Computer Science

In this talk, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution p over a poset is monotone if, for any pair of domain elements x and y such that x \preceq y, p(x) \leq p(y). I am going to present the proof sketch of achieving a lower bound for this problem over a few posets, e.g., matchings, and hypercubes. The main idea of these lower bounds is the following: we introduce a new property called *bigness* over a finite domain, where the distribution is T-big if the minimum probability for any domain element is at least T. Relying on the framework of [Wu-Yang'15], we establish a lower bound of \Omega(n/\log n) for testing bigness of distributions on domains of size n. We then build on these lower bounds to give \Omega(n/\log{n}) lower bounds for testing monotonicity over a matching poset of size n and significantly improved lower bounds over the hypercube poset. Although finding a tight upper bound for testing monotonicity remains open, I will also discuss the steps we took towards the upper bound as well. Joint work with: Themis Gouleakis, John Peebles, Ronitt Rubinfeld, and Anak Yodpinyanee.

Modern supervised machine learning algorithms are at their best when provided with large datasets and large, high-capacity models. This kind of data-driven paradigm has driven remarkable progress in fields ranging from computer vision to natural language processing and speech recognition. However, reinforcement learning algorithms have proven difficult to scale to such large data regimes without the use of simulation. Online reinforcement learning algorithms require recollecting data in each experiment -- when the dataset is on the scale of ImageNet and MS-COCO, this becomes infeasible to do in the real world. Offline reinforcement learning algorithms have so far been difficult to integrate with deep neural networks. In this talk, I will discuss some recent advances in offline reinforcement learning that I think represent a step toward bridging this gap, and in addition carry a number of appealing theoretical properties. I will discuss the theoretical reasons why offline reinforcement learning is challenging, discuss the solutions that have been proposed in the literature, and describe our recent advances in developing conservative Q-learning methods that provide theoretical guarantees in the face of distributional shift, providing not only a practical way of deploying offline deep RL, but also a degree of confidence that the resulting solution will avoid common pitfalls of overestimation. I will also describe how the ideas in offline RL can be applied more broadly to data-driven optimization problems that do not necessarily involve sequential decision making, such as designing protein sequences or robot morphologies from prior experimental data. This emerging field shares many of the theoretical foundations with offline RL, but represents the possibility to broaden the impact of data-driven decision making to a wider range of applications.

Model-free reinforcement learning attempts to find an optimal control action for an unknown dynamical system by directly searching over the parameter space of controllers. The convergence behavior and statistical properties of these approaches are often poorly understood because of the nonconvex nature of the underlying optimization problems and the lack of exact gradient computation. In this talk, we discuss performance and efficiency of such methods by focusing on the standard infinite-horizon linear quadratic regulator  problem for continuous-time systems with unknown state-space parameters. We establish exponential stability for the ordinary differential equation (ODE) that governs the gradient-flow dynamics over the set of stabilizing feedback gains and show that a similar result holds for the gradient descent method that arises from the forward Euler discretization of the corresponding ODE. We also provide theoretical bounds on the convergence rate and sample complexity of the random search method with two-point gradient estimates. We prove that the required simulation time for achieving $\epsilon$-accuracy in the model-free setup and the total number of function evaluations both scale as $\log (1/\epsilon)$.

Most literature on policy evaluation, bandit methods, etc., is focused on settings where actions taken on one unit do not affect other units. Such lack of interference, however, fails to hold in many applications of interest. For example, in a vaccine study, one person getting vaccinated also protects others; in a microcredit study, loans given to one person may stimulate the economy and indirectly benefit others; or, in a jobs-training study, training more people to perform a given task may create over-supply of qualified workers, thus reducing the market value of the training. In this talk, I'll survey various approaches to modeling cross-unit interference, and discuss associated methods for policy evaluation.

