text string | source string |
|---|---|
ˆUcand ˇU∗:= (U∗):,2:r∈Rn×(r−1) be the 2-th tor-th column of U∗. Define ˇR∈R(r−1)×(r−1)as the rotation matrix aligns ˇUcand ˇU∗, i.e., ˇR:= arg minL∈O(r−1)×(r−1)/vextenddouble/vextenddouble/vextenddoubleˇUcL−ˇU∗/vextenddouble/vextenddouble/vextenddouble F. Moreover, without loss of generality, we choose the direction o... | https://arxiv.org/abs/2502.06671v1 |
the last inequality follows from (134). Note that ∥(ˆQ−Q)w∗ i∥2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubler/summationdisplay ℓ=1w∗ i(ℓ)/bracketleftbiggˇRˆvℓ−v∗ ℓ 0/bracketrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble 2 ≤r/summationdisplay ℓ=1w∗ i(ℓ)∥ˇRˆvℓ−v∗... | https://arxiv.org/abs/2502.06671v1 |
J Technical lemmas Lemma J.1. For matrix A∈Rn1×m,B∈Rn2×m, we have max{∥A∥,∥B∥}≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg A B/bracketrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤∥A∥+∥B∥. Proof of Lemma J.1. GivenA∈Rn1×m,B∈Rn2×m. We have /vextenddouble/vextenddouble/vex... | https://arxiv.org/abs/2502.06671v1 |
Neumann eigenmaps for landmark embedding Shashank Sule Department of Mathematics University of Maryland, College Park College Park, United States of America ssule25@umd.eduWojciech Czaja Department of Mathematics University of Maryland, College Park College Park, United States of America wojtek@math.umd.edu Abstract —W... | https://arxiv.org/abs/2502.06689v1 |
thus additionally robust to perturbations due to data subsampling and emphasizes cluster structure due to the added within-cluster diffusion probability via reflection. A. Related work The Neumann spectrum of subgraphs and the Neumann Laplacian were established in the seminal works by Chung, Graham, and Yau [3]–[5] and... | https://arxiv.org/abs/2502.06689v1 |
the following properties: 1)FixxPVS. Then LV˚fpxq“λ1 Ndpxqfpxq. (II.4) 2)FixxPδS. Then ÿ tx,yuPBSwpx, yqpfpxq´fpyqq“0. (II.5) Remark II.1.The condition (II.4) shows that fN 1satisfies an eigenvalue equation at xPVSfor the ambient graph G, whilenot necessarily being either a Laplacian eigenvector of the graph SorG. More... | https://arxiv.org/abs/2502.06689v1 |
by replacing LD Swith any user-chosen operator such as the Schr ¨odinger or transport operator on the subgraph S[7], [23], [24]. Remark III.2.The Neumann condition gives a natural way to extend functions from VStoVSYδS. In particular, if Ng“λgthen defining f“T1{2gwe have LN Sg“λTSg. To make fa Neumann eigenvector, we s... | https://arxiv.org/abs/2502.06689v1 |
data. In particular, we demonstrate that they may learn the underlying collective vari- able more accurately than Fokker-Planck (FP) eigenfunctions using diffusion maps. We empirically illustrate this in the case of the butane ( C4H10) molecule, widely used as a toy model for configurational changes in small molecules.... | https://arxiv.org/abs/2502.06689v1 |
[2] Chung, F.R. and Yau, S.T., 1997. Eigenvalue inequalities for graphs and convex subgraphs. Communications in Analysis and Geometry, 5(4), pp.575-623. [3] Chung, F.R.K., Graham, R.L. and Yau, S.T., 1996. On sampling with Markov chains. Random Structures & Algorithms, 9(1-2), pp.55-77. [4] Chung, F.R. and Yau, S.T., 1... | https://arxiv.org/abs/2502.06689v1 |
[23] Cahill, N.D., Czaja, W. and Messinger, D.W., 2014, June. Schroedinger eigenmaps with nondiagonal potentials for spatial-spectral clustering of hyperspectral imagery. In Algorithms and technologies for multispectral, hyperspectral, and ultraspectral imagery XX (V ol. 9088, pp. 27-39). SPIE. [24] Czaja, W., Dong, D.... | https://arxiv.org/abs/2502.06689v1 |
Leli `evre, T., Pigeon, T., Stoltz, G. and Zhang, W., 2024. Analyzing multimodal probability measures with autoencoders. The Journal of Physical Chemistry B, 128(11), pp.2607-2631. [42] Rogal, J., Schneider, E. and Tuckerman, M.E., 2019. Neural-network- based path collective variables for enhanced sampling of phase tra... | https://arxiv.org/abs/2502.06689v1 |
follows because Nadmits T1{21as a zero-eigenvector: R1“T´1{2 SpI´NqT1{2 S1“I1´T´1{2 SNT1{2 S1“1. To see the non-negativity of the entries, we expand Nin terms of the Dirichlet and Boundary operators: R“T´1{2 SpI´NqT1{2 S “I´T´1{2 SNT1{2 S “I´T´1 SpLGrVS, VSs´BJpTδ Sq´1BT´1{2 Sq “I´T´1 SLGrVS, VSs`T´1 SBJpTδ Sq´1B “I´T´... | https://arxiv.org/abs/2502.06689v1 |
arXiv:2502.06719v1 [stat.ML] 10 Feb 2025Proceedings of Machine Learning Research vol ???: 1–49, 2025 Under Review for COLT 2025 Gaussian Approximation and Multiplier Bootstrap for Stoch astic Gradient Descent Marina Sheshukova * MSHESHUKOVA @HSE.RU HSE University Sergey Samsonov∗SVSAMSONOV @HSE.RU HSE University Denis ... | https://arxiv.org/abs/2502.06719v1 |
