442 research outputs found
Resource Allocation for Sequential Decision Making Under Uncertainaty : Studies in Vehicular Traffic Control, Service Systems, Sensor Networks and Mechanism Design
No full textA fundamental question in a sequential decision making setting under uncertainty is “how to allocate resources amongst competing entities so as to maximize the rewards accumulated in the long run?”. The resources allocated may be either abstract quantities such as time or concrete quantities such as manpower. The sequential decision making setting involves one or more agents interacting with an environment to procure rewards at every time instant and the goal is to find an optimal policy for choosing actions. Most of these problems involve multiple (infinite) stages and the objective function is usually a long-run performance objective. The problem is further complicated by the uncertainties in the sys-tem, for instance, the stochastic noise and partial observability in a single-agent setting or private information of the agents in a multi-agent setting. The dimensionality of the problem also plays an important role in the solution methodology adopted. Most of the real-world problems involve high-dimensional state and action spaces and an important design aspect of the solution is the choice of knowledge representation.
The aim of this thesis is to answer important resource allocation related questions in different real-world application contexts and in the process contribute novel algorithms to the theory as well. The resource allocation algorithms considered include those from stochastic optimization, stochastic control and reinforcement learning. A number of new algorithms are developed as well. The application contexts selected encompass both single and multi-agent systems, abstract and concrete resources and contain high-dimensional state and control spaces. The empirical results from the various studies performed indicate that the algorithms presented here perform significantly better than those previously proposed in the literature. Further, the algorithms presented here are also shown to theoretically converge, hence guaranteeing optimal performance.
We now briefly describe the various studies conducted here to investigate problems of resource allocation under uncertainties of different kinds:
Vehicular Traffic Control The aim here is to optimize the ‘green time’ resource of the individual lanes in road networks that maximizes a certain long-term performance objective. We develop several reinforcement learning based algorithms for solving this problem. In the infinite horizon discounted Markov decision process setting, a Q-learning based traffic light control (TLC) algorithm that incorporates feature based representations and function approximation to handle large road networks is proposed, see Prashanth and Bhatnagar [2011b]. This TLC algorithm works with coarse information, obtained via graded thresholds, about the congestion level on the lanes of the road network. However, the graded threshold values used in the above Q-learning based TLC algorithm as well as several other graded threshold-based TLC algorithms that we propose, may not be optimal for all traffic conditions. We therefore also develop a new algorithm based on SPSA to tune the associated thresholds to the ‘optimal’ values (Prashanth and Bhatnagar [2012]). Our thresh-old tuning algorithm is online, incremental with proven convergence to the optimal values of thresholds. Further, we also study average cost traffic signal control and develop two novel reinforcement learning based TLC algorithms with function approximation (Prashanth and Bhatnagar [2011c]). Lastly, we also develop a feature adaptation method for ‘optimal’ feature selection (Bhatnagar et al. [2012a]). This algorithm adapts the features in a way as to converge to an optimal set of features, which can then be used in the algorithm.
Service Systems The aim here is to optimize the ‘workforce’, the critical resource of any service system. However, adapting the staffing levels to the workloads in such systems is nontrivial as the queue stability and aggregate service level agreement (SLA) constraints have to be complied with. We formulate this problem as a constrained hidden Markov process with a (discrete) worker parameter and propose simultaneous perturbation based simulation optimization algorithms for this purpose. The algorithms include both first order as well as second order methods and incorporate SPSA based gradient estimates in the primal, with dual ascent for the Lagrange multipliers. All the algorithms that we propose are online, incremental and are easy to implement. Further, they involve a certain generalized smooth projection operator, which is essential to project the continuous-valued worker parameter updates obtained from the SASOC algorithms onto the discrete set. We validate our algorithms on five real-life service systems and compare their performance with a state-of-the-art optimization tool-kit OptQuest. Being ��times faster than OptQuest, our scheme is particularly suitable for adaptive labor staffing. Also, we observe that it guarantees convergence and finds better solutions than OptQuest in many cases.
