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An Efficient Batch Constrained Bayesian Optimization Approach for Analog Circuit Synthesis via Multi-objective Acquisition Ensemble

Shuhan Zhang, Fan Yang, Changhao Yan, Dian Zhou, Xuan Zeng

arXiv:2106.15412v1cs.LGcs.SYeess.SY

TL;DR

Analog circuit synthesis through Bayesian optimization is constrained by sequential decisions that limit parallel use of computational resources. The paper proposes MACE, a batch method using acquisition-function Pareto fronts and a two-stage constrained search. At batch size 15, it reports up to 74× less simulation time than DE for unconstrained optimization and up to 15× faster optimization than WEIBO for constrained optimization.

  • Problem

    Sequential Bayesian optimization limits parallel use of computational resources, while batch methods have rarely addressed constrained optimization.

    Source: §I · p. 2

  • Method

    MACE samples from the Pareto front of PI, EI, and LCB, and handles constrained optimization through a two-stage search followed by penalization favoring feasible regions.

    Source: B. Search for the Global Optimum · p. 7, §VII · p. 13

  • Results

    At batch size 15, MACE reduces simulation time by up to 74× versus DE for unconstrained optimization and speeds optimization by up to 15× versus WEIBO for constrained optimization.

    Source: §I · p. 2

  • Takeaways & Limitations

    The reported results cover both unconstrained and constrained optimization, with the constrained procedure first seeking a feasible design before searching for a feasible optimum.

    Source: §I · p. 2, §VII · p. 13

Abstract

from arXiv · show

Bayesian optimization is a promising methodology for analog circuit synthesis. However, the sequential nature of the Bayesian optimization framework significantly limits its ability to fully utilize real-world computational resources. In this paper, we propose an efficient parallelizable Bayesian optimization algorithm via Multi-objective ACquisition function Ensemble (MACE) to further accelerate the optimization procedure. By sampling query points from the Pareto front of the probability of improvement (PI), expected improvement (EI) and lower confidence bound (LCB), we combine the benefits of state-of-the-art acquisition functions to achieve a delicate tradeoff between exploration and exploitation for the unconstrained optimization problem. Based on this batch design, we further adjust the algorithm for the constrained optimization problem. By dividing the optimization procedure into two stages and first focusing on finding an initial feasible point, we manage to gain more information about the valid region and can better avoid sampling around the infeasible area. After achieving the first feasible point, we favor the feasible region by adopting a specially designed penalization term to the acquisition function ensemble. The experimental results quantitatively demonstrate that our proposed algorithm can reduce the overall simulation time by up to 74 times compared to differential evolution (DE) for the unconstrained optimization problem when the batch size is 15. For the constrained optimization problem, our proposed algorithm can speed up the optimization process by up to 15 times compared to the weighted expected improvement based Bayesian optimization (WEIBO) approach, when the batch size is 15.

I. INTRODUCTION

Analog circuit sizing is costly and difficult to automate, while existing optimization approaches face limitations in modeling accuracy or convergence. The paper proposes batch Bayesian optimization for unconstrained and constrained sizing and reports faster simulation than named baselines.

I. INTRODUCTION

  • Analog circuit sizing is increasingly difficult to perform manually as device complexity and parasitic effects grow.The paper focuses on device sizing rather than topology selection.Source: §I · p. 1
  • Circuit simulations can be prohibitively expensive, and inaccessible derivatives and convexity make sizing costly, noisy, and multimodal to optimize.Model-based methods can require hard-to-derive or data-intensive models that may be inaccurate across the design space; simulation-based methods have relatively low convergence rates.Source: §I · p. 1
  • Bayesian optimization combines a probabilistic surrogate with an acquisition function to guide sequential sampling from observed data.The surrogate estimates a posterior distribution, while the acquisition function balances exploration and exploitation.Source: §I · p. 2
  • Sequential acquisition decisions limit parallelism, leaving available multi-core resources underused when circuit simulations are computationally intensive.Batch methods must avoid redundant regional sampling while maximizing information from each query.Source: §I · p. 2
  • Prior batch Bayesian optimization work largely focused on unconstrained problems, while many methods also rely on a single acquisition function.The paper identifies constrained optimization as rarely considered in this literature.Source: §I · p. 2
  • Up to 74× less simulation time than DE for unconstrained optimization and up to 15× faster than WEIBO for constrained optimization, both at batch size 15.The evaluation covers four real-world analog circuits.Source: §I · p. 2
  • MACE handles unconstrained and constrained optimization, using Pareto-front sampling and a two-stage search for constrained problems.The constrained search first seeks a feasible point, then searches for a feasible global optimum.Source: §I · p. 2

