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estimation. More formality is given in Section 5. Typicality Principle (intuitive version) .If data xis sufficiently atypical relative to the posited model with parameter θ, then the hypothesis H={θ}is unwarranted. The thoughtful reader is surely tempted to make a comparison between the typicality principle as stated a... | https://arxiv.org/abs/2501.14860v1 |
divergence (Lehmann, 1983; Pardo, 2018)) to the empirical distribution of the data. Since the latter empirical distribution necessarily represents the data, it might come as a surprise to hear that xmight not look like a typical sample from the “best approximation” Pˆθ(x)in the posited model. The point is that the maxi... | https://arxiv.org/abs/2501.14860v1 |
common penalty functions do not address the question of whether xis a “typical” sample from Pθ, and nor can they; any measure that is designed to quantify typicality must depend on both xandθ. Since the focus is on determining if xis atypical relative to Pθ, Tukey’s insights on model-checking suggest considering some v... | https://arxiv.org/abs/2501.14860v1 |
sense. To see this, set µequal to any of the observed data points, say, x1, and consider the likelihood 7 as a function of σ2alone: Lx(x1, σ2)∝nY i=1h (1−α) expn −(xi−x1)2 2o +ασ−1expn −(xi−x1)2 2σ2oi ={(1−α) +ασ−1}nY i=2h (1−α) expn −(xi−x1)2 2o +ασ−1expn −(xi−x1)2 2σ2oi . The second factor, i.e., the product over i= ... | https://arxiv.org/abs/2501.14860v1 |
take a look at “a more disturbing example” (Le Cam 1990), due to Neyman and Scott (1948). In this case, the model Pθ, indexed by θ= (ξ1, . . . , ξ n, σ2)∈Rn×(0,∞), posits that the data consists of independent pairs, Xi= (Xi1, Xi2), where Xi1, Xi2ind∼N(ξi, σ2), i= 1, ..., n. 8 (a) Effect of σ2 (b) Small σ2 (c) Large σ2 ... | https://arxiv.org/abs/2501.14860v1 |
typicality principle resolves the Neyman–Scott paradox and, more general, the shortcomings of the principle of maximum likelihood. 4.3 Stein’s mean vector length Consider the classical problem in which Xis an-dimensional normal random vector with unknown mean vector Θ and identity covariance matrix. Inference on the me... | https://arxiv.org/abs/2501.14860v1 |
error compared to the maximum marginal likelihood estimator. The Bayesian literature offers a non-informative prior for Θ designed specifically for inference on Φ (Tibshirani 1989). This penalizes large ϕval- ues, which makes the corresponding (profiled) maximum a posteriori estimator appealing. But this prior has a st... | https://arxiv.org/abs/2501.14860v1 |
Fisher’s fiducial inference (e.g., Fisher 1933, 1935a; Zabell 14 1992) and its generalizations (e.g., Fraser 1968; Hannig et al. 2016; Xie and Singh 2013), Dempster–Shafer theory (e.g., Dempster 1966, 2008; Shafer 1976, 1982), and inferential models (e.g., Martin 2015, 2021a, 2024; Martin and Liu 2013, 2015a). One thin... | https://arxiv.org/abs/2501.14860v1 |
course, it’s possible that τX(θ) is small, and this latter event should likewise have small probability. So, when we say that “ xis atypical relative to θ,” so that τx(θ) is small, we mean that a Pθ-rare has occurred. Then it makes sense to require that τX(θ) have a Unif(0,1) distribution, or at least be stochastically... | https://arxiv.org/abs/2501.14860v1 |
ways to the IM’s assessment of the Θ-dependent, to-be-incurred loss associated with an action. Again, the details are beyond our present scope. 16 5.2 Putting the principle into practice In light of the developments discussed in Section 3, there is a conceptually straightforward path to getting from the typicality prin... | https://arxiv.org/abs/2501.14860v1 |
consult Jiang et al. (2023) and Martin (2025). 17 5.3 Relation to other statistical principles The most familiar statistical principle is the likelihood principle (e.g., Basu 1975; Berger and Wolpert 1984; Birnbaum 1962), which states that all of what is relevant in the data for inference on Θ is captured by the shape ... | https://arxiv.org/abs/2501.14860v1 |
deciding when to stop collecting data and begin the analy- sis; the effect of the stopping rule cancels in the likelihood ratio, but clearly each stopping rule determines its own unique probability model.) The above modification makes crys- tal clear the cost of satisfying the likelihood principle: since τx(θ)≤τlp x(θ)... | https://arxiv.org/abs/2501.14860v1 |
