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61 E.3 An alternative expression of GoF-SCORE . . . . . . . . . . . . . . . . . . . . 62 E.4 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 E.5 Proof of auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 E.5.1 Proof of Lemma E.2 . . . . . . . . . . . .... | https://arxiv.org/abs/2502.08609v1 |
. . . . . . . . . . . . . . . . . . . . . . . . . 105 H.4 Proof of Lemma 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 H.5 Theoretical details for Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . 109 H.5.1 Regularity conditions for Theorem 3.7 . . . . . . . . . . . . . . . . . . 1... | https://arxiv.org/abs/2502.08609v1 |
that have more edges within than across; e.g., see Example 1, Section 4, and [41, 28, 4]). The table below presents 12 frequently seen networks, where ( n, K) are as above, and dmin, dmax,¯dare the minimum, maximum, and average degrees, respectively. These networks are not hand-picked for our favor and provide a solid ... | https://arxiv.org/abs/2502.08609v1 |
i, and let πi∈RKbe the membership vector of node i, where πi(k) = the weight node iputs on community k, 1≤k≤K. Also, for a symmetrical non-negative matrix P∈RK,K, letP(k, ℓ) be the baseline connecting probability between communities kandℓ, 1≤k, ℓ≤K. DCMM is a special rank- Kmodel where we assume Ω( i, j) =θiθjπ′ iPπj, ... | https://arxiv.org/abs/2502.08609v1 |
,|τK|}=o(1) [23, 28]. In such cases, the conditions of Theorem 1.1 7 hold [19, Section 3]. Therefore, the DCMM is nearly as broad as the rank- Kmodel. Example 1 . In [16], DCMM was used to model 21 citee networks of the same 2830 nodes and three communities “Bayesian”, “Biostatisitcs”, and “non-parametrics”. For each n... | https://arxiv.org/abs/2502.08609v1 |
are rank- Kmodels, those approaches have no power in telling whether DCBM or DCMM is more appropriate for a given data set. Also, since a rank- Kmodel is not necessarily a DCMM model, we cannot use their methods as GoF metrics for DCMM. See Section 3.5 for more comparisons. Our GoF problem is also related to global tes... | https://arxiv.org/abs/2502.08609v1 |
flexible idea: we extend it to GoF metrics for SBM, DCBM, and MMSBM, denoted by Tn(bΩSBM),Tn(bΩDCBM), and Tn(bΩMMSBM), respectively. In all these cases, we have successfully circumvented the analytical hurdle and showed: Tn(bΩ∗)→N(0,1) in law, if ∗is the correct model (here ∗= SBM, DCBM, MMSBM, and DCMM). Although the ... | https://arxiv.org/abs/2502.08609v1 |
msigned-cycles [5, 23] (e.g.,1 6Cn,3and1 6Un,3(bΩ) 11 are numbers of triangles and signed triangles;1 24Cn,4and1 24Un,4(bΩ) are numbers of quadrilat- erals and signed quadrilaterals). Define the Self-normalized Cycle Count (SCC) statistic by ψn,m(bΩ) = Un,m(bΩ)/p 2mC n,m. The following results are proved in the supplem... | https://arxiv.org/abs/2502.08609v1 |
test statistics with m≤2. 2.2 Analytical strategies and hurdles Theorem 2.1 is for the oracle case. To use the idea for the real case, we need to construct anbΩ and show that |Tn(bΩ)−Tn(Ω)| →0 in probability. Write Tn(bΩ)−Tn(Ω) = [ Un,3(bΩ)− Un,3(Ω)]/p 6Cn,3. To show the claim, the key is to analyze |Un,3(bΩ)−Un,3(Ω)|.... | https://arxiv.org/abs/2502.08609v1 |
there is a key difference when mixed membership presents: it is merely impossible to have an estimate bΠ for Π such that P(bΠ̸= Π) = o(1). This is because: Under DCMM, a row of Π is not discrete but a continuous variable which may take any value in the set {x∈RK:PK i=1xi= 1, xi≥0}. For this reason, we face an analytica... | https://arxiv.org/abs/2502.08609v1 |
of His a weight vector. Write η= Π′Θ1n∈RKandGH= Π′ΘH∈RK,K. Introduce RH= [r1, . . . , r n]′:= diag(Ω 1n)−1ΩH, V H= [v1, . . . , v K]′:= [diag( Pη)]−1PGH.(3.1) For 1≤i≤n, letwi= (V−1 H)′ri. For simplicity, we drop the subscript “ H” invk,riandwi, but bear in mind that they all depend on H. Write ZH=VH(H′ΩH)−1V′ H. Recal... | https://arxiv.org/abs/2502.08609v1 |
ˆπi=∥ˆπ∗ i∥−1 1ˆπ∗ i. 4.(Estimate Θ,Pand Ω). Let bP= [diag( bZH)]−1/2bZH[diag(bZH)]−1/2andbΘ = diag( ˆθ1,ˆθ2, . . . , ˆθn), where for each 1 ≤i≤n,ˆθi=e′ iA1n/∥ˆπi◦cPη∥1. LetbΩ =bΘbΠbPbΠ′bΘ. 5.(GoF for DCMM) . Obtain the GoF metric for DCMM by Tn(bΩ) as in (2.1). Output: bΠ = [ˆ π1, . . . , ˆπn]′,bΘ,bP,bΩ, and Tn(bΩ) (o... | https://arxiv.org/abs/2502.08609v1 |
ˆπMS imay take infinitely many values. Hence, bH=bΠMSdoes not satisfy Requirement (i). Fortunately, we can use bΠMSto construct a desirable bHbynet-rounding . Let S0={x∈ RK +:PK k=1xk= 1}be the standard simplex in RK. A net on S0is a finite-size subset NofS0. One example is the ϵ-netNϵ, where for any a∈S0, there is b∈ ... | https://arxiv.org/abs/2502.08609v1 |
