Multitask Online Mirror Descent

We introduce and analyze MT-OMD , a multitask generalization of Online Mirror1 Descent ( OMD ) which operates by sharing updates between tasks . We prove that2 the regret of MT-OMD is of order √ 1 + σ2 ( N − 1 ) √ T , where σ2 is the task3 variance according to the geometry induced by the regularizer , N is the number4 of tasks , and T is the time horizon . Whenever tasks are similar , that is , σ2 ≤ 1,5 this improves upon the √ NT bound obtained by running independent OMDs on6 each task . Our multitask extensions of Online Gradient Descent and Exponentiated7 Gradient , two important instances of OMD , are shown to enjoy closed-form updates,8 making them easy to use in practice . Finally , we provide numerical experiments on9 several real-world datasets which support our theoretical findings.10 1 Introduction11 In multitask learning [ 9 ] , one faces a set of tasks to solve , and tries to leverage their similarities to learn12 faster . Task similarity is often formalized in terms of Euclidean distances among the best performing13 models for each task ( see [ 16 ] for an example ) . However , in online convex optimization , and Online14 Mirror Descent ( OMD ) in particular , it is well known that using different geometries to measure15 distances in the model space can bring substantial advantages — see , e.g. , [ 22 , 29 ] . For instance,16 when the model space is the probability simplex in Rd , running OMD with the KL divergence17 ( corresponding to an entropic regularizer ) allows to learn at a rate depending only logarithmically18 on d. It is thus natural to investigate to what extent measuring task similarities using geometries that19 are possibly non-Euclidean could improve the analysis of online multitask learning.20 In this work , we introduce MT-OMD , a multitask generalization of OMD which applies to any21 strongly convex regularizer . We present a regret analysis establishing that MT-OMD outperforms22 OMD ( run independently on each task ) whenever tasks are similar according to the geometry induced23 by the regularizer . Our work is inspired by the multitask extension of the Perceptron algorithm24 developed in [ 10 ] , where prior knowledge about task similarities is expressed through a symmetric25 positive definite interaction matrix A . Typically , A = I + L , where L is the Laplacian of a task26 relatedness graph with adjacency matrix W . The authors then show that the number of mistakes27 depends on ∑ i ‖ui‖22 + ∑ i , jWij‖ui − uj‖22 , where each ui denotes the best model for task i. This28 expression can be seen as a measure of the task dispersion with respect to matrix W and norm ‖ · ‖2.29 The Euclidean norm appears because the Perceptron is an instance of OMD for the hinge loss with the30 Euclidean regularizer , so that distances in the model space are measured through the corresponding31 Bregman divergence , which is the Euclidean squared norm.32 For an arbitrary strongly convex regularizer ψ , the regret of OMD is controlled by a Bregman33 divergence and a term inversely proportional to the curvature of the regularizer . The key challenge34 we face is how to extend the OMD regularizer to the multitask setting so that the dispersion term35 captures task similarities . A natural strategy would be to choose a regularizer whose Bregman36 divergence includes ∑ i , jWijBψ ( ui , uj ) . Although this mimics the Euclidean dispersion term of the37 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . Perceptron , the associated regularizer has a small curvature , compromising the divergence-curvature38 balance which , as we said , controls the regret . Observing that the Perceptron ’ s dispersion term can be39 rewritten as ‖A1/2u‖22 , where A is a block version ( across tasks ) of A and u is the concatenation40 of the task vectors ui , our solution consists in using the regularizer ψ ( A1/2 · ) , where ψ is the41 compound version of any base regularizer ψ defined on the model space . While exhibiting the right42 curvature , this regularizer has still the drawback thatA1/2u might be outside the domain of ψ. To43 get around this difficulty , we introduce a notion of variance aligned with the geometry induced by44 ψ , such that the corresponding Bregman divergence Bψ ( A1/2u , A1/2v ) is always defined for sets45 of tasks with small variance . We then show that the Bregman divergence can