Sylvain Arlot
email: sylvain.arlot@math.u-psud.fr

Model selection by resampling penalization

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Model selection by resampling penalization

Introduction

Choosing between the outputs of many learning algorithms, from the prediction viewpoint, remains to estimate their generalization abilities. A classical method for this is penalization, that comes from model selection theory. Basically, it states that a good choice can be made by minimizing the sum of the empirical risk (how does the algorithm fits the data) and some complexity measure of the algorithm (called the penalty). The ideal penalty for prediction is of course the difference between the true and empirical risks of the output, but it is unknown in general. It is thus crucial to obtain tight estimates of such a quantity. Many penalties or complexity measures have been proposed, both in the classification and regression frameworks. Consider for instance regression and least-square estimators on finite-dimensional vector spaces (the models). When the design is fixed and the noise-level constant equal to σ, Mallows' C p penalty [START_REF] Mallows | Some comments on C p[END_REF] (equal to 2n -1 σ 2 D for a D-dimensional space, and it can be modified according to the number of models [START_REF] Birgé | Gaussian model selection[END_REF][START_REF] Sauvé | Histogram selection in non gaussian regression[END_REF]) has some optimality properties [START_REF] Shibata | An optimal selection of regression variables[END_REF][START_REF] Li | Asymptotic optimality for C p , C L , cross-validation and generalized cross-validation: discrete index set[END_REF][START_REF] Baraud | Model selection for regression on a random design[END_REF]. However, such a penalty linear in the dimension may be terrible in an heteroscedastic framework (as shown by [START_REF] Baraud | Model selection for regression on a random design[END_REF] and experiment HSd2 in Sect. 6).

In classification, the VC-dimension has the drawback of being independent of the underlying measure, so that it is adapted to the worst case. It has been improved with data-dependent complexity estimates, such as Rademacher complexities [START_REF] Koltchinskii | Rademacher penalties and structural risk minimization[END_REF][START_REF] Bartlett | Model selection and error estimation[END_REF] (generalized by Fromont with resampling ideas [START_REF] Fromont | Model selection by bootstrap penalization for classification[END_REF]), but they may be too large because they are still global complexity measures. The localization idea then led to local Rademacher complexities [START_REF] Peter | Local Rademacher complexities[END_REF][START_REF] Koltchinskii | IMS Medallion Lecture : Local Rademacher Complexities and Oracle Inequalities in Risk Minimization[END_REF] which are tight estimates of the ideal penalty, but involve unknown constants and may be very difficult to compute in practice. On the other hand, the V -fold cross-validation (VFCV) is very popular for such purposes, but it is still poorly understood from the non-asymptotic viewpoint.

In this article, we propose a new family of penalties, based on Efron's bootstrap heuristics [START_REF] Efron | Bootstrap methods: another look at the jackknife[END_REF] (and its generalization to weighted bootstrap, i.e. resampling). It is a localized version of Fromont's penalties, which does not involve any unknown constant, and is easy to compute (at the price of some loss in accuracy) in its V -fold cross-validation version. We define it in a much general framework, so that it has a wide range of application. As a first theoretical step, we prove the efficiency of these algorithms in the case example of least-square regression on histograms, under reasonable assumptions. Indeed, they satisfy oracle inequalities with constant almost one, asymptotic optimality and adaptivity to the regularity of the regression function. This comes from explicit computations that allow us to deeply understand why these penalties are working well. Then, we compare the "classical" VFCV with the V -fold penalties, enlightening how V should be chosen. Finally, we illustrate these results with a few simulation experiments. In particular, we show that resampling penalties are competitive with classical methods for "easy problems", and may be much better for some harder ones (e.g. with a variable noise-level).

A general model selection algorithm

We consider the following general setting : X × Y is a measurable space, P an unknown probability measure on it and (X 1 , Y 1 ), . . . , (X n , Y n ) ∈ X × Y some data of common law P . Let S be the set of predictors (measurable functions X → Y) and γ : S × (X × Y) → R a contrast function. Given a family ( s m ) m∈Mn of data-dependent predictors, our goal is to find the one minimizing the prediction loss P γ(t). We will extensively use this functional notation Qγ(t) := E (X,Y )∼Q [γ(t, (X, Y ))], for any probability measure Q on X × Y. Notice that the expectation here is only taken w.r.t. (X, Y ), so that Qγ(t) is random when t = s m is random. Assuming that there exists a minimizer s ∈ S of the loss (the Bayes predictor), we will often consider the excess loss l(s, t) = P γ(t) -P γ(s) ≥ 0 instead of the loss.

Assume that each predictor s m may be written as a function s m (P n ) of the empirical distribution of the data

P n = n -1 n i=1 δ (X i ,Y i ) .
The ideal choice for m is the one which minimizes over M n the true prediction risk P γ( s m (P n )) = P n γ( s m (P n )) + pen id (m) where the ideal penalty is equal to

pen id (m) = (P -P n )γ( s m (P n )) .
The resampling heuristics (introduced by Efron [START_REF] Efron | Bootstrap methods: another look at the jackknife[END_REF]) states that the expectation of any functional F (P, P n ) is close to its resampling counterpart E W F (P n , P W n ), where

P W n = n -1 n i=1 W i δ (X i ,Y i )
is the empirical distribution P n weighted by an independent random vector W ∈ [0; +∞) n , with i E[W i ] = n. The expectation E W [•] means that we only integrate w.r.t. the weights W .

We suggest here to use this heuristics for estimating pen id (m), and plug it into the penalized criterion P n γ( s m ) + pen(m). This defines m ∈ M n as follows.

Algorithm 1 (Resampling penalization).

1. Choose a resampling scheme, i.e. the law of a weight vector W . 1. There is a constant C = 1 in front of the penalty, although there isn't any in Efron's heuristics, because we did not normalize W . The asymptotical value of the right normalizing constant C W may be derived from Theorem 3.6.13 in [START_REF] Van Der | Weak convergence and empirical processes[END_REF]. In the case example of histograms, we give a non-asymptotic expression for it [START_REF] Bartlett | Model selection and error estimation[END_REF]. In general, we suggest to use some data-driven method to choose C (see algorithm 3), whereas the resampling penalty only estimates the shape of the ideal one.

Choose a constant

C ≥ C W ≈ n -1 n i=1 E (W i -1) 2 -1 . 3.
2. We allowed C to be larger than C W because overpenalizing may be fruitful in a non-asymptotic viewpoint, e.g. when there is few noisy data.

3. Because of this plug-in method, algorithm 1 seems to be reasonable only if M n is not too large, i.e. if it has a polynomial complexity : Card(M n ) ≤ c M n α M . Otherwise, we can for instance group the models of similar complexities and reduce M n to a polynomial family.