θ∈T, so that our proposed notion of typicality here can be directly tied to the prediction of predictable quantities emphasized in Martin and Liu (2014). 5.4 Stein’s mean vector length, again To illustrate the broader, typicality-motivated uncertainty quantification strategy de- scribed in Section 5, we revisit Stein’s... | https://arxiv.org/abs/2501.14860v1 |
structure in (3), is that the proposed typicality contour merges with the high-quality marginal likelihood-based solution in Martin (2023b) as λincreases. 6 Conclusion Motivated by the deep philosophical considerations and scientific attitudes of Popper and Tukey, here we advanced a new typicality principle that has a ... | https://arxiv.org/abs/2501.14860v1 |
parameter λin (1). While there are so many now-standard tuning parameter selection strategies available, a relevant question is if the data-dependence inherent in our typicality-based penalty war- rants new tuning parameter selection considerations. After all, compared to the usual sparsity-encouraging penalties, p-val... | https://arxiv.org/abs/2501.14860v1 |
J. Amer. Statist. Assoc. , 65(329):395–398. Edwards, A. W. F. (1992). Likelihood . Johns Hopkins University Press, Baltimore, MD, expanded edition. Revised reprint of the 1972 original. Eschker, S. J. and Liu, C. (2024). Towards strong ai: Transformational beliefs and scientific creativity. arXiv preprint arXiv:2412.19... | https://arxiv.org/abs/2501.14860v1 |
First results. arXiv:2203.06703 . Martin, R. (2022b). Valid and efficient imprecise-probabilistic inference with partial priors, II. General framework. arXiv:2211.14567 . Martin, R. (2023a). Fiducial inference viewed through a possibility-theoretic inferential model lens. In Miranda, E., Montes, I., Quaeghebeur, E., an... | https://arxiv.org/abs/2501.14860v1 |
Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA. Philosophy and principles of data analysis: 1949–1964, Edited and with comments by Lyle V. Jones, With a biography of Tukey by Frederick Mosteller. van der Vaart, A. (2002). The statistical work of Lucien Le Cam. Ann. Statist. , 30(3):631– 682. De... | https://arxiv.org/abs/2501.14860v1 |
arXiv:2501.14928v1 [cs.LG] 24 Jan 2025Decision Making in Changing Environments: Robustness, Query-Based Learning, and Differential Privac y Fan Chen fanchen@mit.eduAlexander Rakhlin rakhlin@mit.edu January 28, 2025 Abstract We study the problem of interactive decision making in which the underlying environment changes o... | https://arxiv.org/abs/2501.14928v1 |
Huber ,1965]) or mis-specification (as in agnostic learning) directly as a constraint on the environment to be c lose to a ground-truth model. In turn, the local privacy constraint can be formulated as a restrict ion on the decision-maker to only observe information through differentially private channels. Our approach b... | https://arxiv.org/abs/2501.14928v1 |
optimal query complexity of SQ learning ofdistribution search problems . Not surprisingly, we show that there is quantitative equiva lence between the SQ dimension 2 and our SQ DEC ( Section 4.2 ). Therefore, our results extend the characterizations of Feldman [2017] to general query-based learning problems. •Local-min... | https://arxiv.org/abs/2501.14928v1 |
locally private learning for various statistical estimati on problems [ Duchi et al. ,2013,2018,Duchi and Rogers ,2019], including mean estimation [ Asi et al. ,2022,2024], functional estimation [ Rohde and Steinberger ,2020,Butucea and Issartel ,2021,Butucea et al. ,2023,Duchi and Ruan ,2024], hypothesis testing [ Ber... | https://arxiv.org/abs/2501.14928v1 |
We start this section by formulating the hybrid DMSO framework ( Section 2.1 ), a generalization of the Decision Making with Structured Observation (DMSO) f ramework proposed by Foster et al. [2021]. We then show how this generalization encompasses query-b ased learning ( Section 2.2 ), locally differentially private le... | https://arxiv.org/abs/2501.14928v1 |
be specified by constraint P⋆={M⋆}and Psto={{M⋆}:M⋆∈M} , and •adversarial DMSO framework [ Foster et al. ,2022b ] (detailed in Appendix A.2 ), where the environment is fully adversarial, i.e., the constraint is P⋆=MandPadv={M} . Further examples of hybrid DMSO include SQ DMSO (Section 2.2 ), where the environment is all... | https://arxiv.org/abs/2501.14928v1 |
Let us now discuss the qualitative behavior of p-decH ε(P)with respect to the constraint class P. To start, consider stochastic DMSO, where each constraint i s given by a singleton P={M}. In this case, the infimum over M∈co(P)disappears, recovering the definition of the original PAC DEC in Foster et al. [2023b ] (see als... | https://arxiv.org/abs/2501.14928v1 |
