Several applications of Reinforcement Learning suffer from instability due to high variance. This is especially prevalent in high dimensional domains. Regularization is a commonly used technique in machine learning to reduce variance, at the cost of introducing some bias. Most existing regularization techniques focus on spatial (perceptual) regularization. Yet in reinforcement learning, due to the nature of the Bellman equation, there is an opportunity to also exploit temporal regularization based on smoothness in value estimates over trajectories. This paper explores a class of methods for temporal regularization. We formally characterize the bias induced by this technique using Markov chain concepts. We illustrate the various characteristics of temporal regularization via a sequence of simple discrete and continuous MDPs, and show that the technique provides improvement even in high-dimensional Atari games.
Regularization in RL has been considered via several different perspectives. One line of investigation focuses on regularizing the features learned on the state space . In particular backward bootstrapping method's can be seen as regularizing in feature space based on temporal proximity @cite_2 @cite_1 @cite_7 . These approaches assume that nearby states in the state space have similar value. Other works focus on regularizing the changes in policy directly. Those approaches are often based on entropy methods . Explicit regularization in the temporal space has received much less attention. Temporal regularization in some sense may be seen as a backward'' multi-step method . The closest work to ours is possibly , where they define natural value approximator by projecting the previous states estimates by adjusting for the reward and @math . Their formulation, while sharing similarity in motivation, leads to different theory and algorithm. Convergence properties and bias induced by this class of methods were also not analyzed in .
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We propose a general framework for entropy-regularized average-reward reinforcement learning in Markov decision processes (MDPs). Our approach is based on extending the linear-programming formulation of policy optimization in MDPs to accommodate convex regularization functions. Our key result is showing that using the conditional entropy of the joint state-action distributions as regularization yields a dual optimization problem closely resembling the Bellman optimality equations. This result enables us to formalize a number of state-of-the-art entropy-regularized reinforcement learning algorithms as approximate variants of Mirror Descent or Dual Averaging, and thus to argue about the convergence properties of these methods. In particular, we show that the exact version of the TRPO algorithm of (2015) actually converges to the optimal policy, while the entropy-regularized policy gradient methods of (2016) may fail to converge to a fixed point. Finally, we illustrate empirically the effects of using various regularization techniques on learning performance in a simple reinforcement learning setup. We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x,y)=f(x)+Q(x,y)+g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to “metrically regular” problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to @math and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with “regular” intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l1 algorithm for compressive sensing and an application to rank reduction problems. In this paper we consider approximate policy-iteration-based reinforcement learning algorithms. In order to implement a flexible function approximation scheme we propose the use of non-parametric methods with regularization, providing a convenient way to control the complexity of the function approximator. We propose two novel regularized policy iteration algorithms by adding L2-regularization to two widely-used policy evaluation methods: Bellman residual minimization (BRM) and least-squares temporal difference learning (LSTD). We derive efficient implementation for our algorithms when the approximate value-functions belong to a reproducing kernel Hilbert space. We also provide finite-sample performance bounds for our algorithms and show that they are able to achieve optimal rates of convergence under the studied conditions. We present a novel attribute learning framework named Hypergraph-based Attribute Predictor (HAP). In HAP, a hypergraph is leveraged to depict the attribute relations in the data. Then the attribute prediction problem is casted as a regularized hypergraph cut problem in which HAP jointly learns a collection of attribute projections from the feature space to a hypergraph embedding space aligned with the attribute space. The learned projections directly act as attribute classifiers (linear and kernelized). This formulation leads to a very efficient approach. By considering our model as a multi-graph cut task, our framework can flexibly incorporate other available information, in particular class label. We apply our approach to attribute prediction, Zero-shot and @math -shot learning tasks. The results on AWA, USAA and CUB databases demonstrate the value of our methods in comparison with the state-of-the-art approaches. We establish a new connection between value and policy based reinforcement learning (RL) based on a relationship between softmax temporal value consistency and policy optimality under entropy regularization. Specifically, we show that softmax