arXiv subjects

Tianbao Yang

Publications and source records attributed to Tianbao Yang.

At least 145 records · Page 8Linked to original sources

arXiv2018

How Local is the Local Diversity? Reinforcing Sequential Determinantal Point Processes with Dynamic Ground Sets for Supervised Video Summarization

The large volume of video content and high viewing frequency demand automatic video summarization algorithms, of which a key property is the capability of modeling diversity. If videos are lengthy like hours-long egocentric videos, it is necessary to track the temporal structures of the videos and enforce local diversity. The local diversity refers to that the shots selected from a short time duration are diverse but visually similar shots are allowed to co-exist in the summary if they appear far apart in the video. In this paper, we propose a novel probabilistic model, built upon SeqDPP, to dynamically control the time span of a video segment upon which the local diversity is imposed. In particular, we enable SeqDPP to learn to automatically infer how local the local diversity is supposed to be from the input video. The resulting model is extremely involved to train by the hallmark maximum likelihood estimation (MLE), which further suffers from the exposure bias and non-differentiable evaluation metrics. To tackle these problems, we instead devise a reinforcement learning algorithm for training the proposed model. Extensive experiments verify the advantages of our model and the new learning algorithm over MLE-based methods.

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arXiv2018

Dynamic Regret of Strongly Adaptive Methods

To cope with changing environments, recent developments in online learning have introduced the concepts of adaptive regret and dynamic regret independently. In this paper, we illustrate an intrinsic connection between these two concepts by showing that the dynamic regret can be expressed in terms of the adaptive regret and the functional variation. This observation implies that strongly adaptive algorithms can be directly leveraged to minimize the dynamic regret. As a result, we present a series of strongly adaptive algorithms that have small dynamic regrets for convex functions, exponentially concave functions, and strongly convex functions, respectively. To the best of our knowledge, this is the first time that exponential concavity is utilized to upper bound the dynamic regret. Moreover, all of those adaptive algorithms do not need any prior knowledge of the functional variation, which is a significant advantage over previous specialized methods for minimizing dynamic regret.

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arXiv2018

An Aggressive Genetic Programming Approach for Searching Neural Network Structure Under Computational Constraints

Recently, there emerged revived interests of designing automatic programs (e.g., using genetic/evolutionary algorithms) to optimize the structure of Convolutional Neural Networks (CNNs) for a specific task. The challenge in designing such programs lies in how to balance between large search space of the network structures and high computational costs. Existing works either impose strong restrictions on the search space or use enormous computing resources. In this paper, we study how to design a genetic programming approach for optimizing the structure of a CNN for a given task under limited computational resources yet without imposing strong restrictions on the search space. To reduce the computational costs, we propose two general strategies that are observed to be helpful: (i) aggressively selecting strongest individuals for survival and reproduction, and killing weaker individuals at a very early age; (ii) increasing mutation frequency to encourage diversity and faster evolution. The combined strategy with additional optimization techniques allows us to explore a large search space but with affordable computational costs. Our results on standard benchmark datasets (MNIST, SVHN, CIFAR-10, CIFAR-100) are competitive to similar approaches with significantly reduced computational costs.

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arXiv2018

Learning with Non-Convex Truncated Losses by SGD

Learning with a {\it convex loss} function has been a dominating paradigm for many years. It remains an interesting question how non-convex loss functions help improve the generalization of learning with broad applicability. In this paper, we study a family of objective functions formed by truncating traditional loss functions, which is applicable to both shallow learning and deep learning. Truncating loss functions has potential to be less vulnerable and more robust to large noise in observations that could be adversarial. More importantly, it is a generic technique without assuming the knowledge of noise distribution. To justify non-convex learning with truncated losses, we establish excess risk bounds of empirical risk minimization based on truncated losses for heavy-tailed output, and statistical error of an approximate stationary point found by stochastic gradient descent (SGD) method. Our experiments for shallow and deep learning for regression with outliers, corrupted data and heavy-tailed noise further justify the proposed method.

