This is our newest area of research, with a number of papers on the way. I. Ryzhov, W.B. OCBA is new. Policy for Correlated Normal Beliefs,” Informs Journal on Computing, 180-195 (2012). Ryzhov, I. O., Awais Tariq, W. B. Powell, “May the Best Man Win: Simulation Optimization for Match-Making in E-Sports,” Proceedings of the Winter Simulation Conference, Phoenix, Arizona, December 11-14. Observations of the function, which might involve simulations, laboratory or field experiments, are both expensive and noisy. regression parameters. M. D. Rossetti, R. R. Hill, B. Johansson, A. Dunkin, and R. G. Ingalls, eds, 2009, pp. Ryzhov, I. O. and W. B. Powell, “Bayesian Active Learning With Basis Functions,” SSCI 2011 ADPRL - 2011 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning, Paris, April, 2011. The story that was originally used to motivate the problem (and gave the problem its name) is not really an important application, but is useful for understanding the basic idea behind the problem. Applying the knowledge gradient We do this by developing a continuous approximate of the knowledge gradient. The work is described in, D. Negoescu, P. Frazier and W. B. Powell, “The Knowledge Gradient Algorithm for Sequencing Experiments in Drug Discovery”, Informs Journal on Computing, Vol. the left (below), we have to find the maximum of the knowledge gradient surface 23, No. We can choose the weights in the linear combination, a process we refer to as information blending. 21, No. The problem is closely related to learning in the presence of a physical state, since the initial decision (size and shape) set the stage for the second decision (density) that is run in batch. SIAM Journal on Optimization 21, No. 3, pp. asymptotically optimal. Instead of maximizing the expected value of a measurement, we can adapt the knowledge gradient to maximize the worst outcome. have a budget of N measurements to evaluate each choice to refine your distribution an impact. Brown, C. A. Mirkin, W. B. Powell, “Nested Batch Mode Learning and Stochastic Optimization with an Application to Sequential Multi-Stage Testing in Materials Science,” SIAM J. Optimal Learning for Stochastic Optimization with Nonlinear Parametric Belief Models Powell, W. B. gradient can be viewed as a method of steepest ascent). We research how to help laboratory scientists discover new science through the use of computers, data analysis, machine learning and decision theory. We do this by developing a continuous approximate of the knowledge gradient. We consider the optimal learning problem of optimizing an expensive function with a known parametric form but unknown parameters. ORF 418, Optimal Learning, is an undergraduate course taught in the department of Operations Research and Financial Engineering at Princeton University. a machine for airport security that can sense explosives and it works poorly, The knowledge gradient using a linear belief model, D. Negoescu, P. Frazier and W. B. Powell, “The Knowledge Gradient The KG policy also works on problems where the beliefs about different alternatives are correlated. Syllabus (2012) - Princeton enjoys 12 week semesters, so this syllabus may look a bit short to many faculty. Global Optimization (to appear). 208.1 (2013): 337-370. how to compute the knowledge gradient for problems with correlated beliefs. One of the most famous problems in information collection is the multiarmed bandit problem, where make a choice (out of a discrete set of choices), observe a reward, and use this observation to update estimates of the future value of rewards. Most of my exercises are included in the book, but I continue to revise. Ryzhov, I. O. and W. B. Powell, “Bayesian Active Learning With Basis Functions,” SSCI 2011 ADPRL - 2011 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning, Paris, April, 2011. on problems where the beliefs about different alternatives are correlated. This paper uses a discrete, lookup table representation of the belief model. Below is a summary of research papers that we have produced while pursuing this work. Experimental work shows that it can produce a much higher rate of convergence than the knowledge gradient with independent beliefs, in addition to outperforming other more classical information collection mechanisms. Operations Research, Vol 59, No. of each are given below. A product with a specific set of features might see sales steadily improve as word of mouth gets around. M is not too large (say less than 1000). Learning when the alternatives are continuous. Tutorial: Optimal Learning for the laboratory sciences, An optimal learning video tutorial (by Warren Powell), The knowledge gradient for online and offline learning, Learning with continuous alternatives (parameter tuning), Learning with a robust objective function, P. Frazier, W. B. Powell, S. Dayanik, “The Knowledge-Gradient information is time consuming and expensive. we want to evaluate the alternative that offers the greatest chance of improving Control and Optimization, Vol. Software. Scott, Warren, P. I. Frazier, and W. B. Powell. 346-363, 2011. This work is based on the paper above (Mes a belief about each alternative (known as a "lookup table belief model"), testing different densities) that can be run in batch model. Of course, we include an is to say that trying one alternative can teach us something about other alternatives. Some sample applications include: Each of these problems require making observations (measurements) to as, and often better, than other standard learning policies. trying to maximize. The value of information can be a concave function in the number of Our first effort used an approximation method based on estimating It actually slightly outperforms the best available approximation of Gittins the consistency result for OCBA is new. The paper develops a knowledge gradient policy for guiding an initial design decision (e.g. This article shows Instead of creating you have a normally distributed belief about the value of each choice. collection. Princeton University. A fresh perspective of learning is to introduce a mini-max objective. A Bayesian model is set up to capture the uncertainty in our beliefs about the convergence of the model. Optimal Learning Optimal learning addresses the challenge of how to collect information as efficiently as possible, primarily for settings where collecting information is time consuming and expensive. This condition is useful for verifying consistency of adaptive sequential sampling policies that do not do forced random exploration, making consistency difficult to verify by other means. Finding the best team to compete in an invent. 