Decentralized Online Learning: Take Benefits from Others’ Data without Sharing Your Own to Track Global Trend Yawei Zhao∗, 2 Chen Yu∗ , 3 Peilin Zhao, 2 Hanlin Tang, 4 Shuang Qiu, and 2 Ji Liu 1 National University of Defense Technology, Changsha, China 2 University of Rochester, Rochester, USA 3 Tencent AI Lab, Shenzhen, China zhaoyawei@nudt.edu.cn, yuchen92328@gmail.com, peilinzhao@hotmail.com, htang14@ur.rochester.edu, ji.liu.uwisc@gmail.com arXiv:1901.10593v4 [cs.LG] 28 May 2019 1,2 May 30, 2019 Abstract Decentralized Online Learning (online learning in decentralized networks) attracts more and more attention, since it is believed that Decentralized Online Learning can help the data providers cooperatively better solve their online problems without sharing their private data to a third party or other providers. Typically, the cooperation is achieved by letting the data providers exchange their models between neighbors, e.g.,recommendation model. However, the best regret bound for a decentralized online learning √ algorithm is O n T , where n is the number of nodes (or users) and T is the number of iterations. This is clearly insignificant since this bound can be achieved without any communication in the networks. This reminds us to ask a fundamental question: Can people really get benefit from the decentralized online learning by exchanging information? In this paper, we studied when and why the communication can help the decentralized online learning to reduce the regret. Specifically, each loss function is characterized by two components: the adversarial component and the stochastic component. Under √ this characterization, we show that decentralized online gradient (DOG) enjoys a regret bound O n2 T G2 + nT σ 2 , where G measures the magnitude of the adversarial component in the private data (or equivalently the local loss function) and σ measures the randomness within the private data. This regret suggests that people can get benefits from the randomness in the private data by exchanging private information. Another important contribution of this paper is to consider the dynamic regret – a more practical regret to track users’ interest dynamics. Empirical studies are also conducted to validate our analysis. 1 Introduction Decentralized online learning receives extensive attentions in recent years [Shahrampour and Jadbabaie, 2018, Kamp et al., 2014, Koppel et al., 2018, Zhang et al., 2018a, 2017b, Xu et al., 2015, Akbari et al., 2017, Lee et al., 2016, Nedić et al., 2015, Lee et al., 2018, Benczúr et al., 2018, Yan et al., 2013]. It assumes that computational nodes in a network can communicate between neighbors to minimize an overall cumulative regret. Each computational node, which could be a user in practice, will receive a stream of online losses that are usually determined by a sequence of examples that arrive sequentially. Formally, we can denote fi,t as the loss received by the i-th computational node among the networks at the t-th iteration. The goal of decentralized online learning usually is to minimize its static regret, which is defined as the difference between the cumulative loss (the sum of all the online loss over all the nodes and steps ) suffered by the learning algorithm and that of the best model which can observe all the loss functions beforehand. ∗ Equal contribution. 1 Decentralized online learning attracts more and more attentions recently, mainly because it is believed by the community that it enjoys the following advantages for real-world large-scale applications: • (Utilize all computational resource) It can utilize the computational resource (of edging devices) by avoiding collecting all the loss functions (or equivalently data) to one central node and putting all computational burden on a single node. • (Protect data privacy) It can help many data providers collaborate to better minimize their cumulative loss, while at the same time protecting the data privacy as much as possible. However, the current theoretical study does not explain why people need to use decentralized online √ learning, since the currently best regret result for decentralized online learning O n T for convex loss functions [Hosseini et al., 2013, Yan et al., 2013]) is equal to the overall regret if each node (user) only runs local online gradient without any communication with others1 . Therefore, this reminds us to ask a fundamental question: Can people really get benefit with respect to the regret from the decentralized online learning by exchanging information? In this paper, we mainly study when can the communication really help decentralized online learning to minimize its regret. Specifically, we distinguish two components in the loss function fi,t : the adversary component and the √ stochastic compoent. Then we prove that decentralized online gradient can achieve a static regret bound of O n2 T G2 + nT σ 2 (G represents the bound of gradient. σ measures the randomness of the private data), where the first component of the bound is due to the adversary loss while the second component is due to the stochastic loss. Moreover, if a dynamic sequence of models with pa budget M is used as the reference points, the dynamic regret of the decentralized online gradient is O (n2 T G2 + nT σ) (M + 1) . The dynamic regret is a more suitable performance metric for real-world applications where the optimal model changes over time, such as people’s favorite style of pop musics usually change along with time as the global environment. This shows the communication can help to minimize the stochastic losses, rather than the adversary losses. This result is further verified empirically by extensive experiments. Notations. In the paper, we make the following notations. • For any i ∈ [n] and t ∈ [T ], the random variable ξi,t is subject to a distribution Di,t , that is, ξi,t ∼ Di,t . A set of random variables Ξn,T and their corresponding distributions are defined by Ξn,T = {ξi,t }1≤i≤n,1≤t≤T , Dn,T = {Di,t }1≤i≤n,1≤t≤T , respectively. For math brevity, we use the notation Ξn,T ∼ Dn,T to represent that ξi,t ∼ Di,t holds for any i ∈ [n] and t ∈ [T ]. E represents mathematical expectation. • ‘∇’ represents gradient operator. ‘k·k’ represents the `2 norm in default. ‘.’ represents “less than equal up to a constant factor". ‘A’ represents the set of all online algorithms. ‘1’ and ‘0’ represent all the elements of a vector is 1 and 0, respectively. 