The histogram regression case

As studying algorithm 1 in general is a rather difficult question, we focus in this article on the case example of least-square regression on histograms. Although we do not consider histograms as a final goal, this first theoretical step will be useful to derive heuristics making the general algorithm 1 work. We first precise the framework and some notations. The data (X i , Y i ) ∈ X × R are i.i.d. of common law P . Denoting s the regression function, we have

Y i = s(X i ) + σ(X i )ǫ i (1)
where σ : X → R is the heteroscedastic noise-level and ǫ i are i.i.d. centered noise terms, possibly dependent from X i , but with variance 1 conditionally to X i . The feature space X is typically a compact set of R d . We use the least-square contrast γ : (t, (x, y)) → (t(x)y) 2 to measure the quality of a predictor t : X → Y. As a consequence, the Bayes predictor is the regression function s, and the excess loss is l(s

, t) = E (X,Y )∼P (t(X) -s(X)) 2 .
To each model S m , we associate the empirical risk minimizer s m = s m (P n ) = arg min t∈Sm {P n γ(t)} (when it exists and is unique). Each model in (S m ) m∈Mn is the set of piecewise constant functions (histograms) on some partition (I λ ) λ∈Λm of X . It is thus a vector space of dimension D m = Card(Λ m ), spanned by the family (½ I λ ) λ∈Λm . As this basis is orthogonal in L 2 (µ) for any probability measure on X , we can make explicit computations that will be useful to understand algorithm 1. The following notations will be useful throughout this article.

p λ := P (X ∈ I λ ) p λ := P n (X ∈ I λ ) p W λ = p λ W λ := P W n (X ∈ I λ ) s m := arg min t∈Sm P γ(t) = λ∈Λm β λ ½ I λ β λ = E P [Y |X ∈ I λ ] s m := arg min t∈Sm P n γ(t) = λ∈Λm β λ ½ I λ β λ = 1 n p λ X i ∈I λ Y i s W m := arg min t∈Sm P W n γ(t) = λ∈Λm β W λ ½ I λ β W λ = 1 n p W λ X i ∈I λ W i Y i
Remark that s m is uniquely defined if and only if each I λ contains at least one of the X i , and the same problem arises for s W m . This is why we will slightly modify the general algorithm for histograms. Before this, we compute the ideal penalty (assuming that min λ∈Λm p λ > 0 ; otherwise, the model m should clearly not be chosen) :

pen id (m) = (P -P n )γ( s m ) = λ∈Λm (p λ + p λ ) β λ -β λ 2 + (P -P n )γ(s m ) .
The last term in the sum being centered, it is estimated as zero by the resampling version of pen id . The first term is a sum of D m terms, each one depending only on the restrictions of P and P n to I λ . Thus, if we assume that p λ > 0 and if we compute separately all those terms, conditionally to p W λ > 0, we can define the resampling version of pen id (m). This leads to the following algorithm.

Algorithm 2 (Resampling penalization for histograms).

0. Choose a threshold A n ≥ 1 and replace M n by

M n = m ∈ M n s.t. min λ∈Λm {n p λ } ≥ A n .
1. Choose a resampling scheme L(W ). {P n γ( s m (P n )) + pen(m)} .

Choose a constant

C ≥ C W (A n ) where C W is defined by (3).

3'. Compute the following resampling penalty for each

m ∈ M n : pen(m) = C λ∈Λm E W p λ + p W λ β W λ -β λ 2 W λ > 0 .
Remark 2.

1. The two modifications of the algorithm for histograms do not affect much the result if A n is of the order ln(n). Indeed, models with very few data are not relevant in general, and if min λ∈Λm {n p λ } ≥ A n is not too small, the event {W λ = 0} has a very small probability.

2. We allow C to depend on A n since the "optimal" constant C W may depend on it, but this dependence is mild according to our computations.

When the resampling weights are exchangeable (see definition below), we are able to compute pen explicitly. It is enlightening to compare it with pen id in expectation, conditionally to ( p λ ) λ∈Λm (we denote by E m [•] this conditional expectation) :

E m [pen id (m)] = 1 n λ∈Λm 1 + p λ p λ (σ r λ ) 2 + σ d λ 2 (2)
E m [pen(m)] = C n λ∈Λm (R 1,W (n, p λ ) + R 2,W (n, p λ )) (σ r λ ) 2 + σ d λ 2 with (σ r λ ) 2 := E[σ(x) 2 |X ∈ I λ ] ; σ d λ 2 := E[(s(X) -s m (X)) 2 |X ∈ I λ ] and for k = 1, 2 R k,W (n, p λ ) = E (W i -W λ ) 2 W 3-k λ W λ > 0 .
Hence, contrary to Mallows' penalty (with σ 2 known or estimated), resampling penalties really take into account the heteroscedasticity of the noise (σ r λ depends on λ) and the bias terms σ d λ 2 . We then define

C W (A n ) := sup n p λ ≥An 2 R 1,W (n, p λ ) + R 2,W (n, p λ ) (3)
and

C ′ W (A n ) is the infimum of the same quantity.

Examples of resampling weights

In this article, we consider resampling weights

W = (W 1 , . . . , W n ) ∈ [0; +∞) n such that E[W i ] = 1 for all i and E[W 2 i ] < ∞.
We mainly consider the following exchangeable weights (i.e. such that for any permutation τ , (W τ (1) , . . . , W τ (n) )

(d) = (W 1 , . . . , W n )).
1. Efron (q) : multinomial vector with parameters (q; n -1 , . . . , n -1 ). Then, R 2,W (n, p λ ) = (n/q) × (1 -(n p λ ) -1 ). A classical choice is q = n.

2.

Rademacher :

W i i.i.d., 2 times Bernoulli(1/2). Then, R 2,W (n, p λ ) = 1.
3. Random hold-out (q) (or cross-validation) :

W i = n q ½ i∈I with I uniform random subset (of cardinality q) of {1, . . . , n}. R 2,W (n, p λ ) = (n/q) -1. A classical choice is q = n/2. 4. Leave-one-out = Random hold-out (n -1). Then, R 2,W (n, p λ ) = (n - 1) -1 .
In each case, we can show that

R 1,W = R 2,W (1 + δ (W )
n, p λ ) for some explicit small term δ (W ) n, p λ (numerically of the same order as E[p λ / p λ | p λ > 0] -1 in expectation for the three first resamplings, and slightly smaller in the Leaveone-out case). Thus,

C W ≈ C ′ W ≈ R -1 2,W (asymptotically in A n ).
For computational reasons, it is also convenient to introduce the following V -fold cross-validation resampling weights : given a partition (B j ) 1≤j≤V of {1, . . . , n} and W B ∈ R V leave-one-out weights, we define W i = W B j for each i ∈ B j . The partition should be taken as regular as possible, and then we can compute E[pen(m)] and show that C W ≈ V -1.

The Rademacher weights lead to penalties close in spirit to local Rademacher complexities (the link between global Rademacher complexities and global resampling penalties with Rademacher weights can be found in [START_REF] Fromont | Model selection by bootstrap penalization for classification[END_REF]). The links with the classical leave-one-out and VFCV algorithms are given in Sect. 5.

Main results

In this section, we prove that algorithm 2 has some optimality properties under the following restrictions for some non-negative constants α M , c M , c A , c rich : (P4) The threshold is large enough :

(P1) Polynomial complexity of M n : Card(M n ) ≤ c M n α M . (P2) Richness of M n : ∀x ∈ [1, nc -1 rich ], ∃m ∈ M n s.t. D m ∈ [x; c rich x]. (
C A ln(n) ≥ A n ≥ (26 + 7α M ) ln(n).
Assumption (P1) is almost necessary, since too large families of models need larger penalties than polynomial families [START_REF] Birgé | Gaussian model selection[END_REF][START_REF] Baraud | Model selection for regression on a random design[END_REF][START_REF] Sauvé | Histogram selection in non gaussian regression[END_REF]. Assumption (P2) is necessary but it is always satisfied in practice. Assumption (P3) is only here to ensure that we have an explicit formula for the penalty, and sharp bounds on R 1,W and R 2,W . The constant (26 + 7α M ) in (P4) is quite large due to technical reasons, but much smaller values (larger than 2) should suffice in practice.

Theorem 1. Assume that the (X i , Y i )'s satisfy the following assumptions :

(Ab) Bounded data : Y i ∞ ≤ A < ∞.
(An) Noise-level bounded from below : σ(X i ) ≥ σ min > 0 a.s.