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arXiv2018

Fast Rates of ERM and Stochastic Approximation: Adaptive to Error Bound Conditions

Error bound conditions (EBC) are properties that characterize the growth of an objective function when a point is moved away from the optimal set. They have recently received increasing attention in the field of optimization for developing optimization algorithms with fast convergence. However, the studies of EBC in statistical learning are hitherto still limited. The main contributions of this paper are two-fold. First, we develop fast and intermediate rates of empirical risk minimization (ERM) under EBC for risk minimization with Lipschitz continuous, and smooth convex random functions. Second, we establish fast and intermediate rates of an efficient stochastic approximation (SA) algorithm for risk minimization with Lipschitz continuous random functions, which requires only one pass of $n$ samples and adapts to EBC. For both approaches, the convergence rates span a full spectrum between $\widetilde O(1/\sqrt{n})$ and $\widetilde O(1/n)$ depending on the power constant in EBC, and could be even faster than $O(1/n)$ in special cases for ERM. Moreover, these convergence rates are automatically adaptive without using any knowledge of EBC. Overall, this work not only strengthens the understanding of ERM for statistical learning but also brings new fast stochastic algorithms for solving a broad range of statistical learning problems.

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arXiv2018

First-order Stochastic Algorithms for Escaping From Saddle Points in Almost Linear Time

Two classes of methods have been proposed for escaping from saddle points with one using the second-order information carried by the Hessian and the other adding the noise into the first-order information. The existing analysis for algorithms using noise in the first-order information is quite involved and hides the essence of added noise, which hinder further improvements of these algorithms. In this paper, we present a novel perspective of noise-adding technique, i.e., adding the noise into the first-order information can help extract the negative curvature from the Hessian matrix, and provide a formal reasoning of this perspective by analyzing a simple first-order procedure. More importantly, the proposed procedure enables one to design purely first-order stochastic algorithms for escaping from non-degenerate saddle points with a much better time complexity (almost linear time in terms of the problem's dimensionality). In particular, we develop a {\bf first-order stochastic algorithm} based on our new technique and an existing algorithm that only converges to a first-order stationary point to enjoy a time complexity of {$\widetilde O(d/ε^{3.5})$ for finding a nearly second-order stationary point $\bf{x}$ such that $\|\nabla F(bf{x})\|\leq ε$ and $\nabla^2 F(bf{x})\geq -\sqrtεI$ (in high probability), where $F(\cdot)$ denotes the objective function and $d$ is the dimensionality of the problem. To the best of our knowledge, this is the best theoretical result of first-order algorithms for stochastic non-convex optimization, which is even competitive with if not better than existing stochastic algorithms hinging on the second-order information.

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arXiv2018

NEON+: Accelerated Gradient Methods for Extracting Negative Curvature for Non-Convex Optimization

Accelerated gradient (AG) methods are breakthroughs in convex optimization, improving the convergence rate of the gradient descent method for optimization with smooth functions. However, the analysis of AG methods for non-convex optimization is still limited. It remains an open question whether AG methods from convex optimization can accelerate the convergence of the gradient descent method for finding local minimum of non-convex optimization problems. This paper provides an affirmative answer to this question. In particular, we analyze two renowned variants of AG methods (namely Polyak's Heavy Ball method and Nesterov's Accelerated Gradient method) for extracting the negative curvature from random noise, which is central to escaping from saddle points. By leveraging the proposed AG methods for extracting the negative curvature, we present a new AG algorithm with double loops for non-convex optimization~\footnote{this is in contrast to a single-loop AG algorithm proposed in a recent manuscript~\citep{AGNON}, which directly analyzed the Nesterov's AG method for non-convex optimization and appeared online on November 29, 2017. However, we emphasize that our work is an independent work, which is inspired by our earlier work~\citep{NEON17} and is based on a different novel analysis.}, which converges to second-order stationary point $\x$ such that $\|\nabla f(\x)\|\leq ε$ and $\nabla^2 f(\x)\geq -\sqrtε I$ with $\widetilde O(1/ε^{1.75})$ iteration complexity, improving that of gradient descent method by a factor of $ε^{-0.25}$ and matching the best iteration complexity of second-order Hessian-free methods for non-convex optimization.

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arXiv2017

Improved Dynamic Regret for Non-degenerate Functions

Recently, there has been a growing research interest in the analysis of dynamic regret, which measures the performance of an online learner against a sequence of local minimizers. By exploiting the strong convexity, previous studies have shown that the dynamic regret can be upper bounded by the path-length of the comparator sequence. In this paper, we illustrate that the dynamic regret can be further improved by allowing the learner to query the gradient of the function multiple times, and meanwhile the strong convexity can be weakened to other non-degenerate conditions. Specifically, we introduce the squared path-length, which could be much smaller than the path-length, as a new regularity of the comparator sequence. When multiple gradients are accessible to the learner, we first demonstrate that the dynamic regret of strongly convex functions can be upper bounded by the minimum of the path-length and the squared path-length. We then extend our theoretical guarantee to functions that are semi-strongly convex or self-concordant. To the best of our knowledge, this is the first time that semi-strong convexity and self-concordance are utilized to tighten the dynamic regret.