We consider the problem of reinforcement learning (RL) with unbounded state space, motivated by the classical problem of scheduling in a queueing network. Traditional policies as well as error metric that are designed for finite, bounded or compact state space, require infinite samples for providing any meaningful performance guarantee (e.g. ℓ_∞ error) for unbounded state space. We need a new notion of performance metric. Inspired by the literature in queuing systems and control theory, we propose stability as the notion of “goodness”: the state dynamics under the policy should remain in a bounded region with high probability. As a proof of concept, we propose an RL policy using Sparse-Sampling-based Monte Carlo Oracle and argue that it satisfies the stability property as long as the system dynamics under the optimal policy respects a Lyapunov function. The assumption of existence of a Lyapunov function is not restrictive as it is equivalent to the positive recurrence or stability property of any Markov chain. And, our policy does not utilize the knowledge of the specific Lyapunov function. To make our method sample efficient, we provide an improved, sample efficient Monte Carlo Oracle with Lipschitz value. We also design an adaptive version of the algorithm, based on carefully constructed statistical tests, which finds the correct tuning parameter automatically. This work is joint with Devavrat Shah and Zhi Xu.

We consider the question of learning Q-function in a sample efficient manner for reinforcement learning with continuous state and action spaces under a generative model. If Q-function is Lipschitz continuous, then the minimal sample complexity for estimating œµ-optimal Q-function is known to scale as O(eps^{-(d1+d2+2)}) per classical non-parametric learning theory, where d1 and d2 denote the dimensions of the state and action spaces respectively. The Q-function, when viewed as a kernel, induces a Hilbert-Schmidt operator and hence possesses square-summable spectrum. This motivates us to consider a parametric class of Q-functions parameterized by its "rank" r, which contains all Lipschitz Q-functions as r‚Üí‚àû. As our key contribution, we develop a simple, iterative learning algorithm that finds œµ-optimal Q-function with sample complexity of O(eps^{-max(d1,d2)+2}) when the optimal Q-function has low rank r and the discounting factor is below a certain threshold. Thus, this provides an exponential improvement in sample complexity. To enable our result, we develop a novel Matrix Estimation algorithm that faithfully estimates an unknown low-rank matrix with respect to max-norm sense even in the presence of arbitrary bounded noise, which might be of interest in its own right. Empirical results on several stochastic control tasks confirm the efficacy of our "low-rank" algorithms. This is based on joint work with Dogyoon Song, Zhi Xu, Yuzhe Yang. The manuscript is available at https://arxiv.org/abs/2006.06135.

We study the reward-free reinforcement learning framework, which is particularly suitable for batch reinforcement learning and scenarios where one needs policies for multiple reward functions. This framework has two phases. In the exploration phase, the agent collects trajectories by interacting with the environment without using any reward signal. In the planning phase, the agent needs to return a near-optimal policy for arbitrary reward functions. We give a new efficient algorithm which interacts with the environment at most $O\left( \frac{S^2A}{\epsilon^2}\poly\log\left(\frac{SAH}{\epsilon}\right) \right)$ episodes in the exploration phase, and guarantees to output a near-optimal policy for arbitrary reward functions in the planning phase. Here, $S$ is the size of state space, $A$ is the size of action space, $H$ is the planning horizon, and $\epsilon$ is the target accuracy relative to the total reward. Our sample complexity scales only logarithmically with $H$, in contrast to all existing results which scale polynomially with $H$.  Furthermore, this bound matches the minimax lower bound $\Omega\left(\frac{S^2A}{\epsilon^2}\right)$ up to logarithmic factors.

Offline RL is crucial in applications where experimentation is limited, such as medicine, but it is also notoriously difficult because the similarity between the trajectories observed and those generated by any proposed policy diminishes exponentially as horizon grows, known as the curse of horizon. To better understand this limitation, we study the statistical efficiency limits of two central tasks in offline reinforcement learning: estimating the policy value and the policy gradient from off-policy data. The efficiency bounds reveal that the curse is generally insurmountable without assuming additional structure and as such plagues many standard estimators that work in general problems, but it may be overcome in Markovian settings and even further attenuated in stationary settings. We develop the first estimators achieving the efficiency limits in finite- and infinite-horizon MDPs using a meta-algorithm we term Double Reinforcement Learning (DRL). We provide favorable guarantees for DRL and for off-policy policy optimization via efficiently-estimated policy gradient ascent.

