DOCSLIB.ORG

Visualizing and Understanding the Effectiveness of BERT

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Visualizing and Understanding the Effectiveness of BERT Visualizing and Understanding the Effectiveness of BERT Yaru Haoy∗, Li Dongz, Furu Weiz, Ke Xuy yBeihang University zMicrosoft Research fhaoyaru@,[email protected] flidong1,[email protected] Abstract is effective on downstream tasks in terms of both trainability and generalization capability. In this Language model pre-training, such as BERT, has achieved remarkable results in many NLP work, we take BERT (Devlin et al., 2018) as an tasks. However, it is unclear why the pre- example to understand the effectiveness of pre- training-then-fine-tuning paradigm can im- training. We visualize the loss landscapes and prove performance and generalization capabil- the optimization procedure of fine-tuning on spe- ity across different tasks. In this paper, we pro- cific datasets in three ways. First, we compute the pose to visualize loss landscapes and optimiza- one-dimensional (1D) loss curve, so that we can tion trajectories of fine-tuning BERT on spe- inspect the difference between fine-tuning BERT cific datasets. First, we find that pre-training reaches a good initial point across downstream and training from scratch. Second, we visualize tasks, which leads to wider optima and eas- the two-dimensional (2D) loss surface, which pro- ier optimization compared with training from vides more information about loss landscapes than scratch. We also demonstrate that the fine- 1D curves. Third, we project the high-dimensional tuning procedure is robust to overfitting, even optimization trajectory of fine-tuning to the ob- though BERT is highly over-parameterized for tained 2D loss surface, which demonstrate the downstream tasks. Second, the visualization learning properties in an intuitive way. results indicate that fine-tuning BERT tends to generalize better because of the flat and wide The main findings are summarized as follows. optima, and the consistency between the train- First, visualization results indicate that BERT pre- ing loss surface and the generalization error training reaches a good initial point across down- surface. Third, the lower layers of BERT are stream tasks, which leads to wider optima on the more invariant during fine-tuning, which sug- 2D loss landscape compared with random initial- gests that the layers that are close to input learn ization. Moreover, the visualization of optimiza- more transferable representations of language. tion trajectories shows that pre-training results in 1 Introduction easier optimization and faster convergence. We also demonstrate that the fine-tuning procedure is Language model pre-training has achieved strong robust to overfitting. Second, loss landscapes of performance in many NLP tasks (Peters et al., fine-tuning partially explain the good generaliza- 2018; Howard and Ruder, 2018a; Radford et al., tion capability of BERT. Specifically, pre-training 2018; Devlin et al., 2018; Baevski et al., 2019; obtains more flat and wider optima, which indi- Dong et al., 2019). A neural encoder is trained cates the pre-trained model tends to generalize bet- on a large text corpus by using language model- ter on unseen data (Chaudhari et al., 2017; Li et al., ing objectives. Then the pre-trained model either 2018; Izmailov et al., 2018). Additionally, we is used to extract vector representations for input, find that the training loss surface correlates well or is fine-tuned on the specific datasets. with the generalization error. Third, we demon- Recent work (Tenney et al., 2019b; Liu et al., strate that the lower (i.e., close to input) layers 2019a; Goldberg, 2019; Tenney et al., 2019a) of BERT are more invariant across tasks than the has shown that the pre-trained models can en- higher layers, which suggests that the lower layers code syntactic and semantic information of lan- learn transferable representations of language. We guage. However, it is unclear why pre-training verify the point by visualizing the loss landscape ∗Contribution during internship at Microsoft Research. with respect to different groups of layers. 