¯θn=1 nn−1/summationdisplay i=0θi, n∈N. (3) It has been established (see ( Polyak and Juditsky ,1992 , Theorem 3)) that under appropriate conditions on the objective function f, the noisy gradient estimates F, and the step sizes αk, the sequence of averaged iterates {¯θn}n∈Nsatisfies the central limit theorem: √n(¯θn−θ⋆... | https://arxiv.org/abs/2502.06719v1 |
of matrix A. Given a function f:Rd→R, we write ∇f(θ)and∇2f(θ)for its gradient and Hessian at a point θ. Additionally, we use the standard abbreviations ”i.i.d.” for ”independent and identically d istributed” and ”w.r.t.” for ”with respect to”. Literature review Asymptotic properties of the SGD algorithm, including the ... | https://arxiv.org/abs/2502.06719v1 |
nonconvex problems. However, these papers typically provide recovery rates Σ∞, but only show asymptotic validity of the proposed confidence intervals. A notable exception is the recent pape rWu et al. (2024 ), where the temporal difference (TD) learning algorithm was studied. The author s of Wu et al. (2024 ) provided p... | https://arxiv.org/abs/2502.06719v1 |
a neigh- borhood of θ∗. Similar assumptions have been previously considered in Shao and Zhang (2022 ) andAnastasiou et al. (2019 ), as well as in other works on first-order optimization metho ds, see, e.g., Li et al. (2022a ). In contrast, several studies on the non-asymptotic analy sis of SGD im- pose even stronger smo... | https://arxiv.org/abs/2502.06719v1 |
(2024 ) and Srikant (2024 ), although in a slightly different problem of temporal diff erence learning Sutton and Barto (2018 ), an algorithm which does not correspond to gradient dynami cs. For this reason, we adopt the decomposition of WandDas given in ( 12). The result of ( Shao and Zhang , 2022 , Theorem 3.4) estab... | https://arxiv.org/abs/2502.06719v1 |
an image of a convex set under non- degenerate linear mapping is a convex set, we have dC(√nΣ−1/2 n(¯θn−θ⋆),Y) =dC(√n(¯θn−θ⋆),Σ1/2 nY). 2.3. Convergence rate in CLT for averaged SGD iterates (7) Next, we demonstrate how the result of Theorem 1can be utilized to quantify the convergence rate in (7). The key step in esta... | https://arxiv.org/abs/2502.06719v1 |
samples from the conditional distribution of ¯θb ngiven the data Ξn−1. The core principle behind the bootstrap procedure ( 17) is that the ”bootstrap world” probabilities Pb/parenleftbig√n(¯θb n−¯θn)∈B/parenrightbig are close toP/parenleftbig√n(¯θn−θ⋆)∈B/parenrightbig forB∈C(Rd). Remark 2. While an analytical expressio... | https://arxiv.org/abs/2502.06719v1 |
Ξn−1. In fact, the approach of Shao and Zhang (2022 ) would require one to control the second moments of DbandDb−{Db}(i)with respect to a bootstrap measure Pb, 10 GAR AND BOOTSTRAP FOR SGD on the high-probability event with respect to a measure P. At the same time, we loose a martingale structure of the summands in Db,... | https://arxiv.org/abs/2502.06719v1 |
al. (2018 ) also considered positive bootstrap weights Wi. We have to impose boundedness of Widue to our high-probability bound on Lemma 15. A particular example of a distribution satisfying A 7is provided in Appendix E.1. We also consider the following bound for step sizes αkand sample size n: A8 Letαk=c0{k0+k}−γ, whe... | https://arxiv.org/abs/2502.06719v1 |
, Theorem 3.4 and 3.5)). The authors in Chen et al. (2020 ) constructed a plug-in estimator ˆΣnofΣ∞and showed guaran- tees of the form E[/⌊a∇d⌊lˆΣn−Σ∞/⌊a∇d⌊l]/lessorsimilarCn−γ/2,γ∈(1/2,1)under weaker assumptions than those considered in the current section. The result in this partic ular form is not sufficient to prove... | https://arxiv.org/abs/2502.06719v1 |
. Xi Chen, Zehua Lai, He Li, and Yichen Zhang. Online statistic al inference for contextual bandits via stochastic gradient descent. arXiv preprint arXiv:2212.14883 , 2022. Aaron Defazio, Francis Bach, and Simon Lacoste-Julien. SAG A: A fast incremental gradient method with support for non-strongly convex composite obj... | https://arxiv.org/abs/2502.06719v1 |
inequalities , volume 72. Springer Science & Business Media, 2012. Boris T Polyak and Anatoli B Juditsky. Acceleration of stoch astic approximation by averaging. SIAM journal on control and optimization , 30(4):838–855, 1992. Ning Qian. On the momentum term in gradient descent learning algorithms. Neural networks , 12 ... | https://arxiv.org/abs/2502.06719v1 |