first, SP is tuning-free, and second, aside from the VH steps, GoF-MSCORE is tuning-free. To apply KNN-SP to data points x1, . . . , x n, we fix α > 0 and an integer 19 N≥1. For each xi, letSibe the set of Nnearest neighbors of xifalling within a distance of (max j,k∥xi−xk∥)/αtoxi(including xiitself). If |Si| ≤2, we pr... | https://arxiv.org/abs/2502.08609v1 |
identifiability). For any η∈Rn, let∥η∥,∥η∥1,ηmaxandηmindenote its Euclidean norm, ℓ1-norm (absolute sum of entries), maximum entry, and minimum entry, respectively. For any M∈RK×K, let∥M∥and∥M∥maxbe its spectral norm and entry-wise maximum norm, respectively. Define G=∥θ∥−2Π′Θ2Π∈RK×K. Let λk(PG) be the kth largest (in ... | https://arxiv.org/abs/2502.08609v1 |
∞ in Condition 3.1, as we need to guarantee that bHis concentrated at Π 0. Moreover, (d) is also a mild regularity condition, which excludes the cases where the sine-theta distance between the column spaces of Θ1/2Π and Θ1/2Π0is large. The condition (e) guarantees that the output of net-rounding is unique: Define ˆ gis... | https://arxiv.org/abs/2502.08609v1 |
inappropriate but also not required for identifiability. Instead, letting αn= (1/K)trace( P), we write Ω = αnΠ(α−1 nP)Π′, where trace( α−1 nP) =K, soα−1 nPis on the same scale as the Pin DCMM. In light of this, we use α−1 nPas the new Pand rewrite MMSBM as A= Ω−diag(Ω) + W, and Ω = αnΠPΠ′, trace( P) =K. (3.2) In such a... | https://arxiv.org/abs/2502.08609v1 |
(3.2), where Condition 3.1 and (a) and (c)-(e) of Condition 3.2 hold with Θ =√αnIn. LetbHandTn(bΩ)be as in GoF-MSCORE- rev. As n→ ∞ ,P(bH= Π 0) = 1−O(n−3), and Tn(bΩ)→N(0,1). 3.4 GoF-SCORE for DCBM and SBM The SBM and DCBM are special cases of MMSBM and DCMM, respectively. In MMSBM and DCMM, a row of Π is a continuous ... | https://arxiv.org/abs/2502.08609v1 |
different models (not limited to DCBM). To achieve this broad scope, we need methods and analysis more sophisticated than those in [25] (see Section 2). We now consider SBM. This is a special DCBM, where Ω = αnΠPΠ′. It can also be viewed a special MMSBM where each row of Π is a degenerate weight vector. Similarly as in... | https://arxiv.org/abs/2502.08609v1 |
asn→ ∞ , if SNR n,m(Ω)→ ∞ , then ψn,m(eΩ)→ ∞ in probability.4 Theorem 3.6 studies the power of the oracle SCC metric ψn,m(eΩ), but two questions remain: (i) For each specific null-alternative hypothesis pair, when does the SNR tend to infinity? (ii) Can we show that the real SCC metric also satisfies that ψn,m(bΩ)→ ∞ i... | https://arxiv.org/abs/2502.08609v1 |
special case with K= 1 and Ω = θθ′. The quantity v(θ) defined below measures the level of degree heterogeneity. It shows that our GoF metric for SBM has power in detecting degree heterogeneity. Lemma 3.4 (SBM versus DCBM) .Suppose that the true model is a 1-community DCMM with Ω =θθ′, and the assumed model is a 1-commu... | https://arxiv.org/abs/2502.08609v1 |
constant 6The results are extendable to the case that the true model has K0+jcommunities, for j >1. In fact, by Lemma 3.3, regardless of j, the SNR tends to infinity as long as the ( K0+ 1)th eigenvalue of Ω is large. 30 c0∈(0,1)such that for any given test we can find a pair (Ω0,Ω1)∈ M K0(γn)for which the sum of type-... | https://arxiv.org/abs/2502.08609v1 |
Consider Tn(bΩDCMM) first. To use it for real data analysis, we use Algorithm A with 3 minor regularization steps: (a) after obtaining bPin Algorithm A, we set its negative entries to zero; (b) after obtaining bΩ in Algorithm A, we add a regularization step where we setbΩ(i, j) = 1 if bΩ(i, j)>1 and similarly set bΩ(i,... | https://arxiv.org/abs/2502.08609v1 |
algorithms do not use VH). The VH algorithm has tuning parameters ( N, α), which are set as in Section 3.1 (last two paragraphs). To avoid over-fitting and for a fair comparison, we fix ( N, α, t ) as recommended above and do not change them from setting to setting (e.g., data set, algorithm). The results are relativel... | https://arxiv.org/abs/2502.08609v1 |
as follows. In its original form, Weblog is a directed network with 1 ,494 nodes [1], where each node is a blog, and each directional edge is a hyperlink. As first suggested by [26], we may use brute-force symmetriza- tionto construct an undirected network as follows: define an undirected edge between iandj as long as ... | https://arxiv.org/abs/2502.08609v1 |
