arXiv subjects

Mengdi Wang

Publications and source records attributed to Mengdi Wang.

At least 217 records · Page 12Linked to original sources

arXiv2023

Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data

Diffusion models achieve state-of-the-art performance in various generation tasks. However, their theoretical foundations fall far behind. This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. Our result provides sample complexity bounds for distribution estimation using diffusion models. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. Furthermore, the generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution. The convergence rate depends on the subspace dimension, indicating that diffusion models can circumvent the curse of data ambient dimensionality.

cs.LG↗

arXiv2023

On the Sample Complexity and Metastability of Heavy-tailed Policy Search in Continuous Control

Reinforcement learning is a framework for interactive decision-making with incentives sequentially revealed across time without a system dynamics model. Due to its scaling to continuous spaces, we focus on policy search where one iteratively improves a parameterized policy with stochastic policy gradient (PG) updates. In tabular Markov Decision Problems (MDPs), under persistent exploration and suitable parameterization, global optimality may be obtained. By contrast, in continuous space, the non-convexity poses a pathological challenge as evidenced by existing convergence results being mostly limited to stationarity or arbitrary local extrema. To close this gap, we step towards persistent exploration in continuous space through policy parameterizations defined by distributions of heavier tails defined by tail-index parameter alpha, which increases the likelihood of jumping in state space. Doing so invalidates smoothness conditions of the score function common to PG. Thus, we establish how the convergence rate to stationarity depends on the policy's tail index alpha, a Holder continuity parameter, integrability conditions, and an exploration tolerance parameter introduced here for the first time. Further, we characterize the dependence of the set of local maxima on the tail index through an exit and transition time analysis of a suitably defined Markov chain, identifying that policies associated with Levy Processes of a heavier tail converge to wider peaks. This phenomenon yields improved stability to perturbations in supervised learning, which we corroborate also manifests in improved performance of policy search, especially when myopic and farsighted incentives are misaligned.

cs.LG↗

arXiv2022

Near Sample-Optimal Reduction-based Policy Learning for Average Reward MDP

This work considers the sample complexity of obtaining an $\varepsilon$-optimal policy in an average reward Markov Decision Process (AMDP), given access to a generative model (simulator). When the ground-truth MDP is weakly communicating, we prove an upper bound of $\widetilde O(H \varepsilon^{-3} \ln \frac{1}δ)$ samples per state-action pair, where $H := sp(h^*)$ is the span of bias of any optimal policy, $\varepsilon$ is the accuracy and $δ$ is the failure probability. This bound improves the best-known mixing-time-based approaches in [Jin & Sidford 2021], which assume the mixing-time of every deterministic policy is bounded. The core of our analysis is a proper reduction bound from AMDP problems to discounted MDP (DMDP) problems, which may be of independent interests since it allows the application of DMDP algorithms for AMDP in other settings. We complement our upper bound by proving a minimax lower bound of $Ω(|\mathcal S| |\mathcal A| H \varepsilon^{-2} \ln \frac{1}δ)$ total samples, showing that a linear dependent on $H$ is necessary and that our upper bound matches the lower bound in all parameters of $(|\mathcal S|, |\mathcal A|, H, \ln \frac{1}δ)$ up to some logarithmic factors.

cs.LG↗

arXiv2022

Offline Reinforcement Learning with Differentiable Function Approximation is Provably Efficient

Offline reinforcement learning, which aims at optimizing sequential decision-making strategies with historical data, has been extensively applied in real-life applications. State-Of-The-Art algorithms usually leverage powerful function approximators (e.g. neural networks) to alleviate the sample complexity hurdle for better empirical performances. Despite the successes, a more systematic understanding of the statistical complexity for function approximation remains lacking. Towards bridging the gap, we take a step by considering offline reinforcement learning with differentiable function class approximation (DFA). This function class naturally incorporates a wide range of models with nonlinear/nonconvex structures. Most importantly, we show offline RL with differentiable function approximation is provably efficient by analyzing the pessimistic fitted Q-learning (PFQL) algorithm, and our results provide the theoretical basis for understanding a variety of practical heuristics that rely on Fitted Q-Iteration style design. In addition, we further improve our guarantee with a tighter instance-dependent characterization. We hope our work could draw interest in studying reinforcement learning with differentiable function approximation beyond the scope of current research.

cs.LG↗

arXiv2022

Energy System Digitization in the Era of AI: A Three-Layered Approach towards Carbon Neutrality