olds that p-decτ-SQ ε(T)(M)/lessorsimilarinf Algsup EnvEEnv,Alg[Risk DM(T)]/lessorsimilarp-decτ-SQ ¯ε(T)(M), where supEnvis taken over all environments satisfying query correctnes s with tolerance τfor a modelM⋆∈M ,ε(T)≍1√ T,¯ε(T)≍/radicalBig log|M| T. 1The class of all stochastic models is given by ( Π→∆(V)), correspo... | https://arxiv.org/abs/2501.14928v1 |
we define private PAC-DEC at ĎMas p-decLDP ε(M,ĎM):= inf p∈∆(Π) q∈∆(Π×L)sup M∈M/braceleftbig Eπ∼p[L(M,π)]|E(π,ℓ)∼qD2 ℓ(M(π),ĎM(π))≤ε2/bracerightbig ,(7) and the private PAC-DEC of Masp-decLDP ε(M) = sup ĎM∈co(M)p-decLDP ε(M,ĎM). Theℓ-divergenceis a measure of closeness of two distributions that is weaker th an the Helli... | https://arxiv.org/abs/2501.14928v1 |
the ground truth model M⋆belongs to a given model class M⊆ (Π→∆(O)). In the formulation above, the environment is allowed to be a daptive, mak- ing the learning task harder than the Huber contamination mo del [Huber ,1965,1992], where the environment is stationary , i.e.,M1=···=MT= (1−β)M⋆+βM′for an arbitrary but fixed ... | https://arxiv.org/abs/2501.14928v1 |
regret DEC ofMis then defined as r-decc ε(M) = sup ĎM∈co(M)r-decc ε(M∪{ ĎM},ĎM). Next, to define the regret DEC of a constraint class P, we define MP:=/uniondisplay P∈Pco(P), r-decH ε(P):=r-decc ε(MP). We show that the regret DEC of Pprovides both lower and upper bound for the minimax regret. Theorem 5 (Regret lower and u... | https://arxiv.org/abs/2501.14928v1 |
adaptive environment Envis specified by a sequence of mappings {µt}t∈[T], where the t-th mapping µt(·|H′ t−1)specifies the distribution of the model Mtbased on the full-information history H′ t−1= (Ms,πs,os)s≤t−1. An environment is constrained by Pif 12 there existsP⋆∈Psuch thatµt(·|H′ t−1)is always supported on P⋆for al... | https://arxiv.org/abs/2501.14928v1 |
as sup EnvEEnv,Alg[Risk DM(T)]≥1 2p-decH ε(T)(P), whereTε(T)2≍p-decH ε(T)(P). (16) For a problem with p-decH ε(P)≍ε,Eq. (16) gives a lower bound of Ω/parenleftbig1 T/parenrightbig . While this is worse than the Ω/parenleftBig 1√ T/parenrightBig lower bound provided by (14)under metric-based loss, such a worse lower bou... | https://arxiv.org/abs/2501.14928v1 |
constraint class. Suppose that Assumption 3 holds for the value function V. Then, for any T-round algorithm Alg, sup EnvEEnv,Alg[RegDM(T)]≥sup Envsup π⋆∈ΠEEnv,Alg/bracketleftBiggT/summationdisplay t=1VMt(π⋆)−VMt(πt)/bracketrightBigg (19) ≥T 8/parenleftbigg r-decc ε(T)(MP)−6CVε(T)−Vmax T/parenrightbigg (20) where the su... | https://arxiv.org/abs/2501.14928v1 |
i∈[m] such thatM∈M(i). While we can frame the hypothesis selection problem within s tochastic DMSO (with Pstocorre- sponding toM), the upper bound provided by DEC theory scales with log|M|, the complexity of model class, which is undesirable. On the other hand, when th e subclassesM(1),···,M(m)are con- vex, we can alte... | https://arxiv.org/abs/2501.14928v1 |
of MΠ. Following this idea and using a slightly more careful insta ntiation of ExO+, we have the following upper bounds. Proposition 13. LetT≥1,δ∈(0,1),∆≥0, and we consider the reward-based no-regret learning task ( Example 1 ) with a model class M. Suppose thatMis compact ( Assumption 2 ), and the regret DEC r-decc ε(... | https://arxiv.org/abs/2501.14928v1 |
guarantees for SQ lea rning, as we discuss in Section 4.1 andSection 4.2 . 4.1 General query oracles and DEC theory for query-based lea rning Extending our discussion on SQ learning, we can formulate an y SQ DMSO problem as a learning problem under certain query oracles. Specifically, given a m easurement class Φand a m... | https://arxiv.org/abs/2501.14928v1 |
Appendix G.2 ). By instantiating Theorem 8 , we also have the upper bound of ExO+. Theorem 15 (Query-based upper bound) .LetT≥1,δ∈(0,1), model classM⊆ (Π→V ). Then, for any model M∈Mdand given access to any (possibly adaptive) GQ oracle GQτ MofM, theSQ-E2D (Algorithm 4 ) achieves with probability at least 1−δthat Risk ... | https://arxiv.org/abs/2501.14928v1 |
above char- acterization, we first show that the SQ dimension is quantita tively equivalent to the SQ DEC of M, as long as the Minimax theorem applies. Proposition 17. Suppose thatZis finite, andMd⊆∆(Z)is a distribution class. Then for any success probability β∈[0,1], reference model ĎM∈∆(Z), we have p-decτ-SQ ε(Md,ĎM)>1... | https://arxiv.org/abs/2501.14928v1 |