SHESHUKOVA SAMSONOV BELOMESTNY MOULINES SHAO ZHANG NAUMOV Appendix A. Technical bounds Lemma 2. Assume A 4. Then the following bounds holds: (a) for any p≥2 k/summationdisplay i=1αp i≤cp 0 pγ−1, (b) for any m∈ {0,...,k} k/summationdisplay i=m+1αi≥c0 2(1−γ)((k+k0)1−γ−(m+k0)1−γ), Proof To proof (a), note that k/summation... | https://arxiv.org/abs/2502.06719v1 |
exp/braceleftbigg2c2 0L2 2 2γ−1+3µc0 4(1−γ)k1−γ 0/bracerightbigg . (29) Now we estimate A2,k. Letk1be the largest index ksuch that 4α2 kL2 2≥αkµ. Then, for i > k1, we have that 1−(3/2)αiµ+2α2 iL2 2≤1−αiµ . Thus, using the definition of A2,kin (28), we obtain that A2,k≤k/summationdisplay i=1α2 ik/productdisplay j=i+1(1−α... | https://arxiv.org/abs/2502.06719v1 |
for l≥1orj≥2(that is,2l+j≥2), we use Cauchy-Schwartz inequality |/a\}⌊∇a⌋ketle{tθk−1−θ⋆,∇f(θk−1)+ζk/a\}⌊∇a⌋ket∇i}ht)j| ≤ /⌊a∇d⌊lθk−1−θ⋆/⌊a∇d⌊lj/⌊a∇d⌊l∇f(θk−1)+ζk/⌊a∇d⌊lj, moreover, applying A 1and A 2(2p) together with the Lyapunov inequality, we get E[/⌊a∇d⌊l∇f(θk−1)+ζk/⌊a∇d⌊l2l+j|Fk−1] =E[/⌊a∇d⌊l∇f(θk−1)+g(θk−1,ξk)+η... | https://arxiv.org/abs/2502.06719v1 |
i+j+l=p; i,j,l∈{0,...p}p! i!j!l!δ′2i k−1(−2αk/a\}⌊∇a⌋ketle{tθk−1−θ′ k−1,∇f(θk−1)−∇f(θ′ k−1)+g(θk−1,ξk)−g(θ′ k−1,ξk)/a\}⌊∇a⌋ket∇i}ht)j(αk(L1+L2)δ′ k−1)2l Now we bound each term in the sum above. 1. First, for i=p,j= 0,l= 0, the corresponding term in the sum equals δ′2p k−1. 2. Second, for i=p−1,j= 1,l= 0, we obtain, app... | https://arxiv.org/abs/2502.06719v1 |
C3=M3,4.(38) Define T1(A) = 1+1 A1/(1−γ)(1−γ)Γ(1 1−γ), T2(A) = 1+max/parenleftbigg exp/braceleftbigg1 1−γ/bracerightbigg1 A1/(1−γ)(1−γ)Γ(1 1−γ),1 A(1−γ)2/parenrightbigg .(39) Lemma 7. Assume A 1,A2(4), A3and A 4. Then it holds that E1/2[/⌊a∇d⌊lDn/⌊a∇d⌊l2]≤M1,1√n(/⌊a∇d⌊lθ0−θ⋆/⌊a∇d⌊l+/⌊a∇d⌊lθ0−θ⋆/⌊a∇d⌊l2+σ2+σ2 4)+M1,2σ2 4... | https://arxiv.org/abs/2502.06719v1 |
similar to ( 36), we obtain E[{δ(i) k}4|Fk−1]≤(1−4µαk+4α2 k(L1+L2)2(1+3c0(L1+L2))2){δ(i) k−1}4. Using Lemma 2(a), we obtain E[{δ(i) k}4]≤exp/braceleftbigg4(L1+L2)2(1+3c0(L1+L2))2) 2γ−1/bracerightbigg exp/braceleftbigg −4µk/summationdisplay j=i+1αj/bracerightbigg E[/⌊a∇d⌊lθ(i) i−θi/⌊a∇d⌊l4]. Combining the above inequali... | https://arxiv.org/abs/2502.06719v1 |
(j+k0+1)γ/parenleftbigg/parenleftbiggj+k0+1 i+k0/parenrightbiggγ −1/parenrightbigg exp{−µc0mj i+1} Following the proof of ( Wu et al. ,2024 , Lemma A.5), we have /parenleftbiggj+k0+1 i+k0/parenrightbiggγ −1≤(i+k0)γ−1/parenleftbigg 1+(1−γ)mj i/parenrightbiggγ/(1−γ) 37 SHESHUKOVA SAMSONOV BELOMESTNY MOULINES SHAO ZHANG N... | https://arxiv.org/abs/2502.06719v1 |
this section we prove ( 23). We start from the definition of an isoperimetric constant. Define Aε={x∈Rd:ρA(x)≤ε}andA−ε={x∈A:Bε(x)⊂A}, whereρA(x) = inf y∈A/⌊a∇d⌊lx−y/⌊a∇d⌊lis the distance between A⊂Rdandx∈Rd, and Bε(x) ={y∈Rd:/⌊a∇d⌊lx−y/⌊a∇d⌊l ≤ε}. For some class Aof subsets of Rdwe define its isoperimetric constant ad(A)(... | https://arxiv.org/abs/2502.06719v1 |
and Qiu ,1997 , Theo- rem 1.5)). Lemma 15. Assume A 1, A5, A6, A8. Then for any δ∈(0,1)with probability at least 1−δfor any k∈ {1,...n}it holds /⌊a∇d⌊lθk−θ⋆/⌊a∇d⌊l2≤αkK2log/parenleftbiggen δ/parenrightbigg , 43 SHESHUKOVA SAMSONOV BELOMESTNY MOULINES SHAO ZHANG NAUMOV where K2= max/parenleftbigg8(C1,ξ+2C2,ξ)2 µ,kγ 0/⌊a... | https://arxiv.org/abs/2502.06719v1 |
AND BOOTSTRAP FOR SGD E.5. Matrix Bernstein inequality for Σb nand Gaussian comparison Lemma 18. Under assumptions A 1, A6, A8, A9, there is a set Ω1∈ Fn−1, such that P(Ω1)≥ 1−1/nand onΩ1it holds that /⌊a∇d⌊lΣb n−Σn/⌊a∇d⌊l ≤10CQ,ξ/radicalbig log(2dn) 3√n where the constant CQ,ξis given by CQ,ξ:=C2 Q(C2 1,ξ+λmax(Σξ)), (... | https://arxiv.org/abs/2502.06719v1 |
arXiv:2502.06765v1 [math.ST] 10 Feb 2025Are all models wrong? Fundamental limits in distribution-free empirical model falsification Manuel M. M¨ uller∗, Yuetian Luo†and Rina Foygel Barber‡ ∗Statistical Laboratory, University of Cambridge †Data Science Institute, University of Chicago ‡Department of Statistics, Universit... | https://arxiv.org/abs/2502.06765v1 |
hand, a model classFthat is too complex may lead to challenges in selecting a good model f∈ F, if the available sample size is too small. In statistical learning theory, this is o ften referred to as theapproximation–estimation -tradeoff( Shalev-Shwartz andBen-David ,2014). Moreover, an overly rich model class Fcan also... | https://arxiv.org/abs/2502.06765v1 |
ns, and give an overview of our contributions on the problem of inference on the model class risk. 2.1 Setting and notation LetPdenote the unknown distribution on the data space Z, and letℓ:F × Z → [0,∞) denote a loss function, where Fis the model class of interest. As a canonical example, in the setting of a binary cl... | https://arxiv.org/abs/2502.06765v1 |
