{"title": "Online Forecasting of Total-Variation-bounded Sequences", "book": "Advances in Neural Information Processing Systems", "page_first": 11071, "page_last": 11081, "abstract": "We consider the problem of online forecasting of sequences of length $n$ with total-variation at most $C_n$ using observations contaminated by independent $\\sigma$-subgaussian noise. We design an $O(n\\log n)$-time algorithm that achieves a cumulative square error of $\\tilde{O}(n^{1/3}C_n^{2/3}\\sigma^{4/3} + C_n^2)$ with high probability. We also prove a lower bound that matches the upper bound in all parameters (up to a $\\log(n)$ factor). To the best of our knowledge, this is the first **polynomial-time** algorithm that achieves the optimal $O(n^{1/3})$ rate in forecasting total variation bounded sequences and the first algorithm that **adapts to unknown** $C_n$.Our proof techniques leverage the special localized structure of Haar wavelet basis and the adaptivity to unknown smoothness parameters in the classical wavelet smoothing [Donoho et al., 1998]. We also compare our model to the rich literature of dynamic regret minimization and nonstationary stochastic optimization, where our problem can be treated as a special case. We show that the workhorse in those settings --- online gradient descent and its variants with a fixed restarting schedule --- are instances of a class of **linear forecasters** that require a suboptimal regret of $\\tilde{\\Omega}(\\sqrt{n})$. This implies that the use of more adaptive algorithms is necessary to obtain the optimal rate.", "full_text": "Online Forecasting of Total-Variation-bounded\n\nSequences\n\nDepartment of Computer Science\n\nDepartment of Computer Science\n\nDheeraj Baby\n\nUC Santa Barbara\n\ndheeraj@ucsb.edu\n\nYu-Xiang Wang\n\nUC Santa Barbara\n\nyuxiangw@cs.ucsb.edu\n\nAbstract\n\nn \u03c34/3 + C 2\n\nWe consider the problem of online forecasting of sequences of length n with\ntotal-variation at most Cn using observations contaminated by independent \u03c3-\nsubgaussian noise. We design an O(n log n)-time algorithm that achieves a cu-\nmulative square error of \u02dcO(n1/3C 2/3\nn) with high probability. We also\nprove a lower bound that matches the upper bound in all parameters (up to a log(n)\nfactor). To the best of our knowledge, this is the \ufb01rst polynomial-time algorithm that\nachieves the optimal O(n1/3) rate in forecasting total variation bounded sequences\nand the \ufb01rst algorithm that adapts to unknown Cn. Our proof techniques leverage\nthe special localized structure of Haar wavelet basis and the adaptivity to unknown\nsmoothness parameters in the classical wavelet smoothing [Donoho et al., 1998].\nWe also compare our model to the rich literature of dynamic regret minimization\nand nonstationary stochastic optimization, where our problem can be treated as\na special case. We show that the workhorse in those settings \u2014 online gradient\ndescent and its variants with a \ufb01xed restarting schedule \u2014 are instances of a class\nof linear forecasters that require a suboptimal regret of \u02dc\u2126(\u221an). This implies that\nthe use of more adaptive algorithms is necessary to obtain the optimal rate.\n\n1\n\nIntroduction\n\nNonparametric regression is a fundamental class of problems that has been studied for more than half\na century in statistics and machine learning [Nadaraya, 1964, De Boor et al., 1978, Wahba, 1990,\nDonoho et al., 1998, Mallat, 1999, Scholkopf and Smola, 2001, Rasmussen and Williams, 2006]. It\nsolves the following problem:\n\n\u2022 Let yi = f (ui) + Noise for i = 1, ..., n. How can we estimate a function f using data points\n(u1, y1), ..., (un, yn) and the knowledge that f belongs to a function class F?\n\nFunction class F typically imposes only weak regularity assumptions on the function f such as\nboundedness and smoothness, which makes nonparametric regression widely applicable to many\nreal-life applications especially those with unknown physical processes.\nA recent and successful class of nonparametric regression technique called trend \ufb01ltering [Steidl et al.,\n2006, Kim et al., 2009, Tibshirani, 2014, Wang et al., 2014] was shown to have the property of local\nadaptivity [Mammen and van de Geer, 1997] in both theory and practice. We say a nonparametric\nregression technique is locally adaptive if it can cater to local differences in smoothness, hence\nallowing more accurate estimation of functions with varying smoothness and abrupt changes. For\nexample, for functions with bounded total variation (when F is a total variation class), standard\nnonparametric regression techniques such as kernel smoothing and smoothing splines have a mean\nsquare error (MSE) of O(n\u22121/2) while trend \ufb01ltering has the optimal O(n\u22122/3).\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fTrend \ufb01ltering is, however, a batch learning algorithm where one observes the entire dataset ahead\nof the time and makes inference about the past. This makes it inapplicable to the many time series\nproblems that motivate the study of trend \ufb01ltering in the \ufb01rst place [Kim et al., 2009]. These include\nin\ufb02uenza forecasting, inventory planning, economic policy-making, \ufb01nancial market prediction and so\non. In particular, it is unclear whether the advantage of trend \ufb01ltering methods in estimating functions\nwith heterogeneous smoothness (e.g., sharp changes) would carry over to the online forecasting\nsetting. The focus of this work is in developing theory and algorithms for locally adaptive online\nforecasting which predicts the immediate future value of a function with heterogeneous smoothness\nusing only noisy observations from the past.\n\n1.1 Problem Setup\n\n1. Fix action time intervals 1, 2, ..., n\n2. The player declares a forecasting strategy Ai : Ri\u22121 \u2192 R for i = 1, ..., n.\n3. An adversary chooses a sequence \u03b81:n = [\u03b81, \u03b82, . . . , \u03b8n]T \u2208 Rn.\n4. For every time point i = 1, ..., n:\n\nsubgaussian noise.\n\n(a) We play xi = Ai(y1, ..., yi\u22121).