Wireless Sensor Networks The aim here is to allocate the ‘sleep time’ (resource) of the individual sensors in an intrusion detection application such that the energy consumption from the sensors is reduced, while keeping the tracking error to a minimum. We model this sleep–wake scheduling problem as a partially-observed Markov decision process (POMDP) and propose novel RL-based algorithms -with both long-run discounted and average cost objectives -for solving this problem. All our algorithms incorporate function approximation and feature-based representations to handle the curse of dimensionality. Further, the feature selection scheme used in each of the proposed algorithms intelligently manages the energy cost and tracking cost factors, which in turn, assists the search for the optimal sleeping policy. The results from the simulation experiments suggest that our proposed algorithms perform better than a recently proposed algorithm from Fuemmeler and Veeravalli [2008], Fuemmeler et al. [2011].
Mechanism Design The setting here is of multiple self-interested agents with limited capacities, attempting to maximize their individual utilities, which often comes at the expense of the group’s utility. The aim of the resource allocator here then is to efficiently allocate the resource (which is being contended for, by the agents) and also maximize the social welfare via the ‘right’ transfer of payments. In other words, the problem is to find an incentive compatible transfer scheme following a socially efficient allocation. We present two novel mechanisms with progressively realistic assumptions about agent types aimed at economic scenarios where agents have limited capacities. For the simplest case where agent types consist of a unit cost of production and a capacity that does not change with time, we provide an enhancement to the static mechanism of Dash et al. [2007] that effectively deters misreport of the capacity type element by an agent to receive an allocation beyond its capacity, which thereby damages other agents. Our model incorporates an agent’s preference to harm other agents through a additive factor in the utility function of an agent and the mechanism we propose achieves strategy proofness by means of a novel penalty scheme. Next, we consider a dynamic setting where agent types evolve and the individual agents here again have a preference to harm others via capacity misreports. We show via a counterexample that the dynamic pivot mechanism of Bergemann and Valimaki [2010] cannot be directly applied in our setting with capacity-limited alim¨agents. We propose an enhancement to the mechanism of Bergemann and V¨alim¨aki [2010] that ensures truth telling w.r.t. capacity type element through a variable penalty scheme (in the spirit of the static mechanism). We show that each of our mechanisms is ex-post incentive compatible, ex-post individually rational, and socially efficien
Scalable Sprase Bayesian Nonparametric and Matrix Tri-factorization Models for Text Mining Applications
No full textHierarchical Bayesian Models and Matrix factorization methods provide an unsupervised way to learn latent components of data from the grouped or sequence data. For example, in document data, latent component corn-responds to topic with each topic as a distribution over a note vocabulary of words. For many applications, there exist sparse relationships between the domain entities and the latent components of the data. Traditional approaches for topic modelling do not take into account these sparsity considerations. Modelling these sparse relationships helps in extracting relevant information leading to improvements in topic accuracy and scalable solution. In our thesis, we explore these sparsity relationships for di errant applications such as text segmentation, topical analysis and entity resolution in dyadic data through the Bayesian and Matrix tri-factorization approaches, propos-in scalable solutions.
In our rest work, we address the problem of segmentation of a collection of sequence data such as documents using probabilistic models. Existing state-of-the-art Hierarchical Bayesian Models are connected to the notion of Complete Exchangeability or Markov Exchangeability. Bayesian Nonpareil-metric Models based on the notion of Markov Exchangeability such as HDP-HMM and Sticky HDP-HMM, allow very restricted permutations of latent variables in grouped data (topics in documents), which in turn lead to com-mutational challenges for inference. At the other extreme, models based on Complete Exchangeability such as HDP allow arbitrary permutations within each group or document, and inference is significantly more tractable as a result, but segmentation is not meaningful using such models. To over-come these problems, we explored a new notion of exchangeability called Block Exchangeability that lies between Markov Exchangeability and Com-plate Exchangeability for which segmentation is meaningful, but inference is computationally less expensive than both Markov and Complete Exchange-ability. Parametrically, Block Exchangeability contains sparser number of transition parameters, linear in number of states compared to the quadratic order for Markov Exchangeability that is still less than that for Complete Exchangeability and for which parameters are on the order of the number of documents. For this, we propose a nonparametric Block Exchangeable model (BEM) based on the new notion of Block Exchangeability, which we have shown to be a superclass of Complete Exchangeability and subclass of Markov Exchangeability. We propose a scalable inference algorithm for BEM to infer the topics for words and segment boundaries associated with topics for a document using the collapsed Gibbs Sampling procedure. Empirical results show that BEM outperforms state-of-the-art nonparametric models in terms of scalability and generalization ability and shows nearly the same segmentation quality on News dataset, Product review dataset and on a Synthetic dataset. Interestingly, we can tune the scalability by varying the block size through a parameter in our model for a small trade-o with segmentation quality.
