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jthdegree of freedom yj. The hat indicates the matrix representation in the basis where individual qubits are represented as vectors |0⟩≡(1 0)and|1⟩≡(0 1). Then, we can build the operator (a+ j)β(a− j)αas (a+ j)β(a− j)α≡1 4(2mj+α−β+|α−β|)/2X n=αp (n+β−α)!n! (n−α)!(ˆXj,n+iˆYj,n)⊗(ˆXj,n+β−α−iˆYj,n+β−α). (6.7) Equation (6... | https://arxiv.org/abs/2502.14580v1 |
they inherently respect the constraint. This can be achieved by constructing transformations exclusively from the creation and annihilation operators, ensuring that ˆXandˆYare always employed in pairs of the form ˆXj,kˆYj,l−ˆYj,kˆXj,l. Extracting information from quantum states is accomplished through quantum measureme... | https://arxiv.org/abs/2502.14580v1 |
a significant overlap with the target eigenstates. 2) Decomposing the propagator exp(−iˆLFKPt)into a sequence of quantum gates for various times t. 3) Computing and storing the overlaps between the resulting states and the initial state, also referred to as the autocorrelation function, for the chosen time intervals. 4... | https://arxiv.org/abs/2502.14580v1 |
trial wavefunction using a unitary ansatz UVQE(τ), where τ={τα}NVQE α=1are the NVQEparameters to be optimized. Ideally, the ansatz would explore the entire Hilbert space, for example employing operations from the Pauli group. However, the exponential scaling of the number of elements with the number of qubits makes thi... | https://arxiv.org/abs/2502.14580v1 |
superposition, it would destroy C(k) 0|φ(k) 0⟩. Instead, we must apply the second rotation only on the term containing |0mk⟩. This is achieved by using an “anti-controlled” rotation, ¯CˆRk,1;0=(ˆ1k,0+ˆZk,0)⊗ˆRk,1/2+(ˆ1k,0−ˆZk,0)/2, which applies the rotation ˆRk,1;0only if sk,0=0. Indeed, with the controlled rotation, ... | https://arxiv.org/abs/2502.14580v1 |
essential coupling between a quantum computer and a classical computer, we chose to present the Variational Quantum Eigensolver (VQE) approach instead of the Quantum Phase Estimation (QPE) algorithm, given the latter’s greater complexity and non-trivial nature. Furthermore, QPE requires longer quantum circuits, with a ... | https://arxiv.org/abs/2502.14580v1 |
, 1995, pp. 229–252. doi:10.1007/978-1-4612-4202-4˙5. [17] C. Soize, R. Ghanem, Data-driven probability concentration and sampling on manifold, Journal of Computational Physics 321 (2016) 242– 258. doi:10.1016/j.jcp.2016.05.044. [18] C. Soize, R. Ghanem, Probabilistic learning on manifolds, Foundations of Data Science ... | https://arxiv.org/abs/2502.14580v1 |
Fourier-Hermite functionals, Annals of Mathematics 48 (2) (1947) 385–392. doi:10.2307/1969178. [39] P. Kr ´ee, C. Soize, Mathematics of Random Phenomena, Reidel Pub. Co, 1986, (first published by Bordas in 1983 and also published by 25 Springer Science & Business Media in 2012). [40] R. J. Serfling, Approximation theor... | https://arxiv.org/abs/2502.14580v1 |
Correction, Academic Press, Cambridge, MA, USA, 2012, pp. 91–117. doi:10.1016/B978-0-12-385491-9.00003-4. [61] J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y . Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, J. Tennyson, The Variational Quantum Eigensolver: A review of methods and best practices, Phys. Rep. 986 ... | https://arxiv.org/abs/2502.14580v1 |
Asymptotic Analysis and Practical Evaluation of Jump Rate Estimators in Piecewise-Deterministic Markov Processes Romain Azaïs and Solune Denis Abstract Piecewise-deterministicMarkovprocesses(PDMPs)offerapowerfulstochasticmodelingframework that combines deterministic trajectories with random perturbations at random time... | https://arxiv.org/abs/2502.14621v1 |
frameworks can be considered for this purpose. In this paper, we adopt a particularly general framework, aiming not only to develop statistical methods that can be applied to a large variety of problems, but also to better understand what distinguishes PDMPs. Specifically, we focus on the non-parametric estimation of t... | https://arxiv.org/abs/2502.14621v1 |
relies on the multiplicative intensity model developed by Aalen [1] (see also [2]). To es- timate a jump rate λ, this method consists in exhibiting a counting process N(t) whose stochastic intensity is of the form λ(t)Y(t)for some predictable process Y(t), i.e. such that the process M(t)=N(t)−∫t 0λ(s)Y(s)dsis a continu... | https://arxiv.org/abs/2502.14621v1 |