Estimated weight ˆ win NC for some authors ((1 −ˆw) is the weight in CH). the DCMM is a more appropriate model than DCBM for CoAuthor. Using MSCORE, [24, Table 3] estimated the mixed membership vectors π1, π2, . . . , π n. Since K= 2, we write each ˆπi= (1−ˆwi,ˆwi), with ˆ wibeing the estimated weight in the NC communi... | https://arxiv.org/abs/2502.08609v1 |
U-statistics [42] and we can adapt them to settings beyond networks, such as testing and GoF for hypergraph analysis [38], text analysis, factor models [36], and cancer clustering models [33, 43]. We leave these for the future research. A Additional numerical details In this section, we provide details of numerical res... | https://arxiv.org/abs/2502.08609v1 |
N, α). How to choose ( N, α) has been described in Section 3.1. [28] reported that KNN-SP frequently outperforms SP, so 38 Algorithm C The SP Algorithm Input: The number of vertices K, and data points ˆ r1,ˆr2,···,ˆrn∈Rd(d≥K−1). 1. Initialize by taking Pto be the d×dzero matrix. 2. For each k= 1,2, . . . , K : Ifk≤d−1 ... | https://arxiv.org/abs/2502.08609v1 |
the same way as in Experiment 1.1, and each community has n/2 nodes. Experiment 1.4 (SBM) :θi’s are all equal to√αn, where αn= 0.3; and each community has n/2 nodes. For each true model, we plot the histograms of the proposed GoF metrics Tn(bΩDCMM),Tn(bΩMMSBM),Tn(bΩDCBM), and Tn(bΩSBM). The results are shown in Figure ... | https://arxiv.org/abs/2502.08609v1 |
41 in{1200,1600, . . . , 3600}; in setting (b), bvaries in {0.1,0.15, . . . , 0.4}; in setting (c), xvaries in{0.05,0.1, . . . , 0.35}. For each setting, we consider cases, with θiiid∼Unif(0 .1,0.3) (moderate degree heterogeneity) and θ−1 iiid∼Unif(2 ,10) (severe degree heterogeneity), respectively (the average of θi’s... | https://arxiv.org/abs/2502.08609v1 |
are generated in the same manner as in Experiment 3.1. We still generate the networks so that the edges are independent, but the edge probabilities areP(Aij= 1) = f(Ωij), for 1 ≤i < j ≤n. We call it the nonlinear DCMM model. In Figure A6 (middle panel), we plot the histogram of Tn(bΩDCMM) based on 1000 repetitions. It ... | https://arxiv.org/abs/2502.08609v1 |
of K. For example, the GoF-DCMM metrics with K > 3 (i.e., Kis likely over-specified) are all large in magnitude. We observe a similar phenomenon in simulations. This case is however not covered by Lemma 3.3, and we leave it to future exploration. A.4 Robustness to tuning parameters We recall that aside from the vertex ... | https://arxiv.org/abs/2502.08609v1 |
of ∆ = bΩ−Ω and W1=A−Ω. This is stated in Lemma 2.1 of the main paper and proved in Section E.1 Step 2.2. Secondly, as a direct consequence of Step 2.1, we bound Un,3(bΩ)−Un,3(Ω) ≲∥∆∥3+∥diag(Ω) ∥∥∆∥2+∥diag(Ω) ∥2∥∆∥+∥W1∆∥∥∆∥ +∥W2 1∆∥+∥diag( W2 1)∆∥+∥diag(Ω) ∥∥W1∆∥,(B.3) where ∆ = bΩ−Ω, the estimation error, varies acros... | https://arxiv.org/abs/2502.08609v1 |
four terms in (B.4) are reduced to bounding operator norms of matrices that are simple functionals ofW1andH. These matrices are also of rank K. Therefore, it suffices to prove the 47 entry-wise bounds by computing the asymptotic order of the mean and variance of their entries, utilizing some combinatorial techniques. W... | https://arxiv.org/abs/2502.08609v1 |
The results also hold for ¯Ω. Proof of Lemma D.1. First by the conditions in Theorem 2.1, we trivially have tr(Ωm)≥C∥u∥2m/(n¯u)m→ ∞ . Second, we prove that ECn,m= tr(Ωm) 1 +o(1) . We will show that tr(Ωm)−ECn,m= o(∥u∥2m/(n¯u)m). To do this, we first notice that tr(Ωm) =Pn i1,···,im=1Ωi1i2···Ωimi1and we can decomposeP... | https://arxiv.org/abs/2502.08609v1 |
show Sn,n=Un,m/p var(Un,m)d−→N(0,1), we apply the martingale central limit theorem in [10] and verify that nX k=1E(X2 n,k|Fn,k−1)p→1, (D.3) nX k=1E(X2 n,k1{|Xn,k|>ϵ}|Fn,k−1)p→0,for any ϵ >0. (D.4) We first verify (D.3). For simplicity, we denote [ [ a, b] ] ={a, a+ 1, . . . , b}for integers a, b∈N+ and we write for sho... | https://arxiv.org/abs/2502.08609v1 |
a=1uℓa√n¯u2+βa ≲M−2 n2m−2X ˜m=m∥u∥2 n¯u˜m ·umax√n¯uP˜m a=1βα ≲∥u∥2 n¯u−2 =o(1). Here we used Mn≍ECn,m≍tr(Ωm)≥C ∥u∥2/(n¯u)mand the conditions u2 max/(n¯u) =o(1), n¯u/∥u∥2=o(1). Next, for the second term Ibin (D.6), we have var(Ib) =1 M2 nnX k1,k2=mX 1≤i1<j1≤k1−1 1≤i2<j2≤k2−1X ℓ1,eℓ1∈L(k1,i1,j1)dist ℓ2,eℓ2∈L(k2... | https://arxiv.org/abs/2502.08609v1 |
