Probabilistic Optimal Estimation and Filtering

Transcription

1 Probabilistic Optimal Estimation and Filtering Least Squares and Randomized Algorithms Fabrizio Dabbene 1 Mario Sznaier 2 Roberto Tempo 1 1 CNR - IEIIT Politecnico di Torino 2 Northeastern University Boston Workshop on Uncertain Dynamical Systems, Udine, Italy, August 2011 Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

2 Motivation: Identification for Robust Control The classical approach to system identification is based on statistical assumptions about the measurement error, and provides estimates that have stochastic nature Worst-case identification, on the other hand, only assumes the knowledge of deterministic error bounds, and provides guaranteed estimates, thus being in principle better suited for robust control design However, a main limitation of such deterministic bounds lies in the fact that they often turn out being overly conservative, thus leading to estimates of limited use Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

3 Motivation: Identification for Robust Control A re-approahment We propose a re-approachement of the two paradigms: stochastic and worst-case, introducing probabilistically optimal estimates The main idea is to exclude" sets of measure at most ǫ (accuracy) from the set of deterministic estimates We are decreasing the so-called worst-case radius of information at the expense of a probabilistic risk." We compute a trade-off curve which shows how the radius of information decreases as a function of the accuracy Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

4 The IBC setting for systems ID and filtering Estimation problem: Given an unknown element x, find an estimate of the function S(x), based on a priori information K and on measurements of the function I(x) corrupted by additive noise q. Ingredients (sets) A problem element set X, with prior information K X A measurement space Y A solution space Z Ingredients (operators) An information operator I : X Y Additive uncertainty/noise y = Ix + q A solution operator S : X Z Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

5 The IBC setting for systems ID and filtering Estimation problem: Given an unknown element x, find an estimate of the function S(x), based on a priori information K and on measurements of the function I(x) corrupted by additive noise q. Ingredients (sets) A problem element set X, with prior information K X A measurement space Y A solution space Z Ingredients (operators) An information operator I : X Y Additive uncertainty/noise y = Ix + q A solution operator S : X Z Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

6 IBC Setting for System ID Estimation algorithm Estimation algorithm An algorithm A is a mapping (in general nonlinear) from Y into Z, i.e. A : Y Z An algorithm provides an approximation A(y) of Sx using the available information y Y of x K The outcome of such an algorithm is called an estimator and the notation ẑ = A(y) is used Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

7 The IBC setting for systems ID and filtering Illustration of the considered framework Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

8 The setup of this talk The IBC setting for systems ID and filtering Problem element set X is R n The information operator I : X Y is linear The uncertainty q Q R m, where Q is a bounding set The solution set Z is R s and the solution operator S : X Z is linear Assumption (Sufficient information) We assume that the information operator I is a one-to-one mapping, i.e. m n and rank I = n We assume for the sake of simplicity that the three sets X, Y, Z are equipped by the same l p norm. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

9 Example System parameter identification Parameter identification problem which has the objective to identifying a linear system from noisy measurements. The problem elements are the input-output pairs ξ = ξ(t, x) of a dynamic system, parametrized by some unknown parameter vector x K X and with given basis functions ϕ i (t) ξ(t, x) = n x i ϕ i (t) = Φ T (t)x i=1 Suppose then that m noisy measurements of ξ(t, x) are available for t 1 < t 2 < < t m, y = Ix + q = [Φ(t 1 ) Φ(t m )] T x + q. (1) The solution operator is given by the identity, Sx = x and Z X. In this context, one usually assumes unknown but bounded errors q i R, i = 1,...,m, that is Q = B (R) Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

10 The consistency set I 1 (y) A key role is played by following set, which represents the set of all problem elements x K X compatible with (i.e. not invalidated by) the information Ix, the uncertainty q and the bounding set Q Consistency set I 1 (y) For given y Y, define I 1 (y). = {x K there exists q Q : y = Ix + q} (2) Under the sufficient information assumption, the set I 1 (y) is bounded. For instance, in the previous example we have { } I 1 (y) = x K : y [Φ(t 1 ) Φ(t m )] T x R In system identification, I 1 (y) is sometimes referred to as parameter feasible set. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

