Provable Computational and Statistical Guarantees for Efficient Learning of Continuous-Action Graphical Games

Pro v able Computational a nd Stati stical Guaran tees for Efficien t Learning o f Con tin uous-Action Graphical Games Adarsh Barik Departmen t of Computer Science Purdue Univ ersit y W e st Lafa ye tte, Indiana, USA abarik@purd ue.edu Jean Honorio Departmen t of Computer Science Purdue Univ ersit y W e st Lafa ye tte, Indiana, USA jhonorio@pu rdue.edu Abstract In this pap er, we study the proble m o f learning the set of pure stra tegy Nash equilibria and the exa ct structure of a co n tinuous-action gra phical game with quadratic payoffs by o bs erving a small set of per turb ed equilibria. A contin uous-a ction gra phical game ca n p ossibly hav e an uncountable set of Na sh euqilibr ia. W e prop os e a ℓ 12 ´ blo ck regula r ized method which recov er s a gr aphical game, whose Nash eq uilibr ia ar e the ǫ -Nash equilibria of the game fro m which the data was genera ted (true game). Under a slightly stringent condition on the parameter s of the true game, o ur metho d recov ers the exact structure of the graphica l game. O ur metho d has a logarithmic sample c omplexity with respec t to the num ber of play ers. It also runs in polynomia l time. 1 INTR ODUCTION The r eal world is filled with scenarios wh ic h arise due to comp etitiv e actions by selfish individ ual pla y ers who are trying to maximize their o wn utilities or pa yo ffs. Non-coop erativ e game th eory h as b een considered as th e app r opriate mathematical framew ork to formally study str ate gic b eha vior in suc h m ulti-agen t scenarios. In su c h scenarios, eac h agen t decides its action b ased on the actions of other pla y ers. The core solution concept of Nash e qu ilibrium (NE) [24] serve s a descriptiv e role of th e stable outcome of the o veral l b eha vior of s elf-in terested agen ts (e.g., p eople, companies, go v ernments, groups or autonomous systems) int eracting strategically with eac h other in distribu ted settings. Graphical Games. The introduction of compact r epresen tations to game theory o ver the last t wo decades ha v e extended alg orithmic game theory’s p oten tial for large-scale, p ractical applicati ons often encountered in th e real w orld. In tro d uced within the AI communit y ab out t w o decades ago, gr aphic al games [21] constitute an example of one of th e fi rst and arguably one of the most influentia l graphical mo dels for game theory . Indeed, graphical games pla y ed a pr ominen t role in establishing the computational complexit y of compu ting NE in norm al-form games as we ll as in succinctly represent able multipla yer games (see, e.g., [9, 10, 11] and the references therein). Pla y ers can tak e actions in either a d iscr ete s p ace (for example in v oting) or in a contin uous space (for example in sim ultaneous auctions in online advertising). Corresp ond ingly , graphical games can be studied in b oth d omains. In this paper , w e f o cus on contin u ous-action graphical games. 1 Inference in Graphical Games. There h as b een considerable p rogress on c omputing classical equilibrium solution concepts suc h as NE and c orr elate d e quilibria [2] in graph ical games (see, e.g., [5, 21, 20, 26, 27, 34] and the references th erein) as wel l as on computing the pric e of anar chy in graphical games (see, e.g., [4]) . [19] iden tified the most influent ial pla y ers, i.e., a small set of pla y ers whose collect ive b ehavio r forces ev ery other pla y er to a uniqu e c hoice of action. All the w ork ab ov e fo cus on inf erence problems f or graphical games, and fall in the field of algorithmic game theory . Learning Graphical Games. The aforemen tioned problems of computing Nash equilibr iu m, correlated equilibriu m or price of anarc h y often assume that the str u cture and pay offs of the games under consideration are already av ailable. Relativ ely less atten tion has b een p aid to the problem of le arning b oth the structure and pa yoffs of graphical games fr om d ata. Addressing this pr oblem is essentia l to the dev elopmen t, p oten tial u se and su ccess of ga me-theoretic mo dels in practica l applications. In this p ap er, we study the pr oblem of learnin g the complete c haracterization of p ure strategy Nash equilibrium and structure of the graph in a contin uous-action graphical game. Related W ork. There has b een co ns iderable amount of wo rk done for learning games in th e discrete-actio n setting. [18] prop osed a m axim um-lik eliho o d approac h to learn linear influ en ce games - a class of parametric graphical games with binary act ions a nd linear pa yoffs. Ho w ev er, their metho d run s in exp onential time and th e auth ors assumed a sp ecific observ ation mo del for the strategy profiles. F or the same sp ecific ob s erv ation mo del, [15] p rop osed a p olynomial time algorithm, based on ℓ 1 -regularized logistic regression, for learning linear influence games. Their strategy profiles (or joint actions) we re dra wn from a mixture of uniform distrib utions: one o ver the p ure-strategy Nash equ ilibria (PSNE) set, and the other o v er its complement . [16] obtained necessary and sufficient conditions for learning linear in fluence games und er arb itrary observ ation mo del. [14] u se a discr im in ativ e, max-margin based approac h, to learn tree structured p olymatrix games 1 . Their metho d runs in exp onentia l time and the authors sho w th at learning p olymatrix games is NP-hard under this max-margin s etting, ev en wh en the class of graphs is restricted to trees. Finally , [17] prop osed a p olynomial time algorithm for learning sparse p olymatrix games in the discrete-action setting. Regarding in ference for con tin uous -actio n games, [12] and [32] provide a s u rv ey of v ariational inequalit y m etho ds and Gauss-Seidel m etho ds to compute generalized Nash equilibrium for pu re strategy games. [28] and [29] stu d ied a mixed-strategy acto r-critic algorithm which conv erges to a probabilit y distrib ution that assigns most weigh t to equilibrium states. [23] provided sufficien t conditions under whic h no-regret learning conv erges to equilibrium. Con tinuous-ac tion games with quadratic pa y offs ha ve b een used extensively in the game theory literature [3, 7, 13, 1]. [22] prop osed algo rithm s to learn games with quadratic pa y offs, in a simplified setting. Ho w eve r, the authors do not provide any theoretical guaran tees. In th is wo rk, w e fo cus on prov able guaran tees for a far more general class of games with quadr atic pa yo ffs in the high- dimensional regime. Our Contribution. W e aim to p rop ose a no v el metho d to learn graphical games with quadratic pa y offs, with the follo wing prov able guaran tees in mind : 1 . Correctness - W e wan t to dev elop a metho d wh ic h correctly r eco v ers the set of Nash equilibria and th e structure of the graphical games. 2. Computational efficiency - Ou r metho d m us t run fast enou gh to handle the high 1 P olymatrix games are graphical games where eac h play er’s utility is a sum of unary (single pla yer) and pairwise (tw o pla yers) potential fun ctions. 