and more attention in the data science and signal processing communities. In this paper, to estimate the underlying low-dimensional linear subspace (simply called \emph{ground truth}) from available heterogeneous data samples, we consider the associated non-convex maximum-likelihood estimation problem, which involves maximizing a sum of heterogeneous quadratic forms over an orthogonality constraint (HQPOC). We propose a first-order method -- generalized power method (GPM) -- to tackle the problem and establish its \emph{estimation performance} guarantee. Specifically, we show that, given a suitable initialization, the distances between the iterates generated by GPM and the ground truth decrease at least geometrically to some threshold associated with the residual part of certain "population-residual decomposition". In establishing the estimation performance result, we prove a novel local error bound property of another closely related optimization problem, namely quadratic optimization with orthogonality constraint (QPOC), which is new and can be of independent interest. Numerical experiments are conducted to demonstrate the superior performance of GPM in both Gaussian noise and sub-Gaussian noise settings.

math.OC↗
arXiv2023

Universal Gradient Descent Ascent Method for Nonconvex-Nonconcave Minimax Optimization

Nonconvex-nonconcave minimax optimization has received intense attention over the last decade due to its broad applications in machine learning. Most existing algorithms rely on one-sided information, such as the convexity (resp. concavity) of the primal (resp. dual) functions, or other specific structures, such as the Polyak-Łojasiewicz (PŁ) and Kurdyka-Łojasiewicz (KŁ) conditions. However, verifying these regularity conditions is challenging in practice. To meet this challenge, we propose a novel universally applicable single-loop algorithm, the doubly smoothed gradient descent ascent method (DS-GDA), which naturally balances the primal and dual updates. That is, DS-GDA with the same hyperparameters is able to uniformly solve nonconvex-concave, convex-nonconcave, and nonconvex-nonconcave problems with one-sided KŁ properties, achieving convergence with $\mathcal{O}(ε^{-4})$ complexity. Sharper (even optimal) iteration complexity can be obtained when the KŁ exponent is known. Specifically, under the one-sided KŁ condition with exponent $θ\in(0,1)$, DS-GDA converges with an iteration complexity of $\mathcal{O}(ε^{-2\max\{2θ,1\}})$. They all match the corresponding best results in the literature. Moreover, we show that DS-GDA is practically applicable to general nonconvex-nonconcave problems even without any regularity conditions, such as the PŁ condition, KŁ condition, or weak Minty variational inequalities condition. For various challenging nonconvex-nonconcave examples in the literature, including ``Forsaken'', ``Bilinearly-coupled minimax'', ``Sixth-order polynomial'', and ``PolarGame'', the proposed DS-GDA can all get rid of limit cycles. To the best of our knowledge, this is the first first-order algorithm to achieve convergence on all of these formidable problems.

math.OC↗
arXiv2023

Outlier-Robust Gromov-Wasserstein for Graph Data

Gromov-Wasserstein (GW) distance is a powerful tool for comparing and aligning probability distributions supported on different metric spaces. Recently, GW has become the main modeling technique for aligning heterogeneous data for a wide range of graph learning tasks. However, the GW distance is known to be highly sensitive to outliers, which can result in large inaccuracies if the outliers are given the same weight as other samples in the objective function. To mitigate this issue, we introduce a new and robust version of the GW distance called RGW. RGW features optimistically perturbed marginal constraints within a Kullback-Leibler divergence-based ambiguity set. To make the benefits of RGW more accessible in practice, we develop a computationally efficient and theoretically provable procedure using Bregman proximal alternating linearized minimization algorithm. Through extensive experimentation, we validate our theoretical results and demonstrate the effectiveness of RGW on real-world graph learning tasks, such as subgraph matching and partial shape correspondence.

cs.LG↗
arXiv2023

Projected Tensor Power Method for Hypergraph Community Recovery

This paper investigates the problem of exact community recovery in the symmetric $d$-uniform $(d \geq 2)$ hypergraph stochastic block model ($d$-HSBM). In this model, a $d$-uniform hypergraph with $n$ nodes is generated by first partitioning the $n$ nodes into $K\geq 2$ equal-sized disjoint communities and then generating hyperedges with a probability that depends on the community memberships of $d$ nodes. Despite the non-convex and discrete nature of the maximum likelihood estimation problem, we develop a simple yet efficient iterative method, called the \emph{projected tensor power method}, to tackle it. As long as the initialization satisfies a partial recovery condition in the logarithmic degree regime of the problem, we show that our proposed method can exactly recover the hidden community structure down to the information-theoretic limit with high probability. Moreover, our proposed method exhibits a competitive time complexity of $\mathcal{O}(n\log^2n/\log\log n)$ when the aforementioned initialization condition is met. We also conduct numerical experiments to validate our theoretical findings.