11 The IBC setting for systems ID and filtering Our setup for this talk Illustration of the considered framework Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

12 Worst-Case Setting The Worst Case Setting Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

13 Worst-Case Setting Errors and optimal algorithms Given perturbed information y Y, the worst-case error is defined as r wc (A, y) =. max Sx A(y) p. x I 1 (y) This error is based on the available information y Y about x K, and it measures the approximation error between Sx and A(y) An algorithm A wc o is called worst-case optimal if it minimizes the error r wc (A, y) for any y Y r wc o (y) =. r wc (A wc o, y) =. inf r wc (A, y) A A worst-case optimal estimator is given by ẑo wc = A wc o (y) The minimal error ro wc (y) is called the (local) worst-case radius of information Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

14 Chevbychev center and central algorithms The Chebychev center z c (H) of a set H Z and its radius r c (H) are defined as max h z c(h) =. inf max h z =. r c (H) h H z Z h H Optimal algorithms map data y into the Chebychev center of the set SI 1 (y), i.e. z c (SI 1 (y)) = ẑ wc o For this reason they are also called central algorithms For given set H, the l p -Chebychev center x c (H) and radius r c (H) are the center and radius of the smallest l p ball enclosing H. In general z c (H) may not be unique and not necessarily it belongs to H. if H is centrally symmetric then the origin is a Chebychev center of H Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

15 Chevbychev center and central algorithms The Chebychev center z c (H) of a set H Z and its radius r c (H) are defined as max h z c(h) =. inf max h z =. r c (H) h H z Z h H Optimal algorithms map data y into the Chebychev center of the set SI 1 (y), i.e. z c (SI 1 (y)) = ẑ wc o For this reason they are also called central algorithms For given set H, the l p -Chebychev center x c (H) and radius r c (H) are the center and radius of the smallest l p ball enclosing H. In general z c (H) may not be unique and not necessarily it belongs to H. if H is centrally symmetric then the origin is a Chebychev center of H Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

16 Related literature The computation of the worst-case radius of information ro wc (y) and of the derivation of optimal algorithms A wc o have been the focal point of a vaste literature in a system identification setting If the norm is l 2 and K X, then the linear optimal estimator is the least squares algorithm A ls (y) = Sx ls, with Ix ls y 2. = min x X Ix y 2 In this case, A ls (y) is the Chebychev center of the ellipsoid I 1 (y) For l norms, an optimal algorithm and the radius of information can be computed solving 2n linear programs, corresponding to computing the center of the tightest hyperrectangle containing the polytope SI 1 (y) Spline algorithms have also been introduced, defined as follows A sp (y) = Sx sp (y) where Ix sp (y) y. = min x X Ix y. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

17 Illustrating example Consider the problem of estimating the parameters of a MA(3) model ξ(k) = x 1 u(k)+x 2 u(k 1)+x 3 u(k 2)+q(k), k = 1,...,50 where u(k) is known and q is a uniformly distributed noise with q(k) 0.5 WC optimal (central) algorithm: blu, LS estimate: yellow, spline algorithm: red Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

18 Worst-Case Setting Pro and contra The worst-case setting is ideal for robust control, since it provides explicit bound on the parameter uncertainty Unfortunately, in many problem instances, these bound may be very large Probabilistic approach: In recent years, a parallel approach to robust control has emerged, aimed at guranteeing robust performance for most of the parameters values The idea at the basis of this talk is: Why not apply this approach directly to the identification problem, thus providing bounds guaranteed for most of the cases? Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

19 Probabilistic setting A Probabilistic Setting Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

20 Probabilistic setting Random uncertainty We still assume that q Q, but we also assume to have information on the distribution of q (note that this info is usually available) Objective: To derive optimal algorithms and to compute the related errors for when the uncertainty q is random... In this setting, the error of an algorithm is measured in a worst-case sense, but we disregard a set of measure at most ǫ (0, 1) from the consistency set I 1 (y) Assumption (Random measurement uncertainty) We assume that the measurement noise q is a real random vector with given probability density p Q (q) and support set Q R m. Denote by P Q (q) the probability distribution of q, and by µ Q the probability measure generated by p Q (q) over the set Q. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