with DK(x)=Eµ[gZ0(h(x)) 1{h(Z0)≤h(x)} 1{Z1≥h(x)}], (3) where gx(y)=1 (h◦Φx)′((h◦Φx)−1(y)). The numerator is evaluated as a kernel estimator of the invariant distribution µ composed with h, while the denominator is estimated by its empirical version. The main result of [18] states the convergence in L1-norm to 0of the s... | https://arxiv.org/abs/2502.14621v1 |
is to make the first rigorous, both theoretical and nu- merical, comparison of these estimation strategies. To achieve this, we propose to standardize both the frameworks, the methods and the theoretical convergence re- sults obtained. That is why, in the present paper, we only consider one-dimensional processes observ... | https://arxiv.org/abs/2502.14621v1 |
we aim to evaluate their rates of convergence and, if the rates are equal, their asymptotic variances. For this purpose, we shall rely on the following limit result. Theorem 1.1. [5, Corollary 3.10] Under an ergodicity assumption and regularity conditions, when ngoes to infinity, √ nhs nht n[˚λ♠ n(x)−λ(x)]d⟶N(0, σ2 ♠(x... | https://arxiv.org/abs/2502.14621v1 |
Subsection 4.3. This approach allows us to complement the theoretical and numerical investigations with an evaluation of the estimators in a real-world application. The conclusion of our study, encompassing the theoretical, numerical, and real- world data application aspects of the estimators under consideration, is pr... | https://arxiv.org/abs/2502.14621v1 |
with compact support. In addition, we denote τ2=∫RK2(x)dx. The following result establishes the strong consistency of the two estimators under consideration. Theorem 2.1. If the model satisfies Assumptions 2.1, 2.2 and 2.3, and if the kernel Ksatisfies Assumptions 2.5, then, for any bandwidth hn∝n−γ,0<γ<1, when n goes ... | https://arxiv.org/abs/2502.14621v1 |
named after its role in modeling the well-known Transmission Control Protocol, a key mechanism for data transmission over the Internet (see [7] and the references therein). In that particular case, the invariant distributions (of the continuous-time process Xtand of the embedded chain Zn) are fully explicit [15], µCT(x... | https://arxiv.org/abs/2502.14621v1 |
State space0123456Standard deviation in CLT κ=0.6 κ=0.7 κ=0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 State space0123456Standard deviation in CLT 0.0 0.5 1.0 1.5 2.0 2.5 3.0 State space0123456Standard deviation in CLT 0.0 0.5 1.0 1.5 2.0 2.5 3.0 State space0123456Standard deviation in CLT Figure 2: Standard deviations√ σ2 ♣(x)in p... | https://arxiv.org/abs/2502.14621v1 |
noted that this analysis falls outside the scope of the theorems, as the bandwidths werechosennumericallyanddonotfollowthetheoreticallyprescribeddecayscheme. A slight bias is observable when n=10 000, which highlights (as is well-known) that asymptotic variance alone is insufficient to fully characterize the desirable ... | https://arxiv.org/abs/2502.14621v1 |
with optimal bandwidth parameters h♣ n,h♦ n,h♠s n andh♠t n, from trajectories of size n=1 000(top) and of size n=10 000(bottom) generated from the TCP model with parameter κ=0.4, and referenced Gaussian distributions with mean λ(2)=2and variance σ2 ♣(x)/(nh♣ n)in pink line (left), with variance σ2 ♦(x)/(nh♦ n)in purple... | https://arxiv.org/abs/2502.14621v1 |
are presented in Subsection 4.3. 18 n=1 000 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 State space0.000.050.100.150.200.250.300.350.40Estimation error n=10 000 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 State space0.0000.0250.0500.0750.1000.1250.1500.1750.200Estimation error Figure 7: Pointwise estimation error over ... | https://arxiv.org/abs/2502.14621v1 |
to play a significant role), especially on the cell in question. For each of the temperature condition, for each of the thousands of cells along the hundreds of lineages measured, we fit a linear model to their growth. Some results, gathered by temperature condition, are given in Figure 9. We observe that the histogram... | https://arxiv.org/abs/2502.14621v1 |
27 °C, lower panel: 37 °C), logarithm of cell size measurements before the fifth division event and fitted linear growth (left), distribution of estimated slope (top center), distribution of time between two consecutive division events (bottom center), distribution of logarithm of cell size at division time (top right)... | https://arxiv.org/abs/2502.14621v1 |
and Krell and Schmisser [16, 20], making them comparable with Azaïs and Muller-Gueudin’s approach [5]. The proof involves the study of vector martingales constructed along the embedded chain of the PDMP. The theoretical results prove that the three standardized estimators can not in general be ordered based on their as... | https://arxiv.org/abs/2502.14621v1 |