returning zero: since a valid lo wer bound can have errorαaccording to Definition 1, we may return a trivial lower bound ˆLα(F,Dn) =/braceleftBigg 0,with probability 1 −α, ∞,with probability α.(2) In other words, any meaningful answer to this question must have PP/braceleftbigˆLα(F,Dn)>0} substantially larger than α; we... | https://arxiv.org/abs/2502.06765v1 |
defined via the interpolation capacity of the model class F—the largest sample size Nfor whichˆR(F,DN) = 0. Of course, the definitions and results will be formalized below, b ut here we give an overview of our findings: (i)The low-complexity regime. IfFisnotabletointerpolateourdata(i.e., ˆR(F,Dn)> 0), then we can always c... | https://arxiv.org/abs/2502.06765v1 |
on the model class risk RP(F). Theorem 2. Fixα∈(0,1),n≥1, and model class F. Then ˆLERM α(F,Dn) :=α·ˆR(F,Dn) is a valid distribution-free lower bound on RP(F). Proof.IfRP(F) = 0 then we must have ˆR(F,Dn) = 0 almost surely. If instead RP(F)> 0, then by Markov’s inequality, we have PP{ˆLERM α(F,Dn)≤RP(F)}=PP{ˆR(F,Dn)≤ α... | https://arxiv.org/abs/2502.06765v1 |
bound on the model class risk. Then, for all N≥n, PP/braceleftBig ˆLα(F,Dn)>ˆR(F,DN)/bracerightBig ≤α+n2 2N. To interpret this theorem, let us return to the question of whethe r we can determine that “all models are wrong”: can a valid distribution-free lower bound sat isfyˆLα(F,Dn)>0, certifying that the model class r... | https://arxiv.org/abs/2502.06765v1 |
allx∈Rd), we can calculate the interpolation complexity of this model class as N+(F(m) pwc,P)≥N(F(m) pwc,P)≥m, since, for any ( X1,Y1),...,(Xm,Ym) with distinct values X1,...,X m, we can construct a functionf∈ F(m) pwcwithf(Xi) =Yifor eachi∈[m]. In particular, the lower bound ˆLERM α(F(m) pwc,Dn) =α·ˆR(F(m) pwc,Dn) 9 c... | https://arxiv.org/abs/2502.06765v1 |
when will this be the case? The next result establishes that, in a high-dimensional setting where d≥n, the quantity λn,d(P) is small for a broad class of distributions P. Proposition8. LetPbe adistribution on Rd×Rwith a density. Let (X1,Y1),...,(Xn,Yn)iid∼ P, and define X∈Rn×das the matrix with rows Xi. Ifd≥n, then for ... | https://arxiv.org/abs/2502.06765v1 |
under only mild distributional assumptions. For instance, parametr ic convergence rates of theexcess risk RP(ˆf)−inff∈FRP(f)canbeachieved whentheloss satisfies certainregularity conditions and the algorithm used to select ˆffromFsatisfies stability properties ( Klochkov and Zhivotovskiy ,2021).Mourtada and Ga¨ ıffas (2022... | https://arxiv.org/abs/2502.06765v1 |
being investigated, but is still substantially different: whileLuo and Barber (2024) study the problem of evaluating the risk of a model fitted by a spec ific algorithm A, here we instead ask about the infimum risk inf f∈FRP(f) regardless of whether it is possible to construct an algorithm Athat will identify an (approxima... | https://arxiv.org/abs/2502.06765v1 |
inference. Information and Inference: A Journal of the IMA , 10(2):455–482. Barron, A. R. (1994). Approximation and estimation bounds for ar tificial neural networks. Machine learning , 14:115–133. 14 Bartlett, P. L., Boucheron, S., and Lugosi, G. (2002). Model selec tion and error estimation. Machine Learning , 48:85–1... | https://arxiv.org/abs/2502.06765v1 |
preprint arXiv:2402.07388 . McDiarmid, C. (1998). Concentration. In Habib, M., McDiarmid, C., Ra mirez-Alfonsin, J., and Reed, B., editors, Probabilistic Methods for Algorithmic Discrete Mathemati cs, volume 16, pages 195–248. Springer. Medarametla, D. and Cand` es, E. (2021). Distribution-free cond itional median infe... | https://arxiv.org/abs/2502.06765v1 |
small, it therefore holds that P/braceleftbigg ˆR(F,Dn)≤(1+δ)RP(F)/bracerightbigg ≥1−exp/parenleftbigg −h(δ)·nRP(F) B/parenrightbigg . Next, letδn>0 be the unique solution to h(δn) =Blog(1/α) nRP(F), so that we have P/braceleftbigg ˆR(F,Dn)≤(1+δn)RP(F)/bracerightbigg ≥1−α. To complete the proof, we therefore only need ... | https://arxiv.org/abs/2502.06765v1 |
same lower bound as constructed in Theorem 2, but computed with a much lower complexity model class (if n−1≪m). We are now ready to formally prove the theorem. Proof of Theorem 6.Fixanyε>0,andchoosesome fε∈ F(m) pwcwithRP(fε)≤inff∈F(m) pwcRP(f)+ ε=RP(F(m) pwc)+ε. By definition of the model class, we can express fεas fε(... | https://arxiv.org/abs/2502.06765v1 |