DEC formulation to analyze pri vate DMSO and characterize the complexity of LDP learning. Problems encompassed by private DMSO. Before diving into details, we first discuss several common settings of private learning that are encompassed by private DMSO (page 8). Recall that in this setting, the learner selects, on roun... | https://arxiv.org/abs/2501.14928v1 |
in Section 2.3 , private PAC- DEC can be viewed as a special case of the hybrid DEC, based on t he following characterization of the data-processing under DP channels. We recall that for an y channel Q∈Qand any distribution P∈∆(Z), we denote Q◦Pto be the marginal distribution of ounderz∼P,o∼Q(·|z). The proof of the fol... | https://arxiv.org/abs/2501.14928v1 |
function for this task is reward-based, in the sens e ofDefinition 9 , if we set the reward function as R((x,y),f) = 1−L(y,f(x)). The choices of loss function Lof interest include (1) squared loss: Lsq(y,y′) = (y−y′)2, and (2) absolute loss: Labs(y,y′) =|y−y′|. We also note that the classification task is a special case ... | https://arxiv.org/abs/2501.14928v1 |
ell-conditioned [ Duchi et al. ,2018, Duchi and Ruan ,2024]. Otherwise, the convergence rate can degrade to T−1/2in the worst case, as indicated by the following folklore lower bound [ Duchi and Ruan ,2024,Li et al. ,2024]. Lemma 25. Suppose that d= 1, andνis a given distribution over [−1,1]. Then for any T-round α-LDP... | https://arxiv.org/abs/2501.14928v1 |
(34) 27 where ¯ε(T) =/radicalBig log(|M|/δ) α2T. Further, suppose that the private regret-DEC r-decLDP ε(co(M))is of moderate decay. Then an alter- native instantiation of LDP-ExO (as detailed in Appendix F.5.2 ) achieves with probability at least 1−δ 1 TRegDM(T)≤∆ +O(/radicalbig logT)·/bracketleftBig r-decLDP ¯ε′(T)(c... | https://arxiv.org/abs/2501.14928v1 |
channel choice) contextual bandit instances with mean reward function f, and we let Pcxt:={Pf:f∈F} . Then, the contextual bandits problem with function class Fcan be framed within hybrid DMSO with constraint class Pcxt. Regret guarantees. We show that LDP-ExO achieves a regret bound scaling with the private regret-DEC ... | https://arxiv.org/abs/2501.14928v1 |
to a broa der setting, e.g., RL with linear function approximation. 5.5.2 Lipschitz contextual bandits with finite arms As the next example, we consider a standard non-parametric c ontextual bandit problem: Lipschitz contextual bandits, with Xequipped with a metric ρ. The reward function class is FLip={f:for anya∈A,f(·,... | https://arxiv.org/abs/2501.14928v1 |
where the infAlgis taken over all possible T-roundα-LDP algorithms. In words, the local minimax risk measures the best performance the algorithm can achiev e when it is given the knowledge two 31 possible models. This risk is called local because it measur es the difficulty of a particular model M0against a single worst-... | https://arxiv.org/abs/2501.14928v1 |
avoids (1) the complexity of estimation, e.g. the log-cardinality of the model class or the function c lass (cf. Theorem 21 ), and (2) the com- plexity of exploration, because it suffices to pick the best di stinguishing decision πthat maximizes DTV(M1(π),M0(π)). Hence, even though the local-minimax formulation avoids t ... | https://arxiv.org/abs/2501.14928v1 |
of Nfrac(M,∆), we have shown that Nfrac(M,∆)characterizes the sample complexity of LDP learning the mod el class, up to an expo- nential gap: logNfrac(M,2∆) α2/lessorsimilarCLDP ∆(M)/lessorsimilarNfrac(M,∆/2) α2∆2, (41) where we omit poly-logarithmic factors. We remark that the g ap between the lower and upper bounds c... | https://arxiv.org/abs/2501.14928v1 |
given. Suppose that Algis aT-roundα-JDP algorithm, such that it achieves Risk DM(T)≤∆with probability at least1 2underPM,Algfor anyM∈M . Then it holds that T≥logNfrac(M,∆)−log 2 α. For binary classification under pure JDP, Beimel et al. [2013a ] provide both lower and upper bounds of the sample complexity in terms of th... | https://arxiv.org/abs/2501.14928v1 |
and fractional covering number c haracterizes the JDP learnability of classification. Littlestone dimension. It is known that for binary class, RDim (F)≥Ω (LDim (F))[Feldman and Xiao ,2014], and there exists classes with LDim (F) = 2 while RDim (F)arbitrary large. Hence, LDP learnability is a stronger notion of complexi... | https://arxiv.org/abs/2501.14928v1 |
tests are almost equivalent. arXiv preprint arXiv:2009.06107 , 2020. 37 N. H. Bshouty and V. Feldman. On using extended statistical q ueries to avoid membership queries. Journal of Machine Learning Research , 2(Feb):359–395, 2002. S. Bubeck, J. Ding, R. Eldan, and M. Z. Rácz. Testing for high- dimensional geometry in r... | https://arxiv.org/abs/2501.14928v1 |