2 Related work Online learning has been studied √ for decades of years. An online convex optimization method can achieve a static regret bound of order O T and O (log T ) for convex and strongly convex loss functions, respectively [Hazan, 2016, Shalev-Shwartz, 2012, Bubeck, 2011]. Decentralized online learning. Online learning in a decentralized network has been studied in [Shahrampour and Jadbabaie, 2018, Kamp et al., 2014, Koppel et al., 2018, Zhang et al., 2018a, 2017b, Xu et al., 2015, Akbari et al., 2017, Lee et al., 2016, Nedić et al., 2015, Lee et al., 2018, Benczúr et al., 2018, √ is the number of nodes or users and T is the total number of iterations. The regret of an online algorithm is O T for √ convex loss functions [Hazan, 2016, Shalev-Shwartz, 2012]. Therefore, the overall regret is n T if all users do not communicate. 1n 2 Yan et 2013]. al., Shahrampour and Jadbabaie [2018] provides a dynamic regret (defined in Eq. (1)) bound √ of O n nT M for decentralized online mirror descent, where n, T , and M represent the number of nodes in the newtork, the number of iterations, and the budget of dynamics, respectively. When the Bregman divergence in the decentralized online mirror descent is chosen appropriately, the decentralized online mirror descent becomes identical tothe decentralized online gradient descent. In this paper, we achieve a better √ dynamic regret bound of O n T M for a decentralized online gradient descent method, which mainly benefits from a better of network error (see Lemma 5). Moreover, Kamp et al. [2014] presents a √ bound static regret of O nT for decentralized online prediction. However, it assumes that all the loss functions are generated from an unknown identical distribution, this assumption is too strong to be practical in the dynamic environment and be applied for a general online learning task. Additionally, many decentralized online optimization methods are proposed, for example, decentralized online multi-task learning [Zhang et al., 2018a], decentralized online ADMM [Xu et al., 2015], decentralized online gradient descent [Akbari et al., 2017], decentralized continuous-time online saddle-point method [Lee et al., 2016], decentralized online Nesterov’s primal-dual method [Nedić et al., 2015, Lee et al., 2018], and online distributed dual √averaging [Hosseini et al., 2013]. However, these previous work only studied the static regret bounds O T of the decentralized online learning algorithms, while they did not provide any theoretical analysis for dynamic environments. Besides, Yan et al. [2013] provides necessary and sufficient conditions to preserve privacy for decentralized online learning methods, which be studied to extend our method in our future work. Dynamic regret. The dynamic regret of online learning algorithms has been widely studied for decades [Zinkevich, 2003, Hall and Willett, 2015, 2013, Jadbabaie et al., 2015, Yang et al., 2016, Bedi et al., 2018, Zhang et al., 2017a, Mokhtari et al., 2016, Zhang et al., 2018c, György and Szepesvári, 2016, Wei PT et al., 2016, Zhao et al., 2018]. The first dynamic regret is defined as t=1 (ft (xt ) − ft (x∗t )) subject to PT −1 ∗ xt+1 − x∗t ≤ M where M is a budget for the change over the reference points [Zinkevich, 2003]. For t=1 √ √ this definition, the online gradient descent can achieve a dynamic regret of O T M + T , by selecting an appropriate learning rate. Later, other types of dynamic regrets are also introduced, by using different types of reference points. For example, Hall and Willett [2015, 2013] choose the reference points {x∗t }Tt=1 PT −1 satisfying t=1 x∗t+1 − Φ(x∗t ) ≤ M , where Φ(x∗t ) is the predictive optimal model. When the function Φ predicts accurately, the budget M can decrease significantly so that the dynamic regret effectively decreases. Jadbabaie et al. [2015], Yang et al. [2016], Bedi et al. [2018], Zhang et al. [2017a], Mokhtari et al. [2016], Zhang et al. [2018c] chooses the reference points {yt∗ }Tt=1 with yt∗ = argminz∈X ft (z), where ft is the loss function atthe t-th iteration. György and Szepesvári [2016] provides a new analysis framework, which √ achieves O T M dynamic regret2 for all the above reference points. Recently, the lower bound of the √ dynamic regret was shown to be Ω T M [Zhang et al., 2018b, Zhao et al., 2018], which indicates that the above algorithms are optimal in terms of dynamic regret. In this paper, we propose a new definition of dynamic regret, which covers all the previous ones as special cases. In some literatures, the regret in a dynamic environment is measured by the number of changes of the reference points over time, which is usually termed as shifting regret or tracking regret [Herbster and Warmuth, 1998, György et al., 2005, Gyorgy et al., 2012, György and Szepesvári, 2016, Mourtada and Maillard, 2017, Adamskiy et al., 2016, Wei et al., 2016, Cesa-Bianchi et al., 2012, Mohri and Yang, 2018, Jun et al., 2017]. Both the shifting regret and the tracking regret are usually studied in the setting of “learning with expert advice" while the dynamic regret is more often studied in the general setting of online learning. 2 György and Szepesvári [2016] uses the notation of “shifting regret" instead of “dynamic regret". In the paper, we keep using “dynamic regret" as used in most previous literatures. 3 3 Problem formulation In decentralized online learning, the topological structure of the network can be represented by an undirected graph G = (nodes:[n], edges:E) with vertex set [n] = {1, . . . , n} and edge set E ⊂ [n] × [n]. In real applications, each node i ∈ [n] is associated with a separate learner, for example an mobile device of one user, which maintains a local predictive model. Users would like to cooperatively better minimize their regret without sharing their private data. They typically share private models to their neighbors (or friends), which are directly adjacent nodes in G for each node. Let xi,t denote the local model for user i at iteration t. In iteration t user i predicts the local model xi,t for an unknown loss, and then receives the loss fi,t (·; ξi,t ). As a result, the decentralized online learning algorithm suffered a instantaneous loss fi,t (xi,t ; ξi,t ). ξi,t ’s are independent to each other in terms of i and t, charactering the stochastic component in the function fi,t (·; ξi,t ), while the subscripts i and t of f indicate the adversarial component, for example, the user’s profile, location, local time, and etc. The stochastic component in the function is usually caused by the potential relation among local models. For example, users’ perference to music may be impacted by a popular trend in the Internet at the same time. To measure the efficacy of a decentralized online learning algorithm A ∈ A, a commonly used