High-dimensional Bayesian optimization algorithm based on Voronoi graph migration

Abstract

The invention belongs to the field of integrated circuit computer aided design / electronic design automation, and particularly relates to a high-dimensional Bayesian optimization algorithm based on Voronoi graph migration. According to the method, a continuous design space is mapped into an undirected connection graph, and global exploration of the design space is achieved through migration operation of an upper confidence boundary (UCB) function on the graph. For an actual circuit optimization problem or other inherent low-dimensional problems, a local feature extraction method based on slice inverse regression is introduced, a next simulation query point is generated in an effective subspace, and the sampling efficiency of the algorithm is further improved. The VGT algorithm is the first high-dimensional optimization algorithm which is used for popularizing Bayesian optimization to 1000D and has an ultrahigh-dimensional scale at present.

Description

Technical Field

[0001] This invention belongs to the technical field of computer-aided design / electronic design automation (CAD / EDA) for integrated circuits, specifically relating to a high-dimensional Bayesian optimization algorithm based on Voronoi diagram walks in analog integrated circuit design automation. Background Technology

[0002] Due to the curse of dimensionality in high-dimensional design spaces, the performance of high-dimensional Bayesian optimization algorithms degrades exponentially with increasing dimensionality. On one hand, the sparse observation data in high-dimensional design spaces severely limits the accuracy of Gaussian process models. It is difficult to effectively capture the objective function's context with limited observation points, leading to blind exploration of high-dimensional design spaces. On the other hand, the training complexity of Gaussian process models increases cubically with the number of observation points. However, solving high-dimensional problems typically involves a large number of observation points, further exacerbating the computational cost of Bayesian optimization in solving high-dimensional problems.

[0003] The convergence performance of Bayesian optimization algorithms heavily relies on the accuracy of the Gaussian process model. However, building a reliable surrogate model in a high-dimensional space using limited observation data is impractical. Mainstream high-dimensional Bayesian optimization algorithms focus on improving the credibility of the Gaussian process model to enhance sampling efficiency. Dimensional decoupling methods attempt to fit a high-dimensional objective function using a set of low-dimensional additive Gaussian process models, transforming the high-dimensional problem into a sum of low-dimensional functions. Subspace embedding methods project the high-dimensional input space into a low-dimensional latent space through linear or nonlinear mappings for modeling and optimization. Both of these methods fit a low-dimensional manifold of the objective function within a low-dimensional subspace using the Gaussian process model. Trust region methods gradually improve the credibility of the surrogate model within the trust region through local trust constraints. However, for high-dimensional optimization problems of hundreds or even thousands of dimensions, building a reliable local model remains computationally expensive and impractical.

[0004] Multi-starting-point driven local search algorithms are a common approach for solving high-dimensional optimization problems. Starting from a certain initial point, local search algorithms continuously adjust the search direction and step length to find better solutions. Line search and trust regions are two commonly used types of local search algorithms. However, the final convergence performance of these methods heavily depends on the choice of the initial point.

[0005] Based on the current state of the technology, the inventors of this application propose a high-dimensional Bayesian optimization algorithm based on Voronoi graph traversal (VGT). We aim to extend Bayesian optimization to ultra-high-dimensional spaces of hundreds to thousands of dimensions. Since it is unrealistic to accurately describe the high-dimensional functional landscape by simply relying on surrogate models, we change our problem-solving approach. Instead of pursuing improved accuracy of the Gaussian process model, we fully utilize the boundary information and adjacency relationships provided by the Voronoi graph to guide the optimization. Based on this, this chapter proposes a sample-efficient Voronoi graph traversal (VGT) algorithm for solving high-dimensional black-box optimization problems of hundreds to thousands of dimensions. VGT first uses a Voronoi graph to define the design space. The graph is decomposed into multiple convex Voronoi units, and then mapped into an undirected graph based on the adjacency relationships between nodes in the Voronoi graph. By iteratively performing path selection, local Bayesian sampling, and graph expansion and propagation operations in the graph, a path leading to the global optimum along the valley with a better objective function value is continuously found.

