IEEE Solid-States Circuits Magazine - Fall 2019 - 24

The goal of a multiobjective optimization
algorithm is to find a set of solutions that lie
on the Pareto frontier curve.
2) circuit sizing, where the focus is
on parameter-level optimization
for a fixed circuit topology.
Optimization methodologies have
been applied to circuit sizing, including the sizing of RF circuits [1], [2],
operational amplifiers [3], and other
analog circuits [4]-[9]. The applications for automated optimization tools
are not limited to analog circuit sizing.
In general, any system-design task that
requires adjusting a set of hyperparameters can benefit from automated
optimization methodologies.
System design on a high level can
be directly translated to optimizing
a score function F (x):R d " R, where
x is a d-dimensional vector of hyperparameters and F ( $) is an arbitrary
measure of quality. This measure can
be formulated by combining several
design objectives that characterize
system performance, e.g., power consumption, area, and throughput. The
underlying optimization task can be
regarded as a search over the values
of F(x) in a d-dimensional space where
the boundaries are specified by the
design constraints. A good optimization algorithm is one that can be globally applied to arbitrary problems
and can find the global optima of F ( $)
with a low search overhead.
Optimization can be performed
using two different approaches: numerical methods and black-box methods.
Numerical methods assume access to
the underlying algebraic model that
relates design hyperparameters to the
score function [10]. The task of finding
the underlying mathematical expressions is often exhaustive, if not impossible. Black-box methods address this
challenge by assuming that the score
function is not explicitly known and
thus merely resorting to empirical
(noisy) evaluations of F(x). In this article, we explain the fundamentals of
three black-box optimization methods.
For each method, we provide practical

24

FA L L 2 0 19

system-design examples to illustrate
its applicability in real-world tasks. We
conclude by suggesting future research
directions and areas worth exploring.

Problem Formulation
System design can be viewed as an
optimization problem where design
choices represent the optimization
variables (hyperparameters). In this
setting, system constraints and design
requirements represent the objective
functions for optimization. After the
design hyperparameters and corresponding objective functions are determined, the hyperparameters are
translated to a vectorized representation. Various optimization tools can
then be used to explore the underlying hyperparameter vector space
and find the solution that results in
a good performance. For a vector of
hyperparameters (x ! R d ), we assume
access to an oracle, F (x):R d " R, that
serves as an objective function to the
optimization problem. The optimization goal is thus to find the maxima
(or minima) of the objective function,
with the caveat that F ( $) is not known
in advance. To provide a better understanding, let us consider the example
of circuit design.
To perform automated circuit
optimization, we assume an initial expert-designed mapping of circuit elements and seek to enhance the design,
given a set of user-defined objectives.
Toward this goal, the first step is
identifying the design variables, i.e.,
a subset of hyperparameters {x i} di = 1
that can be tuned to improve the design objective. As an example, hyperparameters can be analog component
characteristics, e.g., capacitance, resistance, and transistor dimensions.
The design hyperparameters are appended to form a vector representation x ! R d. Enhancing analog circuit
performance can then be formally defined as a multiobjective, constrained

IEEE SOLID-STATE CIRCUITS MAGAZINE

optimization. Here, the objective functions f1 (x), ffm (x) represent different circuit performance metrics, i.e.,
the figure of merit. Examples of the
objective function include power, noise
resiliency, and linearity.
In reality, the objectives often conflict with one another. The goal of a
multiobjective optimization algorithm is thus to find a set of solutions
that lie on the Pareto frontier curve.
By definition, Pareto optimal solutions
are those that cannot be improved in
any of the objective functions without
harming at least one other objective.
In general, extracting all Pareto optimal solutions can be rather costly for
real-world complex circuits. A relaxed
alternative is to solve a single-objective optimization of a score function
instead. The score F ( $) is a carefully
designed combination of fi ( $)s that
incorporates the pertinent tradeoffs
among the objective functions. More
specifically, the score function F ( $)
represents a goodness measure for a
given vector of circuit hyperparameters. Therefore, the optimization
goal is equivalent to maximizing F ( $)
and obtaining a "good" design that adheres to the desired characteristics. A
simple example of such a score function is a linear combination of objective functions F (x) = R m
i = 1 w i fi (x). The
relaxed optimization objective can,
therefore, be formalized as
maximize
F (x) s.t.
x

)

C i (x) $ 0
6 i ! " 1, f, k ,
,
@ 6 i ! " 1, f, d ,
, b max
x i ! 6b min
i
i
(1)

where C i (x) is a set of constraints
that are either user-defined limits on
certain design metrics, e.g., power,
or rules imposed by analog circuit
laws, e.g., Kirchhoff's current law
and Kirchhoff's voltage law. Another
set of constraints corresponds to
boundaries on design hyperparameters, [b min, b max], as a result of the
fabrication process or specific design
requirements. In the following sections, we elaborate on three contemporary methods for hyperparameter
optimization: greedy algorithms, RL,
IEEE Solid-States Circuits Magazine - Fall 2019

Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Fall 2019