47, No. This problem arose in a business simulator which used approximate dynamic programming to learn a policy, while we were tuning various business parameters. “Asymptotically Optimal Bayesian sequential change detection and identification rules.” Annals of Operations Research (M. Katehakis, ed.) The knowledge gradient is developed for a locally parametric belief model. the website. (c) Informs, For a more theoretical treatment of learning the coefficients of linear programs, see. which measures the marginal value of a measurement in terms of the value of High-dimensional data analysis, mathematical optimization, statistical learning, information theory, and their applications to medical imaging and computational biology Jianqing Fan Professor of Statistics; Frederick L. Moore '18 Professor of Finance You have a budget of N measurements to evaluate each choice to refine your distribution of belief. SDSU has a Climate Action Plan that commits campus to achieving operational carbon neutrality by 2040 and full carbon neutrality by 2050. Local minima are located close to points that have been previously measured, so we use these points to guess at the locations of local maxima and then use a simple gradient search algorithm starting from each of these points. D. Negoescu, P. Frazier and W. B. Powell, “The Knowledge Gradient Algorithm for Sequencing Experiments in Drug Discovery”, Informs Journal on Computing, Vol. Information Collection,” SIAM J. on Control and Optimization, Vol. Imagine that you want to find the shortest path between two points, but you do not know the times on the links. of contamination in one location and it measures high, we are likely to Motivated by a problem in laboratory experimentation, this paper considers the problem where there is an initial choice (e.g. Our decision rule is easy to compute, and performs Example of course work from Hannah Freid '21. Scott, Warren, P. I. Frazier, and W. B. Powell. There is a base compound with a series of sites (indexed This is our first application of the knowledge gradient algorithm with correlated beliefs to the problem of parameter tuning for simulation models. 3, pp. of an observation, taking into account the possible change in the estimate No. Our decision rule is easy to compute, and performs competitively against other learning policies, including a Monte Carlo adaptation of the knowledge gradient policy for ranking and selection. the size and shape of nanoparticles) followed by batch learning of a secondary tunable parameter (e.g. We develop the knowledge gradient for optimizing a function when our belief is represented by constants computed at different levels of aggregation. Click here to go to the website where the code is available. The project has three requirements: initial problem description, a summary of the math model and learning policies, and then the final report. I use the last three lectures (depending on the size of the class) to allow students to present their projects (without numerical results), so that the rest of the class sees the diversity of problems. The paper shows (the edge we measure). At the moment, this website focuses on our work on the knowledge gradient, a simple, elegant concept for collecting information. The paper provides bounds for finite measurement budgets, and provides experimental work that shows that it works as well as, and often better, than other standard learning policies. 1360-1367. here for online supplement). This paper uses the knowledge gradient for dynamic programs where the value function is now approximated using a linear model. 2, 712-731 (2011). 2, pp. This paper makes two contributions. After your N measurements, you have to choose what appears to This makes it possible to compute the knowledge gradient for problems with correlated beliefs. The basics of Optimal Learning In these demos, you will be introduced to the core concepts behind Optimal Learning, the optimization framework that sequentially guides you through the space of experiments in order to achieve some objective. The paper uses the strategy of solving a sampled belief model, where the prior is represented by a sample of possible parameters (rather than our standard use of multivarite normal distributions). ... when you can learn from the best! I am a member of the Computational Stochastic Optimization and Learning (CASTLE) Lab group working with Prof. Warren Powell on Stochastic Optimization and Optimal Learning with applications in Emergency Storm Response and Driverless Fleets of Electric Vehicles. then identify the information that has the highest impact on the economic problem. The knowledge gradient can be computed for each link in the network using at most two shortest path calculations (and often one). Princeton, NJ : Princeton University Abstract: Collecting information in the course of sequential decision-making can be extremely challenging in high-dimensional settings, where the number of measurement budget is much smaller than both the number … The KG policy is also effective on finite horizon problems. B. Defourny, I. O. Ryzhov, W. B. Powell, “Optimal Information Blending with Measurements in the L2 Sphere". We use a Bayesian model that captures expert DOI: 10.1137/090775026. learning Physics & Astronomy A little bit of information may teach you nothing, and you may have to make an investment in information beyond a certain threshold to actually have an impact. The knowledge gradient is not an optimal policy for collecting information, but these properties suggest that it is generally going to work well. Behaving optimally in such problems is also known as optimal learning. exploration, making consistency difficult to verify by other means. 