2···Ai′mi′ 1 −Ωi1i2···Ωimi1Ωi′ 1i′ 2···Ωi′mi′ 1 Each summand on the RHS above is nonzero if and only if the two cycle ( i1,···, im) and (i′ 1,···, i′ m) share at least one edge. This again implies the number of effective indices in each summand bounded by 2 m−2. In the same manner to previous analysis, we can get var(... | https://arxiv.org/abs/2502.08609v1 |
i′j′)−¯Ωij¯Ωi′j′¯Ωki¯Ωkj¯Ωk′i′¯Ωk′j′ = 2M−2 nX i,j,k(dist);i,j,k′(dist) E(W4 ij)−¯Ω2 ij¯Ωki¯Ωkj¯Ωk′i¯Ωk′j ≲∥u∥2 n¯u−6∥u∥4∥u∥6 3 (n¯u)5≤∥u∥2 n¯u−6∥u∥8u2 max (n¯u)5=∥u∥2 n¯u−2u2 max n¯u=o(1), under the assumptions in Corollary D.1. Here in the third step, we used the fact that E(W2 ijW2 i′j′)−¯Ωij¯Ωi′j′= 0 if ( ... | https://arxiv.org/abs/2502.08609v1 |
W(which are independent Bernoulli’s). In Section E.4, we conduct large- deviation analysis for each term in δand prove the main result, Theorem 3.2. E.1 Proof of Lemma 2.1 Define f(M) =P i1,i2,i3(dist)M(i1, i2)M(i2, i3)M(i3, i1) for a symmetric matrix M∈Rn,n. Using this notation, we have Un,3(bΩ)−Un,3(Ω) = f(A−bΩ)−f(A−... | https://arxiv.org/abs/2502.08609v1 |
of GHin (E.5), H′ΩH=H′ΘΠPΠ′ΘH= G′ HPGH. Additionally, VH= [diag( Pη)]−1PGH, or equivalently, PGH= [diag( Pη)]VH. It follows that H′ΩH=G′ HPP−1PGH=V′ H[diag( Pη)]P−1[diag( Pη)]VH. We plug it into the definition of ZHto get ZH=VH(H′ΩH)−1V′ H= [diag( Pη)]−1P[diag( Pη)]−1. (E.8) It yields that diag( ZH) = [diag( Pη)]−2diag... | https://arxiv.org/abs/2502.08609v1 |
any two diagonal matrices are exchangeable in the product. Next, we prove the expression in (E.10) when each estimated vertex is a linear combination ofbri’s. In GoF-SCORE, the input matrix His such that each row is a weight vector. It follows thatH1K=1nand bRH1K= [diag( A1n)]−1AH1K= [diag( A1n)]−1A1n=1n. (E.17) This m... | https://arxiv.org/abs/2502.08609v1 |
we have shown thatbΩ =AH(H′AH)−1H′Awith probability 1 −o(1). Additionally, using either Theorem 3.1 or the assumption in Theorem 3.2 gives that H=H0or Π 0, with probability 1 −o(1). Hence, without loss of generality, we consider bΩ =AH 0(H′ 0AH 0)−1H′ 0A. (E.26) 66 LetW1=W−diag(Ω) and ∆ = bΩ−Ω. Lemma 2.1 gives a decomp... | https://arxiv.org/abs/2502.08609v1 |
equivalent to H0, maintaining its non-stochastic nature. Recall from (E.5) the definition of GH= Π′ΘH. Observe the fact that ( H′ΩH)−1=G−1 HP−1(G′ H)−1 and ( H′ΩH)−1H′Ω =G−1 HΠ′Θ. By Corollary 5.6.16 in [12], it holds that (H′AH)−1= (H′ΩH)−1(H′ΩH)(H′AH)−1 = (H′ΩH)−1h IK+∞X j=1{IK−(H′AH)(H′ΩH)−1}ji = (H′ΩH)−1(IK+T), (E.... | https://arxiv.org/abs/2502.08609v1 |
diag( ξ)Bdiag( η)D′ ≤ sup ξ,η∈Rn; ∥ξ∥=∥η∥=1h rank diag( ξ)Bdiag( η)D′ × ∥diag( ξ)Bdiag( η)D′∥i ≤min{rank( B),rank( D)} × ∥ B∥ × ∥ D∥, where diag( ξ) denotes the diagonal matrix with diagonal entries being those of the n-vector ξ. The proof is therefore complete. E.5.3 Proof of Lemma E.4 We separate the proof of Lemma ... | https://arxiv.org/abs/2502.08609v1 |
7∥θ∥1≲∥θ∥2 1∥θ∥6 3. By Chebyshev’s inequality, we obtain that max 1≤k,ℓ≤K|{H′ΘW2 1H}k,ℓ|≲∥θ∥1∥θ∥2+OP(∥θ∥1∥θ∥3 3p log(n)) =OP(∥θ∥2 1), where we used the assumption that θmaxp log(n)≤cin view of (a) of Condition 3.1. For∥Π′ΘW2 1ΘH∥, we have, for 1 ≤k, ℓ≤K, {Π′ΘW2 1ΘH}k,ℓ=nX r,s,t=1ΠrkHtℓθrθtWrsWst−nX r,s=1ΠrkHsℓθrθsWrsΩs... | https://arxiv.org/abs/2502.08609v1 |
1)W1H=H′ W−diag(Ω) diag W−diag(Ω)2 W−diag(Ω) H =I0+I1+I′ 1+I21+I22+I3+I′ 3+I4, where I0:=H′ diag(Ω)4H, I 1:=−H′W diag(Ω)3H, 77 I21:=H′diag(Ω)diag( W2)diag(Ω) H, I 22:=H′W diag(Ω)2WH, I3:=−H′Wdiag( W2)diag(Ω) H, I 4:=H′Wdiag( W2)WH . (E.41) We proceed to show that each term in (E.41) are of order OP(∥θ∥2 1... | https://arxiv.org/abs/2502.08609v1 |
(E.42) yields that ∥I4∥= OP(∥θ∥2 1∥θ∥2). The proof for ∥H′W1diag( W2 1)W1H∥is therefore complete by combining the bounds the terms in (E.41). Next, we prove the bound for ∥H′W4 1H∥in (E.40). Observe that H′W4 1H=H′ W−diag(Ω)4H=X α1,···,α4∈{0,1}H′4Y u=1Wαu −diag(Ω)1−αuH . 79 IfP4 u=1αu= 0, the summand on the RHS bec... | https://arxiv.org/abs/2502.08609v1 |
matrix O1∈R(K−1)×(K−1) such that simultaneously for all 1≤i≤n, ˆξ1(i)−ωξ1(i) ≲p nθθmaxθi ∥θ∥3+∥θ∥3/2 3√θilogn ∥θ∥3+θmaxlogn ∥θ∥3+θmaxp nθθilogn ∥θ∥4, 81 ∥bΞ1(i)−Ξ1(i)O1∥≲p nθθmaxθi βn∥θ∥3+∥θ∥3/2 3√θilogn βn∥θ∥3+θmaxlogn βn∥θ∥3+θmaxp nθθilogn β2 n∥θ∥4. wherebΞ1(i)represents ith row of bΞ, and similarly for Ξ(i). Remark.... | https://arxiv.org/abs/2502.08609v1 |
proof of Lemma D.1 in [24] and applied (F.1) in the last steps. We further bound |ˆλ−1 1e′ idiag(Ω) ˆξ1|≲θ2 i∥θ∥−2|ˆξ1(i)| ≤θ2 i∥θ∥−2|ξ1(i)|+θ2 i∥θ∥−2|ˆξ1(i)−ξ(i)|. Thereby, we arrive at ˆξ1(i)−ξ1(i) ≲q nθθmaxθi∥θ∥−3+∥θ∥−2 e′ iWˆξ1 +θ2 i∥θ∥−2|ˆξ1(i)−ξ(i)|. Rearranging both sides gives ˆξ1(i)−ξ1(i) ≲q nθθmaxθi∥θ∥−3+∥θ∥−... | https://arxiv.org/abs/2502.08609v1 |