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[0018] The purpose of this invention is to provide a high-dimensional Bayesian optimization algorithm based on Voronoi diagram walking for the size optimization problem in high-dimensional analog integrated circuits.

[0019] Specifically, this method proposes a high-dimensional Bayesian optimization algorithm based on Voronoi graph traversal (VGT). The VGT algorithm maps a continuous design space to an undirected connected graph and achieves global exploration of the design space through traversal operations on the graph using the upper confidence bound (UCB) function. For practical circuit optimization problems or other inherently low-dimensional problems, we introduce a local feature extraction method based on slice inverse regression to generate the next simulation query point within the effective subspace, further improving the algorithm's sampling efficiency. The VGT algorithm is currently the first high-dimensional optimization algorithm to extend Bayesian optimization to an ultra-high-dimensional scale of 1000D.

[0020] Voronoi undirected graph

[0021] We consider building an undirected graph with self-loops based on the Voronoi graph, called the Voronoi graph, denoted as . Among them, the node set It consists of existing observation data. Each node ν i The corresponding Voronoi element Vor(ν) of a convex polygon i Each non-self-loop edge e i,j ∈ε, i≠j, means that we are at the observation point v i and v j Neighborhood relationships are established between nodes based on their similarity. If we use Voronoi neighbors, the resulting Voronoi undirected graph... This can be viewed as an extension of the Delaunay triangulation by adding self-loops. However, in high-dimensional design spaces, the number of simplexes (such as triangles or tetrahedrons) in the Delaunay triangulation increases exponentially with the dimension of the space, making it infeasible to efficiently compute and store the boundaries and neighbors of Voronoi cells. Fortunately, the VGT algorithm proposed in this invention does not rely on accurate Voronoi neighbor relationships; we only need to construct an undirected connectivity graph based on the similarity between observation points, without the need for explicit computation of Voronoi boundaries and neighbors. For the sake of notation simplicity, we will use the node v i The neighbor subset is defined as Right now Includes node v i It itself and all its neighbors.

[0022] slice inverse regression

[0023] In this section, we employ slice inverse regression to extract the effective dimensionality of the objective function locally. Slice inverse regression (SIR) is a supervised data dimensionality reduction method suitable for scenarios where the number of observations is less than the dimension D of the design space. We consider the following regression model:

[0024]

[0025] in, Let represent the low-dimensional effective direction subspace (EDR) containing d feature directions, where the dimension d << D. ∈ represents the regression error. Under the premise of satisfying the linear design condition (LDC), the pattern space... From the center inverse regression curve Extracted from [the original data]. This is achieved by solving the following eigenvalue decomposition problem:

[0026]

[0027] We can then obtain the d eigenvectors with the largest eigenvalues, which constitute the effective linear subspace we need. Where, Σ (·) This represents the covariance matrix estimated from the observation data according to the y-slice[1].

[0028] Due to the limited number of observation points and high computational complexity, we cannot establish a reliable general GP model in high-dimensional space. Therefore, the effective guidance provided by the surrogate model for generating the next candidate point is significantly reduced, leading to blind exploration of Bayesian optimization in high-dimensional space. The Voronoi Graph Walk (VGT) algorithm, however, adopts a different solution approach. We utilize the geometric information in the Voronoi graph to continuously and implicitly update and adjust the step size and search direction of the next iteration, avoiding unfavorable regions with poor objective function values, and ultimately forming a walk path that gradually leads to a better region.