performance measure is the static regret which is defined as # " n T XX A ∗ e R := (fi,t (xi,t ; ξi,t ) − fi,t (x ; ξi,t ) , E T Ξn,T ∼Dn,T Pn i=1 t=1 PT where x∗ = arg minx EΞn,T ∼Dn,T i=1 t=1 fi,t (x; ξi,t ). The static regret essentially assumes that the optimal model would not change over time. However, in many practical online learning application scenarios, the optimal model may evolve over time. For example, when we want to conduct music recommendation to a user, user’s preference to music may change over time as his/her situation. Thus, the optimal model x∗ should change over time. It leads to the dynamics of the optimal recommendation model. Therefore, for any online algorithm A ∈ A, we choose to use the dynamic regret as the metric: " n T # " n T # XX XX A ∗ RT := fi,t (xi,t ; ξi,t ) − min fi,t (xt ; ξi,t ) , (1) E E Ξn,T ∼Dn,T T T {x? t }t=1 ∈LM Ξn,T ∼Dn,T i=1 t=1 where LTM is defined by LTM = {zt }Tt=1 : TP −1 t=1 i=1 t=1 kzt+1 − zt k ≤ M . LTM restricts how much the optimal model eA may change over time. Obviously, RA T degenerates to RT when M = 0. 4 Decentralized Online Gradient (DOG) algorithm In the section, we introduce the DOG algorithm, followed by the analysis for the dynamic regret. Algorithm 1 DOG: Decentralized Online Gradient method. Require: Learning rate η, number of iterations T , and the confusion matrix W. 1: Initialize xi,1 = 0 for all i ∈ [n]; 2: for t = 1, 2, ..., T do 3: // For all users (say the i-th node i ∈ [n]) 4: Query the neighbors’ local models {xj,t }j∈user i’s neighbor set ; 5: Apply the local model and suffer loss fi,t (xi,t ; ξi,t ); 6: Compute the gradient ∇fi,t (xi,t ; ξi,tP ); 7: Update the local model by xi,t+1 = j∈user i’s neightbours Wi,j xj,t − η∇fi,t (xi,t ; ξi,t ); 8: end for 4 4.1 Algorithm description In the DOG algorithm, users exchange their local models periodically. In each iteration, each user runs the following steps: 1) Query the local models from his/her all neighbors; 2) Apply the local model to fi,t (·; ξi,t ) and compute the gradient; 3) Update the local model by taking average with neighbors’ models followed by a gradient step. The detailed description of the DOG algorithm can be found in Algorithm 1. W ∈ Rn×n is the confusion matrix defined on an undirected graph G = (nodes:[n], edges:E). It is assumed to be a doubly stochastic matrix [Wu et al., 2018, Zeng and Yin, 2018, Yuan et al., 2016], but not necessarily symmetric. Given a decentralized network G, there are many approaches to obtain a doubly stochastic W, for example, Sinkhorn matrix scaling [Knight, 2008]. The following are two naive ways to construct such a doubly stochastic matrix W. Let Ni be the number of user i’s neighbors (exclusive itself), and Nmax := maxi Ni . We can obtain a doubly stochastic matrix by: 1) Wi,j = 0, if i and j are not connected (i 6= j) in E; 2) Wi,j = n1 , if j are i are connected (i 6= j) in E; 3) Wi,i = 1 − Nni . When Nmax := maxi∈[n] Ni is known, an 1 alternative method is: 1) Wi,j = 0, if i and j are not connected (i 6= j) in E; 2) Wi,j = Nmax +1 , if j are i Ni are connected (i 6= j) in E; 3) Wi,i = 1 − Nmax +1 . To take a closer look at the algorithm’s updating rule, we Pn Pn define x̄t = n1 i=1 xi,t . It is not hard to verify that x̄t+1 = x̄t − nη i=1 ∇fi,t (xi,t ; ξi,t ) The detailed proofs are provided in Lemma 1 (See Supplemental Materials.) 4.2 Dynamic regret Analysis Next we show the dynamic regret of DOG in the following. Before that, we first make some common assumptions used in our analysis. Assumption 1. We make following assumptions throughout this paper: • For any i ∈ [n], t ∈ [T ], and x, there exist constants G and σ such that Eξi,t ∼Di,t ∇fi,t (x; ξi,t ) ≤ G, and 2 Eξi,t ∼Di,t ∇fi,t (x; ξi,t ) − Eξi,t ∼Di,t ∇fi,t (x) ≤ σ 2 . 2 • For given vectors x and y, we assume kx − yk ≤ R. Besides, for any i ∈ [n] and t ∈ [T ], we assume fi,t is convex, and has L-Lipschitz gradient. • Let W be doubly stochastic and ρ := W − 11> n . Assume ρ < 1. G essentially gives the upper bound for the adversarial component in fi,t (·; ξi,t ). The stochastic component is bounded by σ 2 . Note that if there is no stochastic component, G is nothing but the upper bound of the gradient like the setting in many online learning literature. It is important for our analysis to split these two components, which will be clear very soon. The last assumption about W is an essential assumption for the decentralized setting. The largest eigenvalue for a doubly stochastic matrix is 1. 1 − ρ measures how fast the information can propagate within the network (the larger the faster). Now we are ready to present the dynamic regret for DOG. 2 2 2 2 2 +σ ) 4L (G +σ ) Theorem 1. Denote constants C0 , C1 , and C2 by C0 := 2L(G , C2 := 2 + (1−ρ)2 , C1 := (1−ρ)2 Choosing η > 0 in Algorithm 1, under Assumption 1 we have n √ RDOG ≤ ηT σ 2 + C0 nT η 2 + C1 nT η 3 + 4 RM + R + C2 nηT G2 . T 2η By choosing an appropriate learning rate η, we obtain sublinear regret as follows. r √ (1−ρ)(nM R+nR) Corollary 1. Choosing η = in Algorithm 1, under Assumption 1 we have 2 nT G +T σ 2 RDOG T v u √ √ √ 5 u T M + R (n2 G2 + nσ 2 ) n2 M + R 2 n M + R t . + + p . 1−ρ 1−ρ (1 − ρ)T 5 1 1−ρ . For simpler discussion, let us treat R, G, and 1 − ρ as constants. The dynamic regret can be simp plified into O (n2 T G2 + nT σ 2 ) (M + 1) . If M = 0, the dynamic regret degenerates the static regret √ O n2 T G2 + nT σ 2 . More discussion is conducted in the following aspects. • (Tightness.) To see the tightness, we consider a few special cases: – (σ = 0 and n = 1.) It degenerates √ to the vanilla online learning setting but with dynamic regret. The implied static regret O T M is consistent with the dynamic regret result in Zhao et al. [2018], which is proven to be optimum. √ √ – (M = 0.) It degenerates to the static regret O n2 T G2 + nT σ 2 . When G < σ/ n, that is, the √ stochastic component dominates the adversarial component, the static regret O nT σ implies the √ √ 1 DOG average regret n RT to be σ T / n, and the convergence rate (from the stochastic optimization √ perspective) to be σ/ nT . It is consistent with the ergodic convergence rate in Tang et al. [2018], and √ improves the non-ergodic result σ/ T in Nedic and Ozdaglar [2009]3 . √ • (Insight.) Setting M = 0, we obtain the static regret O n2 T G2 + nT σ 2 . Consider the baseline that all users do not communicate but only √ run local online gradient. It is not hard to verify that the static regret for this baseline approach is O n2 T G2 + n2 T σ 2 . Comparing with the baseline, the improvement of our new bound is only on the stochastic component. Denote