1518–1534. [8]Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet, A., and Yvinec, R. Probabilistic and piecewise deterministic models in biology. ESAIM: Procs 60 (2017), 225–245. [9]Costa, O. L. V., and Dufour, F. Stability and ergodicity of piecewise deterministic Markov processes. SIAM Journal on Control and... | https://arxiv.org/abs/2502.14621v1 |
1{Zi≥h(x)}, where hn=cn−γ,1/3<γ<1(for the sake of readability, c=1in the sequel of the proof).̂λ♣ n(x)isbasicallythequotientof ̂µn(h(x))over̂Dn(x). Inordertoestablish the asymptotic properties of ̂λ♣ n(x), we shall study the vector (̂µn(h(x)),̂Dn(x))⊤. Our main goal is to show that this vector tends to its deterministi... | https://arxiv.org/abs/2502.14621v1 |
T(2) n≤∥P∥2 ∞n. Together with (13) and (15), one gets ⟨M⟩(1,1) n n1+γa.s.⟶τ2µ(h(x)), establishing the expected result for coefficient ⟨M⟩(1,1) n. •Study of ⟨M⟩(2,2) n One has ⟨M⟩(2,2) n=n ∑ k=1Var(A(2) k∣Zk−1). Each term, Var(A(2) k∣Zk−1), of the sum is a function of Zk−1and is independent of n. This contrasts with its... | https://arxiv.org/abs/2502.14621v1 |
variance. Consequently, n(1−γ)/2R(2) ntends to 0, and n(1−γ)/2RnP⟶0. (19) A.4.2 Martingale term To establish the asymptotic normality of the martingale term Mn/nwith rate n(1−γ)/2, we rely on the central limit theorem for vector martingales [14, Corol- lary 2.1.10]. To this end, we need to check its conditions of appli... | https://arxiv.org/abs/2502.14621v1 |
Efficient Multivariate Robust Mean Estimation Under Mean-Shift Contamination Ilias Diakonikolas∗ University of Wisconsin-Madison ilias@cs.wisc.eduGiannis Iakovidis† University of Wisconsin-Madison iakovidis@wisc.edu Daniel M. Kane‡ University of California, San Diego dakane@cs.ucsd.eduThanasis Pittas§ University of Wis... | https://arxiv.org/abs/2502.14772v1 |
model [DKKLMS18; DKPP24]. Huber’s contamination is a rather strong model, as it allows for the unknown distribution of corruptions to be arbitrary. On the one hand, this level of generality makes the model quite powerful — by allowing it to cover a wider range of phenomena. On the other hand, the model definition is in... | https://arxiv.org/abs/2502.14772v1 |
consistency is achievable in the mean-shift model. Intuitively, this happens because now all the samples are convoluted with a Gaussian; thus, given infinitely many samples, one can form the underlying distribution, perform deconvolution, and recover the single spike corresponding to the inliers. More specifically, [KG... | https://arxiv.org/abs/2502.14772v1 |
overview of our techniques and outline the structure of our algorithm. In Section 2, we formally state the algorithm and sketch its analysis. Finally, in Section 3, we discuss directions for future work. 1.2 Our Techniques Given samples of the form x∼ N(m, I), where mis a random variable taking the value m=µwith probab... | https://arxiv.org/abs/2502.14772v1 |
by O(1/γ); this implies that the number of eigenvectors with eigenvalue larger than Cϵ2is at most O(1/(γϵ2)). Thus, if we take Vto be the span of such eigenvectors, we would have that ∥ProjV⊥(µ)∥ ≤ϵ/2anddim(V) =O(1/(γϵ2)). This procedure implements one round of our dimension reduction. For γ= 1/√ d, this reduces the di... | https://arxiv.org/abs/2502.14772v1 |
samples to average these away. To handle this, we need to use a slightly smaller value of γ, such as 1/p dlog(d). Notation We use Z+for the set of positive integers. We denote [n] ={1, . . . , n }. For a vector xwe denote by ∥x∥its Euclidean norm. Let Iddenote the d×didentity matrix (omitting the subscript when it is c... | https://arxiv.org/abs/2502.14772v1 |
{ProjVt(x) :x∈T′ t}. 21:UseTtto find bµ1∈ Vtwith∥bµ1−ProjVt(µ)∥ ≤ϵ. ▷Use Algorithm from Proposition 2.1 22:return ProjV⊥ t(bµ0) +bµ1. 5 2.1 Computationally Inefficient Multivariate Robust Estimator Proposition 2.1 below provides a robust multivariate mean estimator the mean-shift model that uses n=d2O(1/ϵ2)samples and ... | https://arxiv.org/abs/2502.14772v1 |
zz⊤e−∥z∥2 β√ k. This is shown in Lemma 2.3 below. The factor Zβ,kis the appropriate normalization factor that arises. 6 Lemma 2.3. LetV ⊆Rdbe a subspace of Rdof dimension k≤d. For any z∈ V, we have that Ex∼N(z,ΠV)[Fβ,k(x)]Zβ,k=zz⊤e−∥z∥2 β√ k+2, where Fβ,kandZβ,kare as defined in (1). The deterministic conditions that w... | https://arxiv.org/abs/2502.14772v1 |