21 Probabilistic setting Random uncertainty We still assume that q Q, but we also assume to have information on the distribution of q (note that this info is usually available) Objective: To derive optimal algorithms and to compute the related errors for when the uncertainty q is random... In this setting, the error of an algorithm is measured in a worst-case sense, but we disregard a set of measure at most ǫ (0, 1) from the consistency set I 1 (y) Assumption (Random measurement uncertainty) We assume that the measurement noise q is a real random vector with given probability density p Q (q) and support set Q R m. Denote by P Q (q) the probability distribution of q, and by µ Q the probability measure generated by p Q (q) over the set Q. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

22 Induced measure over I 1 (y) and SI 1 (y) Probabilistic setting The probability measure over the set Q induces, by means of equ. (1), a probability measure µ I 1 over the set I 1 (y) For any measurable set B X, we can measure it through the probability measureµ Q as follows: µ I 1(B) = µ Q (q Q x B I 1 (y) : Ix + q = y) This conditional measure is such that points outside the consistency set I 1 (y) have measure zero, and µ I 1( I 1 (y) ) = 1, that is this induced measure is concentrated over I 1 (y) We denote by P I 1 the induced probability distribution P I 1 and by p I 1 the density, both having support over I 1 (y) The measure µ I 1 is mapped into SI 1 (y) to a measure µ SI 1, and a pdf p SI 1 and cdf P SI 1 Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

23 Probabilistic error and optimal algorithms Probabilistic setting Given perturbed information y Y and accuracy ǫ (0, 1), we define the probabilistic error (to level ǫ) as r pr (A, y,ǫ). = inf X 0 : µ I 1(X 0 ) ǫ max Sx A(y) p (3) x {I 1 (y)\x 0 } Clearly, r pr (A, y,ǫ) r wc (A, y) for any algorithm A, data y Y and ǫ (0, 1), which implies a reduction of the approximation error in a probabilistic setting An algorithm A pr o is called probabilistic optimal (to level ǫ) if it minimizes the radius of information r pr (A, y,ǫ) for any y Y and ǫ (0, 1) r pr o (y,ǫ). = r pr (A pr o, y,ǫ) = inf A r pr (A, y,ǫ) The probabilistic optimal estimator is given by ẑo pr (ǫ) =. A pr o (y,ǫ) The minimal error ro pr (y, ǫ) is called the probabilistic radius of information (to level ǫ) Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

24 Probabilistic error and optimal algorithms Probabilistic setting Given perturbed information y Y and accuracy ǫ (0, 1), we define the probabilistic error (to level ǫ) as r pr (A, y,ǫ). = inf X 0 : µ I 1(X 0 ) ǫ max Sx A(y) p (3) x {I 1 (y)\x 0 } Clearly, r pr (A, y,ǫ) r wc (A, y) for any algorithm A, data y Y and ǫ (0, 1), which implies a reduction of the approximation error in a probabilistic setting An algorithm A pr o is called probabilistic optimal (to level ǫ) if it minimizes the radius of information r pr (A, y,ǫ) for any y Y and ǫ (0, 1) r pr o (y,ǫ). = r pr (A pr o, y,ǫ) = inf A r pr (A, y,ǫ) The probabilistic optimal estimator is given by ẑo pr (ǫ) =. A pr o (y,ǫ) The minimal error ro pr (y, ǫ) is called the probabilistic radius of information (to level ǫ) Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

25 Problem definition Probabilistic setting Problem Computation of ro pr (y, ǫ) and the derivation of probabilistic optimal algorithms A pr o for different probability distributions P Q and support sets Q. In particular, we are interested in the cases: µ Q is Gaussian µ Q is uniform on the l 2 norm ball B 2 (R) and p = 2 µ Q is uniform on the l norm ball B (R) and p = Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