that G measures the magnitude of the adversarial component and σ measures the stochastic component. This result reveals an important observation that the communication does not really help improve the adversarial component, only the stochastic component can benefit from the communication. This observation makes quite sense, since if the users’ private data are totally arbitrary, there is no reason they can benefit to each other by exchanging anything. • (Improve existing dynamic regret in decentralized setting.) If we also treat G and σ to be √ constants, our analysis leads to a dynamic regret O n M T . Shahrampour and Jadbabaie [2018] only 3√ considers the adversary loss, and provides O n 2 M T regret for DOG. Compared with their result, our regret enjoys the state-of-the-art dependence on T and M , and meanwhile improves the dependence on n. Next we discuss how close all local models xi,t ’s close to their average at each time. The following result suggests that xi,t ’s are getting closer and closer over iterations. r √ Pn (1−ρ)(nM R+nR) 1 in Algorithm 1, under Assumption Theorem 2. Denote x̄t = n i=1 xi,t . Choosing η = 2 nT G +T σ 2 1 we have 1 nT " E Ξn,T ∼Dn,T T n X X # kxi,t − x̄t k i=1 t=1 2 √ n(M + R) . . (1 − ρ)T The result suggests that xi,t approaches to x̄t roughly in the O (1/T ) (treat M and ρ as constants.), rate √ which is faster than the convergence of the averaged regret O 1/ T from Corollary 1. 5 Empirical studies We consider online logistic regression with squared `2 norm regularization. In the task, fi,t (x; ξi,t ) = γ 2 −3 log 1 + exp(−yi,t A> is a given hyper-parameter. ξi,t is the randomness of the i,t x) + 2 kxk , where γ = 10 function fi,t , which is caused by the randomness of data in the experiment. Under this setting, we compare the performance of the proposed Decentralized Online Gradient method (DOG) and that of the Centralized Online Gradient method (COG). 3 See the non-ergodic convergent rate in Proposition 3(b) in Nedic and Ozdaglar [2009]. 6 Local OGD (M=10) DOG(M=10) 0.67 0.665 200 400 600 800 1000 T (a) β = 0.9 0.62 0.6 0.58 Local OGD (M=10) DOG(M=10) 0.56 200 400 600 800 1000 0.5 0.45 Local OGD (M=10) DOG(M=10) 0.4 200 400 600 800 1000 T T (b) β = 0.7 (c) β = 0.5 Average loss 0.675 Average loss 0.68 Average loss Average loss 0.55 0.685 0.4 0.3 Local OGD (M=10) DOG(M=10) 0.2 200 400 600 800 1000 T (d) β = 0.3 Figure 1: Local OGD vs. DOG on synthetic data with different ratios of the adversarial component. (10000 nodes with ring topology) The dynamic budget M is fixed as 10 to determine the space of reference points. The learning rate η is tuned to be for each dataset separately. We evaluate the learning performance measuring the average loss: Poptimal Pn byP n PT T 1 ∗ f (x ; ξ ), instead of using the dynamic regret E Ξn,T ∼Dn,T i=1 t=1 i,t i,t i,t i=1 t=1 (fi,t (xi,t ; ξi,t ) − fi,t (xt )) nT ∗ T directly, since the optimal reference point {xt }t=1 is the same for both DOG and COG. We perform emperical evaluation on a toy dataset and several real-world datasets, whose details are presented as follows. Synthetic Data. For the i-th node, a data matrix Ai ∈ R10×T is generated, s.t. Ai = β Ãi + (1 − β)Âi , where Ãi represents the adversary part of data, and Âi represents the stochastic part of data. β with 0 < β < 1 is used to make balance between the adversary and stochastic components. A large β represents the adversary component is significant, and the stochastic component becomes significant with the decrease of β. Specifically, elements of Ãi is uniformly sampled from the interval [−0.5 + sin(i), 0.5 + sin(i)]. Note that Ãi and Ãj with i 6= j are drawn from different distributions. Âi,t is generated according to yi,t ∈ {1, −1} which is generated uniformly. When yi,t = 1, Âi,t is generated by sampling from a time-varying distribution N ((1 + 0.5 sin(t)) · 1, I). When yi,t = −1, Âi,t is generated by sampling from another time-varying distribution N ((−1 + 0.5 sin(t)) · 1, I). Due to this correlation, yi,t can be considered as the label of the instance Âi,t . Real Data. The real public datasets include SUSY 4 (5, 000, 000 samples), room-occupancy 5 (20, 560 samples), usenet2 6 (1, 500 samples), and spam 7 (9, 324 samples). SUSY is a large-scale binary classification dataset, and we use the whole dataset to general two kinds of data: the stochastic data and the adversarial data. The stochastic data is generated by using some instances, e.g., 80% of the whole dataset, and then allocating them to nodes randomly and uniformly. The adversarial data is generated by using the other instances, conducting clustering on those data to yield n clusters, and then allocate every cluster to a node. room-occupancy is a time-series dataset, which is from a natural dynamic enviroment. Both usenet2 and spam are “concept drift” [Katakis et al., 2010] datasets, for which the optimal model changes over time. For all dataset, all values of every feature have been normalized to be zero mean and one variance. We present the numerical results for the dataset SUSY here, and place other results in supplementary materials. DOG is effective to reduce the stochastic component of regret. It is compared with the local online gradient descent (Local OGD), where every node trains a local model without communication with others. We vary β to generate different kinds of synthetic data, to obtain different balance between adversary and stochastic components. As shown before, the stochastic component of data becomes significant with the decrease of β. Figure 1 shows that DOG becomes significantly effective to reduce the stochastic component of regret for small β. It validates that exchanging models in a decentralized network is necessary and important to reduce regret, which matches with our theoretical result. In the following empirical studies, we generate synthetic data in the setting of β = 0.1. DOG yields comparable performance with COG. Figure 2 summarizes the performance of DOG compared with COG. For the synthetic dataset, we simulate a network consisting of 10000 nodes, where every node is randomly connected with other 15 nodes. Similarly, we simulate a ring network for the other real 4 https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/binary.html#SUSY 5 https://archive.ics.uci.edu/ml/datasets/Occupancy+Detection+ 6 http://mlkd.csd.auth.gr/concept_drift.html 7 http://mlkd.csd.auth.gr/concept_drift.html 7 0.2 500 1000 1500 0.58 0.56 1000 2000 T 3000 0.68 DOG(M=10) COG(M=10) 0.67 Average loss 0.4 DOG(M=10) COG(M=10) 0.6 Average loss Average loss Average loss DOG(M=10) COG(M=10) 0.6 0.66 0.65 1000 5000 3000 DOG(M=10) COG(M=10) 0.68 0.66 0.64 1000 5000 3000 5000 T T T (a) synthetic, 10000 nodes, (b) SUSY, 100% stochastic (c) SUSY, 80% stochastic data (d) SUSY, 50% stochastic data β = 0.1, random topology data n=5x10 3 , =2 n=1x10 4 , =2 2 1 200 600 n=1000 n=800 n=600 n=500 0.6 0.58 0.56 1000 1000 T 3000 n=1000 n=800 n=600 n=500 0.68 0.66 Average loss n=1x10 4 , =1 Average loss n=5x10 3 , =1 3 Average loss Average loss Figure 2: DOG vs. COG (1000 nodes with ring topolgy for real data). 