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=/integraldisplayx −∞q(y)dy. Hence du dx=q(x), x=Qq(u). TRANSPORT f-DIVERGENCES 7 From the chain rule, we have q(x) =du dx= (dx du)−1=1 Q′q(u). Thus Q′ p(u) Q′q(u)=d duQp(Fq(x))·du dx =d dxQp(Fq(x)) =T′(x), where the second equality holds by the chain rule, and the las t equality holds because T(x) =Qp(Fq(x)). Substitu... | https://arxiv.org/abs/2504.15515v2 |
strictly convex with respect to the variable u. Then the transport f-divergence satisfies DT,f(p/bardblq) = sup Ψ/integraldisplay ΩΨ(x)p(T(x))dx−/integraldisplay Ωˆf∗(Ψ(x))q(x)dx, (10) where the supreme is over all continuous functions Ψ∈C(Ω;R), andTis a monotone mapping function, such that T#q=p. The optimality conditi... | https://arxiv.org/abs/2504.15515v2 |
proof. /square 4.Examples In this section, we list several examples of transport f-divergences. Example 1 (Transport total variation) .Letf(u) =|u−1|, then DT,f(p/bardblq) = DTTV(p/bardblq) =/integraldisplay Ω|T′(x)−1|q(x)dx =/integraldisplay1 0|Q′ p(u) Q′q(u)−1|du. We call DTTVthe transport total variation (TTV). Unfo... | https://arxiv.org/abs/2504.15515v2 |
the matrix refers to the Jacobian matrix of pushforwar d mapping function; see [16]. We shall also investigate the convexity properties, inequa lities, and variational algorithms for transport f-divergences towards generative AI-related samplingprob lemsandBayesian inverse problems. Acknowledgements . W. Li’s work is s... | https://arxiv.org/abs/2504.15515v2 |
arXiv:2504.15556v1 [math.ST] 22 Apr 2025Dynamical mean-field analysis of adaptive Langevin diffusio ns: Propagation-of-chaos and convergence of the linear respon se Zhou Fan∗, Justin Ko†, Bruno Loureiro‡, Yue M. Lu§, Yandi Shen¶ Abstract Motivated by an application to empirical Bayes learning in h igh-dimensional regress... | https://arxiv.org/abs/2504.15556v1 |
. . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Step 2: Discretization error of DMFT equation . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Step 3: Discretization of Langevin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4 Completing the proof . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2504.15556v1 |
with a resulting asymptotic decoupling of low-dimensional m arginals of {θs}s∈[0,t], is commonly referred to as propagation-of-chaos. We refer to the classical monographs [ 3,4] for a detailed treatment of such models, and to [ 5,6] and [7,8] for modern surveys and examples of recent quantitative conver gence results. ... | https://arxiv.org/abs/2504.15556v1 |
this work on the formalization of the limiting dynamics in a general cont ext. We summarize the main contributions of our paper as follows: 1. Adapting and building upon the methods of [ 42], we prove a DMFT limit for the dynamics ( 1) (with a natural extension to the dynamics ( 4–5) to follow). This will take the form... | https://arxiv.org/abs/2504.15556v1 |
ic limit for the model free energy. Acknowledgments This research was initiated during the “Huddle on Learning and Infer ence from Structured Data” at ICTP Trieste in 2023. We’d like to thank the huddle organiers Jean Barbier, Manuel S´ aenz, Subhabrata Sen, and Pragya Sur for their hospitality and many helpful discuss... | https://arxiv.org/abs/2504.15556v1 |
(Asymptotic scaling) lim n,d→∞n d=δ∈(0,∞). (b) (Random design) X= (xij)∈Rn×dhas independent entries satisfying Exij= 0,Ex2 ij=1 d, and /⌊a∇⌈⌊l√ dxij/⌊a∇⌈⌊lψ2≤Cfor a constant C >0 where/⌊a∇⌈⌊l·/⌊a∇⌈⌊lψ2is the sub-gaussian norm. (c) (Linear model and initial conditions) θ0,θ∗,εare independent of X, andy=Xθ∗+ε. For some p... | https://arxiv.org/abs/2504.15556v1 |
, Rη(t,s) =δβE/bracketleftBig∂ηt ∂ws/bracketrightBig .(18) We note that the above process {∂ηt ∂ws}t≥s≥0defined by ( 15) is in fact deterministic, but we keep the expec- tation defining Rη(t,s) for symmetry of notation. The equations ( 17–18) should be understood as fixed-point equations for α,Cθ,Cη,Rθ,Rη, where the laws ... | https://arxiv.org/abs/2504.15556v1 |
which is the usual notion of propagation-of-chaos for interacting particle systems. Corollary 2.7. In the setting of Theorem 2.5, suppose in addition that (θ∗,θ0)∈Rd×2andX∈Rn×dare both invariant in law under permutations of the coordinates {1,...,d}. Fix any J≥1, and let P(θ∗ 1:J,{θt 1:J}t∈[0,T])denote the joint law of... | https://arxiv.org/abs/2504.15556v1 |