consistent action values correspond to optimal entropy regularized policy probabilities along any action sequence, regardless of provenance. From this observation, we develop a new RL algorithm, Path Consistency Learning (PCL), that minimizes a notion of soft consistency error along multi-step action sequences extracted from both on- and off-policy traces. We examine the behavior of PCL in different scenarios and show that PCL can be interpreted as generalizing both actor-critic and Q-learning algorithms. We subsequently deepen the relationship by showing how a single model can be used to represent both a policy and the corresponding softmax state values, eliminating the need for a separate critic. The experimental evaluation demonstrates that PCL significantly outperforms strong actor-critic and Q-learning baselines across several benchmarks. We consider the task of reinforcement learning with linear value function approximation. Temporal difference algorithms, and in particular the Least-Squares Temporal Difference (LSTD) algorithm, provide a method for learning the parameters of the value function, but when the number of features is large this algorithm can over-fit to the data and is computationally expensive. In this paper, we propose a regularization framework for the LSTD algorithm that overcomes these difficulties. In particular, we focus on the case of l1 regularization, which is robust to irrelevant features and also serves as a method for feature selection. Although the l1 regularized LSTD solution cannot be expressed as a convex optimization problem, we present an algorithm similar to the Least Angle Regression (LARS) algorithm that can efficiently compute the optimal solution. Finally, we demonstrate the performance of the algorithm experimentally. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Two of the leading approaches for model-free reinforcement learning are policy gradient methods and @math -learning methods. @math -learning methods can be effective and sample-efficient when they work, however, it is not well-understood why they work, since empirically, the @math -values they estimate are very inaccurate. A partial explanation may be that @math -learning methods are secretly implementing policy gradient updates: we show that there is a precise equivalence between @math -learning and policy gradient methods in the setting of entropy-regularized reinforcement learning, that "soft" (entropy-regularized) @math -learning is exactly equivalent to a policy gradient method. We also point out a connection between @math -learning methods and natural policy gradient methods. Experimentally, we explore the entropy-regularized versions of @math -learning and policy gradients, and we find them to perform as well as (or slightly better than) the standard variants on the Atari benchmark. We also show that the equivalence holds in practical settings by constructing a @math -learning method that closely matches the learning dynamics of A3C without using a target network or @math -greedy exploration schedule. Sparse representation based classification (SRC) has received much attention in computer vision and pattern recognition. SRC codes a testing sample by sparse linear combination of all the training samples and classifies the testing sample into the class with the minimum representation error. Recently, Zhang analyzes the working mechanism of SRC and points out that it is the collaborative representation but not the L1-norm sparsity that makes SRC powerful. Based on the analysis, they propose a very simple and much more efficient classification scheme, called collaborative representation based classification with regularized least square (CRC_RLS). CRC_RLS is a linear method in nature. Here we propose a kernel collaborative representation based classification with regularized least square (Kernel CRC_RLS, KCRC_RLS) by implicitly mapping the sample into high-dimensional space via kernel tricks. Our approach is highly motivated by the kernel methods which can capture the nonlinear similarity among samples and have been successfully applied in pattern recognition and machine learning. The experimental results on the CENPAMI handwritten digital database, ETH80 database, FERET face database, ORL database, AR face database, demonstrate that Kernel CRC_RLS is effective in classification, leading to promising performance. Recurrent Neural Networks (RNNs) are rich models for the processing of sequential data. Recent work on advancing the state of the art has been focused on the optimization or modelling of RNNs, mostly motivated by adressing the problems of the vanishing and exploding gradients. The control of overfitting has seen considerably less attention. This paper contributes to that by analyzing fast dropout, a recent regularization method for generalized linear models and neural networks from a back-propagation inspired perspective. We show that fast dropout implements a quadratic form of an adaptive, per-parameter regularizer, which rewards large weights in the light of underfitting, penalizes them for overconfident predictions and vanishes at minima of an unregularized training loss. The derivatives of that regularizer are exclusively based on the training error signal. One consequence of this is the absense of a global weight attractor, which is particularly appealing for RNNs, since the dynamics are not biased towards a certain regime. We positively test the hypothesis that this improves the performance of RNNs on four musical data sets. Model-free