DOCSLIB.ORG

Optimization Under Uncertainty

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Optimization Under Uncertainty OPTIMIZATION UNDER UNCERTAINTY LECTURE NOTES 2001 R. T. Rockafellar University of Washington CONTENTS 1. INTRODUCTION 1 2. BASIC STOCHASTIC PROGRAMMING 16 3. LINEAR PROGRAMMING MODELS 30 4. EXTENDED LINEAR-QUADRATIC MODELS 46 5. STATES AND CONTROLS 60 6. DYNAMIC PROGRAMMING 75 7. ADVANCED STOCHASTIC PROGRAMMING 96 1 1. INTRODUCTION Problems of optimization under uncertainty are characterized by the necessity of making decisions without knowing what their full effects will be. Such problems appear in many areas of application and present many interesting challenges in concept and computation. For a beginner, it’s most important at the outset to gain some appreciation of just how optimization under uncertainty differs from other branches of optimization, and what the basic modeling issues are. A review of terms will lay the foundation for this. Optimization: This refers to the analysis and solution of problems in which a single choice must be made from a range of “feasible” ones. Feasible choices are modeled as the elements of some set—the feasible set. The goal is to find a “best” choice (not necessarily unique), or at least a “better” one than might readily be detected. Choices are compared according to the values they give to a certain function—the objective function in the problem. The goal may be maximization or minimization of such values. For simplicity, minimization is taken to be the standard. Discrete optimization: There are only finitely many options to compare, but the num- ber of them could be enormous. The feasible set is a finite (discrete) set, hopefully with enough structure to support some degree of effective analysis. Continuous optimization in finite dimensions: The choices are describable in terms of the values of finitely many real variables. The feasible set can therefore be modeled as a subset of a space IRn. Continuous optimization in infinite dimensions: Not just vectors of variables, but entire functions, scalar-valued or vector-valued, have to be chosen. These unknowns, often interpreted as “policies” of response or control to ongoing situations, may be regarded as elements of an infinite-dimensional vector space. Sources of infinite dimensionality: Problems typically have infinitely many “degrees of freedom” when the decisions need to be represented as depending on con- tinuous time or on observations of continuous random variables. Finite-dimensional approximations: Infinite-dimensional problems can’t ordinarily be solved numerically without converting them somehow to problems in finite dimensions. Uncertainty: Decisions must often be taken in the face of the unknown. Actions decided upon in the present will have consequences that can’t fully be determined until a later stage. But there may be openings for corrective action later or even multiple opportunities for recourse as more and more becomes known. 2 Potential applications of optimization under uncertainty: Here are some prime examples. Note that time is almost inevitably involved, because exact knowledge of the future is what most typically is lacking. Generation of electrical power: An electrical utility has to decide each day how much power to produce without yet knowing the demand, which may depend on weather developments (as could affect heating, cooling, etc.) Longer range decisions may concern the amount of coal to purchase or the kinds of buying/selling contracts set up with other utilities. Operation of reservoirs: Systems of water reservoirs and their connecting flows have to be managed in a reasonable manner for drinking water, irrigation, hydropower, flood control, recreation, and the protection of fish runs. As if these conflicting goals didn’t already pose difficulty, there is climate uncertainty affecting both the inflows into the system and the intensity of irrigation needs and the rest. Inventory management: Supplies of retail goods or spare parts must reliably be main- tained despite the vagaries of demand and the costs of restocking. Portfolio selection: Investments must be made in stocks, bonds, foreign currencies and the like without knowing for sure how their values will rise or fall. At the root of the uncertainties are interest rates and political developments. Facility planning: Where should new facilities be located, and what capacity should they have, if they are designed to serve needs that are anticipated from guesswork? Similar: problems of reconfiguration or expansion of existing facilities. Pollution control: What types and sizes of treatment plants should now be built to en- sure water quality in a basin where population growth and future manufacturing requirements can only be estimated roughly, and where anyway a lot depends on what will happen with rainfall over a long term? Stabilization of mechanisms: An antenna must be kept fixed as steadily as possible on the source of signals it’s supposed to receive, despite random wind gusts that can push it out of line. What configuration of forces should be brought into play to restore misalignments as rapidly as possible, without neglecting the potential of additional wind gusts before the process is complete? Analysis of biological systems: Are the feeding and breeding strategies of animals op- timal in the sense of “maximizing survival” in a particular biological niche with respect to its uncertainties of climate, food supply, predation, and so forth? Note that here the issues are theoretical rather than numerical. 