[0029] We will design the space The graph is divided into multiple convex Voronoi elements and mapped to an undirected Voronoi graph based on their Voronoi neighbor relationships, as shown below. Figure 1As shown. The VGT search process mainly includes two stages: global exploration and local development. In the global exploration stage, the algorithm starts from the current node and selects a high-potential neighbor node for local Bayesian optimization, which corresponds to the walk operation on the Voronoi undirected graph. In the local development stage, we use local Bayesian optimization to generate the next sampling point in the high-potential Voronoi cell, and the sampling step size and search direction directly affect the sampling efficiency of the algorithm. We adjust the sampling mechanism of Bayesian optimization. The boundary of the Voronoi cell, that is, the perpendicular bisector between the selected node and its neighbor nodes, has higher exploration value. Sampling near the boundary of the Voronoi cell can maximize the information gain for the current observation point set V [2]. Therefore, we expect to use the boundary of the Voronoi cell to guide the selection of the step size of the next observation point. In addition, the search direction is determined according to the local Gaussian process model established by the target node and its Voronoi neighbors. Since it is not feasible to explicitly calculate and store the Voronoi boundary, we use a Gaussian distribution to approximate the contour of the target Voronoi cell, such as Figure 2 The dashed ellipse in the diagram illustrates this, and new observation points are generated by sampling based on this distribution. Furthermore, in the VGT, we also provide guidance for the sampling direction of local Bayesian optimization through the shape of the Voronoi cells and their geometric boundaries. Figure 2 The paper illustrates how the geometric boundaries of the Voronoi diagram guide the high-dimensional optimization process. For newly generated observation points with poor objective function values, such as v7, the space of the original objective Voronoi unit Vor(v6) is modified, and its boundary shape is changed, thereby adjusting the sampling direction of the next step of local Bayesian optimization. Figure 2 As shown in (b). "Good" observation points that cause the objective function value to decrease will guide the algorithm towards a more optimal region, such as... Figure 2 As shown in v8 in (b), the new observation point ν9 obtained by performing local Bayesian optimization within the newly generated Voronoi unit Vor(v6) has converged to the vicinity of the global optimum. By utilizing the geometric boundaries and adjacency relationships of the Voronoi graph, the VGT algorithm can fully utilize the coordinate information of each observation point and its observation value to guide the generation of the next sample point, thereby significantly improving the sampling efficiency of Bayesian optimization in high-dimensional space.

[0030] The specific process of the VGT algorithm is as follows: Figure 3As shown in Algorithm 1, it mainly includes three basic steps: (1) Path selection generates a movement step from the current node to a high-potential neighbor node. (2) Local Bayesian optimization generates a new observation point within the selected Voronoi cell. (3) Expansion and propagation operations expand the Voronoi undirected graph with the new observation point and update the reward value. The above steps are continuously iterated during the optimization process until the stopping condition is met.

[0031] The VGT algorithm starts from a randomly generated initial point. Begin the search. This initial node covers the entire search space. like Figure 4 As shown in (a). Here, h is the depth of the node, representing the cumulative number of visits to that node and its parent nodes by the algorithm. The update of the node depth h follows... Figure 3 (a) shows the tree structure. The set of all leaf nodes represents the design space at time t. A partition. Then we will describe the search process of the VGT algorithm in detail.

[0032] Path selection

[0033] In each iteration t, the VGT algorithm starts from the target node of the previous iteration. Start by moving to a high-potential neighbor node. This allows the algorithm to continuously advance towards the globally optimal region. For example... Figure 3 As shown in (b), from node v 5,h＝6 Move to node v 6,h＝6 Because of the optimal node in the previous iteration Also included in the neighbor set Therefore, the VGT algorithm may encounter "static" iteration steps during the iteration process. To achieve global exploration of the entire search space, we use an upper confidence bound (UCB) to evaluate the potential of each node. The UCB function used for graph walk is defined as:

[0034]

[0035] Where Q(v) i ) represents node v i The mass function, h(v) i ) is node v i The depth. For ease of implementation, we use Q(v) in the experiment. i )=-f(x i To evaluate node v i The quality of the hyperparameter C. pThis approach balances the exploration of low-depth Voronoi units with the development of high-quality Voronoi units. The path selection strategy in VGT explicitly restricts the candidate nodes in the t-th iteration to a subset of the neighbors of the best node in the previous iteration, rather than all nodes in the undirected graph. This allows the algorithm to focus on the optimal local regions most likely to decrease the objective function value, thereby continuously improving the quality of the solution.