reinforcement learning algorithms, such as Q-learning, perform poorly in the early stages of learning in noisy environments, because much effort is spent unlearning biased estimates of the state-action value function. The bias results from selecting, among several noisy estimates, the apparent optimum, which may actually be suboptimal. We propose G-learning, a new off-policy learning algorithm that regularizes the value estimates by penalizing deterministic policies in the beginning of the learning process. We show that this method reduces the bias of the value-function estimation, leading to faster convergence to the optimal value and the optimal policy. Moreover, G-learning enables the natural incorporation of prior domain knowledge, when available. The stochastic nature of G-learning also makes it avoid some exploration costs, a property usually attributed only to on-policy algorithms. We illustrate these ideas in several examples, where G-learning results in significant improvements of the convergence rate and the cost of the learning process. Low-rank recovery models have shown potential for salient object detection, where a matrix is decomposed into a low-rank matrix representing image background and a sparse matrix identifying salient objects. Two deficiencies, however, still exist. First, previous work typically assumes the elements in the sparse matrix are mutually independent, ignoring the spatial and pattern relations of image regions. Second, when the low-rank and sparse matrices are relatively coherent, e.g., when there are similarities between the salient objects and background or when the background is complicated, it is difficult for previous models to disentangle them. To address these problems, we propose a novel structured matrix decomposition model with two structural regularizations: (1) a tree-structured sparsity-inducing regularization that captures the image structure and enforces patches from the same object to have similar saliency values, and (2) a Laplacian regularization that enlarges the gaps between salient objects and the background in feature space. Furthermore, high-level priors are integrated to guide the matrix decomposition and boost the detection. We evaluate our model for salient object detection on five challenging datasets including single object, multiple objects and complex scene images, and show competitive results as compared with 24 state-of-the-art methods in terms of seven performance metrics. This paper presents a novel form of policy gradient for model-free reinforcement learning (RL) with improved exploration properties. Current policy-based methods use entropy regularization to encourage undirected exploration of the reward landscape, which is ineffective in high dimensional spaces with sparse rewards. We propose a more directed exploration strategy that promotes exploration of under-appreciated reward regions. An action sequence is considered under-appreciated if its log-probability under the current policy under-estimates its resulting reward. The proposed exploration strategy is easy to implement, requiring small modifications to an implementation of the REINFORCE algorithm. We evaluate the approach on a set of algorithmic tasks that have long challenged RL methods. Our approach reduces hyper-parameter sensitivity and demonstrates significant improvements over baseline methods. Our algorithm successfully solves a benchmark multi-digit addition task and generalizes to long sequences. This is, to our knowledge, the first time that a pure RL method has solved addition using only reward feedback. Regularized autoencoders learn the latent codes, a structure with the regularization under the distribution, which enables them the capability to infer the latent codes given observations and generate new samples given the codes. However, they are sometimes ambiguous as they tend to produce reconstructions that are not necessarily a faithful reproduction of the inputs. The main reason is to enforce the learned latent code distribution to match a prior distribution while the true distribution remains unknown. To improve the reconstruction quality and learn the latent space a manifold structure, this paper presents a novel approach using the adversarially approximated autoencoder (AAAE) to investigate the latent codes with adversarial approximation. Instead of regularizing the latent codes by penalizing on the distance between the distributions of the model and the target, AAAE learns the autoencoder flexibly and approximates the latent space with a simpler generator. The ratio is estimated using a generative adversarial network to enforce the similarity of the distributions. In addition, the image space is regularized with an additional adversarial regularizer. The proposed approach unifies two deep generative models for both latent space inference and diverse generation. The learning scheme is realized without regularization on the latent codes, which also encourages faithful reconstruction. Extensive validation experiments on four real-world datasets demonstrate the superior performance of AAAE. In comparison to the state-of-the-art approaches, AAAE generates samples with better quality and shares the properties of a regularized autoencoder with a nice latent manifold structure. Most existing zero-shot learning methods consider the problem as a visual semantic embedding one. Given the demonstrated capability of Generative Adversarial Networks(GANs) to generate images, we instead leverage GANs to imagine unseen categories from