26 Chance constraint formulation Probabilistic setting We introduce the violation probability for given A and radius r v(r,a) =. { µ I 1 x I 1 (y) Sx A(y) > r } Equation (3) can be reformulated as a chance-constrained optimization problem r pr (A, y,ǫ) = min{r v(r,a) ǫ} A probabilistic optimal algorithm can be computed as } ro {r pr (y,ǫ) = min inf v(r,a) ǫ A = min{r v o (r) ǫ} where we the optimal violation probability for given radius r is v o (r) =. inf µ { I x I 1 (y) : Sx A(y) A 1 p > r } Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

27 Chance constraint formulation Related literature Notably, the probabilistic setup and its chance-constraint formulation have already been introduced in the book of Traub In [Traub et al:88] the connection with the average error setting, which has the objective to minimize the expected value of the estimation error E [g( Sx A(y) )], is outlined. To explain this connection, we see that for any r > 0 v(r,a) = I r (Sx A(y)) p X (x)dx I 1 (y) where I r ( Sx A(y) ) is the indicator function of SI 1 (y). Hence, the probabilistic estimator is equivalent to the average estimator that minimizes E [I r ( Sx A(y) )] Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

28 Random Uncertainty Normally Distributed Optimality of least squares We consider the case when the uncertainty q is normally distributed, i.e. Q R m and q N q,w In this case, the probabilistic optimal algorithm A pr o is the least squares algorithm A ls Theorem Letting K = R n, Q = R m, q N q,w and W = H T H. Then, A pr o (y) = A ls(y) = S(I T H T HI) 1 I T H T y for any y Y and ǫ (0, 1). Moreover, the probabilistic radius r pr o (ǫ). = r pr (A pr o, y,ǫ) does not depend on y. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

29 Probabilistic radius for Gaussian measures r pr o (ǫ) =??? We are not aware of explicit ways to compute the radius ro pr (ǫ) In Traub the following bound on ro pr (ǫ) is provided ro pr (ǫ) 2 ln 5 ǫ r o avg where ro avg is the optimal average radius of information, which can be computed in closed form as a function of covariance matrix W Also, the probabilistic radius ro pr (ǫ) seems to have close relation with the work of Campi and Weyer, where they derive non-asymptotic (i.e. based on a finite number of measure, such as in our case) confidence ellipsoids for least-squares estimates Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

30 Random Uncertainty Uniformly Distributed The induced measure is still uniform We study the case when q is uniformly distributed over the set Q, i.e. q U Q and µ Q µ U(Q) We assume that Q is a compact set. In particular, we are interested in the case when Q is the l p norm ball B p (R) Question: If µ Q is the uniform measure over Q, what is the induced measure µ I 1 over the set I 1 (y)? Theorem Let Q be a compact set, if q U(Q), then for any y Y µ I 1 µ U(I 1 (y)) Moreover, the measure µ SI 1 over Z is log-concave Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

31 Random Uncertainty Uniformly Distributed The induced measure is still uniform We study the case when q is uniformly distributed over the set Q, i.e. q U Q and µ Q µ U(Q) We assume that Q is a compact set. In particular, we are interested in the case when Q is the l p norm ball B p (R) Question: If µ Q is the uniform measure over Q, what is the induced measure µ I 1 over the set I 1 (y)? Theorem Let Q be a compact set, if q U(Q), then for any y Y µ I 1 µ U(I 1 (y)) Moreover, the measure µ SI 1 over Z is log-concave Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

32 Weighted l 2 ball Uniform case Optimality of least squares The random uncertainty q is uniformly distributed in a weighted l 2 ball of radius ρ, and the set SI 1 (y) is hence an ellipsoid In this case, the center of SI 1 (y) is a probabilistic optimal estimator, and coincides with the least squares estimator A ls Theorem Letting K = R n and µ q (Q) = U q (Q) where and W = H T H. Then, Q = { q : q T Wq ρ 2} A pr o (y) = A ls (y) = S(I T H T HI) 1 I T H T y Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