be upper bounded in46 terms of the task variance σ2 , and by tuning appropriately the matrix A we obtain a regret bound for47 MT-OMD that scales as √ 1 + σ2 ( N − 1 ) , which is approximately σ √ N for σ > 0 . In contrast , the48 regret of independent OMD scales as √ N , highlighting the advantage brought by MT-OMD when49 tasks have a small variance . We stress that this improvement is independent of the chosen regularizer,50 thereby offering a substantial acceleration in a wide range of scenarios . The rest of the paper is51 organized as follows . In Section 2 , we formally introduce the problem and describe MT-OMD , our52 multitask extension to solve it . In Section 3 , we derive a regret analysis for MT-OMD , highlighting53 its advantage when tasks are similar . In Section 4 , we detail some algorithmic implementations of54 MT-OMD , while numerical experiments are reported in Section 5.55 Related works . Starting from the seminal work [ 9 ] , multitask learning has been intensively studied56 for more than two decades , see [ 40 ] for a recent survey . Similarly to [ 10 ] , our work is inspired57 by the Laplacian multitask framework of [ 15 ] . This framework has been extended to kernel-based58 learning [ 34 ] , kernel-based unsupervised learning [ 21 ] , contextual bandits [ 11 ] , spectral clustering59 [ 38 ] , stratified model learning [ 37 ] , and , more recently , federated learning [ 14 ] . See also [ 24 ] for60 different applications of Laplacians in online learning . A multitask version of OMD has been61 previously proposed in [ 25 ] . Their approach , unlike ours , is cast in terms of matrix learning , and uses62 group norms and Schatten p-norm regularizers . Their bounds scale with the diameter of the model63 space according to these norms ( as opposed to scaling with the task variance , as in our analysis ) .64 Moreover , their learning bias is limited to the choice of the matrix norm regularizer and does not65 explicitly include a notion of task similarity matrix . In [ 1 , 12 ] , different multitask extensions of66 online learning are investigated , see also [ 3 , 17 , 6 , 13 ] for related extensions to meta-learning . Some67 online multitask applications are studied in [ 31 , 26 , 27 ] , but without providing any regret analyses.68 In [ 32 , 39 ] authors extend the results of [ 10 ] to dynamically updated interaction matrices . However,69 no regret bounds are provided . In [ 28 ] they look at distributed online classification in a synchronous70 setting . Other approaches for learning task similarities have been investigated in [ 41 , 30 , 35 ] .71 2 Multitask online learning72 In this section , we introduce the problem and describe our approach to solve it . We use a cooperative73 and asynchronous multiagent formalism , where an online algorithm is run in a distributed fashion by74 agents that communicate to solve their tasks , but make predictions at different time steps.75 Problem formulation and reminders on OMD . We consider an online convex optimization76 setting with a set of N ∈ N agents , each learning a possibly different task on a common convex77 decision set V ⊂ Rd . At each time step t = 1 , 2 , . . . some agent it ≤ N makes a prediction xt ∈ V78 for its task , incurs loss ` t ( xt ) , and observes a subgradient of ` t at xt , where ` t is a convex loss79 function . We say that it is the active agent at time t. Both the sequence i1 , i2 , . . . of active agents and80 the sequence ` 1 , ` 2 , . . . of convex losses are chosen adversarially and hidden from the agents . Our81 goal is to minimize the multitask regret , defined as the sum of the individual regrets:82 RT = N∑ i=1 ( ∑ t : it=i ` t ( xt ) − inf u∈V ∑ t : it=i ` t ( u ) ) = T∑ t=1 ` t ( xt ) − N∑ i=1 inf u∈V ∑ t : it=i ` t ( u ) . ( 1 ) A first natural idea to minimize ( 1 ) is to run N independent OMDs , one for each agent . Recall that83 OMD refers to a family of algorithms , typically used to minimize a regret of the form ∑ t ` t ( xt ) −84 infu∈V ∑ t ` t ( u ) , for any sequence of proper convex loss functions ` t. An instance of OMD is85 parameterized by a λ-strongly convex regularizer ψ : Rd → R , and has the update rule86 xt+1 = argmin x∈V 〈ηtgt , x〉+Bψ ( x , xt ) , ( 2 ) where gt ∈ Rd is a subgradient of ` t at point xt , Bψ ( x , y ) = ψ ( x ) −ψ ( y ) −〈∇ψ ( y ) , x− y〉 denotes87 the Bregman divergence associated to ψ , and ηt > 0 