text descriptions and hence recognize novel classes with no examples being seen. Specifically, we propose a simple yet effective generative model that takes as input noisy text descriptions about an unseen class (e.g.Wikipedia articles) and generates synthesized visual features for this class. With added pseudo data, zero-shot learning is naturally converted to a traditional classification problem. Additionally, to preserve the inter-class discrimination of the generated features, a visual pivot regularization is proposed as an explicit supervision. Unlike previous methods using complex engineered regularizers, our approach can suppress the noise well without additional regularization. Empirically, we show that our method consistently outperforms the state of the art on the largest available benchmarks on Text-based Zero-shot Learning. Most existing zero-shot learning methods consider the problem as a visual semantic embedding one. Given the demonstrated capability of Generative Adversarial Networks(GANs) to generate images, we instead leverage GANs to imagine unseen categories from text descriptions and hence recognize novel classes with no examples being seen. Specifically, we propose a simple yet effective generative model that takes as input noisy text descriptions about an unseen class (e.g. Wikipedia articles) and generates synthesized visual features for this class. With added pseudo data, zero-shot learning is naturally converted to a traditional classification problem. Additionally, to preserve the inter-class discrimination of the generated features, a visual pivot regularization is proposed as an explicit supervision. Unlike previous methods using complex engineered regularizers, our approach can suppress the noise well without additional regularization. Empirically, we show that our method consistently outperforms the state of the art on the largest available benchmarks on Text-based Zero-shot Learning. Iterative methods that calculate their steps from approximate subgradient directions have proved to be useful for stochastic learning problems over large and streaming data sets. When the objective consists of a loss function plus a nonsmooth regularization term, the solution often lies on a low-dimensional manifold of parameter space along which the regularizer is smooth. (When an l1 regularizer is used to induce sparsity in the solution, for example, this manifold is defined by the set of nonzero components of the parameter vector.) This paper shows that a regularized dual averaging algorithm can identify this manifold, with high probability, before reaching the solution. This observation motivates an algorithmic strategy in which, once an iterate is suspected of lying on an optimal or near-optimal manifold, we switch to a "local phase" that searches in this manifold, thus converging rapidly to a near-optimal point. Computational results are presented to verify the identification property and to illustrate the effectiveness of this approach. Current state-of-the-art classification and detection algorithms rely on supervised training. In this work we study unsupervised feature learning in the context of temporally coherent video data. We focus on feature learning from unlabeled video data, using the assumption that adjacent video frames contain semantically similar information. This assumption is exploited to train a convolutional pooling auto-encoder regularized by slowness and sparsity. We establish a connection between slow feature learning to metric learning and show that the trained encoder can be used to define a more temporally and semantically coherent metric. We present an approach to long-range spatio-temporal regularization in semantic video segmentation. Temporal regularization in video is challenging because both the camera and the scene may be in motion. Thus Euclidean distance in the space-time volume is not a good proxy for correspondence. We optimize the mapping of pixels to a Euclidean feature space so as to minimize distances between corresponding points. Structured prediction is performed by a dense CRF that operates on the optimized features. Experimental results demonstrate that the presented approach increases the accuracy and temporal consistency of semantic video segmentation. Cross-modal retrieval has recently drawn much attention due to the widespread existence of multimodal data. It takes one type of data as the query to retrieve relevant data objects of another type, and generally involves two basic problems: the measure of relevance and coupled feature selection. Most previous methods just focus on solving the first problem. In this paper, we aim to deal with both problems in a novel joint learning framework. To address the first problem, we learn projection matrices to map multimodal data into a common subspace, in which the similarity between different modalities of data can be measured. In the learning procedure, the @math -norm penalties are imposed on the projection matrices separately to solve the second problem, which selects relevant and discriminative features from different feature spaces simultaneously. A multimodal graph regularization term is further imposed on the projected data,which preserves the inter-modality and intra-modality similarity relationships.An iterative algorithm is presented to solve the proposed joint learning problem, along with its convergence analysis. Experimental results on cross-modal retrieval tasks demonstrate that the proposed method outperforms the state-of-the-art subspace approaches.