arXiv subjects

Kaiyi Ji

Publications and source records attributed to Kaiyi Ji.

53 records · Page 3Linked to original sources

arXiv2021

A New One-Point Residual-Feedback Oracle For Black-Box Learning and Control

Zeroth-order optimization (ZO) algorithms have been recently used to solve black-box or simulation-based learning and control problems, where the gradient of the objective function cannot be easily computed but can be approximated using the objective function values. Many existing ZO algorithms adopt two-point feedback schemes due to their fast convergence rate compared to one-point feedback schemes. However, two-point schemes require two evaluations of the objective function at each iteration, which can be impractical in applications where the data are not all available a priori, e.g., in online optimization. In this paper, we propose a novel one-point feedback scheme that queries the function value once at each iteration and estimates the gradient using the residual between two consecutive points. When optimizing a deterministic Lipschitz function, we show that the query complexity of ZO with the proposed one-point residual feedback matches that of ZO with the existing two-point schemes. Moreover, the query complexity of the proposed algorithm can be improved when the objective function has Lipschitz gradient. Then, for stochastic bandit optimization problems where only noisy objective function values are given, we show that ZO with one-point residual feedback achieves the same convergence rate as that of two-point scheme with uncontrollable data samples. We demonstrate the effectiveness of the proposed one-point residual feedback via extensive numerical experiments.

math.OC↗

arXiv2021

Bilevel Optimization: Convergence Analysis and Enhanced Design

Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem. For deterministic bilevel optimization, we provide a comprehensive convergence rate analysis for two popular algorithms respectively based on approximate implicit differentiation (AID) and iterative differentiation (ITD). For the AID-based method, we orderwisely improve the previous convergence rate analysis due to a more practical parameter selection as well as a warm start strategy, and for the ITD-based method we establish the first theoretical convergence rate. Our analysis also provides a quantitative comparison between ITD and AID based approaches. For stochastic bilevel optimization, we propose a novel algorithm named stocBiO, which features a sample-efficient hypergradient estimator using efficient Jacobian- and Hessian-vector product computations. We provide the convergence rate guarantee for stocBiO, and show that stocBiO outperforms the best known computational complexities orderwisely with respect to the condition number $κ$ and the target accuracy $ε$. We further validate our theoretical results and demonstrate the efficiency of bilevel optimization algorithms by the experiments on meta-learning and hyperparameter optimization.

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arXiv2021

Bilevel Optimization for Machine Learning: Algorithm Design and Convergence Analysis

Bilevel optimization has become a powerful framework in various machine learning applications including meta-learning, hyperparameter optimization, and network architecture search. There are generally two classes of bilevel optimization formulations for machine learning: 1) problem-based bilevel optimization, whose inner-level problem is formulated as finding a minimizer of a given loss function; and 2) algorithm-based bilevel optimization, whose inner-level solution is an output of a fixed algorithm. For the first class, two popular types of gradient-based algorithms have been proposed for hypergradient estimation via approximate implicit differentiation (AID) and iterative differentiation (ITD). Algorithms for the second class include the popular model-agnostic meta-learning (MAML) and almost no inner loop (ANIL). However, the convergence rate and fundamental limitations of bilevel optimization algorithms have not been well explored. This thesis provides a comprehensive convergence rate analysis for bilevel algorithms in the aforementioned two classes. We further propose principled algorithm designs for bilevel optimization with higher efficiency and scalability. For the problem-based formulation, we provide a convergence rate analysis for AID- and ITD-based bilevel algorithms. We then develop acceleration bilevel algorithms, for which we provide shaper convergence analysis with relaxed assumptions. We also provide the first lower bounds for bilevel optimization, and establish the optimality by providing matching upper bounds under certain conditions. We finally propose new stochastic bilevel optimization algorithms with lower complexity and higher efficiency in practice. For the algorithm-based formulation, we develop a theoretical convergence for general multi-step MAML and ANIL, and characterize the impact of parameter selections and loss geometries on the their complexities.

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arXiv2020

Boosting One-Point Derivative-Free Online Optimization via Residual Feedback

Zeroth-order optimization (ZO) typically relies on two-point feedback to estimate the unknown gradient of the objective function. Nevertheless, two-point feedback can not be used for online optimization of time-varying objective functions, where only a single query of the function value is possible at each time step. In this work, we propose a new one-point feedback method for online optimization that estimates the objective function gradient using the residual between two feedback points at consecutive time instants. Moreover, we develop regret bounds for ZO with residual feedback for both convex and nonconvex online optimization problems. Specifically, for both deterministic and stochastic problems and for both Lipschitz and smooth objective functions, we show that using residual feedback can produce gradient estimates with much smaller variance compared to conventional one-point feedback methods. As a result, our regret bounds are much tighter compared to existing regret bounds for ZO with conventional one-point feedback, which suggests that ZO with residual feedback can better track the optimizer of online optimization problems. Additionally, our regret bounds rely on weaker assumptions than those used in conventional one-point feedback methods. Numerical experiments show that ZO with residual feedback significantly outperforms existing one-point feedback methods also in practice.

