arXiv subjects

Sridhar Mahadevan

Publications and source records attributed to Sridhar Mahadevan.

52 records · Page 3Linked to original sources

arXiv2020

Finite-Sample Analysis of Proximal Gradient TD Algorithms

In this paper, we analyze the convergence rate of the gradient temporal difference learning (GTD) family of algorithms. Previous analyses of this class of algorithms use ODE techniques to prove asymptotic convergence, and to the best of our knowledge, no finite-sample analysis has been done. Moreover, there has been not much work on finite-sample analysis for convergent off-policy reinforcement learning algorithms. In this paper, we formulate GTD methods as stochastic gradient algorithms w.r.t.~a primal-dual saddle-point objective function, and then conduct a saddle-point error analysis to obtain finite-sample bounds on their performance. Two revised algorithms are also proposed, namely projected GTD2 and GTD2-MP, which offer improved convergence guarantees and acceleration, respectively. The results of our theoretical analysis show that the GTD family of algorithms are indeed comparable to the existing LSTD methods in off-policy learning scenarios.

cs.LG↗

arXiv2020

Proximal Gradient Temporal Difference Learning: Stable Reinforcement Learning with Polynomial Sample Complexity

In this paper, we introduce proximal gradient temporal difference learning, which provides a principled way of designing and analyzing true stochastic gradient temporal difference learning algorithms. We show how gradient TD (GTD) reinforcement learning methods can be formally derived, not by starting from their original objective functions, as previously attempted, but rather from a primal-dual saddle-point objective function. We also conduct a saddle-point error analysis to obtain finite-sample bounds on their performance. Previous analyses of this class of algorithms use stochastic approximation techniques to prove asymptotic convergence, and do not provide any finite-sample analysis. We also propose an accelerated algorithm, called GTD2-MP, that uses proximal ``mirror maps'' to yield an improved convergence rate. The results of our theoretical analysis imply that the GTD family of algorithms are comparable and may indeed be preferred over existing least squares TD methods for off-policy learning, due to their linear complexity. We provide experimental results showing the improved performance of our accelerated gradient TD methods.

cs.LG↗

arXiv2020

Regularized Off-Policy TD-Learning

We present a novel $l_1$ regularized off-policy convergent TD-learning method (termed RO-TD), which is able to learn sparse representations of value functions with low computational complexity. The algorithmic framework underlying RO-TD integrates two key ideas: off-policy convergent gradient TD methods, such as TDC, and a convex-concave saddle-point formulation of non-smooth convex optimization, which enables first-order solvers and feature selection using online convex regularization. A detailed theoretical and experimental analysis of RO-TD is presented. A variety of experiments are presented to illustrate the off-policy convergence, sparse feature selection capability and low computational cost of the RO-TD algorithm.

cs.LG↗

arXiv2019

Global Convergence to the Equilibrium of GANs using Variational Inequalities

In optimization, the negative gradient of a function denotes the direction of steepest descent. Furthermore, traveling in any direction orthogonal to the gradient maintains the value of the function. In this work, we show that these orthogonal directions that are ignored by gradient descent can be critical in equilibrium problems. Equilibrium problems have drawn heightened attention in machine learning due to the emergence of the Generative Adversarial Network (GAN). We use the framework of Variational Inequalities to analyze popular training algorithms for a fundamental GAN variant: the Wasserstein Linear-Quadratic GAN. We show that the steepest descent direction causes divergence from the equilibrium, and convergence to the equilibrium is achieved through following a particular orthogonal direction. We call this successful technique Crossing-the-Curl, named for its mathematical derivation as well as its intuition: identify the game's axis of rotation and move "across" space in the direction towards smaller "curling".

cs.LG↗

arXiv2018

A Unified Framework for Domain Adaptation using Metric Learning on Manifolds

We present a novel framework for domain adaptation, whereby both geometric and statistical differences between a labeled source domain and unlabeled target domain can be integrated by exploiting the curved Riemannian geometry of statistical manifolds. Our approach is based on formulating transfer from source to target as a problem of geometric mean metric learning on manifolds. Specifically, we exploit the curved Riemannian manifold geometry of symmetric positive definite (SPD) covariance matrices. We exploit a simple but important observation that as the space of covariance matrices is both a Riemannian space as well as a homogeneous space, the shortest path geodesic between two covariances on the manifold can be computed analytically. Statistics on the SPD matrix manifold, such as the geometric mean of two matrices can be reduced to solving the well-known Riccati equation. We show how the Ricatti-based solution can be constrained to not only reduce the statistical differences between the source and target domains, such as aligning second order covariances and minimizing the maximum mean discrepancy, but also the underlying geometry of the source and target domains using diffusions on the underlying source and target manifolds. A key strength of our proposed approach is that it enables integrating multiple sources of variation between source and target in a unified way, by reducing the combined objective function to a nested set of Ricatti equations where the solution can be represented by a cascaded series of geometric mean computations. In addition to showing the theoretical optimality of our solution, we present detailed experiments using standard transfer learning testbeds from computer vision comparing our proposed algorithms to past work in domain adaptation, showing improved results over a large variety of previous methods.