in Neural Information Processing Systems , 29, 2016. J. C. Duchi, M. I. Jordan, and M. J. Wainwright. Minimax optim al procedures for locally private estimation. Journal of the American Statistical Association , 113(521):182–201, 2018. C. Dwork, F. McSherry, K. Nissim, and A. Smith. Calibrating n oise to sensitivity in... | https://arxiv.org/abs/2501.14928v1 |
Zhang. Generalized linear ba ndits with local differential privacy. Advances in Neural Information Processing Systems , 34:26511–26522, 2021. S. Hanneke, R. Livni, and S. Moran. Online learning with simp le predictors and a combinatorial characterization of minimax in 0/1 games. In Conference on Learning Theory , pages ... | https://arxiv.org/abs/2501.14928v1 |
learnability of mix tures of gaussians. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing , pages 518–531, 2021. M. J. Osborne and A. Rubinstein. A course in game theory . MIT press, 1994. Y. Polyanskiy and Y. Wu. Dualizing le cam’s method for functi onal estimation, with applications to est... | https://arxiv.org/abs/2501.14928v1 |
. . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Robust decision making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Regret guarantees for hybrid DMSO . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5.1 Example: Smooth adversaries . . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2501.14928v1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.5.2 Lipschitz contextual bandits with finite arms . . . . . . . . . . . . . . . . . . 30 5.5.3 Concave-Lipschitz contextual bandits . . . . . . . . . . . . . . . . . . . . . . 31 6 Local Minimaxity, Learnability, and Joint Privacy 31 6.1 Local-minimax optimality .... | https://arxiv.org/abs/2501.14928v1 |
C.1 Joint DP in interactive learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 C.2 Learnability of regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 D Technical Tools 56 E Proofs for Lower Bounds 56 E.1 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2501.14928v1 |
. . . . . . . . . . . . . . . . . . 72 F.4.1 Proof of Theorem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 F.4.2 Proof of Theorem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 F.4.3 Proof of Proposition 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 F.4.... | https://arxiv.org/abs/2501.14928v1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 G Estimation-to-Decision Algorithm and Guarantees 87 G.1 LDP-E2D Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 G.1.1 Online estimation oracle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 G.1.2 LDP... | https://arxiv.org/abs/2501.14928v1 |
. . . . . . 99 I.2.1 Proof of Proposition I.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 I.3 Proof of Lemma 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 I.4 Proof of Theorem 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 I.4.1 Proo... | https://arxiv.org/abs/2501.14928v1 |
. . . . . . . . . . . . . . . . 118 I.10 Proof of Proposition B.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 I.11 Proof of Proposition B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 J Proofs from Section 6 and Appendix C 121 J.1 Proof of Theorem 33 . . . . . . .... | https://arxiv.org/abs/2501.14928v1 |
captures a number of decision making tasks, including reward-based learn- ing [Foster et al. ,2021,2023b ], interactive estimation and preference-based learning [ Chen et al. , 2022], multi-agent decision making and partial monitoring [ Foster et al. ,2023a ]. Constrained DEC and hybrid DEC. Extending Foster et al. [20... | https://arxiv.org/abs/2501.14928v1 |
we can show that fra ctional covering number of co(M) also provides a lower bound. Proposition A.3 below is a direct corollary of the fractional covering number lower bound of Chen et al. [2024] (see also Appendix J.2 ). Thus, we omit its proof for succinctness. Proposition A.3 (Fractional covering number lower bound) ... | https://arxiv.org/abs/2501.14928v1 |
samples z1,···,zTare i.i.d). For general interactive decision making tasks, we believe t he robust DMSO is a natural choice of contamination model. PAC lower and upper bounds. To apply the results of hybrid DMSO, we only need to show that p-decH ε(Pβ-Huber) =p-decR ε(M). By definition, for any M⋆∈M , reference model ĎM,... | https://arxiv.org/abs/2501.14928v1 |