II. PROBLEM FORMULATION

The paper formulates analog circuit device sizing as either minimizing a weighted performance objective or minimizing a figure of merit subject to design constraints. It then reviews Bayesian optimization and the Gaussian process model used to guide expensive evaluations.

II. PROBLEM FORMULATION

A. Unconstrained Optimization Problem

  • For unconstrained sizing, the objective combines multiple circuit performances using weighting parameters to form a figure of merit to minimize.The input vector contains d design variables; each performance metric has a corresponding weighting parameter.Source: A. Unconstrained Optimization Problem · p. 3

B. Constrained Optimization Problem

  • For constrained sizing, the goal is to minimize the figure of merit while satisfying every constraint function.The formulation requires c_i(x) < 0 for each of N_c constraints.Source: B. Constrained Optimization Problem · p. 3

III. REVIEW OF BAYESIAN OPTIMIZATION

  • Bayesian optimization iteratively fits a Gaussian process to observed data, selects a point using an acquisition function, evaluates it, and updates the dataset.It begins with a randomly sampled initial dataset and returns the best observed objective value.Source: A. Bayesian Optimization · p. 3

A. Bayesian Optimization

  • The surrogate model represents the costly simulation using optimization history, while the acquisition function prioritizes candidate points for evaluation.The surrogate posterior provides predictive means and uncertainty estimates.Source: A. Bayesian Optimization · p. 3
  • The Gaussian process models an unknown noisy objective using observed inputs and outputs, a predefined mean function, and a kernel function.The paper sets the mean function to zero and uses a squared exponential covariance function.Source: A. Bayesian Optimization · p. 3
  • The Gaussian-process hyperparameters include observation-noise variance, function variance, and a length scale for each input dimension.The hyperparameters are determined during model training.Source: A. Bayesian Optimization · p. 3

B. Acquisition Function

Acquisition functions use the surrogate posterior to balance exploration and exploitation when selecting evaluations. The section distinguishes improvement-based criteria from a confidence-bound strategy and names additional alternatives.

B. Acquisition Function

  • The acquisition function uses posterior uncertainty to balance exploring unknown regions with exploiting promising areas.It is a cheap-to-evaluate utility function that guides sampling decisions.Source: B. Acquisition Function · p. 3
  • PI measures the probability of exceeding the current best, while EI measures the expected amount of improvement over that best.The paper introduces a positive jitter ξ of 0.001 in PI to encourage exploration.Source: B. Acquisition Function · pp. 3–4
  • LCB guides search from an optimistic perspective; with its coefficient β, cumulative regret is theoretically bounded.The paper sets ν = 0.5 and δ = 0.05 in its β choice.Source: B. Acquisition Function · p. 4
  • Other acquisition approaches named include entropy search, Thompson sampling, predictive entropy search, max-value entropy search, knowledge gradient, and acquisition portfolios.The cited alternatives include ES, TS, PES, MES, KG, and portfolio methods.Source: B. Acquisition Function · p. 4

IV. BATCH BAYESIAN OPTIMIZATION

The section frames batch Bayesian optimization around information gain and within-batch diversity. It reviews shortcomings of single-acquisition and sequentially emulated batch policies, then introduces the paper’s batch design.

