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Chi Jin

Chi Jin appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
Machine LearningArtificial Intelligencemath.OCData Structures and AlgorithmsSystems and Controleess.SYmath.NAComputation and LanguageDatabasesmath.STNeural and Evolutionary ComputingRoboticsStatistics Theory

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Published work

23 published item(s)

preprint2026arXiv

Odysseus: Scaling VLMs to 100+ Turn Decision-Making in Games via Reinforcement Learning

Given the rapidly growing capabilities of vision-language models (VLMs), extending them to interactive decision-making tasks such as video games has emerged as a promising frontier. However, existing approaches either rely on large-scale supervised fine-tuning (SFT) on human trajectories or apply reinforcement learning (RL) only in relatively short-horizon settings (typically around 20--30 turns). In this work, we study RL-based training of VLMs for long-horizon decision-making in Super Mario Land, a visually grounded environment requiring 100+ turns of interaction with coordinated perception, reasoning, and action. We begin with a systematic investigation of key algorithmic components and propose an adapted variant of PPO with a lightweight turn-level critic, which substantially improves training stability and sample efficiency over critic-free methods such as GRPO and Reinforce++. We further show that pretrained VLMs provide strong action priors, significantly improving sample efficiency during RL training and reducing the need for manual design choices such as action engineering, compared to classical deep RL trained from scratch. Building on these insights, we introduce Odysseus, an open training framework for VLM agents, achieving substantial gains across multiple levels of the game and at least 3 times average game progresses than frontier models. Moreover, the trained models exhibit consistent improvements under both in-game and cross-game generalization settings, while maintaining general-domain capabilities. Overall, our results identify key ingredients for making RL stable and effective in long-horizon, multi-modal settings, and provide practical guidance for developing VLMs as embodied agents.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Globally convergent visual-feature range estimation with biased inertial measurements

The design of a globally convergent position observer for feature points from visual information is a challenging problem, especially for the case with only inertial measurements and without assumptions of uniform observability, which remained open for a long time. We give a solution to the problem in this paper assuming that only the bearing of a feature point, and biased linear acceleration and rotational velocity of a robot -- all in the body-fixed frame -- are available. Further, in contrast to existing related results, we do not need the value of the gravitational constant either. The proposed approach builds upon the parameter estimation-based observer recently developed in (Ortega et al., Syst. Control Lett., vol.85, 2015) and its extension to matrix Lie groups in our previous work. Conditions on the robot trajectory under which the observer converges are given, and these are strictly weaker than the standard persistency of excitation and uniform complete observability conditions. Finally, as an illustration, we apply the proposed design to the visual inertial navigation problem.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Provable Meta-Learning of Linear Representations

Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning -- a key tool for performing meta-learning -- learns a data representation that can transfer knowledge across multiple tasks, which is essential in regimes where data is scarce. Despite a recent surge of interest in the practice of meta-learning, the theoretical underpinnings of meta-learning algorithms are lacking, especially in the context of learning transferable representations. In this paper, we focus on the problem of multi-task linear regression -- in which multiple linear regression models share a common, low-dimensional linear representation. Here, we provide provably fast, sample-efficient algorithms to address the dual challenges of (1) learning a common set of features from multiple, related tasks, and (2) transferring this knowledge to new, unseen tasks. Both are central to the general problem of meta-learning. Finally, we complement these results by providing information-theoretic lower bounds on the sample complexity of learning these linear features.

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preprint2022arXiv

Provable Reinforcement Learning with a Short-Term Memory

Real-world sequential decision making problems commonly involve partial observability, which requires the agent to maintain a memory of history in order to infer the latent states, plan and make good decisions. Coping with partial observability in general is extremely challenging, as a number of worst-case statistical and computational barriers are known in learning Partially Observable Markov Decision Processes (POMDPs). Motivated by the problem structure in several physical applications, as well as a commonly used technique known as "frame stacking", this paper proposes to study a new subclass of POMDPs, whose latent states can be decoded by the most recent history of a short length $m$. We establish a set of upper and lower bounds on the sample complexity for learning near-optimal policies for this class of problems in both tabular and rich-observation settings (where the number of observations is enormous). In particular, in the rich-observation setting, we develop new algorithms using a novel "moment matching" approach with a sample complexity that scales exponentially with the short length $m$ rather than the problem horizon, and is independent of the number of observations. Our results show that a short-term memory suffices for reinforcement learning in these environments.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Understanding Domain Randomization for Sim-to-real Transfer

