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5, respectively. In these cases, the chi-square approximation of Tcommonly incurs negative errors because of cα<χ2 K–1,α orPT<α. However, the liberal FRprocedure often has positive errors and such phenomena are especially evident for the 24 cases of K=4 and 5. Similar to the previous results, the performance of the chi... | https://arxiv.org/abs/2503.17179v1 |
note d in Hettmansperger (1984, Exercise 4.5.7.), and interestingly the proof presented in Friedman (1937) was adapting from the original result by Dr. S. S. Wilks. 3.2 Location shift formulation Under the location shift assumption that at least one θidiffers from the others for Fi j(x)=F(x− µ−θi−γj),the properties of ... | https://arxiv.org/abs/2503.17179v1 |
distributions and power functions The current nonnull approximation proposed in Hettmansper ger (1984) has an attractive and sim- plified expression. However, the particular formulation wa s valid only under some special structure and condition. More importantly, the corresponding proper ties in power calculations do no... | https://arxiv.org/abs/2503.17179v1 |
of groups is K=3 and 5. For K=3,the location shifts are set as (θ1,θ2,θ3)=σ·(−1,0,1),σ·(−2/3,0,2/3),σ·(−1/2,0,1/2),andσ·(−1/3,0,1/3). The lo- cations shifts of K=5 are(θ1,θ2,θ3,θ4,θ5)=σ·(−1,−1/2,0,1/2,1),σ·(−2/3,−1/3,0,1/3, 2/3),σ·(−1/2,−1/4,0,1/4,1/2),andσ·(−1/3,−1/6,0,1/6,1/3). Note that the selected uni- form, norma... | https://arxiv.org/abs/2503.17179v1 |
–0.0088, 0.0061, –0.0175, and 0.0111, respectively. To visualize the power d ifferences of the examined procedures, they are plotted in Figures 1 and 2 for the largest location sh ifts(θ1,θ2,θ3)=σ·(−1,0,1)and (θ1,θ2,θ3,θ4,θ5)=σ·(−1,−1/2,0,1/2,1), respectively. In short, the performance of the FLB approach is reasonably... | https://arxiv.org/abs/2503.17179v1 |
the cu rrent study are available in the cited references. Funding This study was funded by Ministry of Science and Technology. 19 Appendix Exact probability evaluations of the uniform, normal, Lapl ace, and exponential distributions Assume the location shifts (θ1,θ2,...,θK)areθ1≤θ2...≤θK. 1. Uniform (−1/2,1/2):max|θi−θ... | https://arxiv.org/abs/2503.17179v1 |
from any populations: III. The analysis of variance test. Biometrika , 29:322–335. Schneider, G., Chicken, E., and Becvarik, R. (2016). NSM3 Pa ckage: Functions and Datasets to Accompany Hollander, Wolfe, and Chicken–Nonparametric St atistical Methods, Third Edition. Serfling, R. J. (1980). Approximation Theorems of Mat... | https://arxiv.org/abs/2503.17179v1 |
.39−21.94−8.60−7.92 5 0.0100 11.52 0 .01130−0.00414−0.00182 0 .00026 113 .02−41.42−18.19 2 .60 6 0.1025 7.60 0 .00488−0.01001−0.00253−0.00463 4 .76−9.76−2.47−4.52 6 0.0550 8.933 0 .00780−0.01010−0.00407−0.00393 14 .17−18.36−7.40−7.14 6 0.0109 11.733 0 .00855−0.00373−0.00162−0.00010 78 .45−34.25−14.90−0.91 7 0.1025 7.65... | https://arxiv.org/abs/2503.17179v1 |
0 .00181−0.00061−0.00061 −7.44 3 .62−1.22−1.22 0.01 −0.00174 0 .00065−0.00005−0.00016 −17.40 6 .50−0.50−1.60 29 Table 7: Estimated power and simulated power for uniform shi ft alternatives and K=3. Shifts Procedure BEstimated power Simulated power Difference 0.2887·(−1,0,1) TH 9 0.9186 0.8178 0 .1007 FMA 11 0.9146 0.87... | https://arxiv.org/abs/2503.17179v1 |
0.9003 0.8825 0 .0178 FMA 32 0.9027 0.8982 0 .0045 FMB 32 0.9091 0.8990 0 .0101 FLA 33 0.9039 0.9091 −0.0052 FLB 33 0.9080 0.9070 0 .0010 (−1/3,−1/6,0,1/6,1/3) TH 70 0.9014 0.8948 0 .0066 FMA 71 0.9026 0.8993 0 .0033 FMB 70 0.9009 0.8968 0 .0041 FLA 72 0.9032 0.9051 −0.0020 FLB 71 0.9005 0.9004 0 .0001 33 Table 11: Est... | https://arxiv.org/abs/2503.17179v1 |
FMB 6 0.9269 0.8424 0 .0845 FLA 8 0.9416 0.9478 −0.0063 FLB 7 0.9185 0.9074 0 .0111 (−2/3,−1/3,0,1/3,2/3) TH 6 0.9237 0.5158 0 .4078 FMA 11 0.9022 0.8830 0 .0192 FMB 11 0.9217 0.8809 0 .0407 FLA 12 0.9060 0.9125 −0.0065 FLB 12 0.9174 0.9115 0 .0059 (−1/2,−1/4,0,1/4,1/2) TH 10 0.9045 0.6023 0 .3022 FMA 17 0.9005 0.8849 ... | https://arxiv.org/abs/2503.17179v1 |
On Privately Estimating a Single Parameter Hilal Asi1John C. Duchi2Kunal Talwar1 1Apple2Stanford University March 24, 2025 Abstract We investigate differentially private estimators for individual parameters within larger parametric models. While generic private estimators exist, the estimators we provide re- pose on ne... | https://arxiv.org/abs/2503.17252v1 |