4143 Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing, pages 4143–4152, Hong Kong, China, November 3–7, 2019. c 2019 Association for Computational Linguistics 2 Background: BERT where α is a scalar parameter, δ1 = θ1 − θ0 is the optimization direction, and J (θ) is the loss func- We use BERT (Bidirectional Encoder Represen- tion under the model parameters θ. In our experi- tations from Transformers; Devlin et al. 2018) as ments, we set the range of α to [−4; 4] and sample an example of pre-trained language models in our 40 points for each axis. Note that we only consider experiments. BERT is pre-trained on a large cor- the parameters of BERT in θ0 and θ1, so δ1 only pus by using the masked language modeling and indicates the updates of the original BERT param- next-sentence prediction objectives. Then we can eters. The effect of the added task-specific layers add task-specific layers to the BERT model, and is eliminated by keeping them fixed to the learned fine-tune all the parameters according to the down- values. stream tasks. BERT employs a Transformer (Vaswani et al., 3.2 Two-dimensional Loss Surface 2017) network to encode contextual information, The one-dimensional loss curve can be extended which contains multi-layer self-attention blocks. to the two-dimensional (2D) loss surface (Li et al., jxj Given the embeddings fxigi=1 of input text, we 2018). Similar as in Equation (1), we need to de- 0 concatenate them into H = [x1; ··· ; xjxj]. Then, fine two directions (δ1 and δ2) as axes to plot the an L-layer Transformer encodes the input: Hl = loss surface: l−1 Transformer blockl(H ), where l = 1; ··· ;L, L L L f(α; β) = J (θ0 + αδ1 + βδ2) (2) and H = [h1 ; ··· ; hjxj]. We use the hidden vec- L where α; β are scalar values, J (·) is the loss func- tor hi as the contextualized representation of the θ input token xi. For more implementation details, tion, and 0 represents the initialized parameters. we refer readers to Vaswani et al.(2017). Similar to Section 3.1, we are only interested in the parameter space of the BERT encoder, with- 3 Methodology out taking into consideration task-specific layers. One of the axes is the optimization direction δ1 = We employ three visualization methods to under- θ1 − θ0 on the target dataset, which is defined in stand why fine-tuning the pre-trained BERT model the same way as in Equation (1). We compute can achieve better performance on downstream the other axis direction via δ2 = θ2 − θ0, where tasks compared with training from scratch. We θ2 represents the fine-tuned parameters on another plot both one-dimensional and two-dimensional dataset. So the other axis is the optimization di- loss landscapes of BERT on the specific datasets. rection of fine-tuning on another dataset. Even Besides, we project the optimization trajectories though the other dataset is randomly chosen, ex- of the fine-tuning procedure to the loss surface. perimental results confirm that the optimization The visualization algorithms can also be used for directions δ1; δ2 are divergent and orthogonal to the models that are trained from random initial- each other because of the high-dimensional pa- ization, so that we can compare the difference be- rameter space. tween two learning paradigm. The direction vectors δ1 and δ2 are projected onto a two-dimensional plane. It is beneficial to 3.1 One-dimensional Loss Curve ensure the scale equivalence of two axes for visu- Let θ0 denote the initialized parameters. For fine- alization purposes. Similar to the filter normal- tuning BERT, θ0 represents the the pre-trained pa- ization approach introduced in (Li et al., 2018), rameters. For training from scratch, θ0 repre- we address this issue by normalizing two direc- sents the randomly initialized parameters. After tion vectors to the same norm. We re-scale δ2 fine-tuning, the model parameters are updated to to kδ1k δ , where k·k denotes the Euclidean norm. kδ2k 2 θ1. The one-dimensional (1D) loss curve aims to We set the range of both α and β to [−4; 4] and quantify the loss values along the optimization di- sample 40 points for each axis. rection (i.e., from θ0 to θ1). The loss curve is plotted by linear interpola- 3.3 Optimization Trajectory tion between θ0 and θ1 (Goodfellow and Vinyals, Our goal is to project the optimization trajectory of 2015). The curve function f(α) is defined as: the fine-tuning procedure onto the 2D loss surface α β T obtained in Section 3.2. Let f(di ; di )gi=1 de- f(α) = J (θ0 + αδ1) (1) note the projected optimization trajectory, where 4144 α β (di ; di ) is a projected point in the loss surface, 1e-5. For MNLI and SST-2, the batch size is 64, and i = 1; ··· ;T represents the i-th epoch of fine- and the learning rate is 3e-5. tuning. For the setting of training from scratch, we As shown in Equation (2), we have known the use the same network architecture as BERT, and optimization direction δ1 = θ1 − θ0 on the tar- randomly initialize the model parameters. Most get dataset. We can compute the deviation degrees hyper-parameters are kept the same. The num- between the optimization direction and the trajec- ber of training epochs is larger than fine-tuning tory to visualize the projection results. Let θi de- BERT, because training from scratch requires note the BERT parameters at the i-th epoch, and more epochs to converge. The number of epochs i i δ = θ − θ0 denote the optimization direction at is set to 8 for SST-2, and 16 for the other datasets, α β the i-th epoch.