system ( 12–18) given by Theorem 2.4. Then for any fixedt≥s≥0, almost surely as n,d→ ∞, d−1TrCθ(t,s)→Cθ(t,s), d−1TrCθ(t,∗)→Cθ(t,∗), n−1TrCη(t,s)→Cη(t,s), d−1TrRθ(t,s)→Rθ(t,s), n−1TrRη(t,s)→Rη(t,s). The proof of Theorem 2.8is provided in Section 5. We note that the convergence of d−1TrCθand n−1TrCηis an immediate consequ... | https://arxiv.org/abs/2504.15556v1 |
Φ Rθ,ΦRηon the right side to ˜ΦRθ,˜ΦRηon the left side, i.e. ˜ΦRθ(t) = 1+/integraldisplayt 0/parenleftBig (δ|β|+C0)ΦRθ(t′)+/integraldisplayt′ 0ΦRη(t′−s)ΦRθ(s)ds/parenrightBig dt′, ˜ΦRη(t) =|β|/parenleftBig/integraldisplayt 0ΦRθ(t−s)ΦRη(s)ds+δ|β|ΦRθ(t)/parenrightBig . If ΦRθ,ΦRη∈E(λ), then writing Lθ(λ) =/integraltext∞ ... | https://arxiv.org/abs/2504.15556v1 |
all 0 ≤t≤T. (31) Furthermore, Cη(t,s) is uniformly continuous over s,t∈Ifor each maximal interval Iof [0,T]\D, and satisfies /vextendsingle/vextendsingleCη(t,t)−2Cη(t,s)+Cη(s,s)/vextendsingle/vextendsingle≤3β2/bracketleftbigg/parenleftBig T3sup r∈[0,T]Φ′ Rθ(r)2+Tsup r∈[0,T]ΦRθ(r)2/parenrightBig sup r∈[0,T]ΦCη(r) +δ/pare... | https://arxiv.org/abs/2504.15556v1 |
of εand(w∗,{wt}t≤T). Then there exist unique Fη t-adapted processes {ηt}t≤Tand{∂ηt ∂ws}s≤t≤Tsolving (14–15). 13 Proof.Conditional on εand (w∗,{wt}), the equations ( 14–15) are linear Volterra integral equations for which the kernel ( s,t)/ma√sto→Rθ(t,s) is continuous on each maximal interval Iof [0,T]\D. Then, for each... | https://arxiv.org/abs/2504.15556v1 |
to 0 uniformly in τas|s−s′| →0, observe that the last term above is o|s−s′|(1) by the uniform continuity of RθonI′×I. For the second term, writing the range of integration as [0 ,τ] =A∪Bwherer∈Aare the values for which s+r,s′+rbelong to a single maximal interval of [0 ,T]\Dandr∈Bare the values for which s+r,s′+rbelong ... | https://arxiv.org/abs/2504.15556v1 |
tE/bracketleftBig −/parenleftBig δβ−∂θs(θr,αr)/parenrightBig∂θr ∂us/bracketrightBig dr+/integraldisplayt′ t/parenleftBig/integraldisplayr sRη(r,r′)Rθ(r′,s)dr′/parenrightBig dr/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤/integraldisplayt′ t/parenleftBig (δ|β|+C0)¯Rθ(r,s)dr+/integraldisplayr sΦRη(r−r′)¯Rθ(r... | https://arxiv.org/abs/2504.15556v1 |
2θs 2] =Cθ(t,s). Moreover, E(θt 1−θt 2)2≤5[(I)+(II)+(III)+(IV)+(V)] where we set (I) =E/parenleftBig/integraldisplayt 0δβ|θs 1−θs 2|ds/parenrightBig2 (II) =E/parenleftBig/integraldisplayt 0|s(θs 1,αs 1)−s(θs 2,αs 2)|ds/parenrightBig2 (III) =E/parenleftBig/integraldisplayt 0|θs′ 1−θs′ 2|/parenleftBig/integraldisplayt s′... | https://arxiv.org/abs/2504.15556v1 |
1−˜αt 2/⌊a∇⌈⌊l ≤C/integraldisplayt 0/parenleftBig /⌊a∇⌈⌊l˜αs 1−˜αs 2/⌊a∇⌈⌊l+W2(P(θs 1),P(θs 2))/parenrightBig ds≤C/integraldisplayt 0/parenleftBig /⌊a∇⌈⌊l˜αs 1−˜αs 2/⌊a∇⌈⌊l+/radicalBig E(θs 1−θs 2)2/parenrightBig ds. Then /⌊a∇⌈⌊l˜αt 1−˜αt 2/⌊a∇⌈⌊l ≤C/integraldisplayt 0eλsdssup s∈[0,T]e−λs/parenleftBig /⌊a∇⌈⌊l˜αs 1−˜αs ... | https://arxiv.org/abs/2504.15556v1 |
θ, so the Banach fixed-point theorem guarantees Tη→ηandTθ→θhave unique fixed points X= (Rη,Cη,α)∈ Scont ηandY= (Rθ,Cθ,α)∈ Scont θ, for which also Tη→θ(X) =Y. These fixed points remain unique in SηandSθ, because Lemmas 3.5and3.6imply that the images of Tη→η,Tθ→θonSη,Sθare contained in Scont η,Scont θ. Then the tuple ( α,Cθ... | https://arxiv.org/abs/2504.15556v1 |
Here, (w∗ γ,{ws γ}0≤s≤t) is a mean-zero Gaussian vector with covariance E[ws γwr γ] =Cγ θ(s,r),E[ws γw∗ γ] =Cγ θ(s,∗),E[(w∗ γ)2] =Cγ θ(∗,∗), (63) and again∂ηt γ ∂wsγis the usual partial derivative computed from the chain rule. These define{Cγ η(s,r)}r≤s≤t, {Rγ η(s,r)}r<s≤tup to time tvia Cγ η(s,r) =δβ2E[(ηs γ+w∗ γ−ε)(ηr... | https://arxiv.org/abs/2504.15556v1 |
i,j≥0. (74) This is a standard form of an AMP algorithm, see e.g. [ 45,60]. The iterations for ( W0,...,WT−1)∈Rn×kT and (U1,...,UT)∈Rd×kTadmit a mapping to the form of [ 60, Eqs. (2.14) and (D.1–D.2)] with kTvector iterates. Then by the AMP state evolution (c.f. [ 60, Theorem 2.21 and Remark 2.2]), under the conditions... | https://arxiv.org/abs/2504.15556v1 |
thus defined are centered Gaussian processes with covariance E[ui γuj γ] =δ3E[fi,1(W0,...,Wi;ε)fj,1(W0,...,Wj;ε)] =δβ2E[(ηi−W∗−ε)(ηj−W∗−ε)], E[wi γwj γ] =E[gi,1(U1,...,Ui;V)gj,1(U1,...,Uj;V)] =E[θiθj], E[wi γw∗ γ] =E[gi,1(U1,...,Ui;V)g0,2(V)] =E[θiθ∗], E[(w∗ γ)2] =E[g0,2(V)2] =E[(θ∗)2], which verifies ( 59) and (63) in l... | https://arxiv.org/abs/2504.15556v1 |
≤T/summationdisplay t=01 n/⌊a∇⌈⌊lηt γ−˜ηt γ/⌊a∇⌈⌊l2≤T/summationdisplay t=01 n/⌊a∇⌈⌊lX/⌊a∇⌈⌊l2 op/⌊a∇⌈⌊lθt γ−˜θt γ/⌊a∇⌈⌊l2 and/⌊a∇⌈⌊lX/⌊a∇⌈⌊lopis almost surely bounded for all large n,d, the above inductive claim together with ( 71) implies (66–68). The base case of t= 0 in (90) holds exactly. Suppose ( 90) holds up to ... | https://arxiv.org/abs/2504.15556v1 |