\n(b) We receive a feedback yi = \u03b8i + Zi, where Zi is a zero-mean, independent\n5. At the end, the player suffers a cumulative error(cid:80)n\n\ni=1 (xi \u2212 \u03b8i)2.\n\nFigure 1: Nonparametric online forecasting model. The focus of the proposed work is to design\na forecasting strategy that minimizes the expected cumulative square error. Note that the problem\ndepends a lot on the choice of the sequence \u03b8i. Our primary interest is on sequences with bounded\ni=2|\u03b8i \u2212 \u03b8i\u22121|\u2264 Cn, but we will also talk about the adaptivity of our\n\nmethod to easier problems such as forecasting Sobolev and Holder functions.\n\ntotal variation (TV) so that(cid:80)n\n\nWe propose a model for nonparametric online forecasting as described in Figure 1. This model can\nbe re-framed in the language of the online convex optimization model with three differences.\n\n1. We consider only quadratic loss functions of the form (cid:96)t(x) = (x \u2212 \u03b8t)2.\n2. The learner receives independent noisy gradient feedback, rather than the exact gradient.\n3. The criterion of interest is rede\ufb01ned as the dynamic regret [Zinkevich, 2003, Besbes et al.,\n\n2015]:\n\nRdynamic(A, (cid:96)1:n) := E(cid:34) n(cid:88)t=1\n\n(cid:96)t(xt)(cid:35) \u2212\n\nn(cid:88)t=1\n\n(cid:96)t(xt).\n\ninf\nxt\n\nThe new criterion is called a dynamic regret because we are now comparing to a stronger dynamic\nbaseline that chooses an optimal x in every round. Of course in general, the dynamic regret will\nbe linear in n [Jadbabaie et al., 2015]. To make the problem non-trivial, we restrict our attention to\nsequences of (cid:96)1, ..., (cid:96)n that are regular, which makes it possible to design algorithms with sublinear\ndynamic regret. In particular, we borrow ideas from the nonparametric regression literature and con-\nsider sequences [\u03b81, ..., \u03b8n] that are discretizations of functions in the continuous domain. Regularity\nassumptions emerge naturally as we consider canonical functions classes such as the Holder class,\nSobolev class and Total Variation classes [see, e.g., Tsybakov, 2008, for a review].\n\n1.2 Assumptions\n\nWe consolidate all the assumptions used in this work and provide necessary justi\ufb01cations for them.\n\u2022 (A1) The time horizon for the online learner is known to be n.\n\u2022 (A2) The parameter \u03c32 of subgaussian noise in the observations is known.\n\u2022 (A3) The ground truth denoted by \u03b81:n = [\u03b81, ..., \u03b8n]T has its total variation bounded by some posi-\ntive Cn, i.e., we take F to be the total variation class TV(Cn) := {\u03b81:n \u2208 Rn : (cid:107)D\u03b81:n(cid:107)1\u2264\nCn} where D is the discrete difference operator. Here D\u03b81:n = [\u03b82 \u2212 \u03b81, . . . , \u03b8n \u2212 \u03b8n\u22121]T .\n\n\u2022 (A4) |\u03b81|\u2264 U.\n\n2\n\n\fThe knowledge of \u03c32 in assumption (A2) is primarily used to get the optimal dependence of \u03c3 in\nminimax rate. This assumption can be relaxed in practice by using the Median Absolute Deviation\nestimator as described in Section 7.5 of Johnstone [2017] to estimate \u03c32 robustly. Assumption (A3)\nfeatures a samples from a large class of functions with spatially inhomogeneous degree of smoothness.\nThe functions residing in this class need not even be continuous. Our goal is to propose a policy\nthat is locally adaptive whose empirical mean squared error converges at the minimax rate for this\nfunction class. We stress that we do not assume that the learner knows Cn. The problem is open and\nnontrivial even when Cn is known. Assumption (A4) is very mild as it puts restriction only to the\n\ufb01rst value of the sequence. This assumption controls the inevitable prediction error for the \ufb01rst point\nin the sequence.\n\n1.3 Our Results\n\nThe major contributions of this work are summarized below.\n\n\u2022 It is known that the minimax MSE for smoothing sequences in the TV class is \u02dc\u2126(n\u22122/3).\nThis implies a lowerbound of \u02dc\u2126(n1/3) for the dynamic regret in our setting. We present a\npolicy ARROWS (Adaptive Restarting Rule for Online averaging using Wavelet Shrinkage)\nwith a nearly minimax dynamic regret \u02dcO(n1/3) and a run-time complexity of O(n log n).\n\u2022 We show that a class of forecasting strategies \u2014 including the popular Online Gradient\nDescent (OGD) with \ufb01xed restarts [Besbes et al., 2015], moving averages (MA) [Box and\nJenkins, 1970] \u2014 are fundamentally limited by \u02dc\u2126(\u221an) regret.\n\n\u2022 We also provide a more re\ufb01ned lower bound that characterized the dependence of U, Cn\nand \u03c3, which certi\ufb01es the adaptive optimality of ARROWS in all regimes. The bound also\nreveals a subtle price to pay when we move from the smoothing problem to the forecasting\nproblem, which indicates the separation of the two problems when Cn/\u03c3 (cid:29) n1/4, a regime\nwhere the forecasting problem is strictly harder (See Figure 3).\n\u2022 Lastly, we consider forecasting sequences in Sobolev classes and Holder classes and establish\nthat ARROWS can automatically adapt to the optimal regret of these simpler function classes\nas well, while OGD and MA cannot, unless we change their tuning parameter (to behave\nsuboptimally on the TV class).\n\n2 Related Work\n\nThe topic of this paper sits well in between two amazing bodies of literature: nonparametric regression\nand online learning. Our results therefore contribute to both \ufb01elds and hopefully will inspire more\ninterplay between the two communities. Throughout this paper when we refer \u02dcO(n1/3) as the optimal\nregret, we assume the parameters of the problem are such that it is acheivable (see Figure 3).\nNonparametric regression. As we mentioned before, our problem \u2014 online nonparametric fore-\ncasting \u2014 is motivated by the idea of using locally adaptive nonparametric regression for time\nseries forecasting [Mammen and van de Geer, 1997, Kim et al., 2009, Tibshirani, 2014]. It is more\nchallenging than standard nonparametric regression because we do not have access to the data in the\nfuture. While our proof techniques make use of several components (e.g., universal shrinkage) from\nthe seminal work in wavelet smoothing [Donoho et al., 1990, 1998], the way we use them to construct\nand analyze our algorithm is new and more generally applicable for converting non-parametric\nregression methods to forecasting methods.