1.1 with the assumption that ∥µ∥=O(1). Denote by z1, . . . , z αnthe adversarial centers in Definition 1.1. Then, with probability at least 1−δ,Tis(η,√logk)-good with respect to µ, z1, . . . , z αn. For the case β=ϵ, we can resort to a simpler proof which consists of calculating the variance of each entry of our random... | https://arxiv.org/abs/2502.14772v1 |
use that exp(−∥x∥2/√klogk)Z√logk,kis roughly less than exp(−∥zi∥2/√klogk). We can thus obtain that ∥T3i∥ ≤ ∥ zi∥2exp(−∥zi∥2/√klogk)≲√klogkand∥zi∥4exp(−2∥zi∥2/√klogk)≲klogk, which results in near-linear in ksample complexity. 2.3 Analysis of Algorithm 1 The lemma below is the core of the analysis for a single iteration ... | https://arxiv.org/abs/2502.14772v1 |
, T 1+T2, we have∥ProjV⊥ t+1(bµ0−µ)∥≲Pt t′=1√ηt′, where ηt′are the values set in lines 10 and 11. 3.Estimator for the remaining subspace : The lines 19-21 of Algorithm 1 find a vector bµ1∈ Vt such that ∥bµ1−ProjVt(µ)∥ ≤ϵ. Here we will sketch how each claim can be proved, with the full details in Appendix E. We start by... | https://arxiv.org/abs/2502.14772v1 |
overall sample complexity of the algorithm is n=n0+n1· T1+n2·(T2+ 1) = O(d) +dpolylog (d)ϵ−(2+o(1))+ 2O(1/ϵ2)polylog (d). Regarding runtime, the most computationally intensive part is the last step of the algorithm (Line 21), which has runtime τ= 2O(k)poly(n2, d), where the k= 1/ϵ2here denotes the dimension of the fina... | https://arxiv.org/abs/2502.14772v1 |
effect with outlier selection”. The Annals of Statistics 49.1 (2021). [CJ10] T. T. Cai and J. Jin. “Optimal rates of convergence for estimating the null density and proportion of nonnul effects in large-scale multiple testing”. The Annals of Statistics 38.1 (2010). [CS09] T. T. Cai and W. Sun. “Simultaneous testing of ... | https://arxiv.org/abs/2502.14772v1 |
STOC 2025. [DKLPP23] I. Diakonikolas, D. Kane, J. C. H. Lee, A. Pensia, and T. Pittas. “A Spectral Algorithm for List-Decodable Covariance Estimation in Relative Frobenius Norm”. Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023 . 2023. [... | https://arxiv.org/abs/2502.14772v1 |
Pittsburgh, PA, 2023. [LL22] A.LiuandJ.Li.“Clusteringmixtureswithalmostoptimalseparationinpolynomial time”.Proc. 54th Annual ACM Symposium on Theory of Computing (STOC) . 2022. [LRV16] K. A. Lai, A. B. Rao, and S. Vempala. “Agnostic Estimation of Mean and Covari- ance”.Proceedings of FOCS’16 . 2016. [LY20] Y. Liang and... | https://arxiv.org/abs/2502.14772v1 |
Additional Related Work Additional Related Work on Mean Shift Contamination The most closely related work to ours is [KG25]. They study the model of Definition 1.1 in one dimension and derive matching information theoretic upper and lower bounds for mean estimation. [KG25] also consider estimating the variance in the c... | https://arxiv.org/abs/2502.14772v1 |
that in the setting where σi≥σ, we can assume without loss of generality that all σiare exactly equal to σ. To see why, consider the random variable for a single outlier sample: xi=zi+N(0, σ2 i). Observe that we can rewrite xiasxi= (zi+N(0, σ2 i−σ2)) +N(0, σ2). Hence, by treating the outlier centers themselves as rando... | https://arxiv.org/abs/2502.14772v1 |
LY20; DLLZ23; CV24; DKLP25b]. This contamination model can also be viewed as a Gaussian mixture model. However, here each sample originates from its own component ( k=N). Importantly, the shared mean assumption enables meaningful results despite the large number of components. 17 B Additional Preliminaries Cover set of... | https://arxiv.org/abs/2502.14772v1 |
2.1. Denote by T={xi}n i=1, xi∈Rdanα-corrupted set of points from N(µ, I) under the model of Definition 1.1 and denote by Cthe the cover set of Corollary B.2. The algorithm is the following: First, using the algorithm from Fact 2.2, calculate a mvfor each v∈ Csuch that |mv−v⊤µ| ≤ε/8(see next paragraph for more details ... | https://arxiv.org/abs/2502.14772v1 |
cz,1 cI)[f(x)] =b E x∼N(1 cz,1 cI)[xx⊤]−1 cI! =b1 cI+1 c2zz⊤−1 cI 20 =b1 c2zz⊤ =e−∥z∥2 β√ k+2 1 1 +2 β√ k!k 2+2 zz⊤. Lemma 2.7. Letη∈(0,1)andk∈Z+denote the dimension, and assume kis bigger than a sufficiently large constant. There exists sample size n=klog3(k)(1/η)2+o(1)1 δsuch that the following holds: Let Tbe a s... | https://arxiv.org/abs/2502.14772v1 |
We decompose the matrix bAifori:xi∈T\Sinto three terms bAi= gig⊤ i−p klog(k)p klog(k) + 2I! + (zig⊤ i+giz⊤ i) +ziz⊤ i! e−∥xi∥2−k√ klog(k)1(Ei) =:bT1i+bT2i+bT3i, where by bT1i,bT2iandbT3iwe denote each of the terms that sum to bAiand by T1i, T2iandT3iwe denote their corresponding expectations. We will show concentration... | https://arxiv.org/abs/2502.14772v1 |