2 dimensional cases. Ideally , w e wan t to ha ve p olynomial time complexit y with resp ect to the num b er of pla y ers. 3. Sample complexit y - W e w ould like to us e as few s amp les as p ossible for reco v ering the set of Nash equilibria. W e wan t to ac hiev e logarithmic samp le complexit y with r esp ect to the n umb er of play ers. T o this end, w e prop ose a blo c k-norm r egularized metho d to learn graphical games w ith quadratic pa y off fun ctions. F or n pla ye rs, at m ost d in-neigh b ors p er p la y er, and k -dimensional action vect ors, we sho w that O p k 5 d 3 log p dnk qq samples are sufficient to reco v er the complete c har- acterizat ion of the set of ǫ -Nash equilibria. Under sligh tly more stringen t conditions, we also r eco v er the true structure of the game. Ou r metho d also runs in p olynomial time complexit y . Regarding the main c hallenges that we add ress, firs t, the set of Nash equilibria for con tin uous - action games is uncountable, while for discrete-act ion games is coun table. O u r metho d pro vides the complete charact erization of su c h un coun table sets. Our metho d is also oblivious to th e exact pro cess un der whic h p la y ers con v erge to Nash equilibria. In fact, Nash equilibria can b e ”c hosen” b y nature in an arbitrary n on-probabilistic fashion. W e also do not assume any particular pro cess that “c ho oses” Nash equilibria, suc h as, for instance, a sto c hastic pro cess. Our metho d only needs access to some small num b er p erturb ed equilibria. 2 PRELIMINARIES In this section, we in tro du ce our notatio n and formally define the problem of learning graphical games w ith quadr atic utilit y functions. C onsider a directed graph G p V , E q , where V and E are set of vertice s and ed ges resp ectiv ely . W e define V fi t 1 , . . . , n u , where eac h vertex corresp onds to one pla yer. W e denote the in-neighbors of a pla ye r i b y S i , i.e., S i “ t j | p j, i q P E u (i.e. , the set of n o des that p oin t to nod e i in the graph). All the other p lay ers are denoted b y S c i , i.e., S c i “ t 1 , . . . , n uzp S i Y i q . Let | S i | ď d and | S c i | ď n . F or eac h play er i P V , there is a set of actions or pur e-str ate gies A i . That is, pla y er i can tak e ac tion x i P A i . Eac h actio n x i consists of m aking k d ecisions on a limited budget b P R . W e consider games with conti nuous actions. Mathematically , x i P R k and } x i } 2 ď b. F or eac h pla y er i , there is also a lo cal p a y off fun ction u i : A i ˆ p Ś j P S i A j q Ñ R mapping the join t action of pla ye r i and its in-neigh b ors S i , to a real n umb er. Later, we will d efine a particular kind of lo cal pa yo ff fu nction wh ic h is of our interest. A joint action x ˚ P Ś i P V A i is a pur e-str ate g y Nash e qui librium (P SN E) of a graph ical game iff, no pla yer i has any in centiv e to unilaterally deviate from the p rescrib ed action x ˚ i P A i , give n the j oin t action of its in-n eigh b ors x ˚ S i P Ś j P S i A j in the equilibrium. W e d enote a game by G , and the set of all PSNE and ǫ -PS NE of G , by NE p G q and NE ǫ p G q resp ectiv ely , for a constan t ǫ ą 0. F ormally , NE p G q fi t x ˚ P ą i P V A i | x ˚ i P arg max x i P A i u i p x i , x ˚ S i q , @ i P V u NE ǫ p G q fi t x ˚ P ą i P V A i | u i p x ˚ i , x ˚ S i q ě ´ ǫ ` max x i P A i u i p x i , x ˚ S i q , @ i P V u P arametric P a y offs. W e are in terested in solving a parametrized ve rsion of the pr oblem. In that, giv en the we ights W ˚ ij P R k ˆ k , @ i, j P V , for eac h play er i , w e defin e the set of in-neighbors of pla y er i as S i “ t j | W ˚ ij ‰ 0 u and the pa yoff f u nction u i p x i , x S i q “ ´} x i ´ ÿ j P S i W ˚ ij x j } 2 3 Consider max x i u i p x i , x ˚ S i q “ 0 , @ i P t 1 , . . . , n u , th en in a PSNE, eac h pla yer i matc hes their action x i to the weigh ted actions of th eir n eigh b ors, i.e., ř j P S i W ˚ ij x ˚ j . Let ǫ ą 0 b e a constant. The set of all ǫ -PS NE of G is NE ǫ p G q “t x ˚ P n ą i “ 1 A i | } x ˚ i ´ ÿ j P S i W ˚ ij x ˚ j } 2 ď ǫ, @ i P V u . Sampling. Giv en the ab o v e c haracterizat ion, the set of ǫ -PSNE is a con vex p olytop e. W e ob- serv e samples from the set of noisy PSNE whic h f ollo w a local noise mec hanism that adds noise indep end en tly p er p la y er. Observed j oin t actions x “ x ˚ ` e where x ˚ is a Nash equilibr ium, that is x ˚ P NE p G q and e is indep enden t zero mean sub-Gaussian noise with v ariance pro xy σ 2 . The class of sub-Gaussian v ariates includ es for ins tance Gaussian v ariables, an y b ou n ded random v ariable (e.g. Bernoulli, multinomial, un iform), an y rand om v ari- able with strictly log-co ncav e density , an d any fin ite mixtu r e of sub -Gaussian v ariables. Norms and Notations. F or a matrix A P R p ˆ q and t wo sets S Ď t 1 , . . . , p u and T Ď t 1 , . . . , q u , A S T denotes A restricted to ro ws in S and columns in T . Similarly , A S. and A .T are ro w and column restricted m atrices resp ectiv ely . F or a vec tor m P R q , the ℓ 8 -norm is defined as } m } 8 “ max i Pt 1 ,...,p u | m i | . The F rob enius n orm f or a matrix A P R p ˆ q is defined as } A } F “ g f f e p ÿ i “ 1 q ÿ j “ 1 | A ij | 2 . The ℓ 8 -op erator norm f or A is defi ned as } A } 8 , 8 “ max i Pt 1 ,...,p u q ÿ j “ 1 | A ij | . The sp ectral norm of A is defi ned as } A } 2 , 2 “ sup } x } 2 “ 1 } Ax } 2 . W e also d efine a block m atrix norm for row-partitio ned blo c k matrices. Let A P R ř k i “ 1 p i ˆ q , @ i P t 1 , . . . , k u b e a r o w-partitioned blo ck matrix defined as follo ws: A “ “ A 1 ¨ ¨ ¨ A k ‰ ⊺ where eac h A i P R p i ˆ q . Then } A } B , 8 , F “ max i Pt 1 ,...,k u } A i } F } A } B , 8 , 1 “ max i Pt 1 ,...,k u l “ p i ,m “ q ÿ l “ 1 ,m “ 1 |r A i s lm | . 4 3 MAIN RESUL T In this section, we d escrib e our m ain theoretical results. But b efore we do that, we discus s some tec hnical assump tions wh ic h are needed for our pro ofs. Assumption 1 (Budgeted actions) . F or al l x i P A i , } x i } 2 ď b , @ i P t 1 , . . . , n u for some b ą 0 . Assumption 2 (Maxim um zero utilit y) . At PSNE , u i p x ˚ i , x ˚ - i q “ 0 , @ i P t 1 , . . . , n u . Assumption 3 (Mutu al Incoherence) . Consider H “ 1 T ř T t “ 1 ` x ˚ - i t x ˚ - i t ⊺ ` σ 2 I ˘ wher e I is the identity matrix, then “ H ‰ S c i S i “ H ‰ -1 S i S i ď 1 ´ α for some α P p 0 , 1 s . Assumption 1 simply states that eac h play er has a limited budget to allo cate for its actions. F or instance, co nsid er sim ultaneous auctions in an online advertising, wh er e a compan y c ho oses ho w to allo cate its budget int o sev eral options. F or a sufficien tly large b udget b, Assump tion 2 is not difficult to fulfill for quadratic pa y offs. W e p rop ose a m utual incoherence assumption (Assumption 3 ) for games. While mutual incoherence is new to graphical games, it has b een a standard assump tion in v arious estimation pr ob lems su c h as compressed sensing [35], Mark o v random fields [31 ], n on-parametric regression [30], diffusion net works [8], among others. No w that all our assump tions are in place, we are ready to setup ou r estimati on problem. Consider th at we h a v e access to T p erturb ed equ ilibria, i.e., we h a v e access to x t i “ x ˚ i t ` e t i where sup ers cript t d enotes the t -th sample and e t i P R k is a vec tor of zero-mean m utually indep end en t s u b- Gaussian noises with v ariance pr o xy σ 2 . W e estimate the parameters W ij for eac h i, j P t 1 , . . . , n u b y solving the follo wing optimization problem: ˆ W i ¨ “ arg min W i ¨ 1 T T ÿ t “ 1 } x t i ´ n ÿ j “ 1 j ‰ i W ij x t j } 2 2 ` λ n ÿ j “ 1 j ‰ i } W ij } F (1) where W i ¨ denotes th e collectio n of all W ij , @ j P t 1 , . . . , n u , j ‰ i . Our next theorem states that th e reco v ered ˆ W i. completely c haracterizes the set of all ǫ -Nash equilibr ia. Theorem 1. Consider a c ontinuous-action gr aphic al game G such that Assumptions 1, 2 and 3 ar e satisfie d for e ach player. L et λ ą max p 24 ? 