cs.LG↗

arXiv2017

Online Monotone Games

Algorithmic game theory (AGT) focuses on the design and analysis of algorithms for interacting agents, with interactions rigorously formalized within the framework of games. Results from AGT find applications in domains such as online bidding auctions for web advertisements and network routing protocols. Monotone games are games where agent strategies naturally converge to an equilibrium state. Previous results in AGT have been obtained for convex, socially-convex, or smooth games, but not monotone games. Our primary theoretical contributions are defining the monotone game setting and its extension to the online setting, a new notion of regret for this setting, and accompanying algorithms that achieve sub-linear regret. We demonstrate the utility of online monotone game theory on a variety of problem domains including variational inequalities, reinforcement learning, and generative adversarial networks.

cs.GT↗

arXiv2017

Inverting Variational Autoencoders for Improved Generative Accuracy

Recent advances in semi-supervised learning with deep generative models have shown promise in generalizing from small labeled datasets ($\mathbf{x},\mathbf{y}$) to large unlabeled ones ($\mathbf{x}$). In the case where the codomain has known structure, a large unfeatured dataset ($\mathbf{y}$) is potentially available. We develop a parameter-efficient, deep semi-supervised generative model for the purpose of exploiting this untapped data source. Empirical results show improved performance in disentangling latent variable semantics as well as improved discriminative prediction on Martian spectroscopic and handwritten digit domains.

cs.LG↗

arXiv2017

A Manifold Approach to Learning Mutually Orthogonal Subspaces

Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging. One solution is to use Riemannian optimization methods that enforce such constraints implicitly, leveraging the fact that the feasible parameter values form a manifold. While Riemannian methods exist for some specific problems, such as learning a single subspace, there are more general subspace constraints that offer additional flexibility when setting up an optimization problem, but have not been formulated as a manifold. We propose the partitioned subspace (PS) manifold for optimizing matrices that are constrained to represent one or more subspaces. Each point on the manifold defines a partitioning of the input space into mutually orthogonal subspaces, where the number of partitions and their sizes are defined by the user. As a result, distinct groups of features can be learned by defining different objective functions for each partition. We illustrate the properties of the manifold through experiments on multiple dataset analysis and domain adaptation.

cs.LG↗

arXiv2017

Generative Multi-Adversarial Networks

Generative adversarial networks (GANs) are a framework for producing a generative model by way of a two-player minimax game. In this paper, we propose the \emph{Generative Multi-Adversarial Network} (GMAN), a framework that extends GANs to multiple discriminators. In previous work, the successful training of GANs requires modifying the minimax objective to accelerate training early on. In contrast, GMAN can be reliably trained with the original, untampered objective. We explore a number of design perspectives with the discriminator role ranging from formidable adversary to forgiving teacher. Image generation tasks comparing the proposed framework to standard GANs demonstrate GMAN produces higher quality samples in a fraction of the iterations when measured by a pairwise GAM-type metric.

cs.LG↗

arXiv2016

Online Monotone Optimization

This paper presents a new framework for analyzing and designing no-regret algorithms for dynamic (possibly adversarial) systems. The proposed framework generalizes the popular online convex optimization framework and extends it to its natural limit allowing it to capture a notion of regret that is intuitive for more general problems such as those encountered in game theory and variational inequalities. The framework hinges on a special choice of a system-wide loss function we have developed. Using this framework, we prove that a simple update scheme provides a no-regret algorithm for monotone systems. While previous results in game theory prove individual agents can enjoy unilateral no-regret guarantees, our result proves monotonicity sufficient for guaranteeing no-regret when considering the adjustments of multiple agent strategies in parallel. Furthermore, to our knowledge, this is the first framework to provide a suitable notion of regret for variational inequalities. Most importantly, our proposed framework ensures monotonicity a sufficient condition for employing multiple online learners safely in parallel.

cs.LG↗

arXiv2016

Deep Reinforcement Learning With Macro-Actions

Deep reinforcement learning has been shown to be a powerful framework for learning policies from complex high-dimensional sensory inputs to actions in complex tasks, such as the Atari domain. In this paper, we explore output representation modeling in the form of temporal abstraction to improve convergence and reliability of deep reinforcement learning approaches. We concentrate on macro-actions, and evaluate these on different Atari 2600 games, where we show that they yield significant improvements in learning speed. Additionally, we show that they can even achieve better scores than DQN. We offer analysis and explanation for both convergence and final results, revealing a problem deep RL approaches have with sparse reward signals.

cs.LG↗

arXiv2015

Reasoning about Linguistic Regularities in Word Embeddings using Matrix Manifolds