3 The modeling of uncertainty: The uncertainties in a problem have to be represented in such a manner that their effects on present decision-making can properly be taken into account. This is an interesting and challenging subject. Stochastic modeling: The uncertain elements in a problem can often be modeled as random variables to which the theory of probability can be applied. For this purpose such elements have to have a “known” probability distribution. Degrees of knowledge: Such a distribution may be available from statistical data (as with weather variables), or it may be no more than an educated guess (as with interest rates or election prospects)—“subjective” probabilities. Either way, the mathematical treatment is the same. Deterministic modeling: This refers to mathematical formulations in which uncertainty plays no role. In practice, such modeling often prevails even in situations with obviously important uncertainties, because the modelers don’t know how to cope with those features, or don’t have adequate data to work with, or don’t yet have good software available for getting numerical solutions. Deterministic versus stochastic: These terms are often contrasted with each other in the description of mathematical models. Range modeling: Sometimes, when a deterministic model is clearly inappropriate, yet there are few clues to the probabilities that would support a stochastic model, it’s useful to work with ranges of uncertainty. Various quantities that would otherwise be data parameters are neither given specific values nor probability distributions but merely viewed as restricted to particular intervals. One tries to guard against whatever might happen by thinking of the actual values to be faced as chosen from these intervals by an adversary, as in a game setting. Uncertain probabilities: This notion can be combined with stochastic modeling by supposing that a probability distribution is present but is incompletely known. The actual distribution to be faced will be chosen by an adversary— like Mother Nature—from some limited set of distributions, described perhaps by ranges on statistical parameters. The role of scenarios: A common tool in planning for the future is to work with scenar- ios, which are particular representations of how the future might unfold. Some kind of probabilistic model or simulation is used to generate a batch of such scenarios. The challenge then, though, is how to make good use of the scenarios in coming up with an effective decision. 4 A common but faulty approach: Often, planners just solve, for each scenario that is generated, an optimization problem which arises from taking that scenario to be the path the future truly will follow. These problems are themselves deterministic in character. Although each yields a prescription of what should be done here and now, there’s no theoretical guidance about the compromise between those prescriptions that should actually be adopted, even when probabilities can be assigned to the individual scenarios. Indeed, the separate prescriptions obtained for the individual scenario problems may be inconsistent with each other and very fragile—not adequately hedged. They’re optimal only in a context where one can act with perfect foresight. The need for modeling the evolution of information: The crucial feature demanded for serious applications is the effective modeling of how observations at various future stages increase the store of knowledge on which decisions can properly be based, not only in the present, but in subsequent stages as well. In the setting of discrete probability, we’ll formalize this later in terms of a “scenario tree.” Problem data in an elementary framework: It will help in understanding the main issues if we first look at an uncluttered formulation. A basic problem of optimization in IRn takes the form: minimize f0(u) over all u ∈ C n n for a function f0 : IR → IR and a set C ⊂ IR . This set might be specified as consisting of the points u in an underlying subset U ⊂ IRn that satisfy some constraints on the values of additional expressions fi(u), say ≤ 0 for i = 1, . , s, f (u) i = 0 for i = s + 1, . , m, but for now it will be better not to focus on such a representation and to think instead of the condition u ∈ C as expressing feasibility more abstractly. Essential objective function: Even simpler notation, very convenient in theoretical dis- cussions, results from the device of enforcing constraints by infinite penalties. For the basic problem just described, the essential objective function is n f(u) := f0(u) if u ∈ C, ∞ if u∈ / C.