( δnβn∥θ∥)−1p log(n). Since the remaining proof is simply a copy of the proof in Section F.1 of [24], we skip the details and conclude the proof. Moreover, under (e) of Condition 3.2, under the high probability event that the rate of Mixed-SCORE holds, we can claim that ˆ π0i=π0isimultaneously for all 1 ≤i≤n. To see 85... | https://arxiv.org/abs/2502.08609v1 |
bH= Π 0with probability 1 −o(1). Recall the procedures by which we get bHin Section 3.3. Using Corollary F.1, we get that max 1≤i≤n∥bΞ(i)−Ξ(i)O∥≲√logn βn√nαn·1√n. By our vertex hunting algorithm, max 1≤k≤K∥Oˆv∗ k−v∗ k∥≲√logn/(βnn√αn) (for simplicity, we assume a perfect alignment between {ˆv∗ k}and{v∗ k}, thereby exclu... | https://arxiv.org/abs/2502.08609v1 |
later analysis. These estimates can be deduced from Lemma E.4 by letting ∥θ∥1≍n√αn,∥θ∥ ≍√nαnand∥θ∥3≍n1/3√αn. Therefore, we skip their proofs. ∥Π′W1Π∥=OP nα1/2 np log(n) ,∥Π′W1H∥=OP nα1/2 np log(n) , ∥H′W2 1H∥=OP(n2αn), ∥Π′W2 1H∥=OP(n2αn), ∥Π′W2 1Π∥=OP(n2αn), ∥Π′diag( W2 1)Π∥=OP(n2αn), ∥H′W3 1H∥=OP(n2αn),∥Π′W3 1H∥=o... | https://arxiv.org/abs/2502.08609v1 |
n· n2αn,Γ =In, W1 n3α2 n,Γ =W2 1,diag( W2 1) T22(Γ)≲αn∥(PGH)−1∥∥Π′ΓW1H∥≲αn nβn· nα1/2 n√logn,Γ =In n2αn, Γ =W1 oP(n5/2α3/2 n),Γ =W2 1,diag( W2 1) T23(Γ)≤α2 n∥Π′ΓΠ∥≲α2 n· n, Γ =In nα1/2 n√logn ,Γ =W1 n2αn, Γ =W2 1,diag( W2 1)(G.12) which follows from (G.4). Combining these together with the condi... | https://arxiv.org/abs/2502.08609v1 |
diag( A1n)Π diag Π′A1n−1Π′AΠ diag Π′A1n−1Π′diag( A1n),(G.13) with probability 1 −o(1), since the clustering step in the algorithm achieves exact recovery P(bΠ = Π) = 1 −o(1). The proof of exact recovery follows from the arguments for DCMM and SBM in Sections F.2 and G.3. We need first have ∥ˆri−ri∥≲(δnβn∥θ∥)−1p... | https://arxiv.org/abs/2502.08609v1 |
have [diag(Π′A1n)]−1=D0− D 1, where D0= [diag(Π′Ω1n)]−1,D1= [diag(Π′Ω1n)]−1(IK+T3)diag(Π′W11n)[diag(Π′Ω1n)]−1, T3=∞X j=1 diag(Π′(−W1)1n)[diag(Π′Ω1n)]−1j. Applying (G.15), we get ∥diag(Π′W11n)∥=OP(∥θ∥1p log(n)), which, together with (G.18), leads to ∥diag(Π′(−W1)1n)[diag(Π′Ω1n)]−1∥=oP(1), since ∥θ∥1≫p log(n). Conseque... | https://arxiv.org/abs/2502.08609v1 |
1,diag( W1). However, this difference does not affect the analysis significantly thanks to the special form of 98 In,k. Consider 1′ nW1In,kW11nfor an example, and the other quantities in (G.16) of Lemma G.1 can be treated similarly. We write 1′ nW1In,kW11n=nX i,r,j=1δk(r, r)·W1(i, r)W1(r, j) where δk(r, r) = 1 if r∈ C ... | https://arxiv.org/abs/2502.08609v1 |
of Theorem 3.6, Lemmas 3.3 - 3.5 in Sections H.1-H.4, respectively. The theoretical details for Theorem 3.7 of the main paper are presented in Section H.5 where an auxiliary lemma (Lemma H.2) is also introduced. Finally, in Section H.6, we revisit the setting DCBM versus DCMM, as discussed in Lemma 3.5, and extend the ... | https://arxiv.org/abs/2502.08609v1 |
compute their variance below. var(R1)≲P i̸=jΩij P k,ℓ(dist)̸=i,je∆∗ jke∆∗ kℓe∆∗ ℓi2 8tr(Ω4)≲α6 n∥u∥8∥u∥6 3/(n¯u)7 8tr(Ω4) var(R2)≲P i,j,k(dist)ΩijΩjk P ℓ̸=i,j,ke∆∗ kℓe∆∗ ℓi2 8tr(Ω4)≲α4 n∥u∥6∥u∥6 3/(n¯u)6 8tr(Ω4)=o(var(R1)) var(R3)≲P i,j,k,ℓ (dist)ΩijΩkℓ e∆∗ jke∆∗ ℓi2+e∆∗ jke∆∗ ℓie∆∗ jℓe∆∗ ki 8tr(Ω4)≲α4 n∥u∥12 3/(n... | https://arxiv.org/abs/2502.08609v1 |
with K= 4. By computing π′ iPπjfor 4×4 different cases, we obtain a matrix P∗= 1 (1 −a) +ab b a + (1−a)b (1−a) +ab[(1−a)2+a2] + 2a(1−a)b a+ (1−a)b2a(1−a) + [a2+ (1−a)2]b b a + (1−a)b 1 (1 −a) +ab a+ (1−a)b2a(1−a) + [a2+ (1−a)2]b(1−a) +ab[(1−a)2+a2] + 2a(1−a)b . (H.8) Define Π∗by π∗ i= ... | https://arxiv.org/abs/2502.08609v1 |
=λ3 1+λ3 2= 8a4(1−b)3[a2+ 3(1−a)2]. Furthermore, since Π′Θ2Π =∥θ∥2I4/4 in this example, it can be easily derived that SNR n,3(Ω) =∥θ∥2 43trace([ P−eP]3)p 6Cn,3≥C−1a4(1−b)3∥θ∥3. The proof is complete. H.5 Theoretical details for Theorem 3.7 H.5.1 Regularity conditions for Theorem 3.7 Given any integers 1 ≤L≤K, letUL(K... | https://arxiv.org/abs/2502.08609v1 |
H= ΠQ,e∆̸= 0. We employ Lemma 2.1 and write Un,3(bΩ)−Un,3(Ω) = f(W1,∆). In particular, the analysis of f(W1,b∆) is the same as that of f(W1,∆) in the null case. What remains is to study the additional terms in f(W1,∆)−f(W1,b∆) that contain e∆. By elementary derivations, it is easy to obtain the additional terms contain... | https://arxiv.org/abs/2502.08609v1 |
∥θ∥4). Note that Cn,3= tr(Ω3)(1+ o(1))≍ ∥θ∥6. We also recall SNR n,3(Ω) = tr [Ω−eΩ]3 /p 6tr(Ω3). If SNR n,3(Ω)≥γn≫p log(n), then it follows from the condition tr [eΩ−Ω]3 ≫ ∥P−eP∥ · ∥θ∥4that Tn(bΩ) = Tn(Ω) +tr [Ω−eΩ]3 p 6Cn,3(1 +oP(1)) + o(1) =Tn(Ω) + SNR n,3(Ω)(1 + oP(1)) + o(1)≫p log(n) (H.12) with probability 1... | https://arxiv.org/abs/2502.08609v1 |