180-195 (2012). work shows that it can produce a much higher rate of convergence than the D. Negoescu, P. Frazier and W. B. Powell, “The Knowledge Gradient Algorithm for Sequencing Experiments in Drug Discovery”, Mes, M., P. I. Frazier and W. B. Powell, “Hierarchical Knowledge Gradient for Sequential Sampling,”, DC-RBF (Dirichlet Clouds with Radial Basis Functions), I. Ryzhov, W. B. Powell, P. I. Frazier, “The knowledge gradient algorithm for a general class of online learning problems,”, I. Ryzhov, W.B. The model assumes that the set of potential alternatives to be evaluated is finite. We have found that most applications exhibit correlated beliefs, which knowledge gradient with independent beliefs, in addition to outperforming Consistency of the knowledge-gradient policy was shown previously, while the consistency result for This new stopping rule significantly improves the performance of LL1 as compared to its performance under the best other generally known adaptive stopping rule, EOC Bonf, outperforming it in every case tested. infinite-horizon versions of the problem. to find the best of five or ten alternatives with independent beliefs can be 58, pp. We investigate the economic implications of the S-curve effect, showing that it is possible to have too many choices. This problem can be solved by choosing the option with the highest index (known as the Gittins index). The method is illustrated in on a graph, in which we use sequential measurements to rene Bayesian estimates The paper establishes asymptotic optimality for off-line versions of the problem and proposes a computationally tractable algorithm. While the theory behind optimal learning is fairly deep and could only be taught at the graduate level, the modeling concepts and techniques of optimal learning can easily be taught at the undergraduate level to serious students. here to download main paper) (Click Offline learning arises when we have a budget for finding the best possible solution, after which have to use the solution in a production setting. We consider the optimal learning problem of optimizing an expensive function with a known parametric form but unknown parameters. P., W. B. Powell and S. Dayanik, “A Knowledge Gradient Policy for Sequential than a day, so the paper also introduces methods to product results without This theory paper describes an adaptation of the knowledge gradient for general linear programs, extending our previous paper on learning the costs on arcs of a graph for a shortest path problem. 40, No. It uses a biophysical model to develop the structure that is used in developing the prior and the underlying belief model. MOLTE – Modular, Optimal Learning Testing Environment – This is a Matlab-based environment for comparing algorithms for offline and online learning with discrete alternatives. Brown, C. A. Mirkin, W. B. Powell, “Nested Batch Mode Learning and Stochastic Optimization with an Application to Sequential Multi-Stage Testing in Materials Science,” SIAM J. You may want to minimize costs, minimize delays or find the best match between a model and historical metrics. The KG policy is also effective on finite horizon problems. But there are situations where it can work poorly, as we demonstrate in Section 5.2 below. (click here to download main paper) (Click here for online supplement). (click here to download paper) (Click here for online supplement). the ranking and selection problem, which is an off-line version of the multiarmed This problem 23, No. 47, The KG policy also works We derive a one-period look-ahead policy for online subset selection problems, where learning about one subset also gives us information about other subsets. The knowledge gradient policy Local minima are located close to points that have been previously measured, so we use these points to guess at the locations of local maxima and then use a simple gradient search algorithm starting from each of these points. for Sequential Sampling,” J. gradient policy for on-line problems, and show that it very closely matches of thousands of different ads to determine the ones that are best to put on We recently derived the knowledge gradient when using a local parametric approximation called DC-RBF (Dirichlet Clouds with Radial Basis Functions): B. Cheng, A. Jamshidi, W. B. Powell, The Knowledge Gradient using Locally Parametric Approximations, Winter Simulation Conference, 2013. A common challenge in the calibration of simulation model is that we have to tune several continuous parameters. in Operations Research, Chapter 10, pp. 1492-1502. This paper develops and tests a knowledge gradient algorithm when the underlying belief model is nonparametric, using a broad class of kernel regression models. size and shape) followed by a series of experiments (e.g. (Click The KG policy with independent beliefs is extremely easy to compute (we provide closed-form expressions for the case with normal rewards), and requires a simple numerical algorithm for the case with correlated beliefs. We are developing methods to handle problems where the number of potential In some settings, these costs may be approximate, and getting more information can be expensive. Mes, M., P. I. Frazier and W. B. Powell, “Hierarchical Knowledge Gradient for Sequential Sampling,” J. 3 (2011): 996-1026. A common challenge in the calibration of simulation model is that we If we evaluate the level under which measurement policies sample each measurement type infinitely (c) Informs. S Dayanik, W. B. Powell and K. Yamazaki “An Asymptotically Optimal Strategy in Sequential Change Detection and Identification Applied to Problems in Biosurveillance” Proceedings of the 3rd INFORMS Workshop on Data Mining and Health Informatics, (J. Li, D. Aleman, R. Sikora, eds. Control and Optimization, Vol. We develop the knowledge gradient for optimizing a function when our belief is represented by constants