2 1 ´ α α σ b W max b k | S i | log p 2 k 2 | S i |q T , 192 1 ´ α α σ 2 W max b k log p k 2 | S i |q T , 192 1 ´ α α σ 2 ? k W max b W max | S i | log p| S i | k q T W min , 192 1 ´ α α k 1 4 σ b log p 2 k 2 | S i |q T , 24 ? 2 α k σ b W max b | S c i log p 2 k 2 | S c i |q| T , 192 α σ 2 k W max b log p k 2 | S c i |q T , 192 α σ 2 k W max c W max | S c i | 1 2 log p| S c i | k q T , 24 ? 2 α k σ b b log p 2 k 2 | S c i |q T , 192 α σ b k log p 2 k 2 | S c i |q T , 24 p 1 ´ α q σ 2 ? k m ax ij | W ij | α , 24 σ 2 k max ij | W ij | α q , then the f ol lowing claims hold. 1. We c an r e c over NE ǫ p G q by estimating W fr om the optimization pr oblem (1) . 2. F urthermor e, for e ach player i P t 1 , . . . , n u , if min j P S i } W ˚ ij } F ą 2 δ p k , | S i | , C min , α, λ, σ, W max q , then we r e c over the exact structur e of the gr aphic al game G . wher e C min is the minimum eigenvalue of “ H ‰ S i S i , W max fi max i,j | W ij | , ǫ “ | S i | δ p k , | S i | , C min , α, λ, σ, W max q b and δ p k , | S i | , C min , α, λ, σ, W max q “ k a k | S i | 2 C min p αλ 24 p 1 ´ α q ` σ 2 p 1 ` αλ 24 p 1 ´ α q σ 2 ? k W max ` αλ 24 p 1 ´ α q σ 2 ? k max ij W max q ? k W max q ` k a k | S i | 2 C min p αλ 24 p 1 ´ α q ` αλ 24 p 1 ´ α q q ` λ 2 k a k | S i | 2 C min . 5 Pr o of. W e w ill mak e use of the primal-dual witness method to p ro v e Th eorem 1. By using the definition of F rob enious n orm, op timization p roblem (1) can b e equ iv alen tly written as ˆ W i ¨ “ arg min W i. 1 T T ÿ t “ 1 } x t i ´ n ÿ j “ 1 ,j ‰ i W ij x t j } 2 2 ` λ n ÿ j “ 1 ,j ‰ i sup } Z ij } F ď 1 x Z ij , W ij y (2) Consider the term sup } Z ij } F ď 1 x Z ij , W ij y . W e can assign sp ecific v alues to Z ij to get the maxim um p ossible v alue of x Z ij , W ij y . In particular, w e can tak e if W ij ‰ 0 then Z ij “ W ij } W ij } F , and if W ij “ 0 then } Z ij } F ď 1. Note that in th e fi rst case } Z ij } F “ 1 and th us it give s the maximum v alue for x Z ij , W ij y and no further impr o v ement is p ossible. In the second case, since W ij “ 0 , Z ij can tak e an y v alue suc h that } Z ij } F ď 1 without affecting x Z ij , W ij y . W e fix Z ij to one suc h v alue and rewrite equation (2) as ˆ W i ¨ “ arg min W i. 1 T T ÿ t “ 1 } x t i ´ n ÿ j “ 1 ,j ‰ i W ij x t j } 2 2 ` λ n ÿ j “ 1 ,j ‰ i x Z ij , W ij y (3) where the last equalit y comes by keeping in m ind that Z ij are c hosen as describ ed ab o ve. W e can rewrite equation (3) as, ˆ W i ¨ “ arg min W i ¨ 1 T T ÿ t “ 1 ` ´ 2 n ÿ j “ 1 ,j ‰ i x t i ⊺ W ij x t j ` n ÿ j “ 1 ,j ‰ i k “ 1 ,k ‰ i x t j ⊺ W ⊺ ij W ik x t k ˘ ` λ n ÿ j “ 1 ,j ‰ i x Z ij , W ij y Using the s tationarit y Karush-Kuh n-T u c k er condition at the optim um , for eac h W ij w e can write, 1 T T ÿ t “ 1 p´ 2 x t i x t j ⊺ ` 2 n ÿ k “ 1 ,k ‰ i W ik x t k x t j q ` λZ ij “ 0 (4) F urther n ote that x t i “ x ˚ i t ` e t i , x ˚ i t “ ř n j “ 1 ,j ‰ i W ˚ ij x ˚ j t and x ˚ j t “ x t j ´ e t j where e t i , e t j P R k are zero mean sub-Gaussian vecto rs with v ariance p ro xy σ 2 . Th erefore, x t i can b e written as a fu nction of W ˚ ij , x t j , e t i and e t j . Thus, by substituting x t i and writing the system of equations in vect or form, we get 1 T T ÿ t “ 1 ` 2 x t - i x t - i ⊺ p W i ¨ ´ W ˚ i ¨ q ´ 2 x t - i e t - i ⊺ W ˚ i. ´ 2 x t - i e t i ⊺ ˘ ` λ Z i ¨ “ 0 (5) where W ˚ i ¨ “ » — – W ˚ i 1 ⊺ . . . W ˚ in ⊺ fi ffi fl , W i ¨ “ » — – W ⊺ i 1 . . . W ⊺ in fi ffi fl , Z i ¨ “ » — – Z ⊺ i 1 . . . Z ⊺ in fi ffi fl , x t - i “ » — – x t 1 . . . x t n fi ffi fl , e t - i “ » — – e t 1 . . . e t n fi ffi fl (6) with W ˚ i ¨ , W i ¨ , Z i ¨ P R p n ´ 1 q k ˆ k and x t - i , e t - i P R p n ´ 1 q k ˆ 1 . If we denote in-neigh b ors of i by a set S i then W ˚ ij ⊺ “ 0 for all j R S i . W e assume th at W ij ⊺ “ 0 for all j R S i . This choice will b e ju stified later. Th us, the s tationarit y condition can b e wr itten as, 1 T T ÿ t “ 1 ` 2 “ x t - i x t - i ⊺ ‰ ¨ S i “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ ´ 2 “ x t - i e t - i ⊺ ‰ S i ¨ “ W ˚ i. ‰ S i ¨ ´ 2 x t - i e t i ⊺ ˘ ` λ Z i ¨ “ 0 (7) 6 Equation (7) can b e decomp osed in t wo separate equations. On e for the play ers in S i and other for pla y ers n ot in S i whic h we denote by S c i . 1 T T ÿ t “ 1 ` 2 “ x t - i x t - i ⊺ ‰ S i S i “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ ´ 2 “ x t - i e t - i ⊺ ‰ S i S i “ W ˚ i. ‰ S i ¨ ´ 2 “ x t - i ‰ S i ¨ e t i ⊺ ˘ ` λ “ Z i ¨ ‰ S i ¨ “ 0 (8) and 1 T T ÿ t “ 1 ` 2 “ x t - i x t - i ⊺ ‰ S c i S i “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ ´ 2 “ x t - i e t - i ⊺ ‰ S c i S i “ W ˚ i. ‰ S i ¨ ´ 2 “ x t - i ‰ S c i ¨ e t i ⊺ ˘ ` λ “ Z i ¨ ‰ S c i ¨ “ 0 (9) Let ˆ E ` . ˘ denote the emp irical exp ectation. Then equation (8) can b e written as, 2 “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ ´ 2 “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ ´ 2 ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ ` λ “ Z i ¨ ‰ S i ¨ “ 0 (10) Before we mov e ahead, we pro vide some pr op erties of the finite-sample regime whic h hold with high probabilit y . The detaile d pro ofs of these lemmas are a v ailable in App endix A . W e define ˆ H fi ˆ E ` x - i x - i ⊺ ˘ , then Lemma 1 (Posit ive minimum eigen v alue) . Λ min p “ ˆ H ‰ S i S i q ą 0 with pr ob ability at le ast 1 ´ exp p´ cT σ 4 ` O p k | S i |qq ´ exp p´ σ 2 T 128b 2 ` O p k | S i |qq for some c onstant c ą 0 wher e Λ min denotes the minimum eigen- value. Next, w e show that the mutual incoherence condition also holds in th e finite-sample regime with high probabilit y . Lemma 2 (Mutual in coherence in sample) . If } “ H ‰ S c i S i “ H ‰ ´ 1 S i S i } B , 8 , 1 ď 1 ´ α for α P p 0 , 1 s then } “ ˆ H ‰ S c i S i “ ˆ H ‰ ´ 1 S i S i } B , 8 , 1 ď 1 ´ α 2 with pr ob ability at le ast 1 ´ O p exp p ´ K T k 5 | S i | 3 ` log k | S c i | ` log k | S i |qq for some K ą 0 . No w we can use Lemma 1 and 2 to pro ve our main result. W e can rewrite equ ation (10) as, “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ “ “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ ` “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ ´ λ 2 “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ Z i ¨ ‰ S i ¨ (11) This is p ossible b ecause λ min p “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i q ą 0 f rom Lemma 1 . Using equ ation (11), w e can write equation (9) as, “ ˆ E ` x - i x - i ⊺ ˘‰ S c i S i `“ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ ` “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ ´ λ 2 “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ Z i ¨ ‰ S i ¨ ˘ ´ ˆ E `“ x - i e - i ⊺ ‰ S c i S i “ W ˚ i. ‰ S i ¨ ˘ ´ ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ ` λ 2 “ Z i ¨ ‰ S c i ¨ “ 0 (12) Let M “ “ ˆ E ` x - i x - i ⊺ ˘‰ S c i S i `“ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 , then λ 2 “ Z i ¨ ‰ S c i ¨ “ ´ M “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ ´ M ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ ` λ 2 M “ Z i ¨ ‰ S i ¨ ` ˆ E `“ x - i e - i ⊺ ‰ S c i S i “ W ˚ i. ‰ S i ¨ ˘ ` ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ 7 By taking the B , 8 , F -norm on b oth sides and using the norm triangle inequalit y , λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ď} M “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ ´ M ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ ` λ 2 M “ Z i ¨ ‰ S i ¨ } B , 8 , F ` } ˆ E `“ x - i e - i ⊺ ‰ S c i S i ˘“ W ˚ i. ‰ S i ¨ } B , 8 , F ` } ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F Using the in equalit y } AB } B , 8 , F ď } A } B , 8 , 1 } B } 8 , 2 form Lemma 10, we get λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ď} M } B , 8 , 1 ` } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ ´ ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ ` λ 2 “ Z i ¨ ‰ S i ¨ } 8 , 2 ˘ ` } ˆ E `“ x - i e - i ⊺ ‰ S c i S i ˘“ W ˚ i. ‰ S i ¨ } B , 8 , F ` } ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F Again u sing the norm triangle inequalit y , λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ď} M } B , 8 , 1 ` } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ` } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ` λ 2 } “ Z i ¨ ‰ S i ¨ } 8 , 2 ˘ ` } ˆ E `“ x - i e - i ⊺ ‰ S c i S i ˘“ W ˚ i. ‰ S i ¨ } B , 8 , F ` } ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F (13) Next, w e p ro vide some tec hnical lemmas (d etailed pro ofs in App endix A) to b ound all th e terms in righ t hand side of equation (13). Lemma 3 (Bound on } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ) . F or some ǫ 1 ą 0 , 0 ă ǫ 2 ă 8 and ǫ 3 ă 8 ? | S i | max ij | W ˚ ij | min ij | W ˚ ij | , Pr ` } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ą ǫ 1 ` σ 2 p 1 ` ǫ 2 ` ǫ 3 q max i P ind p l q ,l P S i g f f e k ÿ j “ 1 | W ˚ ij | 2 ˘ ď exp p´ ǫ 2 1 T 2 k σ 2 b 2 max j ř | S i | k “ 1 W ˚ k j 2 ` log p 2 k 2 | S i |qq ` k 2 | S i | exp p´ T ǫ 2 2 64 q ` ÿ i P ind p l q l P S i k ÿ j “ 1 exp p´ T ǫ 2 3 | W ˚ ij | 64 | c ř | S i | k “ 1 k ‰ i W ˚ k j 2 | q (14) Lemma 4 (Bound on } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ) . F or some ǫ 4 ą 0 and ǫ 5 ă 8 ? k σ 2 , Pr p} ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ě ǫ 4 ` ǫ 5 q ď exp p´ ǫ 2 4 T 2 k σ 2 b 2 ` log p 2 k 2 | S i |qq ` exp p´ ǫ 2 5 T 64 ? k σ 2 ` log p 2 k 2 | S i |qq (15) Lemma 5 (Bound on } ˆ E `“ x - i e - i ⊺ ‰ S c i S i ˘“ W ˚ i. ‰ S i ¨ } B , 8 , F ) . F or some ǫ 6 ą 0 , 0 ă ǫ 7 ă 8 and ǫ 8 ă 8 ? | S c i | max ij | W ˚ ij | min ij | W ˚ ij | , Pr ` } “ ˆ E ` x - i e - i ⊺ ˘‰ S c i S i “ W ˚ i. ‰ S i ¨ } B , 8 , F ą ǫ 6 ` σ 2 p 1 ` ǫ 7 ` ǫ 8 q max l P S c i g f f e ÿ i P ind p l q k ÿ j “ 1 | W ˚ ij | 2 ˘ ď exp p´ ǫ 2 6 T 2 k 2 σ 2 b 2 max j ř | S c i | k “ 1 W ˚ k j 2 ` log p 2 k 2 | S c i |qq ` k 2 | S c i | exp p´ T ǫ 2 7 64 q` ÿ i P ind p l q ,l P S c i k ÿ j “ 1 exp p´ T ǫ 2 8 | W ˚ ij | 64 | c ř | S c i | k “ 1 k ‰ i W ˚ k j 2 | q (16) 8 Lemma 6 (Bound on } ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F ) . F or some ǫ 9 ą 0 and ǫ 10 ă 8 k σ 2 , Pr p} ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F ě ǫ 9 ` ǫ 10 q ď exp p´ ǫ 2 9 T 2 k 2 σ 2 b 2 ` log p 2 k 2 | S c i |qq ` exp p´ ǫ 2 10 T 64 kσ 2 ` log p 2 k 2 | S c i |qq (17) Recall that, λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ď} M } B , 8 , 1 ` } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ` } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ` λ 2 } “ Z i ¨ ‰ S i ¨ } 8 , 2 ˘ ` } ˆ E `“ x t - i e t - i ⊺ ‰ S c i S i ˘“ W ˚ i. ‰ S i ¨ } B , 8 , F ` } ˆ E `“ x t - i ‰ S c i ¨ e t i ⊺ ˘ } B , 8 , F (18) W e already show ed that m utu al incoherence h olds in the fin ite-sample regime, i.e., } M } B , 8 , 1 ď 1 ´ α for some 0 ă α ă 1. Also note that } “ Z i ¨ ‰ S i ¨ } 8 , 2 ď 1. It follo ws that, λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ď p 1 ´ α q ` } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ` } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ` λ 2 ˘ ` } ˆ E `“ x - i e - i ⊺ ‰ S c i S i ˘“ W ˚ i. ‰ S i ¨ } B , 8 , F ` } ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F Using b oun ds from Lemmas 3, 4, 5, and 6, we get λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ďp 1 ´ α q ` ǫ 1 ` σ 2 p 1 ` ǫ 2 ` ǫ 3 q max i P ind p l q ,l P S i g f f e k ÿ j “ 1 | W 2 ij | ` ǫ 4 ` ǫ 5 ` λ 2 ˘ ` ǫ 6 ` σ 2 p 1 ` ǫ 7 ` ǫ 8 q max l P S c i g f f e ÿ i P ind p l q k ÿ j “ 1 | W ˚ ij | 2 ` ǫ 9 ` ǫ 10 where, ǫ 1 ą 0 , 0 ă ǫ 2 ă 8 , ǫ 3 ă 8 a | S i | max ij | W ˚ ij | min ij | W ˚ ij | ǫ 4 ą 0 , ǫ 5 ă 8 ? k σ 2 , ǫ 6 ą 0 , 0 ă ǫ 7 ă 8 , ǫ 8 ă 8 a | S c i | max ij | W ˚ ij | min ij | W ˚ ij | , ǫ 9 ą 0 , ǫ 10 ă 2 kσ 2 If these conditions hold, then λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ďp 1 ´ α q ` ǫ 1 ` σ 2 p 1 ` ǫ 2 ` ǫ 3 q ? k max ij | W ij | ` ǫ 4 ` ǫ 5 ` λ 2 ˘ ` ǫ 6 ` σ 2 p 1 ` ǫ 7 ` ǫ 8 q k max ij | W ˚ ij | ` ǫ 9 ` ǫ 10 After rearranging the terms, w e get, λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ď p 1 ´ α q ǫ 1 ` p 1 ´ α q σ 2 ? k max ij | W ij | ` p 1 ´ α q σ 2 ǫ 2 ? k max ij | W ij | ` p 1 ´ α q σ 2 ǫ 3 ? k max ij | W ij | ` p 1 ´ α q ǫ 4 ` p 1 ´ α q ǫ 5 ` p 1 ´ α q λ 2 ` ǫ 6 ` σ 2 k max ij | W ˚ ij | ` ǫ 7 σ 2 k max ij | W ˚ ij | ` σ 2 ǫ 8 k max ij | W ˚ ij | ` ǫ 9 ` ǫ 10 (19) 9 Choice of λ . T o k eep the RHS of equation (19) less than λ 2 , we need to set ǫ 1 ă αλ 24 p 1 ´ α q , ǫ 2 ă αλ 24 p 1 ´ α q σ 2 ? k max ij | W ij | , ǫ 3 ă αλ 24 p 1 ´ α q σ 2 ? k m ax ij | W ij | , ǫ 4 ă αλ 24 p 1 ´ α q , ǫ 5 ă αλ 24 p 1 ´ α q , ǫ 6 ă αλ 24 , ǫ 7 ă αλ 24 σ 2 k max ij | W ij | , ǫ 8 ă αλ 24 σ 2 k max ij | W ij | , ǫ 9 ă αλ 24 and ǫ 10 ă αλ 24 . W e also wan t to mak e sure that claim in Lemma 3, 4, 5 and 6 hold with high prob ab ility . This can b e ac h iev ed b y keeping a λ suc h that λ ą max p 24 ? 2 1 ´ α α σ b W max b k | S i | log p 2 k 2 | S i |q T , 192 1 ´ α α σ 2 W max b k log p k 2 | S i |q T , 192 1 ´ α α σ 2 ? k W max b W max | S i | log p| S i | k q T W min , 192 1 ´ α α k 1 4 σ b log p 2 k 2 | S i |q T , 24 ? 2 α k σ b W max b | S c i log p 2 k 2 | S c i |q| T , 192 α σ 2 k W max b log p k 2 | S c i |q T , 192 α σ 2 k W max c W max | S c i | 1 2 log p| S c i | k q T , 24 ? 2 α k σ b b log p 2 k 2 | S c i |q T , 192 α σ b k log p 2 k 2 | S c i |q T , 24 p 1 ´ α q σ 2 ? k max ij | W ij | α , 24 σ 2 k max ij | W ij | α q . This particular c hoice of λ implies th at λ 2 } “ Z i ¨ ‰ S c i ¨ } B , 8 , F ă λ 2 with high probabilit y whic h in turn ensures th at W ij are zero for all j P S c i with high probabilit y . No w, “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ “ “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ ` “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ ´ λ 2 “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ Z i ¨ ‰ S i ¨ (20) By taking the B , 8 , F -norm on b oth sides, } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F “} “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ ` “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ λ 2 “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ Z i ¨ ‰ S i ¨ } B , 8 , F Using the n orm triangle in equalit y , } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F ď } “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } B , 8 , F ` } “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } B , 8 , F ` } λ 2 “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 “ Z i ¨ ‰ S i ¨ } B , 8 , F Using the in equalit y } AB } B , 8 , F ď } A } B , 8 , 1 } B } 8 , 2 from Lemma 10, we get } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F ď } “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } B , 8 , 1 } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ` } “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } B , 8 , 1 } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ` } λ 2 “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } B , 8 , F } “ Z i ¨ ‰ S i ¨ } 8 , 2 Using the