maps is a piecewise c onstant embedding of the discrete-time DMFT system in Section 4.1, and that furthermore this fixed point belongs to S. Lemma 4.3. (a) Given any Xγ∈ Dγ ηand realization of θ∗,θ0,{bt}t∈[0,T], and{¯ut γ}t∈[0,T], the processes (93–94) have a unique solution, and this solution is piecewise cons tant and... | https://arxiv.org/abs/2504.15556v1 |
γ−θ∗)+s(¯θs γ,¯αs γ)+/integraldisplay⌊s⌋ 0¯Rγ η(s,r)(¯θr γ−θ∗)dr+ ¯us γ/bracketrightBig ds+√ 2(b(k+1)γ−bkγ) =¯θkγ γ+γ/parenleftBig −δβ(¯θkγ γ−θ∗)+s(¯θkγ γ,¯αkγ γ)+γk−1/summationdisplay ℓ=0¯Rγ η(kγ,ℓγ)(¯θℓγ γ−θ∗)+ ¯ukγ γ/parenrightBig +√ 2(b(k+1)γ−bkγ), ∂¯θ(k+1)γ γ ∂¯u(k+1)γ γ= 1, and for any j≤k, ∂¯θ(k+1)γ γ ∂¯ujγ γ=∂¯... | https://arxiv.org/abs/2504.15556v1 |
moduli-of-continuity of Tγ η→θandTγ θ→ηin the metricsd(·) of Section 3.2, implying that Tγ η→ηandTγ θ→θare contractive for sufficiently large λ >0 defining d(·). These metrics induce the topologies of uniform convergence on t he spaces Dγ η∩SηandDγ θ∩Sθ, which are equivalent to closed subsets of finite-dimensional vector s... | https://arxiv.org/abs/2504.15556v1 |
λsup 0≤s≤t≤Te−λtE|rθ(t,s)−¯rγ θ(t,s)|+C√γ, and choosing large enough λ >0 and rearranging gives d(Rθ,¯Rγ θ)≤sup 0≤s≤t≤Te−λtE|rθ(t,s)−¯rγ θ(t,s)| ≤C√γ. Bound of d(˜α,¯˜αγ).By definition, ˜αt=α0+/integraldisplayt 0G(˜αs,P(θs))ds,¯˜αt γ=α0+/integraldisplay⌊t⌋ 0G(¯˜αs γ,P(¯θs γ))ds, so/⌊a∇⌈⌊l˜αt−¯˜αt γ/⌊a∇⌈⌊l ≤(I)+(II) wher... | https://arxiv.org/abs/2504.15556v1 |
92). Asimple induction showsthat this is equivalently the solution to a modification of the dynamics ( 4–5), ¯θt γ=θ0+/integraldisplay⌊t⌋ 0/bracketleftBig −βX⊤(X¯θs γ−y)+/parenleftbig s(¯θs γ,j,¯/hatwideαs γ)/parenrightbigd j=1/bracketrightBig ds+√ 2b⌊t⌋, ¯/hatwideαt γ=/hatwideα0+/integraldisplay⌊t⌋ 0G/parenleftBig ¯/ha... | https://arxiv.org/abs/2504.15556v1 |
from instance to instance. We restrict to the almost-sur e event where Lemmas 4.7and4.8hold, and (107) holds for all large n,d. Then, coupling ( 4) and (105) by the same Brownian motion, for any 0≤t≤T, /⌊a∇⌈⌊lθt−¯θt γ/⌊a∇⌈⌊l ≤C/integraldisplay⌊t⌋ 0/parenleftBig /⌊a∇⌈⌊lX⊤X(θs−¯θs γ)/⌊a∇⌈⌊l+/⌊a∇⌈⌊ls(θs;/hatwideαs)−s(θs γ... | https://arxiv.org/abs/2504.15556v1 |
j=1/⌊a∇⌈⌊lθj−˜θj/⌊a∇⌈⌊l2 ∞/parenrightBig1/2/parenleftBig 1+1 dd/summationdisplay j=1/⌊a∇⌈⌊lθj/⌊a∇⌈⌊l2 ∞/parenrightBig1/2 .(113) Set F(θ,/hatwideα) =−βX⊤(Xθ−y)+s(θ,/hatwideα) so that by definition, θt j=θ0 j+/integraltextt 0e⊤ jF(θs,/hatwideαs)ds+√ 2bt j. Hence sup t∈[0,T](θt j)2≤C/parenleftBig (θ0 j)2+/integraldisplayT ... | https://arxiv.org/abs/2504.15556v1 |
We recall that {wt}t∈[0,T]has covariance satisfying |Cθ(t,s)| ≤C|t−s|, soP[/⌊a∇⌈⌊lw/⌊a∇⌈⌊lα> C+x]≤2e−cx2for someC,c >0 (c.f. [62, Theorem 5.32]). Thus E/⌊a∇⌈⌊lη/⌊a∇⌈⌊l2 α≤C. Applying these bounds, the same arguments as above show the almost-sure convergence 1 nn/summationdisplay i=1δη∗ i,εi,{ηt i}t∈[0,T])W2→P(η∗,ε,{ηt}... | https://arxiv.org/abs/2504.15556v1 |
note that for convenience of the proof, we define θt,(j)to be of the same dimension as θ, where one may check from ( 124) that the dynamics of θt,(j) −jdo not involve θt,(j) j. Similarly, for i∈[n], let X[i]= (Xkj1k/n⌉}ationslash=i)k,j∈Rn×d,y[i]=X[i]θ∗+ε, 43 whereX[i]sets theithrow ofXto 0. Define θt+1,[i]=θt,[i]−γ/brack... | https://arxiv.org/abs/2504.15556v1 |
using ( θ0,(j),/hatwideα0,(j)) = (θ0,/hatwideα0), for any t≤T/γ, /⌊a∇⌈⌊lθt−θt,(j)/⌊a∇⌈⌊l+√ d/⌊a∇⌈⌊l/hatwideαt−/hatwideαt,(j)/⌊a∇⌈⌊l ≤t−1/summationdisplay s=0(1+Cγ)st−1max s=0Cγ∆s,j≤C′t−1max s=0∆s,j. Let us now bound ∆ t,j. Writing xj∈Rnfor thejthcolumn of X, we have X(j)=X−xje⊤ j, hence X⊤X−X(j)⊤X(j)=X⊤xje⊤ j+ejx⊤ jX−e... | https://arxiv.org/abs/2504.15556v1 |
This implies 1 dd/summationdisplay j=1∂ε|ε=0θt,(s,j),ε j=γ dTr/parenleftbig Ωt−1...Ωs+1/parenrightbig +γ dt−1/summationdisplay ℓ=s+1d/summationdisplay j=1e⊤ jΩt−1...Ωℓ+1rℓ,(s,j). (137) On an event where /⌊a∇⌈⌊lX/⌊a∇⌈⌊lop≤C0for all large n,d(which holds almost surely), by the Lipschitz continuity ofs(·) in Assumption 2.... | https://arxiv.org/abs/2504.15556v1 |
135), for each s < t, almost surely lim n,d→∞/braceleftbigg γ·1 dTr/parenleftBig Ωt−1...Ωs+1/parenrightBig ,1 dd/summationdisplay j=1∂ε|ε=0θt,(s,j),ε j/bracerightbigg =Rγ θ(t,s), lim n,d→∞/braceleftbigg γ·1 nTr/parenleftBig XΩt−1...Ωs+1X⊤/parenrightBig ,1 nn/summationdisplay i=1∂ε|ε=0ηt,[s,i],ε i/bracerightbigg =β−1rη(... | https://arxiv.org/abs/2504.15556v1 |