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arXiv2017

Stochastic Non-convex Optimization with Strong High Probability Second-order Convergence

In this paper, we study stochastic non-convex optimization with non-convex random functions. Recent studies on non-convex optimization revolve around establishing second-order convergence, i.e., converging to a nearly second-order optimal stationary points. However, existing results on stochastic non-convex optimization are limited, especially with a high probability second-order convergence. We propose a novel updating step (named NCG-S) by leveraging a stochastic gradient and a noisy negative curvature of a stochastic Hessian, where the stochastic gradient and Hessian are based on a proper mini-batch of random functions. Building on this step, we develop two algorithms and establish their high probability second-order convergence. To the best of our knowledge, the proposed stochastic algorithms are the first with a second-order convergence in {\it high probability} and a time complexity that is {\it almost linear} in the problem's dimensionality.

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arXiv2017

On Noisy Negative Curvature Descent: Competing with Gradient Descent for Faster Non-convex Optimization

The Hessian-vector product has been utilized to find a second-order stationary solution with strong complexity guarantee (e.g., almost linear time complexity in the problem's dimensionality). In this paper, we propose to further reduce the number of Hessian-vector products for faster non-convex optimization. Previous algorithms need to approximate the smallest eigen-value with a sufficient precision (e.g., $ε_2\ll 1$) in order to achieve a sufficiently accurate second-order stationary solution (i.e., $λ_{\min}(\nabla^2 f(\x))\geq -ε_2)$. In contrast, the proposed algorithms only need to compute the smallest eigen-vector approximating the corresponding eigen-value up to a small power of current gradient's norm. As a result, it can dramatically reduce the number of Hessian-vector products during the course of optimization before reaching first-order stationary points (e.g., saddle points). The key building block of the proposed algorithms is a novel updating step named the NCG step, which lets a noisy negative curvature descent compete with the gradient descent. We show that the worst-case time complexity of the proposed algorithms with their favorable prescribed accuracy requirements can match the best in literature for achieving a second-order stationary point but with an arguably smaller per-iteration cost. We also show that the proposed algorithms can benefit from inexact Hessian by developing their variants accepting inexact Hessian under a mild condition for achieving the same goal. Moreover, we develop a stochastic algorithm for a finite or infinite sum non-convex optimization problem. To the best of our knowledge, the proposed stochastic algorithm is the first one that converges to a second-order stationary point in {\it high probability} with a time complexity independent of the sample size and almost linear in dimensionality.

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arXiv2017

A Simple Analysis for Exp-concave Empirical Minimization with Arbitrary Convex Regularizer

In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledge, this result is the first of its kind. As a byproduct, we can directly obtain the fast rate with {\it high probability} for exponential concave empirical risk minimization with and without any convex regularization, which not only extends existing results of empirical risk minimization but also provides a unified framework for analyzing exponential concave empirical risk minimization with and without {\it any} convex regularization. Our proof is very simple only exploiting the covering number of a finite-dimensional bounded set and a concentration inequality of random vectors.

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arXiv2017

SEP-Nets: Small and Effective Pattern Networks

While going deeper has been witnessed to improve the performance of convolutional neural networks (CNN), going smaller for CNN has received increasing attention recently due to its attractiveness for mobile/embedded applications. It remains an active and important topic how to design a small network while retaining the performance of large and deep CNNs (e.g., Inception Nets, ResNets). Albeit there are already intensive studies on compressing the size of CNNs, the considerable drop of performance is still a key concern in many designs. This paper addresses this concern with several new contributions. First, we propose a simple yet powerful method for compressing the size of deep CNNs based on parameter binarization. The striking difference from most previous work on parameter binarization/quantization lies at different treatments of $1\times 1$ convolutions and $k\times k$ convolutions ($k>1$), where we only binarize $k\times k$ convolutions into binary patterns. The resulting networks are referred to as pattern networks. By doing this, we show that previous deep CNNs such as GoogLeNet and Inception-type Nets can be compressed dramatically with marginal drop in performance. Second, in light of the different functionalities of $1\times 1$ (data projection/transformation) and $k\times k$ convolutions (pattern extraction), we propose a new block structure codenamed the pattern residual block that adds transformed feature maps generated by $1\times 1$ convolutions to the pattern feature maps generated by $k\times k$ convolutions, based on which we design a small network with $\sim 1$ million parameters. Combining with our parameter binarization, we achieve better performance on ImageNet than using similar sized networks including recently released Google MobileNets.