is a tunable learning rate . Standard results allow88 to bound the regret achieved by the sequence of iterates produced by OMD . For example , for a fixed89 learning rate η and for any initial point x1 ∈ V , we have [ 29 , Theorem 6.8 ] :90 ∀u ∈ V , T∑ t=1 ` t ( xt ) − ` t ( u ) ≤ Bψ ( u , x1 ) η + η 2λ T∑ t=1 ‖gt‖2 ? , ( 3 ) with ‖ · ‖ ? the dual norm of the norm with respect to which ψ is strongly convex ( see Definition 4 ) .91 The choice of the regularizer ψ shapes the above bound through the quantities Bψ ( u , x1 ) and ‖gt‖ ? .92 When ψ = 12‖ · ‖ 2 2 , we have Bψ ( x , y ) = 1 2‖x − y‖ 2 2 , ‖ · ‖ ? = ‖ · ‖2 , λ = 1 , and the algorithm93 is known as Online Gradient Descent ( OGD ) . However , depending on the problem , a different94 choice of the regularizer might be better suited to the underlying geometry . A well-known example95 is Exponentiated Gradient ( EG ) , an instance of OMD in which V is the probability simplex in96 Rd , such that V = ∆ : = { x ∈ Rd+ : ∑ j xj = 1 } . EG uses the negative entropy regularizer97 x 7→ ∑ j xj ln ( xj ) , and assuming that ‖gt‖∞ ≤ L , one achieves bounds of order O ( L √ T ln d ) ,98 while OGD yields bounds of orderO ( L √ dT ) . We emphasize that our cooperative extension adapts to99 several types of regularizers , and can thus exploit these improvements with respect to the dependence100 on d , see Proposition 5 . Let C be the generic constant such that C √ T upper bounds the regret101 incurred by the chosen OMD , e.g. , C = L √ ln d , or C = L √ d in the above examples . Using Jensen ’ s102 inequality , one can show that the regret achieved by N independent OMDs satisfies:103 RT ≤ N∑ i=1 C √ Ti ≤ C √ NT , ( 4 ) where Ti = ∑T t=1 I { it = i } denotes the number of times agent i was active . Our goal is to show that104 introducing communication between the agents may significantly improve on ( 4 ) with respect to the105 dependence on N . We now describe the multitask extension we propose.106 A multitask extension . In order to build some intuition about our approach , we first focus on OGD.107 For i ≤ N and t ≤ T , let xi , t ∈ Rd denote the prediction maintained by agent i at time step t. By108 completing the square in Equation ( 2 ) for ψ = ψEuc : = 12‖ · ‖ 2 2 , the independent OGDs updates can109 be rewritten for all i ≤ N and t such that it = i:110 xi , t+1 = ΠV , ‖·‖2 ( xi , t − ηtgt ) , ( 5 ) where ΠV , ‖·‖ denotes the projection operator onto the convex set V according to the norm ‖ · ‖ , such111 that ΠV , ‖·‖ ( x ) = argminy∈V ‖x− y‖ . Our analysis next relies on compound representations , that112 we explain next . We use bold notation to refer to compound vectors , such that for u1 , . . . , uN ∈ Rd,113 the compound vector is u = [ u1 , . . . , uN ] ∈ RNd . For i ≤ N , we use u ( i ) to refer to the ith block of114 u , such that u ( i ) = ui in the above example . So xt is the compound vector of the ( xi , t ) Ni=1 , such that115 xt = x ( it ) t , and the multitask regret rewrites as RT ( u ) = ∑T t=1 ` t ( x ( it ) t ) − ` t ( u ( it ) ) . For any set116 V ⊂ Rd , let V = V ⊗N ⊂ RNd denote the compound set such that u ∈ V is equivalent to u ( i ) ∈ V117 for all i ≤ N . Equipped with this notation , the independent OGD updates ( 5 ) rewrite as:118 xt+1 = ΠV , ‖·‖2 ( xt − ηtḡt ) , ( 6 ) with ḡt ∈ RNd such that ḡ ( i ) t = gt for i = it , and 0Rd otherwise . In other words , only the active agent119 has a non-zero gradient and therefore makes an update . Our goal is to incorporate communication120 into this independent update . To that end , we consider the general idea of sharing updates by121 considering gradients of the formA−1ḡt , whereA−1 ∈ RNd×Nd is a shortcut notation for A−1⊗ Id122 and A ∈ RN×N is any symmetric positive definite interaction matrix . Note that A is a parameter of123 the algorithm playing the role of a learning bias . While our main result ( Theorem 1 ) holds for any124 choice of A , our more specialized bounds ( Propositions 2 to 5 ) apply to a parameterized family of125 matrices A . A simple computation shows that ( A−1ḡt ) ( i ) = A−1iit gt . Thus , every agent i makes an126 update proportional to A−1iit at each time step t. In other words , the active agent ( the only one to suffer127 a loss ) shares its update with the other agents . The crux of the proofs consists in finding a matrix128 A−1 ( or equivalently A ) such that A−1iit captures the similarity