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arXiv2020

Convergence of Meta-Learning with Task-Specific Adaptation over Partial Parameters

Although model-agnostic meta-learning (MAML) is a very successful algorithm in meta-learning practice, it can have high computational cost because it updates all model parameters over both the inner loop of task-specific adaptation and the outer-loop of meta initialization training. A more efficient algorithm ANIL (which refers to almost no inner loop) was proposed recently by Raghu et al. 2019, which adapts only a small subset of parameters in the inner loop and thus has substantially less computational cost than MAML as demonstrated by extensive experiments. However, the theoretical convergence of ANIL has not been studied yet. In this paper, we characterize the convergence rate and the computational complexity for ANIL under two representative inner-loop loss geometries, i.e., strongly-convexity and nonconvexity. Our results show that such a geometric property can significantly affect the overall convergence performance of ANIL. For example, ANIL achieves a faster convergence rate for a strongly-convex inner-loop loss as the number $N$ of inner-loop gradient descent steps increases, but a slower convergence rate for a nonconvex inner-loop loss as $N$ increases. Moreover, our complexity analysis provides a theoretical quantification on the improved efficiency of ANIL over MAML. The experiments on standard few-shot meta-learning benchmarks validate our theoretical findings.

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arXiv2020

History-Gradient Aided Batch Size Adaptation for Variance Reduced Algorithms

Variance-reduced algorithms, although achieve great theoretical performance, can run slowly in practice due to the periodic gradient estimation with a large batch of data. Batch-size adaptation thus arises as a promising approach to accelerate such algorithms. However, existing schemes either apply prescribed batch-size adaption rule or exploit the information along optimization path via additional backtracking and condition verification steps. In this paper, we propose a novel scheme, which eliminates backtracking line search but still exploits the information along optimization path by adapting the batch size via history stochastic gradients. We further theoretically show that such a scheme substantially reduces the overall complexity for popular variance-reduced algorithms SVRG and SARAH/SPIDER for both conventional nonconvex optimization and reinforcement learning problems. To this end, we develop a new convergence analysis framework to handle the dependence of the batch size on history stochastic gradients. Extensive experiments validate the effectiveness of the proposed batch-size adaptation scheme.

math.OC↗

arXiv2020

Theoretical Convergence of Multi-Step Model-Agnostic Meta-Learning

As a popular meta-learning approach, the model-agnostic meta-learning (MAML) algorithm has been widely used due to its simplicity and effectiveness. However, the convergence of the general multi-step MAML still remains unexplored. In this paper, we develop a new theoretical framework to provide such convergence guarantee for two types of objective functions that are of interest in practice: (a) resampling case (e.g., reinforcement learning), where loss functions take the form in expectation and new data are sampled as the algorithm runs; and (b) finite-sum case (e.g., supervised learning), where loss functions take the finite-sum form with given samples. For both cases, we characterize the convergence rate and the computational complexity to attain an $ε$-accurate solution for multi-step MAML in the general nonconvex setting. In particular, our results suggest that an inner-stage stepsize needs to be chosen inversely proportional to the number $N$ of inner-stage steps in order for $N$-step MAML to have guaranteed convergence. From the technical perspective, we develop novel techniques to deal with the nested structure of the meta gradient for multi-step MAML, which can be of independent interest.

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arXiv2020

SpiderBoost and Momentum: Faster Stochastic Variance Reduction Algorithms

SARAH and SPIDER are two recently developed stochastic variance-reduced algorithms, and SPIDER has been shown to achieve a near-optimal first-order oracle complexity in smooth nonconvex optimization. However, SPIDER uses an accuracy-dependent stepsize that slows down the convergence in practice, and cannot handle objective functions that involve nonsmooth regularizers. In this paper, we propose SpiderBoost as an improved scheme, which allows to use a much larger constant-level stepsize while maintaining the same near-optimal oracle complexity, and can be extended with proximal mapping to handle composite optimization (which is nonsmooth and nonconvex) with provable convergence guarantee. In particular, we show that proximal SpiderBoost achieves an oracle complexity of $\mathcal{O}(\min\{n^{1/2}ε^{-2},ε^{-3}\})$ in composite nonconvex optimization, improving the state-of-the-art result by a factor of $\mathcal{O}(\min\{n^{1/6},ε^{-1/3}\})$. We further develop a novel momentum scheme to accelerate SpiderBoost for composite optimization, which achieves the near-optimal oracle complexity in theory and substantial improvement in experiments.