z n}by its associated empirical distribution Pn. We say that two samples Pn, P′ nareneighboring if they differ in only a single observation, equivalently, that their variation distance satisfies Pn−P′ n TV:= sup A|Pn(A)−P′ n(A)| ≤1 n. We develop mechanisms, meaning a randomized functions on Pn, satisfying Definition 1.... | https://arxiv.org/abs/2503.17252v1 |
of possible gradients G:=n ˙ℓθ(z)|z∈ Z, θ∈Rdo . Providing privacy relies on controlling the amount changing a single example can modify a parameter of interest; upon changing a single example, we have θ(P′ n)≈θ(Pn) +1 n(Pn¨ℓθ+ λregI)−1(g0−g1) for gradients g0, g1∈ G. Thus, to within higher order error terms, the most t... | https://arxiv.org/abs/2503.17252v1 |
for each z∈ Z, we assume that θ7→ℓθ(z) isG0-Lipschitz, has G1-Lipschitz gradient, and G2-Lipschitz Hessian, all with respect to the ℓ2-norm, meaning (respectively) that ∥˙ℓθ∥2≤G0,|||¨ℓθ|||op≤G1,|||¨ℓθ−¨ℓθ′|||op≤G2 θ−θ′ 2, 4 where we leave the observation zimplicit. For d-dimensional problems, we typically expect the sc... | https://arxiv.org/abs/2503.17252v1 |
Before returning to Examples 1 and 2, we thus collect a few properties of quasi-self-concordance and self-concordance. Lemma 2.1 (Self-concordance properties) .The following properties hold. (i) If for some c <∞, the function fsatisfies |f′′′(t)| ≤cf′′(t)for all t, then e−c|s|f′′(t)≤f′′(t+s)≤ec|s|f′′(t) for all t, s, a... | https://arxiv.org/abs/2503.17252v1 |
and P(W∈G(Pn)|Pn)≥1−γ for all Pn∈ Pn. Then the composed pair (M(Pn, W), W) is(ε+ε0, δ+δ0+γ)differentially private. See Appendix A.1.1 for a proof of this lemma. As a typical application of Lemma 2.2, we demonstrate a Gaussian mechanism. Letting Φ(·) denote the standard normal c.d.f., define the ( ε, δ)-variance σ2(ε, δ... | https://arxiv.org/abs/2503.17252v1 |
∥θ(Pn)−θ(P′ n)∥2≤(1 +o(1))2G0 λmin(Pn)+λregto release an estimate bθ with appropriate noise. We use this approach in Section 3.1; in the subsequent Section 3.2, we extend the ideas to release individual parameters. In Section 3.3, we discuss the implied dimension dependence and accuracy guarantees of the main results h... | https://arxiv.org/abs/2503.17252v1 |
our discussion and releases (with high-probability) a lower bound on λmin(Pn). Algorithm 2: Privately lower bounding λmin(Pn) for quasi-self-concordant GLMs. Require: Aφ-quasi-self-concordant loss where φlocally satisfies the linear upper bound (8), privacy parameters ε≥0 and δ∈(0,1) i. Set the recursion Ras in (10). i... | https://arxiv.org/abs/2503.17252v1 |
via a derivation and justification completely parallel to that we have done for the lower eigenvalues, so that we find the smallest Nsuch that RN(λmax(Pn))≥G1(recalling that the Lipschitz constant G1of the gradients upper bounds λmax(Pn)), we obtain that the following algorithm is ε-differentially private. Algorithm 4:... | https://arxiv.org/abs/2503.17252v1 |
of uTθ(Pn). 13 Algorithm 5: Releasing a one-dimensional statistic Require: Aφ-quasi-concordant loss where φlocally satisfies the linear upper bound (8), privacy parameters ε≥0 and δ∈(0,1) i. Letbλminandbλmaxbe the outputs of Algorithms 2 and 4, respectively ii. If εfails to satisfy the appropriate inequality (14) retur... | https://arxiv.org/abs/2503.17252v1 |
interesting behavior, which we believe is worth investigating, though leave to future work: once the sample size is large enough, then Algorithm (5) releases uTθ(Pn) with noise scaling exactly (up to the higher-order term) as the local modulus of continuity ∆(Pn, u). But until we have sufficient sample size to dominate... | https://arxiv.org/abs/2503.17252v1 |
for the exper- iments, we fix a sample size nand dimension d, then generate θ⋆∼rUni(Sd−1), varying the radius r=∥θ⋆∥2. We sample xiiid∼Uni[−1,1]d, and draw yi=⟨θ⋆, xi⟩+ziforziiid∼σ·Lap(1), i= 1, . . . , n . With this setting, we consider either estimating θ⋆ 1, the first coordinate of θ⋆, or the vector θ⋆. Each of our ... | https://arxiv.org/abs/2503.17252v1 |
is the non-private idealized version of the methods here (item 2), objective is objective perturbation (15), and naive is the naive output perturbation estimator. Objective perturbation and the methods here exhibit the best performance, with Alg. 5 exhibiting a noticeable improvement at sufficiently large sample size. ... | https://arxiv.org/abs/2503.17252v1 |
17-dimensional problem data, with covariate vectors xi∈[−1,1]d. 1e+05 2e+05 4e+05 8e+05 1.6e+06104 103 102 1e+05 2e+05 4e+05 8e+05 1.6e+06105 104 103 1e+05 2e+05 4e+05 8e+05 1.6e+06105 104 103 Non private Local Objective perturbation n=Error |bθj−θj(Pn)| Figure 6. Estimation error versus sample size nfor the parameter ... | https://arxiv.org/abs/2503.17252v1 |