between tasks i and it . Intuitively , an129 agent i is interested in sharing the update ( gradient ) of agent it to the extent their respective tasks are130 similar . Overall , the MT-OGD update ( our cooperative extension of OGD ) thus writes:131 xt+1 = ΠV , ‖·‖A ( xt − ηtA−1ḡt ) , ( 7 ) where ‖ · ‖M denotes the Mahalanobis norm such that ‖u‖M = √ u > Mu . In comparison to ( 6 ) , the132 need for changing the norm in the projection , although unclear at first sight , can be explained in133 multiple ways . First , it is key to the analysis , as we see by examining the proof of Theorem 1 . Second,134 it can be interpreted as another way of exchanging information between agents , see Remark 1 . Finally,135 observe that update ( 7 ) can be decomposed as:136 x̃t+1 = argmin x∈RNd 〈ηtḡt , x〉+ 1 2 ‖x− xt‖2A , xt+1 = argmin x∈V 1 2 ‖x− x̃t+1‖2A , ( 8 ) showing that it is natural to keep the same norm in both updates . Most importantly , what update ( 8 ) 137 tells us , is that the MT-OGD update rule is actually an OMD update — see , e.g. , [ 29 , Section 6.4 ] 138 — with regularizer x 7→ 12‖x‖ 2 A = ψEuc ( A1/2x ) . This provides a natural path for extending our139 multitask approach to any regularizer . Given a base regularizer ψ : Rd → R , the compound regularizer140 ψ is given byψ : x ∈ RNd 7→ ∑N i=1 ψ ( x ( i ) ) . For base regularizers for which there exists φ : R→ R141 such that ψ ( x ) = ∑ j φ ( xj ) , the compound regularizer is the natural extension of ψ to RNd . Note,142 however , that the relationship can be more complex , e.g. , when the regularizer is the squared p-norm143 x 7→ 12‖x‖ 2 p. Now , using regularizer ψ ( A1/2 · ) , whose Bregman writes Bψ ( A1/2x , A1/2x′ ) , the144 MT-OMD update reads:145 xt+1 = argmin x∈V 〈ηtḡt , x〉+Bψ ( A1/2x , A1/2xt ) . ( 9 ) Clearly , when ψ = ψEuc , we recover the MT-OGD update . Observe also that whenever A = IN ,146 MT-OMD is equivalent to N independent OMDs . We conclude this exposition with an important147 remark shedding light on the way MT-OMD introduces communication between agents.148 Remark 1 . Using the substitution yt = A1/2xt , update ( 9 ) can be rewritten as:149 xt+1 = A −1/2 argmin y∈A1/2 ( V ) 〈ηtA−1/2ḡt , y〉+Bψ ( y , yt ) . ( 10 ) The two occurrences ofA−1/2 reveal that agents communicate in two distinct ways : one through the150 shared update ( the innermost occurrence ofA−1/2 ) , and one through computing the final prediction151 xt+1 as a linear combination of the solution to the optimization problem . Multiplying both sides of152 Equation ( 10 ) byA1/2 , MT-OMD can also be seen as standard OMD on the transformed iterate yt.153 3 Regret analysis154 We now provide a regret analysis for MT-OMD . We start with a general theorem presenting two155 bounds , for constant and time-varying learning rates . These results are then instantiated to different156 types of regularizer and variance in Propositions 1 to 5 . The main difficulty in the proof is to157 characterize the strong convexity of ψ ( A1/2 · ) , see Lemmas 1 and 2 . Note that the set V is always158 assumed to be convex . All technical proofs can be found in Appendix A.159 Theorem 1 . Let V ⊂ Rd , ψ : Rd → R be λ-strongly convex with respect to norm ‖ · ‖ on V , and160 A ∈ RN×N be symmetric positive definite . Let ( ` t ) Tt=1 be a sequence of proper convex loss functions,161 and set x1 ∈ V . Then , MT-OMD with ηt : = η produces a sequence of iterates ( xt ) Tt=1 which satisfies162 for all u ∈ V :163 RT ( u ) ≤ Bψ ( A1/2u , A1/2x1 ) η + max i≤N A−1ii η 2λ T∑ t=1 ‖gt‖2 ? . ( 11 ) Moreover , for any sequence of nonincreasing learning rates ( ηt ) Tt=1 , MT-OMD produces a sequence164 of iterates ( xt ) Tt=1 which satisfies for all u ∈ V :165 RT ( u ) ≤ max t≤T Bψ ( A1/2u , A1/2xt ) ηT + max i≤N A−1ii 1 2λ T∑ t=1 ηt‖gt‖2 ? . ( 12 ) 3.1 Multitask Online Gradient Descent ( MT-OGD ) 166 For ψ = 12‖ · ‖ 2 2 , A = IN ( independent updates ) , unit-norm reference vectors ( u ( i ) ) Ni=1 , L-Lipschitz167 losses , and x1 = 0 , bound ( 11 ) becomes : ND2/2η + ηTL2/2 . Choosing η = D √ N/L √ T , we168 recover the DL √ NT bound of Equation ( 4 ) . Our goal is to design interaction matrices A that make169 the right-hand side of ( 11 ) smaller . In absence of additional assumptions on the set of comparators,170 it is however impossible to get a systematic improvement : the bounds are the sums of two terms,171 and introducing