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arXiv2017

A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates

This paper focuses on convex constrained optimization problems, where the solution is subject to a convex inequality constraint. In particular, we aim at challenging problems for which both projection into the constrained domain and a linear optimization under the inequality constraint are time-consuming, which render both projected gradient methods and conditional gradient methods (a.k.a. the Frank-Wolfe algorithm) expensive. In this paper, we develop projection reduced optimization algorithms for both smooth and non-smooth optimization with improved convergence rates under a certain regularity condition of the constraint function. We first present a general theory of optimization with only one projection. Its application to smooth optimization with only one projection yields $O(1/ε)$ iteration complexity, which improves over the $O(1/ε^2)$ iteration complexity established before for non-smooth optimization and can be further reduced under strong convexity. Then we introduce a local error bound condition and develop faster algorithms for non-strongly convex optimization at the price of a logarithmic number of projections. In particular, we achieve an iteration complexity of $\widetilde O(1/ε^{2(1-θ)})$ for non-smooth optimization and $\widetilde O(1/ε^{1-θ})$ for smooth optimization, where $θ\in(0,1]$ appearing the local error bound condition characterizes the functional local growth rate around the optimal solutions. Novel applications in solving the constrained $\ell_1$ minimization problem and a positive semi-definite constrained distance metric learning problem demonstrate that the proposed algorithms achieve significant speed-up compared with previous algorithms.

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arXiv2017

Adaptive Accelerated Gradient Converging Methods under Holderian Error Bound Condition

Recent studies have shown that proximal gradient (PG) method and accelerated gradient method (APG) with restarting can enjoy a linear convergence under a weaker condition than strong convexity, namely a quadratic growth condition (QGC). However, the faster convergence of restarting APG method relies on the potentially unknown constant in QGC to appropriately restart APG, which restricts its applicability. We address this issue by developing a novel adaptive gradient converging methods, i.e., leveraging the magnitude of proximal gradient as a criterion for restart and termination. Our analysis extends to a much more general condition beyond the QGC, namely the Hölderian error bound (HEB) condition. {\it The key technique} for our development is a novel synthesis of {\it adaptive regularization and a conditional restarting scheme}, which extends previous work focusing on strongly convex problems to a much broader family of problems. Furthermore, we demonstrate that our results have important implication and applications in machine learning: (i) if the objective function is coercive and semi-algebraic, PG's convergence speed is essentially $o(\frac{1}{t})$, where $t$ is the total number of iterations; (ii) if the objective function consists of an $\ell_1$, $\ell_\infty$, $\ell_{1,\infty}$, or huber norm regularization and a convex smooth piecewise quadratic loss (e.g., squares loss, squared hinge loss and huber loss), the proposed algorithm is parameter-free and enjoys a {\it faster linear convergence} than PG without any other assumptions (e.g., restricted eigen-value condition). It is notable that our linear convergence results for the aforementioned problems are global instead of local. To the best of our knowledge, these improved results are the first shown in this work.

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arXiv2017

Doubly Stochastic Primal-Dual Coordinate Method for Bilinear Saddle-Point Problem

We propose a doubly stochastic primal-dual coordinate optimization algorithm for empirical risk minimization, which can be formulated as a bilinear saddle-point problem. In each iteration, our method randomly samples a block of coordinates of the primal and dual solutions to update. The linear convergence of our method could be established in terms of 1) the distance from the current iterate to the optimal solution and 2) the primal-dual objective gap. We show that the proposed method has a lower overall complexity than existing coordinate methods when either the data matrix has a factorized structure or the proximal mapping on each block is computationally expensive, e.g., involving an eigenvalue decomposition. The efficiency of the proposed method is confirmed by empirical studies on several real applications, such as the multi-task large margin nearest neighbor problem.