\nAdaptive Online Learning. Our problem is also connected to a growing literature on adaptive online\nlearning which aims at matching the performance of a stronger time-varying baseline [Zinkevich,\n2003, Hall and Willett, 2013, Besbes et al., 2015, Chen et al., 2018b, Jadbabaie et al., 2015, Hazan\nand Seshadhri, 2007, Daniely et al., 2015, Yang et al., 2016, Zhang et al., 2018a,b, Chen et al., 2018a].\nMany of these settings are highly general and we can apply their algorithms directly to our problem,\nbut to the best of our knowledge, none of them achieves the optimal \u02dcO(n1/3) dynamic regret.\nIn the remainder of this section, we focus our discussion on how to apply the regret bounds in\nnon-stationary stochastic optimization [Besbes et al., 2015, Chen et al., 2018b] to our problem while\nleaving more elaborate discussion with respect to alternative models (e.g. the constrained comparator\n\n3\n\n\fapproach [Zinkevich, 2003, Hall and Willett, 2013], adaptive regret [Jadbabaie et al., 2015, Zhang\net al., 2018a], competitive ratio [Bansal et al., 2015, Chen et al., 2018a]), as well as the comparison\nto the classical time series models to Appendix A.\nRegret from Non-Stationary Stochastic Optimization The problem of non-stationary stochastic\noptimization is more general than our model because instead of considering only the quadratic\nfunctions, (cid:96)t(x) = (x \u2212 \u03b8t)2, they work with the more general class of strongly convex functions and\ngeneral convex functions. They also consider both noisy gradient feedbacks (stochastic \ufb01rst order\noracle) and noisy function value feedbacks (stochastic zeroth order oracle).\nIn particular, Besbes et al. [2015] de\ufb01ne a quantity Vn which captures the total amount of \u201cvariation\u201d\nof the functions (cid:96)1:n using Vn :=(cid:80)n\u22121\ni=1 (cid:107)(cid:96)i+1 \u2212 (cid:96)i(cid:107)\u221e. 1 Chen et al. [2018b] generalize the notion to\nVn(p, q) :=(cid:16)(cid:80)n\u22121\np(cid:17)1/q\nfor any 1 \u2264 p, q \u2264 +\u221e where (cid:107)\u00b7(cid:107)p:= ((cid:82) |\u00b7(x)|pdx)1/p is the\n\ni=1 (cid:107)(cid:96)i+1 \u2212 (cid:96)i(cid:107)q\nstandard Lp norm for functions2. Table 1 summarizes the known results under the non-stationary\nstochastic optimization setting.\n\nTable 1: Summary of known minimax dynamic regret in the non-stationary stochastic optimization\nmodel. Note that the choice of q does not affect the minimax rate in any way, but the choice of p does.\n\u201c-\u201d indicates that the no upper or lower bounds are known for that setting.\n\nNoisy gradient feedback\n\nAssumptions on (cid:96)1:n\nConvex & Lipschitz\n\np = +\u221e\n\u0398(n2/3V 1/3\nStrongly convex & Smooth \u0398(n1/2V 1/2\n\nn ) O(n\nn ) \u0398(n\n\n2p+d\n\n1 \u2264 p < +\u221e\n3p+d Vn(p, q)\n4p+d Vn(p, q)\n\n2p+d\n\np\n\n3p+d )\n2p\n4p+d ) \u0398(n2/3V 1/3\n\n-\n\nNoisy function value feedback\n1 \u2264 p < +\u221e\n\np = +\u221e\n\n-\n\nn ) \u0398(n\n\n4p+d\n\n6p+d Vn(p, q)\n\n2p\n6p+d )\n\nOur assumption on the underlying trend \u03b81:n \u2208 F can be used to construct an upper bound of\nthis quantity of variation Vn or Vn(p, q). As a result, the algorithms in non-stationary stochastic\noptimization and their dynamic regret bounds in Table 1 will apply to our problem (modulo additional\nrestrictions on bounded domain). However, our preliminary investigation suggests that this direct\nreduction does not, in general, lead to optimal algorithms. We illustrate this observation in the\nfollowing example.\nExample 1. Let F be the set of all bounded sequences in the total variation class T V (1). It can be\nworked out that Vn(p, q) = O(1) for all p, q. Therefore the smallest regret from [Besbes et al., 2015,\nChen et al., 2018b] is obtained by taking p \u2192 +\u221e, which gives us a regret of O(n1/2). Note that we\nexpect the optimal regret to be \u02dcO(n1/3) according to the theory of locally adaptive nonparametric\nregression.\n\nIn Example 1, we have demonstrated that one cannot achieve the optimal dynamic regret using known\nresults in non-stationary stochastic optimization. We show in section 3.1 that \u201cRestarting OGD\u201d\nalgorithm has a fundamental lower bound of \u02dc\u2126(\u221an) on dynamic regret in the TV class.\nOnline nonparametric regression. As we \ufb01nalize our manuscript, it comes to our attention that our\nproblem of interest in Figure 1 can be cast as a special case of the \u201conline nonparametric regression\u201d\nproblem [Rakhlin and Sridharan, 2014, Gaillard and Gerchinovitz, 2015]. The general result of\nRakhlin and Sridharan [2014] implies the existence of an algorithm that enjoys a \u02dcO(n1/3) regret for\nthe TV class without explicitly constructing one, which shows that n1/3 is the minimax rate when\nCn = O(1) (see more details in Appendix A). To the best of our knowledge, our proposed algorithm\nremains the \ufb01rst polynomial time algorithm with \u02dcO(n1/3) regret and our results reveal more precise\n(optimal) upper and lower bounds on all parameters of the problem (see Section 3.4).\n\n3 Main results\n\nWe are now ready to present our main results.\n\n1The Vn de\ufb01nition in [Besbes et al., 2015] for strongly convex functions are de\ufb01ned a bit differently, the\n(cid:107)\u00b7(cid:107)\u221e is taken over the convex hull of minimizers. This creates some subtle confusions regarding our results\nwhich we explain in details in Appendix I.\n2We de\ufb01ne Vn(p, q) to be a factor of n\u22121/q times bigger than the original scaling presented in [Chen et al.,\n\n2018b] so the results become comparable to that of [Besbes et al., 2015].