math.OC↗
arXiv2023

Rotation Group Synchronization via Quotient Manifold

Rotation group $\mathcal{SO}(d)$ synchronization is an important inverse problem and has attracted intense attention from numerous application fields such as graph realization, computer vision, and robotics. In this paper, we focus on the least-squares estimator of rotation group synchronization with general additive noise models, which is a nonconvex optimization problem with manifold constraints. Unlike the phase/orthogonal group synchronization, there are limited provable approaches for solving rotation group synchronization. First, we derive improved estimation results of the least-squares/spectral estimator, illustrating the tightness and validating the existing relaxation methods of solving rotation group synchronization through the optimum of relaxed orthogonal group version under near-optimal noise level for exact recovery. Moreover, departing from the standard approach of utilizing the geometry of the ambient Euclidean space, we adopt an intrinsic Riemannian approach to study orthogonal/rotation group synchronization. Benefiting from a quotient geometric view, we prove the positive definite condition of quotient Riemannian Hessian around the optimum of orthogonal group synchronization problem, and consequently the Riemannian local error bound property is established to analyze the convergence rate properties of various Riemannian algorithms. As a simple and feasible method, the sequential convergence guarantee of the (quotient) Riemannian gradient method for solving orthogonal/rotation group synchronization problem is studied, and we derive its global linear convergence rate to the optimum with the spectral initialization. All results are deterministic without any probabilistic model.

math.OC↗
arXiv2023

A Unified Approach to Synchronization Problems over Subgroups of the Orthogonal Group

The problem of synchronization over a group $\mathcal{G}$ aims to estimate a collection of group elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ based on noisy observations of a subset of all pairwise ratios of the form $G^*_i {G^*_j}^{-1}$. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup -- an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.

math.OC↗
arXiv2023

LogSpecT: Feasible Graph Learning Model from Stationary Signals with Recovery Guarantees

Graph learning from signals is a core task in Graph Signal Processing (GSP). One of the most commonly used models to learn graphs from stationary signals is SpecT. However, its practical formulation rSpecT is known to be sensitive to hyperparameter selection and, even worse, to suffer from infeasibility. In this paper, we give the first condition that guarantees the infeasibility of rSpecT and design a novel model (LogSpecT) and its practical formulation (rLogSpecT) to overcome this issue. Contrary to rSpecT, the novel practical model rLogSpecT is always feasible. Furthermore, we provide recovery guarantees of rLogSpecT, which are derived from modern optimization tools related to epi-convergence. These tools could be of independent interest and significant for various learning problems. To demonstrate the advantages of rLogSpecT in practice, a highly efficient algorithm based on the linearized alternating direction method of multipliers (L-ADMM) is proposed. The subproblems of L-ADMM admit closed-form solutions and the convergence is guaranteed. Extensive numerical results on both synthetic and real networks corroborate the stability and superiority of our proposed methods, underscoring their potential for various graph learning applications.

stat.ML↗
arXiv2023

Decoder-Only or Encoder-Decoder? Interpreting Language Model as a Regularized Encoder-Decoder

The sequence-to-sequence (seq2seq) task aims at generating the target sequence based on the given input source sequence. Traditionally, most of the seq2seq task is resolved by the Encoder-Decoder framework which requires an encoder to encode the source sequence and a decoder to generate the target text. Recently, a bunch of new approaches have emerged that apply decoder-only language models directly to the seq2seq task. Despite the significant advancements in applying language models to the seq2seq task, there is still a lack of thorough analysis on the effectiveness of the decoder-only language model architecture. This paper aims to address this gap by conducting a detailed comparison between the encoder-decoder architecture and the decoder-only language model framework through the analysis of a regularized encoder-decoder structure. This structure is designed to replicate all behaviors in the classical decoder-only language model but has an encoder and a decoder making it easier to be compared with the classical encoder-decoder structure. Based on the analysis, we unveil the attention degeneration problem in the language model, namely, as the generation step number grows, less and less attention is focused on the source sequence. To give a quantitative understanding of this problem, we conduct a theoretical sensitivity analysis of the attention output with respect to the source input. Grounded on our analysis, we propose a novel partial attention language model to solve the attention degeneration problem. Experimental results on machine translation, summarization, and data-to-text generation tasks support our analysis and demonstrate the effectiveness of our proposed model.