interactions typically reduces one term but increases the other . To get around this172 difficulty , we introduce a simple condition on the task similarity , that allows to control the increase of173 Bψ ( A1/2u , A1/2x1 ) for a carefully designed class of interaction matrices.174 Definition 1 . Let ‖ · ‖ : Rd → R be any norm . We define the variance of u ∈ RNd w.r.t . ‖ · ‖ as:175 Var‖·‖ ( u ) = 1 N − 1 N∑ i=1 ∥∥u ( i ) − ū∥∥2 , where ū = 1 N N∑ i=1 u ( i ) . Let V ⊂ Rd , D = supu∈V ‖u‖ , and σ > 0 . The set of comparators with variance smaller than176 σ2D2 is denoted by:177 V‖·‖ , σ = { u ∈ V : Var‖·‖ ( u ) ≤ σ2D2 } . ( 13 ) For sets of comparators of the form ( 13 ) , we show that MT-OGD achieves significant improvements178 over its independent counterpart . The rationale behind this gain is fairly natural : the tasks associated179 with comparators in ( 13 ) are similar due to the variance constraint , so that communication may indeed180 help . Note that this variance condition fundamentally involves information about the entire compound181 reference vector u , and is consequently irrelevant when OGD instances are run independently . The182 class of communication matrices we consider is parameterized as follows . Define L = IN − 11 > /N .183 We consider matrices of the form A ( b ) = IN + b L , where b ≥ 0 quantifies the communication184 magnitude . For more intuition about this choice , we refer the reader to Remark 2 . We can now state a185 first result highlighting the advantage of MT-OGD.186 Proposition 1 . Let V ⊂ Rd , D = supx∈V ‖x‖2 , ψ = 12‖ · ‖ 2 2 , and σ ≤ 1 . Let ( ` t ) Tt=1 be a sequence187 of proper convex loss functions such that ‖gt‖2 ≤ L for all gt ∈ ∂ ` t ( x ) , x ∈ V . Set b = N , x1 = 0,188 and η = D √ N ( N + 1 ) ( 1 + ( N − 1 ) σ2 ) /L √ 2T . Then , MT-OGD produces a sequence of iterates189 ( xt ) T t=1 which satisfies for all u ∈ V‖·‖2 , σ:190 RT ( u ) ≤ DL √ 1 + σ2 ( N − 1 ) √ 2T . ( 14 ) Proof sketch . For the particular choice ψ = 12‖ · ‖ 2 2 , x1 = 0 , and a generic b , one can show that191 2Bψ ( A ( b ) 1/2u , A ( b ) 1/20 ) = ‖u‖22 + b ( N − 1 ) Var‖·‖2 ( u ) . Furthermore , it is easy to check that192 [ A ( b ) −1 ] ii = b+N ( 1+b ) N for all i ≤ N . Substituting these values into ( 11 ) , we get the upper bound:193 ‖u‖22 + b ( N − 1 ) Var‖·‖2 ( u ) 2η + ηTL2 2 b+N ( 1 + b ) N ≤ ND2 ( 1 + bN−1N σ 2 ) 2η + ηTL2 2 b+N ( 1 + b ) N . Setting η = NDL √ ( 1+bN−1N σ 2 ) ( 1+b ) ( b+N ) T , and then b = N , yields the desired bound.194 Thus , MT-OGD enjoys a √ 1 + σ2 ( N − 1 ) dependence , which is smaller than √ N whenever tasks195 have a variance smaller than 1 . Considering limit cases is also informative . When σ = 0 ( all tasks are196 equal ) , MT-OGD scales as if there were only one task . When σ ≥ 1 , the analysis suggests to choose197 b = 0 , i.e. , A = IN , and one recovers the performance of independent OGDs . Note that the additional198 √ 2 factor can be removed for limit cases through a better optimization in b : the bound obtained is199 DL √ C ( σ ) T , with C ( 0 ) = 1 and C ( 1 ) = N . However , the function C lacks of interpretability200 outside of the limit cases ( see Appendix A.2 ) motivating our choice to present the looser but more201 interpretable bound ( 14 ) . For large values of N and σ > 0 , we have √ 1 + σ2 ( N − 1 ) ≈ σ √ N . The202 improvement brought by MT-OGD is thus roughly proportional to the square root of the task variance.203 From now on , we refer to this gain as the variance acceleration.204 Another way to gain intuition about this upper bound is to compare it to the independent OGD lower205 bound . Consider the following setting : let N = 2d , u0 ∈ Rd be such that ‖u0‖22 = 1 − σ2 , and206 set for all i ≤ d : u ( 2i−1 ) = u0 − σ′ei , and u ( 2i ) = u0 + σ′ei , where σ′ = σ √ ( N − 1 ) /N , and207 ( ei ) i≤d is an orthonormal basis of Rd . It is easy to check that ‖u ( i ) ‖22 ≥ 1 − 2σ2 for all i ≤ N ,208 and that Var‖·‖2 ( u ) = σ 2 . Then , a standard lower bound for OGD [ 29 , Theorem 5.1 ] applied209 to the independent instances gives RT ≥ ∑ i ‖u ( i ) ‖2 √ 2Ti/4 ≥ √ 2 ( 1− 2σ2 ) NT/4 , assuming210 unit-norm loss gradients and Ti = T/N . This lower bound is strictly greater than the MT-OGD upper211 bound ( 14 ) as soon as √ ( 1− 2σ2 ) N/4 ≥ √ 1 + σ2N , or equivalently σ2 ≤ ( N − 16 ) / ( 18N ) .212 3.2 Extension to other norm regularizers213 Now , a natural question