certify stability of the local modulus of continuity supx∈XuT∇2Ln(θ(Pn))−1x. The plots also make clear that there remains a substantial gap between methods that can explicitly leverage the local modulus of continuity of the estimand of interest and those that cannot. 5 Parameter and eigenvalue stability guarantees The ... | https://arxiv.org/abs/2503.17252v1 |
we have θ(Pn)−θ(P′ n) 2≤1 2G2" λmin(Pn) +λreg−r (λmin(Pn) +λreg)2−8G0G2 n# . Proposition 5.2 is always sharper than Proposition 5.1 and is (for nlarge) asymptotically tight. Indeed, letting λreg= 0 for simplicity and assuming nis large, a Taylor expansion of√ a2+δ=a+δ/2a+O(δ2) gives 1 2G2" λmin(Pn)−r λ2 min(Pn)−8G0G2 n... | https://arxiv.org/abs/2503.17252v1 |
+λreg)2# +G1 n. Proof Letθ=θ(Pn) and θ′=θ(P′ n) for shorthand. We prove the lower bound; the upper bound is completely similar. Then we have the semidefinite ordering inequalities P′ n¨ℓθ′=Pn¨ℓθ′+ (P′ n−Pn)¨ℓθ′⪰Pn¨ℓθ−G2 θ−θ′ 2−G1 nI, because of the assumptions that ¨ℓθisG2-Lipschitz and that θ7→˙ℓθisG1-Lipschitz, so th... | https://arxiv.org/abs/2503.17252v1 |
is possible to find such a tso long asλ G2>12G0 nλ, that is, λ >p 12G0G2/n, which holds per (C2). Restating this, whenever Condition (C2) holds, we necessarily have ∥θ(Pn)−θ(P′ n)∥2≤12G0 nλ as Proposition 5.1 requires. 24 5.3.2 Proof of Proposition 5.2 Proposition 5.1 guarantees the existence of a solution θ(P′ n) mini... | https://arxiv.org/abs/2503.17252v1 |
perturbations. 26 Lemma 5.2. LetPn, P′ nbe neighboring samples and the conditions of Proposition 5.4 hold. Then θ=θ(Pn)andθ′=θ(P′ n)exist, and γ:=α∥θ−θ′∥2rad(X)<1. Additionally, there is a symmetric matrix Dsatisfying −γ 1−γ⪯D⪯γ 1−γfor which θ′−θ= (Pn¨ℓθ+λregI)−1(Pn−P′ n)˙ℓθ′+ (Pn¨ℓθ+λregI)−1/2D(Pn¨ℓθ+λregI)−1/2(Pn−P′ ... | https://arxiv.org/abs/2503.17252v1 |
n TV≤1 n. (20) The upper inverse modulus of continuity lenf(Pn;t):= min( k∈N|kX i=1Ui(Pn)≥ |t−f(Pn)|) (21) then defines the approximate inverse sensitivity mechanism P(M(Pn)∈A) =R Ae−εlenf(Pn;t)/2dµ(t) R Te−εlenf(Pn;t)/2dµ(t). (M.a) Asi and Duchi [3, Thm. 1] show that the mechanism is differentially private: Corollary ... | https://arxiv.org/abs/2503.17252v1 |
induction and the lemma. Inverting Rkprovides a clean approach to releasing lower bounds on λ(Pn). Define the inverse (Rk)−1(γ):= supn λ≥infC|Rk(λ)≤γo . If we have a high probability guarantee on a (random) Nthat RN(λ(Pn))>infC, then the monotonicity of Rguarantees that λ(Pn)≥(RN)−1(infC), leading to the following algo... | https://arxiv.org/abs/2503.17252v1 |
the recursion λ7→R(λ) =λ 2 1 +r 1−a λ2 −b=λ 2 2−a λ2+O(a2/λ4) −b=λ−a λ−b−Oa2 λ3 . Then for some (numerical) constant cthe recursion RN(λ⋆−a λ⋆−b−ca2 λ⋆3) =λreg, so that (RN)−1(λreg)≥λ⋆−a λ⋆−b−O(a2 λ⋆3). Applying ksteps of the recursion with the above linearization, we obtain Rk(λ⋆) =λ⋆−ka λ⋆+b −O ka2 λ⋆3 . 31... | https://arxiv.org/abs/2503.17252v1 |
such that RN(λmax(Pn)) =G1, we have RN−1(λmax(Pn))< G 1and ( RN)−1(G1)≤λmax(Pn)+O(1)G0r n· λmax(Pn) λmin(Pn)+λreg+G1 n. Iterating this k=k(ε, δ) times from λmax(Pn) yields infn λ|RN−k(ε,δ)(λ)≥G1o ≤λmax(Pn) +O(1)k(ε, δ)G0r nλmax(Pn) λmin(Pn) +λreg+k(ε, δ)G1 n. 7 Private algorithms for parameter release When we wish to r... | https://arxiv.org/abs/2503.17252v1 |
we can prove the claimed deviation guarantee on bθrelative to θ(Pn). The privacy guarantee of the theorem follows by combining Observation 2.4 with Proposition 6.1. For the accuracy guarantee, Corollary 3.2 shows that under the conditions on λmin(Pn) and λregin the statement of Theorem 1, we have bλ≥λmin(Pn)−O(ε−1log1 ... | https://arxiv.org/abs/2503.17252v1 |
H:=Pn¨ℓθ+λregI satisfies θ(P′ n)−θ(Pn) =H−1(Pn−P′ n)˙ℓθ′+H−1/2DH−1/2(Pn−P′ n)˙ℓθ′, where we use θ′=θ(P′ n) and θ=θ(Pn). Then for a any ℓ2-unit vector u, inequality (11) holds: uT(θ(P′ n)−θ(Pn)) ≤ω(u|Pn):= ∆( Pn, u) +2G0 n(λmin(Pn) +λreg)·γ(Pn) 1−γ(Pn) where we recall the directional sensitivity (3b) ∆(Pn, u) =1 nsup g0... | https://arxiv.org/abs/2503.17252v1 |
+κλ0 R(λ0)γ′ 1−γ′. Proposition 8.3. Let the conditions of Proposition 8.2 hold, but define dp=d1−2/pand let the gradient set G=n g∈Rd| ∥g∥p≤G0o . Then 1≤ω(u|Pn) ∆(Pn, u)≤1 +p dpκγ 1−γ and 1−p dps1κ−22dps2 λ0≤ω(u|P′ n) ∆(Pn, u)≤1 +p dpκs1+2s2dp λ0+p dpκλ0 R(λ0)γ′ 1−γ′. Propositions 8.2 and 8.3 rely on fairly careful con... | https://arxiv.org/abs/2503.17252v1 |