Reinforcement learning encounters many challenges when applied directly in the real world. Sim-to-real transfer is widely used to transfer the knowledge learned from simulation to the real world. Domain randomization -- one of the most popular algorithms for sim-to-real transfer -- has been demonstrated to be effective in various tasks in robotics and autonomous driving. Despite its empirical successes, theoretical understanding on why this simple algorithm works is limited. In this paper, we propose a theoretical framework for sim-to-real transfers, in which the simulator is modeled as a set of MDPs with tunable parameters (corresponding to unknown physical parameters such as friction). We provide sharp bounds on the sim-to-real gap -- the difference between the value of policy returned by domain randomization and the value of an optimal policy for the real world. We prove that sim-to-real transfer can succeed under mild conditions without any real-world training samples. Our theory also highlights the importance of using memory (i.e., history-dependent policies) in domain randomization. Our proof is based on novel techniques that reduce the problem of bounding the sim-to-real gap to the problem of designing efficient learning algorithms for infinite-horizon MDPs, which we believe are of independent interest.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

When Is Partially Observable Reinforcement Learning Not Scary?

Applications of Reinforcement Learning (RL), in which agents learn to make a sequence of decisions despite lacking complete information about the latent states of the controlled system, that is, they act under partial observability of the states, are ubiquitous. Partially observable RL can be notoriously difficult -- well-known information-theoretic results show that learning partially observable Markov decision processes (POMDPs) requires an exponential number of samples in the worst case. Yet, this does not rule out the existence of large subclasses of POMDPs over which learning is tractable. In this paper we identify such a subclass, which we call weakly revealing POMDPs. This family rules out the pathological instances of POMDPs where observations are uninformative to a degree that makes learning hard. We prove that for weakly revealing POMDPs, a simple algorithm combining optimism and Maximum Likelihood Estimation (MLE) is sufficient to guarantee polynomial sample complexity. To the best of our knowledge, this is the first provably sample-efficient result for learning from interactions in overcomplete POMDPs, where the number of latent states can be larger than the number of observations.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

A Sharp Analysis of Model-based Reinforcement Learning with Self-Play

Model-based algorithms -- algorithms that explore the environment through building and utilizing an estimated model -- are widely used in reinforcement learning practice and theoretically shown to achieve optimal sample efficiency for single-agent reinforcement learning in Markov Decision Processes (MDPs). However, for multi-agent reinforcement learning in Markov games, the current best known sample complexity for model-based algorithms is rather suboptimal and compares unfavorably against recent model-free approaches. In this paper, we present a sharp analysis of model-based self-play algorithms for multi-agent Markov games. We design an algorithm -- Optimistic Nash Value Iteration (Nash-VI) for two-player zero-sum Markov games that is able to output an $ε$-approximate Nash policy in $\tilde{\mathcal{O}}(H^3SAB/ε^2)$ episodes of game playing, where $S$ is the number of states, $A,B$ are the number of actions for the two players respectively, and $H$ is the horizon length. This significantly improves over the best known model-based guarantee of $\tilde{\mathcal{O}}(H^4S^2AB/ε^2)$, and is the first that matches the information-theoretic lower bound $Ω(H^3S(A+B)/ε^2)$ except for a $\min\{A,B\}$ factor. In addition, our guarantee compares favorably against the best known model-free algorithm if $\min \{A,B\}=o(H^3)$, and outputs a single Markov policy while existing sample-efficient model-free algorithms output a nested mixture of Markov policies that is in general non-Markov and rather inconvenient to store and execute. We further adapt our analysis to designing a provably efficient task-agnostic algorithm for zero-sum Markov games, and designing the first line of provably sample-efficient algorithms for multi-player general-sum Markov games.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Near-optimal Representation Learning for Linear Bandits and Linear RL