∈RK×K, where we denote by δ1:(K−2)= (δ1,···, δK−2)′. Combining the above gives Q[diag( Q′GP˜q|)]−1Q′= Y−1 1:(K−2) 1 ˜yK−1+˜yK121′ 2 . by denoting Y1:(K−2)= diag(˜ y1:(K−2)). Recall (˜ x, X, y, Y ) in (H.18). With these notations, we write diag( P˜q) = diag(˜ x) =X. We also recall x= (x1, x2) = (˜ xK−1,˜xK)∈R2and 1... | https://arxiv.org/abs/2502.08609v1 |
sK−1,˜sK) = diag( s) and the short-hand notation γ= [diag(˜ s1:(K−2))]1/2(β1−β2). In addition, T2=(1−b)3 8tr (1−a)s1−(1 +a)√s1s2 −(1 +a)√s1s2 (1−a)s2 3 =(1−b)3 8h (1−a)3(s3 1+s3 2) + 3(1 + a)2(1−a)s1s2(s1+s2)i . Consequently, combining these two terms gives tr (S(P−eP)S)3 =3(1−b)∥γ∥2q2 1q2 2 2(q1+q2)2hs1 q... | https://arxiv.org/abs/2502.08609v1 |
hold and f G1 2PG1 2,eG1 2ePeG1 2,Γ ≫ ∥θ∥−2. If SNR n,3(Ω)→ ∞ , then Tn(bΩDCBM)→ ∞ , and for any fixed α∈(0,1), the power of the level- αGoF-SCORE test tends to 1. 119 H.6.1 Proof of Lemma H.3 We start with claiming bΠ =eΠ with high probability when χn≥(δnβn∥θ∥)−1log(n). Recall the partition ∪K k=1Skand the residual ... | https://arxiv.org/abs/2502.08609v1 |
2PG1 2] + 3trace [eG1 2ePeG1 2]Γ′[G1 2PG1 2]Γ[eG1 2ePeG1 2]i =∥θ∥6f(G1 2PG1 2,eG1 2ePeG1 2,Γ). (H.26) This finishes our proofs. H.6.2 Proof of Theorem H.1 We first recall that eΩ = diag(Ω 1n)eΠ[diag( eΠ′Ω1n)]−1eΠ′ΩeΠ[diag( eΠ′Ω1n)]−1eΠ′diag(Ω 1n). 122 eΘ :=∥θ∥1·Θdiag(Π Pζ)[diag(eΠeΠ′ΘΠPζ)]−1. Similarly to the proof ... | https://arxiv.org/abs/2502.08609v1 |
Eldan, and M. Z. R´ acz (2016). Testing for high-dimensional geometry in random graphs. Random Struct. Algor. 49 (3), 503–532. [6] Chatterjee, S. (2015). Matrix estimation by universal singular value thresholding. Ann. Statist. 43 (1), 177–214. [7] Chatterjee, S., P. Diaconis, and A. Sly (2011). Random graphs with a gi... | https://arxiv.org/abs/2502.08609v1 |
[28] Ke, Z. T. and J. Jin (2023). Special invited paper: The SCORE normalization, especially for heteroge- neous network and text data. Stat 12 (1), e545. [29] Ke, Z. T. and J. Wang (2024). Optimal network membership estimation under severe degree heterogeneity. J. Amer. Statist. Assoc. (to appear) . [30] Lei, J. (2016... | https://arxiv.org/abs/2502.08609v1 |
Robustly Learning Monotone Generalized Linear Models via Data Augmentation Nikos Zarifis ‖∗ UW Madison zarifis@wisc.eduPuqian Wang ‖† UW Madison pwang333@wisc.eduIlias Diakonikolas‡ UW Madison ilias@cs.wisc.edu Jelena Diakonikolas§ UW Madison jelena@cs.wisc.edu Abstract We study the task of learning Generalized Linear ... | https://arxiv.org/abs/2502.08611v1 |
been a resurgence of research interest on learning GLMs in the more challenging agnostic (or adversarial label noise) model [Haussler, 1992, Kearns et al., 1994], where no assumptions are made on the labels and the goal is to compute a hypothesis that is competitive with thebest-fitfunction in the class. The ideal resu... | https://arxiv.org/abs/2502.08611v1 |
even for realizable learning (Theorem C.13). Comparison to Prior Work Gollakota et al. [2023a] gave an efficient GLM learner for monotone Lipschitz activations and marginal distribution with bounded second moment. However, the error of their algorithm scales linearly with Wand the Lipschitz constant. Wang et al. [2023]... | https://arxiv.org/abs/2502.08611v1 |
The effect of data augmentation on the considered GLM learning task is that it simulates the Ornstein–Uhlenbeck semigroup Tρf(t):=Ez∼N(0,1)[f(ρt+p 1−ρ2z)]applied to any function f(w·x). This process smoothens the function fand induces other regularity properties. Unlike the common use of smoothing in the optimization l... | https://arxiv.org/abs/2502.08611v1 |
to the target solution set. “Local” in the name refers to such inequalities being valid only in a local region around the target solution set. Within learning theory and in the context of GLM learning, they have played a crucial role in the analysis of (stochastic) gradient-based algorithms [Mei et al., 2018, Wang et a... | https://arxiv.org/abs/2502.08611v1 |
learning sign(σ(w∗·x)−t), which is an instance of learning halfspaces with adversarial noise.1In particular, we argue that constant approximate solutions to this halfspace learning problem suffice for our initialization. 1.2 Preliminaries Forn∈Z+, let[n]:={1, . . . , n }. We use bold lowercase letters to denote vectors... | https://arxiv.org/abs/2502.08611v1 |
