Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

Kulkarni, Ankur A.; Borkar, Vivek S.

doi:10.1016/j.automatica.2009.09.031

Cited by 3 publications

(5 citation statements)

References 33 publications

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“…This is the first work to look at the classical gradient and accelerated gradient descent, which means we are able to demonstrate convergence using a new range of step sizes. Turning our attention to the previous body of uncertainty quantification for SA algorithms, in [8] a similar spectral approach is taken, but truncated to a finite dimension at all iterations. They truncate by letting x(θ) = u i B i , for some finite family of functions B i , and then perform a standard Stochastic Approximation procedure to calculate the coefficients u i .…”

Section: Previous Work Much Of the Previous Work Surrounding Chaos Ex...mentioning

confidence: 99%

Uncertainty Quantification for Gradient and Accelerated Gradient Descent Methods on Strongly Convex Functions

Conor¹,

Parpas²

2021

Preprint

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We consider the problem of minimizing a strongly convex function that depends on an uncertain parameter θ. The uncertainty in the objective function means that the optimum, x * (θ), is also a function of θ. We propose an efficient method to compute x * (θ) and its statistics. We use a chaos expansion of x * (θ) along a truncated basis and study first-order methods that compute the optimal coefficients. We establish the convergence rate of the method as the number of basis functions, and hence the dimensionality of the optimization problem is increased. We give the first non-asymptotic rates for the gradient descent and the accelerated gradient descent methods. Our analysis exploits convexity and does not rely on a diminishing step-size strategy. As a result, it is much faster than the state-of-the-art both in theory and in our preliminary numerical experiments. A surprising side-effect of our analysis is that the proposed method also acts as a variance reduction technique to the problem of estimating x * (θ).

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Section: Previous Work Much Of the Previous Work Surrounding Chaos Ex...mentioning

confidence: 99%

Uncertainty Quantification for Gradient and Accelerated Gradient Descent Methods on Strongly Convex Functions

Conor¹,

Parpas²

2021

Preprint

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“…In [17], Kulkarni and Borkar provide a finite dimensional procedure to approximate the minimum of ζ → L(ζ(v), v)µ(dv) on some subset of the real valued functions, for some explicit non-negative function L satisfying some strict convexity properties with respect to (w.r.t.) its first argument.…”

Section: Define (13)mentioning

confidence: 99%

“…They analyze the error due to finite dimensional truncation. However, this analysis makes apparent that, in order to achieve actual convergence results (as opposed to error bounds in [17]), one really needs to let m increase during the algorithm. Moreover, even when the function ζ is explicitly known (which is not the case in our setup) and estimating individual coefficients u i in (1.4) is straightforward by MC simulations, the global convergence of a method where more and more coefficients are computed by Monte Carlo is nontrivial, subject to a fine-tuning of the speeds at which the number of coefficients and the number of simulations go to infinity (see [13]).…”

Section: Define (13)mentioning

confidence: 99%

“…First note that since d 0 = 0 and n a 2 n < ∞, then A is bounded. Since A is closed, we prove that A is relatively compact, and to that goal, we check the sufficient condition given in [17,Theorem 3]: since l 2 is spanned by a complete orthonormal basis and A is bounded, A is relatively compact if and only if for any > 0, there exists N such that for any u ∈ A, i≥N |u i | 2 ≤ . For any N and d (N ) such that d (N ) ≤ N , and for any u ∈ A, it holds…”

Section: (B1)mentioning

confidence: 99%

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Uncertainty Quantification for Stochastic Approximation Limits Using Chaos Expansion

Crépey¹,

Fort²,

Gobet³

et al. 2020

SIAM/ASA J. Uncertainty Quantification

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The uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm is analyzed. In our setup, this limit ζ is defined as a zero of an intractable function and is modeled as uncertain through a parameter θ. We aim at deriving the function ζ , as well as the probabilistic distribution of ζ (θ) given a probability distribution π for θ. We introduce the so-called Uncertainty Quantification for SA (UQSA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of θ → ζ (θ) on an orthogonal basis of a suitable Hilbert space. UQSA, run with a finite number of iterations K, returns a finite set of coefficients, providing an approximation ζ K (•) of ζ (•). We establish the almost-sure and L p-convergences in the Hilbert space of the sequence of functions ζ K (•) when the number of iterations K tends to infinity. This is done under mild, tractable conditions, uncovered by the existing literature for convergence analysis of infinite dimensional SA algorithms. For a suitable choice of the Hilbert basis, the algorithm also provides an approximation of the expectation, of the variance-covariance matrix, and of higher order moments of the quantity ζ K (θ) when θ is random with distribution π. UQSA is illustrated and the role of its design parameters is discussed numerically.

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“…In the operations research and control community, there are two applications of random projections we are aware of. First [1], where a low-rank approximation is used to approximately find the zero of a linear equation, and second [18] where a similar approximation is used within a Newton methodbased stochastic approximation. Our work differs from [1] in two fundamental ways.…”

Section: Introductionmentioning

confidence: 99%

Dimensionality reduction of affine variational inequalities using random projections

Prabhakar

Kulkarni

2014

2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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We present a method for dimensionality reduction of an affine variational inequality (AVI) defined over a compact feasible region. Centered around the Johnson Lindenstrauss lemma [16], our method is a randomized algorithm that produces with high probability an approximate solution for the given AVI by solving a lower-dimensional AVI. The algorithm allows the lower dimension to be chosen based on the quality of approximation desired. The algorithm can also be used as a subroutine in an exact algorithm for generating an initial point close to the solution. The lower-dimensional AVI is obtained by appropriately projecting the original AVI on a randomly chosen subspace. The lower-dimensional AVI is solved using standard solvers and from this solution an approximate solution to the original AVI is recovered through an inexpensive process. Our numerical experiments corroborate the theoretical results and validate that the algorithm provides a good approximation at low dimensions and substantial savings in time for an exact solution.

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Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

Cited by 3 publications

References 33 publications

Uncertainty Quantification for Gradient and Accelerated Gradient Descent Methods on Strongly Convex Functions

Uncertainty Quantification for Gradient and Accelerated Gradient Descent Methods on Strongly Convex Functions

Uncertainty Quantification for Stochastic Approximation Limits Using Chaos Expansion

Dimensionality reduction of affine variational inequalities using random projections

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