Recent work has explored methods for learning continuous vector space word representations reflecting the underlying semantics of words. Simple vector space arithmetic using cosine distances has been shown to capture certain types of analogies, such as reasoning about plurals from singulars, past tense from present tense, etc. In this paper, we introduce a new approach to capture analogies in continuous word representations, based on modeling not just individual word vectors, but rather the subspaces spanned by groups of words. We exploit the property that the set of subspaces in n-dimensional Euclidean space form a curved manifold space called the Grassmannian, a quotient subgroup of the Lie group of rotations in n- dimensions. Based on this mathematical model, we develop a modified cosine distance model based on geodesic kernels that captures relation-specific distances across word categories. Our experiments on analogy tasks show that our approach performs significantly better than the previous approaches for the given task.

cs.CL↗

arXiv2014

Proximal Reinforcement Learning: A New Theory of Sequential Decision Making in Primal-Dual Spaces

In this paper, we set forth a new vision of reinforcement learning developed by us over the past few years, one that yields mathematically rigorous solutions to longstanding important questions that have remained unresolved: (i) how to design reliable, convergent, and robust reinforcement learning algorithms (ii) how to guarantee that reinforcement learning satisfies pre-specified "safety" guarantees, and remains in a stable region of the parameter space (iii) how to design "off-policy" temporal difference learning algorithms in a reliable and stable manner, and finally (iv) how to integrate the study of reinforcement learning into the rich theory of stochastic optimization. In this paper, we provide detailed answers to all these questions using the powerful framework of proximal operators. The key idea that emerges is the use of primal dual spaces connected through the use of a Legendre transform. This allows temporal difference updates to occur in dual spaces, allowing a variety of important technical advantages. The Legendre transform elegantly generalizes past algorithms for solving reinforcement learning problems, such as natural gradient methods, which we show relate closely to the previously unconnected framework of mirror descent methods. Equally importantly, proximal operator theory enables the systematic development of operator splitting methods that show how to safely and reliably decompose complex products of gradients that occur in recent variants of gradient-based temporal difference learning. This key technical innovation makes it possible to finally design "true" stochastic gradient methods for reinforcement learning. Finally, Legendre transforms enable a variety of other benefits, including modeling sparsity and domain geometry. Our work builds extensively on recent work on the convergence of saddle-point algorithms, and on the theory of monotone operators.

cs.LG↗

arXiv2013

Decision-Theoretic Planning with Concurrent Temporally Extended Actions

We investigate a model for planning under uncertainty with temporallyextended actions, where multiple actions can be taken concurrently at each decision epoch. Our model is based on the options framework, and combines it with factored state space models,where the set of options can be partitioned into classes that affectdisjoint state variables. We show that the set of decisionepochs for concurrent options defines a semi-Markov decisionprocess, if the underlying temporally extended actions being parallelized arerestricted to Markov options. This property allows us to use SMDPalgorithms for computing the value function over concurrentoptions. The concurrent options model allows overlapping execution ofoptions in order to achieve higher performance or in order to performa complex task. We describe a simple experiment using a navigationtask which illustrates how concurrent options results in a faster planwhen compared to the case when only one option is taken at a time.

cs.AI↗

arXiv2012

Sparse Q-learning with Mirror Descent

This paper explores a new framework for reinforcement learning based on online convex optimization, in particular mirror descent and related algorithms. Mirror descent can be viewed as an enhanced gradient method, particularly suited to minimization of convex functions in highdimensional spaces. Unlike traditional gradient methods, mirror descent undertakes gradient updates of weights in both the dual space and primal space, which are linked together using a Legendre transform. Mirror descent can be viewed as a proximal algorithm where the distance generating function used is a Bregman divergence. A new class of proximal-gradient based temporal-difference (TD) methods are presented based on different Bregman divergences, which are more powerful than regular TD learning. Examples of Bregman divergences that are studied include p-norm functions, and Mahalanobis distance based on the covariance of sample gradients. A new family of sparse mirror-descent reinforcement learning methods are proposed, which are able to find sparse fixed points of an l1-regularized Bellman equation at significantly less computational cost than previous methods based on second-order matrix methods. An experimental study of mirror-descent reinforcement learning is presented using discrete and continuous Markov decision processes.

cs.LG↗

arXiv2012

Representation Policy Iteration

This paper addresses a fundamental issue central to approximation methods for solving large Markov decision processes (MDPs): how to automatically learn the underlying representation for value function approximation? A novel theoretically rigorous framework is proposed that automatically generates geometrically customized orthonormal sets of basis functions, which can be used with any approximate MDP solver like least squares policy iteration (LSPI). The key innovation is a coordinate-free representation of value functions, using the theory of smooth functions on a Riemannian manifold. Hodge theory yields a constructive method for generating basis functions for approximating value functions based on the eigenfunctions of the self-adjoint (Laplace-Beltrami) operator on manifolds. In effect, this approach performs a global Fourier analysis on the state space graph to approximate value functions, where the basis functions reflect the largescale topology of the underlying state space. A new class of algorithms called Representation Policy Iteration (RPI) are presented that automatically learn both basis functions and approximately optimal policies. Illustrative experiments compare the performance of RPI with that of LSPI using two handcoded basis functions (RBF and polynomial state encodings).

cs.AI↗