computed at different levels of aggregation. Policy for Correlated Normal Beliefs,” Informs Journal on Computing, ***** In support of Princeton University’s education and research mission, the University hosts a diverse and highly-motivated group of high school students each summer to conduct research under the mentorship of Princeton here for online supplement). The paper puts a prior on the distribution of indicator variables that capture whether a coefficient is zero or not. classes: Brief discussions In addition, we may also be receiving rewards or incurring costs, which have to be balanced against the value of the information being gained. theta as quickly as possible. This paper describes a method for applying the knowledge gradient to 585-598 (2009) (c) Informs. Classes typically run between 30 and 40 students, all of whom would have taken a course in probability and statistics. function at an arbitrary query point x, we compute a set of weights w^g_x for each level of aggregation g for each query point x based on the total sum of squares error (variance plus bias). P. Frazier, W. B. Powell, S. Dayanik, “The Knowledge-Gradient Policy for Correlated Normal Beliefs,” Informs Journal on Computing, Vol. Click here for a spreadsheet implementation of the knowledge gradient for independent, normally distributed beliefs, (Click "The Knowledge Gradient for Optimal Learning," Encyclopedia This paper uses a discrete, lookup table representation of the belief model. 1344–1368 http://epubs.siam.org/doi/abs/10.1137/12086279X. B361-B381, DOI: 10.1137/140971117, 2015. The method is motivated by the 10,000 molecular compounds after just 100 experiments. ), and is summarized in, E. of the function at each level of aggregation, as well as the possible change than alternatives 3 and 4. here to download main paper). uses adaptive learning from approximate dynamic programming) requires more The paper shows that just as with problems with independent beliefs, the knowledge gradient is both myopically and asymptotically optimal. Note that the later chapters are more advanced. results in the presence of an S-curve. Ryzhov, I. O., W. B. Powell, “Approximate Dynamic Programming with Correlated Bayesian Beliefs,” Forty-Eighth Annual Allerton Conference on Communication, Control, and Computing, September 29 – October 1, 2010, Allerton Retreat Center, Monticello, Illinois., IEEE Press, pp. a particular material or sensor within the device). We consider Bayesian information collection, in which a measurement policy Ilya Ryzhov, Boris Defourny, Warren Powell, “Ranking and Selection Meets Robust Optimization,” Winter Simulation Conference, 2012. "Optimal Learning: Optimization in the Information Age," article in OR/MS Today (2012). Let X_{ij} = 1 if we put substituent i at site j, and let theta_{ij} be the impact of this combination on the performance of the compound. measurements, but for many problems it is not, and instead follows an S-curve. a belief model. If we have independent beliefs, the knowledge gradient In most applications, our belief about mu_x may be correlated W. Scott, P. Frazier, W. B. Powell – “The Correlated Knowledge We propose the KG(*) algorithm, which maximizes the average value of information, and show that it produces good results when there is a significant S-curve effect. is particularly easy to apply. budgets, and provides experimental work that shows that it works as well This model, called DC-RBF, approximates a function by representing the domain using a series of clouds, which avoids storing the history. This is a short, equation-free article introducing the basic concept of optimal learning, which appeared in the Informs news magazine, OR/MS Today. we might lower our evaluation of other devices that might use similar technologies Frazier, P. I., and W. B. Powell, “Paradoxes in Learning: The The knowledge gradient can be adopted to the problem of making a function at different levels of aggregation. We demonstrate the use of this sufficient condition by showing consistency of two previously proposed ranking and selection policies: OCBA for linear loss, and the knowledge-gradient policy with independent normal priors. SDSU has a Climate Action Plan that commits campus to achieving operational carbon neutrality by 2040 and full carbon neutrality by 2050. Wang, Y. W. B. Powell, K. Reyes, R. Schapire, “Finite-time analysis for the knowledge-gradient policy, and a new testing environment for optimal learning,” Working paper, Department of Operations Research and Financial Engineering, Princeton University. This paper derives the knowledge gradient when the belief model is captured using kernel regression, a class of nonparametric statistical models. Ryzhov, I., W. B. Powell, “Information Collection for Linear Programs with Uncertain Objective Coefficients,” SIAM J. Optimization, Vol. You need to use care to make sure they pick good problems. We use the distances between local minima to perform scaling of the steepest descent algorithm. 1, pp. the performance of Gittins indices for discounted infinite horizon problems. It is useful to divide these models into three fundamental By considering the sampling and stopping problems jointly rather than separately, we derive a new composite stopping/sampling rule. bandit problem, for which Gittins indices are known to be optimal for discounted, The student projects performed in the course taught at Princeton (ORF 418-Optimal Learning) produced a wide range of interesting topics. Evaluating the Knowledge 4, pp. Learning in the presence of a physical state. but this requires careful tuning of a parameter. The knowledge gradient can produce poor learning by j) and a series of small sequences of atoms ("substituents") 2931-2974. First, it provides the first finite-time bound on the performance of the knowledge gradient for offline ranking and selection problems. marginal value of information. A single run of the model (which uses adaptive learning from approximate dynamic programming) requires more than a day, so the paper also introduces methods to product results without a full run. indexed by i. Not only will you learn valuable skills, our emphasis on career training leads to the shortest time to hire. The knowledge gradient for nonparametric belief models: Mes, M., P. I. Frazier and W. B. Powell, “Hierarchical Knowledge Gradient need to find the best molecular compound to solve a particular problem (e.g. P. Frazier, W. B. Powell, H. P. Simao, "Simulation Model Calibration Course instructors may order an examination copy directly from Wiley. This paper extends the work on optimal learning with a linear belief model, to the setting where the belief model is a high-dimensional, sparse linear belief model. 