33 l -ball, the case of S = I n Random Uncertainty Uniformly Distributed We now concentrate on the very important case when Q = B (R) In this case, LS is not optimal anymore, and a probabilistic optimal estimator needs be derived In order to simplify our next developments, we start considering the case when S is the identity operator Assumption (Parameter estimation problems) We assume that S = I n This corresponds to the situation when one is interested in parameter estimation The assumption can be relaxed as S being square, S R n,n We recall that in this case SI 1 (y) = I 1 (y) is a polytope equipped by a uniform measure Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

34 Computation of violation probability l -ball, the case of S = I n Theorem (Computation of violation probability) Let Q be a bounded convex set with p Q = U Q, then for given r > 0 (i) The violation probability v o (r) can be computed as the solution of v o (r) = inf x c vol[x 0 (x c, r)]/v X (4) where X 0 (x c, r). = I 1 (y)\b p (x c, r) and V X = vol[x] (ii) The optimization problem (4) is a quasi-convex problem in x c, over the convex set Ω. = { x I 1 (y) B p (x c, r) } (iii) The function v o (r) is a continuous non-increasing function of r Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

35 Computation of violation probability Schetch of proof Reminding that µ Q is the uniform measure over Q, from the definition of v o (r) we write v o (r) = 1 V X inf A vol[{ x I 1 (y) : x A(y) p > r }] = 1 V X inf x c vol [{ x I 1 (y) : x x c p > r }] = 1 V X inf x c vol [{ x I 1 (y) x B p (x c, r) }] = 1 inf vol [ I 1 (y)\b p (x c, r) ] = 1 inf vol[x 0 (x c, r)] V X x c V X x c from which it follows point (i) Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

36 Computation of violation probability Schetch of proof To prove quasi-convexity (point (ii)), note that (4) rewrites v o (r) = 1 1 sup vol[d(x, r)] (5) V X where D(x, r) = I 1 (y) B p (x, r) This is the problem of maximizing the volume of the intersection of two convex sets, one of those can be translated by x This problem can be shown to be quasi-concave in x over the set Ω where the intersection D(x, r) is non-empty More specifically, in [Zalgaller:01] it is shown that the function φ(x). = (vol[d(x, r)]) 1/n is concave over Ω (this is a direct consequence of Brunn-Minkovski ineq.) Hence, it follows immediately that the function φ(x) n is quasi-concave, since φ(x) is a nonnegative function for x Ω and the n-th power function y n is an increasing function for y 0 x Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

37 Computation of probabilistic optimal estimate Corollary For given r > 0, a global minimizer for problem (4) can be computed as the solution of the following maximization problem supφ(x), with φ(x) = (vol[d(x, r)]) 1/n (6) x Ω Moreover, problem (6) is concave over the convex set Ω, and thus any local maximizer is also global. For fixed r > 0, let x o (r) one such maximizers, than v o (r) = 1 vol[d(x o(r), r)] V X and the probabilistic radius of information (to level ǫ) can be found as the (unique) solution of the following one-dimensional inversion problem r pr o (y,ǫ) = min{r v o(r) ǫ} Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

38 Probabilistic radius and probabilistic optimal estimator For given ǫ > 0, the inversion problem can be easily solved by bisection, thus providing a way of computing the optimal probabilistic radius of information ro pr (y,ǫ) The corresponding optimal estimate ˆx o pr (y,ǫ) can be directly computed as a minimizer x o (r), for r = ro pr (y,ǫ), of the following problem Problem (Maximum intersection) (P-max-int) : sup vol[d(x c, r)] 1 n x c If the set I 1 (y) is centrally symmetric, then it is easy to see from symmetry that the intersection with a ball is maximized when the sets are concentric. Hence, the optimal estimator for the probabilistic case coincides with that of the worst-case setting Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