A. Challenges for Batch Bayesian Optimization

  • Batch optimization must maximize each query’s information gain while maintaining diversity among points in the same batch.The section states these as its two fundamental challenges.Source: A. Challenges for Batch Bayesian… · p. 4
  • Relying on one acquisition function can limit optimization efficiency because no single function always outperforms the others.The section notes that existing methods commonly use EI, PI, or LCB individually.Source: A. Challenges for Batch Bayesian… · p. 4
  • Existing batch policies often select points sequentially with repulsion or penalization, but their manually designed assumptions can reduce efficiency or performance.Examples include LP’s Lipschitz-based repulsion, BLCB’s fake observations, GPUCB-PE’s pure exploration, and pHCBO’s penalization.Source: A. Challenges for Batch Bayesian… · p. 4
  • Figure 1 illustrates a Gaussian-process posterior distribution alongside the corresponding Pareto-optimal set of the multi-objective acquisition ensemble.The section says the proposed batch design is described next.Source: A. Challenges for Batch Bayesian… · p. 4, Figure 1 · p. 5

B. Batch Bayesian Optimization via Acquisition Function Ensemble

MACE selects batch candidates from the Pareto front of PI, EI, and LCB, combining their different search biases while maintaining batch diversity. The method uses multi-objective optimization to obtain non-dominated candidate sets.

B. Batch Bayesian Optimization via Acquisition Function Ensemble

  • MACE samples batch candidates from the Pareto front of PI, EI, and LCB to balance exploration and exploitation while naturally encouraging batch diversity.Other acquisition functions, including KG and ES, can also be incorporated.Source: B. Batch Bayesian Optimization via… · p. 5
  • MACE constructs a Gaussian process model, generates a Pareto-optimal dataset, randomly samples B candidates, evaluates them, and updates the dataset.Source: B. Batch Bayesian Optimization via… · p. 5

C. Multi-objective Optimization

  • The multi-objective optimizer seeks a Pareto-optimal set because conflicting objectives generally have no single point that performs best on all objectives.A point dominates another when it is at least as good on every objective and strictly better on at least one.Source: C. Multi-objective Optimization · p. 5
  • The paper uses DEMO to obtain Pareto-optimal acquisition-function sets, with population size 100 and 2,000 evaluations during acquisition optimization.Source: C. Multi-objective Optimization · p. 6

V. CONSTRAINED BATCH BAYESIAN OPTIMIZATION

Constrained MACE first searches for a feasible point, then uses objective and constraint information to search for a feasible optimum. Its first stage combines feasibility and constraint-violation measures to prioritize batch candidates.

V. CONSTRAINED BATCH BAYESIAN OPTIMIZATION

  • The constrained procedure has two stages: find an initial feasible point, then search for the global optimum satisfying the constraints.Source: §V · p. 6

A. Seek the First Feasible Point

  • Before any feasible point is found, the method prioritizes feasibility to find one sooner and learn more about the feasible region.Source: A. Seek the First Feasible Point · p. 6
  • Probability of feasibility favors likely valid regions but can be zero when even one constraint is violated, a weakness that worsens with more constraints.Source: A. Seek the First Feasible Point · p. 6
  • The method adds penalization terms to prioritize candidates and parallelize the search, while predictive-mean-only violation estimates can be overconfident under high uncertainty.It scales each constraint’s predictive mean by its estimated confidence because constraint-violation effort should decrease as uncertainty increases.Source: A. Seek the First Feasible Point · p. 6
  • The first-stage search combines feasibility probability with adaptive and naive constraint-violation measures, sampling a batch from their Pareto front.The adaptive measure helps when feasibility probability is zero; the naive measure encourages exploration, and the batch design increases feasible-region information while enabling parallelism.Source: A. Seek the First Feasible Point · p. 6

B. Search for the Global Optimum

After finding a feasible point, constrained MACE combines its acquisition ensemble with constraint-based penalization and prunes recommendations likely to violate constraints.