α(F(m) pwc,Dn) is likely bounded away from zero—a very informative lower bound. We are now ready to prove this extension. Proof of Theorem B.2.Following an identical argument as in the proof of Theorem 6, and definingfεas in that proof, we have PP/braceleftBig ˆLpwc,r α(F(m) pwc,Dn)≤RP(fε)/bracerightBig ≥PP/braceleftbig... | https://arxiv.org/abs/2502.06765v1 |
( X,Y)∼P, thenλn,d(P′) =λn,d(P). Therefore, without loss of generality, from this point on we can assume Ω = Id, and we will write h(x) =hId(x) = (det( xx⊤))−1/2. First, we define two continuous distributions on ( X,β)∈Rd×n×Rd. Letf(x,y) denote the density of the distribution PonRd×R. Now, fix any constant c>0, and define... | https://arxiv.org/abs/2502.06765v1 |
XX⊤XU U⊤X⊤U⊤U/parenrightbigg/parenrightBigg =/radicaltp/radicalvertex/radicalvertex/radicalbtdet/parenleftBigg/parenleftbigg XX⊤0 0 I d−n/parenrightbigg/parenrightBigg =/radicalbig det(XX⊤). 26 And, gγ|X/parenleftBigg/parenleftbiggX U⊤/parenrightbigg ·b/parenrightBigg =n/productdisplay i=1fY|X(X⊤ ib|Xi)·(2πc)−d−n 2e−/b... | https://arxiv.org/abs/2502.06765v1 |
. Next, we bound each term in the sum. Since ψ(u) =d dulogΓ(u) andd≥n+2, we have log/parenleftbigg Γ/parenleftbiggd−1 2+1−j 2/parenrightbigg/parenrightbigg −log/parenleftbigg Γ/parenleftbiggd 2+1−j 2/parenrightbigg/parenrightbigg +1 2ψ/parenleftBigd 2+1−j 2/parenrightBig =1 2/bracketleftbigg ψ/parenleftBigd 2+1−j 2/par... | https://arxiv.org/abs/2502.06765v1 |
is nonzero if and only if/summationtext iBi≥n, so equivalently we have that E˜PN/bracketleftbig/summationtext 1≤i1<···<in≤Nfi1,...,in· /BDBi1=···=Bin=1/vextendsingle/vextendsingleB1,...,B N/bracketrightbig /parenleftbig/summationtext iBi n/parenrightbig =EQn[f], on the event that/summationtext iBi≥n. To cover both case... | https://arxiv.org/abs/2502.06765v1 |
Learning an Optimal Assortment Policy under Observational Data Yuxuan Han∗Han Zhong†Miao Lu‡Jose Blanchet‡Zhengyuan Zhou∗ March 19, 2025 Abstract We study the fundamental problem of offline assortment optimization under the Multinomial Logit (MNL) model, where sellers must determine the optimal subset of the products t... | https://arxiv.org/abs/2502.06777v2 |
inefficient to engage in time-consuming online learning iterations. In these situations, offline learning provides an effective solution, allowing sellers to make decisions based solely on existing data without requiring any online interactions. To our best knowledge, Dong et al. (2023) appears to be the only work that... | https://arxiv.org/abs/2502.06777v2 |
constants and logarithmic factors. This result is significant because PRB only requires optimal item coverage — that each item in the optimal assortment has appeared individually in some historical assortment — rather than requiring observations of the complete optimal assortment itself, as in Dong et al. (2023). Neces... | https://arxiv.org/abs/2502.06777v2 |
Blanchet et al., 2023) theoretically demonstrates that in offline data-driven decision making, pessimistic algorithms achieve provable efficiency while only requiring good coverage of (the trajectories induced by) the optimal decision policy. Unfortunately, the settings addressed in all these results are not applicable... | https://arxiv.org/abs/2502.06777v2 |
k=11{ik=j}, which then can imply an estimator of vbased on the relation vj=pj/(1−pj). The rank-breaking method achieves optimal sample efficiency with minimal requirement on the observational assortment dataset by addressing two key limitations of other popular approaches for estimating the underlying attraction parame... | https://arxiv.org/abs/2502.06777v2 |
for a detailed proof. Theorem 4.2 provides the finite-sample guarantee for the PRB algorithm. Notably, in the non-trivial setting where (1 + V)Kp log(N/δ)/mini∈S⋆ni=O(1),the second term is dominated by the first term, thus the sub-optimality gap converges in a rate eO(K/√mini∈S⋆ni). In contrast, the best-known previous... | https://arxiv.org/abs/2502.06777v2 |
any sufficiently large integers N, K such that N≥5K, when the reward vector r=1N, there exists a set of observed K-sized assortments DS={Sk}n k=1⊂ [N]nand a set of problem parameters V ⊂[0,1]N. So that for any algorithm Athat takes as input DSand Dv:={i∈S:S∈DS, i∼P(· |S,v)independently }, if we denote S⋆ vthe optimal a... | https://arxiv.org/abs/2502.06777v2 |
assortment is precisely the set consisting of all items in Nopt. The setNccontains items that are sub-optimal but have only a small gap of ϵin attraction values compared to those in Nopt. Lastly, N0contains items that have much larger attractive values than other items but are definitively sub-optimal due to their zero... | https://arxiv.org/abs/2502.06777v2 |