Load more

Recommended publications
  • A Simple Algorithm for Semi-Supervised Learning with Improved Generalization Error Bound
    A Simple Algorithm for Semi-supervised Learning with Improved Generalization Error Bound Ming Ji∗‡ [email protected] Tianbao Yang∗† [email protected] Binbin Lin\ [email protected] Rong Jiny [email protected] Jiawei Hanz [email protected] zDepartment of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA yDepartment of Computer Science and Engineering, Michigan State University, East Lansing, MI, 48824, USA \State Key Lab of CAD&CG, College of Computer Science, Zhejiang University, Hangzhou, 310058, China ∗Equal contribution Abstract which states that the prediction function lives in a low In this work, we develop a simple algorithm dimensional manifold of the marginal distribution PX . for semi-supervised regression. The key idea It has been pointed out by several stud- is to use the top eigenfunctions of integral ies (Lafferty & Wasserman, 2007; Nadler et al., operator derived from both labeled and un- 2009) that the manifold assumption by itself is labeled examples as the basis functions and insufficient to reduce the generalization error bound learn the prediction function by a simple lin- of supervised learning. However, on the other hand, it ear regression. We show that under appropri- was found in (Niyogi, 2008) that for certain learning ate assumptions about the integral operator, problems, no supervised learner can learn effectively, this approach is able to achieve an improved while a manifold based learner (that knows the man- regression error bound better than existing ifold or learns it from unlabeled examples) can learn bounds of supervised learning. We also veri- well with relatively few labeled examples.
    [Show full text]
  • Statistical Mechanics Methods for Discovering Knowledge from Production-Scale Neural Networks
    Statistical Mechanics Methods for Discovering Knowledge from Production-Scale Neural Networks Charles H. Martin∗ and Michael W. Mahoney† Tutorial at ACM-KDD, August 2019 ∗Calculation Consulting, [email protected] †ICSI and Dept of Statistics, UC Berkeley, https://www.stat.berkeley.edu/~mmahoney/ Martin and Mahoney (CC & ICSI/UCB) Statistical Mechanics Methods August 2019 1 / 98 Outline 1 Prehistory and History Older Background A Very Simple Deep Learning Model More Immediate Background 2 Preliminary Results Regularization and the Energy Landscape Preliminary Empirical Results Gaussian and Heavy-tailed Random Matrix Theory 3 Developing a Theory for Deep Learning More Detailed Empirical Results An RMT-based Theory for Deep Learning Tikhonov Regularization versus Heavy-tailed Regularization 4 Validating and Using the Theory Varying the Batch Size: Explaining the Generalization Gap Using the Theory: pip install weightwatcher Diagnostics at Scale: Predicting Test Accuracies 5 More General Implications and Conclusions Outline 1 Prehistory and History Older Background A Very Simple Deep Learning Model More Immediate Background 2 Preliminary Results Regularization and the Energy Landscape Preliminary Empirical Results Gaussian and Heavy-tailed Random Matrix Theory 3 Developing a Theory for Deep Learning More Detailed Empirical Results An RMT-based Theory for Deep Learning Tikhonov Regularization versus Heavy-tailed Regularization 4 Validating and Using the Theory Varying the Batch Size: Explaining the Generalization Gap Using the Theory: pip install weightwatcher Diagnostics at Scale: Predicting Test Accuracies 5 More General Implications and Conclusions Statistical Physics & Neural Networks: A Long History 60s: I J. D. Cowan, Statistical Mechanics of Neural Networks, 1967. 70s: I W. A. Little, “The existence of persistent states in the brain,” Math.