B. Search for the Global Optimum

  • After finding a feasible point, the method penalizes regions likely to violate constraints while balancing objective minimization with exploration and exploitation.Source: B. Search for the Global Optimum · p. 6
  • MACE combines its acquisition ensemble with the penalization strategy used in equation (14) to reduce sampling around infeasible regions, addressing WEIBO’s sequential design.WEIBO weights expected improvement by feasibility probability but is designed for sequential operation.Source: B. Search for the Global Optimum · pp. 6–7
  • Recommendation pruning uses ρ = 0.05 to favor likely valid regions and limit acquisition effort on candidates likely to violate constraints.Source: B. Search for the Global Optimum · p. 7

C. Summary

The paper summarizes constrained MACE as a two-stage batch method that first finds a feasible point and then searches for a feasible optimum using objective and constraint information. The experiments evaluate MACE on analog circuits and compare it with sequential and batch optimization methods.

C. Summary

  • The two-stage constrained method first seeks feasibility, then favors better objective regions while penalizing likely constraint violations; the authors state it can reach a feasible global optimum after limited iterations.After the first feasible point, updated surrogate posteriors provide more information about constraint satisfaction.Source: C. Summary · p. 7

VI. EXPERIMENTAL RESULTS

  • Experiments evaluate MACE on four real-world analog circuits for constrained and unconstrained optimization, using HSPICE simulations and a Linux workstation with two Intel Xeon CPUs and 128 GB memory.Source: §VI · p. 7
  • For unconstrained comparisons, algorithms receive the same simulation budget and are run 20 times, with best, worst, mean, and standard deviation reported.Source: §VI · p. 7
  • The unconstrained experiments compare sequential MACE with DE, EI, and LCB, and batch MACE with six Bayesian-optimization batch policies and three acquisition-function ensemble alternatives.Source: §VI · p. 7
  • The constrained experiments compare MACE with WEIBO, GASPAD, MSP, DE, PSO, and SA, and test the two-stage design against one-stage oMACE.Source: §VI · p. 8
  • Each algorithm is run 12 times for the constrained comparison, with simulation cost, successful feasible runs, and best-design constraint values reported.Source: §VI · p. 8

A. Two-Stage Operational Amplifier

The two-stage operational amplifier experiments evaluate MACE on unconstrained and constrained device sizing. MACE combines acquisition functions for batch search and reports faster simulation with competitive optimization results across the tested settings.

Experimental setup

  • The two-stage amplifier has 10 design variables and is evaluated for maximizing GAIN, UGF and PM; its constrained specification maximizes GAIN subject to UGF > 40MHz and PM > 60o.The unconstrained study uses a weighted objective combining those three metrics.Source: A. Two-Stage Operational Amplifier · pp. 8–9

Unconstrained optimization

  • 74× less simulation time than DE accompanies better optimization results for MACE in sequential unconstrained optimization.MACE also outperforms EI, LCB, PI-EI, EI-LCB and PI-LCB in convergence rate and stability.Source: A. Two-Stage Operational Amplifier · p. 8
  • Acquisition-function ensembles outperform EI and LCB alone, which the authors attribute to combining their benefits and increasing information gain per data point.MACE also outperforms two-function ensembles PI-EI, EI-LCB and PI-LCB in the reported unconstrained comparisons.Source: A. Two-Stage Operational Amplifier · p. 9
  • MACE’s batch design selects query points at once from the Pareto front of PI, EI and LCB, and consistently outperforms state-of-the-art batch policies at matched batch sizes.Unlike iterative greedy selection, this approach maintains within-batch diversity without manually designed penalization.Source: A. Two-Stage Operational Amplifier · p. 9

Constrained optimization

  • 185×, 164×, 12× and 5× are MACE’s respective simulation-time reductions versus DE, MSP, GASPAD and WEIBO in constrained optimization, with relatively better results.Both PSO and SA fail to meet the constraints.Source: A. Two-Stage Operational Amplifier · p. 9
  • MACE consistently outperforms its one-stage counterpart across batch sizes, and at batch size 15 retains better mean, median, best-case and worst-case results despite slight degradation as batch size grows.The authors attribute the two-stage policy’s advantage to reducing sampling around infeasible regions.Source: A. Two-Stage Operational Amplifier · p. 9

B. Class-E Power Amplifier

The class-E power amplifier experiments test MACE on unconstrained and constrained sizing. Results report simulation-time reductions against several methods and compare batch policies across batch sizes.