inequ ality } A } B , 8 , 1 ď k } A } 8 , 8 , wher e k is the m axim um num b er of ro ws in a blo ck of A , w e obtain } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F ď k } “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } 8 , 8 } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ` k } “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } 8 , 8 } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ` λ 2 k } “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } 8 , 8 Since } A } 8 , 8 ď ? p } A } 2 , 2 for A P R p ˆ p , } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F ď k a k | S i |} “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } 2 , 2 } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ` k a k | S i |} “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } 2 , 2 } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ` λ 2 k a k | S i |} “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } 2 , 2 10 Substituting for } “ ˆ E ` x - i x - i ⊺ ˘‰ S i S i -1 } 2 , 2 , } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F ď k a k | S i | 2 C min } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ` k a k | S i | 2 C min } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ` λ 2 k a k | S i | 2 C min Using results from Lemma 3 an d 4, w ith h igh p robabilit y , } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F ď k a k | S i | 2 C min p ǫ 1 ` σ 2 p 1 ` ǫ 2 ` ǫ 3 q ` k a k | S i | 2 C min p ǫ 4 ` ǫ 5 q ` λ 2 k a k | S i | 2 C min Substituting b oun ds on ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 and ǫ 5 , w e get } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F ď k a k | S i | 2 C min p αλ 24 p 1 ´ α q ` σ 2 p 1 ` αλ 24 p 1 ´ α q σ 2 ? k W max ` αλ 24 p 1 ´ α q σ 2 ? k max ij W max q ? k W max q ` k a k | S i | 2 C min p αλ 24 p 1 ´ α q ` αλ 24 p 1 ´ α q q` λ 2 k a k | S i | 2 C min Let δ p k , | S i | , C min , α, λ, σ, W max q “ k a k | S i | 2 C min p αλ 24 p 1 ´ α q ` σ 2 p 1 ` αλ 24 p 1 ´ α q σ 2 ? k W max ` αλ 24 p 1 ´ α q σ 2 ? k max ij W max q ? k W max q ` k a k | S i | 2 C min p αλ 24 p 1 ´ α q ` αλ 24 p 1 ´ α q q` λ 2 k a k | S i | 2 C min Then, } “ W i ¨ ´ W ˚ i ¨ ‰ S i ¨ } B , 8 , F ď δ p k , | S i | , C min , α, λ, σ, W max q No w, we will c haracterize NE ǫ p G q by W i ¨ . In particular, we define NE ǫ p G q “t x ˚ P ą i P V A i | x ˚ i “ ÿ j P S i W ij x ˚ j , @ i P t 1 , . . . , n uu W e explicitly compute the p a y offs to pro ve that equation (21) ind eed reco vers NE ǫ p G q , i.e., for all x ˚ P NE ǫ p G q u i p x ˚ i , x ˚ - i q “ ´} ÿ j P S i W ij x ˚ j ´ ÿ j P S i W ˚ ij x ˚ j } 2 ě ´ ÿ j P S i }p W ij ´ W ˚ ij q x ˚ j } 2 ě ´ ÿ j P S i } W ij ´ W ˚ ij } F } x j } 2 ě ´| S i | δ p k , | S i | , C min , α, λ, σ, W max q b Th us the set defin ed in equation (21) reco v ers ǫ -PSNE for ǫ “ | S i | δ p k , | S i | , C min , α, λ, σ, W max q b. Next, w e show that if for eac h pla y er i P t 1 , . . . , n u , if m in j P S i } W ˚ ij } F ą 2 δ p k , | S i | , C min , α, λ, σ, W max q then w e reco v er the exact structure of the graphical game. Note that if min j P S i } W ˚ ij } F ą 2 δ p k , | S i | , C min , α, λ, σ, W max q then } W ˚ ij } F ą 0 implies that } W ij } F ą 0. W e ha v e already sho wn that we do not reco v er an y extra pla y er in the set of in-neigh b ors S i and this added condition ensures that for all the p la y ers in S i , } W ij } F ą 0. Thus, we reco v er exact set of p la y ers in S i for eac h pla y er i P t 1 , . . . , n u . W e r eco ver the exact graph ical game b y com b ining the results for all th e pla yers. 11 Sample and Time Complexity . If we ha ve T ą O p k 5 | S i | 3 log p k | S c i || S i |qq and all other cond itions men tioned in Theorem 1 are satisfied for ev ery play er then all our high probability statemen ts are v alid f or ev ery pla yer i . T aking a u nion b oun d ov er n pla y ers only add s a factor of log n . Th us the sample complexit y for our metho d is O p k 5 | S i | 3 log p k | S c i || S i |qq . 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A Pro ofs of Theorems and Lemmas A.1 Pro of of Lemma 1 Lemma 1 [Posit ive minimum eigen v alue] Λ min p “ ˆ H ‰ S i S i q ą 0 with pr ob ability at le ast 1 ´ exp p´ cT σ 4 ` O p k | S i |qq ´ exp p´ σ 2 T 128b 2 ` O p k | S i |qq for some c onstant c ą 0 wher e Λ min denotes the minimum eigen- value. Pr o of. W e pro ve the lemma in t wo steps. First, recall th at H “ 1 T ř T t “ 1 ` x ˚ - i t x ˚ - i t ⊺ ` σ 2 I ˘ where I is iden tit y matrix. Then, Λ min p “ H ‰ S i S i q “ Λ min p 1 T T ÿ t “ 1 p “ x ˚ - i t x ˚ - i t ⊺ ‰ S i S i q ` σ 2 I q Using the in equalit y Λ min p A ` B q ě Λ min p A q ` Λ min p B q ě Λ min p 1 T T ÿ t “ 1 “ x ˚ - i t x ˚ - i t ⊺ ‰ S i S i q ` σ 2 ě σ 2 ą 0 (21) Last inequalit y f ollo w s b y noting th at 1 T ř T t “ 1 p “ x ˚ - i t x ˚ - i t ⊺ ‰ S i S i q is a p ositiv e semi-definite m atrix w ith non-negativ e eigenv alues. Ne xt, w e prov e that if T ą O O p 1 σ 2 max p b 2 , 1 σ 2 q k | S i |q , then Λ min p “ ˆ H ‰ S i S i q ą 14 0 with high probabilit y . Λ min p “ ˆ H ‰ S i S i q “ min } y } 2 “ 1 y ⊺ p 1 T T ÿ t “ 1 “ x ˚ t x ˚ t ⊺ ‰ S i S i ` “ x ˚ t e t ⊺ ‰ S i S i ` “ e t x ˚ t ⊺ ‰ S i S i ` “ e t e t ⊺ ‰ S i S i q y ě min } y } 2 “ 1 y ⊺ 1 T T ÿ t “ 1 “ x ˚ t x ˚ t ⊺ ‰ S i S i y ` m in } y } 2 “ 1 y ⊺ 1 T T ÿ t “ 1 p “ x ˚ t e t ⊺ ‰ S i S i ` “ e t x ˚ t ⊺ ‰ S i S i q y ` min } y } 2 “ 1 y ⊺ 1 T T ÿ t “ 1 “ e t e t ⊺ ‰ S i S i y Noting that 1 T T ÿ t “ 1 p “ x ˚ - i t x ˚ - i t ⊺ ‰ S i S i q is a p ositiv e semidefinite matrix ě min } y } 2 “ 1 y ⊺ 1 T T ÿ t “ 1 p “ x ˚ t e t ⊺ ‰ S i S i ` “ e t x ˚ t ⊺ ‰ S i S i q y ` min } y } 2 “ 1 y ⊺ 1 T T ÿ t “ 1 “ e t e t ⊺ ‰ S i S i y W e define a random v ariable R fi “ 1 T ř T t “ 1 y ⊺ p “ x ˚ t e t ⊺ ‰ S i S i ` “ e t x ˚ t ⊺ ‰ S i S i q y . Notice that R is a sub-Gaussian rand om v ariable w ith mean 0 and parameter 4 ř T t “ 1 a 2 t σ 2 T 2 , where a t “ y ⊺ “ x ˚ ‰ S i . ď b. Th us , Pr p R ď ´ ǫ q ď exp p´ ǫ 2 2 4 ř T t “ 1 a 2 t σ 2 T 2 q ď exp p´ ǫ 2 2 4 ř T t “ 1 b 2 σ 2 T 2 q “ exp p ´ T ǫ 2 8b 2 σ 2 q F ollo wing ǫ -nets argument from [33] and co v ariance matrix concen tration for sub-Gaussian random v ariables, we can wr ite Pr p min } y } 2 “ 1 y ⊺ 1 T T ÿ t “ 1 “ e t e t ⊺ ‰ S i S i y ą σ 2 ´ ǫ q ě 1 ´ exp p´ cǫ 2 T ` O p k | S i |qq (22) and Pr p min } y } 2 “ 1 y ⊺ 1 T T ÿ t “ 1 p “ x ˚ t e t ⊺ ‰ S i S i ` “ e t x ˚ t ⊺ ‰ S i S i q y ě ´ ǫ q ď exp p´ T ǫ 2 8b 2 σ 2 ` O p k | S i |qq (23) Th us , c ho osing ǫ “ σ 2 4 and c ho osing T “ O p 1 σ 2 max p b 2 , 1 σ 2 q k | S i |q , we get Λ min p “ ˆ H ‰ S i S i q ě σ 2 2 with high probabilit y . A.2 Pro of of Lemma 2 First, w e pr o v e a tec hnical lemma that will b e used in Lemma 2. Lemma 7. F or any δ ą 0 , the fol lowing holds: Pr p} “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i } B , 8 , 1 ě δ q ď 2 D exp p ´ δ 2 T 8 f p σ, b q k 4 | S i | 2 ` log k | S c i | ` log k | S i |q (24) Pr p} “ ˆ H ‰ S i S i ´ “ H ‰ S i S i } 8 , 8 ě δ q ď 2 D exp p ´ δ 2 T 8 f p σ, b q k 2 | S i | 2 ` 2 log k | S i |q (25) 15 Pr p}p “ ˆ H ‰ S i S i q ´ 1 ´ p “ H ‰ S i S i q ´ 1 } 8 , 8 ě δ q ď 2 D exp p´ δ 2 C 4 T 32 f p σ, b q k 3 | S i | 3 ` 2 log k | S i |q ` 2 D exp p´ C 2 T 32 f p σ, b q k 2 | S i | 2 ` 2 log k | S i |q (26) Pr o of. Note that, r “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i s j k “ Z j k “ r 1 T T ÿ t “ 1 ` x ˚ - i t e - i t ⊺ ` e - i t x ˚ - i t ⊺ ` e - i t e - i t ⊺ ´ σ 2 I ˘ s ij “ 1 T T ÿ t “ 1 Z t j k W e d efine th ree r andom v ariables R 1 fi r 1 T ř T