(Ap) Polynomial decreasing of the bias :

∃β 1 ≥ β 2 > 0, C s , c s > 0 s.t. c s D -β 1 m ≤ l(s, s m ) ≤ C s D -β 2 m . (Ar) (pseudo)-Regular histograms : ∀m ∈ M n , min λ∈Λm {p λ } ≥ c reg D -1 m .
Let m be the model chosen by algorithm 2 (under restrictions (P1-4)),

with η ′ C ′ W (A n ) ≥ C ≥ ηC W (A n ) for some η, η ′ > 1 2 .
It satisfies, with probability at least 1 -L (A),(P) n -2 (L (A),(P) may depend on constants in (A) and

(P), but not on n), l(s, s m ) ≤ K(η, η ′ ) inf m∈Mn {l(s, s m )} . ( 4
)
At the price of enlarging L (A),(P) , the constant K(η, η ′ ) can be taken close to

(1 + 2(η ′ -1) + )(1 -2(1 -η) + ) -1
, where x + := max(x, 0). In particular, K(η, η ′ ) is almost 1 if η and η ′ are close to 1. Moreover, we have the oracle inequality

E [l(s, s m )] ≤ K(η, η ′ )E inf m∈Mn {l(s, s m )} + A 2 L (A),(P) n 2 . ( 5
)
sketch. By definition of m,

∀m ∈ M n , (pen -pen ′ id )( m) + l(s, s m ) ≤ l(s, s m ) + (pen -pen ′ id )(m)
where we replaced pen id by pen ′ id := pen id -(P n -P )γ(s). In order to obtain (4) with M n instead of M n , we show concentration inequalities for pen(m)pen ′ id (m) around zero, with remainders ≪ l(s, s m ) if D m is large (larger than some power of ln(n)). We use the following steps :

1. explicit computation of pen ′ id and pen when W is exchangeable.

2. accurate bounds on R 1,W and R 2,W , so that (1

-δ(A n ))E m [2p 2 (m)] ≤ E m [pen(m)] ≤ (1+δ(A n ))E m [2p 2 (m)] with p 2 (m) = P n (γ(s m ) -γ( s m ))
and lim An→∞ δ = 0. This needs sharp bounds on

E[Z -1 |Z > 0] with L(Z) = L(W λ | p λ )
, for each resampling scheme introduced in Sect. 3.

3. moment inequalities for pen, p 2 and p 1 (m) = P (γ( s m )γ(s m )), conditionally to ( p λ ) λ∈Λm , around their conditional expectations. This step uses results from [START_REF] Boucheron | Moment inequalities for functions of independent random variables[END_REF], or can be derived from [START_REF] Giné | Exponential and moment inequalities for U-statistics[END_REF], since all those quantites are U-statistics of order 2 (this last fact is not true without the conditioning). This implies (unconditional) concentration inequalities.

4. concentration inequality for (P n -P )(γ(s m )γ(s)) (Bernstein's inequality suffices in the bounded case).

since E

m [p 2 (m)] = E [p 2 (m)], it only remains to prove that E m [p 1 (m)] ≈ E [p 1 (m)
] and p 2 ≈ p 1 with high probability. We here use the Cramér-Chernoff method (it can be used since the ( p λ ) λ∈Λm are negatively associated [START_REF] Dubhashi | Balls and bins: a study in negative dependence[END_REF]), together with estimates of the exponential moments of the inverse of a binomial random variable. Controlling the remainder needs a lower bound on min λ∈Λm {np λ } that comes from (P4) (and Bernstein's inequality).

6. using the assumptions, all the remainders in our concentration inequalities are much smaller than E[l(s, s m )] when D m ≥ D 0 (n) = c 1 (ln(n)) c 2 (with c 1 , c 2 depending on the constants in the assumptions).

Let m ⋆ be a minimizer of l(s, s m ) over M n (with an infinite loss when s m is not uniquely defined). It remains to prove that, with large probability, D m ≥ D 0 (n), D m ⋆ ≥ D 0 (n) and m ⋆ ∈ M n . These hold for n large enough thanks to (Ap) and (Ar) (we did not use (Ar) before).

We finally show that (4) implies ( 5) : let Ω n be the event of probability 1 -L (A),(P) n -2 on which (4) occurs. On Ω c n , l(s, s m ) is bounded by A 2 , so that

E [l(s, s m )] = E [l(s, s m )½ Ωn ] + E l(s, s m )½ Ω c n ≤ K(η, η ′ )E inf m∈Mn l(s, s m ) + L (A),(P) A 2 n -2 .
Theorem 1 implies the a.s. asymptotic optimality of algorithm 3 in this framework. This means that if s and σ(X) do not make the model selection problem too hard, the resampling penalization algorithm is working, without any knowledge on the smoothness of s, the heteroscedasticity of σ or any property that the unknown law P may satisfy. In that sense, it is a naturally adaptive algorithm.

The lower bound in assumption (Ap) may seem strange, but it is intuitive that when the bias is decreasing very fast, the optimal model is of quite small dimension. Then, bounds relying on the fact that this dimension is large can not work. The same kind of assumption has already been used in the density estimation framework for the same reason [START_REF] Stone | An asymptotically optimal histogram selection rule[END_REF].

Moreover, we can prove that non-constant hölderian functions satisfy (Ap) when X has a lower-bounded density w.r.t. the Lebesgue measure on X ⊂ R. The following result states that resampling penalization is adaptive to the hölderian smoothness of s in an heteroscedastic framework, since it attains the minimax rate of convergence n -2α/(2α+1) [START_REF] Stone | Optimal rates of convergence for nonparametric estimators[END_REF].

Theorem 2. Let X be a compact interval of R and Y ⊂ R. Assume that (X i , Y i ) satisfy (Ab), (An) and the following assumptions :

(Ad) Density bounded from below : ∃c X min > 0, ∀I ⊂ X , P (X ∈ I) ≥ c X min Leb(I).

(Ah) Hölderian regression function : there exists α ∈ (0; 1] and R > 0 s.t.

s ∈ H(α, R) i.e. ∀x 1 , x 2 ∈ X , |s(x 1 ) -s(x 2 )| ≤ R |x 1 -x 2 | α .
Let M n be the family of regular histograms of dimensions 1 ≤ D ≤ n, m the model chosen by algorithm 2, with (P3-4) satisfied (α M = 0) and C like in Theorem 1. Then, denoting σ max = sup X |σ| ≤ A, there are some constants L 2,(A),(P) (that may depend on all the constants in the assumptions) and L 1 (η, η ′ , α) such that

E [l(s, s m )] ≤ L 1 n -2α/(2α+1) R 2α/(2α+1) σ 4α/(2α+1) max + L 2,(A),(P) n -2 . (6)
Moreover, if σ is K σ -Lipschitz, the constant σ 2 max may be replaced by X σ(t) 2 dt (at the price of enlarging L 2,(A),(P) ). sketch.

1. Since α ∈ (0; 1], any non-constant function s ∈ H(α, R) satisfies (Ap) with β 2 = 2α and β 1 = 1 + α -1 (the lower bound uses (Ad)).

2. Assumptions (P1), (P2) and (Ar) are automatically satisfied by the regular family, so we can use [START_REF] Birgé | Gaussian model selection[END_REF]. From the proof of Theorem 1, we obtain estimations of E [l(s, s m )]. Optimizing in D m gives (6) for nonconstant functions.

3. When s is constant, a direct proof shows that D m is at most of order ln(n) ξ 1 with large probability. This ensures that E [l(s, s m )] is at most of order (ln(n)) ξ 2 n -1 ≪ n -2α/(2α+1) for every α > 0.

Other results like Theorem 1 may be proved under other assumptions : unbounded data (with moment inequalities for the noise, regularity assumptions on s and an upper bound on σ), σ(x) that can vanish (with the unbounded assumptions, E[σ 2 (X)] > 0 and some regularity on σ), etc. We skip their detailed statements in order to focus on the last two sections, where we give a new look on V -fold cross-validation (seen from the penalization viewpoint) and illustrate theoretical results with a simulation study.

Links with V -fold cross-validation

The results of Sect. 4 assume that the weights are exchangeable. However, computing exactly the resampling penalties with such weights may be quite long : without a closed formula for pen, s W m has to be computed for at least n (and up to 2 n ) different weight vectors. Using the V -fold idea, we defined VFCV weights in Sect. 3, that allows to compute each penalty by considering only V different weight vectors. We call the resulting algorithm penVFCV.