\n\n4\n\n\f3.1 Limitations of Linear Forecasters\n\nRestarting OGD as discussed in Example 1, fails to achieve the optimal regret in our setting. A\ncurious question to ask is whether it is the algorithm itself that fails or it is an artifact of a potentially\nsuboptimal regret analysis. To answer this, let\u2019s consider the class of linear forecasters \u2014 estimators\nthat outputs a \ufb01xed linear transformation of the observations y1:n. The following preliminary\nresult shows that Restarting OGD is a linear forecaster . By the results of Donoho et al. [1998],\nlinear smoothers are fundamentally limited in their ability to estimate functions with heterogeneous\nsmoothness. Since forecasting is harder than smoothing, this limitation gets directly translated to the\nsetting of linear forecasters.\nProposition 2. Online gradient descent with a \ufb01xed restart schedule is a linear forecaster. Therefore,\nit has a dynamic regret of at least \u02dc\u2126(\u221an).\n\nProof. First, observe that the stochastic gradient is of form 2(xt \u2212 yt) where xt is what the agent\nplayed at time t and yt is the noisy observation \u03b8t + Independent noise. By the online gradient\ndescent strategy with the \ufb01xed restart interval and an inductive argument, xt is a linear combination of\ny1, ..., yt\u22121 for any t. Therefore, the entire vector of predictions x1:t is a \ufb01xed linear transformation\nof y1:t\u22121. The fundamental lower bound for linear smoothers from Donoho et al. [1998] implies that\nthis algorithm will have a regret of at least \u02dc\u2126(\u221an).\n\nThe proposition implies that we will need fundamentally new nonlinear algorithmic components to\nachieve the optimal O(n1/3) regret, if it is achievable at all!\n\n3.2 Policy\n\nlength is a power of 2. i.e, a re-centered and padded version of observations.\n\nIn this section, we present our policy ARROWS (Adaptive Restarting Rule for Online averaging using\nWavelet Shrinkage). The following notations are introduced for describing the algorithm.\n\u2022 th denotes start time of the current bin and t be the current time point.\n\u2022 \u00afyth:t denotes the average of the y values for time steps indexed from th to t.\n\u2022 pad0(yth , ..., yt) denotes the vector (yth \u2212 \u00afyth:t, ..., yt \u2212 \u00afyth:t)T zero-padded at the end till its\n\u2022 T (x) where x is a sequence of values, denotes the element-wise soft thresholding of the sequence\n\u2022 H denotes the orthogonal discrete Haar wavelet transform matrix of proper dimensions\n\u2022 Let Hx = \u03b1 = [\u03b11, \u03b12, ..., \u03b1k]T where k being a power of 2 is the length of x. Then the vector\n[\u03b12, ..., \u03b1k]T can be viewed as a concatenation of log2 k contiguous blocks represented by\n\u03b1[l], l = 0, ..., log2(k) \u2212 1. Each block \u03b1[l] at level l contains 2l coef\ufb01cients.\n\nwith threshold \u03c3(cid:112)\u03b2 log(n)\n\nARROWS: inputs - observed y values, time horizon n, std deviation \u03c3, \u03b4 \u2208 (0, 1], a hyper-\nparameter \u03b2 > 24\n\n1. Initialize th = 1, newBin = 1, y0 = 0\n2. For t = 1 to n:\n\n(a) If newBin == 1, predict xth\n(b) set newBin = 0, observe yt and suffer loss (xth\n(c) Let \u02dcy = pad0(yth, ..., yt) and k be the padded length.\n(d) Let \u02c6\u03b1(th : t) = T (H \u02dcy)\n\nt = yt\u22121, else predict xth\nt \u2212 \u03b8t)2\n\nt = \u00afyth:t\u22121\n\n(e) Restart Rule: If 1\u221ak(cid:80)log2(k)\u22121\n\ni. set newBin = 1\nii. set th = t + 1\n\nl=0\n\n2l/2(cid:107)\u02c6\u03b1(th : t)[l](cid:107)1> \u03c3\u221ak\n\nthen\n\nOur policy is the byproduct of following question: How can one lift a batch estimator that is minimax\nover the TV class to a minimax online algorithm?\n\n5\n\n\fRestarting OGD when applied to our setting with squared error losses reduces to partitioning the\nduration of game into \ufb01xed size chunks and outputting online averages. As described in Section 3.1,\nthis leads to suboptimal regret. However, the notion of averaging is still a good idea to keep. If within\na time interval, the Total Variation (TV) is adequately small, then outputting sample averages is\nreasonable for minimizing the cumulative squared error. Once we encounter a bump in the variation,\na good strategy is to restart the averaging procedure. Thus we need to adaptively detect intervals with\nlow TV. For accomplishing this, we communicate with an oracle estimator whose output can be used\nto construct a lowerbound of TV within an interval. The decision to restart online averaging is based\non the estimate of TV computed using this oracle. Such a decision rule introduces non-linearity and\nhence breaks free of the suboptimal world of linear forecasters.\nThe oracle estimator we consider here is a slightly modi\ufb01ed version of the soft thresholding estimator\nfrom Donoho [1995]. We capture the high level intuition behind steps (d) and (e) as follows.\nComputation of Haar coef\ufb01cients involves smoothing adjacent regions of a signal and taking difference\nbetween them. So we can expect to construct a lowerbound of the total variation (cid:107)D\u03b81:n(cid:107)1 from these\ncoeffcients. The extra thresholding step T (.) in (d) is done to denoise the Haar coef\ufb01cients computed\nfrom noisy data. In step (e), a weighted L1 norm of denoised coef\ufb01cients is used to lowerbound\nthe total variation of the true signal. The multiplicative factors 2l/2 are introduced to make the\nlowerbound tighter. We restart online averaging once we detect a large enough variation. The \ufb01rst\ncoef\ufb01cient \u02c6\u03b1(th : t)1 is zero due to the re-centering caused by pad0 operation. The hyper-parameter\n\u03b2 controls the degree to which we shrink the noisy wavelet coef\ufb01cients. For suf\ufb01ciently small \u03b2, It is\nalmost equivalent to the universal soft-thresholding of [Donoho, 1995]. The optimal selection of \u03b2 is\ndescribed in Theorem 3.\nWe refer to the duration between two consecutive restarts inclusive of the \ufb01rst restart but exclusive of\nthe second as a bin. The policy identi\ufb01es several bins across time, whose width is adaptively chosen.