is : can the variance acceleration be achieved with other regularizers ? Indeed,214 the proof of Proposition 1 crucially relies on the fact that the Bregman term can be exactly expressed215 in terms of ‖u‖22 and Var‖·‖2 ( u ) . In the following proposition , we show that such an improvement216 is also possible for all regularizers of the form 12‖ · ‖ 2 , for arbitrary norms ‖ · ‖ , up to an additional217 multiplicative constant . A crucial application is the use of the p-norm on the probability simplex,218 which is known to exhibit a logarithmic dependence with respect to d when p = ( 2 ln d ) / ( 2 ln d− 1 ) .219 Proposition 2 . Let ‖·‖ : Rd → R be any norm , V ⊂ Rd , D = supx∈D ‖x‖ , ψ = 12‖·‖ 2 , and σ ≤ 1.220 Let ( ` t ) Tt=1 be a sequence of proper convex loss functions such that ‖gt‖ ? ≤ L for all gt ∈ ∂ ` t ( x ) ,221 x ∈ V . Set b = N , x1 = 0 , and η = D √ N ( N + 1 ) ( 1 + ( N − 1 ) σ2 ) /L √ 2T . Then , MT-OMD222 produces a sequence of iterates ( xt ) Tt=1 which satisfies for all u ∈ V‖·‖ , σ:223 RT ( u ) ≤ DL √ 1 + σ2 ( N − 1 ) √ 8T . In particular , for d ≥ 3 and V = ∆ , choosing ‖ ·‖ = ‖ ·‖p , for p = 2 ln d/ ( 2 ln d−1 ) , and assuming224 that ‖gt‖∞ ≤ L , it holds for all u ∈∆‖·‖p , σ:225 RT ( u ) ≤ L √ 1 + σ2 ( N − 1 ) √ 16e T ln d. In comparison , under the same assumptions , bound ( 14 ) would write as : L √ 1 + σ2 ( N − 1 ) √ 2Td.226 Projecting onto Vσ . The previous results reveal that whenever tasks are similar ( i.e. , u ∈ Vσ ) ,227 then using the regularizer ψ ( A1/2 · ) introduces communication and accelerates the convergence.228 However , introducing communication is not the only way to leverage the small variance condition.229 For instance , one may also use this information to directly project onto Vσ , by considering the update:230 231 xt+1 = argmin x∈Vσ 〈ηtḡt , x〉+Bψ ( A1/2x , A1/2xt ) . ( 15 ) Although not necessary in general ( Propositions 1 and 2 show that gradient communication only is232 sufficient to get an improvement ) , this refinement presents several advantages . First , it might actually233 be simpler to compute in practice , see Section 4 . Second , it allows for adaptive learning rates , that234 preserve the guarantees while being independent from the horizon T ( Proposition 3 ) . Finally , it235 allows to derive L ? bounds with the variance acceleration for smooth loss functions ( Proposition 4 ) .236 The results are stated for arbitrary norms , but bounds sharper by a factor 2 can be obtained for ‖ · ‖2.237 Proposition 3 . Let ‖·‖ : Rd → R be any norm , V ⊂ Rd , D = supx∈D ‖x‖ , ψ = 12‖·‖ 2 , and σ ≤ 1.238 Let ( ` t ) Tt=1 be a sequence of proper convex loss functions . Set ηt = D √ N ( N + 1 ) ( 1 + ( N − 1 ) σ2 ) 239 / √∑t i=1 ‖gi‖2 ? , and b = N . Then , update ( 15 ) produces a sequence of iterates ( xt ) Tt=1 which240 satisfies for all u ∈ V‖·‖ , σ:241 RT ( u ) ≤ 8D √ 1 + σ2 ( N − 1 ) √√√√ T∑ t=1 ‖gt‖2 ? . Proposition 4 . Let ‖·‖ : Rd → R be any norm , V ⊂ Rd , D = supx∈D ‖x‖ , ψ = 12‖·‖ 2 , and σ ≤ 1.242 Let ( ` t ) Tt=1 be a sequence of convexM -smooth loss functions , i.e. , ‖∇ ` t ( x ) −∇ ` t ( y ) ‖ ? ≤M‖x−y‖243 for all t ≤ T , x , y ∈ V . Set b and ηt as in Proposition 3 . Then , update ( 15 ) produces a sequence of244 iterates ( xt ) Tt=1 which satisfies for all u ∈ V‖·‖ , σ:245 RT ( u ) ≤ 16D √ 1 + σ2 ( N − 1 ) 2MD√1 + σ2 ( N − 1 ) + √√√√M T∑ t=1 ` t ( u ( it ) ) . 3.3 Regularizers on the simplex246 As can be seen from Propositions 1 to 4 , MT-OMD induces a variance acceleration for several247 regularizers ( Euclidean norm , p-norms ) , and under different assumptions on the losses ( Lipschitz248 continuity , smoothness ) . This improvement is thus orthogonal to the one achievable by choosing the249 regularizer appropriately , and bounds benefiting from both improvements can be derived , such as the250 second claim of Proposition 2 . However , all regularizers studied so far share a crucial feature : they251 are defined on the entire space Rd . As a consequence , the Bregman divergence Bψ ( A1/2u , A1/2x ) 252 is always well defined , which might not be true in general . Of particular interest is the case where253 regularizers are defined on the probability simplex ∆ . A workaround consists in assigning the value254 +∞ to the Bregman whenever either of the arguments is outside of the compound simplex ∆ = ∆⊗N .255 The choice of the interaction matrixA then becomes