0.64 5000 1000 T 3000 n=1000 n=800 n=600 n=500 0.68 0.67 0.66 0.65 5000 1000 T 3000 5000 T (a) synthetic, β = 0.1, random (b) SUSY, 100% stochastic (c) SUSY, 80% stochastic data, (d) SUSY, 50% stochastic data, topology data, ring topology ring topology ring topology 1 500 0.58 0.56 1000 1500 0.67 Disconnected WattsStrog(0.5) WattsStrog(1) Ring Fully connected 0.66 0.65 Disconnected WattsStrog(0.5) WattsStrog(1) Ring Fully connected 0.68 0.67 0.66 0.65 0.64 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 T 0.68 Average loss 2 Disconnected WattsStrog(0.5) WattsStrog(1) Ring Fully connected 0.6 Average loss Disconnected WattsStrog(0.5) WattsStrog(1) Ring Fully connected 3 Average loss Average loss Figure 3: The robustness of DOG wrt the network size. 1000 2000 3000 4000 5000 T T T (a) synthetic, 10000 nodes, (b) SUSY, 1000 nodes, 100% (c) SUSY, 1000 nodes, 80% (d) SUSY, 1000 nodes, 50% β = 0.1 stochastic data stochastic data stochastic data Figure 4: The sensitivity of DOG wrt the topology of the network. datasets, and every node is randomly connected with other 3 nodes. Under these settings, we can observe that both DOG and COG are effective for the online learning tasks on all the datasets. Besides, DOG achieves slightly worse performance than COG. It is significant when the adversarial component becomes large, which can be verified by Figures 2(c) and 2(d). The performance of DOG is robust to the network size, but is sensitive to the variance of the stochastic data. Figure 3 summarizes the effect of the network size on the performance of DOG. We change the number of nodes different datasets. The synthetic dataset is tested by using the random topology, and those real datasets are tested by using the ring topology. Figure 3 draws the curves of average loss over time steps. We observe that the average loss curves are mostly overlapped with different nodes. It shows that DOG is robust to the network size (or number of users), which validates our theory, that is, the average regret8 does not increase with the number of nodes. Furthermore, we observe that the average loss becomes large with the increase of the variance of stochastic data, which validates our theoretical result nicely. The performance of DOG can be improved in a well-connected network. Figure 4 shows the effect of the topology of the network on the performance of DOG, where five different topologies are used. 8 The ‘average regret’ equals to regret . nT Our theoretical analysis shows that the average regret of DOG is O 8 q G2 T + σ2 nT . Besides, the ring topology, the Disconnected topology means there are no edges in the network, and every node does not share its local model to others. The Fully connected topology means all nodes are connected, where DOG de-generates to be COG. The topology WattsStrogatz represents a Watts-Strogatz small-world graph, for which we can use a parameter to control the number of stochastic edges (set as 0.5 and 1 in this paper). The result shows Fully connected enjoys the best performance, because that ρ = 0 for it, and Disconnected suffers the worst performance due to ρ = 1 for it. Other topologies owns 0 < ρ < 1 for them. 6 Conclusion We investigate decentralized online learning problem, where the loss is incurred by both adversary and stochastic components. We define a new dynamic regret, and analyze a decentralized online gradient method theoretically. It shows that the communication is only effective to decrease the regret caused by the stochastic component, and thus users can benefit from sharing their private models, instead of private data. References D. Adamskiy, W. M. Koolen, A. Chernov, and V. Vovk. A closer look at adaptive regret. Journal of Machine Learning Research, 17(23):1–21, 2016. M. Akbari, B. Gharesifard, and T. Linder. Distributed online convex optimization on time-varying directed graphs. IEEE Transactions on Control of Network Systems, 4(3):417–428, Sep. 2017. A. S. Bedi, P. Sarma, and K. Rajawat. Tracking moving agents via inexact online gradient descent algorithm. IEEE Journal of Selected Topics in Signal Processing, 12(1):202–217, Feb 2018. A. A. Benczúr, L. Kocsis, and R. Pálovics. Online Machine Learning in Big Data Streams. CoRR, 2018. S. Bubeck. Introduction to online optimization, December 2011. N. Cesa-Bianchi, P. Gaillard, G. Lugosi, and G. Stoltz. Mirror Descent Meets Fixed Share (and feels no regret). In NIPS 2012, page Paper 471, 2012. F. Dufossé and B. Uccar. Notes on Birkhoff-von Neumann decomposition of doubly stochastic matrices. Research Report RR-8852, Inria - Research Centre Grenoble – Rhône-Alpes, Feb. 2016. A. György and C. Szepesvári. Shifting regret, mirror descent, and matrices. In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, ICML’16, pages 2943–2951. JMLR.org, 2016. A. György, T. Linder, and G. Lugosi. Tracking the Best of Many Experts. Proceedings of Conference on Learning Theory (COLT), 2005. A. Gyorgy, T. Linder, and G. Lugosi. Efficient tracking of large classes of experts. IEEE Transactions on Information Theory, 58(11):6709–6725, Nov 2012. E. C. Hall and R. Willett. Dynamical Models and tracking regret in online convex programming. In Proceedings of International Conference on International Conference on Machine Learning (ICML), 2013. E. C. Hall and R. M. Willett. Online Convex Optimization in Dynamic Environments. IEEE Journal of Selected Topics in Signal Processing, 9(4):647–662, 2015. E. Hazan. Introduction to online convex optimization. Foundations and Trends in Optimization, 2(3-4): 157–325, 2016. M. Herbster and M. K. Warmuth. Tracking the best expert. Machine Learning, 32(2):151–178, Aug 1998. 9 S. Hosseini, A. Chapman, and M. Mesbahi. Online distributed optimization via dual averaging. In 52nd IEEE Conference on Decision and Control, pages 1484–1489, Dec 2013. A. Jadbabaie, A. Rakhlin, S. Shahrampour, and K. Sridharan. Online Optimization : Competing with Dynamic Comparators. In Proceedings of International Conference on Artificial Intelligence and Statistics (AISTATS), pages 398–406, 2015. K.