Load more

Recommended publications
  • Stochastic Optimization from Distributed, Streaming Data in Rate-Limited Networks Matthew Nokleby, Member, IEEE, and Waheed U
    1 Stochastic Optimization from Distributed, Streaming Data in Rate-limited Networks Matthew Nokleby, Member, IEEE, and Waheed U. Bajwa, Senior Member, IEEE T Abstract—Motivated by machine learning applications in T training samples fξ(t) 2 Υgt=1 drawn independently from networks of sensors, internet-of-things (IoT) devices, and au- the distribution P (ξ), which is supported on a subset of Υ. tonomous agents, we propose techniques for distributed stochastic There are, in particular, two main categorizations of training convex learning from high-rate data streams. The setup involves a network of nodes—each one of which has a stream of data data that, in turn, determine the types of methods that can be arriving at a constant rate—that solve a stochastic convex used to find approximate solutions to the SO problem. These optimization problem by collaborating with each other over rate- are (i) batch training data and (ii) streaming training data. limited communication links. To this end, we present and analyze In the case of batch training data, where all T samples two algorithms—termed distributed stochastic approximation fξ(t)g are pre-stored and simultaneously available, a common mirror descent (D-SAMD) and accelerated distributed stochastic approximation mirror descent (AD-SAMD)—that are based on strategy is sample average approximation (SAA) (also referred two stochastic variants of mirror descent and in which nodes to as empirical risk minimization (ERM)), in which one collaborate via approximate averaging of the local, noisy subgra- minimizes the empirical average of the “risk” function φ(·; ·) dients using distributed consensus. Our main contributions are in lieu of the true expectation.
    [Show full text]
  • Metaheuristics1
    METAHEURISTICS1 Kenneth Sörensen University of Antwerp, Belgium Fred Glover University of Colorado and OptTek Systems, Inc., USA 1 Definition A metaheuristic is a high-level problem-independent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms (Sörensen and Glover, To appear). Notable examples of metaheuristics include genetic/evolutionary algorithms, tabu search, simulated annealing, and ant colony optimization, although many more exist. A problem-specific implementation of a heuristic optimization algorithm according to the guidelines expressed in a metaheuristic framework is also referred to as a metaheuristic. The term was coined by Glover (1986) and combines the Greek prefix meta- (metá, beyond in the sense of high-level) with heuristic (from the Greek heuriskein or euriskein, to search). Metaheuristic algorithms, i.e., optimization methods designed according to the strategies laid out in a metaheuristic framework, are — as the name suggests — always heuristic in nature. This fact distinguishes them from exact methods, that do come with a proof that the optimal solution will be found in a finite (although often prohibitively large) amount of time. Metaheuristics are therefore developed specifically to find a solution that is “good enough” in a computing time that is “small enough”. As a result, they are not subject to combinatorial explosion – the phenomenon where the computing time required to find the optimal solution of NP- hard problems increases as an exponential function of the problem size. Metaheuristics have been demonstrated by the scientific community to be a viable, and often superior, alternative to more traditional (exact) methods of mixed- integer optimization such as branch and bound and dynamic programming.
    [Show full text]
  • Stochastic Optimization
    Stochastic Optimization Marc Schoenauer DataScience @ LHC Workshop 2015 Content 1 Setting the scene 2 Continuous Optimization 3 Discrete/Combinatorial Optimization 4 Non-Parametric Representations 5 Conclusion Content 1 Setting the scene 2 Continuous Optimization 3 Discrete/Combinatorial Optimization 4 Non-Parametric Representations 5 Conclusion Stochastic Optimization Hypotheses Search Space Ω with some topological structure Objective function F assume some weak regularity Hill-Climbing Randomly draw x0 2 Ω and compute F(x0) Initialisation Until(happy) y = Best neighbor(xt) neighbor structure on Ω Compute F(y) If F(y) F (xt) then xt+1 = y accept if improvement else xt+1 = xt Comments Find closest local optimum defined by neighborhood structure Iterate from different x0's Until(very happy) Stochastic Optimization Hypotheses Search Space Ω with some topological structure Objective function F assume some weak regularity Stochastic (Local) Search Randomly draw x0 2 Ω and compute F(x0) Initialisation Until(happy) y = Random neighbor(xt) neighbor structure on Ω Compute F(y) If F(y) F (xt) then xt+1 = y accept if improvement else xt+1 = xt Comments Find one close local optimum defined by neighborhood structure Iterate, leaving current optimum IteratedLocalSearch Stochastic Optimization Hypotheses Search Space Ω with some topological structure Objective function F assume some weak regularity Stochastic (Local) Search { alternative viewpoint Randomly draw x0 2 Ω and compute F(x0) Initialisation Until(happy) y = Move(xt) stochastic variation on
    [Show full text]
  • Adaptive Stochastic Optimization
    Industrial and Systems Engineering Adaptive Stochastic Optimization Frank E. Curtis1 and Katya Scheinberg2 1Department of Industrial and Systems Engineering, Lehigh University 2Department of Operations Research and Information Engineering, Cornell University ISE Technical Report 20T-001 Adaptive Stochastic Optimization The purpose of this article is to summarize re- Frank E. Curtis Katya Scheinberg cent work and motivate continued research on ◦ the design and analysis of adaptive stochastic op- January 18, 2020 timization methods. In particular, we present an analytical framework—new for the context Abstract. Optimization lies at the heart of of adaptive deterministic optimization—that sets machine learning and signal processing. Contem- the stage for establishing convergence rate guar- porary approaches based on the stochastic gradi- antees for adaptive stochastic optimization tech- ent method are non-adaptive in the sense that niques. With this framework in hand, we remark their implementation employs prescribed param- on important open questions related to how it eter values that need to be tuned for each appli- can be extended further for the design of new cation. This article summarizes recent research methods. We also discuss challenges and oppor- and motivates future work on adaptive stochastic tunities for their use in real-world systems. optimization methods, which have the potential to offer significant computational savings when 2 Background training large-scale systems. Many problems in ML and SP are formulated as optimization problems. For example, given a 1 Introduction data vector y Rm from an unknown distribu- tion, one often∈ desires to have a vector of model The successes of stochastic optimization algo- parameters x Rn such that a composite objec- ∈ rithms for solving problems arising machine tive function f : Rn R is minimized, as in learning (ML) and signal processing (SP) are → now widely recognized.