Experimental setup

  • The class-E circuit has 12 design variables and is evaluated for maximizing PAE subject to Pout > 2.0dBm in the constrained setting.The circuit is implemented in a 180nm TSMC process and its performance is simulated with HSPICE.Source: B. Class-E Power Amplifier · pp. 10–11

Unconstrained optimization

  • 42× less simulation time than DE accompanies more competitive optimization results for MACE in sequential unconstrained optimization.Acquisition-function ensembles outperform EI and LCB, and MACE outperforms the two-function ensembles PI-EI, EI-LCB and PI-LCB.Source: Table III · p. 10, B. Class-E Power Amplifier · p. 10
  • At matched batch sizes, MACE consistently outperforms state-of-the-art batch policies and maintains competitive results across batch sizes, supporting a better exploration–exploitation tradeoff.The batch is selected from the Pareto front of the acquisition-function ensemble rather than through a single acquisition function.Source: B. Class-E Power Amplifier · p. 10

Constrained optimization

  • 11×, 9×, 49×, 70× and 69× are MACE’s respective simulation-time reductions versus WEIBO, GASPAD, MSP, DE and PSO in constrained optimization.MACE finds valid designs in all runs, while SA and oMACE fail to find a feasible design in all runs.Source: Table IV · p. 11, B. Class-E Power Amplifier · p. 11

C. Low-Power Three-Stage Amplifier

The constrained amplifier and charge-pump experiments compare MACE with other optimizers under circuit-specific design specifications. MACE reduces simulation time in both experiments, while its performance across batch sizes and the two-stage policy are also discussed.

C. Low-Power Three-Stage Amplifier

  • The amplifier experiment minimizes static current subject to thresholds for gain, frequency, phase and gain margins, and rising and falling slew rates.The specified thresholds include GAIN > 100dB and UGF > 0.92MHz.Source: C. Low-Power Three-Stage Amplifier · p. 11
  • 6×, 13×, 42× and 42× lower simulation-time consumption than WEIBO, GASPAD, DE and PSO, respectively, accompanies higher sampling efficiency for the amplifier.SA and MSP fail to meet the specification in all runs.Source: C. Low-Power Three-Stage Amplifier · p. 12
  • MACE’s optimization results remain relatively similar across batch sizes, while oMACE’s deteriorate as batch size increases.The authors describe this as evidence of the two-stage approach’s robustness and effectiveness.Source: C. Low-Power Three-Stage Amplifier · p. 12

D. Charge Pump

  • The charge-pump experiment minimizes FOM subject to limits on current differences and deviation across process, voltage and temperature.The experiment compares MACE and oMACE at three batch sizes, with 12 runs per algorithm.Source: D. Charge Pump · p. 12
  • 15×, 46×, 31× and 30× simulation-time reductions versus WEIBO, GASPAD, MSP and DE, respectively, accompany better optimization results for the charge pump.PSO and SA find no feasible designs in all 12 runs.Source: D. Charge Pump · p. 12, Table VI · p. 12
  • MACE consistently outperforms oMACE across batch sizes; the authors suggest the two-stage policy provides more information about the feasible region and accelerates optimization.For MACE and oMACE, the charge-pump experiments use an initial dataset of 20 and a simulation budget of 830.Source: D. Charge Pump · pp. 12–13

VII. CONCLUSION

The conclusion presents MACE as a batch Bayesian optimization algorithm for unconstrained and constrained optimization. It combines PI, EI and LCB through Pareto-front sampling and adapts the constrained search with a two-stage policy and a penalization term.

VII. CONCLUSION

  • MACE samples from the Pareto front of PI, EI and LCB to balance exploration and exploitation in unconstrained optimization.For constrained optimization, it first seeks feasible designs, then uses a penalization term to reduce sampling around invalid regions while exploring potential areas.Source: §VII · p. 13
  • The two-stage constrained procedure first gathers information about the feasible region by focusing on finding feasible designs.Afterward, the penalization term favors feasible regions while reducing sampling around invalid regions.Source: §VII · p. 13
  • The experiments demonstrate the proposed algorithm’s robustness and cost-effectiveness.Source: §VII · p. 13

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