the gradients of the square loss applied to the augmented data correlate with a target parameter vector w∗.We use Lρ(w) =E(˜x,y)∼Dρ[(σ(w·˜x)−y)2]to denote the square loss on the augmented data and refer to it as the “augmented loss.” Proposition 2.2 (Main Structural Result) .Fix an activation σ:R→R. Let Dbe a distribut... | https://arxiv.org/abs/2502.08611v1 |
)-Critical Point for some θ0andC >1an absolute constant, L(w)≤O(OPT) + 4 ∥P>(1/θ∗)2σ∥2 L2. To prove Proposition 2.5, we first prove the following technical lemma, which decomposes the error intoO(OPT )and error terms that depend on the properties of the activation σ.A more formal version of Lemma 2.6 is stated as Lemma... | https://arxiv.org/abs/2502.08611v1 |
Sample access to D 2:[w(0),¯θ] =Initialization [σ](Section 4.3); set ρ0= cos ¯θ 3:fort= 0, . . . , Tdo 4:Draw nsamples bDρt={(˜x(i), y(i))}n i=1fromDρtusing Algorithm 1 5:bg(w(t)) =−(1/ρt)E(˜x,y)∼bDρt[yσ′(w(t)·˜x)(˜x)⊥w(t)] 6:ηt=p (1−ρt)/2/(4∥bg(w(t))∥2) 7:w(t+1)= (w(t)−ηtbg(w(t)))/∥w(t)−ηtbg(w(t))∥2 8:ρt+1= 1−(1−1/256... | https://arxiv.org/abs/2502.08611v1 |
∥w(t+1)−w∗∥2 2≤ ∥w(t)−w∗∥2 2−ηt|g(t)·w∗|. Thus, by choosing ηtappropriately, there exists ξ >0 7 such that θt+1≤θt−ξand if we choose ρt+1so that cos−1ρt−cos−1ρt+1< ξ, we ensure ρt+1≤cosθt+1. Alternatively, if sinθt≲ζ(ρt)and|cosθt−ρt| ≥sin2θt, then by the triangle inequality we obtain ∥w(t+1)−w∗∥2≤ ∥w(t)−w∗∥2+ηt∥g(t)∥2.... | https://arxiv.org/abs/2502.08611v1 |
that ∥P>1/θ2σ∥2 L2≲2∥Φk−TcosθΦk∥2 L2+θ2∥TcosθΦ′ k∥2 L2. We further show that TcosθΦ′ k→Tcosθσ′, therefore, it remains to show that ∥Φk−TcosθΦk∥2 L2≲ θ2∥TcosθΦ′ k∥2 L2. Proposition 4.6 in Section 4.2 proves the claim that when ρis not too small, ∥Φ− TρΦ∥2 L2≲θ2∥TρΦ′∥2 L2, for any Φ(z)that is a monotonic staircase functi... | https://arxiv.org/abs/2502.08611v1 |
form of TρΦ(z/ρ), we show that all these three terms can be bounded by ∥TρΦ′(z)∥2 L2, using the properties of TρΦ(z/ρ)provided in Lemma 4.7. We define the centered augmentation as Tρσ(z/ρ) =Eu∼N[σ(z+ (p 1−ρ2/ρ)u)].We show that theL2 2error between the centered augmentation TρΦ(z/ρ)andΦ(z),TρΦ(z)are well controlled, as ... | https://arxiv.org/abs/2502.08611v1 |
this direction is attainable. Another question is whether one can obtain a similarly robust GLM learner (even for the known activation case) for more general marginal distributions, e.g., encompassing all isotropic log-concave distributions. This remains open even for the special case of a single general (i.e., potenti... | https://arxiv.org/abs/2502.08611v1 |
M. J. Wainwright. Randomized smoothing for stochastic optimization. SIAM Journal on Optimization , 22(2):674–701, 2012. H. Federer. Geometric Measure Theory . Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Springer, 1969. ISBN 9780387045054. S. Goel, A. Gollakota, and A. R. Klivans. Statistic... | https://arxiv.org/abs/2502.08611v1 |
Zarifis, I. Diakonikolas, and J. Diakonikolas. Sample and computationally efficient robust learning of gaussian single-index models. The Thirty-Eighth Annual Conference on Neural Information Processing Systems , 2024. D. Yin, R. Gontijo Lopes, J. Shlens, E. D. Cubuk, and J. Gilmer. A fourier perspective on model robust... | https://arxiv.org/abs/2502.08611v1 |
(3) Finally, note that their sample complexity depends on c1, therefore their sample complexity can be even larger if c1is extremely small. Our results address these issues: (1) as discussed in the introduction, this work’s error bound in Theorem 4.1 is independent of all the parameters ∥σ′∥L2,∥σ∥L∞,dandϵ, and therefor... | https://arxiv.org/abs/2502.08611v1 |
(2011) studied the problem of learning GLMs in the realizable setting. They considered monotone 1-Lipschitz activations under any distribution Dthat is supported onB×[0,1]. Their analysis is not applicable to our robust learning setting. B Additional Notation and Preliminaries Additional Notation LetN(µ,Σ)denote the d-... | https://arxiv.org/abs/2502.08611v1 |
B.4), we have that E x∼N[g(x)(Tρf(x)−f(x))] =Z1 ρ(1/t)E x∼N[∇f(x)∇Ttg(x)] dt. Note that using Fact B.2 (f) and Stein’s lemma Fact B.7, we have: ∇Tρg(x) =ρTρ∇g(x) =ρE z∼N[∇g(ρx+p 1−ρ2z)] = (ρ/(p 1−ρ2))E z∼N[g(ρx+p 1−ρ2z)z]. 15 Therefore, since z,xare independent standard Gaussian random vectors, we have that E x∼N[g(x)(... | https://arxiv.org/abs/2502.08611v1 |
(B, L)-regular activation. In our approach, it is without loss of generality to assume that ∥w∥2= 1. This is because, for any nonzero vector w∈B(W), we can always rescale the activation σ(w·x)toσ(∥w∥2(w/∥w∥2)·x):= ¯σ((w/∥w∥2)·x) =¯σ(w′·x)where ∥w′∥2= 1. In other words, we define ¯σ(z) =σ(∥w∗∥2z), where w∗ is one of the... | https://arxiv.org/abs/2502.08611v1 |