    [Show full text]
  • Conservative Stochastic Optimization with Expectation Constraints
    1 Conservative Stochastic Optimization with Expectation Constraints Zeeshan Akhtar,? Amrit Singh Bedi,y and Ketan Rajawat? Abstract This paper considers stochastic convex optimization problems where the objective and constraint functions involve expectations with respect to the data indices or environmental variables, in addition to deterministic convex constraints on the domain of the variables. Although the setting is generic and arises in different machine learning applications, online and efficient approaches for solving such problems have not been widely studied. Since the underlying data distribution is unknown a priori, a closed-form solution is generally not available, and classical deterministic optimization paradigms are not applicable. Existing approaches towards solving these problems make use of stochastic gradients of the objective and constraints that arrive sequentially over iterations. State-of-the-art approaches, such as those using the saddle point framework, are able to ensure that the optimality gap as well as the constraint violation − 1 decay as O T 2 where T is the number of stochastic gradients. The domain constraints are assumed simple and handled via projection at every iteration. In this work, we propose a novel conservative − 1 stochastic optimization algorithm (CSOA) that achieves zero average constraint violation and O T 2 optimality gap. Further, we also consider the scenario where carrying out a projection step onto the convex domain constraints at every iteration is not viable. Traditionally, the projection operation can be avoided by considering the conditional gradient or Frank-Wolfe (FW) variant of the algorithm. The state-of-the-art − 1 stochastic FW variants achieve an optimality gap of O T 3 after T iterations, though these algorithms have not been applied to problems with functional expectation constraints.
    [Show full text]
  • Monte Carlo Methods for Sampling-Based Stochastic Optimization
    Monte Carlo methods for sampling-based Stochastic Optimization Monte Carlo methods for sampling-based Stochastic Optimization Gersende FORT LTCI CNRS & Telecom ParisTech Paris, France Joint works with B. Jourdain, T. Leli`evre,G. Stoltz from ENPC and E. Kuhn from INRA. A. Schreck and E. Moulines from Telecom ParisTech P. Priouret from Paris VI Monte Carlo methods for sampling-based Stochastic Optimization Introduction Simulated Annealing Simulated Annealing (1/2) Let U denote the objective function one wants to minimize. U(x) min U(x) () max exp(−U(x)) () max exp − 8T > 0 x2X x2X x2X T In order to sample from πT? where U(x) π (x) = exp − T T sample successively from a sequence of tempered distributions πT1 , πT2 , ··· with T1 > T2 > ··· > T?. Monte Carlo methods for sampling-based Stochastic Optimization Introduction Simulated Annealing Simulated Annealing (1/2) Let U denote the objective function one wants to minimize. U(x) min U(x) () max exp(−U(x)) () max exp − 8T > 0 x2X x2X x2X T In order to sample from πT? where U(x) π (x) = exp − T T sample successively from a sequence of tempered distributions πT1 , πT2 , ··· with T1 > T2 > ··· > T?. or sample successively from a sequence of (nt-iterated) kernels (PTt (x; ·))t such that πT PT = πT . Monte Carlo methods for sampling-based Stochastic Optimization Introduction Simulated Annealing Simulated Annealing (2/2) Under conditions on X, on the cooling schedule (Tt)t, on the kernels (Pt)t, on the dominating measure and the set of minima, ··· Kirkpatrick, Gelatt and Vecchi. Optimization via Simulated Annealing. Science (1983) Geman and Geman.