This paper studies representation learning for multi-task linear bandits and multi-task episodic RL with linear value function approximation. We first consider the setting where we play $M$ linear bandits with dimension $d$ concurrently, and these bandits share a common $k$-dimensional linear representation so that $k\ll d$ and $k \ll M$. We propose a sample-efficient algorithm, MTLR-OFUL, which leverages the shared representation to achieve $\tilde{O}(M\sqrt{dkT} + d\sqrt{kMT} )$ regret, with $T$ being the number of total steps. Our regret significantly improves upon the baseline $\tilde{O}(Md\sqrt{T})$ achieved by solving each task independently. We further develop a lower bound that shows our regret is near-optimal when $d > M$. Furthermore, we extend the algorithm and analysis to multi-task episodic RL with linear value function approximation under low inherent Bellman error \citep{zanette2020learning}. To the best of our knowledge, this is the first theoretical result that characterizes the benefits of multi-task representation learning for exploration in RL with function approximation.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Near-Optimal Reinforcement Learning with Self-Play

This paper considers the problem of designing optimal algorithms for reinforcement learning in two-player zero-sum games. We focus on self-play algorithms which learn the optimal policy by playing against itself without any direct supervision. In a tabular episodic Markov game with $S$ states, $A$ max-player actions and $B$ min-player actions, the best existing algorithm for finding an approximate Nash equilibrium requires $\tilde{\mathcal{O}}(S^2AB)$ steps of game playing, when only highlighting the dependency on $(S,A,B)$. In contrast, the best existing lower bound scales as $Ω(S(A+B))$ and has a significant gap from the upper bound. This paper closes this gap for the first time: we propose an optimistic variant of the \emph{Nash Q-learning} algorithm with sample complexity $\tilde{\mathcal{O}}(SAB)$, and a new \emph{Nash V-learning} algorithm with sample complexity $\tilde{\mathcal{O}}(S(A+B))$. The latter result matches the information-theoretic lower bound in all problem-dependent parameters except for a polynomial factor of the length of each episode. In addition, we present a computational hardness result for learning the best responses against a fixed opponent in Markov games---a learning objective different from finding the Nash equilibrium.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

On Function Approximation in Reinforcement Learning: Optimism in the Face of Large State Spaces

The classical theory of reinforcement learning (RL) has focused on tabular and linear representations of value functions. Further progress hinges on combining RL with modern function approximators such as kernel functions and deep neural networks, and indeed there have been many empirical successes that have exploited such combinations in large-scale applications. There are profound challenges, however, in developing a theory to support this enterprise, most notably the need to take into consideration the exploration-exploitation tradeoff at the core of RL in conjunction with the computational and statistical tradeoffs that arise in modern function-approximation-based learning systems. We approach these challenges by studying an optimistic modification of the least-squares value iteration algorithm, in the context of the action-value function represented by a kernel function or an overparameterized neural network. We establish both polynomial runtime complexity and polynomial sample complexity for this algorithm, without additional assumptions on the data-generating model. In particular, we prove that the algorithm incurs an $\tilde{\mathcal{O}}(δ_{\mathcal{F}} H^2 \sqrt{T})$ regret, where $δ_{\mathcal{F}}$ characterizes the intrinsic complexity of the function class $\mathcal{F}$, $H$ is the length of each episode, and $T$ is the total number of episodes. Our regret bounds are independent of the number of states, a result which exhibits clearly the benefit of function approximation in RL.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Provable Self-Play Algorithms for Competitive Reinforcement Learning

Self-play, where the algorithm learns by playing against itself without requiring any direct supervision, has become the new weapon in modern Reinforcement Learning (RL) for achieving superhuman performance in practice. However, the majority of exisiting theory in reinforcement learning only applies to the setting where the agent plays against a fixed environment; it remains largely open whether self-play algorithms can be provably effective, especially when it is necessary to manage the exploration/exploitation tradeoff. We study self-play in competitive reinforcement learning under the setting of Markov games, a generalization of Markov decision processes to the two-player case. We introduce a self-play algorithm---Value Iteration with Upper/Lower Confidence Bound (VI-ULCB)---and show that it achieves regret $\tilde{\mathcal{O}}(\sqrt{T})$ after playing $T$ steps of the game, where the regret is measured by the agent's performance against a \emph{fully adversarial} opponent who can exploit the agent's strategy at \emph{any} step. We also introduce an explore-then-exploit style algorithm, which achieves a slightly worse regret of $\tilde{\mathcal{O}}(T^{2/3})$, but is guaranteed to run in polynomial time even in the worst case. To the best of our knowledge, our work presents the first line of provably sample-efficient self-play algorithms for competitive reinforcement learning.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Reward-Free Exploration for Reinforcement Learning