ziz⊤ i opE" e−2∥xi∥2−k√ klog(k)1(Ei)# ≤ ∥zi∥4e−2∥zi∥2√ klog(k)+O ∥zi∥√ log(k/η)√ klog(k) (1/η)o(1)(using Claim D.1) ≤klog(k) log2(k/η)eO(log(k/η)√ klog(k))(1/η)o(1). (by Claim D.2) ≤klog3(k) log2(1/η)eO(log(1/η)√ klog(k))(1/η)o(1) ≤klog3(k)(1/η)o(1). Hence, from Fact B.4, we have that Pr 1 αnX i:xi∈T\S bT3i−T3i ... | https://arxiv.org/abs/2502.14772v1 |
same as in (4). First, we will use bA=1 |S|X i:xi∈SbAiwhere bAi=Fβ,k(xi)e√ k/log(k)1(Ei). The points xi∈Sare all drawn from the same Gaussian component N(µ, I). Fix v∈Rk:∥v∥= 1. We bound the Lpnorm of the random variable v⊤ bAi−E[Ai] vfori:xi∈Sas follows: v⊤ bAi−E[Ai] v Lp(8) ≲ v⊤bAiv Lp(by triangle inequality and ... | https://arxiv.org/abs/2502.14772v1 |
standard basis of Rk. By Lemma 2.3 we have that E[∆kℓ] = 0and by Claim D.3 (applied with β=ϵ), we have that Var[∆st]≤1 (αn)2X i:xi∈T\SVar[e⊤ s(bAi−A)et]≲e4/ϵ2d αn. Thus, by Chebyshev’s inequality, with probability at least 1−τ, we have that |∆st| ≤r Var[∆st] τ≲s e4/ϵ2k ατn The right hand side becomes less than η/kwhen ... | https://arxiv.org/abs/2502.14772v1 |
step above, which claims that (1 +2 β√ k)k+4/(1 +4 β√ k)k/2≲e4/β2: First note that since k≥1/β2we have that (1 +2 β√ k)4≤34=O(1), thus it suffices to prove that (1 +2 β√ k)k/(1 +4 β√ k)k/2≲e4/β2. Towards this end, we will use the fact that ex≤(1 +x/n)n+x/2 for all x, n > 0. Applying this with n=k/2andx= 2√ k/β, we have... | https://arxiv.org/abs/2502.14772v1 |
η e3C2log(k η)√ klog(k). Proof of iii Similarly to the previous inequality, let f(x) =x4e−x2√ klog(k)+Cxr log(k η)√ klog(k). Aslimx→+∞f(x) =−∞andf(0) = 0andfcontinuous on (0,+∞), we have that the maximum must occur on a point in (0,+∞)where the derivative is 0either it is upper bounded than 0. Also, we have that f′(x)... | https://arxiv.org/abs/2502.14772v1 |
10. 3.Phase 2 (1/ϵ2≤k≤Clog4(d)/ϵ5): (a)LetE′ tbe the event that the set Ttfrom line 12, after the transformation of line 13 is (ηt, ϵ)-good (cf. Definition 2.6) with respect to ˜µt,˜z(t) 1, . . . , ˜z(t) α, where ηtis the parameter set in 11, ˜µ=ProjVt(µ−bµ0)and˜z(t) 1, . . . , ˜z(t) αare some vectors in Vt. Then, with... | https://arxiv.org/abs/2502.14772v1 |
of Corollary 2.12 in [DK23]. Without loss of generality, we apply this corollary with the fraction of outliers being α= Ω(1)since we can always treat some of the inliers as outliers in the model of Definition 1.1. That corollary yields that n0=O(d)samples suffice. 34 Proof of Item 2b We claim that each iteration of Pha... | https://arxiv.org/abs/2502.14772v1 |
by an application of Lemma 2.8 with probability of failure δ= 10−5/log(log(d)/ϵ). Note that the requirement of that lemma that k > 1/ϵ2holds since the algorithm would exit entirely the while loop of line 9 otherwise. Since the dimension is k≤Clog4(d)/ϵ5during this phase of the algorithm, and ηt=O(ϵ/√ k), the sample com... | https://arxiv.org/abs/2502.14772v1 |
in one dimension, [KG25] has shown that consistent estimation with arbitrarily small error ϵ using the information theoretic optimal of 2Θ(1/ϵ2)samples is only possible when c >2−Θ(1/ϵ2), thus we only consider that regime in this section. Theorem F.1. (Higher breakdown point) Letd∈Z+denote the dimension, µ∈Rdbe an unkn... | https://arxiv.org/abs/2502.14772v1 |
1/δblowup in the sample complexity, we can follow the analysis done in Appendix E to obtain a high probability conclusion for the final error of the algorithm. 37 Now we will use Claim F.2 along with Lepskii’s method to get an error guarantee that depends on the true parameter γwithout it being known to the algorithm. ... | https://arxiv.org/abs/2502.14772v1 |
An Adversarial Analysis of Thompson Sampling for Full-information Online Learning: from Finite to Infinite Action Spaces Dedicated to the memory of David Draper Alexander Terenin Cornell University Jeffrey Negrea University of Waterloo and the Vector Institute Abstract We develop a form Thompson sampling for online lea... | https://arxiv.org/abs/2502.14790v4 |
online learning with three experts, namely X={1,2,3}andY={y∈ ℓ∞(X) :∥y∥∞≤1}—and discovered the learner’s optimal strategy to be a particular form of Thompson sampling: the learner should choose arms with probability proportional to them being best-in-hindsight under a certain maximin-optimal Bayesian prior distribution... | https://arxiv.org/abs/2502.14790v4 |
this view, our results are closely-related to the relax-and-randomize perspective of Rakhlin et al. (2012), but do not rely on Rademacher-complexity-theoretic assumptions on the perturbations, and lead to an easier-to-check condition relating the adversary’s function class and perturbation distribution. Our results can... | https://arxiv.org/abs/2502.14790v4 |
