In this paper, we present a link between preference-based and multiobjective sequential decision-making. While transforming a multiobjective problem to a preference-based one is quite natural, the other direction is a bit less obvious. We present how this transformation (from preference-based to multiobjective) can be done under the classic condition that preferences over histories can be represented by additively decomposable utilities and that the decision criterion to evaluate policies in a state is based on expectation. This link yields a new source of multiobjective sequential decision-making problems (i.e., when reward values are unknown) and justifies the use of solving methods developed in one setting in the other one.
A reinforcement learning problem is usually defined using the Markov Decision Process (MDP) model. A standard MDP @cite_3 is defined as a tuple @math where: @math is a finite set of states, @math is a finite set of actions, @math is a transition function with @math being the probability of reaching state @math after executing action @math in state @math , @math is a reward function with @math being the immediate numerical environmental feedback received by the agent after performing action @math in state @math . In this framework, a @math -step history @math is a sequence of state-action: where @math and @math . The value of such a history @math is defined as: where @math is a discount factor. A policy specifies how to choose an action in every state. A deterministic policy @math is a function from the set of states to the set of actions, while a randomized policy @math states the probability @math of choosing an action @math in a state @math .
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Consider the problem of approximating the optimal policy of a Markov decision process (MDP) by sampling state transitions. In contrast to existing reinforcement learning methods that are based on successive approximations to the nonlinear Bellman equation, we propose a Primal-Dual @math Learning method in light of the linear duality between the value and policy. The @math learning method is model-free and makes primal-dual updates to the policy and value vectors as new data are revealed. For infinite-horizon undiscounted Markov decision process with finite state space @math and finite action space @math , the @math learning method finds an @math -optimal policy using the following number of sample transitions @math where @math is an upper bound of mixing times across all policies and @math is a parameter characterizing the range of stationary distributions across policies. The @math learning method also applies to the computational problem of MDP where the transition probabilities and rewards are explicitly given as the input. In the case where each state transition can be sampled in @math time, the @math learning method gives a sublinear-time algorithm for solving the averaged-reward MDP. We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of @math where @math is the time horizon, @math the number of states, @math the number of actions and @math the number of time-steps. This result improves over the best previous known bound @math achieved by the UCRL2 algorithm of , 2010. The key significance of our new results is that when @math and @math , it leads to a regret of @math that matches the established lower bound of @math up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in @math ), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in @math ). Consider a Markov decision process (MDP) that admits a set of state-action features, which can linearly express the process's probabilistic transition model. We propose a parametric Q-learning algorithm that finds an approximate-optimal policy using a sample size proportional to the feature dimension @math and invariant with respect to the size of the state space. To further improve its sample efficiency, we exploit the monotonicity property and intrinsic noise structure of the Bellman operator, provided the existence of anchor state-actions that imply implicit non-negativity in the feature space. We augment the algorithm using techniques of variance reduction, monotonicity preservation, and confidence bounds. It is proved to find a policy which is @math -optimal from any initial state with high probability using @math sample transitions for arbitrarily large-scale MDP with a discount factor @math . A matching information-theoretical lower bound is proved, confirming the sample optimality of the proposed method with respect to all parameters (up to polylog factors). We consider the problem of learning an unknown Markov Decision Process (MDP) that is weakly communicating in the infinite horizon setting. We propose a Thompson Sampling-based reinforcement learning algorithm with dynamic episodes (TSDE). At the beginning of each episode, the algorithm generates a sample from the posterior distribution over the unknown model parameters. It then follows the optimal stationary policy for the sampled model for the rest of the episode. The duration of each episode is dynamically determined by two stopping criteria. The first stopping criterion controls the growth rate of episode length. The second stopping criterion happens when the number of visits to any state-action pair is doubled. We establish @math bounds on expected regret under a Bayesian setting, where @math and @math are the sizes of the state and action spaces, @math is time, and @math is the bound of the span. This regret bound matches the best available bound for weakly communicating MDPs. Numerical results show it to perform better than existing algorithms for infinite horizon MDPs. Reinforcement learning (RL) agents have traditionally been tasked with maximizing the value function of a Markov decision process (MDP), either in continuous settings, with fixed discount factor @math , or in episodic settings, with @math . While this has proven effective for specific tasks with well-defined objectives (e.g., games), it has never been established that fixed discounting is suitable for general purpose use (e.g., as a model of human preferences). This paper characterizes rationality in sequential decision making using a set of seven axioms and arrives at a form of discounting that generalizes traditional fixed discounting. In particular, our framework admits a state-action dependent "discount" factor that is not constrained to be less than 1, so long as there is eventual long run discounting. Although this broadens the range of possible preference structures in continuous settings, we show that there exists a unique "optimizing MDP" with fixed @math whose optimal value function matches the true utility of the optimal policy, and we quantify the difference between value and utility for suboptimal policies. Our work can be seen as providing a normative justification for (a slight generalization of) Martha White's RL task formalism (2017) and other recent departures from the traditional RL, and is relevant to task specification in RL, inverse RL and preference-based RL. In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose @math is a lattice in semisimple Lie