In addition to exploring the association between documents and words, we also explore the sparse relationships for dyadic data, where associations between one pair of domain entities such as (documents, words) and as-associations between another pair such as (documents, users) are completely observed. We motivate the analysis of such dyadic data introducing an additional discrete dimension, which we call topics, and explore sparse relation-ships between the domain entities and the topic, such as of user-topic and document-topic respectively.
In our second work, for this problem of sparse topical analysis of dyadic data, we propose a formulation using sparse matrix tri-factorization. This formulation requires sparsity constraints, not only on the individual factor matrices, but also on the product of two of the factors. To the best of our knowledge, this problem of sparse matrix tri-factorization has not been stud-ide before. We propose a solution that introduces a surrogate for the product of factors and enforces sparsity on this surrogate as well as on the individual factors through L1-regularization. The resulting optimization problem is e - cogently solvable in an alternating minimization framework over sub-problems involving individual factors using the well-known FISTA algorithm. For the sub-problems that are constrained, we use a projected variant of the FISTA algorithm. We also show that our formulation leads to independent sub-problems towards solving a factor matrix, thereby supporting parallel implementation leading to a scalable solution. We perform experiments over bibliographic and product review data to show that the proposed framework based on sparse tri-factorization formulation results in better generalization ability and factorization accuracy compared to baselines that use sparse bi-factorization.
Even though the second work performs sparse topical analysis for dyadic data, ending sparse topical associations for the users, the user references with di errant names could belong to the same entity and those with same names could belong to different entities. The problem of entity resolution is widely studied in the research community, where the goal is to identify real users associated with the user references in the documents.
Finally, we focus on the problem of entity resolution in dyadic data, where associations between one pair of domain entities such as documents-words and associations between another pair such as documents-users are ob.-served, an example of which includes bibliographic data. In our nil work, for this problem of entity resolution in bibliographic data, we propose a Bayesian nonparametric `Sparse entity resolution model' (SERM) exploring the sparse relationships between the grouped data involving grouping of the documents, and the topics/author entities in the group. Further, we also exploit the sparseness between an author entity and the associated author aliases. Grouping of the documents is achieved with the stick breaking prior for the Dirichlet processes (DP). To achieve sparseness, we propose a solution that introduces separate Indian Bu et process (IBP) priors over topics and the author entities for the groups and k-NN mechanism for selecting author aliases for the author entities. We propose a scalable inference for SERM by appropriately combining partially collapsed Gibbs sampling scheme in Focussed topic model (FTM), the inference scheme used for parametric IBP prior and the k-NN mechanism. We perform experiments over bibliographic datasets, Cite seer and Rexa, to show that the proposed SERM model imp-proves the accuracy of entity resolution by ending relevant author entities through modelling sparse relationships and is scalable, when compared to the state-of-the-art baselin
Stochastic approximation with set-valued maps and Markov noise: Theoretical foundations and applications
No full textStochastic approximation algorithms produce estimates of a desired solution using noisy real world data.
Introduced by Robbins and Monro, in 1951, stochastic approximation techniques have been instrumental in
the asymptotic analysis of several adaptive algorithms in learning, signal processing and control. A popular
method for the analysis of stochastic approximation schemes is the dynamical systems approach or the o.d.e.
method introduced by Ljung and developed further extensively by Benaim and Hirsch.
We build upon the works of Benaim et.al. and Bhatnagar et.al., and present the asymptotic analysis of
stochastic approximation schemes with set-valued drift functions and nonadditive Markov noise. The frame-
works studied by us are under the weakest set of assumptions and encompass a majority of the methods studied
to date.
We first present the asymptotic analysis of stochastic approximation schemes with set-valued drift function
and non-additive iterate-dependent Markov noise. We show that a linearly interpolated trajectory of such a
recursion is an asymptotic pseudotrajectory for the
flow of a limiting differential inclusion obtained by averaging
the set-valued drift function of the recursion with respect to the stationary distributions of the Markov noise.
The limit set theorem by Benaim is then used to characterize the limit sets of the recursion in terms of the
dynamics of the limiting differential inclusion. The analysis presented allows us characterize the asymptotic
behavior of controlled stochastic approximation, subgradient descent, approximate drift problem and analysis
of discontinuous dynamics all in the presence of non-additive iterate-dependent Markov noise.