-S. Jun, F. Orabona, S. Wright, and R. Willett. Improved strongly adaptive online learning using coin betting. In A. Singh and J. Zhu, editors, Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS), volume 54, pages 943–951, 20–22 Apr 2017. M. Kamp, M. Boley, D. Keren, A. Schuster, and I. Sharfman. Communication-efficient distributed online prediction by dynamic model synchronization. In Proceedings of the 2014th European Conference on Machine Learning and Knowledge Discovery in Databases - Volume Part I, ECMLPKDD’14, pages 623–639, Berlin, Heidelberg, 2014. Springer-Verlag. I. Katakis, G. Tsoumakas, and I. Vlahavas. Tracking recurring contexts using ensemble classifiers: An application to email filtering. Knowledge and Information Systems, 22(3):371–391, 2010. P. Knight. The sinkhorn–knopp algorithm: Convergence and applications. SIAM Journal on Matrix Analysis and Applications, 30(1):261–275, 2008. A. Koppel, S. Paternain, C. Richard, and A. Ribeiro. Decentralized online learning with kernels. IEEE Transactions on Signal Processing, 66(12):3240–3255, June 2018. S. Lee, A. Ribeiro, and M. M. Zavlanos. Distributed continuous-time online optimization using saddle-point methods. In 2016 IEEE 55th Conference on Decision and Control (CDC), pages 4314–4319, Dec 2016. S. Lee, A. Nedić, and M. Raginsky. Coordinate dual averaging for decentralized online optimization with nonseparable global objectives. IEEE Transactions on Control of Network Systems, 5(1):34–44, March 2018. M. Mohri and S. Yang. Competing with automata-based expert sequences. In A. Storkey and F. Perez-Cruz, editors, Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, volume 84, pages 1732–1740, 09–11 Apr 2018. A. Mokhtari, S. Shahrampour, A. Jadbabaie, and A. Ribeiro. Online optimization in dynamic environments: Improved regret rates for strongly convex problems. In Proceedings of IEEE Conference on Decision and Control (CDC), pages 7195–7201. IEEE, 2016. J. Mourtada and O.-A. Maillard. Efficient tracking of a growing number of experts. arXiv.org, Aug. 2017. A. Nedic and A. E. Ozdaglar. Distributed Subgradient Methods for Multi-Agent Optimization. IEEE Trans. Automat. Contr., 54(1):48–61, 2009. A. Nedić, S. Lee, and M. Raginsky. Decentralized online optimization with global objectives and local communication. In 2015 American Control Conference (ACC), pages 4497–4503, July 2015. S. Shahrampour and A. Jadbabaie. Distributed online optimization in dynamic environments using mirror descent. IEEE Transactions on Automatic Control, 63(3):714–725, March 2018. S. Shalev-Shwartz. Online Learning and Online Convex Optimization. Foundations and Trends R in Machine Learning, 4(2):107–194, 2012. H. Tang, S. Gan, C. Zhang, T. Zhang, and J. Liu. Communication Compression for Decentralized Training. arXiv.org, Mar. 2018. 10 C.-Y. Wei, Y.-T. Hong, and C.-J. Lu. Tracking the best expert in non-stationary stochastic environments. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, Proceedings of Advances in Neural Information Processing Systems, pages 3972–3980, 2016. T. Wu, K. Yuan, Q. Ling, W. Yin, and A. H. Sayed. Decentralized consensus optimization with asynchrony and delays. IEEE Transactions on Signal and Information Processing over Networks, 4(2):293–307, June 2018. H.-F. Xu, Q. Ling, and A. Ribeiro. Online learning over a decentralized network through admm. Journal of the Operations Research Society of China, 3(4):537–562, Dec 2015. F. Yan, S. Sundaram, S. V. N. Vishwanathan, and Y. Qi. Distributed autonomous online learning: Regrets and intrinsic privacy-preserving properties. IEEE Transactions on Knowledge and Data Engineering, 25 (11):2483–2493, Nov 2013. T. Yang, L. Zhang, R. Jin, and J. Yi. Tracking Slowly Moving Clairvoyant - Optimal Dynamic Regret of Online Learning with True and Noisy Gradient. In Proceedings of the 34th International Conference on Machine Learning (ICML), 2016. K. Yuan, Q. Ling, and W. Yin. On the Convergence of Decentralized Gradient Descent. SIAM Journal on Optimization, 2016. J. Zeng and W. Yin. On nonconvex decentralized gradient descent. IEEE Transactions on Signal Processing, 66(11):2834–2848, June 2018. C. Zhang, P. Zhao, S. Hao, Y. C. Soh, B. S. Lee, C. Miao, and S. C. H. Hoi. Distributed multi-task classification: a decentralized online learning approach. Machine Learning, 107(4):727–747, Apr 2018a. L. Zhang, T. Yang, J. Yi, R. Jin, and Z.-H. Zhou. Improved Dynamic Regret for Non-degenerate Functions. In Proceedings of Neural Information Processing Systems (NIPS), 2017a. L. Zhang, S. Lu, and Z.-H. Zhou. Adaptive online learning in dynamic environments. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems 31, pages 1323–1333. 2018b. L. Zhang, T. Yang, rong jin, and Z.-H. Zhou. Dynamic regret of strongly adaptive methods. In Proceedings of the 35th International Conference on Machine Learning (ICML), pages 5882–5891, 10–15 Jul 2018c. W. Zhang, P. Zhao, W. Zhu, S. C. H. Hoi, and T. Zhang. Projection-free distributed online learning in networks. In D. Precup and Y. W. Teh, editors, Proceedings of the 34th International Conference on Machine Learning, pages 4054–4062, International Convention Centre, Sydney, Australia, 06–11 Aug 2017b. Y. Zhao, S. Qiu, and J. Liu. Proximal Online Gradient is Optimum for Dynamic Regret. CoRR, cs.LG, 2018. M. Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of International Conference on Machine Learning (ICML), pages 928–935, 2003. 11 Supplementary materials for theoretical analysis For math brevity, we denote the function Fi,t by Fi,t (·) := Eξi,t ∼Di,t fi,t (·; ξi,t ) throughout proofs. Proof to Theorem 1: Proof. From the regret definition, we have n n E 1X 1X (fi,t (xi,t ; ξi,t ) − fi,t (x∗t ; ξi,t )) ≤ h∇fi,t (xi,t ; ξi,t ), xi,t − x∗t i E n i=1 Ξn,t ∼Dn,t n i=1 E 1X (h∇fi,t (xi,t ; ξi,t ), xi,t − x̄t i + h∇fi,t (xi,t ; ξi,t ), x̄t − x̄t+1 i) n i=1 {z } Ξn,t ∼Dn,t n = Ξn,t ∼Dn,t | I1 (t) * + E Ξn,t ∼Dn,t | + n 1X ∗ ∇fi,t (xi,t ; ξi,t ), x̄t+1 − xt . n i=1 {z } I2 (t) Now, we begin to bound I1 (t). n I1 (t) = E Ξn,t ∼Dn,t | 1X h∇fi,t (xi,t ; ξi,t ), xi,t − x̄t i + E n i=1 Ξn,t ∼Dn,t {z } | J1 (t) * + n 1X ∇fi,t (xi,t ; ξi,t ), x̄t − x̄t+1 . n i=1 {z } J2 (t) For J1 (t), we have J1 (t) = n X 1 h∇fi,t (xi,t ; ξi,t ), xi,t − x̄t i E n Ξn,t ∼Dn,t i=1 = n X 1 h∇Fi,t (xi,t ), xi,t − x̄t i E n Ξn,t−1 ∼Dn,t−1 i=1 n n X X 1 1 = h∇Fi,t (xi,t ) − ∇Fi,t (x̄t ), xi,t − x̄t i + h∇Fi,t (x̄t ), xi,t − x̄t i E E n Ξn,t−1 ∼Dn,t−1 i=1 n Ξn,t−1 ∼Dn,t−1 i=1 = n n X X L 1 2 kxi,t − x̄t k + h∇Fi,t (x̄t ), xi,t − x̄t i . E E n Ξn,t−1 ∼Dn,t−1 i=1 n Ξn,t−1 ∼Dn,t−1 i=1 Consider the last term, and we have n X 1 h∇Fi,t (x̄t ), xi,t − x̄t i E n Ξn,t−1 ∼Dn,t−1 i=1 1 1 tr ∇Ft (X̄t )> Xt − X̄t E n Ξn,t−1 ∼Dn,t−1 = !! t−1 1X t−1−s > − ηGs Ws v1 v1 n s=1 s=1 !! t−1 t−1 X 1X t−1−s t−1−s > η∇Fs (Xs )Ws − η∇Fs (Xs )Ws v1 v1 n s=1 s=1 1 = tr ∇Ft (X̄t )> E n Ξn,t−1 ∼Dn,t−1 1 = tr ∇Ft (X̄t )> n t−1 X ηGs Wst−1−s 12 (2) t−1 ηX 1 > t−1−s > = tr ∇Ft (X̄t ) ∇Fs (Xs )Ws In − v 1 v 1 n s=1 n t−1 1 ηX > t−1−s > tr ∇Ft (X̄t ) ∇Fs (Xs ) Ws − v1 v1 = n s=1 n t−1 1 1 η X t−1−s 2 ∇Ft (X̄t ) F + t−1−s ∇Fs (Xs ) Wst−1−s − v1 v1> ρ ≤ 2n s=1 ρ n 2 η 2n t−1 X ρt−1−s ∇Ft (X̄t ) 2 F + ρt−1−s k∇Fs (Xs )kF t−1 η X t−1−s ∇Ft (X̄t ) ρ ≤ 2n s=1 2 F + k∇Fs (Xs )kF ≤ 2 ! F 2 s=1 ≤ηG2 t−1 X 2 ρt−1−s s=1 ηG2 ≤ . 