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Approximation power of random neural networks

arXiv.org Machine Learning

This paper investigates the approximation power of three types of random neural networks: (a) infinite width networks, with weights following an arbitrary distribution; (b) finite width networks obtained by subsampling the preceding infinite width networks; (c) finite width networks obtained by starting with standard Gaussian initialization, and then adding a vanishingly small correction to the weights. The primary result is a fully quantified bound on the rate of approximation of general general continuous functions: in all three cases, a function $f$ can be approximated with complexity $\|f\|_1 (d/\delta)^{\mathcal{O}(d)}$, where $\delta$ depends on continuity properties of $f$ and the complexity measure depends on the weight magnitudes and/or cardinalities. Along the way, a variety of ancillary results are developed: an exact construction of Gaussian densities with infinite width networks, an elementary stand-alone proof scheme for approximation via convolutions of radial basis functions, subsampling rates for infinite width networks, and depth separation for corrected networks.


Offline Neural Contextual Bandits: Pessimism, Optimization and Generalization

arXiv.org Artificial Intelligence

Offline policy learning (OPL) leverages existing data collected a priori for policy optimization without any active exploration. Despite the prevalence and recent interest in this problem, its theoretical and algorithmic foundations in function approximation settings remain under-developed. In this paper, we consider this problem on the axes of distributional shift, optimization, and generalization in offline contextual bandits with neural networks. In particular, we propose a provably efficient offline contextual bandit with neural network function approximation that does not require any functional assumption on the reward. We show that our method provably generalizes over unseen contexts under a milder condition for distributional shift than the existing OPL works. Notably, unlike any other OPL method, our method learns from the offline data in an online manner using stochastic gradient descent, allowing us to leverage the benefits of online learning into an offline setting. Moreover, we show that our method is more computationally efficient and has a better dependence on the effective dimension of the neural network than an online counterpart. Finally, we demonstrate the empirical effectiveness of our method in a range of synthetic and real-world OPL problems.


Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes

arXiv.org Machine Learning

We study reinforcement learning (RL) with linear function approximation where the underlying transition probability kernel of the Markov decision process (MDP) is a linear mixture model (Jia et al., 2020; Ayoub et al., 2020; Zhou et al., 2020) and the learning agent has access to either an integration or a sampling oracle of the individual basis kernels. We propose a new Bernstein-type concentration inequality for self-normalized martingales for linear bandit problems with bounded noise. Based on the new inequality, we propose a new, computationally efficient algorithm with linear function approximation named $\text{UCRL-VTR}^{+}$ for the aforementioned linear mixture MDPs in the episodic undiscounted setting. We show that $\text{UCRL-VTR}^{+}$ attains an $\tilde O(dH\sqrt{T})$ regret where $d$ is the dimension of feature mapping, $H$ is the length of the episode and $T$ is the number of interactions with the MDP. We also prove a matching lower bound $\Omega(dH\sqrt{T})$ for this setting, which shows that $\text{UCRL-VTR}^{+}$ is minimax optimal up to logarithmic factors. In addition, we propose the $\text{UCLK}^{+}$ algorithm for the same family of MDPs under discounting and show that it attains an $\tilde O(d\sqrt{T}/(1-\gamma)^{1.5})$ regret, where $\gamma\in [0,1)$ is the discount factor. Our upper bound matches the lower bound $\Omega(d\sqrt{T}/(1-\gamma)^{1.5})$ proved in Zhou et al. (2020) up to logarithmic factors, suggesting that $\text{UCLK}^{+}$ is nearly minimax optimal. To the best of our knowledge, these are the first computationally efficient, nearly minimax optimal algorithms for RL with linear function approximation.


Approximation of BV functions by neural networks: A regularity theory approach

arXiv.org Machine Learning

In this paper we are concerned with the approximation of functions by single hidden layer neural networks with ReLU activation functions on the unit circle. In particular, we are interested in the case when the number of data-points exceeds the number of nodes. We first study the convergence to equilibrium of the stochastic gradient flow associated with the cost function with a quadratic penalization. Specifically, we prove a Poincar\'e inequality for a penalized version of the cost function with explicit constants that are independent of the data and of the number of nodes. As our penalization biases the weights to be bounded, this leads us to study how well a network with bounded weights can approximate a given function of bounded variation (BV). Our main contribution concerning approximation of BV functions, is a result which we call the localization theorem. Specifically, it states that the expected error of the constrained problem, where the length of the weights are less than $R$, is of order $R^{-1/9}$ with respect to the unconstrained problem (the global optimum). The proof is novel in this topic and is inspired by techniques from regularity theory of elliptic partial differential equations. Finally we quantify the expected value of the global optimum by proving a quantitative version of the universal approximation theorem.


Randomized Policy Learning for Continuous State and Action MDPs

arXiv.org Artificial Intelligence

Recently, for continuous control tasks, reinforcement learning (RL) algorithms based on actor-critic architecture [9] or policy optimization [16] have shown remarkably good performance. The policy and the value function are represented by deep neural networks and then the weights are updated accordingly. However, [7] shows that the performance of these RL algorithms vary a lot with changes in hyperparameters, network architecture etc. Furthermore, [10] showed that a simple linear policy-based method with weights updated by a random search method can outperform some of these state-of-the-art results. A key question is how far we can go by relying almost exclusively on these architectural biases. For Markov decision processes (MDPs) with discrete state and action spaces, model-based algorithms based on dynamic programming (DP) ideas [13] can be used when the model is known. Unfortunately, in many problems (e.g., robotics), the system model is unknown, or simply too complicated to be succinctly stated and used in DP algorithms. Usually, latter is the more likely case.