It is quite enlightening to compare penVFCV to a more classical version of VFCV, where the final estimator is s m with m ∈ arg min

m∈Mn {crit VFCV (m)} = arg min m∈Mn 1 V V j=1 P (j) n γ( s (-j) m ) . (7)
The superscript (j) (resp. (-j)) above means that P n and s m are computed with the data belonging to the block B j (resp. to B c j ). Assuming that the V blocks have the same size (and forgetting unicity issues of s (-j) m , that may be solved as before), we have (for any j)

E [crit VFCV (m)] = P γ(s m ) + E P γ( s (-j) m ) -P γ(s m ) (8) = P γ(s m ) + V (V -1)n λ∈Λm 1 + δ (V ) n,p λ (σ r λ ) 2 + σ d λ 2
where δ

(V )
n,p λ is typically small and non-negative (when np λ is large enough). On the other hand, we can compute exactly the expectation of the pen-VFCV criterion (with a constant C = C W = V -1) when the blocks have the same size :

E [crit penVFCV (m)] = P γ(s m ) + 1 n λ∈Λm 1 + δ (penV ) n,p λ (σ r λ ) 2 + σ d λ 2 (9)
for some typically small non-negative δ (penV )

n,p λ . Comparing ( 8) and ( 9) with (2), one can see that up to small terms, both criterions are in expectation the sum of the bias and a variance term. The main difference between them lies in the constant in front of the variance : it is equal to C/(V -1) = 1 for penVFCV, whereas it is equal to V /(V -1) > 1 for VFCV.

The classical V -fold cross-validation is thus "overpenalizing" within a factor V /(V -1) because it estimates the generalization ability of s (-j) m , which is built upon less data than s m . This enlightens some clues for the choice of V : computational issues (the smaller V , the faster will be the algorithm), stability of the algorithm (V = 2 is known to be quite unstable, and leaveone-out much more stable), and overpenalization (V /(V -1) should not be too far from 1). Our analysis do not quantify the stability issue, but it is sufficient to explain why the asymptotic optimality of leave-p-out needs p ≪ n for a prediction purpose [START_REF] Li | Asymptotic optimality for C p , C L , cross-validation and generalized cross-validation: discrete index set[END_REF] and p ∼ n for an identification purpose [START_REF] Zhang | Model selection via multifold cross validation[END_REF]. Indeed, the overpenalization factor is n/(np) = (1p/n) -1 should go to 1 for optimal prediction and to infinity for a.s. identification. Moreover, from the non-asymptotic viewpoint (n small and σ large, or s irregular), it is known that overpenalization (i.e. positively biased penalties) gives better results. This means that the better V may not always be the largest one for classical V -fold, independently from computational issues.

On the contrary, penVFCV is not overpenalizing, unless we explicitly choose C > C W . We thus do not have to take into account the third factor for choosing V , so that it may be more accurate than VFCV within a smaller computation time. In the non-asymptotic viewpoint (or for an identification purpose), it is also easier to overpenalize when we need to, without destabilizing the algorithm by taking a small V .

A refined analysis of the "negligible" terms such as δ (penV ) n,p λ , compared to the expectation of p λ / p λ , explains why the leave-one-out may be overfitting a little (see the simulations hereafter). We do not detail this phenomenon since it disappears when V /(V -1) stays away from 1.

Simulations

To illustrate the results of Sect. 4 and the analysis of Sect. 5, we compare the performances of algorithm 2 (with several resampling schemes), Mallows' C p and VFCV on some simulated data. We report here four experiments, called S1, S2, HSd1 and HSd2. Data are generated according to (1) with X i i.i.d. uniform on X = [0; 1] and ǫ i ∼ N (0, 1) independent from X i . They differ from the regression function s (smooth for S, see Fig. 1 ; smooth with jumps for HS, see Fig. 4), the noise type (homoscedastic for S1 and HSd1, heteroscedastic for S2 and HSd2), the number n of data, and are repeated N = 1000 times. Instances of data sets are given in Fig. 2-3 and 5-6. Their last difference lies in the families of models M n : S1 regular histograms with 1 ≤ D ≤ n ln(n) pieces.

S2 histograms regular on 0; 1 2 and on 1 2 ; 1 , with D 1 (resp. D 2 ) pieces,

1 ≤ D 1 , D 2 ≤ n 2 ln(n) . The model of constant functions is added to M n .
HSd1 dyadic regular histograms with 2 k pieces, 0

≤ k ≤ ln 2 (n) -1.
HSd2 dyadic regular histograms with bin sizes 2 -k 1 and 2

-k 2 , 0 ≤ k 1 , k 2 ≤ ln 2 (n) -1 (dyadic version of S2
). The model of constant functions is added to M n .

We compare the following algorithms :

Mal Mallows' C p penalty : pen(m) = 2 σ 2 D m n -1 where σ 2 is the variance estimator used in [START_REF] Baraud | Model selection for regression on a fixed design[END_REF], Sect. 6.

VFCV Classical V -fold cross-validation, defined by [START_REF] Boucheron | Moment inequalities for functions of independent random variables[END_REF], with V ∈ {2, 5, 10, 20}. penVFCV V -fold penalty, with V ∈ {2, 5, 10, 20}.

C = C W = V -1.
For each of these except VFCV, we also consider the same penalties multiplied by 5/4 (denoted by a + symbol added after its shortened name). This intends to test for overpenalization.

In each experiment, for each simulated data set, we first remove the models with less than A n = 2 data points in one piece of their associated partition. Then, we compute the least-square estimators s m for each m ∈ M n . Finally, we select m ∈ M n using each algorithm and compute its true excess risk l(s, s m ) (and the excess risk of each model m ∈ M n ). Since we simulate N data sets, we can then estimate the two following benchmarks :

C or = E [l(s, s m )] E [inf m∈Mn l(s, s m )]
C path-or = E l(s, s m ) inf m∈Mn l(s, s m ) Basically, C or is the constant that should appear in an oracle inequality like (4), and C path-or corresponds to a pathwise oracle inequality like [START_REF] Birgé | Gaussian model selection[END_REF]. As C or and C path-or approximatively give the same rankings between algorithms, we only report C or in Tab. 1.

We always observe that penRad and penRHO are competitive with Mal (S1) and much better for more "difficult" problems (S2 is heteroscedastic ; jumps in HSd1 and HSd2 induce much bias). On the hand, VFCV is a little worse than Mal for easy problems (S1) and better for more difficult ones, but never better than penRad or penRHO.

The best resampling schemes (not taking overpenalization into account) are penRad and penRHO, in view of S1 and S2 (dyadic models do not induce much differences between them in HSd1 and HSd2). Then, penLOO is slightly underpenalizing and penEfr strongly overfits. The comparison penRad ≈ penRHO > penLOO ≫ penEfr can also be derived from Sect. 3.

In the four experiments, overpenalizing within a factor 5/4 leads to better results, mainly because n is quite small for the noisy (S1, S2) or irregular (HSd1, HSd2) signals observed. This is no longer the case for some larger n or smaller σ.

We consider now V -fold algorithms. VFCV is slightly better than pen-VFCV, but worse than penVFCV+. The influence of V on C or confirms the discussion of Sect. 5. For VFCV, the best V may be V = 2 (which overpenalizes, HSd1) or V = 20 (which is more stable, HSd2), or even both (S1,S2). On the contrary, penVFCV (and penVFCV+) is always improved when V increases, or at least it does not get worse. Then, the best one is penLOO (or penLOO+), i.e. V = n, the small terms δ (penV ) n,p λ being far less important than stability. This enlightens the interest of defining V -fold penalties, for which it is easier to solve the complexity-accuracy trade-off. The constants C or and C path-or are decreasing to 1 when n increases and σ decreases.