j) =nmin4KX ℓ=1D(Qv(S(ℓ) 1)∥Qv′(S(ℓ) 1)). and the following result on each D(Qv(S(ℓ) 1)∥Qv′(S(ℓ) 1)): Lemma 5.4. For every ℓ,it holds that D(Qv(S(ℓ) 1)∥Qv′(S(ℓ) 1))≤( 5ϵ2,ifℓ∈(Nopt(r,v)∪ Nopt(r′,v′))\(Nopt(r,v)∩ Nopt(r′,v′)), 0, otherwise. 8 Lemma 5.4 indicates that with the selection ϵ=q C1 20nmin,we have max (r,v),... | https://arxiv.org/abs/2502.06777v2 |
+P k∈Steθk! | {z } :=ℓ(θ), requires iterative optimization methods to maximize the objective function ℓ(θ). Here, we compare with the first-order methods, which have the lowest computational cost per iteration. Specifically, at each iteration, computing the gradient incurs a cost of O(nK), while updating the parameter ... | https://arxiv.org/abs/2502.06777v2 |
note that the existing theoretical guarantees for PASTA require observations of the optimal assortment size nS⋆(Dong et al., 2023), our simulations show that PASTA still highly outperforms the non-pessimistic baseline even when nS⋆= 0. This suggests that the proposed effective number, mini∈S⋆ni, may be the true complex... | https://arxiv.org/abs/2502.06777v2 |
theoretical insights and practical tools for real-world implementation. References Agrawal, S. ,Avadhanula, V. ,Goyal, V. andZeevi, A. (2017). Thompson sampling for the mnl-bandit. InConference on learning theory . PMLR. 1, 3, 4 Agrawal, S. ,Avadhanula, V. ,Goyal, V. andZeevi, A. (2019). Mnl-bandit: A dynamic learning ... | https://arxiv.org/abs/2502.06777v2 |
arXiv:2405.09831 . 3, 4, 6, 19 Li, S. ,Luo, Q. ,Huang, Z. andShi, C. (2022). Online learning for constrained assortment optimization under markov chain choice model. Available at SSRN 4079753 . 3 Liu, Z. ,Lu, M. ,Wang, Z. ,Jordan, M. andYang, Z. (2022). Welfare maximization in competitive equilibrium: Reinforcement lea... | https://arxiv.org/abs/2502.06777v2 |
3 Xie, T. ,Cheng, C.-A. ,Jiang, N. ,Mineiro, P. andAgarwal, A. (2021). Bellman-consistent pessimism for offline reinforcement learning. Advances in neural information processing systems 346683–6694. 3 Xiong, W. ,Zhong, H. ,Shi, C. ,Shen, C. ,Wang, L. andZhang, T. (2023). Nearly minimax optimal offline reinforcement lea... | https://arxiv.org/abs/2502.06777v2 |
term in Theorem A.1, we have X i∈S⋆ri 1 +P i∈S⋆vi48 log( N/δ)(1 + vi)Pn k=11{i∈Sk}(1 +P j∈Skvj)−1≤48K(1 +V) log( N/δ) mini∈S⋆ni, as desired, finishing the proof. A.3 Proof of Lemma 4.1 Proof of Lemma 4.1. We first prove a general monotone result of the revenue function Runder v,similar to those in Lemma A.3 of Agrawal ... | https://arxiv.org/abs/2502.06777v2 |
K/8 by⌊K/4⌋,⌊K/8⌋all the analysis still hold. Now for every K-sized subset Sof [4K], define its 17 K/4-neighborhood under ∆ as δ(S, K/ 4) :={S′⊂[4K] :|S′|=K,∆(S, S′)≤K/4}. We have then denote Fas the maximal K/2-packing set consists of K-sized subsets of [4 K], we have |δ(S, K/ 4)| · |F| ≥4K K . Since otherwise there... | https://arxiv.org/abs/2502.06777v2 |
by j∈ N 0andvj=v′ j= 1, |pj−qj|= 1 K+ 1/K+ϵ−1 K+ 1/K =ϵ (K+ 1/K)(K+ 1/K+ϵ)≤ϵ K2, thus then (pj−qj)2 qj≤ϵ2(K+ 1/K) K4≤2ϵ2 K3. ii) For j=ℓ,we have |pj−qj|= 1/K+ϵ K+ 1/K+ϵ−1/K K+ 1/K =(K+ 1/K)(1/K+ϵ)−(K+ 1/K+ϵ)1/K (K+ 1/K)(K+ 1/K+ϵ)≤ϵ K, 19 which then implies (pi−qi)2 qi≤ϵ2 K2·(2K)2≤4ϵ2. Combining above two cases, we get ... | https://arxiv.org/abs/2502.06777v2 |
A Closed-Form Transition Density Expansion for Elliptic and Hypo-Elliptic SDEs Yuga Iguchi1,∗and Alexandros Beskos1 1Department of Statistical Science, University College London, London, UK ∗Address for correspondence. Yuga Iguchi. yuga.iguchi.21@ucl.ac.uk Abstract We introduce a closed-form expansion for the transitio... | https://arxiv.org/abs/2502.07047v1 |
for Mathematical Research.arXiv:2502.07047v1 [math.NA] 10 Feb 2025 2 Y. Iguchi and A. Beskos first class is the elliptic one, where we consider SDEs of the following form: dXt=VR,0(Xt, θ)dt+X 1≤j≤dVR,j(Xt, θ)dBj,t, X 0=x0∈RN,(E) so that Vj=VR,j, 0≤j≤d. We set σR= [VR,1, . . . , V R,d],aR=σRσ⊤ R, and assume that aR=aR(x... | https://arxiv.org/abs/2502.07047v1 |
‘correction term’ is given in closed-form, and includes Hermite polynomials up to a degree J≥1, obtained via working with q∆(x, y;θ). The correction term plays a key role in capturing non-linear/non-Gaussian effects in the true transitions. In detail, A¨ ıt-Sahalia (2002) constructs the CF-expansion by first applying a... | https://arxiv.org/abs/2502.07047v1 |