channel Qinduces anα-LDP algorithm, which at each step t∈[T]selects theα-LDP channel Qtgiven by Qt(o|z) =Q(ot=o|zt=z,o1,···,ot−1), based on the history H(t−1)= (o1,···,ot−1). Conversely, an α-LDP algorithm also induces a sequentialα-private channel. Therefore, sequential α-private channels are equivalent to the α- LDP ... | https://arxiv.org/abs/2501.14928v1 |
and for each M∈M , π∈Π, the loss function is given by L(M,π) =1{π∝\⌉}atio\slash=πM}, whereπMis the unique index i∈[m] such thatM∈M(i). Note that the LDP hypothesis selection problem can be regard ed as a special case of Example 2 (with the measurement class Φ =Qαthe class of all α-DP channels). Therefore, we summarize ... | https://arxiv.org/abs/2501.14928v1 |
of mdistributions{D1,···,Dm}isε-correlated relative to Dif ∀i,j,|ρD(Di,Dj)|≤/braceleftBigg ε2, i∝\⌉}atio\slash=j, mε2, i=j. SupposeM⊆ ∆(Z). For any ∆, we define the minimum correlation cor(M,∆)to be the minimum ofεsuch that there exists a reference model ĎMand a set of models {M1,···,Mm}⊆M , such that (1){M1,···,Mm}isε-... | https://arxiv.org/abs/2501.14928v1 |
for interactive learning under JDP. Proposition C.1 (Fractional covering number lower bound for JDP learning) .LetT≥1, and Alg is a weakα-JDP algorithm. Suppose that with T-round of interactions, Algachieves Risk DM(T)≤ ∆with probability at least1 2underPM,Algfor anyM∈M . Then it holds that T≥logNfrac(M,∆)−log 2 α. Not... | https://arxiv.org/abs/2501.14928v1 |
the Minimax theorem. Theorem D.2 (Ky Fan’s minimax theorem, Fan[1953]).LetXbe a compact Hausdorff space and Y an arbitrary set (not topologized). Let fbe a real-valued function on X×Ysuch that, for every y∈Y,f(·,y)is continuous over X. Then, iffis convex-like on Xand concave-like on Y, then min x∈Xsup y∈Yf(x,y) = sup y∈... | https://arxiv.org/abs/2501.14928v1 |
E.1 yields sup EnvEEnv,Alg[Risk DM(T)]≥δ 2sup ĎMp-decq,H εδ(T),δ(P)≥δ 4sup ĎMp-decH εδ(T)(P,ĎM). 57 Lettingδ→1 2gives the desired lower bound: sup EnvEEnv,Alg[Risk DM(T)]≥1 8sup ĎMp-decH ε(T)(P,ĎM)≥1 8p-decH ε(T)(P). (51) Similarly, we can apply Proposition E.1 to general loss function. Proof of Theorem 7 :Eq. (15) .By... | https://arxiv.org/abs/2501.14928v1 |
e squared Hellinger distance to the “error probability”-sty le quantity in the definition of SQ DEC (4). Lemma E.4. Suppose that P∈∆(O), andO0⊆O is a measurable subset of O. Then it holds that 1 2P(Oc 0)≤ inf P′:supp(P′)⊆O 0D2 H/parenleftbig P′,P/parenrightbig ≤P(Oc 0). Note thatPMconsists of all models M′such that supp... | https://arxiv.org/abs/2501.14928v1 |
need to verify Assumption 3 . For any decision π∈Π, we consider the binary channel Qπ∈Qαgiven by Qπ(+1|z) =1 +cαR(z,π) 2, Qπ(−1|z) =1−cαR(z,π) 2, wherecα= 1−e−αensures that Qπisα-DP (cf. Example 5 ), and we assume without loss of generality that{−1,1}⊆O . Then, by definition, it holds that cα/vextendsingle/vextendsingle... | https://arxiv.org/abs/2501.14928v1 |
distribution µ⋆∈∆(P⋆)such thatM⋆=EM∼µ⋆[M]. Then, for the stationary environment Envspecified by µ⋆(i.e., it selects Mt∼µ⋆independently), it holds thatPEnv,Alg(HT=·) =PM⋆,Alg(HT=·). Therefore, by data-processing inequality, we have 1 2/parenleftBig/radicalbig PEnv,Alg(L(P⋆,πT+1)≥∆)−/radicalbig PĎM,Alg(L(P⋆,πT+1)≥∆)/paren... | https://arxiv.org/abs/2501.14928v1 |
t=1Eπ∼qt/vextendsingle/vextendsingleVM(π)−VĎM(π)/vextendsingle/vextendsingle2/bracketrightBigg =1 (CCVε)2Eπ∼p′ 0/bracketleftBig/vextendsingle/vextendsingleVM(π)−VĎM(π)/vextendsingle/vextendsingle2/bracketrightBig ≤2 C2. Further, by Eq. (56) , we have D2 H/parenleftBig PM,Alg′,PĎM,Alg′/parenrightBig ≤7T·Eπ∼p0D2 H/parenl... | https://arxiv.org/abs/2501.14928v1 |
ε(M,ĎM)≤p-deco γ(M,ĎM) +γε2, (63) and analogous conversions also hold for the regret-DECs and the hybrid DECs. The first inequality inEq. (63) can be loose in general, and a tighter conversion is possible under reward-based loss function ( Proposition F.10 ). 67 F.1 Information set structure Recall that in Section 2.1 ,... | https://arxiv.org/abs/2501.14928v1 |
. It has two options: pacfor PAC learning and regfor no-regret learning. For these two tasks, we specify diffe rent spaces Sof distributions for 68 exploration-exploitation: Spac:= ∆(Π)×∆(Π),Sreg:={(q|Π,q) :q∈∆(Π)}⊂∆(Π)×∆(Π), where we recall that Π:= Π×Φ, and for any distribution q∈∆(Π),q|Π∈∆(Π) is the marginal distribu... | https://arxiv.org/abs/2501.14928v1 |