math.OC↗

arXiv2020

Proximal Gradient Algorithm with Momentum and Flexible Parameter Restart for Nonconvex Optimization

Various types of parameter restart schemes have been proposed for accelerated gradient algorithms to facilitate their practical convergence in convex optimization. However, the convergence properties of accelerated gradient algorithms under parameter restart remain obscure in nonconvex optimization. In this paper, we propose a novel accelerated proximal gradient algorithm with parameter restart (named APG-restart) for solving nonconvex and nonsmooth problems. Our APG-restart is designed to 1) allow for adopting flexible parameter restart schemes that cover many existing ones; 2) have a global sub-linear convergence rate in nonconvex and nonsmooth optimization; and 3) have guaranteed convergence to a critical point and have various types of asymptotic convergence rates depending on the parameterization of local geometry in nonconvex and nonsmooth optimization. Numerical experiments demonstrate the effectiveness of our proposed algorithm.

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arXiv2020

Robust Stochastic Bandit Algorithms under Probabilistic Unbounded Adversarial Attack

The multi-armed bandit formalism has been extensively studied under various attack models, in which an adversary can modify the reward revealed to the player. Previous studies focused on scenarios where the attack value either is bounded at each round or has a vanishing probability of occurrence. These models do not capture powerful adversaries that can catastrophically perturb the revealed reward. This paper investigates the attack model where an adversary attacks with a certain probability at each round, and its attack value can be arbitrary and unbounded if it attacks. Furthermore, the attack value does not necessarily follow a statistical distribution. We propose a novel sample median-based and exploration-aided UCB algorithm (called med-E-UCB) and a median-based $ε$-greedy algorithm (called med-$ε$-greedy). Both of these algorithms are provably robust to the aforementioned attack model. More specifically we show that both algorithms achieve $\mathcal{O}(\log T)$ pseudo-regret (i.e., the optimal regret without attacks). We also provide a high probability guarantee of $\mathcal{O}(\log T)$ regret with respect to random rewards and random occurrence of attacks. These bounds are achieved under arbitrary and unbounded reward perturbation as long as the attack probability does not exceed a certain constant threshold. We provide multiple synthetic simulations of the proposed algorithms to verify these claims and showcase the inability of existing techniques to achieve sublinear regret. We also provide experimental results of the algorithm operating in a cognitive radio setting using multiple software-defined radios.

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arXiv2020

Learning Latent Features with Pairwise Penalties in Low-Rank Matrix Completion

Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent variables can be exploited by pairwise learning using the graph regularized matrix factorization (GRMF) method. However, existing GRMF approaches often use the squared loss to measure the pairwise differences, which may be overly influenced by dissimilar pairs and lead to inferior prediction. To fully empower pairwise learning for matrix completion, we propose a general optimization framework that allows a rich class of (non-)convex pairwise penalty functions. A new and efficient algorithm is developed to solve the proposed optimization problem, with a theoretical convergence guarantee under mild assumptions. In an important situation where the latent variables form a small number of subgroups, its statistical guarantee is also fully considered. In particular, we theoretically characterize the performance of the complexity-regularized maximum likelihood estimator, as a special case of our framework, which is shown to have smaller errors when compared to the standard matrix completion framework without pairwise penalties. We conduct extensive experiments on both synthetic and real datasets to demonstrate the superior performance of this general framework.

stat.ML↗

arXiv2019

Improved Zeroth-Order Variance Reduced Algorithms and Analysis for Nonconvex Optimization

Two types of zeroth-order stochastic algorithms have recently been designed for nonconvex optimization respectively based on the first-order techniques SVRG and SARAH/SPIDER. This paper addresses several important issues that are still open in these methods. First, all existing SVRG-type zeroth-order algorithms suffer from worse function query complexities than either zeroth-order gradient descent (ZO-GD) or stochastic gradient descent (ZO-SGD). In this paper, we propose a new algorithm ZO-SVRG-Coord-Rand and develop a new analysis for an existing ZO-SVRG-Coord algorithm proposed in Liu et al. 2018b, and show that both ZO-SVRG-Coord-Rand and ZO-SVRG-Coord (under our new analysis) outperform other exiting SVRG-type zeroth-order methods as well as ZO-GD and ZO-SGD. Second, the existing SPIDER-type algorithm SPIDER-SZO (Fang et al. 2018) has superior theoretical performance, but suffers from the generation of a large number of Gaussian random variables as well as a $\sqrtε$-level stepsize in practice. In this paper, we develop a new algorithm ZO-SPIDER-Coord, which is free from Gaussian variable generation and allows a large constant stepsize while maintaining the same convergence rate and query complexity, and we further show that ZO-SPIDER-Coord automatically achieves a linear convergence rate as the iterate enters into a local PL region without restart and algorithmic modification.