4, pp. The knowledge gradient policy guides this search by always choosing to measure the choice which would produce the highest value if you only have one more measurement (the knowledge gradient can be viewed as a method of steepest ascent). Often, we do not have time to wait for a process to reach its asymptotic limit, so we can fit a function that tries to guess (imperfectly) this limit. A proof of convergence is provided. A proof of convergence is provided. 213-246 (2008) (c) Informs. This paper develops the knowledge gradient for maximizing the expected value of information when solving linear programs. Cite this reference as: Warren B. Powell, Reinforcement Learning and Stochastic Optimization and Learning: A Unified Framework, Department of Operations Research and Financial Engineering, Princeton University, 2019. Course project - Students are encouraged to work in teams of two. 21, No. Although the page constraints limited the scope, it covers the central dimensions of information collection, along with an overview of a number of the most popular heuristic policies. An easy tutorial is contained in the article. here for online supplement), The S-curve effect - Handling the nonconcavity of information. as a "parametric belief model"). provide closed-form expressions for the case with normal rewards), and requires Uncertainty Quantification (to appear). Students are required to take a total of five courses and earn at least B- for each course: one of the “Foundations of Statistics” courses, one of the “Foundations of Machine Learning” courses, and three elective courses. above, but the original paper on this topic is, P. Frazier, W. B. Powell, S. Dayanik, “The Knowledge-Gradient Finding the optimal solution of a linear program assumes that you have accurate information on costs (among other things). the final solution. a problem with a very large number of alternatives. maximizes the average value of information, and show that it produces good This is our first application The Optimal learning – This research addresses the challenges of collecting information, when information (observations, simulations, laboratory and field experiments) are expensive. Machine Learning Research, Vol. Contact Us! Together they form a unique fingerprint. http://epubs.siam.org/doi/abs/10.1137/12086279X. introduce the dimension of correlated beliefs. The presentation focuses more on the knowledge other more classical information collection mechanisms. Let an alternative x be a discrete number 1, ..., M where Like other Bayesian approaches, the knowledge gradient uses subjective prior beliefs on … a simple numerical algorithm for the case with correlated beliefs. We propose the KG(*) algorithm, which of belief. 5.1.3 The Four Distributions of Learning;˙ The knowledge gradient, using a parametric belief model, was used to sequence experiments while searching for the best compound to cure a form of Ewing's sarcoma. the tuning of two continuous parameters, which required approximately six If you are interested in the real theory, see. Imagine that we have a finite-horizon online learning problem where we have a total of N measurements, and we have already learned n. If v^{off}_x is the offline knowledge gradient for alternative x, then the online knowledge gradient is given by, v^{online}_x = \theta^n_x + (N-n) v^{offline}_x. The knowledge gradient using a nonlinear belief model. Most of the applications that we have considered introduce the dimension of correlated beliefs. A Bayesian model is set up to capture the uncertainty in our 2410-2439 (2008). 37, No. (2012). 2931-2974, 2011. 188-201, 2011. Optimization, Vol. The KG policy with independent beliefs is extremely easy to compute (we band set to maximize DVD sales after a band performance, Competing with Netflix: Recommending the Right Movie, Learning Optimal Tolls for the Lincoln Tunnel: Solving Port Authority Pricing 21, No. From offline learning to online learning: The knowledge-gradient policy was originally derived for off-line learning 585-598 (2009) (c) Informs, (Click (c) Informs. This produces a nonconcave surface that we have to maximize. You of the most powerful advantages of the knowledge gradient over other methods, We then revisit the We have previously developed the knowledge gradient with correlated beliefs for discrete alternatives. 1, pp. Once we know the parameters, we can estimate the value In this paper, we propose an algorithm in the optimal learning framework that learns the shape of the function and finds the optimal design with a limited number of measurements. After your N measurements, you have to choose what appears to be the best based on your current belief. 346-363, 2011. Most of the applications that we have considered However, it is easy to add lectures using material from the book. Scientific Computing, Vol. The knowledge-gradient policy was originally derived for off-line learning problems such as ranking and selection. I.O. (click 1, pp. We consider this one The knowledge gradient with independent beliefs. alternatives might number in the tens of thousands (of molecules), hundreds We decision (the path we choose) is distinct from the measurement decision Imagine that you have M choices (M is not too large) where you have a normally distributed belief about the value of each choice. gradient for different belief models. produce the highest value if you only have one more measurement (the knowledge We model the economic decision we are trying to make, and So alternative 2 may be much more attractive to evaluate 5, pp. This paper investigates a stopping rule based on the knowledge gradient concept. Some sample applications include: How do you discover the best drug to treat a disease, out of the thousands of potential combinations? This makes it possible to provide meaningful guidance to find the best out of The paper shows that this policy is myopically optimal (by construction), but is also asymptotically optimal, making it the only stationary policy that is both myopically and asymptotically optimal. 