can also consider more general sets. Let the set Vbe a symmetric convex body as before. Define the maximal ℓp-inscribed- and ℓp-radii by radp(V):= sup{∥v∥p|v∈ V} and ins p(V):= sup{t|tBd p⊂ V} , so that the ratio rad p(V)/insp(V) gives a type of condition number for V. For example, V= [−1,1]dhas rad 2(V) =√ dand ins 2(... | https://arxiv.org/abs/2503.17252v1 |
which is (nearly) as sharp as possible by Lemma 8.4. We therefore have via Lemma 8.5 that ∆(P′ n, u) =2 nsup g∈GpuTH−1 1g≥ 1−s1p dpκ(H0)−s22dp λmin(H0) ∆(Pn, u). Using that ∆( Pn, u) =2G0√ dpn H−1 0u q≥2G0√ dpnλ1, where q=p p−1<2 is conjugate to p, we rearrange to obtain2G0 nλ0≤p dpκ∆(Pn, u), so that ω(u|Pn)≤∆(Pn, u)... | https://arxiv.org/abs/2503.17252v1 |
and a more direct attempt to implement Asi and Duchi’s [2] inverse sensitivity, which is instance optimal, may be more sensible. Regardless, we hope that continued interest in practicable procedures for private estimation continues. A Technical appendices A.1 Proofs of basic privacy building blocks In this appendix, we... | https://arxiv.org/abs/2503.17252v1 |
where we recall the nuclear norm ∥A∥∗=P iσi(A), Mahalanobis norm ∥v∥2 Σ= vTΣ−1v, and use the distance-like function on positive definite matrices dpd(A, B) = maxn A−1/2(B−A)A−1/2 ∗, B−1/2(A−B)B−1/2 ∗o . We also recall σ2(ε, δ), the variance (7) necessary for Gaussians to provide ( ε, δ)-privacy. Lemma A.1. Letε, δ > 0,... | https://arxiv.org/abs/2503.17252v1 |
any λ0≥0, R(λ):=λ 2−exp b 1−r 1−a λ+λ0 −c is an accelerating recursion, for which it suffices to show that R′(λ)≥1 for all λ+λ0≥a. Taking derivatives and using that∂ ∂λb(1−p 1−a/(λ+λ0)) =−ab 2(λ+λ0)2√ 1−a/(λ+λ0), we have R′(λ) = 2−exp b(1−p 1−a/(λ+λ0)) + exp b(1−p 1−a/(λ+λ0))ba 2(λ+λ0)p 1−a/(λ+λ0) = 2 + exp ... | https://arxiv.org/abs/2503.17252v1 |
and negative eigenvalues. Let U+andU−be the eigenvectors associated with the positive and negative eigenvalues of C, and let Π +=U+UT +and Π −=U−UT −be the associated projection matrices, and let Π 0be the orthogonal projector to the null space of Π ++Π−. IfAandBare invariant in these eigenspaces, in that AΠ+= Π +AΠ+an... | https://arxiv.org/abs/2503.17252v1 |
0v≤κ(H−1 0,V)·sup v∈VuTH−1 0v≤rad2(V) ins2(V)κ(H0)·sup v∈VuTH−1 0v. Noting that the set {v/ins2(V)|v∈ V} ⊃ Bd 2, we have uTH−1 0u∥v∥2≤ ∥v∥2·sup v∈VuTH−1 0v/ins2(V)≤rad2(V) ins2(V)sup v∈VuTH−1 0v. Combining the preceding inequalities then gives that uTE1v≤s1rad2(V) ins2(V)κ(H0) + 1 2 sup v∈VuTH−1 0v. We now control th... | https://arxiv.org/abs/2503.17252v1 |
without private covariance estimation. arXiv:2106.13329 [cs.LG] , 2021. [11] G. Brown, S. B. Hopkins, and A. Smith. Fast, sample-efficient, affine-invariant private mean and covariance estimation for subgaussian distributions. In Proceedings of the Thirty Sixth Annual Conference on Computational Learning Theory , 2023.... | https://arxiv.org/abs/2503.17252v1 |
GLIVENKO–CANTELLI FOR f-DIVERGENCE HAOMING WANG AND LEK-HENG LIM Abstract. We extend the celebrated Glivenko–Cantelli theorem, sometimes called the fundamen- tal theorem of statistics, from its standard setting of total variation distance to all f-divergences. A key obstacle in this endeavor is to define f-divergence o... | https://arxiv.org/abs/2503.17355v2 |
AND L.-H. LIM marginal likelihood that is controlled by the parameter α. The goal of our article is to generalize the Glivenko–Cantelli theorem to any f-divergence, restricted to a Glivenko–Cantelli class defined with respect to this f-divergence. At this point, one may wonder about Wasserstein metrics, another class o... | https://arxiv.org/abs/2503.17355v2 |
for all nandCis not a Glivenko–Cantelli class. So the existence of (nontrivial) Glivenko–Cantelli classes is not obvious. The Glivenko–Cantelli theorem shows that they indeed exists, with the class of rays the original [22, 8] and best known example. Definition 1.3 (Rays) .The class of open rays and the class of closed... | https://arxiv.org/abs/2503.17355v2 |
fthat (a) reduces to the Kolmogorov–Smirnov distance in (3) when f(t) =|t−1|/2; and (b) reduces to the standard f-divergence in (5) when Ris replaced by B? 4 H. WANG AND L.-H. LIM The answer to Question 1 is yes, provided by what we will call an f-divergence over Rand denoted DR f(µ∥ν). We will establish the existence ... | https://arxiv.org/abs/2503.17355v2 |
The metric projection onto Mis the operator projM:L2(ν)→L2(ν) that takes g∈L2(ν) to the closest point g∗∈M, i.e., ∥g−g∗∥= minh∈M∥h−g∥. The existence and uniqueness of g∗is guaranteed by the conditions on Mand so projMis well-defined. It is also well-known that projMis continuous [2, p. 52]. 2.2.Sequences. We will remin... | https://arxiv.org/abs/2503.17355v2 |