    [Show full text]
  • Multitask Learning with Local Attention for Tibetan Speech Recognition
    Hindawi Complexity Volume 2020, Article ID 8894566, 10 pages https://doi.org/10.1155/2020/8894566 Research Article Multitask Learning with Local Attention for Tibetan Speech Recognition Hui Wang , Fei Gao , Yue Zhao , Li Yang , Jianjian Yue , and Huilin Ma School of Information Engineering, Minzu University of China, Beijing 100081, China Correspondence should be addressed to Fei Gao; [email protected] Received 22 September 2020; Revised 12 November 2020; Accepted 26 November 2020; Published 18 December 2020 Academic Editor: Ning Cai Copyright © 2020 Hui Wang et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we propose to incorporate the local attention in WaveNet-CTC to improve the performance of Tibetan speech recognition in multitask learning. With an increase in task number, such as simultaneous Tibetan speech content recognition, dialect identification, and speaker recognition, the accuracy rate of a single WaveNet-CTC decreases on speech recognition. Inspired by the attention mechanism, we introduce the local attention to automatically tune the weights of feature frames in a window and pay different attention on context information for multitask learning. *e experimental results show that our method improves the accuracies of speech recognition for all Tibetan dialects in three-task learning, compared with the baseline model. Furthermore, our method significantly improves the accuracy for low-resource dialect by 5.11% against the specific-dialect model. 1. Introduction recognition, dialect identification, and speaker recognition in a single model.
    [Show full text]
  • Lecture 9: Generalization
    Lecture 9: Generalization Roger Grosse 1 Introduction When we train a machine learning model, we don't just want it to learn to model the training data. We want it to generalize to data it hasn't seen before. Fortunately, there's a very convenient way to measure an algorithm's generalization performance: we measure its performance on a held-out test set, consisting of examples it hasn't seen before. If an algorithm works well on the training set but fails to generalize, we say it is overfitting. Improving generalization (or preventing overfitting) in neural nets is still somewhat of a dark art, but this lecture will cover a few simple strategies that can often help a lot. 1.1 Learning Goals • Know the difference between a training set, validation set, and test set. • Be able to reason qualitatively about how training and test error de- pend on the size of the model, the number of training examples, and the number of training iterations. • Understand the motivation behind, and be able to use, several strate- gies to improve generalization: { reducing the capacity { early stopping { weight decay { ensembles { input transformations { stochastic regularization 2 Measuring generalization So far in this course, we've focused on training, or optimizing, neural net- works. We defined a cost function, the average loss over the training set: N 1 X L(y(x(i)); t(i)): (1) N i=1 But we don't just want the network to get the training examples right; we also want it to generalize to novel instances it hasn't seen before.