\n\nFigure 2: An illustration of ARROWS on a sequence with heterogeneous smoothness. We compare\nqualitatively (on the left) and quantitatively (on the right) to two popular baselines: (a) restarting\nonline gradient descent [Besbes et al., 2015]; (b) the moving averages [Box and Jenkins, 1970] with\noptimal parameter choices. As we can see, ARROWS achieves the optimal \u02dcO(n1/3) regret while the\nbaselines are both suboptimal.\n\n3.3 Dynamic Regret of ARROWS\n\nIn this section, we provide bounds for non-stationary regret and run-time of the policy.\nTheorem 3. Let the feedback be yt = \u03b8t + Zt, t = 1, . . . , n and Zt be independent, \u03c3-subgaussian\nrandom variables. If \u03b2 = 24 + 8 log(8/\u03b4)\n, then with probability at least 1 \u2212 \u03b4, ARROWS achieves\nlog(n)\na dynamic regret of \u02dcO(n1/3(cid:107)D\u03b81:n(cid:107)2/3\n2+\u03c32) where \u02dcO hides a logarithmic\nfactor in n and 1/\u03b4.\n\n1 \u03c34/3 + |\u03b81|2+(cid:107)D\u03b81:n(cid:107)2\n\nProof Sketch. Our policy is similar in spirit to restarting OGD but with an adaptive restart schedule.\nThe key idea we used is to reduce the dynamic regret of our policy in probability roughly to a sum of\nsquared error of a soft thresholding estimator and number of restarts. This was accomplished by using\na Follow The Leader (FTL) reduction. For bounding the squared error part of the sum we modi\ufb01ed\n\n6\n\n0500010000150002000025000300001.00.50.00.51.01.52.0ogdmaarrowstruth102103104105106107n102103104105regretogdmaarrowsn1/3logn line(nlogn)1/2 line\fthe threshold value for the estimator in Donoho [1995] and proved high probability guarantees for\nthe convergence of its empirical mean. To bound the number of times we restart, we \ufb01rst establish a\nconnection between Haar coef\ufb01cients and total variation. This is intuitive since computation of Haar\ncoef\ufb01cients can be viewed as smoothing the adjacent regions of a signal and taking their difference.\nThen we exploit a special condition called \u201cuniform shrinkage\u201d of the soft-thresholding estimator\nwhich helps to optimally bound the number of restarts with high probability.\n\nn \u03c34/3 + U 2 + C 2\n\nTheorem 3 provides an upper bound of the minimax dynamic regret for forecasting the TV class.\nCorollary 4. Suppose the ground truth \u03b81:n \u2208 T V (Cn) and |\u03b81|\u2264 U. Then (cid:107)D\u03b81:n(cid:107)1\u2264 Cn. By\nnoting that (cid:107)D\u03b81:n(cid:107)2\u2264 (cid:107)D\u03b81:n(cid:107)1, under the setup in Theorem 3 ARROWS achieves a dynamic regret\nof \u02dcO(n1/3C 2/3\nRemark 5 (Adaptivity to unknown parameters.). Observe that ARROWS does not require the knowl-\nedge of Cn.It adapts optimally to the unknown TV radius Cn := (cid:107)D\u03b81:n(cid:107)1 of the ground truth \u03b81:n.\nThe adaptivity to n can be achieved by a standard doubling trick. \u03c3, if unknown, can be robustly\nestimated from the \ufb01rst few observations by a Median Absolute Deviation estimator (eg. Section 7.5\nof Johnstone [2017]), thanks to the sparsity of wavelet coef\ufb01cients of TV bounded functions.\n\nn + \u03c32) with probability at-least 1 \u2212 \u03b4.\n\n3.4 A lower bound on the minimax regret\n\nWe now give a matching lower bound of the expected regret, which establishes that ARROWS is\nadaptively minimax.\nProposition 6. Assume min{U, Cn} > 2\u03c0\u03c3 and n > 3, there is a universal constant c such that\n\nn + \u03c32 log n + n1/3C 2/3\n\nn \u03c34/3).\n\ninf\nx1:n\n\nsup\n\n\u03b81:n\u2208TV(Cn)\n\nE(cid:34) n(cid:88)t=1\n\n(xt(y1:t\u22121) \u2212 \u03b8t)2(cid:35) \u2265 c(U 2 + C 2\n\nThe proof is deferred to the Appendix I. The result shows that our result in Theorem 3 is optimal up\nto a logarithmic term in n and 1/\u03b4 for almost all regimes (modulo trivial cases of extremely small\nmin{U, Cn}/\u03c3 and n)3.\nRemark 7 (The price of forecasting). The result also shows that forecasting is strictly harder\nthan smoothing. Observe that a term with C 2\nn is required even if \u03c3 = 0, whereas in the case\nof a one-step look-ahead oracle (or the smoothing algorithm that sees all n observations) does\nnot have this term. This implies that the total amount of variation that any algorithm can handle\nwhile producing a sublinear regret has dropped from Cn = o(n) to Cn = o(\u221an). Moreover, the\nregime where the n1/3C 2/3\nn \u03c34/3 term is meaningful only when Cn = o(n1/4). For the region where\nn. This is illustrated in\n\u03c3n1/4 (cid:28) Cn (cid:28) \u03c3n1/2, the minimax regret is essentially proportional to C 2\nFigure 3.\n\nWe note that in much of the online learning literature, it is conventional to consider a slightly more\nrestrictive setting with bounded domain, which could reduce the minimax regret. The following\nremark summarizes a variant of our results in this setting.\nin bounded domain).\nRemark 8 (Minimax regret\nfrom a subset of\nB for\n\nIf we consider predicting sequences\nthe T V (Cn) ball having an extra boundedness condition |\u03b8i|\u2264\nis\nn \u03c34/3} + B2 + min{nB2, BCn} + \u03c32(cid:17). In particular, forecasting is\n\u02dc\u2126(cid:16)min{nB2, n\u03c32, n1/3C 2/3\nstill strictly harder than smoothing due to the min{nB2, BCn} term in the bound. The discussion in\nAppendix I, shows a way of using ARROWS whose regret can match this lower bound.\n\n(see Appendix I) minimax regret\n\nit can be shown that\n\ni = 1 . . . n,\n\n\u221a\n3When both U and Cn are moderately small relative to \u03c3, the lower bound will depend on \u03c3 a little differently\nbecause the estimation error goes to 0 faster than 1/\nn. We know the minimax risk exactly for that case as\n\u221a\nwell but it is somewhat messy [see e.g. Wasserman, 2006]. When they are both much smaller than \u03c3, e.g., when\nmin{U, Cn} \u2264 \u03c3/\nn, then outputting 0 when we do not have enough information will be better than doing\nonline averages.