39 Solving (P-max-int) : Volume oracle and oracle-polynomial-time algorithm We first consider the case when one assumes to have a volume oracle that, given x, returns the volume of D(x, r) (and a sub-gradient of the function φ(x)) Problem (P-max-int) has been considered in [FukUno:07], where they derive a strongly oracle-polynomial-time algorithm for polytopic sets Indeed, the fact the the problem is NP hard does not make the intersection maximization problem worthless to investigate, since one can compute the volume of a polytope quickly for considerably complex polytopes in modest (say up to n = 10) dimensions Hence, for small dimensional n, one can use the method proposed by [FukUno:07]. This method has also been used in our examples to compare with the other techniques proposed further on Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

40 Solving (P-max-int) : Volume oracle and oracle-polynomial-time algorithm Function v o(r) for a problem with n = 3 and m = 13, and corresponding sequence of optimal boxes maximizing the intersection with the polytope. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

41 Solving (P-max-int) : Example Computation v o(r) for a problem with six parameters and m = For ǫ = 0.05, the probabilistic radius is ro pr (y,ǫ) = 0.068, almost half of the worst case radius which is ro pr (y) = Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

42 Solving (P-max-int) : Randomized algorithms and stochastic optimization We do not have a volume oracle in general! We use a randomized approach. Algorithm (Randomized approximation of φ(x)) 1 Generate N points in the ball B p (x, r) 2 Count how many points N g belong to the set I 1 (y) (this can be done in polynomial-time for polytopes or ellipsoids) 3 An approximation of the function φ(x) is immediately obtained as ( ) 1/n Ng φ(x) N vol[b p(x, r))] Note that the expected value wrt samples E( φ(x)) coincides with φ(x). Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

43 Solving (P-max-int) : Stochastic optimization SPSA We consider the following stochastic problem sup x E[ˆφ(x)], E( φ(x)) = φ(x) For its solution, we can make recourse to classical stochastic optimization algorithms (see Kushner & Yin), as for instance FPSA, SPSA,... Algorithm (Simultaneous Perturbations Stoch. Approx. (SPSA)) Consider a starting point θ 0, and run the following algorithm θ k+1 = θ k +α k [ 1 k ] φ + φ 2c k where [ k ] {0, 1} n is a Bernoulli sequence, [ 1 k ] =. [ T 1 k,1 1 k,n] and φ ±. = φ(θk ± c k k )+η ± k where η ± k is random noise Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

44 Solving (P-max-int) : Stochastic optimization Convergence of SPSA Theorem Assume φ(x) is concave and nondifferentiable (as in our case), and let φ(x) a subgradient of φ. If a k 0, k a k = and c k 0, k c k =, then under mild conditions, θ k converges to a point such that 0 φ(θ) Remarks: (+) Simulations show that the method works for quite large n, m (-) At present, we have convergence only for k. This is contrary to our philosophy, that aim at results valid for finite k -> Working direction: derive (probabilistic) bounds on θ k and φ(θ k ) -> Is it possible to find conservative but guaranteed approaches? Idea: instead of maximizing the volume of the intersection, we maximize an appropriate lower-bound of this volume. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

45 Solving (P-max-int) : An SDP relaxation Ellipsoidal approximation Idea: to construct, for fixed r, the maximal volume ellipsoid contained in the intersection D(x c, r) i.e. to solve the problem sup x c,x ce,p E vol[e(x E, P E )] subject to E(x E, P E ) D(x c, r) where E(x E, P E ). = {x R n x = x E + P E z, z 2 1} The above problem is still non-concave, but a concave formulation is possible. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

46 Solving (P-max-int) : An SDP relaxation Ellipsoidal approximation Theorem Let r > 0 be fixed. A global optimizer for problem (7) can be computed solving inf x c,x ce,p E subject to log det P E [ ] (R + yi I i )x E I n P E Ii T 0, i = 1,...,m (R + y i I i )x E [ ] (R yi +I i )x E I n P E Ii T 0, i = 1,...,m (R y i +I i )x E [ ] (r + e T i (x c x E ))I n P E e i r + ei T 0, i = 1,...,n (x c x E ) [ ] (r e T i (x c x E ))I n P E e i r ei T 0, i = 1,...,n (x c x E ) Let x sdp c v sdp o (r) a solution x c the above SDP, and define (r) =. [ ] vol D(xc sdp (r), r). Then, vo sdp (r) v o (r) Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