adversary’s decision space Y, matches the minimax regret min pmax qR(p, q). (3) For finite expert classes, the minimax expected regret is Θ(√TlogN). In this setting, the most well-known strategy achieving the minimax rate is the exponential weights algorithm, also known as hedge(Littlestone and Warmuth, 1994; Vovk, 199... | https://arxiv.org/abs/2502.14790v4 |
a learning rate term, which makes it trust the data less. In light of this, somewhat surprisingly, Gravin et al. (2016) show that the exact Nash equilibrium of discounted online learning with three experts admits a Bayesian interpretation: the learner’s algorithm is a form of Thompson sampling. At each round, the learn... | https://arxiv.org/abs/2502.14790v4 |
have R(p, y) =R(p, q(γ)) +TX t=1Γ∗ t+1(y1:t)−Γ∗ t(y1:t−1)− ⟨yt|pt⟩+E⟨γt|p(γ) t⟩ Eq(γ)(p,y)(5) where p(γ) tis a copy of the learner’s strategy applied to simulated data γ1:t−1drawn from the prior q(γ)instead of the y1:t−1. We call R(p, q(γ))theprior regret andEq(γ)theexcess regret. All proofs are in Appendix A. A sketch... | https://arxiv.org/abs/2502.14790v4 |
random variables foreachexpertleadsto R(·, q(γ)) = Θ(√TlogN)bystandardarguments—indeed, tightupper bounds on R(·, q(γ))can be proven even in infinite-expert settings by chaining techniques (Talagrand, 2005). Thus, ifonecanprovethat Eq(γ)(p, y)≤ O(√TlogN)—or, inmoregeneral settings, whatever the corresponding rate is—th... | https://arxiv.org/abs/2502.14790v4 |
probabilistic view, has significant consequences from a convex-analytic one: using the integral form of Taylor’s Theorem for convex functions, one can write the resulting Bregman divergences as DΓ∗ t(y1:t||y1:t−1) =1 2Z1 0∂2 yt,ytΓ∗ t(y1:t−1+αyt) dα (10) where ∂2 (·,·)Γ∗ tis the (distributional) second Gâteaux derivati... | https://arxiv.org/abs/2502.14790v4 |
integration by parts, which one can derive in the infinite-dimensional setting from the Cameron–Martin Theorem, one can show ∂2 u,vΓ∗ t(f) =1√ T−t+ 1Eu(x∗ f+γt:T) γt:T K−1v (12) where K−1is the inverse (on appropriate subsets) of the covariance operator K:Ms(X)→ C(X;R)defined by µ→R Xk(x,·) dµ(x), and also known as the... | https://arxiv.org/abs/2502.14790v4 |
functions. In this situation, one can take kto be of equal variance, and immediately obtain the following general bound on the regret of Thompson sampling under a Gaussian process prior, in the sense of Definition 1, as a strategy for the online learning game. Theorem 9. LetX⊆Rdbe compact, and let Yconsists of bounded ... | https://arxiv.org/abs/2502.14790v4 |
are working in the finite-horizon setting. This shows that, even though the game-theoretic analysis of Gravin et al. (2016) may at first seem very limited— applying only to X={1,2,3}, to the infinite-horizon discounted setting, and to exact Nash equilibria—the key idea of using a perturbation sequence derived from an a... | https://arxiv.org/abs/2502.14790v4 |
24 of Appendix A, for this family of kernels on X= [0,1]d, our prior regret is R(p, q)≤16σvuutTdlog 1 +√ d κ! . (20) In the aforementioned random-walk scaling limit, the asymptotic variance is Θ(β2)and the asymptotic length scale is Θ(λ−1), suggesting a lower bound on the minimax regret of Ω βq Tdlog(1 +√ dλ β) . In ... | https://arxiv.org/abs/2502.14790v4 |
techniques of Abernethy et al. (2014, 2016) and Orabona (2019)—and, in the process, removes learning rate schedule restrictions from these approaches, which would prohibit the analysis of Thompson sampling and related algorithms which correspond to FTPL with increasing learning rates. We also extend the techniques for ... | https://arxiv.org/abs/2502.14790v4 |
H∞Optimal Control and Related Minimax Design Problems: a Dynamic Game Approach . Springer, 2008. Cited on page 21. J. F. Bonnans and A. Shapiro. Perturbation Analysis of Optimization Problems . Springer, 2013. Cited on page 21. N. Cesa-Bianchi and G. Lugosi. Prediction, Learning, and Games . Cambridge University Press,... | https://arxiv.org/abs/2502.14790v4 |
Minimax Optimal Quantile and Semi-adversarial Regret via Root-logarithmic Regularizers. Advances in Neural Information Processing Systems , 2021. Cited on page 4. F. Orabona. A Modern Introduction to Online Learning. arXiv:1912.13213 , 2019. Cited on pages 2, 3, 6, 8, 14. A. Rakhlin, K. Sridharan, and A. Tewari. Online... | https://arxiv.org/abs/2502.14790v4 |
