Local continuity of log-concave projection, with applications to estimation under model misspecification

Barber, Rina Foygel; Samworth, Richard J.

doi:10.3150/20-bej1316

Cited by 6 publications

(5 citation statements)

References 49 publications

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“…A further topic for future research could be to seek quantitative versions of the continuity result (Proposition S12) for our

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‐projection onto the class of S‐shaped functions, in the spirit of the recent work of Barber and Samworth (2021) on the log‐concave projection. Such a result could, for instance, provide insight into the rate at which the estimated inflection point converges to the inflection point of the projected regression function under model misspecification.…”

Section: Discussionmentioning

confidence: 99%

Nonparametric, Tuning-Free Estimation of S-Shaped Functions

Feng

Han

Carroll

et al. 2022

Journal of the Royal Statistical Society Series B: Statistical Methodology

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We consider the nonparametric estimation of an Sshaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimization problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm

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“…A further topic for future research could be to seek quantitative versions of the continuity result (Proposition S12) for our

L^{2}

Section: Discussionmentioning

confidence: 99%

Nonparametric, Tuning-Free Estimation of S-Shaped Functions

Feng

Han

Carroll

et al. 2022

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…While a natural statement, arguing this is challenging because this requires us to understand the behaviour of log-concave MLEs 'off-themodel,' i.e., when the data is not drawn from a log-concave distribution itself. With the notable exception of Barber and Samworth [BS21], this task has not been undertaken in the literature, with most works focusing on on-the-model minimax rate bounds [KS16; KDR19; Han21; CDSS18]. Let us consider this in some detail.…”

Section: Challenges and Context From The Theory Of Log-concave Mlesmentioning

confidence: 99%

“…However, these results give quite poor rates. Roughly speaking, Theorem 5 of their paper [BS21] shows that off-the-model, d H (p t , L p ) t −1/4d , and thus any analysis that exploits this result cannot hope to show that σ t is large for t ≪ d H (p, L) −4d . This power of 4d arises since the analysis of [BS21] passes through a reduction to convergence of empirical laws in Wasserstein distance (which gives the relatively benign factor of d), and further suffers a 1/4th power slowdown relative to this convergence (which is both unavoidable, and leads to a 4d exponent).…”

Section: Challenges and Context From The Theory Of Log-concave Mlesmentioning

confidence: 99%

“…which was identified by Barber and Samworth [BS21] as a means to lower bound the covariance of the logconcave projection of P , which in turn can be exploited to upper bound the supremum of L p and (indirectly) pt . Observe that ∆ P roughly corresponds to the minimum eigenvalue of the covariance matrix of P -indeed, it is best seen as a robust version of the same.…”

Section: Bounding Typical Rejection Times For the Ulr Testmentioning

confidence: 99%

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A Sequential Test for Log-Concavity

Gangrade¹,

Rinaldo²,

Ramdas³

2023

Preprint

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On observing a sequence of i.i.d. data with distribution P on R d , we ask the question of how one can test the null hypothesis that P has a log-concave density. This paper proves one interesting negative and positive result: the non-existence of test (super)martingales, and the consistency of universal inference. To elaborate, the set of log-concave distributions L is a nonparametric class, which contains the set G of all possible Gaussians with any mean and covariance. Developing further the recent geometric concept of fork-convexity, we first prove that there do no exist any nontrivial test martingales or test supermartingales for G (a process that is simultaneously a nonnegative supermartingale for every distribution in G), and hence also for its superset L. Due to this negative result, we turn our attention to constructing an eprocess -a process whose expectation at any stopping time is at most one, under any distribution in L -which yields a level-α test by simply thresholding at 1/α. We take the approach of universal inference, which avoids intractable likelihood asymptotics by taking the ratio of a nonanticipating likelihood over alternatives against the maximum likelihood under the null. Despite its conservatism, we show that the resulting test is consistent (power one), and derive its power against Hellinger alternatives. To the best of our knowledge, there is no other e-process or sequential test for L.

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“…It is further known that when d ≤ 3, the log-concave maximum likelihood estimator can adapt to certain subclasses of log-concave densities, including log-concave densities whose logarithms are piecewise affine (Kim et al, 2018;Feng et al, 2021). See also Barber and Samworth (2021) for recent work on extensions to the misspecified setting (where the true distribution from which the data are drawn does not have a log-concave density).…”

Section: Related Workmentioning

confidence: 99%

A new computational framework for log-concave density estimation

Chen¹,

Mazumder²,

Samworth³

2021

Preprint

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In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concave maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov smoothing, combined with an appropriate integral discretization of increasing accuracy. We prove that these methods enjoy, both with high probability and in expectation, a convergence rate of order 1/T up to logarithmic factors on the objective function scale, where T denotes the number of iterations. The benefits of our new computational framework are demonstrated on both synthetic and real data, and our implementation is available in a github repository LogConcComp (Log-Concave Computation).

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Local continuity of log-concave projection, with applications to estimation under model misspecification

Cited by 6 publications

References 49 publications

Nonparametric, Tuning-Free Estimation of S-Shaped Functions

Nonparametric, Tuning-Free Estimation of S-Shaped Functions

A Sequential Test for Log-Concavity

A new computational framework for log-concave density estimation

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