1−ρ 1 holds due to Xt = [x1,t ; x2,t ; · · · ; xn,t ], and X̄t = [x̄t ; x̄t ; · · · ; x̄t ]. 2 holds due to Lemma 3. Substitute it into (2), and we thus have J1 (t) = n X ηG2 L 2 kxi,t − x̄t k + . E n Ξn,t−1 ∼Dn,t−1 i=1 1−ρ For J2 (t), we have J2 (t) * = E Ξn,t ∼Dn,t η ≤ 2 η ≤ 2 n 1X ∇fi,t (xi,t ; ξi,t ), x̄t − x̄t+1 n i=1 n 2 1 2η + E 1X ∇fi,t (xi,t ; ξi,t ) n i=1 E 1X (∇fi,t (xi,t ; ξi,t ) − ∇Fi,t (xi,t ) + ∇Fi,t (xi,t )) n i=1 E 1X (∇fi,t (xi,t ; ξi,t ) − ∇Fi,t (xi,t )) n i=1 Ξn,t ∼Dn,t + 2 E Ξn,t ∼Dn,t kx̄t − x̄t+1 k n Ξn,t ∼Dn,t n ≤η Ξn,t ∼Dn,t 2 + 2 1 2η n +η E Ξn,t ∼Dn,t n n η 1X (∇Fi,t (xi,t ) − ∇Fi,t (x̄t )) ≤ σ 2 + 2η E n Ξn,t−1 ∼Dn,t−1 n i=1 n + 2η E Ξn,t−1 ∼Dn,t−1 η 2η ≤ σ2 + n n E Ξn,t−1 ∼Dn,t−1 n X 2 + 1 2η 2 kx̄t − x̄t+1 k 1X ∇Fi,t (xi,t ) n i=1 1X 1 η 2 σ +η (∇Fi,t (xi,t ) − ∇Fi,t (x̄t ) + ∇Fi,t (x̄t )) E ≤ n Ξn,t−1 ∼Dn,t−1 n i=1 1X ∇Fi,t (x̄t ) n i=1 E Ξn,t ∼Dn,t 2 + 1 2η E 2 Ξn,t ∼Dn,t + 1 2η 2 kx̄t − x̄t+1 k 2 2 kx̄t − x̄t+1 k E Ξn,t ∼Dn,t 2 k∇Fi,t (xi,t ) − ∇Fi,t (x̄t )k + 2ηG2 + i=1 13 1 2η E Ξn,t ∼Dn,t 2 E Ξn,t ∼Dn,t kx̄t − x̄t+1 k 2 kx̄t − x̄t+1 k 2 η 2 2ηL2 σ + ≤ n n n X E Ξn,t−1 ∼Dn,t−1 2 kxi,t − x̄t k + 2ηG2 + i=1 1 2η 2 E Ξn,t ∼Dn,t kx̄t − x̄t+1 k . 1 holds due to n E Ξn,t ∼Dn,t 1X (∇fi,t (xi,t ; ξi,t ) − ∇Fi,t (xi,t )) n i=1 n X 1 = 2 n Ξn,t ∼Dn,t 2 n2 Ξn,t ∼Dn,t + = 1 n2 E E E 2 ! E ξ ∼Di,t i=1 i,t n n X X k∇fi,t (xi,t ; ξi,t ) − ∇Fi,t (xi,t )k 2 h∇fi,t (xi,t ; ξi,t ) − ∇Fi,t (xi,t ), ∇fj,t (xj,t ; ξj,t ) − ∇Fj,t (xj,t )i i=1 j=1,j6=i n X Ξn,t−1 ∼Dn,t−1 i=1 E ξi,t ∼Di,t 2 k∇fi,t (xi,t ; ξi,t ) − ∇Fi,t (xi,t )k + 0 1 ≤ σ2 . n 2 holds due to Fi,t has L Lipschitz gradients. Therefore, we obtain I1 (t) = (J1 (t) + J2 (t)) ! n n X X η 2 2ηL2 L 2 2 kxi,t − x̄t k + σ + kxi,t − x̄t k = E E n Ξn,t−1 ∼Dn,t−1 i=1 n n Ξn,t−1 ∼Dn,t−1 i=1 1 1 2 2 + 2+ ηG + kx̄t − x̄t+1 k E 1−ρ 2η Ξn,t ∼Dn,t n X 1 L 2ηL2 ησ 2 1 2 2 kx − x̄ k + 2 + ≤ + ηG2 + + kx̄t − x̄t+1 k . E E i,t t n n 1 − ρ n 2η Ξn,t−1 ∼Dn,t−1 Ξn,t ∼Dn,t i=1 Therefore, we have T X I1 (t) t=1 ≤ L 2ηL2 + n n E Ξn,t−1 ∼Dn,t−1 n X T X 2 kxi,t − x̄t k + 2 + i=1 t=1 1 1−ρ T ηG2 + T ησ 2 1 + n 2η E Ξn,t ∼Dn,t T X 2 kx̄t − x̄t+1 k . t=1 Now, we begin to bound I2 (t). Denote that the update rule is xi,t+1 = n X Wij xj,t − η∇fi,t (xi,t ; ξi,t ). j=1 According to Lemma 1, we have x̄t+1 = x̄t − η ! n 1X ∇fi,t (xi,t ; ξi,t ) . n i=1 Denote a new auxiliary function φ(z) as * n + 1X 1 2 ∇fi,t (xi,t ; ξi,t ), z + kz − x̄t k . φ(z) = n i=1 2η 14 (3) It is trivial to verify that (3) satisfies the first-order optimality condition of the optimization problem: minz∈Rd φ(z), that is, ∇φ(x̄t+1 ) = 0. We thus have * x̄t+1 = argmin φ(z) = argmin z∈Rd z∈Rd n 1X ∇fi,t (xi,t ; ξi,t ), z n i=1 + + 1 2 kz − x̄t k . 2η Furthermore, denote a new auxiliary variable x̄τ as x̄τ = x̄t+1 + τ (x∗t − x̄t+1 ) , where 0 < τ ≤ 1. According to the optimality of x̄t+1 , we have 0 ≤ φ(x̄τ ) − φ(x̄t+1 ) * n + 1X 1 2 2 = kx̄τ − x̄t k − kx̄t+1 − x̄t k ∇fi,t (xi,t ; ξi,t ), x̄τ − x̄t+1 + n i=1 2η * n + 1X 1 2 2 = ∇fi,t (xi,t ; ξi,t ), τ (x∗t − x̄t+1 ) + kx̄t+1 + τ (x∗t − x̄t+1 ) − x̄t k − kx̄t+1 − x̄t k n i=1 2η * n + 1X 1 2 = ∇fi,t (xi,t ; ξi,t ), τ (x∗t − x̄t+1 ) + kτ (x∗t − x̄t+1 )k + 2 hτ (x∗t − x̄t+1 ) , x̄t+1 − x̄t i . n i=1 2η Note that the above inequality holds for any 0 < τ ≤ 1. Divide τ on both sides, and we have * n + 1X ∗ I2 (t) = ∇fi,t (xi,t ; ξi,t ), x̄t+1 − xt E n i=1 Ξn,t ∼Dn,t 1 2 ∗ ∗ lim τ k(xt − x̄t+1 )k + 2 hxt − x̄t+1 , x̄t+1 − x̄t i ≤ E 2η Ξn,t ∼Dn,t τ →0+ 1 = hx∗ − x̄t+1 , x̄t+1 − x̄t i E η Ξn,t ∼Dn,t t 1 2 2 2 = kx∗t − x̄t k − kx∗t − x̄t+1 k − kx̄t − x̄t+1 k . E 2η Ξn,t ∼Dn,t (4) Besides, we have x∗t+1 − x̄t+1 = x∗t+1 2 2 2 − kx∗t − x̄t+1 k 2 − kx∗t k − 2 x̄t+1 , −x∗t + x∗t+1 = x∗t+1 − kx∗t k x∗t+1 + kx∗t k − 2 x̄t+1 , −x∗t + x∗t+1 ≤ x∗t+1 − x∗t x∗t+1 + kx∗t k + 2 kx̄t+1 k x∗t+1 − x∗t √ ≤4 R x∗t+1 − x∗t . √ √ The last inequality holds due to our assumption, that is, x∗t+1 = x∗t+1 − 0 ≤ R, kx∗t k = kx∗t − 0k ≤ R, √ and kx̄t+1 k = kx̄t+1 − 0k ≤ R. Thus, telescoping I2 (t) over t ∈ [T ], we have ! T T T X X √ X 1 1 2 2 2 ∗ ∗ ∗ ∗ I2 (t) ≤ 4 R xt+1 − xt + kx̄1 − x̄1 k − kx̄T − x̄T +1 k − kx̄t − x̄t+1 k E E 2η 2η Ξn,T ∼Dn,T Ξn,T ∼Dn,T t=1 t=1 t=1 15 ≤ 1 √ 1 4 RM + R − 2η 2η T X E Ξn,T ∼Dn,T 2 kx̄t − x̄t+1 k . t=1 Here, M the budget of the dynamics. Combining those bounds of I1 (t), and I2 (t) together, we finally obtain E Ξn,T ∼Dn,T T X n X fi,t (xi,t ; ξi,t ) − fi,t (x∗t ; ξi,t ) ≤ n t=1 i=1 T X (I1 (t) + I2 (t)) t=1 T X n X nT ηG2 n √ 4 RM + R + 2nηT G2 + 2η 1−ρ Ξn,T ∼Dn,T t=1 i=1 2L + 4ηL2 nT η 2 (G2 + σ 2 ) n √ 1 1 2 + 4 RM + R + 2 + nηT G2 . ≤ ηT σ + 2 (1 − ρ) 2η 1−ρ ≤ηT σ 2 + L + 2ηL2 E 2 kx̄t − xi,t k + 1 holds due to Lemma 5 E n X T X Ξn,T ∼Dn,T 2 kxi,t − x̄t k ≤ i=1 t=1 nT η 2 (2G2 + 2σ 2 ) . (1 − ρ)2 Rearranging items, we finally completes the proof. Pn Lemma 1. Denote x̄t = n1 i=1 xi,t . We have x̄t+1 = x̄t − η ! n 1X ∇fi,t (xi,t ; ξi,t ) . n i=1 Proof. Denote Xt =[x1,t , x2,t , ..., xn,t ] ∈ Rd×n , Gt =[∇f1,t (x1,t ; ξ1,t ), ∇f2,t (x2,t ; ξ2,t ), ..., ∇fn,t (xn,t ; ξn,t )] ∈ Rd×n . Denote that xi,t+1 = n X Wij xj,t − η∇fi,t (xi,t ; ξi,t ). j=1 Equivalently, we re-formulate the update rule as Xt+1 = Xt W − ηGt . Since the confusion matrix W is doublely stochastic, we have W1 = 1. Thus, we have n x̄t+1 = 1X xi,t+1 n i=1 1 n 1 1 =Xt W − ηGt n n =Xt+1 16 1 1 − ηGt n n ! n 1X =x̄t − η ∇fi,t (xi,t ; ξi,t ) . n i=1 =Xt It completes the proof. Lemma 2. For any doubly stochastic matrix W, its norm kWk = 1. Proof. According to Birkhoff-von Neumann theorem [Dufossé and Uccar, 2016], W is a convex combination of some, e.g. k, permutation matrices {Mi }ki=1 , that is, W= k X ζi Mi , (5) i=1 Pk where 0 ≤ ζi ≤ 1 for any 1 ≤ i ≤ k and i=1 ζi = 1. For any a vector u such that kuk = 1, we have kMi k = supkuk=1 kMi uk = supkuk=1 kuk = 1. Therefore, we have kWk ≤ k X ζi kMi k = i=1 k X ζi = 1. (6) i=1 2 Since kWk is the maximal eigenvalue of WW> , and WW> has a eigenvalue 1, thus kWk ≥ 1. Therefore, kWk = 1, and the proof is completed. Lemma 3. Denote v1 = √1 . n Given any matrix X and doubly stochastic matrix W, we have XWt − Xv1 v1> where ρ = kW − Vn k and Vn = 2 F ≤ ρt kXkF 2 , 1 > n 11 . Proof. We start from the left hand side: XWt − XVn 2 F =tr (XWt − XVn )(XWt − XVn )> > > =tr XWt Wt X> − XWt Vn> X> − XVn Wt X> + XVn Vn> X> 1 tr XWt Wt > X> − XV X> . n = ‘tr’ represents the trace operator. 