Exploration is widely regarded as one of the most challenging aspects of reinforcement learning (RL), with many naive approaches succumbing to exponential sample complexity. To isolate the challenges of exploration, we propose a new "reward-free RL" framework. In the exploration phase, the agent first collects trajectories from an MDP $\mathcal{M}$ without a pre-specified reward function. After exploration, it is tasked with computing near-optimal policies under for $\mathcal{M}$ for a collection of given reward functions. This framework is particularly suitable when there are many reward functions of interest, or when the reward function is shaped by an external agent to elicit desired behavior. We give an efficient algorithm that conducts $\tilde{\mathcal{O}}(S^2A\mathrm{poly}(H)/ε^2)$ episodes of exploration and returns $ε$-suboptimal policies for an arbitrary number of reward functions. We achieve this by finding exploratory policies that visit each "significant" state with probability proportional to its maximum visitation probability under any possible policy. Moreover, our planning procedure can be instantiated by any black-box approximate planner, such as value iteration or natural policy gradient. We also give a nearly-matching $Ω(S^2AH^2/ε^2)$ lower bound, demonstrating the near-optimality of our algorithm in this setting.

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preprint2020arXiv

What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?

Minimax optimization has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises---"what is a proper definition of local optima?" Most previous work answers this question using classical notions of equilibria from simultaneous games, where the min-player and the max-player act simultaneously. In contrast, most applications in machine learning, including GANs and adversarial training, correspond to sequential games, where the order of which player acts first is crucial (since minimax is in general not equal to maximin due to the nonconvex-nonconcave nature of the problems). The main contribution of this paper is to propose a proper mathematical definition of local optimality for this sequential setting---local minimax, as well as to present its properties and existence results. Finally, we establish a strong connection to a basic local search algorithm---gradient descent ascent (GDA): under mild conditions, all stable limit points of GDA are exactly local minimax points up to some degenerate points.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Sampling Can Be Faster Than Optimization

Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical understanding of the relationships between these two kinds of methodology, and limited understanding of relative strengths and weaknesses. Moreover, existing results have been obtained primarily in the setting of convex functions (for optimization) and log-concave functions (for sampling). In this setting, where local properties determine global properties, optimization algorithms are unsurprisingly more efficient computationally than sampling algorithms. We instead examine a class of nonconvex objective functions that arise in mixture modeling and multi-stable systems. In this nonconvex setting, we find that the computational complexity of sampling algorithms scales linearly with the model dimension while that of optimization algorithms scales exponentially.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Efficient Algorithms for Large-scale Generalized Eigenvector Computation and Canonical Correlation Analysis

This paper considers the problem of canonical-correlation analysis (CCA) (Hotelling, 1936) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics (Shi and Malik, 2000; Hardoon et al., 2004; Witten et al., 2009). We provide simple iterative algorithms, with improved runtimes, for solving these problems that are globally linearly convergent with moderate dependencies on the condition numbers and eigenvalue gaps of the matrices involved. We obtain our results by reducing CCA to the top-$k$ generalized eigenvector problem. We solve this problem through a general framework that simply requires black box access to an approximate linear system solver. Instantiating this framework with accelerated gradient descent we obtain a running time of $O(\frac{z k \sqrtκ}ρ \log(1/ε) \log \left(kκ/ρ\right))$ where $z$ is the total number of nonzero entries, $κ$ is the condition number and $ρ$ is the relative eigenvalue gap of the appropriate matrices. Our algorithm is linear in the input size and the number of components $k$ up to a $\log(k)$ factor. This is essential for handling large-scale matrices that appear in practice. To the best of our knowledge this is the first such algorithm with global linear convergence. We hope that our results prompt further research and ultimately improve the practical running time for performing these important data analysis procedures on large data sets.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Faster Eigenvector Computation via Shift-and-Invert Preconditioning