Next we present the asymptotic analysis of a stochastic approximation scheme on two timescales with set-
valued drift functions and in the presence of non-additive iterate-dependent Markov noise. It is shown that
the recursion on each timescale tracks the
flow of a differential inclusion obtained by averaging the set-valued
drift function in the recursion with respect to a set of measures which take into account both the averaging
with respect to the stationary distributions of the Markov noise terms and the interdependence between the
two recursions on different timescales.
Finally, we present the analysis of stochastic approximation schemes with set-valued maps in the absence of
a stability guarantee. We prove that after a large number of iterations if the stochastic approximation process
enters the domain of attraction of an attracting set it gets locked into the attracting set with high probability.
We demonstrate that the above result is an effective instrument for analyzing stochastic approximation schemes
in the absence of a stability guarantee, by using it to obtain an alternate criterion for convergence in the presence
of a locally attracting set for the mean field and by using it to show that a feedback mechanism, which involves
resetting the iterates at regular time intervals, stabilizes the scheme when the mean field possesses a globally
attracting set, thereby guaranteeing convergence. Our results build on the works of Borkar, Andrieu et al. and
Chen et al., by allowing for the presence of set-valued drift functions
Approximate Dynamic Programming and Reinforcement Learning - Algorithms, Analysis and an Application
No full textProblems involving optimal sequential making in uncertain dynamic systems arise in domains such as engineering, science and economics. Such problems can often be cast in the framework of Markov Decision Process (MDP). Solving an MDP requires computing the optimal value function and the optimal policy. The idea of dynamic programming (DP) and the Bellman equation (BE) are at the heart of solution methods. The three important exact DP methods are value iteration, policy iteration and linear programming. The exact DP methods compute the optimal value function and the optimal policy. However, the exact DP methods are inadequate in practice because the state space is often large and in practice, one might have to resort to approximate methods that compute sub-optimal policies. Further, in certain cases, the system observations are known only in the form of noisy samples and we need to design algorithms that learn from these samples. In this thesis we study interesting theoretical questions pertaining to approximate and learning algorithms, and also present an interesting application of MDPs in the domain of crowd sourcing.
Approximate Dynamic Programming (ADP) methods handle the issue of large state space by computing an approximate value function and/or a sub-optimal policy. In this thesis, we are concerned with conditions that result in provably good policies. Motivated by the limitations of the PBE in the conventional linear algebra, we study the PBE in the (min, +) linear algebra. It is a well known fact that deterministic optimal control problems with cost/reward criterion are (min, +)/(max, +) linear and ADP methods have been developed for such systems in literature. However, it is straightforward to show that infinite horizon discounted reward/cost MDPs are neither (min, +) nor (max, +) linear. We develop novel ADP schemes namely the Approximate Q Iteration (AQI) and Variational Approximate Q Iteration (VAQI), where the approximate solution is a (min, +) linear combination of a set of basis functions whose span constitutes a subsemimodule. We show that the new ADP methods are convergent and we present a bound on the performance of the sub-optimal policy.
The Approximate Linear Program (ALP) makes use of linear function approximation (LFA) and offers theoretical performance guarantees. Nevertheless, the ALP is difficult to solve due to the presence of a large number of constraints and in practice, a reduced linear program (RLP) is solved instead. The RLP has a tractable number of constraints sampled from the original constraints of the ALP. Though the RLP is known to perform well in experiments, theoretical guarantees are available only for a specific RLP obtained under idealized assumptions. In this thesis, we generalize the RLP to define a generalized reduced linear program (GRLP) which has a tractable number of constraints that are obtained as positive linear combinations of the original constraints of the ALP. The main contribution here is the novel theoretical framework developed to obtain error bounds for any given GRLP.