Conversely, let g∈G(R). Suppose we have x < y with g(x)< g(y). Then there is some awith ( −∞, a)⊆ {g > g (x)} ⊆(−∞, a]. Since g(y)> g(x), it follows that y∈ {g > g (x)}, and so x < y < a . Consequently, x∈ {g > g (x)}, i.e., g(x)> g(x), a contradiction. Hence gmust be nonincreasing. G(R) is a closed convex cone because... | https://arxiv.org/abs/2503.17355v2 |
of ordered simple function hnsuch that hn→ginL2-norm. By passing through a subsequence if necessary, Lemma 2.1 ensures hn→galmost surely as well. □ We next see that the metric projection of an ordered simple function onto G(R) remains ordered simple. Proposition 3.5 (Projection) .Leth=Pn i=1αi 1Ai∈L2 +(ν)be an ordered ... | https://arxiv.org/abs/2503.17355v2 |
measurable space ( R,B) with µ≪ν, there exists a C-measurable function ρCsuch thatµ(A) =R AρCdνfor any A∈C. What happens if Cis not a σ-subalgebra? For us, the most important case to consider is when C=R, which is not a σ-subalgebra. We will show in Corollary 3.11 that Rsatisfies a kind of Radon–Nikodym property: There... | https://arxiv.org/abs/2503.17355v2 |
sup n→∞ν(E∩Ec n) + lim sup n→∞ν(Fn) ≤ν lim sup n→∞E∩Ec n +ν lim sup n→∞Fn = 0, and by H¨ older, ∥h 1E−hn 1En∥1=∥h 1E−h 1En+h 1En−hn 1En∥1 ≤ ∥h·( 1E− 1En)∥1+∥(h−hn)· 1En∥1 ≤ ∥h∥∥ 1E− 1En∥+∥h−hn∥∥ 1En∥ →0. The same argument also yields (projG(R)hn)· 1En→(projG(R)g)· 1E inL1-norm. By Lemma 2.2, we getZ g 1Edν= lim n→∞... | https://arxiv.org/abs/2503.17355v2 |
(a) when fis chosen to be f(t) =|t−1|/2, we recover the Kolmogorov–Smirnov distance in (3); and Definition 3.12 shows that by replacing RbyB, we recover the standard f-divergence in (5). A reminder of our convention in Section 2.3: we assume 12 H. WANG AND L.-H. LIM thatµ≪νwhenever we write d µ/dν. This is not an addit... | https://arxiv.org/abs/2503.17355v2 |
see that just about every property that we know holds for f-divergences also holds for f-divergences over R. Henceforth we will denote the convex cone of functions convex on [0,∞) and both vanishing and strictly convex at 1 by (16) F:={f: [0,∞)→R:fis convex, fis strictly convex at 1, and f(1) = 0 }. The strict convexit... | https://arxiv.org/abs/2503.17355v2 |
of distributions on a three-element set. Evidently, the projection operator projG(R)plays a key role in our definition of f-divergence overR. The next two results show how it interacts with nondecreasing functions and with convex functions. GLIVENKO–CANTELLI FOR f-DIVERGENCE 15 Lemma 4.2. Letf:R→Rbe nondecreasing. If g... | https://arxiv.org/abs/2503.17355v2 |
dλ=Z f(ρ◦T−1) dτ=Z f projG(R)dη dτ dτ= DR f(η∥τ) as required. □ Collectively, Propositions 4.4 and 4.5 lead to the following conclusion: An inequality relation between two divergences hold if and only if that same relation hold for their counterparts over rays. 1In this article, we use the decumulative distribution f... | https://arxiv.org/abs/2503.17355v2 |
DKL(µ∥ν)≤log 1/νmin−1 1−2νmin 1−(1−D2 H(µ∥ν))2 . (e)Kullback–Leibler and χ2: DKL(µ∥ν)≤log 1 + D χ2(µ∥ν) ≤Dχ2(µ∥ν). (f)Le Cam and Hellinger: 1 2D2 H(µ∥ν)≤DLC(µ∥ν)≤D2 H(µ∥ν). (g)Le Cam and Jensen–Shannon: DLC(µ∥ν)≤DJS(µ∥ν)≤2 log 2 ·DLC(µ∥ν). Proof. These are well-known inequalities between the respective f-divergen... | https://arxiv.org/abs/2503.17355v2 |
statement) that lim n→∞DR TV(νn∥ν) = lim n→∞sup A∈R νn(A)−ν(A) = 0 almost surely; taken together with the bounded convergence theorem, 0 = lim n→∞sup A∈R νn(A)−ν(A) = lim n→∞Z1 2 projG(R)dνn dν−1 dν =Z1 2 lim n→∞projG(R)dνn dν−1 dν≥0. Hence the last integrand must converge to zero, or, equivalently, □ (19) lim n→∞p... | https://arxiv.org/abs/2503.17355v2 |
rays Rto more general classes C⊆B. Surprisingly, we found that the original Vapnik–Chervonenkis theorem [55] provides a key. Replace Rby a class C⊆Bin the definition of G(R) in (6). As Section 3 shows, if G(C) satisfies the conclusions of Proposition 3.1 and Theorem 3.15 with Cin place of R, then Definition 3.12 with C... | https://arxiv.org/abs/2503.17355v2 |
the fact thatRis the class of rays — every result therein holds verbatim with Cin place of Ras long as DC f is well-defined, i.e., as long as Cis a pre-Glivenko–Cantelli class. The original Vapnik–Chervonenkis theorem [55], as applied in Propositions 5.1 and 5.3, shows that a pre-Glivenko–Cantelli class C with finite v... | https://arxiv.org/abs/2503.17355v2 |
For the class of rays R, it is easy to see that νR, the restriction of νtoR, is a capacity. So we could simply define f-divergence over Ras DR f(µ∥ν):= D f(µR∥νR) =CZ fdµR dνR dνR. Two immediate problems are that the Radon–Nikodym derivative d µR/dνRcannot be easily defined and the function f(dµR/dνR) may not be R-me... | https://arxiv.org/abs/2503.17355v2 |