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arXiv2017

Empirical Risk Minimization for Stochastic Convex Optimization: $O(1/n)$- and $O(1/n^2)$-type of Risk Bounds

Although there exist plentiful theories of empirical risk minimization (ERM) for supervised learning, current theoretical understandings of ERM for a related problem---stochastic convex optimization (SCO), are limited. In this work, we strengthen the realm of ERM for SCO by exploiting smoothness and strong convexity conditions to improve the risk bounds. First, we establish an $\widetilde{O}(d/n + \sqrt{F_*/n})$ risk bound when the random function is nonnegative, convex and smooth, and the expected function is Lipschitz continuous, where $d$ is the dimensionality of the problem, $n$ is the number of samples, and $F_*$ is the minimal risk. Thus, when $F_*$ is small we obtain an $\widetilde{O}(d/n)$ risk bound, which is analogous to the $\widetilde{O}(1/n)$ optimistic rate of ERM for supervised learning. Second, if the objective function is also $λ$-strongly convex, we prove an $\widetilde{O}(d/n + κF_*/n )$ risk bound where $κ$ is the condition number, and improve it to $O(1/[λn^2] + κF_*/n)$ when $n=\widetildeΩ(κd)$. As a result, we obtain an $O(κ/n^2)$ risk bound under the condition that $n$ is large and $F_*$ is small, which to the best of our knowledge, is the first $O(1/n^2)$-type of risk bound of ERM. Third, we stress that the above results are established in a unified framework, which allows us to derive new risk bounds under weaker conditions, e.g., without convexity of the random function and Lipschitz continuity of the expected function. Finally, we demonstrate that to achieve an $O(1/[λn^2] + κF_*/n)$ risk bound for supervised learning, the $\widetildeΩ(κd)$ requirement on $n$ can be replaced with $Ω(κ^2)$, which is dimensionality-independent.

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arXiv2016

Efficient Non-oblivious Randomized Reduction for Risk Minimization with Improved Excess Risk Guarantee

In this paper, we address learning problems for high dimensional data. Previously, oblivious random projection based approaches that project high dimensional features onto a random subspace have been used in practice for tackling high-dimensionality challenge in machine learning. Recently, various non-oblivious randomized reduction methods have been developed and deployed for solving many numerical problems such as matrix product approximation, low-rank matrix approximation, etc. However, they are less explored for the machine learning tasks, e.g., classification. More seriously, the theoretical analysis of excess risk bounds for risk minimization, an important measure of generalization performance, has not been established for non-oblivious randomized reduction methods. It therefore remains an open problem what is the benefit of using them over previous oblivious random projection based approaches. To tackle these challenges, we propose an algorithmic framework for employing non-oblivious randomized reduction method for general empirical risk minimizing in machine learning tasks, where the original high-dimensional features are projected onto a random subspace that is derived from the data with a small matrix approximation error. We then derive the first excess risk bound for the proposed non-oblivious randomized reduction approach without requiring strong assumptions on the training data. The established excess risk bound exhibits that the proposed approach provides much better generalization performance and it also sheds more insights about different randomized reduction approaches. Finally, we conduct extensive experiments on both synthetic and real-world benchmark datasets, whose dimension scales to $O(10^7)$, to demonstrate the efficacy of our proposed approach.

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arXiv2016

Improved Dropout for Shallow and Deep Learning

Dropout has been witnessed with great success in training deep neural networks by independently zeroing out the outputs of neurons at random. It has also received a surge of interest for shallow learning, e.g., logistic regression. However, the independent sampling for dropout could be suboptimal for the sake of convergence. In this paper, we propose to use multinomial sampling for dropout, i.e., sampling features or neurons according to a multinomial distribution with different probabilities for different features/neurons. To exhibit the optimal dropout probabilities, we analyze the shallow learning with multinomial dropout and establish the risk bound for stochastic optimization. By minimizing a sampling dependent factor in the risk bound, we obtain a distribution-dependent dropout with sampling probabilities dependent on the second order statistics of the data distribution. To tackle the issue of evolving distribution of neurons in deep learning, we propose an efficient adaptive dropout (named \textbf{evolutional dropout}) that computes the sampling probabilities on-the-fly from a mini-batch of examples. Empirical studies on several benchmark datasets demonstrate that the proposed dropouts achieve not only much faster convergence and but also a smaller testing error than the standard dropout. For example, on the CIFAR-100 data, the evolutional dropout achieves relative improvements over 10\% on the prediction performance and over 50\% on the convergence speed compared to the standard dropout.

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