have the following guarantee of ExO+. Theorem F.2 (ExO+upper bound for PAC learning; Type 2) .For PAC learning under hybrid DMSO, suppose that the algorithm ExO+is instantiated with the Type 2 information set struc- ture Ψ =P, andw1= Unif( P). Then for any environment constrained by P,ExO+achieves with probability at l... | https://arxiv.org/abs/2501.14928v1 |
the decision space Πis finite, (2) the latent observation space Zis a compact metric space under a certain metric ρ, and (3) the value function is given by a reward function R(cf.Definition 9 ) withR(z,π)being Lipschitz with respect toz. This is indeed the case for agnostic regression task ( Section 5.2 ). In these assum... | https://arxiv.org/abs/2501.14928v1 |
any environment specified by a model M⋆∈M ,ExO+(when instantiated on Pmandγ=4 ¯ε(T)2) achieves with probability at least 1−δthat EπT+1∼ˆpL(M⋆,πT+1)≤p-deco,H γ/4(Pm) +2γlog(m/δ) T. 73 BecauseM1,···,Mmare convex, we have p-deco,H γ/4(Pm) =p-deco γ/4(M)≤p-decc 2/√γ(M) =p-decc ¯ε(T)(M), where the inequality follows from Eq.... | https://arxiv.org/abs/2501.14928v1 |
reference model ĎM∈co(M), we let p-deco,LDP γ(M,ĎM):= inf p∈∆(Π) q∈∆(Π×L)sup M∈M/braceleftbig Eπ∼p[L(M,π)]−γE(π,ℓ)∼qD2 ℓ(M(π),ĎM(π))/bracerightbig , (72) r-deco,LDP γ(M,ĎM):= inf p∈∆(Π×L)sup M∈M/braceleftbig Eπ∼p[VM(πM)−VM(π)]−γE(π,ℓ)∼qD2 ℓ(M(π),ĎM(π))/bracerightbig ,(73) and we define p-deco,LDP γ(M) = sup ĎM∈co(M)p-de... | https://arxiv.org/abs/2501.14928v1 |
al. [2024] (see also Section 3.3 ), we consider the decision-based (or, “ policy- based ”) information set structure Ψ = Ψ polgiven by Ψpol= Π,Mπ={M:VM(πM)−VM(π)≤∆},∀π∈Π. (79) By definition, MΨmod=/uniondisplay π∈Πco(Mπ) =MΠ, N frac(M,Ψpol; ∆) =Nfrac(M,∆). Therefore, we may instantiate LDP-ExO with Ψpolto obtain the fol... | https://arxiv.org/abs/2501.14928v1 |
that |L(M,π)−L(M′,π)|≤2DTV(M′,M)(becauseL is reward-based) and D2 ℓ(M′,ĎM)≤2D2 ℓ(M,ĎM) + 2 D2 ℓ(M′,M). Taking supremum over ĎM∈co(M)gives the desired result. F.5.4 Contextual Bandits In this section, we work with contextual DMSO (introduced in Section 5.5 ). Note that contextual DMSO is not encompassed by private DMSO,... | https://arxiv.org/abs/2501.14928v1 |
that for any fixed ψ, −logEt−1[exp (Xt(ψ;πt,ot))] = Err(pt,qt,ξt;wt,Mt,ψ). Combining the equations above and applying Cauchy inequali ty, we now have Eψ∼w1EExO+exp/parenleftBigg 1 2T/summationdisplay t=1Err(pt,qt,ξt;wt,Mt,ψ)/parenrightBigg ≤1. Notice that ψ∼w1is independent of the randomness of the T-round interactions ... | https://arxiv.org/abs/2501.14928v1 |
M′(π),ĎM(π)/parenrightbig ≤ max ĎM∈co(M0)inf (p,q)∈Smax (M,ψ):M∈MψEπ∼p[Lψ(M,π)]−γ 4Eπ∼qD2 H/parenleftbig M(π),ĎM(π)/parenrightbig , where the last line follows again from the weak duality. To finalize the proof, we notice that by the arbitrariness of w∈∆(Ψ) , we have already proven exoγ(Ψ)≤3e−A+ max ĎM∈co(M0)inf (p,q)∈S... | https://arxiv.org/abs/2501.14928v1 |
Proof of Lemma F.7 DenoteD:= supε′∈[ε,1]p-decH ε′(P) ε′. We consider γ=6D ε. We fix an arbitrary reference model ĎM. For each j≥0, we define εj= 2−j, and letdj:= p-decH εj(P,ĎM), (pj,qj):= arg min p∈∆(Π) q∈∆(Π)sup P∈P/braceleftbigg Eπ∼pL(P,π)|inf M∈co(P)Eπ∼qD2 H/parenleftbig M(π),ĎM(π)/parenrightbig ≤ε2 j/bracerightbigg ... | https://arxiv.org/abs/2501.14928v1 |
et al. , 2021,2023b ] where the performance is measured in terms of the squared He llinger error. Assumption G.1 (Estimation oracle for M).At each time t∈[T], an online estimation oracle AlgEstforMreturns, given Ht−1= (π1,ℓ1,o1),..., (πt−1,ℓt−1,ot−1) with (πi,ℓi)∼piandoi∼Rad(M⋆(πi)[ℓi]), an estimator /hatwiderMt∈(Π→∆(Z... | https://arxiv.org/abs/2501.14928v1 |
the refining phase, the algorithm proceeds as follow s to identify an index tsuch that/hatwiderMt achieves a small estimation error. Refining phase. At the start of this phase, the algorithm randomly samples t1,···,tK∼ Unif([N]). Then, with probability at least 1−3 4δ, there exists k∈[K]such that E(π,ℓ)∼qtkD2 ℓ(M⋆(π),/ha... | https://arxiv.org/abs/2501.14928v1 |
fact E[X2 t|Ht−1] =Eℓ∼qtD2 ℓ/parenleftBig /hatwiderMt,M⋆/parenrightBig gives the desired upper bound. G.2 Query-based E2D algorithm In the following, we present the E2D algorithm ( SQ-E2D ,Algorithm 4 ) for SQ DMSO. 92 Algorithm 4 Query-based Estimation-to-Decisions (SQ- E2D) Input: RoundT≥1, error probability δ>0, mod... | https://arxiv.org/abs/2501.14928v1 |
case Lis metric-based, i.e., it is given by L(M,π) =ρ(πM,π)for a pseudo-metric ρover Π. We denote ε:=1 2√ Tand∆:=p-decτ-SQ ε(M,ĎM). We first describe any T-round query-based algorithm in the following way (cf. Section 3 ). A T-round algorithm Alg={qt}t∈[T]∪{p}is specified by a sequence of mappings, where the t-th mapping... | https://arxiv.org/abs/2501.14928v1 |