critical to prevent the bound from exploding , and256 calls for a new definition of the variance . Indeed , for i ≤ N we have ( A ( b ) 1/2 ) ( i ) = ( 1+b ) u ( i ) −bū.257 If all u ( i ) are equal ( say to u0 ∈ ∆ ) , then all ( A ( b ) 1/2u ) ( i ) are also equal to u0 andA ( b ) 1/2u ∈∆.258 However , if they are different , by definition of ū , for all j ≤ d , there exists i ≤ N such that259 u ( i ) j ≤ ūj . Then , for b large enough , ( 1 + b ) u ( i ) j − būj becomes negative , and ( A ( b ) 1/2u ) ( i ) is260 out of the simplex . Hopefully , the maximum acceptable value for b can be easily deduced from the261 following variance definition.262 Definition 2 . Let u ∈ Rd . For all j ≤ d , let umaxj = maxi≤N u ( i ) j and u min j = mini≤N u ( i ) j . Then,263 given σ ≤ 1 and with the convention 0/0 = 0 we define:264 Var∆ ( u ) = max j≤d ( umaxj − uminj umaxj ) 2 , and ∆σ = { u ∈∆ : Var∆ ( u ) ≤ σ2 } . Equipped with this new variance definition , we can now analyze regularizers defined on the simplex.265 Proposition 5 . Let σ ≤ 1 , and ψ : ∆→ R be λ-strongly convex w.r.t . norm ‖ · ‖ . Assume that there266 exist x∗ ∈ ∆ and C < +∞ such that for all x ∈ ∆ , Bψ ( x , x∗ ) ≤ C. Let ( ` t ) Tt=1 be a sequence of267 proper convex loss functions such that ‖gt‖ ? ≤L for all gt ∈ ∂ ` t ( x ) , x ∈ ∆ . Set b = ( 1− σ2 ) /σ2,268 x1 = [ x ∗ , . . . , x∗ ] , and η = N √ 2λ ( 1 + b ) C/L √ ( b+N ) T . Then , MT-OMD produces a sequence269 of iterates ( xt ) Tt=1 which satisfies for all u ∈∆σ:270 RT ( u ) ≤ L √ 1 + σ2 ( N − 1 ) √ 2CT/λ . In particular , for the negative entropy we have x∗=1/d and C= ln d. Hence , for subgradients271 satisfying ‖gt‖∞ ≤ L we obtain:272 RT ( u ) ≤ L √ 1 + σ2 ( N − 1 ) √ 2T ln d. Proposition 5 shows that if the variance definition is adapted to the geometry of the problem , then the273 variance acceleration can be extended to many regularizers . We conclude the theoretical analysis by274 discussing more local variance definitions and different classes of interaction matrices.275 Remark 2 . Observe that we can rewrite Var‖·‖2 ( u ) as:276 ( N − 1 ) Var‖·‖2 ( u ) = ∑ i ‖u ( i ) − ū‖22 = 1 N ∑ i , j ‖u ( i ) − u ( j ) ‖22 = u > Lu , where L = IN − 11 > /N is the Laplacian matrix of the weighted clique graph over { 1 , . . . , N } ,277 with edges of 1/N . Therefore , a natural extension consists in considering variances of the form:278 VarW‖·‖2 ( u ) = ∑N i , j=1Wij‖u ( i ) −u ( j ) ‖22 = u > LWu , for any adjacency matrixW and its Laplacian279 LW . For instance , if we expect tasks to be concentrated in clusters , it is natural to consider Wij = 1280 if u ( i ) and u ( j ) ( are thought to ) belong to the same cluster , and 0 otherwise . This local version is281 interesting as it allows to satisfy the variance condition with a smaller σ , which improves the MT-OMD282 regret bound . The proof of Theorem 1 can be readily adapted to this definition by considering the class283 of interaction matrices { A ( b ) = IN + bLW } . The complete analysis however requires computing284 maxi≤N [ A ( b ) −1 ] ii , which depends on W in a nontrivial way , preventing from stating a general285 result for an arbitrary W . Note that local variances can be similarly considered on the simplex.286 Instead of involving the global umaxj , the variance formula then features for each task/node i a local287 maximum ( respectively minimum ) taken over the neighbours N ( i ) of i : umax , ij = maxk∈N ( i ) u ( k ) j .288 4 Algorithmic implementation289 We now show that MT-OGD and MT-EG enjoy closed-form updates , making them easy to implement.290 Computing MT-OGD . Consider V = { u ∈ Rd : ‖u‖2 ≤ D } and σ ≤ 1 . Recall that we have V = V ⊗N = { u ∈ RNd : ‖u‖2 , ∞ ≤ D } , and V‖·‖2 , σ = { u ∈ V : Var‖·‖2 ( u ) ≤ σ } . Solving the first equation in ( 8 ) , the iterate xt+1 produced by MT-OGD is the solution to : min x∈RNd { ∥∥xt − ηtA−1ḡt − x∥∥2A : ‖x‖2 , ∞ ≤ D } . However , computing this update is made difficult by the discrepancy between the norms used in291 the objective and the constraint . A simple solution consists in projecting onto the Mahalanobis292 ball VA = { u ∈ RNd : ‖u‖2A ≤ ( 1 + bσ2 ) ND2 } . It is easy to check that V‖·‖2 , σ ⊂ VA , so that293 every result derived in Section 3 for MT-OGD remains valid ( only the fact that comparators and294 iterates are in VA is actually used ) . Then , using the substitution yt = A1/2xt and the