\n\n7\n\n\fof Cn. The non-trivial regime for forecasting is when Cn lies between \u03c3(cid:113) log(n)\n\nFigure 3: An illustration of the minimax (dynamic) regret of forecasters and smoothers as a function\nand \u03c3 n1/4\nwhere forecasting is just as hard as smoothing. When Cn > \u03c3 n1/4, forecasting is harder than\nsmoothing. The yellow region indicates the extra loss incurred by any minimax forecaster. The\ngreen region marks the extra loss incurred by a linear forecaster compared to minimax forecasting\nstrategy. The \ufb01gure demonstrates that linear forecasters are sub-optimal even in the non-trivial\nregime. When Cn > \u03c3 n1/2, it is impossible to design a forecasting strategy with sub-linear regret.\nn ,\n\nFor Cn > \u03c3 n, identity function is optimal estimator for smoothing and when when Cn < \u03c3(cid:113) log(n)\n\nonline averaging is optimal for both problems.\n\nn\n\n3.5 The adaptivity of ARROWS to Sobolev and Holder classes\n\nIt turns out that ARROWS is also adaptively optimal in forecasting sequences in the discrete Sobolev\nclasses and the discrete Holder classes, which are de\ufb01ned as\n\nS(C(cid:48)n) = {\u03b81:n : (cid:107)D\u03b81:n(cid:107)2\u2264 C(cid:48)n},\n\nH(B(cid:48)n) = {\u03b81:n : (cid:107)D\u03b81:n(cid:107)\u221e\u2264 B(cid:48)n}.\n\nThese classes feature sequences that are more spatially homogeneous than those in the TV\nclass. The minimax cumulative error of nonparametric estimation in the discrete Sobolev class\nis \u0398(n2/3[C(cid:48)n]2/3\u03c34/3) [see e.g., Sadhanala et al., 2016, Theorem 5 and 6].\nCorollary 9. Let the feedback be yt = \u03b8t + Zt where Zt is an independent, \u03c3-subgaussian random\nvariable. Let \u03b81:n \u2208 S(C(cid:48)n) and |\u03b81|\u2264 U. If \u03b2 = 24 + 8 log(8/\u03b4)\n, then with probability at least 1 \u2212 \u03b4,\n(cid:48)\nARROWS achieves a dynamic regret of \u02dcO(n2/3[C(cid:48)n]2/3\u03c34/3 + U 2 + [C\nn]2 + \u03c32) where \u02dcO hides a\nlogarithmic factor in n and 1/\u03b4.\n\nlog(n)\n\nThus despite the fact that ARROWS is designed for total variation class, it adapts to the optimal rates\nof forecasting sequences that are spatially regular. To gain some intuition, let\u2019s minimally expand the\nSobolev ball to a TV ball of radius Cn = \u221anC(cid:48)n. The chosen scaling of Cn activates the embedding\nS(C(cid:48)n) \u2282 T V (Cn) (see the illustration in Table 2) with both classes having same minimax rate in\nthe batch setting. This implies that dynamic regret of ARROWS is simultaneously minimax optimal\nover S(C(cid:48)n) and T V (Cn) wrt the term containing n. It can be shown that ARROWS is optimal wrt to\n(cid:48)\nn]2, U 2, \u03c32 terms as well. Minimaxity in Sobolev class implies minimaxity in Holder\nthe additive [C\nclass since it is known that a Holder ball is sandwiched between two Sobolev balls having the same\nminimax rate [see e.g., Tibshirani, 2015]. A proof of the Corollary and related experiments are\npresented in Appendix F and J.\n\n3.6 Fast computation\n\nLast but not least, we remark that there is a fast implementation of ARROWS that reduces the overall\ntime-complexity for n step from O(n2) to O(n log n).\nProposition 10. The run time of ARROWS is O(n log(n)), where n is the time horizon.\n\nThe proof exploits the sequential structure of our policy and sparsity in wavelet transforms, which\nallows us to have O(log n) incremental updates in all but O(log n) steps. See Appendix G for details.\n\n8\n\n\fClass\n\nMinimax rate for\nForecasting\n\nTV\n\nSobolev\nHolder\n\n(cid:107)D\u03b81:n(cid:107)1\u2264 Cn\n(cid:107)D\u03b81:n(cid:107)2\u2264 C(cid:48)n\n(cid:107)D\u03b81:n(cid:107)\u221e\u2264 Ln\n\nMinimax Algorithm\n\nn1/3C 2/3\n\nn \u03c34/3 + C 2\n\nn + \u03c32\n\nn2/3[C(cid:48)n]2/3\u03c34/3 + [C(cid:48)n]2 + \u03c32\n\nnL2/3\n\nn \u03c34/3 + nL2\nARROWS\n\nn + \u03c32\n\nMinimax rate for\nSmoothing\nn1/3C 2/3\nn \u03c34/3 + \u03c32\nn2/3[C(cid:48)n]2/3\u03c34/3 + \u03c32\n\nn \u03c34/3 + \u03c32\n\nnL2/3\nWavelet Smoothing\n\nTrend Filtering\nForecasting\n\nMinimax rate for\nLinear Forecasting\nn + \u03c32\nn1/2Cn\u03c3 + C 2\n\nn2/3[C(cid:48)n]2/3\u03c34/3 + [C(cid:48)n]2 + \u03c32\n\nnL2/3\n\nn + \u03c32\n\nn \u03c34/3 + nL2\nRestarting OGD\nMoving Averages\n\nTable 2: Minimax rates for cumulative error(cid:80)n\n\ni=1(\u02c6\u03b8i \u2212 \u03b8)2 in various settings and policies that\nachieve those rates. ARROWS is adaptively minimax across all of the described function classes\nwhile linear forecasters fail to perform optimally over the TV class. For simplicity, we assume U is\nsmall and hide a log n factors in all the forecasting rates.\n\nCanonical Scalinga\nTV\nCn (cid:16) 1\nSobolev C(cid:48)n (cid:16) 1/\u221an\nLn (cid:16) 1/n\n\nHolder\n\nSmoothing\n\nLinear Forecasting\n\nn1/3\nn1/3\nn1/3\n\nn1/3\nn1/3\nn1/3\n\nn1/2\nn1/3\nn1/3\n\naThe \u201ccanonical scaling\u201d are obtained by discretizing functions in canon-\nical function classes. Under the canonical scaling, Holder class \u2282 Sobolev\nclass \u2282 TV class, as shown in the \ufb01gure on the left. ARROWS is optimal for\nthe Sobolev and Holder classes inscribed in the TV class. MA and Restarting\nOGD on the other hand require different parameters and prior knowledge of\nvariational budget (i.e Cn or C(cid:48)\nn) to achieve the minimax linear rates for the\nTV class and the Sobolev/Holder class.\n\n3.7 Experimental Results\n\nTo empirically validate our results, we conducted a number of numerical simulations that compares\nthe regret of ARROWS, (Restarting) OGD and MA. Figure 2 shows the results on a function with\nheterogeneous smoothness (see the exact details and more experiments in Appendix B) with the hyper-\nparameters selected according to their theoretical optimal choice for the TV class (See Theorem 11,\n12 for OGD and MA in Appendix C). The left panel illustrates that ARROWS is locally adaptive to\nheterogeneous smoothness of the ground truth. Red peaks in the \ufb01gure signi\ufb01es restarts. During the\ninitial and \ufb01nal duration, the signal varies smoothly and ARROWS chooses a larger window size for\nonline averaging. In the middle, signal varies rather abruptly. Consequently ARROWS chooses a\nsmaller window size. On the other hand, the linear smoothers OGD and MA use a constant width and\ncannot adapt to the different regions of the space. This differences are also re\ufb02ected in the quantitative\nevaluation on the right, which clearly shows that OGD and MA has a suboptimal \u02dcO(\u221an) regret while\nARROWS attains the \u02dcO(n1/3) minimax regret!