p,β)sup ℓ∈LPM∼µ/parenleftbig Dℓ(M,ĎM)>τ/parenrightbig ≤ε2 ⇔ ∀p∈∆(Π),sup µ∈∆(Md p,β)inf ℓ∈L1 PM∼µ/parenleftbig Dℓ(M,ĎM)>τ/parenrightbig≥ε−2 ⇔ inf p∈∆(Π)sup µ∈∆(Md p,β)inf ℓ∈L1 PM∼µ/parenleftbig Dℓ(M,ĎM)>τ/parenrightbig≥ε−2 ⇔ SQDimτ β(Md,ĎM)≥ε−2. This is the desired result. H.4 Proof of Lemma 18 By definition, in interact... | https://arxiv.org/abs/2501.14928v1 |
PAC- DEC provides a lower bound regardless of the structure of the loss function. Proposition I.1 (Quantile-based private PAC-DEC lower bound) .For anyT≥1and constant δ∈[0,1), we denote ε(T):=1 (eα−1)/radicalBig δ 2T. Then, for any T-roundα-LDP algorithm Alg, there exists M⋆∈M such that under PM⋆,Alg, Risk DM(T)≥p-decq... | https://arxiv.org/abs/2501.14928v1 |
of LDP-ExO (byTheorem F.4 ). 101 Proposition I.4. LetT≥1,γ > 0. Then, for linear regression under L1loss, LDP-ExO (instan- tiated on Ψdefined above) achieves with probability at least 1−δ Eˆθ∼/hatwidepL1(M⋆,ˆθ)≤p-deco,LDP cα2γ(MLin) +2γlog(|Θ|/δ) T. Note that for simplicity, we assume Θis finite. By applying the argument... | https://arxiv.org/abs/2501.14928v1 |
least 1−δthat Eπ∼/hatwidepL1(M⋆,π)≤O(1)·/bracketleftbiggd α2γ+2γlog(|Θ|/δ) T/bracketrightbigg . Note thatL1(M⋆,π)is a convex function with respect to π∈Bd(1), and hence we can let LDP- ExO output ˆθ=Eπ∼/hatwidep[π]∈Bd(1). Then, by choosing γ > 0suitably, it is guaranteed that with probability at least 1−δ L1(M⋆,ˆθ)≤˜O/... | https://arxiv.org/abs/2501.14928v1 |
to logarithmic factors). I.6 Proof of Theorem 30 We claim that the private regret-DEC of Mcan be bounded as r-decLDP ε(M) = sup ĎM∈co(M)r-decLDP ε(M∪{ ĎM},ĎM)≤(20d+ 6)ε,∀ε∈[0,1]. (100) With Eq. (100) , we may directly apply Proposition 29 , aslogN∞(FLin,∆)≤O(dlog(1/∆)). In the following, it remains to prove Eq. (100) .... | https://arxiv.org/abs/2501.14928v1 |
. (105) We then define p⋆∈∆(Π×L)to be the distribution of (π,ℓ)underπ∼P,ℓ∼qπ. We summarize the properties of p⋆in the following lemma. Lemma I.12. Suppose that M∈Mp⋆,2ε2(ĎM). Then it holds that Eπ∼P/vextendsingle/vextendsingleVM(π)−VĎM(π)/vextendsingle/vextendsingle≤2ε, (106) Eπ∼P/vextenddouble/vextenddoubleEx∼νnπ(x)/pa... | https://arxiv.org/abs/2501.14928v1 |
Therefore, Eπ∼PEx∼ν∝⌊a∇⌈⌊lφ(x,π)∝⌊a∇⌈⌊lΣ−1 P≤Eπ∼P/radicalBig d·/angbracketleftbig Σ−1 P,U−2π/angbracketrightbig ≤/radicalBig d·Eπ∼P/angbracketleftbig Σ−1 P,U−2π/angbracketrightbig =/radicalBig d/angbracketleftbig Σ−1 P,Eπ∼PU−2π/angbracketrightbig =d, where the last line follows from Eπ∼PU−2 π= ΣPandtr(Id) =d. I.6.2 Pro... | https://arxiv.org/abs/2501.14928v1 |
set structure Ψ = Π +, with Mψ:={Mν,f:ν∈∆(X),f∈Fψ}, πψ=ψ, ψ∈Ψ = Π +. Clearly, Ψis a valid information set structure for the constraint set Pcxtintroduced in Sec- tion 5.5 , and we have log|Ψ|=NXlogNA. Therefore, it remains to upper bound the regret DEC r-deco,LDP γ(MΨ). Theorem I.13 (Learning Lipschitz contextual bandi... | https://arxiv.org/abs/2501.14928v1 |
= 2Eπ∼pEℓ∼QDℓ(M(π),ĎM(π))2 ≥Eπ∼p /summationdisplay x∈X∆|νM(x)−νĎM(x)|2+/vextendsingle/vextendsingleνM(x)˜fM(x,π(x))−νĎM(x)˜fĎM(x,π(x))/vextendsingle/vextendsingle2 116 ≥Eπ∼p 1 2/summationdisplay x∈X∆νĎM(x)2/vextendsingle/vextendsingle˜fM(x,π(x))−˜fĎM(x,π(x))/vextendsingle/vextendsingle2 =1 2/summationdisplay ... | https://arxiv.org/abs/2501.14928v1 |
we know that PM∼µ,π∼p(L(M,π)≥∆)≥inf πPM∼µ(L(M,π)≥∆)≥1 2. Therefore, there must exist M0⊂M such thatµ(M0)≥1 3, and Pπ∼p(L(M,π)≥∆)≥1 4,∀M∈M 0. Then, we also know µ(M0) min M∈M 0Eℓ∼qDℓ(M,ĎM)2≤EM∼µEℓ∼qDℓ(M,ĎM)2≤1 2ε2 0, and hence there exists M∈M 0withEℓ∼qDℓ(M,ĎM)2≤3 2ε2 0. This gives p-decq,LDP√ 2ε0,1/4(M,ĎM)≥∆, which als... | https://arxiv.org/abs/2501.14928v1 |
Lemma J.1. For any 2-point model class M0={M1,M0}⊆M , it holds that p-decLDP ε(M0)≤/braceleftBigg infπ∈Π(L(M1,π) +L(M0,π)),ifsupπDTV(M1(π),M0(π))≤2ε, 0, otherwise. With Lemma J.1 , we know that sup M1∈Mp-decLDP ε({M0,M1})≤p-decloc 2ε(M). Applying Theorem 21 gives MT({M1,M0})≤p-decLDP ¯εδ(T)({M0,M1}) +δ. Therefore, we m... | https://arxiv.org/abs/2501.14928v1 |
Notice that by definition, for any model M∈M , PM,Alg(L(M,πT+1)≤∆)≥1 2, and hence PĎM,Alg(L(M,πT+1)≤∆)≥e−TαPM,Alg(L(M,πT+1)≤∆)≥1 2e−Tα,∀M∈M. Then, by the definition of fractional covering number ( Definition 5 ), we know for the distribution p=PĎM,Alg(πT+1=·)∈∆(Π) , it holds that 2eTα≥sup M∈M1 p(L(M,π)≤∆)≥Nfrac(M,∆). This... | https://arxiv.org/abs/2501.14928v1 |
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