−jX⊤ −jxj−1 dTr/parenleftBig X−jΩt−1,(j) −j...Ωk+1,(j) −jX⊤ −j/parenrightBig . Sincexjis independent of Ωt,(j) −jandX−j, the Hanson-Wright inequality yields P/bracketleftBig |r(j,k) 2,2| ≥max/parenleftBigC√logd d/⌊a∇⌈⌊lW/⌊a∇⌈⌊lF,Clogd d/⌊a∇⌈⌊lW/⌊a∇⌈⌊lop/parenrightBig/bracketrightBig ≤e−cd for some C,c >0, where W=X−jΩt... | https://arxiv.org/abs/2504.15556v1 |
θ(k,s)+Et+1,(s,j) θ =r(θj,/hatwideα) θ(t+1,s)+Et+1,(s,j) θ where lim n,d→∞d−1/summationtextd j=1|Et+1,(s,j) θ|p= 0 a.s. for each p≥1, concluding the proof the inductive claim ( 141) forEt+1,(s,j) θ. Claim for ∂ε|ε=0ηt,[s,i],ε i.We now show the claim ( 141) forEt+1,[s,i] η. Again fixing s∈Z+andi∈[n], let us shorthand ηt,... | https://arxiv.org/abs/2504.15556v1 |
we may apply the dominated convergence theorem to the second statement of ( 141) to get, almost surely, lim n,d→∞n−1TrRη(t,s) = lim n,d→∞δβ2·1 nn/summationdisplay i=1∂ε|ε=0/a\}⌊∇a⌋k⌉tl⌉{tηt,[s,i],ε i/a\}⌊∇a⌋k⌉t∇i}ht= lim n,d→∞δβ·1 nn/summationdisplay i=1/a\}⌊∇a⌋k⌉tl⌉{trη(t,s)/a\}⌊∇a⌋k⌉t∇i}ht=Rγ η(t,s), concluding the p... | https://arxiv.org/abs/2504.15556v1 |
of s(·) in Assumption 2.2thatJt 1−¯Jt γ,1is diagonal with /⌊a∇⌈⌊lJt 1−¯Jt γ,1/⌊a∇⌈⌊lF≤C(/⌊a∇⌈⌊lθt−¯θt γ/⌊a∇⌈⌊l+√ d/⌊a∇⌈⌊l/hatwideαt−¯/hatwideαt γ/⌊a∇⌈⌊l). Next using the arguments that led to ( 109), we have that on the event {/⌊a∇⌈⌊lX/⌊a∇⌈⌊lop≤C0,/⌊a∇⌈⌊ly/⌊a∇⌈⌊l ≤C0√ d}, withx0= (θ0,/hatwideα0), /a\}⌊∇a⌋k⌉tl⌉{t/⌊a∇⌈⌊l... | https://arxiv.org/abs/2504.15556v1 |
0(/a\}⌊∇a⌋k⌉tl⌉{t/⌊a∇⌈⌊lUs−¯Us/⌊a∇⌈⌊lF/a\}⌊∇a⌋k⌉t∇i}htx+√ d/a\}⌊∇a⌋k⌉tl⌉{t/⌊a∇⌈⌊lWs−¯Ws/⌊a∇⌈⌊lF/a\}⌊∇a⌋k⌉t∇i}htx)ds+ι(γ)/parenleftBig/⌊a∇⌈⌊lθ0/⌊a∇⌈⌊l√ d+/⌊a∇⌈⌊l/hatwideα0/⌊a∇⌈⌊l+1/parenrightBig .(175) Combining ( 174) and (175) yields /a\}⌊∇a⌋k⌉tl⌉{t/⌊a∇⌈⌊lUt−¯Ut γ/⌊a∇⌈⌊lF+√ d/⌊a∇⌈⌊lWt−¯Wt γ/⌊a∇⌈⌊lF/a\}⌊∇a⌋k⌉t∇i}htx ≤C... | https://arxiv.org/abs/2504.15556v1 |
(I).NotethatbyProposition A.2(c), TrdPt−s(x)E⊤X⊤XE=/a\}⌊∇a⌋k⌉tl⌉{tTrVt−sE⊤X⊤XE/a\}⌊∇a⌋k⌉t∇i}htx=/a\}⌊∇a⌋k⌉tl⌉{tTrUt−s· X⊤X/a\}⌊∇a⌋k⌉t∇i}htx,where{xt,Vt}t≥0followthedynamics( 160)andUtasbeforeistheupper-leftblockof Vt. Similarly, LemmaA.5yields that Trd Pγ [t]−[s](x)E⊤X⊤XE=/a\}⌊∇a⌋k⌉tl⌉{tTr¯V⌊t⌋−⌊s⌋ γE⊤X⊤XE/a\}⌊∇a⌋k⌉t∇i... | https://arxiv.org/abs/2504.15556v1 |
associated to ( 178). When the initial condition x0=xis clear from context, we will abbreviate /a\}⌊∇a⌋k⌉tl⌉{tf(xt)/a\}⌊∇a⌋k⌉t∇i}ht=/a\}⌊∇a⌋k⌉tl⌉{tf(xt)/a\}⌊∇a⌋k⌉t∇i}htx0=x. We denote the infinitesimal generator L of this semigroup by Lf(x) =u(x)⊤∇f(x)+TrMM⊤∇2f(x). (182) Throughout this section, constants C,C′,c >0 may ... | https://arxiv.org/abs/2504.15556v1 |
in x, that∇Ptf(x) =∇x/a\}⌊∇a⌋k⌉tl⌉{tf(xt)/a\}⌊∇a⌋k⌉t∇i}htx0=x=/a\}⌊∇a⌋k⌉tl⌉{t∇xf(xt(x))/a\}⌊∇a⌋k⌉t∇i}htand∇2Ptf(x) =∇2 x/a\}⌊∇a⌋k⌉tl⌉{tf(xt)/a\}⌊∇a⌋k⌉t∇i}htx0=x= /a\}⌊∇a⌋k⌉tl⌉{t∇2 xf(xt(x))/a\}⌊∇a⌋k⌉t∇i}ht, and that these are also uniformly bounded and continuous over t∈[0,T] andx∈Rm. For the derivative in t, by Itˆ o’... | https://arxiv.org/abs/2504.15556v1 |
formula for any h >0, Pε s−h,sf(x) =/a\}⌊∇a⌋k⌉tl⌉{tf(xs)/a\}⌊∇a⌋k⌉t∇i}htxs−h=x=f(x)+/integraldisplays s−h/a\}⌊∇a⌋k⌉tl⌉{tLε rf(xr)/a\}⌊∇a⌋k⌉t∇i}htxs−h=xdr. The same argument as in Proposition A.2shows that Lε tf(xt(s,x)) is uniformly integrable over compact domains of t≥s≥0 and of x∈Rm, so by dominated convergence we ha... | https://arxiv.org/abs/2504.15556v1 |
unique. A.2 Discrete dynamics We record (elementary) analogues of the preceding results for dis crete dynamics xt+1=xt+u(xt)+√ 2M(bt+1−bt) (194) where{bt}t∈Z+is a Gaussian process with b0= 0 and independent increments bt+1−bt∼ N(0,γI), for someγ >0. The following is an analogue of Proposition A.1. Proposition A.4. Supp... | https://arxiv.org/abs/2504.15556v1 |
[8] Daniel Lacker and Luc Le Flem. Sharp uniform-in-time propagatio n of chaos. Probability Theory and Related Fields , 187(1-2):443–480, 2023. [9] Haim Sompolinsky and Annette Zippelius. Dynamic theory of the spin -glass phase. Physical Review Letters, 47(5):359, 1981. [10] Haim Sompolinsky and Annette Zippelius. Rela... | https://arxiv.org/abs/2504.15556v1 |
stochastic gradient descent in Gaussian mixture cla ssification. J. Stat. Mech. Theory Exp., (12):Paper No. 124008, 23, 2021. [28] FrancescaMignaccoandPierfrancescoUrbani. Theeffectiven oiseofstochasticgradientdescent. Journal of Statistical Mechanics: Theory and Experiment , 2022(8):083405, 2022. [29] Blake Bordelon and... | https://arxiv.org/abs/2504.15556v1 |
pages 2022–04, 2022. [49] Sumit Mukherjee, Bodhisattva Sen, and Subhabrata Sen. A me an field approach to empirical Bayes estimation in high-dimensional linear regression. arXiv preprint arXiv:2309.16843 , 2023. [50] Youngseok Kim, Wei Wang, Peter Carbonetto, and Matthew St ephens. A flexible empirical Bayes approach to ... | https://arxiv.org/abs/2504.15556v1 |