[0036]

[0037] Local Bayesian optimization

[0038] We select high-potential neighbor nodes corresponding to Voronoi units. We perform local Bayesian optimization to generate new observation points. To reduce the high computational complexity of the Gaussian process model, we propose a Voronoi neighbored Gaussian process (VNGP) model, which uses only a subset of the target node's neighbors. Modeling is then performed. Since the correlation defined by the kernel function of the Gaussian process model decays exponentially with increasing distance between observation points, observation points far from the target Voronoi cell contribute very little to the local modeling within that Voronoi cell. The VNGP model, on the other hand, fully utilizes the adjacency relationships provided by the Voronoi undirected graph, significantly reducing computational costs while maintaining local modeling accuracy. Then, we model the Voronoi cell... The internal optimization function is used to generate a new round of observation points. To achieve efficient sampling, we adopted a point-based approach. The objective function is sampled using a Gaussian distribution centered at the center, such as... Figure 3 The black dashed ellipse in (c) is shown. We use the ellipse falling on the target cell. The hyperparameters of the sampling distribution are adjusted using internal sample points, while sample points falling outside the target Voronoi unit are discarded.

[0039] Expansion and dissemination

[0040] Each time at a depth of Voronoi unit Internal sampling will generate a depth of The new Voronoi unit, while the original Voronoi unit The depth is increased by 1. Then, the algorithm adds the new node. Add to Voronoi undirected graph In the middle, and update the UCB function, such as Figure 3As shown in (d), the expansion and propagation operations improve the Voronoi undirected graph by adding new observation points, thereby continuously adjusting and optimizing the next search path and improving the convergence speed of the algorithm.

[0041] By iteratively executing the above steps, the VGT algorithm continuously adjusts its trajectory, guiding it towards the optimal region. During the optimization process, the VNGP model is also continuously updated and incorporates new observation points as the algorithm progresses, maintaining the model's adaptability to the local trends of the objective function. Through the combined effect of the UCB-based path selection strategy and local Bayesian optimization, the algorithm finds a path along the valley with the smallest objective function value towards the global optimum, such as... Figure 5 As shown by the red path in the diagram. To more intuitively illustrate the working mechanism of the VGT algorithm, we... Figure 4 The iterative and convergence process of the VGT algorithm is illustrated below. The algorithm starts its search from a random initial point v0, as shown below. Figure 4 As shown in (a), Vor(v0) represents the entire space. "Good" observation points that allow the function value to decrease, such as Figure 4 V4, V5, V6, and V7, guided by UCB, will lead the algorithm towards a more performant region, as shown by the red arrows in the figure. Meanwhile, "bad" observations with poor objective function values, such as ν1, ν2, ν3, and ν8, will help the algorithm avoid unfavorable regions and adjust the sampling direction for the next step of local Bayesian optimization. As iterations proceed, high-potential target Voronoi units gradually shrink and converge to the vicinity of the global optimum.

[0042] Next, we introduce two key techniques to extend the VGT algorithm to high-dimensional input spaces: the approximation of Voronoi undirected graphs and subspace sampling techniques.

[0043] Approximation of Voronoi undirected graphs

[0044] In scenarios involving high-dimensional space and large sample budgets, the cost of explicitly calculating and storing Voronoi graphs and Delaunay triangulations is prohibitive. Therefore, for high-dimensional scenarios, we no longer explicitly calculate the geometric boundaries of Voronoi cells and Delaunay adjacency relationships. We introduce similarity search methods, such as K-nearest neighbor search (K-NNS)[4] or approximate nearest neighbor search (ANNS)[5,6], to approximate the neighbor relationships between observation points and establish an approximate Voronoi undirected graph. In this section, we use K-nearest neighbor search to estimate the neighbor nodes of the target node. That is, for each observation point, we connect it to the K nearest neighbor nodes in terms of Euclidean distance, thus forming an undirected connected graph in the search space. Here, K is a hyperparameter, and the value of K is usually related to the dimension D of the problem. For a sufficiently large K, all Voronoi neighbor nodes of the node will be included in the K nearest neighbor nodes. For a medium value of K, the Voronoi neighbors of the target node tend to be consistent with its K nearest neighbors. Generally, in our experiments we take... Furthermore, in the high-dimensional case, we no longer explicitly compute Voronoi boundaries, but instead discard candidate points that fall outside the target Voronoi cell. Therefore, the approximation problem of high-dimensional Voronoi undirected graphs is ultimately reduced to using K-nearest neighbor search. This approximation effectively captures the similarity relationships between observation points while avoiding the high computational and storage costs associated with explicitly computing Voronoi boundaries and Delaunay adjacency relationships.