The overpenalization factor 5/4 is generally not optimal, and even not always better than 1 (in particular when n is large or σ small). We have for instance C or (penLOO) < C or (penRHO) < C or (penRHO+) in S1 with σ ≡ 0.1 (with only small differences).

On the tuning parameters

The above simulations confirm that the best weights (for accuracy) are Random hold-out (n/2) and Rademacher, whereas V -fold or leave-one-out weights may be of interest for computational purposes. The second tuning parameter, A n , may be taken equal to 2 (its "minimal" value because terms of the penalty with n p λ = 1 would be zero) without serious consequences on C or in practice.

On the contrary, the constant C ≥ C W is quite important, and the best ratio C/C W strongly depends on n, σ, s and M n . Moreover, there is no reason for C W (histograms) to be the right non-asymptotic constant in the general algorithm 1. Our suggest is to choose C with the so-called "slope heuristics", proposed by Birgé and Massart [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF] for penalties linear in dimension. Their claim is that the optimal penalty is twice the minimal penalty, i.e. the one under which the selected model is obviously too large. This leads to estimating the shape of pen id by resampling, and the constant C with the slope heuristics, as follows. 1. Choose a resampling scheme, i.e. the law of a weight vector W . Step 4 may need to artificially introduce huge models in M n , all the other ones being considered as "reasonably small". Finally, notice that C → m(C) is piecewise constant with at most Card(M n ) jumps, so that steps 3-4 have a complexity O(Card(M n )). As a consequence, the V -fold algorithm 3 is fastly computable.

. 4 .

 4 Compute the following resampling penalty for each m ∈ M n : pen(m) = CE W P n γ s m P W n -P W n γ s m P W n Minimize the penalized empirical criterion to choose m and thus s m : m ∈ arg min m∈Mn {P n γ( s m (P n )) + pen(m)} . Remark 1.

4 '.

 4 Minimize the penalized empirical criterion to choose m and thus s m : m ∈ arg min m∈ Mn

  P3) The weights are exchangeable, among the examples given in Sect. 3.

Figure 1 :Figure 4 :

 14 Figure 1: s(x) = sin(πx)

penEfr

  Efron (n) penalty, C = C W = 1. penRad Rademacher penalty, C = C W = 1. penRHO Random hold-out (n/2) penalty, C = C W = 1. penLOO Leave-one-out penalty, C = C W = n -1.

Remark 3 .

 3 We only report here the result of 4 experiments, but several other ones (with n larger, σ smaller, σ(x) = ½ x∈[ 1 2 ;1] or other regression functions s such as Doppler, √ • and a regular histogram) give the same kind of results.

2 .. 3 .

 23 Compute the following resampling penalty for each m ∈ M n : pen 0 (m) = E W P n γ s m P W n -P W n γ s m P W n Compute the selected model m(C) as a function of C > 0 m(C) ∈ arg min m∈Mn {P n γ( s m (P n )) + C pen 0 (m)} . 4. Choose the minimal C = C such that D m(C) is "reasonably small", and take m = m(2 C).

Table 1 :

 1 Accuracy indexes C or for each algorithm in four experiments, ± a rough estimate of uncertainty of the value reported (i.e. the empirical standard deviation divided by √ N). In each column, the more accurate algorithms (taking the uncertainty into account) are bolded. ± 0.04 3.864 ± 0.02 1.606 ± 0.015 1.487 ± 0.011 Mal+ 1.800 ± 0.03 4.047 ± 0.02 1.606 ± 0.015 1.487 ± 0.011 2-FCV 2.078 ± 0.04 2.542 ± 0.05 1.002 ± 0.003 1.184 ± 0.004 5-FCV 2.137 ± 0.04 2.582 ± 0.06 1.014 ± 0.003 1.115 ± 0.005 10-FCV 2.097 ± 0.05 2.603 ± 0.06 1.021 ± 0.003 1.109 ± 0.004 20-FCV 2.088 ± 0.04 2.578 ± 0.06 1.029 ± 0.004 1.105 ± 0.004 penEfr 2.597 ± 0.07 3.152 ± 0.07 1.067 ± 0.005 1.114 ± 0.005 penRad 1.973 ± 0.04 2.485 ± 0.06 1.018 ± 0.003 1.102 ± 0.004 penRHO 1.982 ± 0.04 2.502 ± 0.06 1.018 ± 0.003 1.103 ± 0.004 penLOO 2.080 ± 0.05 2.593 ± 0.06 1.034 ± 0.004 1.105 ± 0.004 pen2-FCV 2.578 ± 0.06 3.061 ± 0.07 1.038 ± 0.004 1.103 ± 0.005 pen5-FCV 2.219 ± 0.05 2.750 ± 0.06 1.037 ± 0.004 1.104 ± 0.004 pen10-FCV 2.121 ± 0.05 2.653 ± 0.06 1.034 ± 0.004 1.104 ± 0.004 pen20-FCV 2.085 ± 0.04 2.639 ± 0.06 1.034 ± 0.004 1.105 ± 0.004 penEfr+ 2.016 ± 0.05 2.605 ± 0.06 1.011 ± 0.003 1.097 ± 0.004 penRad+ 1.799 ± 0.03 2.137 ± 0.05 1.002 ± 0.003 1.095 ± 0.004 penRHO+ 1.798 ± 0.03 2.142 ± 0.05 1.002 ± 0.003 1.095 ± 0.004 penLOO+ 1.844 ± 0.03 2.215 ± 0.05 1.004 ± 0.003 1.096 ± 0.004 pen2-FCV+ 2.175 ± 0.05 2.748 ± 0.06 1.011 ± 0.003 1.106 ± 0.004 pen5-FCV+ 1.913 ± 0.03 2.378 ± 0.05 1.006 ± 0.003 1.102 ± 0.004 pen10-FCV+ 1.872 ± 0.03 2.285 ± 0.05 1.005 ± 0.003 1.098 ± 0.004 pen20-FCV+ 1.898 ± 0.04 2.254 ± 0.05 1.004 ± 0.003 1.098 ± 0.004 Algorithm 3 (Resampling penalization with slope heuristics).

	Experiment	S1	S2	HSd1	HSd2
	s	sin	sin	HeaviSine	HeaviSine
	σ(x)	1	x	1	x
	n (data)	200	200	2048	2048
	M n	regular	2 bin sizes	dyadic, regu-lar	dyadic, 2 bin sizes
	Mal	1.928

Acknowledgements

I gratefully thank Pascal Massart for many fruitful discussions.


Davit Hakobyan

Etienne Brasselet
email: etienne.brasselet@u-bordeaux.fr

Optical torque reversal and spin-orbit rotational Doppler shift experiments

Keywords:

We report on optical rotational Doppler frequency shift experiments in the context of a counter-intuitive optomechanical phenomenon that is the angular analog of so-called negative optical radiation forces, which involves spin-orbit scattering of light. In practice, spin-orbit opto-mechanical effects arising from the interaction between polarized light and azimuthally varying birefringent optical elements are retrieved from mechano-optical experiments that involve spatial of the medium. Two kinds of experiments (single-beam and two-beam geometries) are performed and both approaches are discussed in the framework of previous dynamical geometric phase and rotational Doppler shift experiments based on spin and/or orbital angular momentum of light.

Introduction

It is a common knowledge that light can exert torque on matter, which is associated with lightmatter angular momentum exchange following the principles of mechanics. The founding experiment on optical radiation torque is that reported by Beth in 1936, who measured the spin angular momentum carried by circularly polarized light by means of a torsion pendulum made of a transparent birefringent crystal slab [START_REF] Beth | Mechanical detection and measurement of the angular momentum of light[END_REF]. Several decades later, the spin angular momentum of light became a powerful tool in the development of optical manipulation techniques. Indeed, the angular degree of freedom of microparticles can be all-optically controlled via dissipative [START_REF] Friese | Optical torque controlled by elliptical polarization[END_REF] or non-dissipative [START_REF] Friese | Optical alignment and spinning of laser-trapped microscopic particles[END_REF] transfer of spin angular momentum to matter, the direction of the optical torque being that of the incident spin angular momentum.

