Simultaneous perturbation stochastic approximation

Simultaneous perturbation stochastic approximation (SPSA) is an

. A comprehensive book on the subject is Bhatnagar et al. (2013). An early paper on the subject is Spall (1987) and the foundational paper providing the key theory and justification is Spall (1992).

SPSA is a descent method capable of finding global minima, sharing this property with other methods such as simulated annealing. Its main feature is the gradient approximation that requires only two measurements of the objective function, regardless of the dimension of the optimization problem. Recall that we want to find the optimal control ${\displaystyle u^{*}}$ with loss function ${\displaystyle J(u)}$:

${\displaystyle u^{*}=\arg \min _{u\in U}J(u).}$

Both Finite Differences Stochastic Approximation (FDSA) and SPSA use the same iterative process:

${\displaystyle u_{n+1}=u_{n}-a_{n}{\hat {g}}_{n}(u_{n}),}$

where ${\displaystyle u_{n}=((u_{n})_{1},(u_{n})_{2},\ldots ,(u_{n})_{p})^{T}}$ represents the ${\displaystyle n^{th}}$ iterate, ${\displaystyle {\hat {g}}_{n}(u_{n})}$ is the estimate of the gradient of the objective function ${\displaystyle g(u)={\frac {\partial }{\partial u}}J(u)}$ evaluated at ${\displaystyle {u_{n}}}$, and ${\displaystyle \{a_{n}\}}$ is a positive number sequence converging to 0. If ${\displaystyle u_{n}}$ is a p-dimensional vector, the ${\displaystyle i^{th}}$ component of the

finite difference gradient estimator is:

FD: ${\displaystyle ({\hat {g_{n}}}(u_{n}))_{i}={\frac {J(u_{n}+c_{n}e_{i})-J(u_{n}-c_{n}e_{i})}{2c_{n}}},}$

1 ≤i ≤p, where ${\displaystyle e_{i}}$ is the unit vector with a 1 in the ${\displaystyle i^{th}}$ place, and ${\displaystyle c_{n}}$is a small positive number that decreases with n. With this method, 2p evaluations of J for each ${\displaystyle g_{n}}$ are needed. When p is large, this estimator loses efficiency.

Let now ${\displaystyle \Delta _{n}}$ be a random perturbation vector. The ${\displaystyle i^{th}}$ component of the stochastic perturbation gradient estimator is:

SP: ${\displaystyle ({\hat {g_{n}}}(u_{n}))_{i}={\frac {J(u_{n}+c_{n}\Delta _{n})-J(u_{n}-c_{n}\Delta _{n})}{2c_{n}(\Delta _{n})_{i}}}.}$

Remark that FD perturbs only one direction at a time, while the SP estimator disturbs all directions at the same time (the numerator is identical in all p components). The number of loss function measurements needed in the SPSA method for each ${\displaystyle g_{n}}$ is always 2, independent of the dimension p. Thus, SPSA uses p times fewer function evaluations than FDSA, which makes it a lot more efficient.

Simple experiments with p=2 showed that SPSA converges in the same number of iterations as FDSA. The latter follows

estimator of the gradient, as shown in the following lemma.

Convergence lemma

Denote by

${\displaystyle b_{n}=E[{\hat {g}}_{n}|u_{n}]-\nabla J(u_{n})}$

the bias in the estimator ${\displaystyle {\hat {g}}_{n}}$. Assume that ${\displaystyle \{(\Delta _{n})_{i}\}}$ are all mutually independent with zero-mean, bounded second moments, and ${\displaystyle E(|(\Delta _{n})_{i}|^{-1})}$ uniformly bounded. Then ${\displaystyle b_{n}}$→0 w.p. 1.

Sketch of the proof

The main idea is to use conditioning on ${\displaystyle \Delta _{n}}$ to express ${\displaystyle E[({\hat {g}}_{n})_{i}]}$ and then to use a second order Taylor expansion of ${\displaystyle J(u_{n}+c_{n}\Delta _{n})_{i}}$ and ${\displaystyle J(u_{n}-c_{n}\Delta _{n})_{i}}$. After algebraic manipulations using the zero mean and the independence of ${\displaystyle \{(\Delta _{n})_{i}\}}$, we get

${\displaystyle E[({\hat {g}}_{n})_{i}]=(g_{n})_{i}+O(c_{n}^{2})}$

The result follows from the hypothesis that ${\displaystyle c_{n}}$→0.