    [Show full text]
  • Stochastic Optimization,” in Handbook of Computational Statistics: Concepts and Methods (2Nd Ed.) (J
    Citation: Spall, J. C. (2012), “Stochastic Optimization,” in Handbook of Computational Statistics: Concepts and Methods (2nd ed.) (J. Gentle, W. Härdle, and Y. Mori, eds.), Springer−Verlag, Heidelberg, Chapter 7, pp. 173–201. dx.doi.org/10.1007/978-3-642-21551-3_7 STOCHASTIC OPTIMIZATION James C. Spall The Johns Hopkins University Applied Physics Laboratory 11100 Johns Hopkins Road Laurel, Maryland 20723-6099 U.S.A. [email protected] Stochastic optimization algorithms have been growing rapidly in popularity over the last decade or two, with a number of methods now becoming “industry standard” approaches for solving challenging optimization problems. This chapter provides a synopsis of some of the critical issues associated with stochastic optimization and a gives a summary of several popular algorithms. Much more complete discussions are available in the indicated references. To help constrain the scope of this article, we restrict our attention to methods using only measurements of the criterion (loss function). Hence, we do not cover the many stochastic methods using information such as gradients of the loss function. Section 1 discusses some general issues in stochastic optimization. Section 2 discusses random search methods, which are simple and surprisingly powerful in many applications. Section 3 discusses stochastic approximation, which is a foundational approach in stochastic optimization. Section 4 discusses a popular method that is based on connections to natural evolution—genetic algorithms. Finally, Section 5 offers some concluding remarks. 1 Introduction 1.1 General Background Stochastic optimization plays a significant role in the analysis, design, and operation of modern systems. Methods for stochastic optimization provide a means of coping with inherent system noise and coping with models or systems that are highly nonlinear, high dimensional, or otherwise inappropriate for classical deterministic methods of optimization.
    [Show full text]
  • Contextual Covariance Matrix Adaptation Evolutionary Strategies
    Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) Contextual Covariance Matrix Adaptation Evolutionary Strategies Abbas Abdolmaleki1;2;3, Bob Price5, Nuno Lau1, Luis Paulo Reis2;3, Gerhard Neumann4∗ 1: IEETA, DETI, University of Aveiro 2: DSI, University of Minho 3: LIACC, University of Porto, Portugal 4: CLAS, TU Darmstadt, Germany 5: PARC, A Xerox Company, USA Abstract the ball to its team mates which are positioned on differ- ent locations on the field. Here, the soccer robot should Many stochastic search algorithms are designed to learn to kick the ball to any given target location, which is optimize a fixed objective function to learn a task, specified by the context vector, on the field. In such cases, i.e., if the objective function changes slightly, for learning for every possible context is clearly inefficient or example, due to a change in the situation or con- even infeasible. Therefore our goal is to generalize learned text of the task, relearning is required to adapt to tasks from similar contexts to a new context. To do so, we the new context. For instance, if we want to learn learn a context-dependent policy for a continuous range of a kicking movement for a soccer robot, we have to contexts without restarting the learning process. In this pa- relearn the movement for different ball locations. per, we consider stochastic search algorithms for contextual Such relearning is undesired as it is highly inef- learning. Stochastic search algorithms [Hansen et al., 2003; ficient and many applications require a fast adap- Sun et al., 2009; Ruckstieߨ et al., 2008] optimize an objec- tation to a new context/situation.