Dynamical mean-field analysis of adaptive Langevin diffusions: Replica-symmetric fixed point and empirical Bayes Zhou Fan∗, Justin Ko†, Bruno Loureiro‡, Yue M. Lu§, Yandi Shen¶ Abstract In many applications of statistical estimation via sampling, one may wish to sample from a high- dimensional target distribution that ... | https://arxiv.org/abs/2504.15558v1 |
Langevin dynamics . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.1 A general condition for dimension-free convergence . . . . . . . . . . . . . . . . . . . . 13 2.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Analysis of approximately-TTI DMFT systems 19... | https://arxiv.org/abs/2504.15558v1 |
. . . . . . . . . . . 35 4.1.2 Interpretation of the DMFT correlation and response . . . . . . . . . . . . . . . . . . 37 4.2 Posterior bounds and Wasserstein-2 convergence . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3 Properties of the correlation and response . . . . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2504.15558v1 |
of g∗in settings of high- dimensional regression designs X∈Rn×d, asn, d→ ∞ . However, direct computation of this maximum marginal-likelihood estimate is intractable for general regression designs, motivating approaches based on approximate posterior inference schemes. We will investigate in this work a parametric analo... | https://arxiv.org/abs/2504.15558v1 |
these results as follows: 1. In the setting of a non-adaptive Langevin diffusion, we formalize a condition of approximate time- translation-invariance (TTI) for the DMFT system. We perform an analysis of the dynamical fixed- point equations for the DMFT correlation and response functions under this condition, and show ... | https://arxiv.org/abs/2504.15558v1 |
takes a step towards filling this gap, by providing a rigorous analysis of the DMFT approximation to Langevin dynamics in a more general model without a rotationally invariant prior, in settings where approximate-TTI holds. As a by-product of our analyses, we obtain a new proof of a replica formula [57] for the free en... | https://arxiv.org/abs/2504.15558v1 |
2030 program with the reference “ANR-23-IACL-0008” and the Choose France - CNRS AI Rising Talents program. Y. M. Lu was supported in part by the Harvard FAS Dean’s Competitive Fund for Promising Scholarship and by a Harvard College Professorship. Notational conventions In the context of the posterior law Pg(θ|X,y) for ... | https://arxiv.org/abs/2504.15558v1 |
continuous-time evolution of the prior parameter α∈RKthat is coupled to the Langevin diffusion (9) of the posterior sample, and R:RK→Ris a possible smooth regularizer. (In this work, we will be interested mostly in the behavior of these dynamics when R(α)≡0, and we introduce R(α) for theoretical purposes to confine the... | https://arxiv.org/abs/2504.15558v1 |
each other and of X, and y=Xθ∗+ε. The entries of θ∗,εare distributed as θ∗ 1, . . . , θ∗ diid∼g∗, ε 1, . . . , ε niid∼ N(0, σ2) (16) for some σ2>0 and probability density g∗(both fixed and independent of n, d), where g∗satisfies the log-Sobolev inequality Entθ∗∼g∗[f(θ∗)2]≤CLSIEθ∗∼g∗[(f′(θ∗))2] for all f∈C1(R). (17) (d)... | https://arxiv.org/abs/2504.15558v1 |
dtαt=G(αt,P(θt)),G(α,P) =Eθ∼P[∇αlogg(θ, α)]− ∇R(α) (27) 8 with initial condition αt|t=0=α0given in Assumption 2.2, where P(θt) is the law of θtin (23). The covariance and response functions Cθ, Cη, Rθ, Rηare also defined for all t≥s≥0 self-consistently via the above processes by Cθ(t, s) =E[θtθs], C θ(t,∗) =E[θtθ∗], C ... | https://arxiv.org/abs/2504.15558v1 |
and response functions of the DMFT system that characterize an approximate time-translation-invariance (TTI) property. Under these conditions, we establish convergence of the joint law of ( θ∗, θt) in the DMFT equations to a replica-symmetric fixed point ast→ ∞ . Definition 2.4. In the setting of a fixed prior g(·) [i.... | https://arxiv.org/abs/2504.15558v1 |
(45) Remark 2.6. Let⟨f(θ)⟩and⟨f(θ,θ′)⟩denote the expectation over independent samples θ,θ′∼Pg(· |X,y) from the posterior law (5) with a fixed prior g(·). Then the asymptotic overlaps lim n,d→∞d−1⟨θ⊤θ⟩, lim n,d→∞d−1⟨θ⊤θ′⟩, lim n,d→∞d−1⟨θ⊤θ∗⟩ are predicted in the DMFT system, respectively, by ctti θ(0) = lim t→∞Cθ(t, t),... | https://arxiv.org/abs/2504.15558v1 |
approximately-TTI, where the statements of Definition 2.4 hold with ε(t) =Ce−ctand some constants C, c > 0. As a consequence, we obtain the following corollary showing that the asymptotic free energy and mean- squared-errors associated to the posterior distribution Pg(θ|X,y) in the linear model (with a possibly misspec... | https://arxiv.org/abs/2504.15558v1 |