The transition towards carbon-neutral electricity is one of the biggest game changers in addressing climate change since it addresses the dual challenges of removing carbon emissions from the two largest sectors of emitters: electricity and transportation. The transition to a carbon-neutral electric grid poses significant challenges to conventional paradigms of modern grid planning and operation. Much of the challenge arises from the scale of the decision making and the uncertainty associated with the energy supply and demand. Artificial Intelligence (AI) could potentially have a transformative impact on accelerating the speed and scale of carbon-neutral transition, as many decision making processes in the power grid can be cast as classic, though challenging, machine learning tasks. We point out that to amplify AI's impact on carbon-neutral transition of the electric energy systems, the AI algorithms originally developed for other applications should be tailored in three layers of technology, markets, and policy.

cs.AI↗

arXiv2022

Representation Learning for General-sum Low-rank Markov Games

We study multi-agent general-sum Markov games with nonlinear function approximation. We focus on low-rank Markov games whose transition matrix admits a hidden low-rank structure on top of an unknown non-linear representation. The goal is to design an algorithm that (1) finds an $\varepsilon$-equilibrium policy sample efficiently without prior knowledge of the environment or the representation, and (2) permits a deep-learning friendly implementation. We leverage representation learning and present a model-based and a model-free approach to construct an effective representation from the collected data. For both approaches, the algorithm achieves a sample complexity of poly$(H,d,A,1/\varepsilon)$, where $H$ is the game horizon, $d$ is the dimension of the feature vector, $A$ is the size of the joint action space and $\varepsilon$ is the optimality gap. When the number of players is large, the above sample complexity can scale exponentially with the number of players in the worst case. To address this challenge, we consider Markov games with a factorized transition structure and present an algorithm that escapes such exponential scaling. To our best knowledge, this is the first sample-efficient algorithm for multi-agent general-sum Markov games that incorporates (non-linear) function approximation. We accompany our theoretical result with a neural network-based implementation of our algorithm and evaluate it against the widely used deep RL baseline, DQN with fictitious play.

cs.LG↗

arXiv2022

1xN Pattern for Pruning Convolutional Neural Networks

Though network pruning receives popularity in reducing the complexity of convolutional neural networks (CNNs), it remains an open issue to concurrently maintain model accuracy as well as achieve significant speedups on general CPUs. In this paper, we propose a novel 1xN pruning pattern to break this limitation. In particular, consecutive N output kernels with the same input channel index are grouped into one block, which serves as a basic pruning granularity of our pruning pattern. Our 1xN pattern prunes these blocks considered unimportant. We also provide a workflow of filter rearrangement that first rearranges the weight matrix in the output channel dimension to derive more influential blocks for accuracy improvements and then applies similar rearrangement to the next-layer weights in the input channel dimension to ensure correct convolutional operations. Moreover, the output computation after our 1xN pruning can be realized via a parallelized block-wise vectorized operation, leading to significant speedups on general CPUs. The efficacy of our pruning pattern is proved with experiments on ILSVRC-2012. For example, given the pruning rate of 50% and N=4, our pattern obtains about 3.0% improvements over filter pruning in the top-1 accuracy of MobileNet-V2. Meanwhile, it obtains 56.04ms inference savings on Cortex-A7 CPU over weight pruning. Our project is made available at https://github.com/lmbxmu/1xN.

cs.CV↗

arXiv2022

Communication Efficient Distributed Learning for Kernelized Contextual Bandits

We tackle the communication efficiency challenge of learning kernelized contextual bandits in a distributed setting. Despite the recent advances in communication-efficient distributed bandit learning, existing solutions are restricted to simple models like multi-armed bandits and linear bandits, which hamper their practical utility. In this paper, instead of assuming the existence of a linear reward mapping from the features to the expected rewards, we consider non-linear reward mappings, by letting agents collaboratively search in a reproducing kernel Hilbert space (RKHS). This introduces significant challenges in communication efficiency as distributed kernel learning requires the transfer of raw data, leading to a communication cost that grows linearly w.r.t. time horizon $T$. We addresses this issue by equipping all agents to communicate via a common Nyström embedding that gets updated adaptively as more data points are collected. We rigorously proved that our algorithm can attain sub-linear rate in both regret and communication cost.

cs.LG↗

arXiv2022

Efficient Reinforcement Learning in Block MDPs: A Model-free Representation Learning Approach