sum of observations. Ann. Math. Statistics , 23:493–507, 1952. [11] G. Choquet. Theory of capacities. Ann. Inst. Fourier (Grenoble) , 5:131–295, 1953/54. [12] D. L. Cohn. Measure theory . Birkh¨ auser Advanced Texts: Basel Textbooks. Birkh¨ auser/Springer, New York, second edition, 2013. [13] I. Csiszar. Eine informati... | https://arxiv.org/abs/2503.17355v2 |
Auto-encoding variational bayes. In 2nd International Conference on Learning Representations (ICLR) , 2014. [32] L. Le Cam. Asymptotic methods in statistical decision theory . Springer Series in Statistics. Springer, New York, 1986. [33] Y. Li and R. E. Turner. R´ enyi divergence variational inference. In D. Lee, M. Su... | https://arxiv.org/abs/2503.17355v2 |
I. Vajda. Note on discrimination information and variation. IEEE Trans. Inform. Theory , IT-16:771–773, 1970. [54] V. N. Vapnik. Statistical learning theory . Adaptive and Learning Systems for Signal Processing, Communications, and Control. John Wiley & Sons, Inc., New York, 1998. [55] V. N. Vapnik and A. Y. Chervonenk... | https://arxiv.org/abs/2503.17355v2 |
Efficient Knowledge Distillation via Curriculum Extraction Shivam Gupta University of Texas at Austin∗ shivamgupta@utexas.eduSushrut Karmalkar Microsoft Research, Cambridge† skarmalkar@microsoft.com March 25, 2025 Abstract Knowledge distillation is a technique used to train a small student network using the output gene... | https://arxiv.org/abs/2503.17494v1 |
during the training of the teacher network act as an implicit curriculum for the training of the student network, with earlier checkpoints emphasizing simpler patterns (e.g., local syntax in the case of language models), and later checkpoints capturing complex abstractions (e.g., long-range semantics). Please see Secti... | https://arxiv.org/abs/2503.17494v1 |
stages, only a subset of student and teacher layers are active, reducing memory and FLOPs per iteration. 1.1 Our Results Sparse Parity Learning. We show that our curriculum extraction scheme is significantly more efficient than one-shot distillation for the task of learning sparse parities using a two-layer MLP, and pr... | https://arxiv.org/abs/2503.17494v1 |
paradigm has been widely adopted in modern language models for inference cost reduction. Examples of models trained via distillation include ChatGPT O1-mini [Ope24], Gemini Flash [Tea24], and the Phi series of models [ AAB+24]. Despite the success of distillation, there have been a number of works [ MFL+20,CH19,HRM+23,... | https://arxiv.org/abs/2503.17494v1 |
weights. 2.Layer-Wise Training: For each i∈[ℓ, L−1], such that ti>0: (a) Define a random projection matrix Pi:Rmi→Rnifor layer i. (b)Train Sifortiiterations to reduce the MSE loss between Si(x)andPi(Ti(x)):Li= 1 TPT t=1∥Si(x(t))−Pi(Ti(x(t)))∥2 2where Tis the number of training samples. 3.Train the entire student networ... | https://arxiv.org/abs/2503.17494v1 |
the first layer of the student (i.e. f(1) s(x) :=σ(Wsx+bs)) is trained using the a similarly regularized version of the following distillation loss: ℓDL(x, f(1) s, Af(1) t) =−f(1) s(x)·(Af(1) t(x)), i.e. we use the loss ℓDL(x, f(1) s, Af(1) t)−λ∥Ws∥2for a carefully chosen value of λ. The second layer of the student ( a... | https://arxiv.org/abs/2503.17494v1 |
shows that the k-sparse parity can be accurately approximated by training only the top layer of the student, provided that the hidden layer satisfies some mild conditions, such as being large enough (at least ˜Ω(2kk)) and satisfying a clear gap between the in-support and out-of-support coordinates of the weights at the... | https://arxiv.org/abs/2503.17494v1 |
Lemma D.3. Lemma 4.4 (Correlation Gap (Informal)) .Letd≥Ω(k4)andmt≥Ω(k2). Then, with proba- bility 99%over the randomness of initialization, an independently drawn subset of at least ˜Ω(2kk) coordinates of the projected teacher network satisfy |Ex[(Aft)i(x)xj]|>Ω((k2√mt)−1)for all jin the support of the unknown sparse ... | https://arxiv.org/abs/2503.17494v1 |
is done using the Cross-Entropy loss. We measure performance of our model by looking at the accuracy. 4.3.1 Discussion Multi-Layer Perceptron: In Figure 3 (a), we compare the performance of MLPs trained using one-shot distillation, layer-wise curriculum extraction, and the progressive distillation approach from [PLM+24... | https://arxiv.org/abs/2503.17494v1 |