\n\n4 Concluding Discussion\n\nIn this paper, we studied the problem of online nonparametric forecasting of bounded variation\nsequences. We proposed a new forecasting policy ARROWS and proved that it achieves a cumulative\nsquare error (or dynamic regret) of \u02dcO(n1/3C 2/3\nn) with total runtime of O(n log n).\nWe also derived a lower bound for forecasting sequences with bounded total variation which matches\nthe upper bound up to a logarithmic term which certi\ufb01es the optimality of ARROWS in all parameters.\nThrough connection to linear estimation theory, we assert that no linear forecaster can achieve the\noptimal rate. ARROWS is highly adaptive and has essentially no tuning parameters. We show that it is\nadaptively minimax (up to a logarithmic factor) simultaneously for all discrete TV classes, Sobolev\nclasses and Holder classes with unknown radius. Future directions include generalizing to higher\norder TV class and other convex loss functions.\n\nn \u03c34/3+\u03c32+U 2+C 2\n\nAcknowledgement\n\nDB and YW were supported by a start-up grant from UCSB CS department and a gift from Amazon\nWeb Services. The authors thank Yining Wang for a preliminary discussion that inspires the work,\nand Akshay Krishnamurthy and Ryan Tibshirani for helpful comments to an earlier version of the\npaper.\n\n9\n\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AAAB/XicdVDJSgNBEO1xN25xuXlpDIKnMIsavYlePEYwCyQh9PTUxMaehe4aMQ7BX/HiQRGv/oc3/8bOIqjog4LHe1XdVc9PpdBo2x/W1PTM7Nz8wmJhaXllda24vlHXSaY41HgiE9X0mQYpYqihQAnNVAGLfAkN//ps6DduQGmRxJfYT6ETsV4sQsEZGqlb3Goj3GJ+nsgAFOWSaT0odIslu+xVDr2DCrXLh+6x5zqGuPsVzz2gTtkeoUQmqHaL7+0g4VkEMY6eaDl2ip2cKRRcwqDQzjSkjF+zHrQMjVkEupOPth/QXaMENEyUqRjpSP0+kbNI637km86I4ZX+7Q3Fv7xWhuFRJxdxmiHEfPxRmEmKCR1GQQOhgKPsG8K4EmZXyq+YYhxNYMMQvi6l/5O6W3YMv9gvnZxO4lgg22SH7BGHVMgJOSdVUiOc3JEH8kSerXvr0XqxXsetU9ZkZpP8gPX2CdW0lXc=AAAB/XicdVDJSgNBEO1xN25xuXlpDIKnMIsavYlePEYwCyQh9PTUxMaehe4aMQ7BX/HiQRGv/oc3/8bOIqjog4LHe1XdVc9PpdBo2x/W1PTM7Nz8wmJhaXllda24vlHXSaY41HgiE9X0mQYpYqihQAnNVAGLfAkN//ps6DduQGmRxJfYT6ETsV4sQsEZGqlb3Goj3GJ+nsgAFOWSaT0odIslu+xVDr2DCrXLh+6x5zqGuPsVzz2gTtkeoUQmqHaL7+0g4VkEMY6eaDl2ip2cKRRcwqDQzjSkjF+zHrQMjVkEupOPth/QXaMENEyUqRjpSP0+kbNI637km86I4ZX+7Q3Fv7xWhuFRJxdxmiHEfPxRmEmKCR1GQQOhgKPsG8K4EmZXyq+YYhxNYMMQvi6l/5O6W3YMv9gvnZxO4lgg22SH7BGHVMgJOSdVUiOc3JEH8kSerXvr0XqxXsetU9ZkZpP8gPX2CdW0lXc=AAAB/XicdVDJSgNBEO1xN25xuXlpDIKnMIsavYlePEYwCyQh9PTUxMaehe4aMQ7BX/HiQRGv/oc3/8bOIqjog4LHe1XdVc9PpdBo2x/W1PTM7Nz8wmJhaXllda24vlHXSaY41HgiE9X0mQYpYqihQAnNVAGLfAkN//ps6DduQGmRxJfYT6ETsV4sQsEZGqlb3Goj3GJ+nsgAFOWSaT0odIslu+xVDr2DCrXLh+6x5zqGuPsVzz2gTtkeoUQmqHaL7+0g4VkEMY6eaDl2ip2cKRRcwqDQzjSkjF+zHrQMjVkEupOPth/QXaMENEyUqRjpSP0+kbNI637km86I4ZX+7Q3Fv7xWhuFRJxdxmiHEfPxRmEmKCR1GQQOhgKPsG8K4EmZXyq+YYhxNYMMQvi6l/5O6W3YMv9gvnZxO4lgg22SH7BGHVMgJOSdVUiOc3JEH8kSerXvr0XqxXsetU9ZkZpP8gPX2CdW0lXc=f|f(x)f(y)|\uf8ff|xy| 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kD\u2713k1\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\u2713k2\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Z10(f0(x))2dx\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\u2713k1\uf8ff1AAAB/HicbZDLSgNBEEV7fMb4imbppjEIrsKMCLoM6sJlBPOAzDD0dCpJk56H3TVCmMRfceNCEbd+iDv/xk4yC0280HC4VUVV3yCRQqNtf1srq2vrG5uFreL2zu7efungsKnjVHFo8FjGqh0wDVJE0ECBEtqJAhYGElrB8Hpabz2C0iKO7nGUgBeyfiR6gjM0ll8qu+MbFweAzB37jivhgTp+qWJX7ZnoMjg5VEiuul/6crsxT0OIkEumdcexE/QyplBwCZOim2pIGB+yPnQMRiwE7WWz4yf0xDhd2ouVeRHSmft7ImOh1qMwMJ0hw4FerE3N/2qdFHuXXiaiJEWI+HxRL5UUYzpNgnaFAo5yZIBxJcytlA+YYhxNXkUTgrP45WVonlUdw3fnldpVHkeBHJFjckocckFq5JbUSYNwMiLP5JW8WU/Wi/VufcxbV6x8pkz+yPr8AfeflE4=AAAB/HicbZDLSgNBEEV7fMb4imbppjEIrsKMCLoM6sJlBPOAzDD0dCpJk56H3TVCmMRfceNCEbd+iDv/xk4yC0280HC4VUVV3yCRQqNtf1srq2vrG5uFreL2zu7efungsKnjVHFo8FjGqh0wDVJE0ECBEtqJAhYGElrB8Hpabz2C0iKO7nGUgBeyfiR6gjM0ll8qu+MbFweAzB37jivhgTp+qWJX7ZnoMjg5VEiuul/6crsxT0OIkEumdcexE/QyplBwCZOim2pIGB+yPnQMRiwE7WWz4yf0xDhd2ouVeRHSmft7ImOh1qMwMJ0hw4FerE3N/2qdFHuXXiaiJEWI+HxRL5UUYzpNgnaFAo5yZIBxJcytlA+YYhxNXkUTgrP45WVonlUdw3fnldpVHkeBHJFjckocckFq5JbUSYNwMiLP5JW8WU/Wi/VufcxbV6x8pkz+yPr8AfeflE4=AAAB/HicbZDLSgNBEEV7fMb4imbppjEIrsKMCLoM6sJlBPOAzDD0dCpJk56H3TVCmMRfceNCEbd+iDv/xk4yC0280HC4VUVV3yCRQqNtf1srq2vrG5uFreL2zu7efungsKnjVHFo8FjGqh0wDVJE0ECBEtqJAhYGElrB8Hpabz2C0iKO7nGUgBeyfiR6gjM0ll8qu+MbFweAzB37jivhgTp+qWJX7ZnoMjg5VEiuul/6crsxT0OIkEumdcexE/QyplBwCZOim2pIGB+yPnQMRiwE7WWz4yf0xDhd2ouVeRHSmft7ImOh1qMwMJ0hw4FerE3N/2qdFHuXXiaiJEWI+HxRL5UUYzpNgnaFAo5yZIBxJcytlA+YYhxNXkUTgrP45WVonlUdw3fnldpVHkeBHJFjckocckFq5JbUSYNwMiLP5JW8WU/Wi/VufcxbV6x8pkz+yPr8AfeflE4=AAAB/HicbZDLSgNBEEV7fMb4imbppjEIrsKMCLoM6sJlBPOAzDD0dCpJk56H3TVCmMRfceNCEbd+iDv/xk4yC0280HC4VUVV3yCRQqNtf1srq2vrG5uFreL2zu7efungsKnjVHFo8FjGqh0wDVJE0ECBEtqJAhYGElrB8Hpabz2C0iKO7nGUgBeyfiR6gjM0ll8qu+MbFweAzB37jivhgTp+qWJX7ZnoMjg5VEiuul/6crsxT0OIkEumdcexE/QyplBwCZOim2pIGB+yPnQMRiwE7WWz4yf0xDhd2ouVeRHSmft7ImOh1qMwMJ0hw4FerE3N/2qdFHuXXiaiJEWI+HxRL5UUYzpNgnaFAo5yZIBxJcytlA+YYhxNXkUTgrP45WVonlUdw3fnldpVHkeBHJFjckocckFq5JbUSYNwMiLP5JW8WU/Wi/VufcxbV6x8pkz+yPr8AfeflE4=supn2N+0\uf8ffx1<...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\fReferences\nNikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Cliff\nStein. A 2-competitive algorithm for online convex optimization with switching costs. In LIPIcs-\nLeibniz International Proceedings in Informatics, volume 40. Schloss Dagstuhl-Leibniz-Zentrum\nfuer Informatik, 2015.\n\nLeonard E Baum and Ted Petrie. Statistical inference for probabilistic functions of \ufb01nite state markov\n\nchains. The annals of mathematical statistics, 37(6):1554\u20131563, 1966.\n\nOmar Besbes, Yonatan Gur, and Assaf Zeevi. Non-stationary stochastic optimization. Operations\n\nresearch, 63(5):1227\u20131244, 2015.\n\nPJ Bickel et al. Minimax estimation of the mean of a normal distribution when the parameter space is\n\nrestricted. The Annals of Statistics, 9(6):1301\u20131309, 1981.\n\nLucien Birge and Pascal Massart. Gaussian model selection. Journal of the European Mathematical\n\nSociety, 3(3):203\u2013268, 2001.