(2019) in the context of elliptic SDEs. The aforementioned works also demonstrate the effective use of a CF-expansion within parameter inference procedures. In particular, the approaches provide an approximate Maximum Likelihood Estimator (MLE). Obtained numerical results showcase that: the proxy MLEs stays close to th... | https://arxiv.org/abs/2502.07047v1 |
by the work of one of the co-authors in Iguchi and Yamada (2021). This latter work lies in the area of numerical methods for SDEs and looks at the development of approximation schemes for elliptic SDEs of improved weak order of convergence. To the best of our knowledge, this is the first time in the literature that a C... | https://arxiv.org/abs/2502.07047v1 |
CF transition density expansion for a wide class of Itˆ o processes in (1.1), including the family of hypo-elliptic SDEs specified in (H). We write the transition density of Xt+∆given Xt=x∈RNas y7→pX ∆(x, y;θ) :=P(Xt+∆∈dy|Xt=x)/dy, with t≥0, ∆ >0. Closed-Form Transition Density Expansion for SDEs 5 1.2 1.0 0.8 0.6 0.4 ... | https://arxiv.org/abs/2502.07047v1 |
>0, ∆ >0, so that: ¯XEM,θ t+∆:=x+VR,0(x, θ)∆ + σR(x, θ) Bt+∆−Bt . (2.1) Under regularity conditions on x7→VR,j(x, θ), 0≤j≤d, and the requirement that the matrix aR(x, θ) = (σRσ⊤ R)(x, θ) is positive definite for all ( x, θ)∈RN×Θ, the EM scheme gives rise to a well-defined baseline Gaussian transition density, y7→p¯XE... | https://arxiv.org/abs/2502.07047v1 |
Density Expansion for SDEs 7 w.r.t. the state argument and Hermite polynomials now defined via partial derivatives of the non-degenerate Gaussian density p¯X ∆(x, y;θ). Remark 2.1 Iguchi and Yamada (2021) work in an elliptic setting to develop Monte-Carlo estimators of improved weak order of convergence for E[φ(XT)],φ:... | https://arxiv.org/abs/2502.07047v1 |
we introduce z∈R2to represent the initial condition for (2.3), thus distinguish the latter from argument x∈R2upon which the linear drift in (2.3) applies. For the above vector 8 Y. Iguchi and A. Beskos fields, H¨ ormander’s condition holds via: ¯V1(x) = 0, σ]⊤, [¯Vz 0,¯V1](x) =−¯V1¯Vz 0(x) = −σ, σα⊤. (2.10) Thus, th... | https://arxiv.org/abs/2502.07047v1 |
x, y;θ) =Z I(s1:j)¯Pθ,z sjfLz θ¯Pθ,z sj−1−sj···fLz θ¯Pθ,z s1−s2fLz θp¯Xz ∆−s1(·, y;θ)(x) z=xdsj···ds1; RM 1(∆, x, y;θ) =Z I(s1:M)Pθ sMfLz θ¯Pθ,z sM−1−sM···fLz θ¯Pθ,z s1−s2fLz θp¯Xz ∆−s1(·, y;θ)(x) z=xdsM···ds1, (2.19) I(s1:k) :={s1:k= (s1, . . . , s k) : 0≤sk≤ ··· ≤ s1≤∆}, k≥0, with the convention s0≡∆. Step 2. SinceTj... | https://arxiv.org/abs/2502.07047v1 |
by a few earlier works in the case of the elliptic class (E). We also describe that the obtained expansion (3.4) can be given in the form (1.3), namely a series in powers of ∆. As the error estimates vary for classes (E), (H), we make use of the notation w∈ {E,H}and write pX,(w)≡pX,p¯X,(w)≡p¯X,RM,(w) 1 ≡RM 1 andRj,β[j]... | https://arxiv.org/abs/2502.07047v1 |
∈ {E,H}, Theorem 1 (Bound for RM,(w) 1 )Letx∈RNbe the initial state of the transition dynamics and M≥1. Under Assumptions 1–5, there exist constants C1, C2>0such that for all (∆, y, θ)∈(0,1)×RN×Θ: RM,(w) 1 (∆, x, y;θ) ≤C1∆M 2× G(w)(∆, x, y, θ ) ≤C2∆M−m(w) 2, w ∈ {E,H}. Theorem 2 (Bound for Rj,β[j],(w) 2 )Letx∈RNbe the ... | https://arxiv.org/abs/2502.07047v1 |
the term ∆−k 2). Based upon Lemma 3.1, the form of the CF-expansion is determined from the expression of the differential operator Dz,θ α and the number of derivatives involved therein. A detailed characterisation for the differential operator Dz,θ α is provided in Supplementary Material. In particular, Lemma C.3 in Su... | https://arxiv.org/abs/2502.07047v1 |
(3.11). For the replacement by the Taylor approximation to only affect terms of size O(∆(J+1)/2), one should select J′as the smallest even integer so that J′≥J. An even J′guarantees integrability of the density proxy. 4. Numerical Experiments We focus on the bivariate FitzHugh-Nagumo (FHN) SDE used in neuroscience. Thi... | https://arxiv.org/abs/2502.07047v1 |
∆ ,ι∈ {I,II}. We try J= 2,3,4,5, and for ep∆we set J′= 2, as the correction term includes powers ∆3/2, . . . , ∆J/2,J≤5, and the transform can only affect terms of size O((∆3/2)(J′+1)) = O(∆9/2). We find the benchmark ‘true’ density via a simulation that: (i) uses 2 ×107samples from the FHN SDE at ∆ via an EM scheme wi... | https://arxiv.org/abs/2502.07047v1 |
0.0 V1.00 0.75 0.50 0.25 0.000.250.500.751.00U(,J)=(0.1,4) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 V1.00 0.75 0.50 0.25 0.000.250.500.751.00U(,J)=(0.1,5) 0.5 0.4 0.3 0.2 0.1 V0.4 0.2 0.00.20.40.60.8U(,J)=(0.05,2) 0.5 0.4 0.3 0.2 0.1 V0.4 0.2 0.00.20.40.60.8U(,J)=(0.05,3) 0.5 0.4 0.3 0.2 0.1 V0.4 0.2 0.00.20.40.60.8U(,J)=(0.05,4) 0... | https://arxiv.org/abs/2502.07047v1 |