The model is then trained via a cross-entropy objective to predict the original tokens at those masked positions. For sequence-to-sequence modelling our teacher and student networks map from sequences to real vectors ft, fs:vh→Rh×C. The teacher’s output distribution at position iis given by p(i) T(x;τ) = softmax [ft(x... | https://arxiv.org/abs/2503.17494v1 |
the top linear layer for iterations 501 to 1500, and finally trains the entire network for iterations 1501 through 4000. The three-stage curriculum skips two encoder blocks for iterations 1 through 400, skips one encoder block for iterations 401 through 1200, and skips the final projection layer for iterations 1201 thr... | https://arxiv.org/abs/2503.17494v1 |
Wang, Liyue Zhang, Lei Xu, Leyi Xia, Mingchuan Zhang, Minghua Zhang, Minghui Tang, Meng Li, Miaojun Wang, Mingming Li, Ning Tian, Panpan Huang, Peng Zhang, Qiancheng Wang, Qinyu Chen, Qiushi Du, Ruiqi Ge, Ruisong Zhang, Ruizhe Pan, Runji Wang, R. J. Chen, R. L. Jin, Ruyi Chen, Shanghao Lu, Shangyan Zhou, Shanhuang Chen... | https://arxiv.org/abs/2503.17494v1 |
Farajtabar, Ang Li, Nir Levine, Akihiro Matsukawa, and Hassan Ghasemzadeh. Improved knowledge distillation via teacher assistant. InProceedings of the AAAI conference on artificial intelligence , volume 34, pages 5191–5198, 2020. [Ope24] OpenAI. Openai o1-mini, 2024. 12 [PLM+24]Abhishek Panigrahi, Bingbin Liu, Sadhika ... | https://arxiv.org/abs/2503.17494v1 |
sums of scaled Rademacher random variables. Theorem B.3 (Berry–Esseen) .LetX1, X2, . . . , X nbe independent random variables satisfying E[Xi] = 0,E[X2 i] =σ2 i>0,andE[|Xi|3]<∞,fori= 1, . . . , n. Define Sn=nX i=1Xi, σ2=nX i=1σ2 i,and ρn=nX i=1E[|Xi|3]. Then for all x∈R, we have PrSn σ≤x −Φ(x) ≤Cρn σ3, where Φ(x)is t... | https://arxiv.org/abs/2503.17494v1 |
2. Thus, setting t=np 2in Hoeffding’s inequality, we obtain Pr"nX i=1Xi≤np 2# ≤exp −2 np 22 n! = exp −np2 2! . 17 To guarantee that this probability is at most δ, we require exp −np2 2! ≤δ. Taking logarithms on both sides gives −np2 2≤ln(δ), which is equivalent to n≥2 ln(1 /δ) p2.Thus, if n≥2 ln(1 /δ) p2,then with pr... | https://arxiv.org/abs/2503.17494v1 |
error as a function of the batch size. Claim C.1 (Gradient Concentration [ PLM+24]).Letfbe a two-layer network initialized using the symmetric initialization in Definition 4.2 with mbeing its hidden dimension. Fix δ, τg>0. For alli∈[m], j∈[d], for a randomly sampled batch of size B1,{(xk, yk)}B1 k=1, with probability a... | https://arxiv.org/abs/2503.17494v1 |
:=−f(x)·g(x). Here, f=f(1) s∈Rmsis the first layer of the student network, and g=Af(1) t(x)∈Rmsis the first layer of the teacher network projected to the student’s hidden layer dimension. 20 Algorithm 2 2-stage training for student Require: Number of iterations T2, Learning rates η1, η2, batch sizes B1, B2, weight deca... | https://arxiv.org/abs/2503.17494v1 |
the randomness of bℓ, |Ex[ϕbℓ(wℓ·x)xj]| ≥1 4k−O(τgd|ζk−1|−1). Applying Hoeffding’s inequality to this event, we conclude that if mt≥Ω(log(1/δ)), then with probability 1−δ, at least mt/8neurons satisfy |Ex[ϕbℓ(wℓ·x)xj]| ≥1 16k−O(τgd|ζk−1|−1). Since|wij| ≤1/(2k)±(τg/|ξk−1|)forallj∈[k]andi∈[mt], we can show an upper bound... | https://arxiv.org/abs/2503.17494v1 |
the at-least mt/16indices with Ex[ϕbi(wi·x)xj]≥Ω(1/k), we have each contributing at least Ω (1/mt)2·(1/k2) . Hence, σ2≥mt/16 m2 t·Ω 1 k2 = Ω 1 mtk2 .This yields a standard deviation σ≥ Ω 1√mtk .Now, applying the anti-concentration inequality for Rademacher sums (see Theorem B.4), for any fixed j∈[k]andℓ∈[ms]we ... | https://arxiv.org/abs/2503.17494v1 |
below. Lemma D.5 (Correlation within-support variables) .Under the event that the conditions in Lemma C.2 are satisfied by each neuron, which occurs with probability at least 99%w.r.t. the randomness of initialization as long as mt≥Ω(k2)andd≥Ω(k4), the output of the teachernetwork after the first phase satisfies the fo... | https://arxiv.org/abs/2503.17494v1 |
a randomly sampled batch of samples B={(xt, yt)}|B| t=1with size |B| ≥Ω((kd)2log(mtd/δ)).Then, with probability at least 1−δ, for every index i∈[ms]and every coordinate j∈[d], we have the empirical gradient satisfies bEx∼Bh ∇wijℓDL x, f(1) s(x), Af(1) t(x)i = Ω 1√mtk2 ifj∈[k], ˜O 1√mtd ifj /∈[k]. Proof.Recall... | https://arxiv.org/abs/2503.17494v1 |
may then set η=√mtandT= Θ( k2klog(k/ϵ))andms=O(T)to ensure that with probability 99%, the gap between at least Trandomly selected student weights out of a total of msstudent weights matches the gap in [PLM+24] and is bounded below by Ω(1/k2). 28 This sets up the Tweights exhibiting a gap in the bottom layer to satisfy ... | https://arxiv.org/abs/2503.17494v1 |