We give faster algorithms and improved sample complexities for estimating the top eigenvector of a matrix $Σ$ -- i.e. computing a unit vector $x$ such that $x^T Σx \ge (1-ε)λ_1(Σ)$: Offline Eigenvector Estimation: Given an explicit $A \in \mathbb{R}^{n \times d}$ with $Σ= A^TA$, we show how to compute an $ε$ approximate top eigenvector in time $\tilde O([nnz(A) + \frac{d*sr(A)}{gap^2} ]* \log 1/ε)$ and $\tilde O([\frac{nnz(A)^{3/4} (d*sr(A))^{1/4}}{\sqrt{gap}} ] * \log 1/ε)$. Here $nnz(A)$ is the number of nonzeros in $A$, $sr(A)$ is the stable rank, $gap$ is the relative eigengap. By separating the $gap$ dependence from the $nnz(A)$ term, our first runtime improves upon the classical power and Lanczos methods. It also improves prior work using fast subspace embeddings [AC09, CW13] and stochastic optimization [Sha15c], giving significantly better dependencies on $sr(A)$ and $ε$. Our second running time improves these further when $nnz(A) \le \frac{d*sr(A)}{gap^2}$. Online Eigenvector Estimation: Given a distribution $D$ with covariance matrix $Σ$ and a vector $x_0$ which is an $O(gap)$ approximate top eigenvector for $Σ$, we show how to refine to an $ε$ approximation using $ O(\frac{var(D)}{gap*ε})$ samples from $D$. Here $var(D)$ is a natural notion of variance. Combining our algorithm with previous work to initialize $x_0$, we obtain improved sample complexity and runtime results under a variety of assumptions on $D$. We achieve our results using a general framework that we believe is of independent interest. We give a robust analysis of the classic method of shift-and-invert preconditioning to reduce eigenvector computation to approximately solving a sequence of linear systems. We then apply fast stochastic variance reduced gradient (SVRG) based system solvers to achieve our claims.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences

We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with $M \geq 3$ components. Our first main result shows that the population likelihood function has bad local maxima even in the special case of equally-weighted mixtures of well-separated and spherical Gaussians. We prove that the log-likelihood value of these bad local maxima can be arbitrarily worse than that of any global optimum, thereby resolving an open question of Srebro (2007). Our second main result shows that the EM algorithm (or a first-order variant of it) with random initialization will converge to bad critical points with probability at least $1-e^{-Ω(M)}$. We further establish that a first-order variant of EM will not converge to strict saddle points almost surely, indicating that the poor performance of the first-order method can be attributed to the existence of bad local maxima rather than bad saddle points. Overall, our results highlight the necessity of careful initialization when using the EM algorithm in practice, even when applied in highly favorable settings.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Provable Efficient Online Matrix Completion via Non-convex Stochastic Gradient Descent

Matrix completion, where we wish to recover a low rank matrix by observing a few entries from it, is a widely studied problem in both theory and practice with wide applications. Most of the provable algorithms so far on this problem have been restricted to the offline setting where they provide an estimate of the unknown matrix using all observations simultaneously. However, in many applications, the online version, where we observe one entry at a time and dynamically update our estimate, is more appealing. While existing algorithms are efficient for the offline setting, they could be highly inefficient for the online setting. In this paper, we propose the first provable, efficient online algorithm for matrix completion. Our algorithm starts from an initial estimate of the matrix and then performs non-convex stochastic gradient descent (SGD). After every observation, it performs a fast update involving only one row of two tall matrices, giving near linear total runtime. Our algorithm can be naturally used in the offline setting as well, where it gives competitive sample complexity and runtime to state of the art algorithms. Our proofs introduce a general framework to show that SGD updates tend to stay away from saddle surfaces and could be of broader interests for other non-convex problems to prove tight rates.

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preprint2016arXiv

Robust Shift-and-Invert Preconditioning: Faster and More Sample Efficient Algorithms for Eigenvector Computation