1 holds due to Vn = Vn> , Wt Vn> = 1 1 t−1 W W11> = Wt−1 11> = · · · = Vn , n n similarly Vn W t > = > > 1 > 1 11 Wt = 11> Wt−1 = Vn , n n and Vn Vn> = Vn . Additionally, since W is a doubly stochastic matrix, we have Wt Wt > 1 t−1 > 1 t−1 1 1 √ = Wt W> √ = ··· = √ . W √ = Wt W> n n n n 17 > (1) Thus, 1 is one of eigenvalues of Wt (Wt ) , and its largest eigenvalue λWt (Wt )> ≥ 1. According to Lemma 2, we have (1) λWt (Wt )> = Wt W> t > (1) Thus, λWt (Wt )> = 1 is its largest eigenvalue, and v1 = t t ≤ W (W) t W> ≤ kWk √1 1 n t ≤ 1. (7) is the corresponding eigenvector. Since t > > > W (W ) is a symmetric and positive semi-definite matrix, we decompose Wt (Wt ) as Wt (Wt ) = Pn (i) > > n×n . vi is the normalized eigenvector i=1 λWt (Wt )> vi vi = PΛP , where P = [v1 , v2 , ..., vn ] ∈ R (i) corresponding to the i-th eigenvalue λWt (Wt )> . Denote the absolute value of the i-th largest eigenvalue of Wt (Wt ) > (i) (1) (i) (n) by λWt (Wt )> , that is, λWt (Wt )> ≥ · · · ≥ λWt (Wt )> ≥ · · · ≥ λWt (Wt )> . Λ is a diagonal matrix, (i) and λWt (Wt )> is its i-th element. Pn > (i) Due to Wt (Wt ) = Vn + i=2 λWt (Wt )> vi vi> , we have XWt − 2 XVn F n X =tr X ! (i) λWt (Wt )> vi vi> ! X> i=2 = n X (i) λWt (Wt )> tr X vi vi> X> i=2 = n X (i) λWt (Wt )> kXvi k 2 i=2 (2) ≤λWt (Wt )> n X 2 kXvi k i=1 (2) =λWt (Wt )> kXPk2F (2) =λWt (Wt )> tr XPP> X (2) =λWt (Wt )> tr XX> (due to P’s definition) > (2) =λWt (Wt )> kXk2F . > Recall that Vn = Vn> , Wt Vn> = Vn (Wt ) = Vn , and Vn Vn = Vn . We thus have (W − Vn )t (W> − Vn )t = (W − Vn )t−1 (WW> − WVn − Vn W> + Vn Vn )(W> − Vn )t−1 =(W − Vn )t−1 (WW> − Vn )(W> − Vn )t−1 = · · · = Wt (W> )t − Vn . Since ρ = kW − Vn k = W> − Vn , we have (2) t λWt (Wt )> = Wt (W> )t − Vn ≤ kW − Vn k W > − Vn We finally have XWt − XVn 2 F It completes the proof. 18 ≤ ρt kXkF 2 . t = ρ2t . ∞ Lemma 4 (Lemma 6 in [Tang et al., 2018]). Given two non-negative sequences {at }∞ t=1 and {bt }t=1 that satisfying at = t X ρt−s bs , s=1 with ρ ∈ [0, 1), we have k X a2t ≤ t=1 k X 1 b2 . (1 − ρ)2 s=1 s Shahrampour and Jadbabaie [2018] investigates the dynamic regret of DOG, and provide the following sublinear regret. Theorem 3 (Implied by Theorem 3 and Corollary 4 in Shahrampour and Jadbabaie [2018]). Choose q 3q (1−ρ)M MT DOG 2 η= in Algorithm 1. Under Assumption 1, the dynamic regret R is bounded by O n T T 1−ρ . √ As illustrated in Theorem 3, Shahrampour and Jadbabaie [2018] has provided a O n nT M regret for DOG. Comparing with the regret in Shahrampour and Jadbabaie [2018], our analysis improves the dependence on n, which benefits from the following better bound of difference between xi,t and x̄t . Lemma 5. Setting η > 0 in Algorithm 1, under Assumption 1 we have E Ξn,T ∼Dn,T n X T X 2 kxi,t − x̄t k ≤ i=1 t=1 2nT η 2 (G2 + σ 2 ) . (1 − ρ)2 Proof. Denote that xi,t+1 = n X Wij xj,t − η∇fi,t (xi,t ; ξi,t ), j=1 and according to Lemma 1, we have x̄t+1 = x̄t − η ! n 1X ∇fi,t (xi,t ; ξi,t ) . n i=1 Denote Xt =[x1,t , x2,t , ..., xn,t ] ∈ Rd×n , Gt =[∇f1,t (x1,t ; ξ1,t ), ∇f2,t (x2,t ; ξ2,t ), ..., ∇fn,t (xn,t ; ξn,t )] ∈ Rd×n . By letting xi,1 = 0 for any i ∈ [n], the update rule is re-formulated as Xt+1 = Xt W − ηGt = − t X ηGs Wt−s . s=1 Similarly, denote Ḡt = 1 n Pn i=1 ∇fi,t (xi,t ; ξi,t ), and we have x̄t+1 = x̄t − η ! n t X 1X ∇fi,t (xi,t ; ξi,t ) = − η Ḡs . n i=1 s=1 19 Therefore, we obtain n X 2 kxi,t − x̄t k =1 i=1 n t−1 X X s=1 i=1 t−1 X 2 2 η Ḡs − ηGs Wt−s−1 ei = 2 ηGs v1 v1> − ηGs W t−s−1 s=1 =η 2 F t−1 X !2 Gs v1 v1> − Gs W t−s−1 s=1 ≤η 2 F !2 t−1 X Gs v1 v1> − Gs W t−s−1 F s=1 3 ≤ η 2 t−1 X !2 ρ t−s−1 kGs kF s=1 1 holds due to ei is a unit basis vector, whose i-th element is 1 and other elements are 0s. 2 holds due to 1n v1 = √ . 3 holds due to Lemma 3. n Thus, we have T n X X E Ξn,T ∼Dn,T ≤ E ≤ t=1 !2 ηρt−s−1 kGs kF s=1 T X 2 η (1 − ρ)2 η2 = (1 − ρ)2 ≤ i=1 t=1 T t−1 X X Ξn,T ∼Dn,T 1 2 kxi,t − x̄t k 2η 2 (1 − ρ)2 E Ξn,T ∼Dn,T E t=1 n T X X Ξn,T ∼Dn,T E Ξn,T ∼Dn,T ! 2 kGt kF ! k∇fi,t (xi,t ; ξi,t ) − ∇Fi,t (xi,t ) + ∇Fi,t (xi,t )k 2 t=1 i=1 n T X X 2 k∇fi,t (xi,t ; ξi,t ) − ∇Fi,t (xi,t )k + t=1 i=1 2η 2 (1 − ρ)2 E Ξn,T ∼Dn,T n T X X k∇Fi,t (xi,t )k 2 t=1 i=1 nT η 2 (2G2 + 2σ 2 ) ≤ . (1 − ρ)2 1 holds due to Lemma 4. It completes the proof. Proof to Theorem 2: r √ (1−ρ)(nM R+nR) Proof. Setting η = into Lemma 5, we finally complete the proof. 2 nT G +T σ 2 Supplementary materials for empirical studies The dynamics of time-varying distributions are illustrated in Figure 5, which shows the change of the optimal learning model over time and the importance of studying the dynamic regret. We use such time-varying distribution to simulate the dynamics of the synthetic data. More numerical results are presented as Figures 6-8. 20 Density 0.4 0.2 0 -5 0 5 10 15 0.3 0.2 0.1 500 1000 20 DOG(M=10) COG(M=10) 15 10 5 1500 Average loss DOG(M=10) COG(M=10) Average loss Average loss Variable Figure 5: An illustration of the dynmaics caused by the time-varying distributions of data. Data distributions 1 and 2 satisify N (1 + sin(t), 1) and N (−1 + sin(t), 1), respectively. Suppose we want to conduct classification between data drawn from distributions 1 and 2, respectively. The optimal classification model should change over time. 250 200 150 100 50 500 50 100 150 200 250 T DOG(M=10) COG(M=10) T T (a) room-occupancy, 5 nodes, ring topology 1000 1500 (b) usenet2, 5 nodes, ring topology (c) spam, 5 nodes, ring topology n=5 n=10 n=20 1.5 1 0.5 100 200 300 n=5 n=10 n=20 20 15 10 20 40 (a) room-occupancy, ring topology 60 T T (b) usenet2, ring topology 250 Average loss Average loss 2 Average loss Figure 6: The average loss yielded by DOG is comparable to that yielded by COG. n=5 n=10 n=20 200 150 100 50 100 200 (c) spam, ring topology Figure 7: The average loss yielded by DOG is insensitive to the network size. 21 300 T 400 0 150 200 250 300 350 Disconnected WattsStrogatz(0.5) WattsStrogatz(1) Ring Fully connected 40 20 20 40 T T (a) room-occupancy, 20 nodes 60 (b) usenet2, 20 nodes Average loss 2 60 Average loss Average loss 4 Disconnected WattsStrogatz(0.5) WattsStrogatz(1) Ring Fully connected 80 60 40 Disconnected WattsStrogatz(0.5) WattsStrogatz(1) Ring Fully connected 20 100 200 300 400 T (c) spam, 20 nodes Figure 8: The average loss yielded by DOG is insensitive to the topology of the network. 22