Intriguingly, a few years ago, it has been predicted that a tightly focused circularly polarized Gaussian beam could exert a torque directed oppositely to that of the incident angular momentum on wavelength-sized prolate particles made of transparent isotropic media [START_REF] Simpson | Optical trapping of spheroidal particles in gaussian beams[END_REF]. Much more recently, the case of arrays of dielectric microspheres with wavelength size has also been investigated numerically [START_REF] Chen | Negative optical torque[END_REF] whereas the experimental identification of such counter-intuitive optical torque has been reported by using transparent space-variant form-birefringent optical elements illuminated by paraxial circularly polarized Gaussian beam [START_REF] Hakobyan | Left-handed optical radiation torque[END_REF]. Noticeably, the angular analog of negative optical forces [START_REF] Hakobyan | Left-handed optical radiation torque[END_REF] has been introduced as a 'lefthanded' effect rather than a 'negative' one, hence avoiding potential confusion with the sign of the incident angular momentum, which can be positive or negative, irrespective of whether it induces usual or counter-intuitive mechanical effects. Also, this echoes the fact that physicists usually refer to a phenomenon as being left-handed to emphasize its counter-intuitive nature. The concept is summarized in Fig. 1 where both right-handed and left-handed manifestations of the optical torque per photon τ z , are illustrated in the case of an incident Gaussian beam propagating along the z axis and carrying a (spin) angular momentum per photon s z . Namely the former case corresponds to s z τ z > 0 whereas the latter situation refers to s z τ z < 0, whatever is the value of s z .

In fact, unconventional manifestations of mechanical effects of light attracted some attention during the past few years. In particular, several studies have been devoted to the concept of socalled negative optical forces, namely optical forces that push objects upstream of an incident photon flux, see review paper [START_REF] Dogariu | Optically induced 'negative forces[END_REF], whose first experimental demonstrations have been reported quite recently [START_REF] Brzobohaty | Experimental demonstration of optical transport, sorting and self-arrangement using a 'tractor beam[END_REF][START_REF] Kajorndejnukul | Linear momentum increase and negative optical forces at dielectric interface[END_REF]. Other examples of non-trivial optomechanics have also been observed, such as optical forces driven by the spin momentum density [START_REF] Angelsky | Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams[END_REF], and predicted, such as transverse optical forces and torques [START_REF] Bliokh | Extraordinary momentum and spin in evanescent waves[END_REF][START_REF] Wang | Lateral optical force on chiral particles near a surface[END_REF][START_REF] Canaguier-Durand | Transverse spinning of a sphere in a plasmonic field[END_REF][START_REF] Bekshaev | Transverse Spin and Momentum in Two-Wave Interference[END_REF] that are directed perpendicular to the incident photon flux. Regarding the angular momentum of light, it is likely that its transverse and longitudinal components of both spin and orbital nature [START_REF] Bliokh | Transverse and longitudinal angular momenta of light[END_REF] will bring other optomechanical surprises in the future.

Here we present complementary experimental analysis with respect to the original report on optical torque reversal that takes place in the case of space-variant optically anisotropic transparent media [START_REF] Hakobyan | Left-handed optical radiation torque[END_REF]. Indeed, in the latter work, the observations consisted in the detection of rotational Doppler frequency shifts when such spin-orbit optical couplers are rotated around the propagation axis of the incident beam, which is associated to optical torque exerted on the sample. This was done by observing interference pattern obtained from the coherent superposition of the output beam and a non-collinear reference beam. In present study, we report on the two-beam (output and reference fields) collinear variant and single-beam (output field alone) option. Both approaches are compared and discussed in the framework of previous dynamical geometric phase and rotational Doppler shift experiments based on spin and/or orbital angular momentum of light. 2. Spin-orbit optomechanics of space variant birefringent media 2.1. Materials: dielectric q-plates Following previous work [START_REF] Hakobyan | Left-handed optical radiation torque[END_REF], optical torque reversal experiments reported here are performed using nanostructured glass slabs having a uniform birefringent phase retardation of π (i.e. halfwave plate condition) for the considered wavelength (here 532 nm) and azimuthally varying optical axis orientation angle ψ of the form ψ = mφ with m half-integer and φ the usual polar angle in the (x, y) plane, as illustrated in Figs. 2(a)-2(c). These anisotropic inhomogeneous optical elements, often called 'q-plates' in reference to the work by Marrucci et al. [START_REF] Marrucci | Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media[END_REF], have been fabricated by Altechna R&D using femtosecond laser writing of self-assembled nanostructures in silica glass. Indeed, subwavelength structuring induces form birefringence with in-plane slow/fast axis oriented perpendicular/parallel to the nanograting wavevector [START_REF] Shimotsuma | Self-organized nanogratings in glass irradiated by ultrashort light pulses[END_REF]. This is assessed experimentally by observing the samples in between crossed linear polarizers, as shown in Figs. 2(d)-2(f).

Optical consequences of spin-orbit interaction

As detailed in [START_REF] Beresna | Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass[END_REF], the magnitude of the induced effective optical anisotropy can be precisely controlled by adjusting the various writing parameters. By specifying the half-wave plate condition, an incident circularly polarized paraxial Gaussian beam carrying (spin) angular momentum per photon σ h (σ = ±1) along the propagation axis z will emerge from the sample with a total angular momentum that is the sum of a -σ h spin contribution and a 2σ mh orbital contribution [START_REF] Marrucci | Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media[END_REF], which accounts for a complex representation for the electric field of the form exp(-iωt + ikz) with ω the angular frequency of light, k the wavevector and time t.

In other words, an incident circularly polarized paraxial Gaussian beam is transformed into a contra-circularly polarized optical vortex beam carrying on-axis optical phase singularity with topological charge 2σ m. In fact, this can be easily grasped by evaluating the Jones matrix of a static q-plate of order m in the circular polarization basis (c + , c -), with c σ = (x + iσ y)/ √ 2. Namely, up to an unimportant phase factor,

J (m) static = 0 e +2imφ e -2imφ 0 . (1)
Optical vortex generation is therefore described by applying J (m) static on the incident Jones vector (1, 0) T when σ = 1 or (0, 1) T when σ = -1, X T being the transpose of X.

This is experimentally illustrated in Figs. 2(g)-2(i) where the output doughnut-shaped intensity pattern is shown for m = 1/2, 1 and 3/2. Such optical spin-orbit interaction phenomenon that couples the spin and orbital angular momentum contributions of light finds its origin in the geometric phase arising from the spatial rotation of the local coordinate frame attached to the anisotropy axis [START_REF] Biener | Formation of helical beams by use of Pancharatnam-Berry phase optical elements[END_REF].