7, No. have to tune several continuous parameters. 1, pp. Yan Li, Kristopher G. Reyes, Jorge Vazquez-Anderson, Yingfei Wang, Lydia M Contreras, Warren B. Powell, “A Knowledge Gradient Policy for Sequencing Experiments to Identify the Structure of RNA Molecules Using a Sparse Additive Belief Model,” Working paper, Department of Operations Research and Financial Engineering, Princeton University, 2015. Policy for Sequential Information Collection,” SIAM J. on Control and This makes it very easy for others to add new problems, and new algorithms. on Computing, Vol. 2009. competitively against other learning policies, including a Monte Carlo adaptation be the best based on your current belief. from ORF 418 - Optimal Learning. This paper addresses the problem of learning when the belief model is nonlinear in the parameters, motivated by a problem in materials science. et al. This paper introduces the idea of using the knowledge gradient within a dyamic program, which effectively means in the presence of a physical state. Click here. Support Princeton Splash Equal Opportunity and Nondiscrimination at Princeton University: Princeton University believes that commitment to principles of fairness and respect for all is favorable to the free and open exchange of ideas, and the University seeks to reach out as widely as possible in order to attract the ablest individuals as students, faculty, and staff. 4, pp. Which links should you learn about to have the greatest impact on your ability to find the shortest path? Numerical examples are provided to verify the asymptotic optimality and the speed of convergence. Decision Analysis, Vol. Our estimate of the function at any point is given by a weighted sum of estimates at different levels of aggregation. 346-363, 2011. Ryzhov, I. O. and W. B. Powell, “Bayesian Active Learning With Basis Functions,” SSCI 2011 ADPRL - 2011 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning, Paris, April, 2011. here for online supplement). 378-403, 2010. The method is illustrated in the tuning of two continuous parameters, which required approximately six runs of the model. We have previously developed the knowledge gradient with correlated beliefs for discrete alternatives. Discovery). As the website evolves, we will provide a more complete representation of the different frameworks and methods that have evolved for solving this important problem class. The knowledge gradient can produce poor learning results in the presence of an S-curve. often, ensuring consistency, i.e., that a globally optimal future decision In addition to general nonlinear models, we study special cases such as logistics regression. optimal, making it the only stationary policy that is both myopically and We would like to predict how many ad clicks an ad will receive based on attributes Ryzhov,I.O., W. B. Powell, “Information Collection in a Linear Program,” SIAM J. Optimization (to appear). The project requires that they pick a problem where the collection of information is time-consuming or expensive. The visual graph tracks the occurrence of the word "romantic" in OKCupid essays by age and gender. The work is motivated by a problem involving learning the structure of RNA molecules. P. Frazier and W. B. Powell, “Consistency of Sequential Bayesian Sampling Policies” SIAM J. 12, pp. Optimal Learning develops the needed principles for gathering information to make decisions, especially when collecting information is time-consuming and expensive. For more on this project, click here. 7, No. The paper develops an approximation of the knowledge gradient for batch learning to guide the initial discrete decision (size and shape). This work is summarized in. Powell, W.B. These two cases are characterized by a fundamental combinatorial parameter of a learning problem: the VC (Vapnik-Chervonenkis) dimension. If you have any questions, please email us at splash@princeton.edu. It actually slightly outperforms the best available approximation of Gittins indices (by Gans and Chick) on problems for which Gittins indices should be optimal. We then revisit the knowledge gradient algorithm, which allocates measurements based on the marginal value of information. Frazier, P. I., and W. B. Powell, “Paradoxes in Learning: The Marginal Value of Information and the Problem of Too Many Choices,” Decision Analysis, Vol. You have a way of collecting information, but it is expensive, and you have a limited amount of time to learn the best path. We may have a belief mu_x about each x. (c) Informs. The knowledge gradient has to compute the expected value 4, pp. This often arises when we have to find the set of parameters that will produce the best results for a model. 1, pp. Learning nonlocal constitutive models with neural networks, Xu-Hui Zhou, Jiequn Han, Heng Xiao, … Optimal Learning Optimal learning represents the problem of making observations (or measurements) in an efficient way to achieve some objective. including the classical bandit theory. 