Reinforcement Learning (RL) algorithms can be viewed as sample trajectory based solution methods for solving MDPs. Typically, RL algorithms that make use of stochastic approximation (SA) are iterative schemes taking small steps towards the desired value at each iteration. Actor-Critic algorithms form an important sub-class of RL algorithms, wherein, the critic is responsible for policy evaluation and the actor is responsible for policy improvement. The actor and critic iterations have deferent step-size schedules, in particular, the step-sizes used by the actor updates have to be generally much smaller than those used by the critic updates. Such SA schemes that use deferent step-size schedules for deferent sets of iterates are known as multitimescale stochastic approximation schemes. One of the most important conditions required to ensure the convergence of the iterates of a multi-timescale SA scheme is that the iterates need to be stable, i.e., they should be uniformly bounded almost surely. However, the conditions that imply the stability of the iterates in a multi-timescale SA scheme have not been well established. In this thesis, we provide veritable conditions that imply stability of two timescale stochastic approximation schemes. As an example, we also demonstrate that the stability of a widely used actor-critic RL algorithm follows from our analysis.
Crowd sourcing (crowd) is a new mode of organizing work in multiple groups of smaller chunks of tasks and outsourcing them to a distributed and large group of people in the form of an open call. Recently, crowd sourcing has become a major pool for human intelligence tasks (HITs) such as image labeling, form digitization, natural language processing, machine translation evaluation and user surveys. Large organizations/requesters are increasingly interested in crowd sourcing the HITs generated out of their internal requirements. Task starvation leads to huge variation in the completion times of the tasks posted on to the crowd. This is an issue for frequent requesters desiring predictability in the completion times of tasks specified in terms of percentage of tasks completed within a stipulated amount of time. An important task attribute that affects the completion time of a task is its price. However, a pricing policy that does not take the dynamics of the crowd into account might fail to achieve the desired predictability in completion times. Here, we make use of the MDP framework to compute a pricing policy that achieves predictable completion times in simulations as well as real world experiments
Stochastic Optimization And Its Application In Reinforcement Learning
No full textNumerous engineering fields, such as transportation systems, manufacturing, communication networks, healthcare, and finance, frequently encounter problems requiring optimization in the presence of uncertainty. Simulation-based optimization is a workable substitute for accurate analytical solutions because of the numerous input variables and the need for a system model. Smoothed functional (SF) algorithms
belong to the class of simultaneous perturbation methods that have been found useful for stochastic optimization problems, particularly in high-dimensional parameter spaces. SF methods update the gradient of the objective using function measurements involving parameters
that are perturbed simultaneously along all component directions. \cite{katkul} originally developed the SF gradient procedure. This results in the objective function
getting smoothed because of the convolution. The objective function smoothing
that results from the convolution with a smoothing density function can help the algorithm converge to a global minimum or a point close to it.
First, we present a stochastic gradient algorithm for minimizing a smooth objective function that is an expectation over
noisy cost samples and only the latter are observed for any given parameter. Our algorithm employs a gradient estimation scheme with random perturbations, which are formed using the truncated Cauchy distribution from the sphere. We analyze the bias and variance of the proposed gradient estimator. Our algorithm is found to be particularly useful in the case when the objective function is non-convex and the parameter dimension is high. From an asymptotic convergence analysis, we establish that our algorithm converges almost surely to the set of stationary points of the objective function and obtains the asymptotic convergence rate. We also show that our algorithm avoids unstable equilibria, implying convergence to local minima. Further, we perform a non-asymptotic convergence analysis of our algorithm. In particular, we establish here a non-asymptotic bound for finding an -stationary point of the non-convex objective function. Finally, we demonstrate numerically through simulations that our algorithm outperforms GSF, SPSA, and RDSA by a significant margin over a few non-convex settings, and we further validate its performance over convex (noisy) objectives.
Next, we consider the problem of control in the setting of reinforcement learning (RL), where model information is not available. Policy gradient algorithms are a popular solution approach for this problem and are usually shown to converge to a stationary point of the value function. We propose two policy Newton algorithms that incorporate cubic regularization. Both algorithms employ the likelihood ratio method to form estimates of the gradient and Hessian of the value function using sample trajectories. The first algorithm requires an exact solution of the cubic regularized problem in each iteration, while the second algorithm employs an efficient gradient descent-based approximation to the cubic regularized problem. We establish convergence of our proposed algorithms to a second-order stationary point (SOSP) of the value function, which results in the avoidance of traps in the form of saddle points. In particular, the sample complexity of our algorithms towards finding an -SOSP is , and this is a significant improvement over the previous state-of-the-art sample complexity of
Algorithms for Challenges to Practical Reinforcement Learning
No full textReinforcement learning (RL) in real world applications faces major hurdles - the foremost being safety of the physical system controlled by the learning agent and the varying environment conditions in which the autonomous agent functions.