We present BRIEE (Block-structured Representation learning with Interleaved Explore Exploit), an algorithm for efficient reinforcement learning in Markov Decision Processes with block-structured dynamics (i.e., Block MDPs), where rich observations are generated from a set of unknown latent states. BRIEE interleaves latent states discovery, exploration, and exploitation together, and can provably learn a near-optimal policy with sample complexity scaling polynomially in the number of latent states, actions, and the time horizon, with no dependence on the size of the potentially infinite observation space. Empirically, we show that BRIEE is more sample efficient than the state-of-art Block MDP algorithm HOMER and other empirical RL baselines on challenging rich-observation combination lock problems that require deep exploration.

cs.LG↗

arXiv2022

Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic $L$-functions

We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic $L$-function on $GL_m (m \geq 2)$ and polynomial nilsequences has logarithmic decay; the other is that the mean value of the Möbius function, coefficients of automorphic $L$-function and polynomial nilsequences also has logarithmic decay.

math.NT↗

arXiv2022

Sample Complexity of Nonparametric Off-Policy Evaluation on Low-Dimensional Manifolds using Deep Networks

We consider the off-policy evaluation problem of reinforcement learning using deep convolutional neural networks. We analyze the deep fitted Q-evaluation method for estimating the expected cumulative reward of a target policy, when the data are generated from an unknown behavior policy. We show that, by choosing network size appropriately, one can leverage any low-dimensional manifold structure in the Markov decision process and obtain a sample-efficient estimator without suffering from the curse of high data ambient dimensionality. Specifically, we establish a sharp error bound for fitted Q-evaluation, which depends on the intrinsic dimension of the state-action space, the smoothness of Bellman operator, and a function class-restricted $χ^2$-divergence. It is noteworthy that the restricted $χ^2$-divergence measures the behavior and target policies' {\it mismatch in the function space}, which can be small even if the two policies are not close to each other in their tabular forms. We also develop a novel approximation result for convolutional neural networks in Q-function estimation. Numerical experiments are provided to support our theoretical analysis.

cs.LG↗

arXiv2022

Waring-Goldbach problem in short intervals

Let $k\geq2$ and $s$ be positive integers. Let $θ\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $θ>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions, can be written as $$ n=p_1^k+\cdots+p_s^k, $$ where $p_i(1\leq i\leq s)$ are primes in the interval $((\frac{n}{s})^{\frac{1}{k}}-n^{\fracθ{k}},(\frac{n}{s})^{\frac{1}{k}}+n^{\fracθ{k}}]$. The second result of this paper is to show that if $s>\frac{k(k+1)}{2}$ and $θ>0.55$, then almost all integers $n$, subject to certain congruence conditions, have above representation.

math.NT↗

arXiv2022

Provably Efficient Reinforcement Learning for Online Adaptive Influence Maximization

Online influence maximization aims to maximize the influence spread of a content in a social network with unknown network model by selecting a few seed nodes. Recent studies followed a non-adaptive setting, where the seed nodes are selected before the start of the diffusion process and network parameters are updated when the diffusion stops. We consider an adaptive version of content-dependent online influence maximization problem where the seed nodes are sequentially activated based on real-time feedback. In this paper, we formulate the problem as an infinite-horizon discounted MDP under a linear diffusion process and present a model-based reinforcement learning solution. Our algorithm maintains a network model estimate and selects seed users adaptively, exploring the social network while improving the optimal policy optimistically. We establish $\widetilde O(\sqrt{T})$ regret bound for our algorithm. Empirical evaluations on synthetic network demonstrate the efficiency of our algorithm.

cs.LG↗

arXiv2022

Decentralized Gossip-Based Stochastic Bilevel Optimization over Communication Networks

Bilevel optimization have gained growing interests, with numerous applications found in meta learning, minimax games, reinforcement learning, and nested composition optimization. This paper studies the problem of distributed bilevel optimization over a network where agents can only communicate with neighbors, including examples from multi-task, multi-agent learning and federated learning. In this paper, we propose a gossip-based distributed bilevel learning algorithm that allows networked agents to solve both the inner and outer optimization problems in a single timescale and share information via network propagation. We show that our algorithm enjoys the $\mathcal{O}(\frac{1}{K ε^2})$ per-agent sample complexity for general nonconvex bilevel optimization and $\mathcal{O}(\frac{1}{K ε})$ for strongly convex objective, achieving a speedup that scales linearly with the network size. The sample complexities are optimal in both $ε$ and $K$. We test our algorithm on the examples of hyperparameter tuning and decentralized reinforcement learning. Simulated experiments confirmed that our algorithm achieves the state-of-the-art training efficiency and test accuracy.