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arXiv2019

Momentum Schemes with Stochastic Variance Reduction for Nonconvex Composite Optimization

Two new stochastic variance-reduced algorithms named SARAH and SPIDER have been recently proposed, and SPIDER has been shown to achieve a near-optimal gradient oracle complexity for nonconvex optimization. However, the theoretical advantage of SPIDER does not lead to substantial improvement of practical performance over SVRG. To address this issue, momentum technique can be a good candidate to improve the performance of SPIDER. However, existing momentum schemes used in variance-reduced algorithms are designed specifically for convex optimization, and are not applicable to nonconvex scenarios. In this paper, we develop novel momentum schemes with flexible coefficient settings to accelerate SPIDER for nonconvex and nonsmooth composite optimization, and show that the resulting algorithms achieve the near-optimal gradient oracle complexity for achieving a generalized first-order stationary condition. Furthermore, we generalize our algorithm to online nonconvex and nonsmooth optimization, and establish an oracle complexity result that matches the state-of-the-art. Our extensive experiments demonstrate the superior performance of our proposed algorithm over other stochastic variance-reduced algorithms.

math.OC↗

arXiv2018

Minimax Estimation of Neural Net Distance

An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance based on samples drawn from the distributions. We develop the first known minimax lower bound on the estimation error of the neural net distance, and an upper bound tighter than an existing bound on the estimator error for the empirical neural net distance. Our lower and upper bounds match not only in the order of the sample size but also in terms of the norm of the parameter matrices of neural networks, which justifies the empirical neural net distance as a good approximation of the true neural net distance for training GANs in practice.

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arXiv2018

When Will Gradient Methods Converge to Max-margin Classifier under ReLU Models?

We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show that there can exist spurious asymptotic local minima besides asymptotic global minima. We then show that gradient descent (GD) can converge to either a global or a local max-margin direction, or may diverge from the desired max-margin direction in a general context. For stochastic gradient descent (SGD), we show that it converges in expectation to either the global or the local max-margin direction if SGD converges. We further explore the implicit bias of these algorithms in learning a multi-neuron network under certain stationary conditions, and show that the learned classifier maximizes the margins of each sample pattern partition under the ReLU activation.

cs.LG↗

arXiv2018

Asymptotic Miss Ratio of LRU Caching with Consistent Hashing

To efficiently scale data caching infrastructure to support emerging big data applications, many caching systems rely on consistent hashing to group a large number of servers to form a cooperative cluster. These servers are organized together according to a random hash function. They jointly provide a unified but distributed hash table to serve swift and voluminous data item requests. Different from the single least-recently-used (LRU) server that has already been extensively studied, theoretically characterizing a cluster that consists of multiple LRU servers remains yet to be explored. These servers are not simply added together; the random hashing complicates the behavior. To this end, we derive the asymptotic miss ratio of data item requests on a LRU cluster with consistent hashing. We show that these individual cache spaces on different servers can be effectively viewed as if they could be pooled together to form a single virtual LRU cache space parametrized by an appropriate cache size. This equivalence can be established rigorously under the condition that the cache sizes of the individual servers are large. For typical data caching systems this condition is common. Our theoretical framework provides a convenient abstraction that can directly apply the results from the simpler single LRU cache to the more complex LRU cluster with consistent hashing.

cs.PF↗

arXiv2017

On Resource Pooling and Separation for LRU Caching

Caching systems using the Least Recently Used (LRU) principle have now become ubiquitous. A fundamental question for these systems is whether the cache space should be pooled together or divided to serve multiple flows of data item requests in order to minimize the miss probabilities. In this paper, we show that there is no straight yes or no answer to this question, depending on complex combinations of critical factors, including, e.g., request rates, overlapped data items across different request flows, data item popularities and their sizes. Specifically, we characterize the asymptotic miss probabilities for multiple competing request flows under resource pooling and separation for LRU caching when the cache size is large. Analytically, we show that it is asymptotically optimal to jointly serve multiple flows if their data item sizes and popularity distributions are similar and their arrival rates do not differ significantly; the self-organizing property of LRU caching automatically optimizes the resource allocation among them asymptotically. Otherwise, separating these flows could be better, e.g., when data sizes vary significantly. We also quantify critical points beyond which resource pooling is better than separation for each of the flows when the overlapped data items exceed certain levels. Technically, we generalize existing results on the asymptotic miss probability of LRU caching for a broad class of heavy-tailed distributions and extend them to multiple competing flows with varying data item sizes, which also validates the Che approximation under certain conditions. These results provide new insights on improving the performance of caching systems.

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