\n\nGeorge EP Box and Gwilym M Jenkins. Time series analysis: forecasting and control. John Wiley &\n\nSons, 1970.\n\nNicolo Cesa-Bianchi and Gabor Lugosi. Prediction, Learning, and Games. Cambridge University\n\nPress, New York, NY, USA, 2006. ISBN 0521841089.\n\nNiangjun Chen, Gautam Goel, and Adam Wierman. Smoothed online convex optimization in high\ndimensions via online balanced descent. In Conference on Learning Theory (COLT-18), 2018a.\n\nXi Chen, Yining Wang, and Yu-Xiang Wang. Non-stationary stochastic optimization under lp,\n\nq-variation measures. 2018b.\n\nAmit Daniely, Alon Gonen, and Shai Shalev-Shwartz. Strongly adaptive online learning.\n\nInternational Conference on Machine Learning, pages 1405\u20131411, 2015.\n\nIn\n\nCarl De Boor, Carl De Boor, Etats-Unis Math\u00e9maticien, Carl De Boor, and Carl De Boor. A practical\n\nguide to splines, volume 27. Springer-Verlag New York, 1978.\n\nDavid Donoho, Richard Liu, and Brenda MacGibbon. Minimax risk over hyperrectangles, and\n\nimplications. Annals of Statistics, 18(3):1416\u20131437, 1990.\n\nDavid L Donoho. De-noising by soft-thresholding. IEEE transactions on information theory, 41(3):\n\n613\u2013627, 1995.\n\nDavid L Donoho, Iain M Johnstone, et al. Minimax estimation via wavelet shrinkage. The annals of\n\nStatistics, 26(3):879\u2013921, 1998.\n\nPierre Gaillard and S\u00e9bastien Gerchinovitz. A chaining algorithm for online nonparametric regression.\n\nIn Conference on Learning Theory, pages 764\u2013796, 2015.\n\nEric Hall and Rebecca Willett. Dynamical models and tracking regret in online convex programming.\n\nIn International Conference on Machine Learning (ICML-13), pages 579\u2013587, 2013.\n\nElad Hazan and Comandur Seshadhri. Adaptive algorithms for online decision problems. In Electronic\n\ncolloquium on computational complexity (ECCC), volume 14, 2007.\n\nRobert J Hodrick and Edward C Prescott. Postwar us business cycles: an empirical investigation.\n\nJournal of Money, credit, and Banking, pages 1\u201316, 1997.\n\nJan-Christian Hutter and Philippe Rigollet. Optimal rates for total variation denoising. In Conference\n\non Learning Theory (COLT-16), 2016.\n\nAli Jadbabaie, Alexander Rakhlin, Shahin Shahrampour, and Karthik Sridharan. Online optimization:\nCompeting with dynamic comparators. In Arti\ufb01cial Intelligence and Statistics, pages 398\u2013406,\n2015.\n\n10\n\n\fIain M. Johnstone. Gaussian estimation: Sequence and wavelet models. 2017.\nSeung-Jean Kim, Kwangmoo Koh, Stephen Boyd, and Dimitry Gorinevsky. (cid:96)1 trend \ufb01ltering. SIAM\n\nReview, 51(2):339\u2013360, 2009.\n\nWouter M Koolen, Alan Malek, Peter L Bartlett, and Yasin Abbasi. Minimax time series prediction.\n\nIn Advances in Neural Information Processing Systems (NIPS\u201915), pages 2557\u20132565. 2015.\n\nWojciech Kot\u0142owski, Wouter M. Koolen, and Alan Malek. Online isotonic regression. In Annual\n\nConference on Learning Theory (COLT-16), volume 49, pages 1165\u20131189. PMLR, 2016.\n\nSt\u00e9phane Mallat. A wavelet tour of signal processing. Elsevier, 1999.\n\nEnno Mammen and Sara van de Geer. Locally apadtive regression splines. Annals of Statistics, 25(1):\n\n387\u2013413, 1997.\n\nElizbar A Nadaraya. On estimating regression. Theory of Probability & Its Applications, 9(1):\n\n141\u2013142, 1964.\n\nAlexander Rakhlin and Karthik Sridharan. Online non-parametric regression. In Conference on\n\nLearning Theory, pages 1232\u20131264, 2014.\n\nAlexander Rakhlin and Karthik Sridharan. Online nonparametric regression with general loss\n\nfunctions. CoRR, abs/1501.06598, 2015.\n\nCarl Edward Rasmussen and Christopher KI Williams. Gaussian processes for machine learning.\n\nMIT Press, 2006.\n\nVeeranjaneyulu Sadhanala, Yu-Xiang Wang, and Ryan Tibshirani. Total variation classes beyond 1d:\nMinimax rates, and the limitations of linear smoothers. Advances in Neural Information Processing\nSystems (NIPS-16), 2016.\n\nBernhard Scholkopf and Alexander J Smola. Learning with kernels: support vector machines,\n\nregularization, optimization, and beyond. MIT press, 2001.\n\nGabriel Steidl, Stephan Didas, and Julia Neumann. Splines in higher order TV regularization.\n\nInternational Journal of Computer Vision, 70(3):214\u2013255, 2006.\n\nRyan Tibshirani. Nonparametric Regression: Statistical Machine Learning, Spring 2015, 2015. URL:\nhttp://www.stat.cmu.edu/~larry/=sml/nonpar.pdf. Last visited on 2019/04/29.\n\nRyan J Tibshirani. Adaptive piecewise polynomial estimation via trend \ufb01ltering. Annals of Statistics,\n\n42(1):285\u2013323, 2014.\n\nAlexandre B. Tsybakov. Introduction to Nonparametric Estimation. Springer Publishing Company,\n\nIncorporated, 1st edition, 2008.\n\nGrace Wahba. Spline models for observational data, volume 59. Siam, 1990.\n\nYu-Xiang Wang, Alex Smola, and Ryan Tibshirani. The falling factorial basis and its statistical\napplications. In International Conference on Machine Learning (ICML-14), pages 730\u2013738, 2014.\n\nLarry Wasserman. All of Nonparametric Statistics. Springer, New York, 2006.\n\nTianbao Yang, Lijun Zhang, Rong Jin, and Jinfeng Yi. Tracking slowly moving clairvoyant: optimal\ndynamic regret of online learning with true and noisy gradient. In International Conference on\nMachine Learning (ICML-16), pages 449\u2013457, 2016.\n\nLijun Zhang, Shiyin Lu, and Zhi-Hua Zhou. Adaptive online learning in dynamic environments. In\n\nAdvances in Neural Information Processing Systems (NeurIPS-18), pages 1323\u20131333, 2018a.\n\nLijun Zhang, Tianbao Yang, Zhi-Hua Zhou, et al. Dynamic regret of strongly adaptive methods. In\n\nInternational Conference on Machine Learning (ICML-18), pages 5877\u20135886, 2018b.\n\nMartin Zinkevich. Online convex programming and generalized in\ufb01nitesimal gradient ascent. In\n\nInternational Conference on Machine Learning (ICML-03), pages 928\u2013936, 2003.\n\n11\n\n\f", "award": [], "sourceid": 5938, "authors": [{"given_name": "Dheeraj", "family_name": "Baby", "institution": "UC Santa Barbara"}, {"given_name": "Yu-Xiang", "family_name": "Wang", "institution": "UC Santa Barbara"}]}