notation295 Proj ( x , τ ) = min { 1 , τ‖x‖2 } x , the MT-OGD update rewrites ( see Appendix B.1 for details ) :296 yt+1 = Proj ( yt − ηtA−1/2ḡt , √ ( 1 + bσ2 ) ND ) . ( 16 ) Note that ( 16 ) can be easily turned back into an update on the xt ’ s by making the inverse substitution.297 However , ( 16 ) is closer to practice , xt being actually only computed to make predictions.298 Computing MT-EG . Using the two-step formulation ( 8 ) with yt = A1/2xt , MT-EG is given by:299 ỹt+1 = argmin y∈RNd 〈ηtA−1/2gt , y〉+Bψ ( y , yt ) , ( 17 ) yt+1 = argmin y∈A1/2 ( ∆ ) Bψ ( y , ỹt+1 ) , ( 18 ) where ψ is the compound negative entropy regularizer such that ψ ( x ) = ∑N i=1 ∑d j=1 x ( i ) j ln ( x ( i ) j ) .300 One can show ( see Appendix B.2 for details ) that the update can be rewritten as:301 ∀i ≤ N , j ≤ d , ỹ ( i ) t+1 , j = y ( i ) t , j exp ( −ηtA−1/2iit gt , j − 1 ) , y ( i ) t+1 , j = ỹ ( i ) t+1 , j∑d k=1 ỹ ( i ) t+1 , k . Combining both equations , we finally obtain that302 ∀i ≤ N , j ≤ d , y ( i ) t+1 , j = y ( i ) t , j e −ηtA−1/2iit gt , j∑d k=1 y ( i ) t , k e −ηtA−1/2iit gt , k . ( 19 ) Update ( 19 ) enjoys a natural interpretation . Each block y ( i ) is operating an individual standard EG303 update , but with gradient A−1/2iit gt . When A = IN , only the active block is updated . Otherwise , the304 update of block i is proportional to A−1/2iit , which quantifies the similarity between tasks i and it.305 Note that the derivation of ( 19 ) works for any matrix A , as long as A−1/2 is stochastic . This is true306 in particular for matrices of the form A = IN − LW , see Appendix B.2 for technical details.307 5 Experiments308 In this section , we empirically compare the cumulative performance of MT-OGD and MT-EG against309 their task-independent versions , I-OGD and I-EG . All experiments show that MT-OMD outperforms310 I-OMD for all values of T in the range of the data , see Figure 1.311 OGD . We run OGD using the parameterized clique A ( b ) = ( 1 + b ) IN − 11 > /N and the square312 loss ` t ( w ) = ( w > xt − yt ) 2 . The paramenters b ( MT-OGD ) and η ( MT-OGD and I-OGD ) were set313 according to their optimal theoretical values . The values D = 10 ( diameter ) and σ = 0.01 ( variance ) ,314 which are required to tune η in MT-OGD , were found through grid-search ( oracle tuning ) and then315 used in all experiments . We use the two following datasets . Lenk [ 4 ] consists of 2880 computer316 ratings made by 180 individuals ( the tasks ) on the basis of 14 binary features . Each computer is rated317 on a discrete scale from 0 to 10 , expressing the likelihood of an individual buying that computer.318 School [ 5 ] consists of 15362 student performances based on 28 features and collected from 139319 schools representing the tasks . Features and labels are in the range [ 0 , 1 ] . Both plots are averaged320 over 3 independent runs with respect to the random choice of the initial vector for OGD ( note that321 Lenk is simpler than School , hence the lack of variance in the plot ) .322 EG . For EG we considered the two following datasets . Extended Yale B [ 19 ] , a face classification323 dataset containing 64 images ( 1024 grey levels ) for each one of 38 individuals . We took a subset324 consisting of the first 15 individuals and ran random forest to select 10 features . We generated325 ( 15 2 ) = 105 binary classification tasks via the all-pairs reduction and used the logistic loss ` t ( w ) =326 log ( 1 + e−ytw > xt ) . The interaction matrix A used by MT-EG is a regular graph , connecting pairs of327 tasks sharing the same person . We set η in MT-EG using oracle tuning , whereas η in I-EG was set328 according to its optimal theoretical value . NYSE [ 23 ] is a portfolio selection dataset consisting of329 5651 price relatives for 36 stocks . The goal at each round is to select a portfolio ( i.e. , a probability330 vector w ) . We created 100 tasks by assigning uniformly at random the data to the tasks . The reward331 incurred at each round is the log wealth log ( w > xt ) . 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Summary: Inspired by the analysis in [10], this paper shows bounds on multitask online convex optimization w.r.t. an arbitrary strongly convex regularizer. The task relatedness is expressed I.t.o. task variance, defined by means of a task relatedness matrix. Experiments on real data are presented at the end.