[0045] Subspace sampling technology

[0046] In high-dimensional design spaces, Gaussian process models are typically highly underdetermined and exhibit large uncertainties in most regions, especially when the number of available observations is much smaller than the problem dimension D. To address this issue, we employ a slice inverse regression method to extract the effective dimension of the practical problem, thereby achieving dimensionality reduction. For the target node... We observe data adjacent to it Localized SIR (LSIR)[7] analysis was performed to extract Voronoi units. Local effective subspace within Furthermore, we follow the direction s of the last descent of the objective function. t-1 Searching on this subspace has a higher probability of yielding a better solution. Utilizing this property, we construct a pattern subspace. The dimension of the subspace is much smaller than the original dimension of the problem, and Therefore, within the effective subspace, we can more accurately model the trend of the objective function with fewer observation points. Then, in the subspace... Internal optimization obtains the function α(x) t +s) to generate the next observation point x t +s:

[0047]

[0048] Subspace methods based on slice inverse regression can capture the effective low-dimensional manifold of the objective function in a high-dimensional space, improving the sampling efficiency of local Bayesian optimization. However, since newly generated observations using subspace techniques all lie within this low-dimensional manifold, the diversity of subsequently generated observations decreases. To alleviate this problem, we employ an alternating strategy of subspace sampling and full-dimensional sampling. We introduce a hyperparameter R... p R is used to represent the ratio between subspace sampling and full-dimensional sampling. For inherently low-dimensional problems with a small effective dimension, a larger R value can be set. p The value can be used to allocate more resources for subspace sampling, thereby improving the convergence speed. For complex problems with a high intrinsic effective dimension, a smaller R value can be set. p This is to better maintain the diversity of observational data. Attached Figure Description

[0049] Figure 1 .Voronoi undirected graph Schematic diagram

[0050] Figure 2 A schematic diagram of high-dimensional optimization guided by the geometric boundaries of the Voronoi diagram.

[0051] Figure 3 .Diagram of the VGT algorithm's search process

[0052] Figure 4 A schematic diagram of the convergence process of the VGT algorithm.

[0053] Figure 5 .VGT algorithm search path diagram

[0054] Figure 6 Schematic diagram of OPAMP circuit

[0055] Figure 7 Experimental results of the implementation example

[0056] Figure 8 Schematic diagram of PLL circuit Detailed Implementation

[0057] We use two practical engineering application examples to evaluate the performance of the VGT algorithm in real-world problems. We mainly consider two analog circuit optimization problems: an amplifier circuit OPAMP36D and a phase-locked loop (PLL) circuit PLL77D. The following describes the implementation process of the present invention through specific examples.

[0058] Implementation Example 1

[0059] OPAMP circuit: The schematic diagram of the OPAMP circuit is as follows Figure 6 As shown, this circuit is designed based on a 350nm open-source process library. Considering topological symmetry and constraints such as current mirrors, the circuit contains 36 free design variables, consisting of dimensions such as transistor width and length, and capacitor area. For the optimization metrics, we minimize its quiescent current Iddq, while imposing strict design constraints on its delay@1MHz, Vpp@1MHz, and Voffset. We transform the circuit optimization problem into an unconstrained single-objective optimization problem with constraint penalties, where the circuit performance and constraints are provided by the SPICE simulator. Figure 7 The average experimental results and variance of 10 repeated experiments are given. Since the device model of transistors in the circuit is generally piecewise continuous, this leads to the heterogeneity of the performance function of the circuit. In this test case, the convergence speed of the VGT algorithm proposed in this invention continues to lead other comparative methods. The convergence performance of the Monte Carlo Tree Descent Method (MCTD) is very close to that of the Trust Region Bayesian Optimization Method (TuRBO) [9], which shows that the random three-point descent method proposed in MCTD does not produce significant benefits in this practical problem. The spatial decomposition method LaMCTS based on SVM boundary performs poorly in circuit problems, even worse than the random search method Random Search. The classic simplex method Nelder Mead cannot find a feasible solution that satisfies the circuit constraints in this problem.