    [Show full text]
  • Arxiv:2005.11011V2 [Quant-Ph] 11 Aug 2020 Error Is Fundamental to Measuring the Values on a Quan- Heuristics and Test Them Numerically on Example Prob- Tum Device
    Using models to improve optimizers for variational quantum algorithms Kevin J. Sung,1, 2, ∗ Jiahao Yao,3 Matthew P. Harrigan,1 Nicholas C. Rubin,1 Zhang Jiang,1 Lin Lin,3, 4 Ryan Babbush,1 and Jarrod R. McClean1, y 1Google Research, Venice, CA 2Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 3Department of Mathematics, University of California, Berkeley, CA 4Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA (Dated: August 13, 2020) Variational quantum algorithms are a leading candidate for early applications on noisy intermediate-scale quantum computers. These algorithms depend on a classical optimization outer- loop that minimizes some function of a parameterized quantum circuit. In practice, finite sampling error and gate errors make this a stochastic optimization with unique challenges that must be ad- dressed at the level of the optimizer. The sharp trade-off between precision and sampling time in conjunction with experimental constraints necessitates the development of new optimization strate- gies to minimize overall wall clock time in this setting. In this work, we introduce two optimization methods and numerically compare their performance with common methods in use today. The methods are surrogate model-based algorithms designed to improve reuse of collected data. They do so by utilizing a least-squares quadratic fit of sampled function values within a moving trusted region to estimate the gradient or a policy gradient. To make fair comparisons between optimization methods, we develop experimentally relevant cost models designed to balance efficiency in testing and accuracy with respect to cloud quantum computing systems. The results here underscore the need to both use relevant cost models and optimize hyperparameters of existing optimization meth- ods for competitive performance.
    [Show full text]
  • A Review of Simheuristics: Extending Metaheuristics to Deal with Stochastic Combinatorial Optimization Problems
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Juan, Angel A.; Faulin, Javier; Grasman, Scott E.; Rabe, Markus; Figueira, Gonçalo Article A review of simheuristics: Extending metaheuristics to deal with stochastic combinatorial optimization problems Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Juan, Angel A.; Faulin, Javier; Grasman, Scott E.; Rabe, Markus; Figueira, Gonçalo (2015) : A review of simheuristics: Extending metaheuristics to deal with stochastic combinatorial optimization problems, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 2, pp. 62-72, http://dx.doi.org/10.1016/j.orp.2015.03.001 This Version is available at: http://hdl.handle.net/10419/178250 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence.
    [Show full text]
  • Kafisto: a Kalman Filtering Framework for Stochastic Optimization
    KaFiStO: A Kalman Filtering Framework for Stochastic Optimization Aram Davtyan, Sepehr Sameni, Llukman Cerkezi, Givi Meishvilli, Adam Bielski, Paolo Favaro Computer Vision Group University of Bern faram.davtyan, sepehr.sameni, llukman.cerkezi givi.meishvilli, adam.bielski, [email protected] Abstract tic gradient descent (SGD) [61]. SGD owes its performance gains to the adoption of an approximate version of the ob- Optimization is often cast as a deterministic problem, jective function at each iteration step, which, in turn, yields where the solution is found through some iterative proce- an approximate or noisy gradient. dure such as gradient descent. However, when training neu- While SGD seems to benefit greatly (e.g., in terms of rate ral networks the loss function changes over (iteration) time of convergence) from such approximation, it has also been due to the randomized selection of a subset of the samples. shown that too much noise hurts the performance [83,7]. This randomization turns the optimization problem into a This suggests that, to further improve over SGD, one could stochastic one. We propose to consider the loss as a noisy attempt to model the noise of the objective function. We observation with respect to some reference optimum. This consider the iteration-time varying loss function used in interpretation of the loss allows us to adopt Kalman filter- SGD as a stochastic process obtained by adding the ex- ing as an optimizer, as its recursive formulation is designed pected risk to zero mean Gaussian noise. A powerful ap- to estimate unknown parameters from noisy measurements. proach designed to handle estimation with such processes Moreover, we show that the Kalman Filter dynamical model is Kalman filtering [36].
    [Show full text]
  • CMA-ES and Advanced Adaptation Mechanisms Youhei Akimoto, Nikolaus Hansen
    CMA-ES and Advanced Adaptation Mechanisms Youhei Akimoto, Nikolaus Hansen To cite this version: Youhei Akimoto, Nikolaus Hansen. CMA-ES and Advanced Adaptation Mechanisms. GECCO ’18 Companion: Proceedings of the Genetic and Evolutionary Computation Conference Companion, Jul 2018, Kyoto, Japan. hal-01959479 HAL Id: hal-01959479 https://hal.inria.fr/hal-01959479 Submitted on 18 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. CMA-ES and Advanced Adaptation Mechanisms Youhei Akimoto1 & Nikolaus Hansen2 1. University of Tsukuba, Japan 2. Inria, Research Centre Saclay, France [email protected] [email protected] Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author. GECCO '18 Companion, July 15–19, 2018, Kyoto, Japan © 2018 Copyright is held by the owner/author(s). ACM ISBN 978-1-4503-5764-7/18/07.
    [Show full text]