well-defined. Throughout this work, all topological vector spaces we consider are assumed Hausdorff. Definition 12 (General-state online learning game) .LetXbe second-countable compact Hausdorff. Let Y⊂ Y ⊆ C(X;R). Let Ms(X)be the space of signed finite measures on X, and let M1(X)⊆ M s(X)be the corresponding subset of... | https://arxiv.org/abs/2502.14790v4 |
Now, we add and subtract E⟨γt|p(γ) t⟩from both sides to get R(p, y) =E" Γ∗ 1(0)−TX t=1⟨γt|p(γ) t⟩# R(p,q(γ))+TX t=1Γ∗ t+1(y1:t)−Γ∗ t(y1:t−1)− ⟨yt|pt⟩+E⟨γt|p(γ) t⟩ Eq(γ)(p,y). (30) The claim follows. 19 Terenin and Negrea The above claim generalizes to the case where, instead of the best-in-hindsight point, one instead ... | https://arxiv.org/abs/2502.14790v4 |
our Gaussian process priors—the function Γ∗ tof interest defines a bounded linear Gâteaux derivative. Compare this with the situation that would occur for exponential- weights-type algorithms in our setting: in contrast with Γ∗ t, the usual Kullback–Leibler divergence p7→DKL(p||p0), for a given p0∈ M 1(X), is not smoot... | https://arxiv.org/abs/2502.14790v4 |
it is non-decreasing—for all y∈ Y. We will take our rule to be equivalent to α•ifsis a continuity point of V(α•) y,v(s). We now need only define the rule for discontinuity points of V(α•) y,v. Letsdbe a discontinuity point of V(α•) y,v. Since Xis sequentially compact, every sequence inXhas a convergent subsequence: thu... | https://arxiv.org/abs/2502.14790v4 |
is to establish a form of Fenchel duality induced by Γ∗ t. To focus attention on regret-theoretic aspects, and given the generality of our setting, we adopt a minimalist approachwhichconsistsofderivingjust-enoughconsequencesofthisdualityforourcalculation to go through. To do so, let C(X;R)∗be the topological dual of C(... | https://arxiv.org/abs/2502.14790v4 |
An Adversarial Analysis of Thompson Sampling Proof.By definition Eq(γ)(p, y) =TX t=1Γ∗ t+1(y1:t)−Γ∗ t(y1:t−1)− ⟨yt|pt⟩+E⟨γt|p(γ) t⟩ Et. (57) We bound this term-by-term. Recall that, Using this, write Et= Γ∗ t+1(y1:t)−Γ∗ t(y1:t−1)− ⟨yt|pt⟩+E⟨γt|p(γ) t⟩ (58) (i)= Γ∗ t+1(y1:t)−Γ∗ t(y1:t−1)−E⟨yt−γt|pt⟩ (59) =E γt∼q(γ) t⟨yt... | https://arxiv.org/abs/2502.14790v4 |
and use this assumption to take limits of both sides: this gives ∂vV(θ)≤ L 0(x∗ v(θ), v) =∂vL(x∗ v(θ), θ) (83) where the final equality is the expression for the Gâteaux derivative of an affine function. Combining this expression with the Lemma 20, we conclude that the respective inequality is tight, and that each x∗ v... | https://arxiv.org/abs/2502.14790v4 |
< µ≤u. Lemma 23. Letz∼TN(µ,Σ;−∞, α)where α∈Rd, and Σis strictly positive definite. Then E(z) =µ−Σ(pzi(αi))n i=1 (92) where pziis the marginal probability density of zi. Proof.Kotz et al. (2000, Chapter 45). We are now ready to prove the key Hessian bound. Theorem 8. Suppose that q(γ)=GP(0, k)is IID over time, where kis... | https://arxiv.org/abs/2502.14790v4 |
assumption, and final equality follows by the derived identity together with positivity of p(f+γ)(·)(α). Collecting terms, we obtain DΓ∗ t(y1:t||y1:t−1)≤1 2√ T−t+ 1Z1 0E(y(xi)2+CY,k|y(xi)|)γ(xi) k(xi, xi)dθ(113) +O r hlog1 h! +O(ωt(h)) (114) and the claim follows by taking h→0. We are now ready to prove the main regret... | https://arxiv.org/abs/2502.14790v4 |
with variance σ2 and length scale κ. Then Esup x∈[0,1]dγ(x)≤16σvuutdlog 1 +√ d κ! . (120) Moreover, γadmits a expected global modulus of continuity of ψ(h) =Oq hlog 1 h . Proof.Note first that the result for general σ >0follows immediately from the case σ= 1, so assume this without loss of generality. By the Dudley ... | https://arxiv.org/abs/2502.14790v4 |
a few key modifications worth noting: 1.The space over which the supremum is taken is the rectangular strip defined by ∥x−x′∥ ≤h, therefore the upper limit of the Dudley entropy integral is given by half the maximal standard deviation, which, for ∆, isq 1−exp −h κ ≤q h 2κ. 33 Terenin and Negrea 2.The covariance satis... | https://arxiv.org/abs/2502.14790v4 |
arXiv:2502.14941v1 [math.CT] 20 Feb 2025Categorical algebra of conditional probability Mika Bohinen and Paolo Perrone University of Oxford 2025 In the field of categorical probability, one uses concepts an d techniques from category theory, such as monads and monoidal categorie s, to study the structures of probability ... | https://arxiv.org/abs/2502.14941v1 |
zero- one laws [FR20], and the ergodic decomposition theorem [EP2 3]. In statistics, Markov categories have formalized concepts like sufficient statist ics [Fri20] and, through their connection with probability monads [Jac18], led to new insi ghts into Blackwell’s theo- rem on statistical experiments [FGPR23]. A central ... | https://arxiv.org/abs/2502.14941v1 |