23, No. Vol. Powell, W. B. and P. Frazier, "Optimal Learning," TutORials Our estimate of the function at any point is given by a weighted sum of estimates at different levels of aggregation. This paper introduces the idea of using the knowledge gradient within a dyamic program, which effectively means in the presence of a physical state. This was an invited tutorial on the topic of optimal learning, and represents a fairly easy introduction to the general field of information collection. here for online supplement). Below are some general purpose routines that we have developed. The challenge is that measurements take A review of the book by Steve Chick appeared in the November 2012 issue of Informs Journal on Computing. central dimensions of information collection, along with an overview of problems such as ranking and selection. There are many applications that require models that are nonlinear in the parameters. Ryzhov, I., W. B. Powell, “Information Collection for Linear Programs with Uncertain Objective Coefficients,” SIAM J. Optimization, Vol. (as shown to the right) with different levels of uncertainty about each alternative, Manage knowledge with Bayesian Statistics time and/or cost money, which means we have to collect this information carefully. (e.g. Although the page constraints limited the scope, it covers the 1360-1367. 60, No. The raise our belief about the level of toxin in nearby locations. regression to estimate a function. Experimental Optimal learning addresses the challenge of how to collect This is our newest area of research, with a number of papers on the way. Non-Parametric Belief Models,” J. Vol. Parametric models - We can further divide these according to: Low-dimensional (small number of parameters), High-dimensional - Here we use a sparse-additive belief model. I. Ryzhov, W. B. Powell, P. I. Frazier, “The knowledge gradient algorithm for a general class of online learning problems,” Operations Research, Vol. P. Frazier and W. B. Powell, “Consistency of Sequential Bayesian Sampling Policies” SIAM J. "The Knowledge Gradient for Optimal Learning," 2, 712-731 (2011). I give weekly problem sets and a midterm, after which the students take on a course project. applied to a wide range of settings. with our belief about another alternative, x'. W. B. gradient. Powell, "Information collection on a graph,". Of course, we include an introduction to the knowledge gradient concept. We study the joint problem of sequential change detection and multiple hypothesis testing. of the knowledge gradient policy for ranking and selection. This paper describes a method for applying the knowledge gradient to a problem with a very large number of alternatives. It turns out that there is a very simple, elegant relationship between the knowledge gradient for offline learning, and the knowledge gradient for online learning. beliefs about the convergence of the model. of individual arc costs in order to learn about the best path. Powell, “The Knowledge Gradient Policy using a Sparse Additive Belief Model,” Working paper, Department of Operations Research and Financial Engineering, Princeton University, 2015. We have extended the knowledge gradient to two classes of nonparametric Algorithm for Sequencing Experiments in Drug Discovery”, Informs Journal Barut, W. B. Powell, “Optimal Learning for Sequential Sampling with Frazier, in the weights w^g_x which have to be recomputed after each observation. A very short presentation illustrating the jungle of stochastic optimization (updated April 12, 2019). Ryzhov, I. O., W. B. Powell, “Approximate Dynamic Programming with Correlated Bayesian Beliefs,” Forty-Eighth Annual Allerton Conference on Communication, Control, and Computing, September 29 – October 1, 2010, Allerton Retreat Center, Monticello, Illinois., IEEE Press, pp. 5, pp. information as efficiently as possible, primarily for settings where collecting Second, it describes the first general-purpose testing environment, MOLTE, which provides a large library of problems, each implemented in its own .m file, and a library of algorithms that can be applied to these problems (each of which is also provided in its own .m file). Problem sets (2012) - This zipped file includes latex files and associated software (spreadsheets and matlab code). The knowledge gradient policy is introduced here as a method for solving the ranking and selection problem, which is an off-line version of the multiarmed bandit problem. runs of the model. the information gained by the measurement. If we have five alternatives This is a rule that tells us which action xwe should take next in order to observe something new. Optimal Learning is a rich field that includes contributions from different communities. The method is motivated by the need to find the best molecular compound to solve a particular problem (e.g. We compare the method against Huang's adaptation of sequential kriging to problems with noisy measurements. We show that the resulting decision rule is easily computable, and present experimental evidence that the policy is competitive against other online learning policies. Let X_{ij} = 1 if we put substituent i at site j, and let of the ad (the topic, number of words, graphics, ...). The knowledge gradient policy is a method for determining which of which will do the most to identify the best choice. We also computed the knowledge gradient when we are using kernel Optimal learning represents the problem of making observations (or measurements) in an efficient way to achieve some objective. here for online supplement), (click Semidefinite programming relaxations are used to create efficient convex approximations to the nonconvex blending problem. Gradient Algorithm with Linear Beliefs for the Street Cart Vendor Problem, Optimal Tuning of a Particle Swarm Algorithm, The Ultimate Set List – Using the knowledge gradient to find the best The only policy which is competitive with KG seems to be interval estimation, but this requires careful tuning of a parameter. An initial investigation of this idea is. showing that it is possible to have too many choices. of adaptive sequential sampling policies that do not do forced random of parameter tuning for simulation models. 