We provide faster algorithms and improved sample complexities for approximating the top eigenvector of a matrix. Offline Setting: Given an $n \times d$ matrix $A$, we show how to compute an $ε$ approximate top eigenvector in time $\tilde O ( [nnz(A) + \frac{d \cdot sr(A)}{gap^2}]\cdot \log 1/ε)$ and $\tilde O([\frac{nnz(A)^{3/4} (d \cdot sr(A))^{1/4}}{\sqrt{gap}}]\cdot \log1/ε)$. Here $sr(A)$ is the stable rank and $gap$ is the multiplicative eigenvalue gap. By separating the $gap$ dependence from $nnz(A)$ we improve on the classic power and Lanczos methods. We also improve prior work using fast subspace embeddings and stochastic optimization, giving significantly improved dependencies on $sr(A)$ and $ε$. Our second running time improves this further when $nnz(A) \le \frac{d\cdot sr(A)}{gap^2}$. Online Setting: Given a distribution $D$ with covariance matrix $Σ$ and a vector $x_0$ which is an $O(gap)$ approximate top eigenvector for $Σ$, we show how to refine to an $ε$ approximation using $\tilde O(\frac{v(D)}{gap^2} + \frac{v(D)}{gap \cdot ε})$ samples from $D$. Here $v(D)$ is a natural variance measure. Combining our algorithm with previous work to initialize $x_0$, we obtain a number of improved sample complexity and runtime results. For general distributions, we achieve asymptotically optimal accuracy as a function of sample size as the number of samples grows large. Our results center around a robust analysis of the classic method of shift-and-invert preconditioning to reduce eigenvector computation to approximately solving a sequence of linear systems. We then apply fast SVRG based approximate system solvers to achieve our claims. We believe our results suggest the general effectiveness of shift-and-invert based approaches and imply that further computational gains may be reaped in practice.

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preprint2016arXiv

Streaming PCA: Matching Matrix Bernstein and Near-Optimal Finite Sample Guarantees for Oja's Algorithm

This work provides improved guarantees for streaming principle component analysis (PCA). Given $A_1, \ldots, A_n\in \mathbb{R}^{d\times d}$ sampled independently from distributions satisfying $\mathbb{E}[A_i] = Σ$ for $Σ\succeq \mathbf{0}$, this work provides an $O(d)$-space linear-time single-pass streaming algorithm for estimating the top eigenvector of $Σ$. The algorithm nearly matches (and in certain cases improves upon) the accuracy obtained by the standard batch method that computes top eigenvector of the empirical covariance $\frac{1}{n} \sum_{i \in [n]} A_i$ as analyzed by the matrix Bernstein inequality. Moreover, to achieve constant accuracy, our algorithm improves upon the best previous known sample complexities of streaming algorithms by either a multiplicative factor of $O(d)$ or $1/\mathrm{gap}$ where $\mathrm{gap}$ is the relative distance between the top two eigenvalues of $Σ$. These results are achieved through a novel analysis of the classic Oja's algorithm, one of the oldest and most popular algorithms for streaming PCA. In particular, this work shows that simply picking a random initial point $w_0$ and applying the update rule $w_{i + 1} = w_i + η_i A_i w_i$ suffices to accurately estimate the top eigenvector, with a suitable choice of $η_i$. We believe our result sheds light on how to efficiently perform streaming PCA both in theory and in practice and we hope that our analysis may serve as the basis for analyzing many variants and extensions of streaming PCA.

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preprint2015arXiv

Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition

We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict saddle property for non-convex problem that allows for efficient optimization. Using this property we show that stochastic gradient descent converges to a local minimum in a polynomial number of iterations. To the best of our knowledge this is the first work that gives global convergence guarantees for stochastic gradient descent on non-convex functions with exponentially many local minima and saddle points. Our analysis can be applied to orthogonal tensor decomposition, which is widely used in learning a rich class of latent variable models. We propose a new optimization formulation for the tensor decomposition problem that has strict saddle property. As a result we get the first online algorithm for orthogonal tensor decomposition with global convergence guarantee.

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preprint2014arXiv

Differentially Private Data Releasing for Smooth Queries with Synthetic Database Output

We consider accurately answering smooth queries while preserving differential privacy. A query is said to be $K$-smooth if it is specified by a function defined on $[-1,1]^d$ whose partial derivatives up to order $K$ are all bounded. We develop an $ε$-differentially private mechanism for the class of $K$-smooth queries. The major advantage of the algorithm is that it outputs a synthetic database. In real applications, a synthetic database output is appealing. Our mechanism achieves an accuracy of $O (n^{-\frac{K}{2d+K}}/ε)$, and runs in polynomial time. We also generalize the mechanism to preserve $(ε, δ)$-differential privacy with slightly improved accuracy. Extensive experiments on benchmark datasets demonstrate that the mechanisms have good accuracy and are efficient.

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