bounded. First, we show that we can truncate the labels y without loss of generality. Claim C.6. Letσbe a(B, L)-Regular activation. Let ¯y=sign(y)min{|y|, B}. Then, E(x,y)∼D[(¯y− σ(w∗·x))2]≤OPT. Furthermore, for any bwsuch that E(x,y)∼D[(¯y−σ(bw·x))2]≤O(OPT ), we have E(x,y)∼D[(y−σ(bw·x))2]≤O(OPT). Hence it is w.l.o.g.... | https://arxiv.org/abs/2502.08611v1 |
have that Pr[|σ(z)| ≥T]≤Ez∼N[σ(z)2+ζ] T2+ζ≤Bσ T2+ζ. Note that E[σ2(z)] =R∞ 0Pr[σ2(z)≥t]dt=R∞ 02uPr[σ2(z)≥u2]du(the last part is after change of variables to u2=t). Therefore, we have that 19 E[σ2(z)1{|σ(z)| ≥D}] =Z∞ 02uPr[σ2(z)1{|σ(z)| ≥D} ≥u2]du =Z∞ 02uPr[|σ(z)|1{|σ(z)| ≥D} ≥u]du =Z∞ D2uPr[|σ(z)| ≥u]du≤Z∞ D2uBσ u2+ζ≤4... | https://arxiv.org/abs/2502.08611v1 |
by E z∼N[(σ(z)−¯σ(z))2]≤s Bσ,4ϵ2 Bσ,4≤ϵ. Therefore, σis an Extended (p Bσ,4/ϵ, L)-Regular activation. 2. Next, assume that σis(R, r)-Sub-exponential. Similarly, let ¯σ(z) = sign( σ(z)) min{|σ(z)|, eR(4rlog(4) + log( R4/ϵ2))r} Denote for simplicity Bσ:=eR(4rlog(4) + log(R4/ϵ2))rThen, ¯σis a(Bσ, L)-Regular activation. Us... | https://arxiv.org/abs/2502.08611v1 |
1−2sin2(θ/2)≤7/8. Note that for any u,v∈V, with u̸=v, it holds that ∥σ(v·x)−σ(u·x)∥2 L2= 2(1 −E[σ(v·x)σ(u·x)]) = 2X k≥0(1−coskθ)ˆσ(i)2 ≥2(1−ˆσ(0)2)−cosθX k≥1ˆσ(i)2≥2(1−ˆσ(0)2)−2 cosθ∥σ∥2 L2, where ˆσ(i)are the Hermite coefficients of σ. Furthermore, note that ˆσ(0) = E[σ(z)] =√ δ. Hence, we have that ∥σ(v·x)−σ(u·x)∥2 L... | https://arxiv.org/abs/2502.08611v1 |
Proof.Using the definition of Dρ,we have that E ˜x∼(Dρ)˜x[f(w·˜x)] = E x∼Dx[E z∼N(0,I)[f(ρw·x+p 1−ρ2w·z)]] =E x∼Dx[E ζ∼N(0,1)[f(ρw·x+p 1−ρ2ζ)]] = E x∼Dx[Tρf(w·x)], where we have used that w·zis distributed according to the standard normal distribution. Lemma D.2 shows that our data augmentation technique is equivalent ... | https://arxiv.org/abs/2502.08611v1 |
w·xandv·xare independent standard Gaussian random variables noted above. To bound the first term on the right-hand side of (11), we again use that w·xandv·xare independent standard Gaussian random variables and apply Stein’s lemma (Fact B.7) to obtain E x∼Dx[σ(w∗·x)Tρσ′(w·x)v·x] =E x∼Dx[σ(cosθw·x+ sin θv·x)Tρσ′(w·x)v·x... | https://arxiv.org/abs/2502.08611v1 |
any θ≤¯θ, it holds ψ′ σ(z)≥0. Furthermore, if σ′′is in L2(N)and∥σ′′∥L2≤L′, then since (Tρf(z))′=ρTρf′(z)andTρis a non-expansive operator (Fact B.2), we have h(θ) =∥Tcosθσ′∥2 L2−sin2θ∥Tcosθσ′′∥2 L2≥ ∥Tcosθσ′∥2 L2−sin2θ(L′)2. Assuming θ≤π/3, we have h(θ)≥0as long as θ≤¯θ= min( π/3,∥T1/2σ′∥2 L2/(L′)2). The significance of... | https://arxiv.org/abs/2502.08611v1 |
then σ′(t).=P i=1ai√ iHei−1(t)is the Hermite expansion of σ′. Therefore, we have that E t∼N[(σ(t))2]−E t∼N[σ(t)Tcosθσ(t)]≤θ2∥Pkσ′∥2 L2+∥P>kσ∥2 L2, which, combined with Equation (15), completes the proof. To complete the proof, it remains to bound ∥Pkσ′∥2 L2above, which is done in the following claim. Claim D.10. When k... | https://arxiv.org/abs/2502.08611v1 |
to w∗is no longer be 29 Algorithm 4 SGD−VA: SGD with Variable Augmentation 1:Input:Parameters ϵ, T; Sample access to D. 2:[w(0),¯θ] =Initialization [σ](Appendix F.3); set ρ0= cos ¯θ. 3:fort= 0, . . . , Tdo 4:Draw nsamples {(˜x(i), y(i))}n i=1fromDρtusing Algorithm 3 and construct the empirical distri- bution bDρt. 5:bg... | https://arxiv.org/abs/2502.08611v1 |
overcome these hurdles, let us study the event Et:={|cosθt−ρt| ≤sin2θt,sinθt≤Cζ(ρt)}. We first observe that when Etis satisfied, then, since sinθt≤Cζ(ρt)the algorithm may not be converging anymore, as discussed in Case 1 above. However, since Etalso satisfies |cosθt−ρt| ≤sin2θt, one can show that ζ(ρt)≈ζ(cosθt), theref... | https://arxiv.org/abs/2502.08611v1 |
lower bound oncosθt, which implies that sinθtshrinks at a linear rate and hence the event Etwill eventually be satisfied at some iteration t∗. Claim E.2. Lett′be the maximum t∈[0, T]such that for all t= 0, . . . , t′,Etis not satisfied. Then, for all t≤t′, it holds that ρt≤cosθt. Proof of Claim E.2. Weuseinductiontosho... | https://arxiv.org/abs/2502.08611v1 |
OPT/∥Tρtσ′∥L2, applying triangle inequality and the non-expansiveness of projection operator, it holds 2 sin( θt+1/2) =∥w(t+1)−w∗∥2=∥projB(w(t)−ηtbg(w(t)))−w∗∥2≤ ∥w(t)−ηtbg(t)−w∗∥2 ≤ ∥w(t)−w∗∥2+η2∥bg(t)∥2= 2 sin( θt/2) +φt/4. Using the assumption cosθt−ρt≥sin2θt, we observe that 1−sin2(θt/2)−(1−2φ2 t)≥sin2θt≥2 sin2(θt/... | https://arxiv.org/abs/2502.08611v1 |
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