which is still finer or equal than the “partition” induced by a statistical experiment . (This idea of “partition induced by a function” may remind the read er ofdescent, but we leave a more thorough investigation of this analogy to future work .) 3 Thestructuresand techniques of categorical algebra reach far beyondweak... | https://arxiv.org/abs/2502.14941v1 |
Example 2.2. The category Stochis specified via the following data. •Objects are measurable spaces, i.e., pairs ( X,A) whereX∈SetandAis a sigma- algebra on X; •Morphisms ( X,A)→(Y,B) are Markov kernels of entries k(B|x), forx∈Xand B∈B. That is to say,, k(B|−) is a measurable function X→Rfor allB∈B, andk(−|x) is a probab... | https://arxiv.org/abs/2502.14941v1 |
2.10. A monad (P,µ,δ)onDis said to be affineifPI∼=I. Thus, ifPis affine and Iis terminal, then PI∈DPis also terminal and we get the following result. Corollary 2.11. Let(P,µ,δ)be a symmetric monoidal affine monad on a Markov cat- egoryD. Then the Kleisli category DPis again a Markov category in a canonical way. 7 Example 2.1... | https://arxiv.org/abs/2502.14941v1 |
morphism p:S→Xfor someS∈C. By abuse of notation we will also write p∈Xwhenpis a generalized element. PuttingC=SetandS={∗}recovers the usual notion of elements in a set. Indeed, we are particularly interested in the case where Cis a Markov category and pis a state. Definition 2.21. Let(T,µ,η)be a monad on some category C... | https://arxiv.org/abs/2502.14941v1 |
the Kleisli category CTis a weak pullback if and only if its image under the right-adj oint R:CT→Cis. 11 Proof.Consider the bijection given by the Kleisli adjunction, CT(X,A) =CT(LX,A)C(X,RA) =C(X,TA) f f#∼= whereLandRdenote the left- and right-adjoints. (Note that on objects, LX=Xand RA=A.) By naturality of the biject... | https://arxiv.org/abs/2502.14941v1 |
these mixtures of measures: 13 1 3[N(−2,1)] +1 3[N(0,1)] +1 3[N(2,1)]1 3[N(−2,1)] +2 3/bracketleftBigg 1 2N(0,1) +1 2N(2,1)/bracketrightBigg ∈PPX ∈PPX In particular, given a morphism p:S→PX(or a Kleisli morphism p♭:S→X), we can view a morphism d:S→PPXsuch thatµ◦d=p(or equivalently a Kleisli morphism d♭:S→PXsuch that sa... | https://arxiv.org/abs/2502.14941v1 |
therefo re update our belief to the following distribution, p(θ|x= 1): pp(−|x= 1) 1✓2✗3✗ f 15 •The excluded regions (2) and (3) have now probability zero; •The confirmed region (1) has now probability one; •Within region (1), and only within there, we keep the probabi lity proportional to the old measure p– but we need ... | https://arxiv.org/abs/2502.14941v1 |
partial generalization, with a similar intuition. Definition 2.35. Let(Θ,p)be a probability space in a representable Markov category, and letf: Θ→Xbe an a.s. deterministic statistical experiment. The hypernormal- ization ofpwith respect to f, if it exists, is the standard measure ˆfponPΘ. The construction given in [Jac1... | https://arxiv.org/abs/2502.14941v1 |
f: Θ→Xandg: Θ→Yon(Θ,p)with ˆfp=π,ˆgp=τ, andf≥gin the Blackwell order. We will use the following auxiliary statement. Lemma 2.41. Letπ:I→PΘ, denotesamp◦π:I→Θbyp, and consider the Bayesian inversesamp† π: Θ→PΘas a statistical experiment on (Θ,p). Then its standard measure is exactlyπ. Proof of Lemma 2.41. Using the usual... | https://arxiv.org/abs/2502.14941v1 |
so, by the right equality in (14), we get exactly the sampling cancellation property (9). Conversely, supposethat Cis a.s.-compatibly representable (andhas conditionals). Letp:A→X,f,g:X⊗A→Y,π:A→PXsuch that samp◦π=p, and suppose that the right equality in (14) holds. To prove the le ft equality, by the sampling cancella... | https://arxiv.org/abs/2502.14941v1 |
( X,p) are almost surely equal if and only if they agree on a set of probability one. This condition is equivalent to say that the equalizer of fandghas full measure, or again equivalently, that the measure p, seen as a kernel I→X, factors through the equalizer of fandg. With this example in mind,let’s definethe followi... | https://arxiv.org/abs/2502.14941v1 |
map r:A→X×ZY inCsuch that ρ= (f∗g,g∗f)◦r, i.e. such (f∗g)◦r=pand (g∗f)◦r=q. 3.3. The universal property of hypernormalizations The purpose of this section is to make the following intuitio n precise: •Consider a deterministic experiment fon (Θ,p), which we can consider a “parti- tion” or “coarse-graining”; •Form the st... | https://arxiv.org/abs/2502.14941v1 |
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