t “ 1 ` x ˚ - i t e - i t ⊺ ˘ s j k , R 2 “ r 1 T ř T t “ 1 ` e - i t x ˚ - i t ⊺ ˘ s j k and R 3 fi r 1 T ř T t “ 1 ` e - i t e - i t ⊺ ´ σ 2 I ˘ s j k . W e will pro vide a separate b ound on these random v ariables. Note that R 1 is a sub-Gaussian r andom v ariable with 0 mean and parameter σ 2 ř T t “ 1 x - i ˚ j t 2 T 2 and R 2 is a su b-Gaussian random v ariable with 0 mean and parameter σ 2 ř T t “ 1 x - i ˚ k t 2 T 2 . Th us, Pr p| R 1 | ě ǫ q ď exp p´ ǫ 2 σ 2 ř T t “ 1 x - i ˚ j t 2 T 2 q ď 2 exp p´ T ǫ 2 b 2 σ 2 q and Pr p| R 2 | ě ǫ q ď exp p´ ǫ 2 σ 2 ř T t “ 1 x - i ˚ k t 2 T 2 q ď 2 exp p´ T ǫ 2 b 2 σ 2 q R 3 is a su b-exp onentia l random v ariable (chec k Lemm a 3 and 9). Th us, for 0 ă ǫ ă σ 2 . Pr p| R 3 | ě ǫ q ď 2 exp p´ T ǫ 2 8 σ 4 q No w, } “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i } B , 8 , 1 “ max i P S c i p ÿ j P ind p i q k ÿ k “ 1 | Z j k |q Com bining the resu lts for rand om v ariables R 1 , R 2 and R 3 and applying union b ound, w e get Pr r| Z j k | ě ǫ s ď 2 D exp p ´ ǫ 2 T 8 f p σ, b q q where D ą 0 is a constan t and f p σ , b q “ max p b 2 σ 2 8 , σ 4 q . T aking ǫ “ δ k 2 | S i | for any i P S c i . Pr p| Z j k | ě δ k 2 | S i | q ď 2 D exp p ´ δ 2 T 8 f p σ, b q k 4 | S i | 2 q 16 Using the u nion b ound o v er @ i P S c i , j P ind p i q , @ l P S i , k P ind p l q we can write, Pr p} “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i } B , 8 , 1 ě δ q ď | S c i || S i | k 2 2 D exp p ´ δ 2 T 8 f p σ, b q k 4 | S i | 2 q ď 2 D exp p ´ δ 2 T 8 f p σ, b q k 4 | S i | 2 ` log k 2 | S c i || S i |q Similarly we can p ro v e equation (25), Pr p} “ ˆ H ‰ S i S i ´ “ H ‰ S i S i } 8 , 8 ě δ q ď k 2 | S i | 2 Pr p| Z j k | ě δ k | S i | q ď 2 D exp p ´ δ 2 T 8 f p σ, b q k 2 | S i | 2 ` 2 log k | S i |q No w we prov e equation (26 ). No te that, }p “ ˆ H ‰ S i S i q ´ 1 ´ p “ H ‰ S i S i q ´ 1 } 8 , 8 “ }p “ H ‰ S i S i q ´ 1 r “ H ‰ S i S i ´ “ ˆ H ‰ S i S i sp “ ˆ H ‰ S i S i q ´ 1 } 8 , 8 ď a k | S i |}p “ H ‰ S i S i q ´ 1 r “ H ‰ S i S i ´ “ ˆ H ‰ S i S i sp “ ˆ H ‰ S i S i q ´ 1 } 2 , 2 ď a k | S i |}p “ H ‰ S i S i q ´ 1 } 2 , 2 }r “ H ‰ S i S i ´ “ ˆ H ‰ S i S i s} 2 , 2 }p “ ˆ H ‰ S i S i q ´ 1 } 2 , 2 ď a k | S i | C min }r “ H ‰ S i S i ´ “ ˆ H ‰ S i S i s} 2 , 2 }p “ ˆ H ‰ S i S i q ´ 1 } 2 , 2 Note that, Pr p Λ min p “ ˆ H ‰ S i S i q ě C min ´ δ s ě 1 ´ 2 exp p´ δ 2 T 8 k 2 | S i | 2 ` 2 log k | S i |q . T aking δ “ C min 2 , w e get Pr p Λ min p “ ˆ H ‰ S i S i q ě C min 2 q ě 1 ´ 2 exp p´ C 2 min T 32 k 2 | S i | 2 ` 2 log k | S i |q . T h is m eans that, Pr p}p “ ˆ H ‰ S i S i q ´ 1 } 2 , 2 ď 2 C min q ě 1 ´ 2 exp p´ C 2 min T 32 k 2 | S i | 2 ` 2 log k | S i |q . (27) F urtherm ore, Pr p} “ H ‰ S i S i ´ “ ˆ H ‰ S i S i } 2 , 2 ě ǫ q ď 2 D exp p´ ǫ 2 T 8 f p σ, b q k 2 | S i | 2 ` 2 log k | S i |q T aking ǫ “ δ C 2 min 2 ? k | S i | , w e get: Pr p} H S i S i ´ ˆ H S i S i } 2 , 2 ě δ C 2 min 2 a k | S i | q ď 2 D exp p´ δ 2 C 4 min T 32 k 3 f p σ, b q| S i | 3 ` 2 log k | S i |q It follo ws that, Pr p}p “ ˆ H ‰ S i S i q ´ 1 ´ p “ H ‰ S i S i q ´ 1 } 8 , 8 ď δ q ě 1 ´ 2 exp p´ δ 2 C 4 min N 32 k 3 | S i | 3 ` 2 log k | S i | ´ 2 D exp p´ C 2 min T 32 f p σ, b q k 2 | S i | 2 ` 2 log k | S i |q No w, we provi de th e detailed pro of of Lemma 2 . 17 Lemma 2 [Mutual incoherence in sample] If } “ H ‰ S c i S i “ H ‰ ´ 1 S i S i } B , 8 , 1 ď 1 ´ α for α P p 0 , 1 s then } “ ˆ H ‰ S c i S i “ ˆ H ‰ ´ 1 S i S i } B , 8 , 1 ď 1 ´ α 2 with pr obabilit y at least 1 ´ O p exp p ´ K T k 5 | S i | 3 ` log k | S c i | ` log k | S i |qq for some K ą 0. Pr o of. W e can rewrite ˆ H S c i S i p ˆ H S i S i q ´ 1 as the su m of four terms d efi ned as: “ ˆ H ‰ S c i S i p “ ˆ H ‰ S i S i q ´ 1 “ T 1 ` T 2 ` T 3 ` T 4 } “ ˆ H ‰ S c i S i p “ ˆ H ‰ S i S i q ´ 1 } B , 8 , 1 ď } T 1 } B , 8 , 1 ` } T 2 } B , 8 , 1 ` } T 3 } B , 8 , 1 ` } T 4 } B , 8 , 1 (28) where, T 1 fi “ H ‰ S c i S i rp “ ˆ H ‰ S i S i q ´ 1 ´ “ H ‰ ´ 1 S i S i s T 2 fi r “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i s “ H ‰ ´ 1 S i S i T 3 fi r “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i srp “ ˆ H ‰ S i S i q ´ 1 ´ “ H ‰ ´ 1 S i S i s T 4 fi “ H ‰ S c i S i “ H ‰ ´ 1 S i S i and eac h T i is treated as a ro w-partitioned blo ck m atrix of | S c i | blo c ks with eac h blo c k con taining k ro ws. F rom Mutual incoherence Assumption, it is clear that } T 4 } B , 8 , 1 ď 1 ´ α . W e con trol the other three terms by using results from Lemma 7 . Con trolling the first term of equation (28) . W e can wr ite T 1 as, T 1 “ ´ “ H ‰ S c i S i p “ H ‰ S i S i q ´ 1 r “ ˆ H ‰ S i S i ´ “ H ‰ S i S i sp “ ˆ H ‰ S i S i q ´ 1 then, } T 1 } B , 8 , 1 “ } “ H ‰ S c i S i p “ H ‰ S i S i q ´ 1 r “ ˆ H ‰ S i S i ´ “ H ‰ S i S i sp “ ˆ H ‰ S i S i q ´ 1 } B , 8 , 1 ď } “ H ‰ S c i S i p “ H ‰ S i S i q ´ 1 } B , 8 , 1 }r “ ˆ H ‰ S i S i ´ “ H ‰ S i S i s} 8 , 8 }p “ ˆ H ‰ S i S i q ´ 1 } 8 , 8 ď p 1 ´ α q}r “ ˆ H ‰ S i S i ´ “ H ‰ S i S i s} 8 , 8 a k | S i |}p “ ˆ H ‰ S i S i q ´ 1 } 2 , 2 No w using equation (27) and equation (25) with δ “ αC min 12 ? k | S i |p 1 ´ α q w e can say that, Pr r} T 1 } B , 8 , 1 ď α 6 s ě 1 ´ 2 exp p´ C 2 min T 32 k 2 | S i | 2 ` 2 log k | S i |q ´ 2 exp p´ K T α 2 C 2 min 144 p 1 ´ α q 2 k 3 | S i | 3 ` 2 log k | S i |q Con trolling the second term of equation (28) . W e can wr ite } T 2 } B , 8 , 1 as, } T 2 } B , 8 , 1 “ }r “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i s “ H ‰ ´ 1 S i S i } B , 8 , 1 ď }r “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i s} B , 8 , 1 } “ H ‰ ´ 1 S i S i } 8 , 8 ď }r “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i s} B , 8 , 1 a k | S i |} “ H ‰ ´ 1 S i S i } 2 , 2 ď a k | S i | C min }r “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i s} B , 8 , 1 18 Using equation (24) with δ “ αC min 6 ? k | S i | w e get, Pr p} T 2 } B , 8 , 1 ď α 6 q ě 1 ´ 2 D exp p ´ α 2 C 2 min T 288 f p σ, b q k 5 | S i | 3 ` log k | S c i | ` log k | S i |q Con trolling the third term of equation (28) . W e can write } T 3 } B , 8 , 1 as, } T 3 } B , 8 , 1 ď }r “ ˆ H ‰ S c i S i ´ “ H ‰ S c i S i s} B , 8 , 1 }rp “ ˆ H ‰ S i S i q ´ 1 ´ “ H ‰ ´ 1 S i S i s} 8 , 8 Using equation (24) and (26 ) b oth with δ “ a α 6 , w e get Pr p} T 3 } B , 8 , 1 ď α 6 q ě 1 ´ 2 D exp p´ δ 2 C 4 min T 32 f p σ, b q k 3 | S i | 3 ` 2 log k | S i |q ´ 2 D exp p´ C 2 min T 32 f p σ, b q k 2 | S i | 2 ` 2 log k | S i |q ´ 2 D exp p ´ αT 48 f p σ, b qp k 3 | S i |q 2 ` log k | S c i | ` log k | S i |q Putting ev erything together we get, Pr r} “ ˆ H ‰ S c i S i “ ˆ H ‰ ´ 1 S i S i } B , 8 , 1 ď 1 ´ α 2 s ě 1 ´ O p exp p ´ K T f p σ, b q k 5 | S i | 3 ` log k | S c i | ` log k | S i |q whic h approac hes 1 as long as we hav e N ą O p k 5 d 3 log nk q A.3 Pro of of Lemma 3 Lemma 3 [Bound on } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ] F or some ǫ 1 ą 0 , 0 ă ǫ 2 ă 8 and ǫ 3 ă 8 ? | S i | max ij | W ˚ ij | min ij | W ˚ ij | , Pr ` } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ą ǫ 1 ` σ 2 p 1 ` ǫ 2 ` ǫ 3 q max i P ind p l q ,l P S i g f f e k ÿ j “ 1 | W ˚ ij | 2 ˘ ď exp p´ ǫ 2 1 T 2 k σ 2 b 2 max j ř | S i | k “ 1 W ˚ k j 2 ` log p 2 k 2 | S i |qq ` k 2 | S i | exp p´ T ǫ 2 2 64 q ` ÿ i P ind p l q l P S i k ÿ j “ 1 exp p´ T ǫ 2 3 | W ˚ ij | 64 | c ř | S i | k “ 1 k ‰ i W ˚ k j 2 | q (29) Pr o of. Note that, } “ ˆ E ` x - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 “ } “ ˆ E ` x ˚ - i e - i ⊺ ` e - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ď } “ ˆ E ` x ˚ - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ` } “ ˆ E ` e - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 (30) Again, we will b ound b oth terms sep arately . 