Mechanical consequences of spin-orbit interaction

Besides optical vortex generation from spin-orbit interaction, mechanical effects may also occur. Indeed, angular momentum conservation of the light-matter system implies that an optical radiation torque τ z per photon is exerted on the spin-orbit coupler. More precisely, under incident circularly polarized Gaussian beam with helicity σ , one has

τ (σ ) z = τ (σ ) z, spin + τ (σ ) z, orbital , (2)
where the spin and orbital contribution to the total torque are respectively τ

z, spin = 2σ h and τ (σ ) z, orbital = -2σ mh, which gives τ

(σ ) z = 2σ (1 -m)h . (3)
More generally, in the case of an elliptically polarized incident beam characterized by the ellipticity angle χ [START_REF] Born | Principles of Optics[END_REF] (-π/4 ≤ χ ≤ π/4), which carries h sin 2χ spin angular momentum per photon, the optical radiation torque can be found by decomposing the incident field on the circular polarization basis. One obtains

τ (χ) z = 1 2 (1 + sin 2χ) τ (+) z + (1 -sin 2χ) τ (-) z , (4) hence τ
(χ) z = 2 sin 2χ (1 -m)h . (5)
Right-handed optical torque, τ

z sin 2χ > 0, is thus obtained for m < 1 whereas left-handed situation, τ (χ) z sin 2χ < 0, occurs for m > 1 when the radiation torque is dominated by the orbital contribution rather than that of the spin. On the other hand, the particular case m = 1 gives zero torque, which is a mere consequence of the rotational invariance of the medium around the z axis. The case of an incident linearly polarized Gaussian beam (χ = 0) also leads to zero net torque but this results from the exact compensation between the respective contributions of the two circular components of the field.

Experimental approach

Optical radiation torque reversal is here addressed by considering two distinct rotational Doppler shift experiments performed using a continuous-wave laser source operating at 532 nm wavelength. Both experiments, which consist to rotate the sample around the z axis at constant angular frequency Ω, are presented and analyzed in the next section.

The main idea is to probe the opto-mechanical effect of light on the q-plate from the assessment of the mechano-optical effect of the rotating q-plate on light, thereby circumventing the practical difficulty associated with the direct observation of light-induced rotation. Indeed, as previously discussed in [START_REF] Hakobyan | Left-handed optical radiation torque[END_REF], the fact that a circularly polarized incident beam exerts a nonzero optical radiation torque (which implies m = 1) on the sample is associated with a rotational Doppler frequency shift δ ω imparted to the output contra-circularly polarized light field as the q-plate is rotated. The expression of the frequency shift can be derived following energy conservation principle for the 'light+matter' system considered as isolated, namely w + hδ ω = 0 where w = τ z Ω is the work per photon produced on the slab, which gives

δ ω = 2σ Ω(m -1) . (6)
It is also instructive to retrieve Eq. ( 6) from Jones formalism by evaluating the time-dependent Jones matrix that describes the rotating q-plate. First, we note that the optical axis orientation angle ψ, in the fixed reference frame (x, y, z), is rotated by an angle (1m)ϕ when the sample is rotated by an angle ϕ around the z axis. The dynamical Jones matrix in the circular polarization basis is thus obtained from Eq. ( 1) by applying the transformation ψ → ψ + (1m)Ωt, assuming that the static q-plate starts to rotate with angular velocity Ω at t = 0. This gives

J (m) dynamic = 0 e +2i[mφ +(1-m)Ωt] e -2i[mφ +(1-m)Ωt] 0 . (7)
According to the adopted complex representation mentioned above that implies angular frequency definition of the form -∂ Φ/∂t, where Φ denotes the phase of the optical field, one thus retrieves from Eq. ( 7) the expression for the frequency shift given by Eq. ( 6).

Spin-orbit rotational Doppler shift experiments

Two-beam experiment

In a first set of experiments we have accessed to rotational Doppler shift by using the setup shown in Fig. 3. In practice, the temporal behavior of interferograms obtained using a Mach-Zehnder interferometer is video recorded. Such interferogram consists of the collinear coherent superposition of the output light field with a reference Gaussian beam whose circular polarization state is orthogonal to that of the incident Gaussian beam. The q-plate is rotated at constant angular velocity Ω ≈ 0.4 rad/s by placing the samples on a motor-controlled rotating stage. The results are shown in Fig. 4 for m = (1/2, 1, 3/2).

For each m, a snapshot of the 2|m|-arm spiraling intensity pattern observed in the (x, y) plane is shown in Figs. 4(a)-(c). The dynamics of the latter pattern is quantitatively retrieved by looking at the time dependence of the correlation coefficient C between the intensity patterns at time 0 and t, see Figs. 4(d)-(f) that correspond to 120 seconds acquisition time duration. The observed time dependence of C(t) as a function of m follows the predicted dependence of the frequency shift δ ω given by Eq. ( 6). Indeed, overall static behavior is found for m = 1, which is basically associated with the rotational invariance around the z axis of the q-plate. In contrast, when m = 1, C(t) exhibits sinusoidal behavior with angular frequency Ω C that satisfies the expected relationship Ω C = |δ ω| = 2|(m -1)Ω|. Indeed, data fitting gives Ω C = 0.36 rad/s for m = 1/2 and Ω C = 0.40 rad/s for m = 3/2. -σ . Optical elements: P is linear polarizer, HWP is half-wave plate, QWP is quarter-wave plate, QP is q-plate, M is mirror, PBS is polarizing beam splitter, NPBS is non-polarizing beam splitter, CCD is imaging device. Remarkably, the reminiscence of optical torque reversal from a right-handed to a left-handed situation while the input angular momentum is kept fixed as m increases from 1/2 to 3/2 [see the factor m -1 in Eq. ( 5)] is retrieved from the space-time representation of the interferograms shown in Figs. 4(g)-(i). Indeed, handedness reversal of the screw-shaped spatiotemporal interferograms is observed when passing from m = 1/2 [Fig. 4(g)] to m = 3/2 [Fig. 4(i)].

Single-beam experiment

Following the sketch shown in Fig. 5, reference-free interference pattern can also be obtained by using a linearly polarized (here along the x axis) incident Gaussian beam instead of a circularly polarized one. Indeed, the incident field can be described in that case as a coherent superposition of orthogonally polarized circularly polarized identical Gaussian beams, each of them exerting on the sample an optical radiation torque with equal magnitude and opposite sign. Although the total torque exerted on the sample is zero, the individual optomechanical contributions of each circularly polarized component of the incident field can nevertheless be retrieved by rotating the sample at constant angular frequency Ω.

Practically, this is done by projecting the output field along the y axis by placing a linear polarizer after the sample, as depicted in Fig. 5. This allows orthogonal circularly polarized output components to interfere and produce a 4|m|-fold rotationally symmetric patterns in the (x, y) plane, as illustrated in the insets of Fig. 6 that correspond to the situation at rest. More precisely, the output field just after the q-plate is described by

E (m,Ω=0) after QP = J (m) static (1/ √ 2, 1/ √ 2) T = cos(2mφ ) x + sin(2mφ ) y . (8)
The corresponding beam is characterized by an azimuthally varying linear polarization state that results from the superposition of two contra-circularly polarized vortex beams of opposite topological charges ±2m and is known as a vector beam, see for instance [START_REF] Maurer | Tailoring of arbitrary optical vector beams[END_REF]. From Eq. ( 8), the projection on the output linear polarizer thus gives a beam with azimuthally modulated intensity proportional to sin 2 (2mφ ). When the sample is rotated around the z axis, the azimuthally modulated intensity pattern acquires a dynamical behavior for m = 1, as shown in Figs. 6(a)-6(c) that present the volumetric rendering of the spatiotemporal intensity pattern for m = (1/2, 1, 3/2) over two full rotations of the sample; here the angular velocity is Ω = 0.44 rad/s with corresponding time duration T ≈ 30 s. Such a behavior is reminiscent of oppositely frequency shifted circular components that emerge from non-axisymmetric samples. The underlying optical torque reversal for the situations m = 1/2 and m = 3/2 is grasped from spatiotemporal patterns with opposite handedness. In fact these observations can be retrieved by evaluating the projection on the output polarizer of the dynamical Jones vector that emerges from the rotating q-plate,

E (m,Ω) after QP = J (m) dynamic (1/ √ 2, 1/ √ 2) T , = cos(2[mφ + (1 -m)Ωt)] x + sin(2[mφ + (1 -m)Ωt)] y , (9)
which gives E , owing to the output polarizer oriented along the y axis for linearly polarized incident Gaussian beam along the x axis with waist radius w ≈ 1 mm. Optical elements: P in/out are input/output linear polarizers, QP is q-plate, and CCD is imaging device. Assuming a radial dependence of the output beam of the form of a paraxial Laguerre-Gaussian beam with azimuthal index l = 2m and radial index p = 0 (which formally neglects the radial modal content of the field generated by the q-plate, but does not affect qualitatively the predictions) the corresponding spatiotemporal intensity pattern is described as

I(r,t) = (r/w) 4|m| exp(-2r 2 /w 2 ) sin 2 (2[mφ + (1 -m)Ωt]) , (11)
where w is the beam waist radius, see Figs. 6(d)-6(f) and a fair agreement with experimental data is obtained.