A RL agent learns to control a system by exploring available actions. In some operating states, when the RL agent exercises an exploratory action, the system may enter unsafe operation, which can lead to safety hazards both for the system as well as for humans supervising the system. RL algorithms thus need to respect these safety constraints and must do so with limited available information. Additionally, RL autonomous agents learn optimal decisions in the presence of a stationary environment. However, the stationary assumption on the environment is very restrictive. In many real world problems like traffic signal control, robotic applications, etc., one often encounters situations with non-stationary environments, and in these scenarios, RL algorithms yield sub-optimal decisions.
In this thesis, the first part develops algorithmic solutions to the challenges of safety and non-stationary environmental conditions. In order to handle safety restrictions and facilitate safe exploration during learning, this thesis proposes a cross-entropy method based sample efficient learning algorithm. This algorithm is developed on constrained optimization framework and utilizes very limited information for the learning of feasible policies. Also during the learning iterations, the exploration is guided in a manner that minimizes safety violations. In the first part, another algorithm for the second challenge is also described. The goal of this algorithm is to maximize the long-term discounted reward accrued
when the latent model of the environment changes with time. To achieve this, the algorithm leverages a change point detection algorithm to find change in the statistics of the environment. The results from this statistical algorithm are used to reset learning of policies.
The second part of this thesis describes the application of RL in networked intelligent systems. We consider two such systems - aerial quadrotor navigation and industrial internet of things system. In quadrotor navigation problem, with improved usage of machine learning computational frameworks, our proposed method is able to improve upon previously proposed obstacle avoidance algorithms in aerial vehicles. Obstacle avoidance in quadrotor aerial vehicle navigation brings in additional challenges when compared to ground vehicles. This is because, an aerial vehicle has to navigate across more types of obstacles - for e.g., objects like decorative items, furnishings, ceiling fans, sign-boards, tree branches, etc., are also potential obstacles for a quadrotor aerial vehicle. Thus, methods of obstacle avoidance developed for ground robots are clearly inadequate for UAV navigation. Our algorithm improves the efficiency of learning by inferring navigation decisions from temporal information of the ambient surroundings. This information is represented using monocular camera images collected by the quadrotor aerial vehicle.
An industrial internet-of-things (IIoT) system has multiple IoT devices, a user equipment (UE), together with a base station (BS) that receives the UE and IoT data. To circumvent the issue of numerous IoT-to-BS connections and to conserve IoT devices' energies, the UE serves as a relay to forward the IoT data to the BS. In this thesis, we consider a specific problem of multiple objective optimization that arises in this simple IIoT setup. The UE employs frame-based uplink transmissions, wherein it shares few slots of every frame to relay the IoT data. The IIoT system experiences a transmission failure called outage when IoT data is not transmitted. The unsent UE data is stored in the UE's buffer and is discarded after the storage time exceeds the age threshold. As the UE and IoT devices share the transmission slots, trade-offs exist between system outages and aged UE data loss. To resolve system outage-data ageing challenge, we adapt the Q-learning algorithm for slot-sharing between UE and IoT data and show numerical results for the same
Optimization Algorithms for Deterministic, Stochastic and Reinforcement Learning Settings
No full textOptimization is a very important field with diverse applications in physical, social and biological sciences and in various areas of engineering. It appears widely in ma-chine learning, information retrieval, regression, estimation, operations research and a wide variety of computing domains. The subject is being deeply studied both theoretically and experimentally and several algorithms are available in the literature. These algorithms which can be executed (sequentially or concurrently) on a computing machine explore the space of input parameters to seek high quality solutions to the optimization problem with the search mostly guided by certain structural properties of the objective function. In certain situations, the setting might additionally demand for “absolute optimum” or solutions close to it, which makes the task even more challenging.