[0060] Implementation Example 2

[0061] PLL circuit: Circuit schematic shown Figure 8As shown, the circuit mainly consists of a Phase Frequency Detector (PFD), a charge pump, a low-pass filter (LPF), a ring oscillator, and a 4-bit feedback circuit. For this circuit, we primarily optimized the charge pump, low-pass filter, and voltage-controlled oscillator (VCO) modules. For the other digital logic circuit parts, we used behavioral models to reduce simulation time. Considering the symmetry of the circuit topology and current mirroring, the circuit contains 77 free design variables consisting of transistor width and length, resistor length, and capacitor area. For this PLL circuit, we minimized its average current consumption Iddavg and ensured that the peak-to-peak value of its output voltage Vclkout was higher than 2.8V. Experimental results are shown below. Figure 7 As shown. In this PLL circuit test case, the convergence performance of the VGT algorithm is still significantly better than other comparative methods. The HesBO[8] method based on random subspace has good convergence performance in the early stage of iteration, but performs poorly in the later stage. This also shows that the subspace method can effectively improve the sample efficiency for actual low-dimensional problems, but its dependence on subspace technology can easily lead to a decrease in the sample diversity of the observation data, thus affecting the final convergence result of the algorithm. LaMCTS based on SVM boundary again shows poor convergence performance for this circuit problem, which shows that its applicability and generalization ability of SVM boundary for actual problems are weak, especially for nonlinear and heterogeneous high-dimensional analog circuit optimization problems, its objective function descent speed is even difficult to exceed that of random search.

Claims

1. A high-dimensional Bayesian optimization algorithm based on Voronoi diagram walks, characterized in that, The VGT algorithm maps a continuous design space to an undirected connected graph and achieves global exploration of the design space through walks on the graph using the upper confidence bound (UCB) function. For practical circuit optimization problems or other inherently low-dimensional problems, a local feature extraction method based on slice inverse regression is introduced to generate the next simulation query point in the effective subspace, further improving the sampling efficiency of the algorithm; The VGT algorithm is the first high-dimensional optimization algorithm to extend Bayesian optimization to an ultra-high-dimensional scale of 1000D.

2. The method according to claim 1, characterized in that, The VGT algorithm first decomposes the design space x into multiple convex Voronoi units using a Voronoi graph. Then, it maps the Voronoi graph to an undirected graph based on the adjacency relationships between nodes. By iteratively performing path selection, local Bayesian optimization, and graph expansion and propagation operations on the graph, it continuously finds a path leading to the global optimum along the valley with a relatively good objective function value.

3. The method according to claim 2, characterized in that, In the aforementioned local Bayesian optimization operation, the Voronoi unit corresponding to the selected high-potential neighbor node is... Local Bayesian optimization is performed within the scope to generate new observation points; To reduce the high computational complexity of Gaussian process models, a Voronoi-neighbored Gaussian process (VNGP) model is proposed, which uses only a subset of the target node's neighbors. Modeling; Since the correlation defined by the kernel function of the Gaussian process model decays exponentially with increasing distance between observation points, observation points far from the target Voronoi cell contribute very little to the local modeling within that Voronoi cell. The VNGP model, on the other hand, fully utilizes the adjacency relationships provided by the Voronoi undirected graph, significantly reducing computational costs while maintaining local modeling accuracy. Then, within the Voronoi cell... The internal optimization function is used to generate a new round of observation points. To achieve efficient sampling, a point-based sampling method is adopted. The objective function is sampled using a Gaussian distribution centered at the target cell, as shown by the black dashed ellipse in Figure 3(c); the sampled data is taken from the target cell. The hyperparameters of the sampling distribution are adjusted using internal sample points, and sample points falling outside the target Voronoi unit are discarded.

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