21, No. determine which choice works the best. knowledge gradient algorithm, which allocates measurements based on the choices to learn a regression model. This paper can handle low-dimensional vectors of continuous parameters. a particular material or sensor within the device). Optimal learning (OL) addresses these issues in a systematic way to navigate experiment space and achieve your objective. The paper provides bounds for finite measurement that this policy is myopically optimal (by construction), but is also asymptotically Click here for research paper describing the MOLTE environment and initial tests. indices (by Gans and Chick) on problems for which Gittins indices should than the tutorial listed next. Global Optimization (to appear). For example, if we are trying to find the hot spot (in red) of the surface to (click We give a sufficient condition 2410-2439 (2008). P. Frazier, W. B. Powell, H. P. Simao, “Simulation Model Calibration with Correlated Knowledge-Gradients,” Winter Simulation Conference, December, 2009. The proposed method outperforms several other heuristics in numerical experiments conducted on two broad problem classes. as quickly as possible. 4, pp. Wiley and Sons. killing cancer cells). 22(4), pp. here to download paper) (Click This condition is useful for verifying consistency We may pose a regression We compare the method against Huang’s adaptation of sequential kriging to problems with noisy measurements. We consider the ranking and selection of normal means in a fully sequential Bayesian context. Our certified teachers will get to know you on a personal basis for the optimal learning experience. "The Correlated Knowledge Gradient for Simulation Optimization of Continuous Parameters Using Gaussian Process Regression." 377-400 (2008). with Correlated Knowledge-Gradients," Winter Simulation Conference, December, (c) Informs. Princeton Training is considered a top technical training institution. We propose a new exploration strategy based on the knowledge gradient concept from the optimal learning literature, which is currently the only method capable of handling correlated belief structures. The knowledge gradient with correlated beliefs (offline learning, discrete alternatives), P. Frazier, W. B. Powell, S. Dayanik, “The Knowledge-Gradient demonstrate the use of this sufficient condition by showing consistency This is a shorter but more up-to-date tutorial on optimal learning Powell, "Information collection on a graph," Operations Research, Vol 59, No. Health sciences – Projects in health have included drug discovery, drug delivery, blood management, dosage decisions, personal health, and health policy. The measurement may require field Powell, Here, we combine the frequentist Lasso regularization methodology to identify the most important parameters: Yan Li, Han Liu, W.B. 3, pp. The knowledge gradient policy is a method for determining which of a discrete set of measurements we should make to determine which of a discrete set of choices we should make. The paper presents two optimal blending strategies: an active learning method that maximizes uncertainty reduction, and an economic approach that maximizes an expected improvement criterion. This paper extends this idea to problems with continuous alternatives. Central to the concept of optimal learning is a measurement policy. The goal is to choose compounds to test that allow us to estimate the parameters theta as quickly as possible. We derive a knowledge gradient policy for an optimal learning problem on a graph, in which we use sequential measurements to rene Bayesian estimates of individual arc costs in order to learn about the best path. 585-598 (2009) (c) Informs (Click We consider a class of optimal learning problems in which sequential measurements are used to gradually improve estimates of unknown quantities. The campus has a dedication to green buildings, reducing its impact on the environment and providing optimal space for learning, teaching, researching, and working. There is a base compound with a series of sites (indexed by j) and a series of small sequences of atoms (“substituents”) indexed by i. E. Barut and W. B. Powell, “Optimal Learning for Sequential Sampling with Non-Parametric Beliefs,” Journal of Global Optimization, Vol. collects information to support a future decision. I am an Associate Research Scholar at the Operations Research and Financial Engineering Department at Princeton University. theta_{ij} be the impact of this combination on the performance of the compound. bandit problem. Observations of the function, which might involve simulations, laboratory or field experiments, are both expensive and noisy. Consistency of the knowledge-gradient policy was shown previously, while In this study, we focus on a Bayesian approach known as optimal learning with knowledge gradient, which selects alternatives that maximizes the expected value of information. 188-201, 2011. 585-598 (2009) (c) Informs. 5, pp. 47, No. We propose computationally efficient sequential decision rules that are asymptotically either Bayes-optimal or optimal in a Bayesian fixed-error formulation, as the unit detection delay cost or the misdiagnosis and false alarm probabilities go to zero, respectively. The sampling component of the derived composite rule is the same as the previously introduced LL1 sampling rule, but the stopping rule is new. 4, pp. Encyclopedia for Operations Research and Management Science, 2011 (c) John For example, a problem in logistics might require including costs that reflect the quality of service provided by a vendor, but it may be necessary to use the vendor for a period of time, or collect historical information from other manufacturers, to refine these costs. We investigate the economic implications of the S-curve effect, We formulate the problem as a dynamic program, provide the optimality condition using Bellman’s equation, and propose a multiperiod lookahead policy to overcome the nonconcavity in the value of information. Considerable attention has been given to the on-line version of this problem, known popularly as the multiarmed bandit problem, for which Gittins indices are known to be optimal for discounted, infinite-horizon versions of the problem. experimentation or running a time consuming simulation (some business simulators In order to provide an optimal learning environment for the students, we ask that parents do not attend classes with their children.