19 Bound on } “ ˆ E ` x ˚ - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 . F or simplicit y , let “ x ˚ - i ‰ S i “ y ˚ P R | S i | k ˆ 1 , “ e - i ‰ S i “ u P R | S i | k ˆ 1 and “ W ˚ i. ‰ S i ¨ “ W ˚ P R | S i | k ˆ k . W e d efine a random v ariable R and then, R fi ˆ E ` y ˚ u ⊺ W ˚ ˘ ij “ ˆ E ` | S i | ÿ k “ 1 y ˚ i u k W ˚ k j ˘ “ 1 T T ÿ t “ 1 ` | S i | ÿ k “ 1 y ˚ i t u t k W ˚ k j ˘ (31) F or a given y t i ˚ , @ t P t 1 , . . . , T u , R is a sub-Gaussian random v ariable with 0 mean and parameter σ 2 T 2 ř T t “ 1 ř | S i | k “ 1 ` y ˚ i t W ˚ k j ˘ 2 . Th us, for some ǫ 1 ą 0, w e can u se a tail b ound on a sub-Gaussian random v ariable: Pr ¨| y t i ˚ p| R | ą ǫ 1 q ď 2 exp p´ ǫ 2 1 T 2 2 σ 2 ř T t “ 1 ř | S i | k “ 1 ` y ˚ i t W ˚ k j ˘ 2 q ď 2 exp p´ ǫ 2 1 T 2 σ 2 b 2 ř | S i | k “ 1 W ˚ k j 2 q (32) where last inequalit y holds b ecause y ˚ i t ď b , @ t P t 1 , . . . , T u . Therefore, Pr p| R | ą ǫ 1 q “ E y t i ˚ ` Pr ¨| y t i ˚ p| R | ą ǫ 1 q ˘ ď 2 exp p´ ǫ 2 1 T 2 σ 2 b 2 ř | S i | k “ 1 W ˚ k j 2 q (33) T aking un ion b ound across i P ind p l q , @ l P S i and j P t 1 , . . . , k u , we get Pr p} “ ˆ E ` x ˚ - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ą ǫ 1 q ď 2 k 2 | S i | exp p´ ǫ 2 1 T 2 k σ 2 b 2 max j ř | S i | k “ 1 W ˚ k j 2 q “ exp p´ ǫ 2 1 T 2 k σ 2 b 2 max j ř | S i | k “ 1 W ˚ k j 2 ` log p 2 k 2 | S i |qq (34) Bound on } “ ˆ E ` e - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 . Again for simp licit y , let “ e - i ‰ S i “ u P R | S i | k ˆ 1 and “ W ˚ i. ‰ S i ¨ “ W ˚ P R | S i | k ˆ k .W e defi n e a random v ariable R and then, R fi ˆ E ` uu ⊺ W ˚ ˘ ij “ 1 T T ÿ t “ 1 ` | S i | ÿ k “ 1 u t i u t k W ˚ k j ˘ “ 1 T T ÿ t “ 1 u t i u t i W ˚ ij ` 1 T T ÿ t “ 1 ` | S i | ÿ k “ 1 k ‰ i u t i u t k W ˚ k j ˘ (35) 20 W e will b ound the random v ariable R in t wo steps. First, we will b ound the r andom v ariable R 1 fi 1 T ř T t “ 1 u t i u t i W ˚ ij and then we will b ound the random v ariable R 2 fi 1 T ř T t “ 1 ` ř | S i | k “ 1 k ‰ i u t i u t k W ˚ k j ˘ . Bound on R 1 W e observe that R 1 “ σ 2 W ˚ ij T T ÿ t “ 1 u t i σ u t i σ (36) W e d efi ne a random v ariable y fi u t i σ . Note that y is a su b-Gaussian random v ariable with 0 mean and parameter 1. W e pro ve in Lemma 8 that y 2 is a sub -exp onent ial r andom v ariable w ith parameter p 4 ? 2 , 4 q . Therefore, we can use a Bernstein t yp e b ound on p u t i σ q 2 , th us for some ǫ 2 P p 0 , 8 q , Pr p| 1 T T ÿ t “ 1 p u t i σ u t i σ ´ E ` u t i σ u t i σ ˘ q |ą ǫ 2 q ď 2 exp p´ T ǫ 2 2 64 q (37) F urtherm ore, note th at E ` u t i σ u t i σ ˘ ď 1, thus Pr p| 1 T T ÿ t “ 1 p u t i σ u t i σ ´ 1 q |ą ǫ 2 q ď 2 exp p´ T ǫ 2 2 64 q (38) Pr p σ 2 | W ˚ ij | | 1 T T ÿ t “ 1 u t i σ u t i σ ´ 1 |ą σ 2 | W ˚ ij | ǫ 2 q ď 2 exp p´ T ǫ 2 2 64 q (39) In other words, Pr p R 1 ă σ 2 | W ˚ ij | ´ σ 2 | W ˚ ij | ǫ 2 _ R 1 ą σ 2 | W ˚ ij | ` σ 2 | W ˚ ij | ǫ 2 q ď 2 exp p´ T ǫ 2 2 64 q (40) Bound on R 2 W e observe that R 2 “ σ 2 g f f f e | S i | ÿ k “ 1 k ‰ i W ˚ k j 2 1 T T ÿ t “ 1 u t i σ ř | S i | k “ 1 k ‰ i u t k W ˚ k j σ c ř | S i | k “ 1 k ‰ i W ˚ k j 2 (41) Here u t i σ and ř | S i | k “ 1 k ‰ i u t k W ˚ kj σ d ř | S i | k “ 1 k ‰ i W ˚ kj 2 are indep enden t sub -Gaussian r andom v ariables with 0 mean and pa- rameter 1. Thus, similar to Lemma 4 , w e can us e a Berns tein t yp e tail b ound for the sum of sub-exp onential random v ariables. F or s ome ǫ 3 ă 8, Pr p| 1 T T ÿ t “ 1 u t i σ ř | S i | k “ 1 k ‰ i u t k W ˚ k j σ c ř | S i | k “ 1 k ‰ i W ˚ k j 2 | ą ǫ 3 q ď 2 exp p´ T ǫ 2 3 64 q (42) 21 Or for some ǫ 3 ă 8 σ 2 | c ř | S i | k “ 1 k ‰ i W ˚ k j 2 | , Pr p| R 2 | ą ǫ 3 q ď 2 exp p´ T ǫ 2 3 64 σ 4 ř | S i | k “ 1 k ‰ i W ˚ k j 2 q (43) Com bining the b ounds on R 1 and R 2 and taking a un ion b ound, we get Pr p R ă σ 2 | W ˚ ij | ´ σ 2 | W ˚ ij | ǫ 2 ´ ǫ 3 _ R ą σ 2 | W ˚ ij | ` σ 2 | W ˚ ij | ǫ 2 ` ǫ 3 q ď 2 exp p´ T ǫ 2 2 8 q ` 2 exp p´ T ǫ 2 3 64 σ 4 | ř | S i | k “ 1 k ‰ i W ˚ k j 2 | q (44) Or for some ǫ 3 ă 8 | d ř | S i | k “ 1 k ‰ i W ˚ kj 2 | | W ˚ ij | , Pr p R ă σ 2 | W ˚ ij |p 1 ´ ǫ 2 ´ ǫ 3 q _ R ą σ 2 | W ˚ ij |p 1 ` ǫ 2 ` ǫ 3 q ď 2 exp p´ T ǫ 2 2 8 q ` 2 exp p´ T ǫ 2 3 | W ˚ ij | 64 | ř | S i | k “ 1 k ‰ i W ˚ k j 2 | q (45) T aking one sided u nion b ound across i P ind p l q , @ l P S i and j P t 1 , . . . , k u , we get Pr p} “ ˆ E ` e - i e - i ⊺ ˘‰ S i S i “ W ˚ i. ‰ S i ¨ } 8 , 2 ą σ 2 p 1 ` ǫ 2 ` ǫ 3 q max i P ind p l q ,l P S i g f f e k ÿ j “ 1 | W ˚ ij | 2 q ď k 2 | S i | exp p´ T ǫ 2 2 8 q ` ÿ i P ind p l q ,l P S i k ÿ j “ 1 exp p´ T ǫ 2 3 | W ˚ ij | 64 | ř | S i | k “ 1 k ‰ i W ˚ k j 2 | q (46) A.4 Pro of of Lemma 4 Lemma 4 [Bound on } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ] F or some ǫ 4 ą 0 and ǫ 5 ă 8 ? k σ 2 , Pr p} ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ě ǫ 4 ` ǫ 5 q ď exp p´ ǫ 2 4 T 2 k σ 2 b 2 ` log p 2 k 2 | S i |qq ` exp p´ ǫ 2 5 T 64 ? k σ 2 ` log p 2 k 2 | S i |qq (47) Pr o of. Note that “ x - i ‰ S i ¨ “ “ x ˚ - i ‰ S i ¨ ` “ e - i ‰ S i ¨ . Th us, } ˆ E `“ x - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 “ } ˆ E ` p “ x ˚ - i ‰ S i ¨ ` “ e - i ‰ S i ¨ q e i ⊺ ˘ } 8 , 2 ď } ˆ E ` p “ x ˚ - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ` } ˆ E `“ e - i ‰ S i ¨ q e i ⊺ ˘ } 8 , 2 (48) W e will b ound b oth the terms separately . 22 Bound on } ˆ E `“ x ˚ - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 . F or simp licit y , let “ x ˚ - i ‰ S i ¨ “ y ˚ P R | S i | k ˆ 1 and e i “ u P R k and w e defin e a random v ariable R R fi ˆ E ` y ˚ u ⊺ ˘ ij “ ˆ E ` y ˚ i 1 u j ˘ “ 1 T T ÿ t “ 1 p y ˚ i 1 t u t j q “ T ÿ t “ 1 R t (49) Then for a given y t i 1 , random v ariable R t is a sub-Gaussian r an d om v ariable with 0 mean and parameter y t i 1 2 σ 2 T 2 . Corr esp ondin gly R is a su b -Gaussian random v ariable with 0 mean and parameter σ 2 ř T t “ 1 y t i 1 2 T 2 . Using the tail b ound for the sub -Gaussian v ariable for some ǫ 4 ą 0, we can wr ite Pr . | y t i 1 p| R |ą ǫ 4 q ď 2 exp p´ ǫ 2 4 2 σ 2 ř T t “ 1 y t i 1 2 T 2 q ď 2 exp p´ ǫ 2 4 T 2 σ 2 b 2 q (50) where last inequalit y follo ws by noting that y t i 1 2 ď b 2 . Thus, Pr p| ˆ E ` y ˚ u ⊺ ˘ ij |ą ǫ 4 q “ E y t i 1 ` Pr . | y t i 1 p| ˆ E ` y ˚ u ⊺ ˘ ij |ą ǫ 4 q ˘ ď 2 exp p´ ǫ 2 4 T 2 σ 2 b 2 q (51) No w, } ˆ E ` y ˚ u ⊺ ˘ } 8 , 2 “ max l P S i max i P ind p l q } ˆ E ` y ˚ l 1 u ⊺ ˘ } 2 (52) T aking un ion b ound across i P ind p l q , @ l P S i and j P t 1 , . . . , k u , we get Pr p} ˆ E `“ x ˚ - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ě ǫ 4 q ď 2 k 2 | S i | exp p´ ǫ 2 4 T 2 k σ 2 b 2 q “ exp p´ ǫ 2 4 T 2 k σ 2 b 2 ` log p 2 k 2 | S i |qq (53) Bound on } ˆ E `“ e - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 . Again for s implicit y , let “ e - i ‰ S i ¨ “ v P R | S i | k ˆ 1 and e i “ u P R k and w e defin e a random v ariable R R fi ˆ E ` vu ⊺ ˘ ij “ 1 T T ÿ t “ 1 v t i 1 u t j (54) 23 Note that v t i 1 and u t j are ind ep endent sub-Gaussian random v ariables w ith 0 m ean and σ 2 param- eter. W e will use Lemma 9 to get a tail b ound on the random v ariable R . No w, E ` R ˘ “ 0 and for some ǫ 5 ą 0, Pr p| 1 T T ÿ t “ 1 v t i 1 u t j |ą ǫ 5 q “ Pr p σ 2 | 1 T T ÿ t “ 1 v t i 1 σ u t j σ |ą ǫ 5 q (55) Here v t i 1 σ and u t j σ are sub-Gaussian random v ariables with parameter 1. Thus using resu lt from Lemma 9, w e can use a Bernstein tail b ound for the sum of su b-exp onenti al random v ariables an d