Next we resume some of the hypotheses under which ${\displaystyle u_{t}}$ converges in probability to the set of global minima of ${\displaystyle J(u)}$. The efficiency of the method depends on the shape of ${\displaystyle J(u)}$, the values of the parameters ${\displaystyle a_{n}}$ and ${\displaystyle c_{n}}$ and the distribution of the perturbation terms ${\displaystyle \Delta _{ni}}$. First, the algorithm parameters must satisfy the following conditions:

• ${\displaystyle a_{n}}$ >0, ${\displaystyle a_{n}}$→0 when n→∝ and ${\displaystyle \sum _{n=1}^{\infty }a_{n}=\infty }$. A good choice would be ${\displaystyle a_{n}={\frac {a}{n}};}$ a>0;
• ${\displaystyle c_{n}={\frac {c}{n^{\gamma }}}}$, where c>0, ${\displaystyle \gamma \in \left[{\frac {1}{6}},{\frac {1}{2}}\right]}$;
• ${\displaystyle \sum _{n=1}^{\infty }({\frac {a_{n}}{c_{n}}})^{2}<\infty }$
• ${\displaystyle \Delta _{ni}}$ must be mutually independent zero-mean random variables, symmetrically distributed about zero, with ${\displaystyle \Delta _{ni}<a_{1}<\infty }$. The inverse first and second moments of the ${\displaystyle \Delta _{ni}}$ must be finite.

A good choice for ${\displaystyle \Delta _{ni}}$ is the Rademacher distribution, i.e. Bernoulli +-1 with probability 0.5. Other choices are possible too, but note that the uniform and normal distributions cannot be used because they do not satisfy the finite inverse moment conditions.

The loss function J(u) must be thrice continuously differentiable and the individual elements of the third derivative must be bounded: ${\displaystyle |J^{(3)}(u)|<a_{3}<\infty }$. Also, ${\displaystyle |J(u)|\rightarrow \infty }$ as ${\displaystyle u\rightarrow \infty }$.

In addition, ${\displaystyle \nabla J}$ must be Lipschitz continuous, bounded and the ODE ${\displaystyle {\dot {u}}=g(u)}$ must have a unique solution for each initial condition. Under these conditions and a few others, ${\displaystyle u_{k}}$

in probability to the set of global minima of J(u) (see Maryak and Chin, 2008).

It has been shown that differentiability is not required: continuity and convexity are sufficient for convergence.^[1]

Extension to second-order (Newton) methods

It is known that a stochastic version of the standard (deterministic) Newton-Raphson algorithm (a “second-order” method) provides an asymptotically optimal or near-optimal form of stochastic approximation. SPSA can also be used to efficiently estimate the Hessian matrix of the loss function based on either noisy loss measurements or noisy gradient measurements (stochastic gradients). As with the basic SPSA method, only a small fixed number of loss measurements or gradient measurements are needed at each iteration, regardless of the problem dimension p. See the brief discussion in Stochastic gradient descent.

References

• Bhatnagar, S., Prasad, H. L., and Prashanth, L. A. (2013), Stochastic Recursive Algorithms for Optimization: Simultaneous Perturbation Methods, Springer [1].
• Hirokami, T., Maeda, Y., Tsukada, H. (2006) "Parameter estimation using simultaneous perturbation stochastic approximation", Electrical Engineering in Japan, 154 (2), 30–3 [2]
• Maryak, J.L., and Chin, D.C. (2008), "Global Random Optimization by Simultaneous Perturbation Stochastic Approximation," IEEE Transactions on Automatic Control, vol. 53, pp. 780-783.
• Spall, J. C. (1987), “A Stochastic Approximation Technique for Generating Maximum Likelihood Parameter Estimates,” Proceedings of the American Control Conference, Minneapolis, MN, June 1987, pp. 1161–1167.
• Spall, J. C. (1992), “Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation,” IEEE Transactions on Automatic Control, vol. 37(3), pp. 332–341.
• Spall, J.C. (1998). "Overview of the Simultaneous Perturbation Method for Efficient Optimization" 2. Johns Hopkins APL Technical Digest, 19(4), 482–492.
• Spall, J.C. (2003) Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control, Wiley.
  ISBN 0-471-33052-3
  (Chapter 7)