cs.CL↗
arXiv2023

Decentralized Weakly Convex Optimization Over the Stiefel Manifold

We focus on a class of non-smooth optimization problems over the Stiefel manifold in the decentralized setting, where a connected network of $n$ agents cooperatively minimize a finite-sum objective function with each component being weakly convex in the ambient Euclidean space. Such optimization problems, albeit frequently encountered in applications, are quite challenging due to their non-smoothness and non-convexity. To tackle them, we propose an iterative method called the decentralized Riemannian subgradient method (DRSM). The global convergence and an iteration complexity of $\mathcal{O}(\varepsilon^{-2} \log^2(\varepsilon^{-1}))$ for forcing a natural stationarity measure below $\varepsilon$ are established via the powerful tool of proximal smoothness from variational analysis, which could be of independent interest. Besides, we show the local linear convergence of the DRSM using geometrically diminishing stepsizes when the problem at hand further possesses a sharpness property. Numerical experiments are conducted to corroborate our theoretical findings.

math.OC↗
arXiv2023

A Convergent Single-Loop Algorithm for Relaxation of Gromov-Wasserstein in Graph Data

In this work, we present the Bregman Alternating Projected Gradient (BAPG) method, a single-loop algorithm that offers an approximate solution to the Gromov-Wasserstein (GW) distance. We introduce a novel relaxation technique that balances accuracy and computational efficiency, albeit with some compromises in the feasibility of the coupling map. Our analysis is based on the observation that the GW problem satisfies the Luo-Tseng error bound condition, which relates to estimating the distance of a point to the critical point set of the GW problem based on the optimality residual. This observation allows us to provide an approximation bound for the distance between the fixed-point set of BAPG and the critical point set of GW. Moreover, under a mild technical assumption, we can show that BAPG converges to its fixed point set. The effectiveness of BAPG has been validated through comprehensive numerical experiments in graph alignment and partition tasks, where it outperforms existing methods in terms of both solution quality and wall-clock time.

cs.CG↗
arXiv2023

Testing Stationarity Concepts for ReLU Networks: Hardness, Regularity, and Robust Algorithms

We study the computational problem of the stationarity test for the empirical loss of neural networks with ReLU activation functions. Our contributions are: Hardness: We show that checking a certain first-order approximate stationarity concept for a piecewise linear function is co-NP-hard. This implies that testing a certain stationarity concept for a modern nonsmooth neural network is in general computationally intractable. As a corollary, we prove that testing so-called first-order minimality for functions in abs-normal form is co-NP-complete, which was conjectured by Griewank and Walther (2019, SIAM J. Optim., vol. 29, p284). Regularity: We establish a necessary and sufficient condition for the validity of an equality-type subdifferential chain rule in terms of Clarke, Fréchet, and limiting subdifferentials of the empirical loss of two-layer ReLU networks. This new condition is simple and efficiently checkable. Robust algorithms: We introduce an algorithmic scheme to test near-approximate stationarity in terms of both Clarke and Fréchet subdifferentials. Our scheme makes no false positive or false negative error when the tested point is sufficiently close to a stationary one and a certain qualification is satisfied. This is the first practical and robust stationarity test approach for two-layer ReLU networks.

math.OC↗
arXiv2022

Incremental Methods for Weakly Convex Optimization

Incremental methods are widely utilized for solving finite-sum optimization problems in machine learning and signal processing. In this paper, we study a family of incremental methods -- including incremental subgradient, incremental proximal point, and incremental prox-linear methods -- for solving weakly convex optimization problems. Such a problem class covers many nonsmooth nonconvex instances that arise in engineering fields. We show that the three said incremental methods have an iteration complexity of $O(\varepsilon^{-4})$ for driving a natural stationarity measure to below $\varepsilon$. Moreover, we show that if the weakly convex function satisfies a sharpness condition, then all three incremental methods, when properly initialized and equipped with geometrically diminishing stepsizes, can achieve a local linear rate of convergence. Our work is the first to extend the convergence rate analysis of incremental methods from the nonsmooth convex regime to the weakly convex regime. Lastly, we conduct numerical experiments on the robust matrix sensing problem to illustrate the convergence performance of the three incremental methods.

