Learning Adversarial MDPs with Bandit Feedback and Unknown Transition

We consider the problem of learning in episodic finite-horizon Markov decision processes with an unknown transition function, bandit feedback, and adversarial losses. We propose an efficient algorithm that achieves $\mathcal{\tilde{O}}(L|X|\sqrt{|A|T})$ regret with high probability, where $L$ is the horizon, $|X|$ is the number of states, $|A|$ is the number of actions, and $T$ is the number of episodes. To the best of our knowledge, our algorithm is the first to ensure $\mathcal{\tilde{O}}(\sqrt{T})$ regret in this challenging setting; in fact it achieves the same regret bound as (Rosenberg & Mansour, 2019a) that considers an easier setting with full-information feedback. Our key technical contributions are two-fold: a tighter confidence set for the transition function, and an optimistic loss estimator that is inversely weighted by an $\textit{upper occupancy bound}$.

Provably Efficient Exploration in Policy Optimization

While policy-based reinforcement learning (RL) achieves tremendous successes in practice, it is significantly less understood in theory, especially compared with value-based RL. In particular, it remains elusive how to design a provably efficient policy optimization algorithm that incorporates exploration. To bridge such a gap, this paper proposes an Optimistic variant of the Proximal Policy Optimization algorithm (OPPO), which follows an "optimistic version" of the policy gradient direction. This paper proves that, in the problem of episodic Markov decision process with linear function approximation, unknown transition, and adversarial reward with full-information feedback, OPPO achieves $\tilde{O}(\sqrt{d^3 H^3 T})$ regret. Here $d$ is the feature dimension, $H$ is the episode horizon, and $T$ is the total number of steps. To the best of our knowledge, OPPO is the first provably efficient policy optimization algorithm that explores.

Provably Efficient Reinforcement Learning with Linear Function Approximation

Modern Reinforcement Learning (RL) is commonly applied to practical problems with an enormous number of states, where function approximation must be deployed to approximate either the value function or the policy. The introduction of function approximation raises a fundamental set of challenges involving computational and statistical efficiency, especially given the need to manage the exploration/exploitation tradeoff. As a result, a core RL question remains open: how can we design provably efficient RL algorithms that incorporate function approximation? This question persists even in a basic setting with linear dynamics and linear rewards, for which only linear function approximation is needed. This paper presents the first provable RL algorithm with both polynomial runtime and polynomial sample complexity in this linear setting, without requiring a "simulator" or additional assumptions. Concretely, we prove that an optimistic modification of Least-Squares Value Iteration (LSVI)---a classical algorithm frequently studied in the linear setting---achieves $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret, where $d$ is the ambient dimension of feature space, $H$ is the length of each episode, and $T$ is the total number of steps. Importantly, such regret is independent of the number of states and actions.

On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems

We consider nonconvex-concave minimax problems, $\min_{x} \max_{y\in\mathcal{Y}} f(x, y)$, where $f$ is nonconvex in $x$ but concave in $y$. The standard algorithm for solving this problem is the celebrated gradient descent ascent (GDA) algorithm, which has been widely used in machine learning, control theory and economics. However, despite the solid theory for the convex-concave setting, GDA can converge to limit cycles or even diverge in a general setting. In this paper, we present a nonasymptotic analysis of GDA for solving nonconvex-concave minimax problems, showing that GDA can find a stationary point of the function $\Phi(\cdot) :=\max_{y\in\mathcal{Y} }f(\cdot, y)$ efficiently. To the best our knowledge, this is the first theoretical guarantee for GDA in this setting, shedding light on its practical performance in many real applications.

Stochastic Gradient Descent Escapes Saddle Points Efficiently

This paper considers the perturbed stochastic gradient descent algorithm and shows that it finds $\epsilon$-second order stationary points ($\left\|\nabla f(x)\right\|\leq \epsilon$ and $\nabla^2 f(x) \succeq -\sqrt{\epsilon} \mathbf{I}$) in $\tilde{O}(d/\epsilon^4)$ iterations, giving the first result that has linear dependence on dimension for this setting. For the special case, where stochastic gradients are Lipschitz, the dependence on dimension reduces to polylogarithmic. In addition to giving new results, this paper also presents a simplified proof strategy that gives a shorter and more elegant proof of previously known results (Jin et al. 2017) on perturbed gradient descent algorithm.

A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm

In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.

Minmax Optimization: Stable Limit Points of Gradient Descent Ascent are Locally Optimal

Minmax optimization, especially in its general nonconvex-nonconcave formulation, has found extensive applications in modern machine learning frameworks such as generative adversarial networks (GAN), adversarial training and multi-agent reinforcement learning. Gradient-based algorithms, in particular gradient descent ascent (GDA), are widely used in practice to solve these problems. Despite the practical popularity of GDA, however, its theoretical behavior has been considered highly undesirable. Indeed, apart from possiblity of non-convergence, recent results (Daskalakis and Panageas, 2018; Mazumdar and Ratliff, 2018; Adolphs et al., 2018) show that even when GDA converges, its stable limit points can be points that are not local Nash equilibria, thus not game-theoretically meaningful. In this paper, we initiate a discussion on the proper optimality measures for minmax optimization, and introduce a new notion of local optimality---local minmax---as a more suitable alternative to the notion of local Nash equilibrium. We establish favorable properties of local minmax points, and show, most importantly, that as the ratio of the ascent step size to the descent step size goes to infinity, stable limit points of GDA are exactly local minmax points up to degenerate points, demonstrating that all stable limit points of GDA have a game-theoretic meaning for minmax problems.

Sampling Can Be Faster Than Optimization

Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical understanding of the relationships between these two kinds of methodology, and limited understanding of relative strengths and weaknesses. Moreover, existing results have been obtained primarily in the setting of convex functions (for optimization) and log-concave functions (for sampling). In this setting, where local properties determine global properties, optimization algorithms are unsurprisingly more efficient computationally than sampling algorithms. We instead examine a class of nonconvex objective functions that arise in mixture modeling and multi-stable systems. In this nonconvex setting, we find that the computational complexity of sampling algorithms scales linearly with the model dimension while that of optimization algorithms scales exponentially.

On the Local Minima of the Empirical Risk

Population risk is always of primary interest in machine learning; however, learning algorithms only have access to the empirical risk. Even for applications with nonconvex nonsmooth losses (such as modern deep networks), the population risk is generally significantly more well-behaved from an optimization point of view than the empirical risk. In particular, sampling can create many spurious local minima. We consider a general framework which aims to optimize a smooth nonconvex function $F$ (population risk) given only access to an approximation $f$ (empirical risk) that is pointwise close to $F$ (i.e., $\|F-f\|_{\infty} \le \nu$). Our objective is to find the $\epsilon$-approximate local minima of the underlying function $F$ while avoiding the shallow local minima---arising because of the tolerance $\nu$---which exist only in $f$. We propose a simple algorithm based on stochastic gradient descent (SGD) on a smoothed version of $f$ that is guaranteed to achieve our goal as long as $\nu \le O(\epsilon^{1.5}/d)$. We also provide an almost matching lower bound showing that our algorithm achieves optimal error tolerance $\nu$ among all algorithms making a polynomial number of queries of $f$. As a concrete example, we show that our results can be directly used to give sample complexities for learning a ReLU unit.