single adult female rat with time length 250ms and equidistant step-size 0 .02ms. In our study we choose a particular dataset, specifically the file 1554.mat from the above URL, which was obtained while the 5th lumber dermatome was stimulated. We subsample the first 40ms of data with a step-size 0 .08ms, i.e. we have (... | https://arxiv.org/abs/2502.07047v1 |
centred-parametrisation led to MCMC chains with very poor convergence performance. 510152025 0.150.300.450.60 246810 0.81.62.43.24.0 510152025 0.15 0.30 0.45 0.60 0.81.62.43.24.0 (a)P1 (Partial LDL scheme) 510152025 0.150.300.450.60 246810 0.81.62.43.24.0 510152025 0.15 0.30 0.45 0.60 0.81.62.43.24.0 (b) P2 (CF-expansi... | https://arxiv.org/abs/2502.07047v1 |
main text, i.e., Theorems 1 and 2, error estimates for the proposed CF expansion. Section C studies the expression of the differential operator Dz,θ αgiven in (2.21). Section D provides supporting information for the implementation of density expansion used in the numerical experiments and additional experiments of Bay... | https://arxiv.org/abs/2502.07047v1 |
θ,fLz θ]¯Pθ,z t−sφ(ξ1)p¯Xz s(x, ξ1;θ)dξ1 =g[L0,z θ,fLz θ](s), where we made use of (A.7) and integration by parts in the second and third lines, respectively. Thus, the higher-order derivatives of gfLz θ(s) are given as: ∂kgfLz θ(s) =g(adL0,z θ)k(fLz θ)(s) =¯Pθ,z s adL0,z θk(fLz θ) ¯Pθ,z t−sφ(x), 0≤k≤J, (A.10) and t... | https://arxiv.org/abs/2502.07047v1 |
1,2 with mean ai∈RNand covariance Ai∈RN×N, it follows (see e.g. Vinga (2004)) that: Z RNφ1(ξ;a1, A1)φ2(ξ;a2, A2)dξ=1r (2π)Ndet A1+A2exp −1 2(a1−a2)⊤(A1+A2)−1(a1−a2) .(B.9) III. Let λ1>0. For any α∈ZN ≥0, there exist constants c, λ2>0 such that for all ξ∈RN, |ξα| ×exp −λ1|ξ|2 ≤Cexp −λ2|ξ|2 . (B.10) We note that ... | https://arxiv.org/abs/2502.07047v1 |
z, θ )≥c×( tNunder model class (E); t4dunder model class (H) .(B.18) Thus, it follows from (B.14), (B.15) and (B.18) that: H(w),z (j)(∆, x, y, θ )×p¯Xz,(w) ∆ (x, y;θ) ≤C1 ν(w),z(∆, y, x, θ ) ×∆−m(w) 2exp −λ1 ν(w),z(∆, y, x, θ ) 2 ≤C2∆−m(w) 2exp −λ2 ν(w),z(∆, y, x, θ ) 2 ∵(B.10) for some constants C1, C2, λ1, λ2>0... | https://arxiv.org/abs/2502.07047v1 |
A. Beskos Proof of Lemma B.1. We first show the bound (B.27). We set eΓ(w) t≡(Γt,(w))⊤Γt,(w), t∈(0,1). Proposition B.1 with the multi-index α=0gives: Z RNG(w),x(t−s, ξ, y, θ )G(w)(s, x, ξ, θ )dξ ≤C1Z RN(t−s)−m(w) 2exp −λ1 ν(w),x(t−s, y, ξ, θ ) 2 ×s−m(w) 2exp −λ2 Γ−1 s,(w) ξ−x−V0(x, θ)s 2 dξ ≤C2q det c1eΓ(w) t−s+... | https://arxiv.org/abs/2502.07047v1 |
= 0; 1 2⌊3 2J⌋+ 1, J = 1,2; 1 2⌊3 2J⌋+3 2, J≥3.(B.37) Furthermore, (B.35) holds with the differential operator Dz,θ (J)being replaced with (Dz,θ (J))∗. Lemma B.4 Let Assumptions 1, 3 and 4 hold. Let x∈RNbe the initial state of SDE and ψ(w)∈F(w), w∈ {E,H}. Then for any α∈Zj ≥0, j≥2, there exists a constant C >0such that... | https://arxiv.org/abs/2502.07047v1 |
appropriate bound, we again consider the cases s1∈[0,∆/2] and s1∈[∆/2,∆] separately. Step II-1. Consider the case s1∈[0,∆/2]. We have ∂α y Pθ,(w) skfLz θ¯Pθ,z,(w) sk−1−skfLz θ···¯Pθ,z,(w) s1−s2fLz θ p¯Xz,(w) ∆−s1(·, y;θ) (x)|z=x =Z RN∂α yfLz θ p¯Xz,(w) ∆−s1(·, y;θ) (ξ) Pθ,(w) skfLz θ¯Pθ,z,(w) sk−1−skfLz θ···fLz θ... | https://arxiv.org/abs/2502.07047v1 |
(∆, ξ, y, z, θ )∈(0,1)×RN×RN×Zκ×Θand for any 0≤sj≤sj−1≤ ··· ≤ s1≤∆, j∈N, ∂β ξ∂γ yF[J],z,(w) s1:j(ξ, y;θ) ≤C∆−∥β+γ∥w−j−1 2−Qw(J)· 1 + ξ−z√ ∆ q ·G(w),z(∆, ξ, y, θ ), (B.46) where Qw(J)is defined in (B.37). Proof of Proposition B.3 . We exploit the mathematical induction on the integer j. In what follows, we make use of... | https://arxiv.org/abs/2502.07047v1 |
for s1∈[0,∆/2] ∂β ξ∂γ yF[J],z,(w) s1:i+1(ξ, y;θ) ≤C∆−∥β+γ∥w−Qw(J)−i 2× 1 + ξ−z√ ∆ q ×G(w),z(∆, ξ, y, θ ). Step II-2 .s1∈[∆/2,∆]. IBP together with (B.8) gives ∂β ξ∂γ yF[J],z,(w) s1:i+1(ξ, y;θ) ≤CZ RN p¯Xz,(w) ∆−s1(·, y;θ)(ξ) × ∂γ η(fLz θ)∗ η ∂β ξ¯Pθ,z,(w) si+1Dz,θ (J)¯Pθ,z,(w) si−si+1fLz θ···fLz θp¯Xz,(w) s1−s2(·, η... | https://arxiv.org/abs/2502.07047v1 |
= H, ∵(B.37) (B.58) where C >0 is a constant independent of ∆ , yandθ. For the case of w= H in (B.58), we have further β[j] i−1 2⌊3 2(β[j] i+ 1)⌋+j 2+1 2×1β[j] i=0,1=1 2 ⌊β[j] i 2⌋ −1 +j 2+1 2×1β[j] i=0,1=1 2 ⌊β[j] i 2⌋ −1β[j] i≥2 +j 2, and the proof of Theorem 2 is now complete. □ C. Expression of the differenti... | https://arxiv.org/abs/2502.07047v1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.