math.OC↗
arXiv2022

Fast and Provably Convergent Algorithms for Gromov-Wasserstein in Graph Data

In this paper, we study the design and analysis of a class of efficient algorithms for computing the Gromov-Wasserstein (GW) distance tailored to large-scale graph learning tasks. Armed with the Luo-Tseng error bound condition~\citep{luo1992error}, two proposed algorithms, called Bregman Alternating Projected Gradient (BAPG) and hybrid Bregman Proximal Gradient (hBPG) enjoy the convergence guarantees. Upon task-specific properties, our analysis further provides novel theoretical insights to guide how to select the best-fit method. As a result, we are able to provide comprehensive experiments to validate the effectiveness of our methods on a host of tasks, including graph alignment, graph partition, and shape matching. In terms of both wall-clock time and modeling performance, the proposed methods achieve state-of-the-art results.

cs.LG↗
arXiv2022

On the Effectiveness of Parameter-Efficient Fine-Tuning

Fine-tuning pre-trained models has been ubiquitously proven to be effective in a wide range of NLP tasks. However, fine-tuning the whole model is parameter inefficient as it always yields an entirely new model for each task. Currently, many research works propose to only fine-tune a small portion of the parameters while keeping most of the parameters shared across different tasks. These methods achieve surprisingly good performance and are shown to be more stable than their corresponding fully fine-tuned counterparts. However, such kind of methods is still not well understood. Some natural questions arise: How does the parameter sparsity lead to promising performance? Why is the model more stable than the fully fine-tuned models? How to choose the tunable parameters? In this paper, we first categorize the existing methods into random approaches, rule-based approaches, and projection-based approaches based on how they choose which parameters to tune. Then, we show that all of the methods are actually sparse fine-tuned models and conduct a novel theoretical analysis of them. We indicate that the sparsity is actually imposing a regularization on the original model by controlling the upper bound of the stability. Such stability leads to better generalization capability which has been empirically observed in a lot of recent research works. Despite the effectiveness of sparsity grounded by our theory, it still remains an open problem of how to choose the tunable parameters. To better choose the tunable parameters, we propose a novel Second-order Approximation Method (SAM) which approximates the original problem with an analytically solvable optimization function. The tunable parameters are determined by directly optimizing the approximation function. The experimental results show that our proposed SAM model outperforms many strong baseline models and it also verifies our theoretical analysis.

cs.CL↗
arXiv2022

A Linearly Convergent Algorithm for Rotationally Invariant $\ell_1$-Norm Principal Component Analysis

To do dimensionality reduction on the datasets with outliers, the $\ell_1$-norm principal component analysis (L1-PCA) as a typical robust alternative of the conventional PCA has enjoyed great popularity over the past years. In this work, we consider a rotationally invariant L1-PCA, which is hardly studied in the literature. To tackle it, we propose a proximal alternating linearized minimization method with a nonlinear extrapolation for solving its two-block reformulation. Moreover, we show that the proposed method converges at least linearly to a limiting critical point of the reformulated problem. Such a point is proved to be a critical point of the original problem under a condition imposed on the step size. Finally, we conduct numerical experiments on both synthetic and real datasets to support our theoretical developments and demonstrate the efficacy of our approach.

math.OC↗
arXiv2022

No Dimension-Free Deterministic Algorithm Computes Approximate Stationarities of Lipschitzians

We consider the computation of an approximately stationary point for a Lipschitz and semialgebraic function $f$ with a local oracle. If $f$ is smooth, simple deterministic methods have dimension-free finite oracle complexities. For the general Lipschitz setting, only recently, Zhang et al. [47] introduced a randomized algorithm that computes Goldstein's approximate stationarity [25] to arbitrary precision with a dimension-free polynomial oracle complexity. In this paper, we show that no deterministic algorithm can do the same. Even without the dimension-free requirement, we show that any finite time guaranteed deterministic method cannot be general zero-respecting, which rules out most of the oracle-based methods in smooth optimization and any trivial derandomization of Zhang et al. [47]. Our results reveal a fundamental hurdle of nonconvex nonsmooth problems in the modern large-scale setting and their infinite-dimensional extension.

math.OC↗