Discussion and conclusion

Early rotational Doppler frequency shift experiments were performed with uniform uniaxial optical retarders, namely rotating half-wave plates [START_REF] Somargren | Up/down frequency shifter for optical heterodyne interferometry[END_REF][START_REF] Garetz | Variable frequency shifting of circularly polarized laser radiation via rotating halfwave retardation plate[END_REF] where the frequency shift imparted to light has been treated using Jones formalism. Later, variants have been also reported, for instance using a rotating quarter-wave plate, albeit described from the point of view of the geometric phase [START_REF] Simon | Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment[END_REF]. Still, in [START_REF] Simon | Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment[END_REF], Simon et al. have emphasized the equivalence of the rotational Doppler effect and geometric phase in systems with rotating elements. Such an equivalence has been rediscussed later in the framework of energy conservation considerations in [START_REF] Bretenaker | Energy exchange between a rotating retardation plate and a laser beam[END_REF] by Bretenaker and Le Floch, who also provided identical conclusions supported by experimental observations. Rotational Doppler shift experiments have also been extended to the case of orbital angular momentum [START_REF] Courtial | Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum[END_REF], which has provided experimental foundations to earlier theoretical predictions [START_REF] Allen | Azimuthal doppler shift in light beams with orbital angular momentum[END_REF][START_REF] Nienhuis | Doppler effect induced by rotating lenses[END_REF][START_REF] Bialynicki-Birula | Rotational frequency shift[END_REF]. On the other hand, present work should also be replaced in the context of vector beams since they are involved in the single-beam experiment. Indeed, when rotated, a vector beam acquires a geometric phase that is related to its rotational symmetry [START_REF] Milione | Higher order Pancharatnam-Berry phase and the angular momentum of light[END_REF]. Since in our case the generated vector beam is put into rotation, see Eq. ( 9), the measured rotational frequency shift manifests as a dynamical geometric phase for vector beams. From a general point of view, the equivalence mentioned above is not restricted to propagating fields and also holds for surface waves as reported for instance by Bliokh et al. in [START_REF] Bliokh | Coriolis effect in optics: unified geometric phase and spin-hall effect[END_REF] in the case of visible domain radiation (λ ∼ 0.5 µm) and by Dahan et al. in [27] in the case of thermal radiation (λ ∼ 10 µm).

Noticeably, the connection between optical radiation torque driven by the spin angular momentum of light and rotation-induced frequency shift has been discussed a long time before above mentioned works, in the case of a microwave experiment [START_REF] Allen | A radiation torque experiment[END_REF].

In such context, present experiments carried out with space-variant optically anisotropic structures, which are basically spin-orbit scattering transparent optical elements, constitute an optomechanical spin-orbit approach to the long-lasting topic of optical rotational Doppler frequency shifts, dynamical geometric phase and angular momentum of light. In particular we mention the unified picture in terms of the Coriolis effect reported in [START_REF] Bliokh | Coriolis effect in optics: unified geometric phase and spin-hall effect[END_REF] and that is associated to observations in a noninertial reference frame, noting that in our case it is the medium that rotates instead of the observer; in turn one passes from one situation to another by mere change of sign of the involved angular velocity.

We also note that the spin-orbit version proposed here shares some analogy with previous experimental work regarding the additive contribution of the spin and orbital angular momentum components of light to the frequency shift [START_REF] Courtial | Rotational frequency shift of a light beam[END_REF]. Moreover, our results bring an an experimental playground to previous theoretical study that discussed the use of spin-orbit rotational Doppler frequency shifts for encoding an extra degree of freedom into a light field [START_REF] Chen | Sorting photons of different rotational doppler shifts (rds) by orbital angular momentum of single-photon with spin-orbit-rds entanglement[END_REF]. Finally, as far as optomechanics itself is concerned, we stress that the direct experimental observation of a left-handed optical radiation torque is still an open issue at present day on which we should hopefully report in a near future.

Fig. 1 .

 1 Fig. 1. Illustration of intuitive and counter-intuitive mechanical manifestations of an optical torque per photon τ z exerted on matter by an incident Gaussian beam carrying (spin) optical angular momentum s z per photon along its propagation direction z. (a) Right-handed situation, s z τ z > 0 whatever s z . (b) Left-handed situation, s z τ z < 0 whatever s z .

Fig. 2 .

 2 Fig. 2. Design and characterization of azimuthally varying half-wave plate retarders of azimuthal order m = 1/2 (upper row), 1 (middle row) and 3/2 (bottom row) used in this work, with 532 nm operating wavelength. For each m: the left panel shows slow-axis spatial distribution in the (x, y) plane, where segments refer to the local slow-axis local orientation; the middle panel is the image of the inhomogeneous retarder observed between crossed linear polarizers whose direction are indicated as white cross in panel (d); the right panel displays output intensity profile for circularly polarized incident Gaussian beam, which corresponds to an optical vortex beam of topological charge ±2m where the sign depends on the incident polarization state handedness.

Fig. 3 .

 3 Fig. 3. Sketch of the two-beam rotational Doppler experimental set-up used to videorecord the interference pattern between (i) the frequency shifted output beam emerging from the sample, labeled c (ω+2σ Ω(m-1)) -σ , illuminated by collimated circularly polarized Gaussian beam with waist radius w ≈ 1 mm and helicity σ , c (ω) σ , and (ii) a collinear reference beam with helicity -σ , c (ω)

  (0),I(t)) m = 1/2. Fitting line 0.45+ 0.51sin(0.36t+0,61) (0),I(t)) m = 3/2 fitting line 0.668+0.3184sin(0.4t+1.19) line 0.668+0.3184sin(0.4t+1.19)

Fig. 4 .

 4 Fig. 4. Two-beam rotational Doppler experimental results for m = 1/2 (upper row), m = 1 (middle row) and m = 3/2 (bottom row) by using the set-up depicted in Fig. 3. For each m: left panel displays a snapshot of the interference intensity pattern observed in the (x, y) plane at t = 0; middle panel shows the time dependence of the correlation coefficient C between intensity patterns at t = 0 and t where the solid line (for m = 1/2 and 3/2) refers to fit by sinusoidal function, see text for details; right panel is a volumetric representation of the recorded interferogram in the (x, y,t) spatiotemporal frame during T = 120 s.

Fig. 5 .

 5 Fig. 5. Single-beam rotational Doppler experimental sketch that aims at videorecording the interference pattern between the two frequency shifted circular components c (ω+2σ Ω(1-m)) σ

Fig. 6 .

 6 Fig. 6. Single-beam rotational Doppler experiment for m = 1/2 (left column), m = 1 (middle column) and m = 3/2 (right column) following the set-up shown in Fig. 5. Snapshot of the intensity profile in the (x, y) plane of the linearly polarized output component that is orthogonal to that of a linearly polarized incident Gaussian beam is shown in the inset of each panel whereas the spatiotemporal behavior are represented with volumetric rendering over a duration that corresponds to two full rotations of the sample, T ≈ 30 s. Upper row: experiment. Bottom row: model.

Acknowledgment

This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of "the Investments for the future" Programme IdEx Bordeaux -LAPHIA (ANR-10-IDEX-03-02).