arXiv:2503.17521v1 [math.ST] 21 Mar 2025The Entropy and Crossentropy of Generalized Mallows Models Marina Meil˘ a Department of Statistics University of Washington mmp@stat.washington.edu March 25, 2025 Abstract The Generalized Mallows Model (GMM) is a well known family of models for ranking data. A GMM is a distributi... | https://arxiv.org/abs/2503.17521v1 |
in a survey. A ranker (typically a person, natural process, or autom ated system) produces a ranking of the elements in the set, according to their pr eference (e.g. the best candidate first, the statement they most agreewith firs t). The set of all permutations ofEis denoted SE, or simply Sn, by identifying eiwithi= 1,.... | https://arxiv.org/abs/2503.17521v1 |
the rows and columns are sorted by π, or equivalently d(π,σ) =d(σ,π) =d(π◦σ−1,id) =/summationdisplay e′∈E/summationdisplay e≺σe′Qe′e. (3) Example 1 Consider the permutations π= [c,a,d,b,f,e ], andid = [a,b,c,d,e,f ]. We can calculate the number of inversions of πeasily from the inversion matrix Q, shown below. Q(π;σ) =... | https://arxiv.org/abs/2503.17521v1 |
[Meil˘ a and Bao, 2010] that the sr(π|σ) terms can be calculated, like the Kendall’s tau distance, from the inversion matr ixQ(π,σ) by sr(π|σ) =/summationdisplay e′≺σeQee′(π|σ) wheree=π(r). (9) Hencesrare row sums, in which only entries in the lower triangle of Qare counted. Example 3 Letπ= [c,a,d,b,f,e ]as before, and... | https://arxiv.org/abs/2503.17521v1 |
the close relationship between the GMMmodel and the inversionmatrix Q. At first sight, computing the expectation in (13) for theGMMrequires summation over a sample space of size n!. We bypass this challenge by showing that the crossentropydepends on matrix - ve ctor products with submatrices of Qwhich can be averaged tr... | https://arxiv.org/abs/2503.17521v1 |
=/summationtextn j=1(Qup A,A)ij.MMP: recheck thisThe vector of these values is obtained by Qup A,A1k. Under the sampling 9 distribution, the item at rank ihas probability αi−1 r/Zn−r+1(αr) to be sampled. This probability is the same for any set A, since it depends only on the sampled rank,not on the item at that rank. ... | https://arxiv.org/abs/2503.17521v1 |
the previous rank are given, we have to enumerate all the positions ( i0,j0) and their respective probabilities, that can lead to the event Eab,ij. If ranka=i0, and rankb=j0before the (r−1)-th sample, then after sampling an item, these ranks can only sta y the same decrease. Namely, i←i0,j←j0if the deleted element is a... | https://arxiv.org/abs/2503.17521v1 |
the entropy and crossentropy. Acknowledgements The author thanks the organizers of the Interactions of Statist ics and Geometry (ISAG) II Workshop, and especially to Jun Zhang, for the interest ing presen- tations that are the impetus for this work. The author acknowledg es partial support from the NSF MMS 2019901 awar... | https://arxiv.org/abs/2503.17521v1 |
arXiv:2503.17532v1 [math.ST] 21 Mar 2025Modeling of stochastic processes in Lp(T)using orthogonal polynomials Universal Journal of Applied Mathematics 2(3): 141-147, 2014 DOI: 10.13189/ujam.2014.020304 Oleksandr Mokliachuk1,∗ 1National Technical University of Ukraine “Igor Sicorsky Ky iv Polytechnic Institute”, Kyiv, U... | https://arxiv.org/abs/2503.17532v1 |
series can be con- sidered: GF(x,ω) =∞/summationdisplay n=0Pn(x) n!ωn. Under some minimal conditions, this series has pos- itive convergence radius. In such a case, the function GF(x,ω) is called generating function of the polynomial set{Pn(x)}[3]. If for some functional basis {gk(λ)}a generating func- tion exists, nam... | https://arxiv.org/abs/2503.17532v1 |
we obtain ak(t)≤/parenleftbigg1 (k+1)(k+2)/integraldisplay∞ −∞Z2 f(t,λ)dλ/parenrightbigg1 2 , 3 where Zf(t,λ) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2f(t,λ) ∂λ2−λ∂f(t,λ) ∂λ+λ2−2 4f(t,λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Therefore, cN=/integraldisplayT 0/parenleftBiggN/summation... | https://arxiv.org/abs/2503.17532v1 |
dλ/parenrightBigg1/2 . Lets calculate the second integral of the last expres- sion separetely. /integraldisplay1 −1/parenleftbigg1−ωλ 2−ωλ+ω2/parenrightbigg2 dλ= = 21 ω(4+3ω2+ω4)/parenleftbig ω(5+5ω2+2ω2)+ + (4+7ω2+4ω4+ω6)ln{(ω2−ω+2)/(ω2+ +ω+2)}) :=DT(ω). Then the estimator of τϕ(X(t)) will take the form: τϕ(X(t))≤/rad... | https://arxiv.org/abs/2503.17532v1 |
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