In this thesis, we propose an optimization algorithm which is “gradient-free”, i.e., does not employ any knowledge of the gradient or higher order derivatives of the objective function, rather utilizes objective function values themselves to steer the search. The proposed algorithm is particularly effective in a black-box setting, where a closed-form expression of the objective function is unavailable and gradient or higher-order derivatives are hard to compute or estimate. Our algorithm is inspired by the well known cross entropy (CE) method. The CE method is a model based search method to solve continuous/discrete multi-extremal optimization problems, where the objective function has minimal structure. The proposed method seeks, in the statistical manifold of the parameters which identify the probability distribution/model defined over the input space to find the degenerate distribution concentrated on the global optima (assumed to be finite in quantity). In the early part of the thesis, we propose a novel stochastic approximation version of the CE method to the unconstrained optimization problem, where the objective function is real-valued and deterministic. The basis of the algorithm is a stochastic process of model parameters which is probabilistically dependent on the past history, where we reuse all the previous samples obtained in the process till the current instant based on discounted averaging. This approach can save the overall computational and storage cost. Our algorithm is incremental in nature and possesses attractive features such as stability, computational and storage efficiency and better accuracy. We further investigate, both theoretically and empirically, the asymptotic behaviour of the algorithm and find that the proposed algorithm exhibits global optimum convergence for a particular class of objective functions.
Further, we extend the algorithm to solve the simulation/stochastic optimization problem. In stochastic optimization, the objective function possesses a stochastic characteristic, where the underlying probability distribution in most cases is hard to comprehend and quantify. This begets a more challenging optimization problem, where the ostentatious nature is primarily due to the hardness in computing the objective function values for various input parameters with absolute certainty. In this case, one can only hope to obtain noise corrupted objective function values for various input parameters. Settings of this kind can be found in scenarios where the objective function is evaluated using a continuously evolving dynamical system or through a simulation. We propose a multi-timescale stochastic approximation algorithm, where we integrate an additional timescale to accommodate the noisy measurements and decimate the effects of the gratuitous noise asymptotically. We found that if the objective function and the noise involved in the measurements are well behaved and the timescales are compatible, then our algorithm can generate high quality solutions.
In the later part of the thesis, we propose algorithms for reinforcement learning/Markov decision processes using the optimization techniques we developed in the early stage. MDP can be considered as a generalized framework for modelling planning under uncertainty. We provide a novel algorithm for the problem of prediction in reinforcement learning, i.e., estimating the value function of a given stationary policy of a model free MDP (with large state and action spaces) using the linear function approximation architecture. Here, the value function is defined as the long-run average of the discounted transition costs. The resource requirement of the proposed method in terms of computational and storage cost scales quadratically in the size of the feature set. The algorithm is an adaptation of the multi-timescale variant of the CE method proposed in the earlier part of the thesis for simulation optimization. We also provide both theoretical and empirical evidence to corroborate the credibility and effectiveness of the approach.
In the final part of the thesis, we consider a modified version of the control problem in a model free MDP with large state and action spaces. The control problem most commonly addressed in the literature is to find an optimal policy which maximizes the value function, i.e., the long-run average of the discounted transition payoffs. The contemporary methods also presume access to a generative model/simulator of the MDP with the hidden premise that observations of the system behaviour in the form of sample trajectories can be obtained with ease from the model. In this thesis, we consider a modified version, where the cost function to be optimized is a real-valued performance function (possibly non-convex) of the value function. Additionally, one has to seek the optimal policy without presuming access to the generative model. In this thesis, we propose a stochastic approximation algorithm for this peculiar control problem. The only information, we presuppose, available to the algorithm is the sample trajectory generated using a priori chosen behaviour policy. The algorithm is data (sample trajectory) efficient, stable, robust as well as computationally and storage efficient. We provide a proof of convergence of our algorithm to a high performing policy relative to the behaviour policy
Algorithms for Product Pricing and Energy Allocation in Energy Harvesting Sensor Networks
No full textIn this thesis, we consider stochastic systems which arise in different real-world application contexts. The first problem we consider is based on product adoption and pricing. A monopolist selling a product has to appropriately price the product over time in order to maximize the aggregated profit. The demand for a product is uncertain and is influenced by a number of factors, some of which are price, advertising, and product technology. We study the influence of price on the demand of a product and also how demand affects future prices. Our approach involves mathematically modelling the variation in demand as a function of price and current sales. We present a simulation-based algorithm for computing the optimal price path of a product for a given period of time. The algorithm we propose uses a smoothed-functional based performance gradient descent method to find a price sequence which maximizes the total profit over a planning horizon.