    [Show full text]
  • Issues in Using Function Approximation for Reinforcement Learning
    To appear in: Proceedings of the Fourth Connectionist Models Summer School Lawrence Erlbaum Publisher, Hillsdale, NJ, Dec. 1993 Issues in Using Function Approximation for Reinforcement Learning Sebastian Thrun Anton Schwartz Institut fÈur Informatik III Dept. of Computer Science UniversitÈat Bonn Stanford University RÈomerstr. 164, D±53225 Bonn, Germany Stanford, CA 94305 [email protected] [email protected] Reinforcement learning techniques address the problem of learning to select actions in unknown, dynamic environments. It is widely acknowledged that to be of use in complex domains, reinforcement learning techniques must be combined with generalizing function approximation methods such as arti®cial neural networks. Little, however, is understood about the theoretical properties of such combinations, and many researchers have encountered failures in practice. In this paper we identify a prime source of such failuresÐnamely, a systematic overestimation of utility values. Using Watkins' Q-Learning [18] as an example, we give a theoretical account of the phenomenon, deriving conditions under which one may expected it to cause learning to fail. Employing some of the most popular function approximators, we present experimental results which support the theoretical ®ndings. 1 Introduction Reinforcement learning methods [1, 16, 18] address the problem of learning, through experimentation, to choose actions so as to maximize one's productivity in unknown, dynamic environments. Unlike most learning algorithms that have been studied in the ®eld of Machine Learning, reinforcement learning techniques allow for ®nding optimal action sequences in temporal decision tasks where the external evaluation is sparse, and neither the effects of actions, nor the temporal delay between actions and its effects on the learner's performance is known to the learner beforehand.
    [Show full text]
  • Effective Dimensionality Revisited
    Under review as a conference paper at ICLR 2021 RETHINKING PARAMETER COUNTING: EFFECTIVE DIMENSIONALITY REVISITED Anonymous authors Paper under double-blind review ABSTRACT Neural networks appear to have mysterious generalization properties when using parameter counting as a proxy for complexity. Indeed, neural networks often have many more parameters than there are data points, yet still provide good general- ization performance. Moreover, when we measure generalization as a function of parameters, we see double descent behaviour, where the test error decreases, increases, and then again decreases. We show that many of these properties become understandable when viewed through the lens of effective dimensionality, which measures the dimensionality of the parameter space determined by the data. We relate effective dimensionality to posterior contraction in Bayesian deep learning, model selection, width-depth tradeoffs, double descent, and functional diversity in loss surfaces, leading to a richer understanding of the interplay between parameters and functions in deep models. We also show that effective dimensionality compares favourably to alternative norm- and flatness- based generalization measures. 1 INTRODUCTION Parameter counting pervades the narrative in modern deep learning. “One of the defining properties of deep learning is that models are chosen to have many more parameters than available training data. In light of this capacity for overfitting, it is remarkable that simple algorithms like SGD reliably return solutions with low test error” (Dziugaite and Roy, 2017). “Despite their massive size, successful deep artificial neural networks can exhibit a remarkably small difference between training and test performance” (Zhang et al., 2017). “Increasing the number of parameters of neural networks can give much better prediction accuracy” (Shazeer et al., 2017).
    [Show full text]
  • Generalizing to Unseen Domains: a Survey on Domain Generalization
    PREPRINT 1 Generalizing to Unseen Domains: A Survey on Domain Generalization Jindong Wang, Cuiling Lan, Member, IEEE, Chang Liu, Yidong Ouyang, Wenjun Zeng, Fellow, IEEE, Tao Qin, Senior Member, IEEE Abstract—Machine learning systems generally assume that the Sketch Cartoon Art painting Photo training and testing distributions are the same. To this end, a key requirement is to develop models that can generalize to unseen distributions. Domain generalization (DG), i.e., out-of- distribution generalization, has attracted increasing interests in recent years. Domain generalization deals with a challenging setting where one or several different but related domain(s) are given, and the goal is to learn a model that can generalize to Training set Test set an unseen test domain. Great progress has been made in the area of domain generalization for years. This paper presents Fig. 1. Examples from the dataset PACS [1] for domain generalization. The the first review of recent advances in this area. First, we training set is composed of images belonging to domains of sketch, cartoon, provide a formal definition of domain generalization and discuss and art paintings. DG aims to learn a generalized model that performs well several related fields. We then thoroughly review the theories on the unseen target domain of photos. related to domain generalization and carefully analyze the theory behind generalization. We categorize recent algorithms into three classes: data manipulation, representation learning, and learning datasets) that will generalize well on unseen testing domains. strategy, and present several popular algorithms in detail for For instance, given a training set consisting of images coming each category.
