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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In Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence , UAI438 2010 , page 733 , 2010.439 Checklist440 1 . For all authors ... 441 ( a ) Do the main claims made in the abstract and introduction accurately reflect the paper ’ s442 contributions and scope ? [ Yes ] 443 ( b ) Did you describe the limitations of your work ? [ Yes ] For every theoretical claim444 the required assumptions are clearly stated . In general , the acceleration provided by445 MT-OMD is conditioned upon the fact that tasks do present some similarity ( i.e. , have446 a variance σ2 smaller than 1 ) , what is reflected by the reduced sets of competitors Vσ .447 ( c ) Did you discuss any potential negative societal impacts of your work ? [ No ] This is a448 theoretically oriented paper , without any foreseeable impact on society.449 ( d ) Have you read the ethics review guidelines and ensured that your paper conforms to450 them ? [ Yes ] 451 2 . 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Summary: This paper studies the online multi-task learning setting and proposes a multi-task variant of online mirror descent that works with convex losses, a general class of Bregman divergences, and any positively weighted task-interaction matrix. They provide regret guarantees that, on two different instantiations, improve over the baseline of running independent mirror descent algorithms. The algorithms have straightforward implementations that are used on experiments on four datasets.