both viewed as a function of α∈O⊂RK. Here, the fixed points ( ω, ω∗)≡(ω(α), ω∗(α)) implicitly depend on αand are well-defined by Theorem 2.9 for all α∈O. Recalling the law Pg∗,ω∗;g,ω(θ∗, θ) from (40), let us abbreviate this law with g≡g(·, α) and fixed points (ω(α), ω∗(α)) as Pα≡Pg∗,ω∗(α);g(·,α),ω(α). (50) We write θ∼P... | https://arxiv.org/abs/2504.15558v1 |
simulation the convergence of ( θt,bαt), the landscape of the replica- symmetric free energy F(α), and the nature of its critical point set Crit in a few settings where a posterior log-Sobolev inequality may not hold. 14 Example 2.14. Consider the Gaussian prior g(θ, α) =rω0 2πexp −ω0 2(θ−α)2 with varying mean α∈Rand... | https://arxiv.org/abs/2504.15558v1 |
show in Proposition 5.2 of Section 5.2 that adding such a regularizer indeed confines the dynamics of {αt}t≥0to a bounded domain. We will study analytically a large- δlimit under a reparametrization of the noise variance σ2bys2=σ2/δ, corresponding to a rescaling of the regression design Xto have entries of variance 1 /... | https://arxiv.org/abs/2504.15558v1 |
a single instance ( X,y) with max( n, d) = 5000, initialization θ0 jiid∼ N(0,1), and an Euler-Maruyama discretization of the dynamics. (a–e) Landscape of the replica-symmetric free energy F(α) plotted (for visual clarity) as log( F(α)−F(α∗)+10−3), for δ∈ {4,2,1,0.5,0.25}. Two stable fixed points of 0 = ∇F(α) are depict... | https://arxiv.org/abs/2504.15558v1 |
Mean- squared-error1 d∥θt−θ∗∥2 2across iterations for these same three initial states, at δ= 0.75. The predicted value of mse( α∗) + mse ∗(α∗) for a posterior sample is depicted by the dashed line. Example 2.18. Consider a K+ 1-component Gaussian mixture prior g(θ, α) =KX k=0pk(α)rωk 2πexp −ωk 2(θ−µk)2 , p k(α) =eαk ... | https://arxiv.org/abs/2504.15558v1 |
from random initializations of θ0, but does converge to α∗under a ground-truth initialization θ0=θ∗andbα0close to α∗. 3 Analysis of approximately-TTI DMFT systems In this section, we prove Theorem 2.5 on the equilibrium properties of the solution to the DMFT equations under an assumption of approximate time-translation... | https://arxiv.org/abs/2504.15558v1 |
m=1c2 me−amτ, C(M) η(t, s) =MX m=1c2 m am(e−am|t−s|−e−am(t+s)) +ctti η(∞). A direct calculation of the covariance shows that C(M) η(t, s) =E[ut Mus M] for ut M=z+MX m=1cmZt 0e−am(t−s)√ 2 dbs m, (62) where z∼ N(0, ctti η(∞)) and {bt 1}t≥0, . . . ,{bt M}t≥0are standard Brownian motions independent of each other and of z.... | https://arxiv.org/abs/2504.15558v1 |
j=1X i∈IjZt 0˜c2 i e−˜ais−e−ajs2ds, II =MX j=1X i∈IjZt 0˜c2 i 1−cjqP ℓ∈Ij˜c2 ℓ2 e−2ajsds. Let ∆ = 1 /√ Mbe the spacing of {aj}M j=0. Then, since |˜ai−aj| ≤∆ and ˜ ai≤ajfor all i∈Ij, I≤MX j=1X i∈IjZt 0˜c2 ie−2aj−1ss2∆2ds≤MX j=1Zt 0X i∈Ij˜c2 i ˜ai aje−2aj−1ss2∆2ds(∗)=MX j=1c2 jZt 0e−2aj−1ss2∆2ds where we useP i∈Ij˜... | https://arxiv.org/abs/2504.15558v1 |
We define a coupling of {θt}t≥T0and{˜θt}t≥T0by the joint evolutions, for t≥T0, dθt=h U(θt, θ∗) +Zt 0Rη(t, s)(θs−θ∗)ds+uti dt+h(|θt−˜θt|)√ 2 dbt+q 2(1−h(|θt−˜θt|)2) d˜bt, d˜θt=h U(˜θt, θ∗) +Zt 0R(M) η(t−s)(˜θs−θ∗)ds+ ˜uti dt−h(|θt−˜θt|)√ 2 dbt+q 2(1−h(|θt−˜θt|)2) d˜bt. Thus the coupling of the Brownian motions defining ... | https://arxiv.org/abs/2504.15558v1 |
A′(τ) =−c0A(τ)−PM m=1c2 me−amτ, i.e. A′(τ) +R(M) η(τ) =−c0A(τ). (71) We will require that A(τ)≥0 for all τ≥0. To check this condition, observe that explicitly evaluating the integral defining A(τ) yields ec0τA(τ) =A(0)−MX m=1c2 m am−c0 1−e−(am−c0)τ ≥δ σ2−c0−MX m=1c2 m am−c0≥δ σ2−c0−MX m=1c2 m am·ι ι−c0, the last ineq... | https://arxiv.org/abs/2504.15558v1 |
= 0. Recalling the sequences {am}M m=0,{cm}M m=1defining R(M) η, rtti η(τ)−R(M) η(τ) =Z∞ ιae−aτµη(da)−MX m=1c2 me−amτ =MX m=1Zam am−1(ae−aτ−ame−amτ)µη(da) +Z a:a>aMae−aτµη(da), 25 hence using the fact that h(a) =ae−aτsatisfies |h′(a)| ≤2e−aτ/2and|am−am−1|= 1/√ M, Zt 0 rtti η(τ)−R(M) η(τ) dτ≤MX m=1Zam am−1Zt 0|ae−aτ−am... | https://arxiv.org/abs/2504.15558v1 |
for some C, T, R > 0 sufficiently large depending on C′, c0, the above implies d dtE(rt)4≤CE(rt)4,d dtE(rt)4≤ −4c0E(rt)4+c0E(rt)4<0 whenever t≥TandE(rt)4≥R. This implies that supt≥0E(rt)4is bounded by a constant depending only on C, T, R . Then supt≥0E(θt)4is also bounded since θt=¯θt+θ∗and|¯θt| ≤f(¯θt)≤rt. The argumen... | https://arxiv.org/abs/2504.15558v1 |