The second system we consider is in the domain of sensor networks. A sensor network is a collection of autonomous nodes, each of which senses the environment. Sensor nodes use energy for sensing and communication related tasks. We consider the problem of finding optimal energy sharing policies that maximize the network performance of a system comprising of multiple sensor nodes and a single energy harvesting(EH) source. Nodes periodically sense a random field and generate data, which is stored in their respective data queues. The EH source harnesses energy from ambient energy sources and the generated energy is stored in a buffer. The nodes require energy for transmission of data and and they receive the energy for this purpose from the EH source. There is a need for efficiently sharing the stored energy in the EH source among the nodes in the system, in order to minimize average delay of data transmission over the long run. We formulate this problem in the framework of average cost infinite-horizon Markov Decision Processes[3],[7]and provide algorithms for the same
Algorithms For Stochastic Games And Service Systems
No full textThis thesis is organized into two parts, one for my main area of research in the field of stochastic games, and the other for my contributions in the area of service systems. We first provide an abstract for my work in stochastic games.
The field of stochastic games has been actively pursued over the last seven decades because of several of its important applications in oligopolistic economics. In the past, zero-sum stochastic games have been modelled and solved for Nash equilibria using the standard techniques of Markov decision processes. General-sum stochastic games on the contrary have posed difficulty as they cannot be reduced to Markov decision processes. Over the past few decades the quest for algorithms to compute Nash equilibria in general-sum stochastic games has intensified and several important algorithms such as stochastic tracing procedure [Herings and Peeters, 2004], NashQ [Hu and Wellman, 2003], FFQ [Littman, 2001], etc., and their generalised representations such as the optimization problem formulations for various reward structures [Filar and Vrieze, 1997] have been proposed. However, they suffer from either lack of generality or are intractable for even medium sized problems or both. In our venture towards algorithms for stochastic games, we start with a non-linear optimization problem and then design a simple gradient descent procedure for the same. Though this procedure gives the Nash equilibrium for a sample problem of terrain exploration, we observe that, in general, it need not be true. We characterize the necessary conditions and define KKT-N point. KKT-N points are those Karush-Kuhn-Tucker (KKT) points which corresponding to Nash equilibria. Thus, for a simple gradient based algorithm to guarantee convergence to Nash equilibrium, all KKT points of the optimization problem need to be KKT-N points, which restricts the applicability of such algorithms.
We then take a step back and start looking at better characterization of those points of the optimization problem which correspond to Nash equilibria of the underlying game. As a result of this exploration, we derive two sets of necessary and sufficient conditions. The first set, KKT-SP conditions, is inspired from KKT conditions itself and is obtained by breaking down the main optimization problem into several sub-problems and then applying KKT conditions to each one of those sub-problems. The second set, SG-SP conditions, is a simplified set of conditions which characterize those Nash points more compactly. Using both KKT-SP and SG-SP conditions, we propose three algorithms, OFF-SGSP, ON-SGSP and DON-SGSP, respectively, which we show provide Nash equilibrium strategies for general-sum discounted stochastic games. Here OFF-SGSP is an off-line algorithm while ONSGSP and DON-SGSP are on-line algorithms. In particular, we believe that DON-SGSP is the first decentralized on-line algorithm for general-sum discounted stochastic games. We show that both our on-line algorithms are computationally efficient. In fact, we show that DON-SGSP is not only applicable for multi-agent scenarios but is also directly applicable for the single-agent case, i.e., MDPs (Markov Decision Processes).
The second part of the thesis focuses on formulating and solving the problem of minimizing the labour-cost in service systems. We define the setting of service systems and then model the labour-cost problem as a constrained discrete parameter Markov-cost process. This Markov process is parametrized by the number of workers in various shifts and with various skill levels. With the number of workers as optimization variables, we provide a detailed formulation of a constrained optimization problem where the objective is the expected long-run averages of the single-stage labour-costs, and the main set of constraints are the expected long-run average of aggregate SLAs (Service Level Agreements). For this constrained optimization problem, we provide two stochastic optimization algorithms, SASOC-SF-N and SASOC-SF-C, which use smoothed functional approaches to estimate gradient and perform gradient descent in the aforementioned constrained optimization problem. SASOC-SF-N uses Gaussian distribution for smoothing while SASOC-SF-C uses Cauchy distribution for the same. SASOC-SF-C is the first Cauchy based smoothing algorithm which requires a fixed number (two) of simulations independent of the number of optimization variables. We show that these algorithms provide an order of magnitude better performance than existing industrial standard tool, OptQuest. We also show that SASOC-SF-C gives overall better performance
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