group, all of whose factors have rank @math or higher. Let @math be a smooth @math -action on a compact nilmanifold @math that lifts to an action on the universal cover. If the linear data @math of @math contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of @math and @math , on a finite-index subgroup of @math . If @math is a @math action and contains an Anosov element, then the semiconjugacy is a @math conjugacy. As a corollary, we obtain @math global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank @math or higher. We also obtain global rigidity of Anosov actions of @math on @math for @math and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds. Many practitioners of reinforcement learning problems have observed that oftentimes the performance of the agent reaches very close to the optimal performance even though the estimated (action-)value function is still far from the optimal one. The goal of this paper is to explain and formalize this phenomenon by introducing the concept of the action-gap regularity. As a typical result, we prove that for an agent following the greedy policy ( ) with respect to an action-value function @math , the performance loss @math is upper bounded by @math , in which ζ ≥ = 0) is the parameter quantifying the action-gap regularity. For ζ > 0, our results indicate smaller performance loss compared to what previous analyses had suggested. Finally, we show how this regularity affects the performance of the family of approximate value iteration algorithms. The goal of search result diversification is to select a subset of documents from the candidate set to satisfy as many different subtopics as possible. In general, it is a problem of subset selection and selecting an optimal subset of documents is NP-hard. Existing methods usually formalize the problem as ranking the documents with greedy sequential document selection. At each of the ranking position the document that can provide the largest amount of additional information is selected. It is obvious that the greedy selections inevitably produce suboptimal rankings. In this paper we propose to partially alleviate the problem with a Monte Carlo tree search (MCTS) enhanced Markov decision process (MDP), referred to as M @math Div. In M @math Div, the construction of diverse ranking is formalized as an MDP process where each action corresponds to selecting a document for one ranking position. Given an MDP state which consists of the query, selected documents, and candidates, a recurrent neural network is utilized to produce the policy function for guiding the document selection and the value function for predicting the whole ranking quality. The produced raw policy and value are then strengthened with MCTS through exploring the possible rankings at the subsequent positions, achieving a better search policy for decision-making. Experimental results based on the TREC benchmarks showed that M @math Div can significantly outperform the state-of-the-art baselines based on greedy sequential document selection, indicating the effectiveness of the exploratory decision-making mechanism in M @math Div. This paper presents the temporal logic of rewriting @math . Syntactically, @math is a very simple extension of @math which just adds action atoms, in the form of spatial action patterns, to @math . Semantically and pragmatically, however, when used together with rewriting logic as a "tandem" of system specification and property specification logics, it has substantially more expressive power than purely state-based logics like @math , or purely action-based logics like A- @math . Furthermore, it avoids the system property mismatch problem experienced in state-based or action-based logics, which makes many useful properties inexpressible in those frameworks without unnatural changes to a system's specification. The advantages in expresiveness of @math are gained without losing the ability to use existing tools and algorithms to model check its properties: a faithful translation of models and formulas is given that allows verifying @math properties with @math model checkers. Motivated by applications in blockchains and sensor networks, we consider a model of @math nodes trying to reach consensus on their majority bit. Each node @math is assigned a bit at time zero, and is a finite automaton with @math bits of memory (i.e., @math states) and a Poisson clock. When the clock of @math rings, @math can choose to communicate, and is then matched to a uniformly chosen node @math . The nodes @math and @math may update their states based on the state of the other node. Previous work has focused on minimizing the time to consensus and the probability of error, while our goal is minimizing the number of communications. We show that when @math , consensus can be reached at linear communication cost, but this is impossible if @math . We also study a synchronous variant of the model, where our upper and lower bounds on @math for achieving linear communication cost are @math and @math , respectively. A key step is to distinguish when nodes can become aware of knowing the majority bit and stop communicating. We show that this is impossible if their memory is too low. Deterministic finite automata DFA have long served as a fundamental computational model in the study of theoretical computer science, and the problem of learning a DFA from given input data is a classic topic in computational learning theory. In this paper we study the learnability of a random DFA and propose a computationally efficient algorithm for learning and recovering a random DFA from uniform input strings and state information in the statistical query model. A random DFA is uniformly generated: for each state-symbol pair @math , we choose a state @math with replacement uniformly and independently at random and let @math , where Q is the state space, @math is the alphabet and @math is the transition function. The given data are string-state pairs x,i¾?q where x is a string drawn uniformly at random and q is the state of the DFA reached on input x starting from the start state @math . A theoretical guarantee on the maximum absolute error of the algorithm in the statistical query model is presented. Extensive experiments demonstrate the efficiency and accuracy of the algorithm. Many real-world problems face the dilemma of choosing best @math out of @math options at a given time instant. This setup can be modelled as combinatorial bandit which chooses @math out of @math arms at each time, with an aim to achieve an efficient tradeoff between exploration and exploitation. This is the first work for combinatorial bandit where the reward received can be a non-linear function of the chosen @math arms. The direct use of multi-armed bandit requires choosing among @math -choose- @math options making the state space large. In this paper, we present a novel algorithm which is computationally efficient and the storage is linear in @math . The proposed algorithm is a divide-and-conquer based strategy, that we call CMAB-SM. Further, the proposed algorithm achieves a regret bound of @math for a time horizon @math , which is sub-linear in all parameters @math , @math , and @math . The evaluation results on different reward functions and arm distribution functions show significantly improved performance as compared to standard multi-armed bandit approach with @math choices. We seek a global minimum of @math . The solution to @math will find local minima. Using the idea of simulated annealing, we consider the diffusion process, @math , @math , where @math is the n-dimensional standard Brownian motion and @math is the annealing rate which decreases to zero as t goes to @math . Under suitable condition on @math , we prove that @math converges weakly to a probability measure @math if for large t, @math with @math , where @math has a simple expression involving the action function of the dynamical system @math , @math concentrates on the global minima of U and is the weak limit of the Gibbs densities @math .The above result can also be formulated as follows: consider the Fokker–Planck equation (forward equation) [ t V(t,y) = ... Let @math denote a set of polynomials with complex coefficients. Let @math denote any set of sample points . For any @math , the discrete polynomial transform of @math (with respect to @math and @math ) is defined as the collection of sums, @math , where @math for some associated weight function @math . These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of @math orthogonal polynomials of degree at most @math , we give an @math algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the @math operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a three-term recurrence and thus it may be applied to any such set of discretely sampled functions. In particular, sampled orthogonal polynomials generate the vector space of functions on a distance transitive graph. As a direct application of our work, we are able to give a fast algorithm for computing subspace decompositions of this vector space which respect the action of the symmetry group of such a graph. This has direct applications to treating computational bottlenecks in the spectral analysis of data on distance transitive graphs, and we discuss this in some detail. Optimal search for targets with conditionally deterministic motion is investigated. The target motion takes place in @math , a copy of Euclidean n-space, and depends on a stochastic parameter @math which takes values in @math , another copy of Euclidean n-space. The target motion is deterministic given knowledge of @math . That is, there is a function @math , where T is a time interval, such that @math gives the target motion conditioned on @math . Search plans are specified by functions @math . A functional P is defined so that @math gives the probability of detecting the target by time t using plan @math .Let @math be the absolute value of the Jacobian of @math evaluated at x. If there exist functions @math and @math such that @math for @math , the target motion is called factorable Let $ _2 :T ... Deep Reinforcement Learning methods have achieved state of the art performance in learning control policies for the games in the Atari 2600 domain. One of the important parameters in the Arcade Learning Environment (ALE) is the frame skip rate. It decides the granularity at which agents can control game play. A frame skip value of @math allows the agent to repeat a selected action @math number of times. The current state of the art architectures like Deep Q-Network (DQN) and Dueling Network Architectures (DuDQN) consist of a framework with a static frame skip rate, where the action output from the network is repeated for a fixed number of frames regardless of the current state. In this paper, we propose a new architecture, Dynamic Frame skip Deep Q-Network (DFDQN) which makes the frame skip rate a dynamic learnable parameter. This allows us to choose the number of times an action is to be repeated based on the current state. We show empirically that such a setting improves the performance on relatively harder games like Seaquest. Let @math be a finite lattice and let @math be a positive integer. A function @math is said to be submodular if @math for all @math . In this paper we study submodular functions when @math is a diamond. Given oracle access to @math we are interested in finding @math such that @math as efficiently as possible. We establish a min--max theorem, which states that the minimum of the submodular function is equal to the maximum of a certain function defined over a certain polyhedron; and a good characterisation of the minimisation problem, i.e., we show that given an oracle for computing a submodular @math and an integer @math such that @math , there is a proof of this fact which can be verified in time polynomial in @math and @math ; and a pseudo-polynomial time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular @math one can find @math in time bounded by a polynomial in @math and @math . We prove that any smooth action of @math , @math , on an @math -dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e., isomorphic up to a finite permutation to an affine action on the torus or on its factor by @math . Furthermore this isomorphism has nice geometric properties; in particular, it is smooth in the sense of Whitney on a set whose complement has arbitrarily small measure. We further derive restrictions on topology of manifolds that may admit such actions, for example, excluding spheres and obtaining lower estimate on the first Betti number in the odd-dimensional case. We present a new algorithm for the contextual bandit learning problem, where the learner repeatedly takes one of @math actions in response to the observed context, and observes the reward only for that chosen action. Our method assumes access to an oracle for solving fully supervised cost-sensitive classification problems and achieves the statistically optimal regret guarantee with only @math oracle calls across all @math rounds, where @math is the number of policies in the policy class we compete against. By doing so, we obtain the most practical contextual bandit learning algorithm amongst approaches that work for general policy classes. We further conduct a proof-of-concept experiment which demonstrates the excellent computational and prediction performance of (an online variant of) our algorithm relative to several baselines. Consider the problem of choosing a string of actions to optimize an objective function that is string submodular. It was shown in previous papers that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a @math -approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvature, namely, total backward curvature, total forward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. If the objective function has total forward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also introduce the notion of string-matroid and consider the problem of maximizing the objective function subject to a string-matroid constraint. We investigate two applications of string submodular functions with curvature constraints: 1) choosing a string of actions to maximize the expected fraction of accomplished tasks; and 2) designing a string of measurement matrices such that the information gain is maximized.