g)′(θ′)) + ( x−x′)⊤L(x−x′) where L= δ σ2−c1. . .−cM −c1a1 ...... −cM aM . By the positivity of the Schur complement ω(M)=δ/σ2−PM m=1c2 m/am≥δ/σ2−(ctti η(0)−ctti η(∞)) and of am≥ι, this matrix Lis strictly positive-definite, with smallest eigenvalue bounded away from 0 independently of M. Then, recalling the f... | https://arxiv.org/abs/2504.15558v1 |
Pg,ω(θ|y) in the scalar Gaussian convolution model (37) where y=θ∗+ω∗−1/2zandz∼ N(0,1). Let ⟨·⟩g,ωbe average over θ, θ′conditional on θ∗, z, and let Fbe the class of 1-Lipschitz functions f(θ∗, θ, θ′). Then, for any f∈ F, Eg∗,ω∗⟨f(θ∗, θ, θ′)⟩g,ω=ER f(θ∗, θ, θ′) exp(−ω 2[(θ∗+ω∗−1/2z−θ)2+ (θ∗+ω∗−1/2z−θ′)2])g(θ)g(θ′)d(θ, ... | https://arxiv.org/abs/2504.15558v1 |
simple Gronwall argument to bound the propagation of the discretization error ε(M) over time. 3.2.1 Comparison with an auxiliary process We again fix a positive integer M, and define {am}M m=0and{cm}M m=1by am=ι+m√ Mform= 0, . . . , M,c2 m am=µθ([am−1, am)) with µθnow instead of µη. For convenience, let us introduce ξt... | https://arxiv.org/abs/2504.15558v1 |
=A+cc⊤/σ2, and consider the 2-dimensional Gaussian law N(0,ΣM)with ΣM= ρ2 MκM κMρ2 M , κ M=σ2 Z·h 1−c⊤Λ−1c/σ2i2 , ρ2 M=κM+c⊤Λ−1c, where σ2 Z=E(θ∗)2+ctti θ(∞)−2cθ(∗) +σ2. Then there exists an error ε(T)not depending on T0, Mand satisfying limT→∞ε(T) = 0 , such that for any M, T 0, T, T′>0, W2(P(ξT0+T M,T 0, ξT0+T+T′ M... | https://arxiv.org/abs/2504.15558v1 |
all t≥0 under Definition 2.4, this gives ∥E[UU⊤]∥op≤ C′PM m=1c2 m/a2 m≤C′µθ([ι,∞))/ι. Combining these bounds shows ∥E[xT0(xT0)⊤]∥op≤Cfor a constant C >0 not depending on M, T 0. Then, combining with the previous bounds ∥Λ−1/2u∥2≤Candλmin(Λ)≥ι, this shows |I| ≤ε(T), so|E[(˜ξT0+T)2]−ρ2 M| ≤ε(T). The bound for E[(˜ξT0+T+T... | https://arxiv.org/abs/2504.15558v1 |
of its Markov semigroup. For bounded observables, similar fluctuation-dissipation theorems have been stated and shown in [77, 78] and Bismut-Elworthy-Li formulae in [79, 80]. We give versions of these results here for a class of unbounded observables which may have linear growth A={f∈C2(Rd,R) :∇f,∇2fare globally bounde... | https://arxiv.org/abs/2504.15558v1 |
to g(s,θ) =Pt−sf(θ) gives f(θt) =g(t,θt) =g(0,θ0) +Zt 0∂sg(s,θs)ds+Zt 0∇θg(s,θs)⊤dθs+Zt 0Tr∇2 θg(s,θs)ds =Ptf(x) +Zt 0(∂s+ L)Pt−sf(θs)ds+√ 2Zt 0∇Pt−sf(θs)⊤dbs =Ptf(x) +√ 2Zt 0∇Pt−sf(θs)⊤dbs. Since ∇2Uis bounded, {Vt}t∈[0,T]is bounded over finite time horizons, soRt 0Vs⊤dbsis a martingale. Multiplying both sides by this... | https://arxiv.org/abs/2504.15558v1 |
>0for which ⟨∥θ∥2 2⟩ ≤C(d+∥y∥2 2), (106) ⟨∥∇logq(θ)∥2 2⟩ ≤C(d+∥y∥2 2), (107) ∥∇2logq(θ)∥op≤C. (108) In particular, on E(C0, CLSI), for a constant C′>0we have ⟨∥θ∥2 2⟩ ≤C′dand⟨∥∇logq(θ)∥2 2⟩ ≤C′d. Proof. (108) is immediate from the form of log q(θ), the bound ∥X∥op≤C0, and Assumption 2.2(a). For (106), write Eg,Pgfor th... | https://arxiv.org/abs/2504.15558v1 |
t≥1. (114) We have also the T2-transportation inequality (c.f. [83, Theorem 9.6.1]) W2(qt, q)2≤CLSIDKL(qt∥q), (115) and (111) for t≥1 follows follows from combining (113), (114), and (115). 39 4.3 Properties of the correlation and response In this section, on the event E(C0, CLSI), we now show approximate time-translat... | https://arxiv.org/abs/2504.15558v1 |
(126) shows (116). For (117), note that TrRθ(s+τ, s) =dX j=1⟨(∂jPτej)[θs]⟩θ0, TrR∞ θ(τ) =dX j=1⟨(∂jPτej)[θ]⟩. (127) Let d Pt(x)∈Rd×dbe the Jacobian of the vector map x7→Pt(x). By (98) of Lemma 4.2 applied with f=ej for each j= 1, . . . , d , we have dPt(x) =⟨Vt⟩x (128) where (with slight extension of the notation) we w... | https://arxiv.org/abs/2504.15558v1 |
TrR∞ η(τ) =δ σ4 TrX[dPτ(θ)]X⊤ . (136) Let{θt,Vt}t≥0and{˜θt,˜Vt}t≥0be the solutions of (97) with initial conditions ( x,I) and ( ˜x,I). Ifτ∈[0,1], we apply (128) to obtain |TrX[dPτ(x)]X⊤−TrX[dPτ(˜x)]X⊤| ≤ ∥Vτ−˜Vτ∥F x,˜x· ∥X⊤X∥F≤√ d∥X∥2 op· ∥Vτ−˜Vτ∥F x,˜x, which leads to the bound (132) up to a different constant dependi... | https://arxiv.org/abs/2504.15558v1 |
Next, (141) of Lemma 4.8 implies for some C, c > 0, lim sup n,d→∞ d−1Ps(θ0)⊤⟨θ⟩ −d−1∥⟨θ⟩∥2 2 ≤Ce−csfor all s≥0. (145) 44 Then lim sup n,d→∞d−1∥⟨θ⟩∥2 2≤˜cθ(s) +Ce−cs, lim inf n,d→∞d−1∥⟨θ⟩∥2 2≥˜cθ(s)−Ce−cs. Taking s→ ∞ on the right side of both statements shows that there exists a limit ctti θ(∞) := lim n,d→∞d−1∥⟨θ⟩∥2 2=... | https://arxiv.org/abs/2504.15558v1 |
(146), applying Theorem 4.3 and (139) and (142) shows that there exist limits ˜cη(s) := lim n,d→∞δ nσ4(XPs(θ0)−y)⊤(X⟨θ⟩ −y) = lim t→∞Cη(t, s), (154) ctti η(∞) := lim n,d→∞δ nσ4∥X⟨θ⟩ −y∥2 2= lim s→∞˜cη(s). (155) Note that n−1TrC∞ η(τ) =n−1nX i=1 xi(θ)Pτxi(θ) =δ nσ4∥X⟨θ⟩ −y∥2 2+1 nnX i=1Z∞ ιe−aτd⟨xi(θ)Eaxi(θ)⟩. Defining ... | https://arxiv.org/abs/2504.15558v1 |