write, Pr p| 1 T T ÿ t “ 1 v t i 1 u t j |ą ǫ 5 q ď 2 exp p´ T ǫ 2 5 64 σ 4 q , @ ǫ 5 ă 8 σ 2 (56) Again, taking union b ound across i P in d p l q , @ l P S i and j P t 1 , . . . , k u , for all ǫ 5 ă 8 ? k σ 2 w e get Pr p} ˆ E `“ e - i ‰ S i ¨ e i ⊺ ˘ } 8 , 2 ě ǫ 5 q ď 2 k 2 | S i | exp p´ ǫ 2 5 T 64 ? k σ 4 q “ exp p´ ǫ 2 5 T 64 ? k σ 4 ` log p 2 k 2 | S i |qq (57) A.5 Pro of of Lemma 5 Lemma 5 [Bound on } ˆ E `“ x - i e - i ⊺ ‰ S c i S i ˘“ W ˚ i. ‰ S i ¨ } B , 8 , F ] F or some ǫ 6 ą 0 , 0 ă ǫ 7 ă 8 and ǫ 8 ă 8 ? | S c i | max ij | W ˚ ij | min ij | W ˚ ij | , Pr p} “ ˆ E ` x - i e - i ⊺ ˘‰ S c i S i “ W ˚ i. ‰ S i ¨ } B , 8 , F ą ǫ 6 ` σ 2 p 1 ` ǫ 7 ` ǫ 8 q max l P S c i g f f e ÿ i P ind p l q k ÿ j “ 1 | W ˚ ij | 2 q ď exp p´ ǫ 2 6 T 2 k 2 σ 2 b 2 max j ř | S c i | k “ 1 W ˚ k j 2 ` log p 2 k 2 | S c i |qq ` k 2 | S c i | exp p´ T ǫ 2 7 64 q` ÿ i P ind p l q ,l P S c i k ÿ j “ 1 exp p´ T ǫ 2 8 | W ˚ ij | 64 | c ř | S c i | k “ 1 k ‰ i W ˚ k j 2 | q (58) Pr o of. Note that, } ˆ E `“ x t - i e t - i ⊺ ‰ S c i S i ˘“ W ˚ i. ‰ S i ¨ } B , 8 , F ď } “ ˆ E ` x ˚ - i e - i ⊺ ˘‰ S c i S i “ W ˚ i. ‰ S i ¨ } B , 8 , F ` } “ ˆ E ` e - i e - i ⊺ ˘‰ S c i S i “ W ˚ i. ‰ S i ¨ } B , 8 , F (59) 24 W e can follo w the exact same argumen t of Lemma 3 to b ound the ab ov e tw o terms unt il we tak e the union b ound . This time we will take u nion b ound across i P S c i and j P t 1 , . . . , k ˆ k u , w e get Pr p} “ ˆ E ` x ˚ - i e - i ⊺ ˘‰ S c i S i “ W ˚ i. ‰ S i ¨ } B , 8 , F ą ǫ 6 q ď 2 k 2 | S c i | exp p´ ǫ 2 6 T 2 k 2 σ 2 b 2 max j ř | S c i | k “ 1 W ˚ k j 2 q “ exp p´ ǫ 2 6 T 2 k 2 σ 2 b 2 max j ř | S c i | k “ 1 W ˚ k j 2 ` log p 2 k 2 | S c i |qq (60) and Pr p} “ ˆ E ` e - i e - i ⊺ ˘‰ S c i S i “ W ˚ i. ‰ S i ¨ } B , 8 , F ą σ 2 p 1 ` ǫ 7 ` ǫ 8 q max l P S c i g f f e ÿ i P ind p l q k ÿ j “ 1 | W ˚ ij | 2 q ď k 2 | S c i | exp p´ T ǫ 2 7 64 q ` ÿ i P ind p l q ,l P S c i k ÿ j “ 1 exp p´ T ǫ 2 8 | W ˚ ij | 64 | c ř | S c i | k “ 1 k ‰ i W ˚ k j 2 | q (61) for s ome ǫ 6 , ǫ 7 and ǫ 8 ă 2 ? | S c i | max ij | W ˚ ij | min ij | W ˚ ij | . A.6 Pro of of Lemma 6 Lemma 6 [Bound on } ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F ] F or some ǫ 9 ą 0 and ǫ 10 ă 8 kσ 2 , Pr p} ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F ě ǫ 9 ` ǫ 10 q ď exp p´ ǫ 2 9 T 2 k 2 σ 2 b 2 ` log p 2 k 2 | S c i |qq ` exp p´ ǫ 2 10 T 64 kσ 2 ` log p 2 k 2 | S c i |qq (62) Pr o of. Again note that “ x - i ‰ S c i ¨ “ “ x ˚ - i ‰ S c i ¨ ` “ e - i ‰ S c i ¨ . Thus, } ˆ E `“ x - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F “ } ˆ E ` p “ x ˚ - i ‰ S c i ¨ ` “ e - i ‰ S c i ¨ q e i ⊺ ˘ } B , 8 , F ď } ˆ E ` p “ x ˚ - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F ` } ˆ E `“ e - i ‰ S c i ¨ q e i ⊺ ˘ } B , 8 , F (63) Lik e in Lemma 4 , we can b ound b oth the terms separately using similar argument s. Th e only c hange would b e that this time we will tak e union b ound across i P S c i and j P t 1 , . . . , k ˆ k u , we get Pr p} ˆ E `“ x ˚ - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F ě ǫ 9 q ď 2 k 2 | S c i | exp p´ ǫ 2 9 T 2 k 2 σ 2 b 2 q “ exp p´ ǫ 2 9 T 2 k 2 σ 2 b 2 ` log p 2 k 2 | S c i |qq (64) and similarly for ǫ 10 ă 2 kσ 2 , Pr p} ˆ E `“ e - i ‰ S c i ¨ e i ⊺ ˘ } B , 8 , F ě ǫ 10 q ď 2 k 2 | S c i | exp p´ ǫ 2 10 T 64 kσ 2 q “ exp p´ ǫ 2 10 T 64 kσ 2 ` log p 2 k 2 | S c i |qq (65) 25 B Pro ofs of Auxiliary Lemmas B.1 Sub exp onen tialit y of square of sub-Gaussian random v ariables Lemma 8. If y is a sub-Gaussian r andom varia ble with 0 me an and p ar ameter 1 , then y 2 is a sub-exp onential r andom variable with p ar ameters p 4 ? 2 , 4 q . Pr o of. Sin ce y is a 0 mean su b-Gaussian rand om v ariable with parameter 1, w e can wr ite p@ λ P R q E ` exp p λy q ˘ ď exp p λ 2 2 q Let Γ p r q b e the Gamma fu nction, then momen ts of the sub-Gaussian v ariable y are b oun ded as follo ws: p@ r ě 0 q E ` | y | r ˘ ď r 2 r 2 Γ p r 2 q Let v fi y 2 and µ v fi E ` v ˘ . Using p o wer series expansion and noting that Γ p r q “ p r ´ 1 q ! for an in teger r , we hav e: E ` exp p λ p v ´ µ v qq ˘ “ 1 ` λ E ` v ´ µ v ˘ ` 8 ÿ r “ 2 λ r E ` p v ´ µ v q r ˘ r ! ď 1 ` 8 ÿ r “ 2 λ r E ` | y | 2 r ˘ r ! ď 1 ` 8 ÿ r “ 2 λ r 2 r 2 r Γ p r q r ! “ 1 ` 8 ÿ r “ 2 λ r 2 r ` 1 “ 1 ` 8 λ 2 1 ´ 2 λ W e take λ ď 1 4 . Th us, E ` exp p λ p v ´ µ v qq ˘ ď 1 ` 16 λ 2 ď exp p 16 λ 2 q ď exp p p 4 ? 2 q 2 λ 2 2 q It follo ws that v “ y 2 is a su b exp onen tial rand om v ariable with parameters p 4 ? 2 , 4 q . B.2 Sub exp onen tialit y of pro duct of independent sub-Gaussian random v ari- ables Lemma 9. L e t p and q b e two indep endent sub-Gaussian r andom variables with 0 me an and p ar ameter 1 , then pq is a sub-exp onential r andom variable with p ar ameters p 4 ? 2 , 4 q . 26 Pr o of. Sin ce p and q are b oth 0 mea n su b-Gaussian random v ariable with parameter 1, w e can write p@ λ P R q E ` exp p λp q ˘ ď exp p λ 2 2 q p@ λ P R q E ` exp p λq q ˘ ď exp p λ 2 2 q Let Γ p r q b e the Gamma f unction, then momen ts of the sub-Gaussian v ariable p an d q are b ounded as follo ws: p@ r ě 0 q E ` | p | r ˘ ď r 2 r 2 Γ p r 2 q p@ r ě 0 q E ` | q | r ˘ ď r 2 r 2 Γ p r 2 q Let v fi pq . Note th at E ` v ˘ “ E ` pq ˘ “ E ` p ˘ E ` q ˘ “ 0 du e to ind ep endence. Usin g p o we r series expansion and noting that Γ p r q “ p r ´ 1 q ! for an integ er r , we h a v e: E ` exp p λv q ˘ “ 1 ` λ E ` v ˘ ` 8 ÿ r “ 2 λ r E ` v r ˘ r ! ď 1 ` 8 ÿ r “ 2 λ r E ` | p | r | q | r ˘ r ! ď 1 ` 8 ÿ r “ 2 λ r E ` | p | r ˘ E ` | q | r ˘ r ! ď 1 ` 8 ÿ r “ 2 λ r r 2 2 r Γ p r 2 q 2 r ! Note that Γ p r 2 q 2 ď Γ p r q . Thus, E ` exp p λv q ˘ ď 1 ` 8 ÿ r “ 2 λ r r 2 2 r Γ p r q r ! “ 1 ´ 8 p λ ´ 1 q λ 2 p 1 ´ 2 λ q 2 ď exp p 16 λ 2 q ď exp p p 4 ? 2 q 2 2 λ 2 q where last inequalit y holds for | λ | ď 1 4 . Thus, pq is s ub exp onen tial with p arameters p 4 ? 2 , 2 q . B.3 Norm Inequalities Here w e will deriv e some n orm in equ alities whic h we will use in ou r p ro ofs. Lemma 10 (Norm Inequalities) . L et A b e a r ow-p artitione d blo ck matrix which c onsists of p blo cks wher e b lo ck A i P R m i ˆ n , @ i P t 1 , . . . , p u and B P R n ˆ o . Then the f ol lowing ine qu alities hold: } AB } B , 8 , F ď } A } B , 8 , 1 } B } 8 , 2 } AB } B , 8 , 1 ď } A } B , 8 , 1 } B } 8 , 8 27 Pr o of. Let v ec p . q b e an op erator w hic h fl attens the matrix and con v erts it to a ve ctor. Let Y b e a ro w-partitioned blo ck matrix with same size and blo ck stru cture as A . } AB } B , 8 , F “ max i Pt 1 ,...,p u } v ec pp AB q i q} 2 “ max i Pt 1 ,...,p u , } ve c p Y i q} 2 ď 1 v ec pp AB q i q ⊺ v ec p Y i q “ max i Pt 1 ,...,p u , } ve c p Y i q} 2 ď 1 rp A i q 1 . B . . . p A i q m i . B s v ec p Y i q “ max i Pt 1 ,...,p u , } ve c p Y i q} 2 ď 1 rp A i q 1 . B p Y i q 1 . ` ¨ ¨ ¨ ` p A i q m i . B p Y i q m i . s ď max i Pt 1 ,...,p u , } v ec p Y i q} 2 ď 1 }p A i q 1 . } 1 } B p Y i q 1 . } 8 ` ¨ ¨ ¨ ` } p A i q m i . } 1 } B p Y i q m i . } 8 ď } A } B , 8 , 1 } B } 8 , 2 W e follo w a s imilar pro cedure for the last norm inequalit y . } AB } B , 8 , 1 “ max i Pt 1 ,...,p u } v ec pp AB q i q} 1 “ max i Pt 1 ,...,p u , } ve c p Y i q} 8 ď 1 v ec pp AB q i q ⊺ v ec p Y i q “ max i Pt 1 ,...,p u , } ve c p Y i q} 8 ď 1 rp A i q 1 . B . . . p A i q m i . B s v ec p Y i q “ max i Pt 1 ,...,p u , } ve c p Y i q} 8 ď 1 rp A i q 1 . B p Y i q 1 . ` ¨ ¨ ¨ ` p A i q m i . B p Y i q m i . s ď max i Pt 1 ,...,p u , } ve c p Y i q} 8 ď 1 }p A i q 1 . } 1 } B p Y i q 1 . } 8 ` . . . }p A i q m i . } 1 } B p Y i q m i . } 8 ď } A } B , 8 , 1 } B } 8 , 8 28