    [Show full text]
  • Batch Policy Learning Under Constraints
    Batch Policy Learning under Constraints Hoang M. Le 1 Cameron Voloshin 1 Yisong Yue 1 Abstract deed, many such real-world applications require the primary objective function be augmented with an appropriate set of When learning policies for real-world domains, constraints (Altman, 1999). two important questions arise: (i) how to effi- ciently use pre-collected off-policy, non-optimal Contemporary policy learning research has largely focused behavior data; and (ii) how to mediate among dif- on either online reinforcement learning (RL) with a focus on ferent competing objectives and constraints. We exploration, or imitation learning (IL) with a focus on learn- thus study the problem of batch policy learning un- ing from expert demonstrations. However, many real-world der multiple constraints, and offer a systematic so- settings already contain large amounts of pre-collected data lution. We first propose a flexible meta-algorithm generated by existing policies (e.g., existing driving behav- that admits any batch reinforcement learning and ior, power grid control policies, etc.). We thus study the online learning procedure as subroutines. We then complementary question: can we leverage this abundant present a specific algorithmic instantiation and source of (non-optimal) behavior data in order to learn se- provide performance guarantees for the main ob- quential decision making policies with provable guarantees jective and all constraints. As part of off-policy on both primary objective and constraint satisfaction? learning, we propose a simple method for off- We thus propose and study the problem of batch policy policy policy evaluation (OPE) and derive PAC- learning under multiple constraints.
    [Show full text]
  • Learning Better Structured Representations Using Low-Rank Adaptive Label Smoothing
    Published as a conference paper at ICLR 2021 LEARNING BETTER STRUCTURED REPRESENTATIONS USING LOW-RANK ADAPTIVE LABEL SMOOTHING Asish Ghoshal, Xilun Chen, Sonal Gupta, Luke Zettlemoyer & Yashar Mehdad {aghoshal,xilun,sonalgupta,lsz,mehdad}@fb.com Facebook AI ABSTRACT Training with soft targets instead of hard targets has been shown to improve per- formance and calibration of deep neural networks. Label smoothing is a popular way of computing soft targets, where one-hot encoding of a class is smoothed with a uniform distribution. Owing to its simplicity, label smoothing has found wide- spread use for training deep neural networks on a wide variety of tasks, ranging from image and text classification to machine translation and semantic parsing. Complementing recent empirical justification for label smoothing, we obtain PAC- Bayesian generalization bounds for label smoothing and show that the generaliza- tion error depends on the choice of the noise (smoothing) distribution. Then we propose low-rank adaptive label smoothing (LORAS): a simple yet novel method for training with learned soft targets that generalizes label smoothing and adapts to the latent structure of the label space in structured prediction tasks. Specifi- cally, we evaluate our method on semantic parsing tasks and show that training with appropriately smoothed soft targets can significantly improve accuracy and model calibration, especially in low-resource settings. Used in conjunction with pre-trained sequence-to-sequence models, our method achieves state of the art performance on four semantic parsing data sets. LORAS can be used with any model, improves performance and implicit model calibration without increasing the number of model parameters, and can be scaled to problems with large label spaces containing tens of thousands of labels.