stat.ML↗

arXiv2022

Optimal Estimation of Off-Policy Policy Gradient via Double Fitted Iteration

Policy gradient (PG) estimation becomes a challenge when we are not allowed to sample with the target policy but only have access to a dataset generated by some unknown behavior policy. Conventional methods for off-policy PG estimation often suffer from either significant bias or exponentially large variance. In this paper, we propose the double Fitted PG estimation (FPG) algorithm. FPG can work with an arbitrary policy parameterization, assuming access to a Bellman-complete value function class. In the case of linear value function approximation, we provide a tight finite-sample upper bound on policy gradient estimation error, that is governed by the amount of distribution mismatch measured in feature space. We also establish the asymptotic normality of FPG estimation error with a precise covariance characterization, which is further shown to be statistically optimal with a matching Cramer-Rao lower bound. Empirically, we evaluate the performance of FPG on both policy gradient estimation and policy optimization, using either softmax tabular or ReLU policy networks. Under various metrics, our results show that FPG significantly outperforms existing off-policy PG estimation methods based on importance sampling and variance reduction techniques.

stat.ML↗

arXiv2022

Offline Stochastic Shortest Path: Learning, Evaluation and Towards Optimality

Goal-oriented Reinforcement Learning, where the agent needs to reach the goal state while simultaneously minimizing the cost, has received significant attention in real-world applications. Its theoretical formulation, stochastic shortest path (SSP), has been intensively researched in the online setting. Nevertheless, it remains understudied when such an online interaction is prohibited and only historical data is provided. In this paper, we consider the offline stochastic shortest path problem when the state space and the action space are finite. We design the simple value iteration-based algorithms for tackling both offline policy evaluation (OPE) and offline policy learning tasks. Notably, our analysis of these simple algorithms yields strong instance-dependent bounds which can imply worst-case bounds that are near-minimax optimal. We hope our study could help illuminate the fundamental statistical limits of the offline SSP problem and motivate further studies beyond the scope of current consideration.

cs.LG↗

arXiv2022

Bandit Theory and Thompson Sampling-Guided Directed Evolution for Sequence Optimization

Directed Evolution (DE), a landmark wet-lab method originated in 1960s, enables discovery of novel protein designs via evolving a population of candidate sequences. Recent advances in biotechnology has made it possible to collect high-throughput data, allowing the use of machine learning to map out a protein's sequence-to-function relation. There is a growing interest in machine learning-assisted DE for accelerating protein optimization. Yet the theoretical understanding of DE, as well as the use of machine learning in DE, remains limited. In this paper, we connect DE with the bandit learning theory and make a first attempt to study regret minimization in DE. We propose a Thompson Sampling-guided Directed Evolution (TS-DE) framework for sequence optimization, where the sequence-to-function mapping is unknown and querying a single value is subject to costly and noisy measurements. TS-DE updates a posterior of the function based on collected measurements. It uses a posterior-sampled function estimate to guide the crossover recombination and mutation steps in DE. In the case of a linear model, we show that TS-DE enjoys a Bayesian regret of order $\tilde O(d^{2}\sqrt{MT})$, where $d$ is feature dimension, $M$ is population size and $T$ is number of rounds. This regret bound is nearly optimal, confirming that bandit learning can provably accelerate DE. It may have implications for more general sequence optimization and evolutionary algorithms.

cs.LG↗

arXiv2022

Byzantine-Robust Online and Offline Distributed Reinforcement Learning

We consider a distributed reinforcement learning setting where multiple agents separately explore the environment and communicate their experiences through a central server. However, $α$-fraction of agents are adversarial and can report arbitrary fake information. Critically, these adversarial agents can collude and their fake data can be of any sizes. We desire to robustly identify a near-optimal policy for the underlying Markov decision process in the presence of these adversarial agents. Our main technical contribution is Weighted-Clique, a novel algorithm for the robust mean estimation from batches problem, that can handle arbitrary batch sizes. Building upon this new estimator, in the offline setting, we design a Byzantine-robust distributed pessimistic value iteration algorithm; in the online setting, we design a Byzantine-robust distributed optimistic value iteration algorithm. Both algorithms obtain near-optimal sample complexities and achieve superior robustness guarantee than prior works.

cs.LG↗