We discuss the solution of multistage decision problems using methods that are based on the idea of policy iteration (PI for short), i.e., start from some base policy and generate an improved policy. Rollout is the simplest method of this type, where just one improved policy is generated. We can view PI as repeated application of rollout, where the rollout policy at each iteration serves as the base policy for the next iteration. In contrast with PI, rollout can be applied on-line and is suitable for on-line replanning. Moreover, rollout can use as base policy one of the policies produced by PI, thereby improving on that policy. This is the type of scheme underlying the prominently successful AlphaZero chess program. In this paper we focus on rollout and PI-like methods for multiagent problems, where the control consists of multiple components each selected (conceptually) by a separate agent. We discuss an approach, whereby at every stage, the agents sequentially (one-at-a-time) execute a local rollout algorithm that uses a base policy, together with some coordinating information from the other agents. The amount of total computation required at every stage grows linearly with the number of agents. By contrast, in the standard rollout algorithm, the amount of total computation grows exponentially with the number of agents. Despite the dramatic reduction in required computation, we show that our multiagent rollout algorithm has the fundamental cost improvement property of standard rollout: it guarantees an improved performance relative to the base policy.  We first develop our agent-by-agent policy improvement approach  for finite horizon problems, and then we extend it to exact and approximate PI for discounted and other infinite horizon problems. We prove that the cost improvement property steers the algorithm towards convergence to an agent-by-agent optimal policy, thus establishing a connection with the theory of teams. We also discuss autonomous multiagent rollout schemes that allow the agents to make decisions autonomously through the use of precomputed signaling information, which  is sufficient to maintain the cost improvement property, without any on-line coordination of control selection between the agents.

We consider the batch (off-line) policy learning problem in the infinite horizon Markov Decision Process. Motivated by mobile health applications, we focus on learning a policy that  maximizes the long-term average reward. We propose a doubly robust estimator for the average reward for a given policy and show that it achieves semi-parametric efficiency. The proposed estimator requires estimation of two policy dependent nuisance functions.  We develop an optimization algorithm to compute the optimal policy in a parameterized stochastic policy class. The performance of the estimated policy is measured by the regret, i.e., the difference between the optimal average reward in the policy class and the average reward of the estimated policy and we establish a finite-sample regret guarantee for our proposed method.

We establish a theoretical comparison between the asymptotic mean-squared error of Double Q-learning and Q-learning. Using prior work on the asymptotic mean-squared error of linear stochastic approximation based on Lyapunov equations, we show that the asymptotic mean-squared error of Double Q-learning is exactly equal to that of Q-learning if Double Q-learning uses twice the learning rate of Q-learning and outputs the average of its two estimators. We also present some practical implications of this theoretical observation using simulations.

Sample complexity bounds are a common performance metric in the RL literature. In the discounted cost, infinite horizon setting, all of the known bounds can be arbitrarily large, as the discount factor approaches unity. For a large discount factor, these bounds seem to imply that a very large number of samples is required to achieve an epsilon-optimal policy. In this talk, we will discuss a new class of algorithms that have sample complexity uniformly bounded for all discount factors.  One may argue that this is impossible, due to a recent min-max lower bound. The explanation is that this previous lower bound is for a specific problem, which we modify, without compromising the ultimate objective of obtaining an epsilon-optimal policy.  Specifically, we show that the asymptotic covariance of the Q-learning algorithm with an optimized step-size sequence is a quadratic function of a factor that goes to infinity, as discount factor gets close to 1; an expected, and essentially known result. The new relative Q-learning algorithm proposed here is shown to have asymptotic covariance that is uniformly bounded for all discount factors.