    [Show full text]
  • Learning Universal Graph Neural Network Embeddings with Aid of Transfer Learning
    Learning Universal Graph Neural Network Embeddings With Aid Of Transfer Learning Saurabh Verma Zhi-Li Zhang Department of Computer Science Department of Computer Science University of Minnesota Twin Cities University of Minnesota Twin Cities [email protected] [email protected] Abstract Learning powerful data embeddings has become a center piece in machine learn- ing, especially in natural language processing and computer vision domains. The crux of these embeddings is that they are pretrained on huge corpus of data in a unsupervised fashion, sometimes aided with transfer learning. However currently in the graph learning domain, embeddings learned through existing graph neural networks (GNNs) are task dependent and thus cannot be shared across different datasets. In this paper, we present a first powerful and theoretically guaranteed graph neural network that is designed to learn task-independent graph embed- dings, thereafter referred to as deep universal graph embedding (DUGNN). Our DUGNN model incorporates a novel graph neural network (as a universal graph encoder) and leverages rich Graph Kernels (as a multi-task graph decoder) for both unsupervised learning and (task-specific) adaptive supervised learning. By learning task-independent graph embeddings across diverse datasets, DUGNN also reaps the benefits of transfer learning. Through extensive experiments and ablation studies, we show that the proposed DUGNN model consistently outperforms both the existing state-of-art GNN models and Graph Kernels by an increased accuracy of 3% − 8% on graph classification benchmark datasets. 1 Introduction Learning powerful data embeddings has become a center piece in machine learning for producing superior results. This new trend of learning embeddings from data can be attributed to the huge success of word2vec [31, 35] with unprecedented real-world performance in natural language processing (NLP).
    [Show full text]
  • 1 Introduction 2 the Learning Problem
    9.520: Statistical Learning Theory and Applications February 8th, 2010 The Learning Problem and Regularization Lecturer: Tomaso Poggio Scribe: Sara Abbasabadi 1 Introduction Today we will introduce a few basic mathematical concepts underlying conditions of the learning theory and generalization property of learning algorithms. We will dene empirical risk, expected risk and empirical risk minimization. Then regularization algorithms will be introduced which forms the bases for the next few classes. 2 The Learning Problem Consider the problem of supervised learning where for a number of input data X, the output labels Y are also known. Data samples are taken independently from an underlying distribution µ(z) on Z = X × Y and form the training set S: (x1; y1);:::; (xn; yn) that is z1; : : : ; zn. We assume independent and identically distributed (i.i.d.) random variables which means the order does not matter (this is the exchangeability property). Our goal is to nd the conditional probability of y given a data x. µ(z) = p(x; y) = p(yjx) · p(x) p(x; y) is xed but unknown. Finding this probabilistic relation between X and Y solves the learning problem. XY P(y|x) P(x) For example suppose we have the data xi;Fi measured from a spring. Hooke's law F = ax gives 2 2 p(F=x) = δ(F −ax). For additive Gaussian noise Hooke's law F = ax gives p(F=x) = e−(F −ax) =2σ . 2-1 2.1 Hypothesis Space Hypothesis space H should be dened in every learning algorithm as the space of functions that can be explored.
    [Show full text]
  • Principles and Algorithms for Forecasting Groups of Time Series: Locality and Globality
    Principles and Algorithms for Forecasting Groups of Time Series: Locality and Globality Pablo Montero-Manso Rob J Hyndman Discipline of Business Analytics Department of Econometrics and Business Statistics University of Sydney Monash University Darlington, NSW 2006, Australia Clayton, VIC 3800, Australia [email protected] [email protected] March 30, 2021 Abstract Forecasting of groups of time series (e.g. demand for multiple products offered by a retailer, server loads within a data center or the number of completed ride shares in zones within a city) can be approached locally, by considering each time series as a separate regression task and fitting a function to each, or globally, by fitting a single function to all time series in the set. While global methods can outperform local for groups composed of similar time series, recent empirical evidence shows surprisingly good performance on heterogeneous groups. This suggests a more general applicability of global methods, potentially leading to more accurate tools and new scenarios to study. However, the evidence has been of empirical nature and a more fundamental study is required. Formalizing the setting of forecasting a set of time series with local and global methods, we provide the following contributions: • We show that global methods are not more restrictive than local methods for time series forecasting, a result which does not apply to sets of regression problems in general. Global and local methods can produce the same forecasts without any assumptions about similarity of the series in the set, therefore global models can succeed in a wider range of problems than previously thought.
    [Show full text]