47 Solving (P-max-int) : An SDP relaxation Ellipsoidal approximation This SDP relaxation thus leads to a easily computable suboptimal violation function vo sdp (r) Ellipsoid-based SDP relaxation vo sdp (r) (red) and optimal one vo sdp (r) (blu) for an example with m = 200 and n = 4. The convex relaxation (red), is always above the optimal one, as expected. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

48 Recovering tight bounds From our developements, it follows that the probabilist radius of information ro pr (y,ǫ), and the optimal estimator ˆx o pr (y,ǫ) can be computed as the solution of a convex optimization problem These quantities are to be interpreted as the radius and center of a l ball guaranteed to contain 1 ǫ of the total volume of I 1 (y) Theorem (Tight bound: ǫ-enclosing orthotope) Let ˆx o pr (y, ǫ) be a probabilistic optimal estimator guaranteeing a probabilistic radius of information of ro pr (y,ǫ). Then, tight bounds on the parameters h i,h + i, i = 1,...,n, can be computed as the solution of the following 2n linear programs: h i ( ) h + i = inf (sup) xi i = 1,...,n subject to x ˆx o pr pr (y,ǫ) ro (y,ǫ) Iˆx o pr (y,ǫ) y R, Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

49 The case of S = I n When S is not the identity, our approach cannot be applied directly, since the measure on SI 1 (y) is not uniform anymore However, this measure is log-concave, and hence a results similar to the one proved in the previous section can still be proved Theorem Let µ SI 1 the measure induced by µ Q over Z, which is logconcave. Then φ(z) = µ SI 1(D S (z)) 1/n where D S (z) = SI 1 (y) B p (z, r) (7) is still a concave function. This results shows that our problem is still well posed Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

50 The case of S = I n However, it is hard to compute the measure µ SI 1(D(x, r)), and even a direct stochastic approximation approach would require generating samples according to this measure A solution is to go back to space X, and maximize the volume of the intersection B 1 S (z) I 1 (y) where B 1 S (z). = {x X Sx B p (z, r)} is the back-image of the ball B p (z, r) through S Simple geometric reasoning show that B 1 S (z) is a cylinder, thus the computation of D S (z) can be performed using the same techniques discussed in the previous sections Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

51 Future work and extensions A different approach is to approximately solve the chance-constraint problem using the discarded-constraints scenario approach of Calafiore Campi Generate N vectors x (i) uniformly distributed in SI 1 (y) Construct the randomized optimization problem ˆr o (N, L) = min zc Sx (i) z c r for all i = 1,...,N but D discarded ones For given ǫ 1 < ǫ, one obtains Prob{r pr o (ǫ, y) ˆr o(n, L) r pr o (ǫ 1, y)} 1 δ(n, L) Problems: i) not easy to sample in I 1 (y), ii) not easy to construct an optimal discard procedure Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

52 Future work and extensions Reduced complexity estimates In various work, Garulli et al. consider the conditional estimation problem. In this case, they seek an estimate of x using a lower-dimensional representation, described as a l dimensional linear manifold, with l < n, i.e. ˆx M = { z X x = x o + Mz, z R l}. That is, they consider the class of algorithms A M that map Y M X. Then, an optimal conditional central estimate is given by the conditional Chebychev center x M o = arg inf sup x x c p. x c M x I 1 (y) This amounts at finding the point x c lying on the manifold M such that its distance from the farthest point of I 1 (y) is minimized. Nonlinear systems Extensions to the case of nonlinear operators is under study. Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48

53 The end...for now... THANK YOU! Dabbene, Sznaier, Tempo (CNR-IEIIT, Northeastern) Probabilistic Optimal Estimation and Filtering Udine, WUDS / 48