holds }. Note that by assumption, this event holds a.s. for all large n, dand does not depend on θ∗,ε. For the first statement, let us consider Z(θ∗,ε) = logZ exp −1 2σ2∥Xθ∗+ε−Xθ∥2 2+dX j=1logg(θj) dθ (which coincides with log Pg(y|X) up to an additive constant) as a function of ( θ∗,ε). Then ∇θ∗Z(θ∗,ε) =−1 σ2X⊤(Xθ∗+... | https://arxiv.org/abs/2504.15558v1 |
η(∞) and YMSE =σ4 δµη,n([ι,∞))→σ4 δ(ctti η(0)− ctti η(∞)) = ymse as defined in (42). For YMSE ∗, writing E[· |X] for the expectation over ( θ∗,ε) as in Proposition 4.11, observe first that n−1E[∥X⟨θ⟩ −y∥2 2|X] =n−1E[∥X⟨θ⟩ −Xθ∗∥2 2|X]−2n−1E[ε⊤(X⟨θ⟩ −Xθ∗) +σ2|X], and Gaussian integration-by-parts gives E[ε⊤(X⟨θ⟩ −Xθ∗)|X]... | https://arxiv.org/abs/2504.15558v1 |
the squared Wasserstein-2 distance in (166) is uniformly bounded in Lpand hence uniformly integrable with respect to ⟨·⟩for all large n, d, so dominated convergence implies, almost surely, lim n,d→∞* W2 1 ddX j=1δ(θ∗ j,θt j),P(θ∗, θt)!2+ = 0. (169) Combining (165) and (169) shows that for any fixed t≥0, almost surely, ... | https://arxiv.org/abs/2504.15558v1 |
shows, for some constants C, c > 0 depending only on CLSIand for any T >0, DKL(qT∥˜q)≤C sup t∈[0,T]∆(t) +e−cTDKL(q0∥˜q) . (170) We now specialize (170) to the initialization q0= ˜q, and bound ∆( t). We have ∆(t)≤* 1 s2X⊤(Xθ∗+sz−Xθt)−1 ˜s2X⊤(Xθ∗+ ˜sz−Xθt) 2 2+dX j=1 ∂θlogg(θt j, α)−∂θlogg(θt j,˜α)2+ . LetC, C′, C′′>... | https://arxiv.org/abs/2504.15558v1 |
law Pg(θ|X,y)∝exp −s 2∥y−Xθ∥2 2+dX j=1logg(θj) andE[· |X] is the expectation over ( θ∗,ε) where εalso has variance s−1. We write also I[s],DKL[s] for the above quantities I(y,θ∗) and D KL(Pg∗(y|X)∥Pg(y|X)) in this model with noise variance s−1. Then [87, Theorem 2] and [88, Eq. (24)] show the I-MMSE relations d dsI[s... | https://arxiv.org/abs/2504.15558v1 |
=1 2E⟨(θ∗−θ)2⟩g,ω−ω ω∗E⟨(θ− ⟨θ⟩g,ω)2⟩g,ω, ∂ω∗I =ω2 2ω2∗E⟨(θ− ⟨θ⟩g,ω)2⟩g,ω. Then at the fixed points ( ω, ω∗) = (ω(s), ω∗(s)), we have ∂ωI|(ω,ω∗)=(ω(s),ω∗(s))=1 2(mse( s) + mse ∗(s))−ω(s) ω∗(s)mse(s) ∂ω∗I|(ω,ω∗)=(ω(s),ω∗(s))=ω(s)2 2ω∗(s)2mse(s). Applying mse( s) =δ/ω(s)−σ2and mse ∗(s) =δ/ω∗(s)−σ2by (43) and comparing wi... | https://arxiv.org/abs/2504.15558v1 |
DMFT system is approximately-TTI for each α∈Oby Assumption 2.11 and Theorem 2.9, hence ω(α), ω∗(α) are well-defined.) Then F(α) =f(ω(α), ω∗(α), α) +δ 2(1 + log 2 πσ2). (184) By the same calculations as (178), at the fixed points ( ω(α), ω∗(α)), we have ∂ωf(ω(α), ω∗(α), α) = 0 and ∂ω∗f(ω(α), ω∗(α), α) = 0. By Lemma 4.12... | https://arxiv.org/abs/2504.15558v1 |
every unit vector v∈RKandα∈S, VarαdX j=1v⊤∇αlogg(θj, α) ≤(CLSI/2) dX j=1 v⊤∂θ∇αlogg(θj, α)2 α by the Poincar´ e inequality for qαimplied by its LSI. Since ∂θ∇αlogg(θ, α) is bounded over α∈S, the second term of (186) is also bounded on E(C0, CLSI). Thus ∇2bF(α) is uniformly bounded over α∈Sfor all large n, d. This i... | https://arxiv.org/abs/2504.15558v1 |
the evolution of αtis given by d dtαt=Eθt∼P(θt)∇αlogg(θt, αt)− ∇R(αt) (191) where P(θt) is the law of the DMFT variable θt. The law qtof˜θtsatisfies the Fokker-Planck equation d dtqt(˜θ) =∇˜θ· qt(˜θ)∇˜θ1 2σ2∥y−X˜θ∥2 2−dX j=1logg(˜θj, αt) + log qt(˜θ) . (192) Then, using (191) and (192) to differentiate V(qt, αt) +R... | https://arxiv.org/abs/2504.15558v1 |
for all large n, d, and lim n,d→∞d−1∥˜θt∥2 2= (θt)2a.s. by Theorem 2.3(b) and (190) where θthere is the θ-component of the limiting DMFT system, this implies also E(θt)2≤C (199) for all t≥0. Furthermore, for any s≤t, applying ˜θt−˜θs=Zt sh1 σ2X⊤(y−X˜θr) + ∂θlogg(˜θr j, αr)d j=1i dr+√ 2(bt−bs) and uniform Lipschitz co... | https://arxiv.org/abs/2504.15558v1 |
all t∈[t0−cδ, t0+cδ] and some constant c >0. ThenRt0+cδ t0−cδ∥∇F(αt) + ∇R(αt)∥2 2dt≥2cδ3. The condition (204) must hold for infinitely many times t0because α∞is a limit point of{αt}t≥0, but this contradicts (201). Thus we must have α∞∈Crit. Since this holds for every limit point α∞of{αt}t≥0, and Sis compact, this impli... | https://arxiv.org/abs/2504.15558v1 |
j=1logg(θj, αt) dθ ≥ −1 dlogZ (2πσ2)−n/2exp −1 2σ2∥y−Xθ∥2 2−f(0)d−dX j=1c0 2(θj−αt)2 dθ 62 Applying ∥X∥op≤Ca.s. for all large n, d, it is readily checked by explicit evaluation of this integral over θ that V(qt, αt)≥C+c0 2(αt)2−1 2dX⊤y σ2+c0αt1⊤X⊤X σ2+c0I−1X⊤y σ2+c0αt1 a.s. for all large n, dand a constant C∈R... | https://arxiv.org/abs/2504.15558v1 |
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