tall 配列を使用したベイズ最適化

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この例では、名前と値のペアの引数 'OptimizeHyperparameters' を使用してカーネル分類器の学習に最適なパラメーターをベイズ最適化によって選択する方法を示します。標本データセット airlinesmall.csv は、飛行機のフライトデータについての表形式ファイルが含まれている大規模なデータセットです。この例では、データが含まれている tall table を作成して、最適化手順を実行するために使用します。

MATLAB® へのデータの取得

データがあるフォルダーの場所を参照するデータストアを作成します。データは、単一のファイル、ファイルの集まり、またはフォルダー全体のどれに含まれていてもかまいません。一連のファイルが格納されているフォルダーについて、フォルダー全体の場所を指定するか、'*.csv' のようにワイルドカード文字を使用して、同じファイル拡張子をもつ複数のファイルをデータストアに含めることができます。処理する変数のサブセットを選択します。datastore で NaN 値に置き換えるため、'NA' 値を欠損データとして扱います。データストア内のデータを含む tall table を作成します。

ds = datastore('airlinesmall.csv');
ds.SelectedVariableNames = {'Month','DayofMonth','DayOfWeek',...
                            'DepTime','ArrDelay','Distance','DepDelay'};
ds.TreatAsMissing = 'NA';
tt  = tall(ds) % Tall table

Starting parallel pool (parpool) using the 'local' profile ...
connected to 6 workers.

tt =

  M×7 tall table

    Month    DayofMonth    DayOfWeek    DepTime    ArrDelay    Distance    DepDelay
    _____    __________    _________    _______    ________    ________    ________

     10          21            3          642          8         308          12   
     10          26            1         1021          8         296           1   
     10          23            5         2055         21         480          20   
     10          23            5         1332         13         296          12   
     10          22            4          629          4         373          -1   
     10          28            3         1446         59         308          63   
     10           8            4          928          3         447          -2   
     10          10            6          859         11         954          -1   
      :          :             :           :          :           :           :
      :          :             :           :          :           :           :

tall 配列の計算を実行する場合、既定の実行環境ではローカルの MATLAB セッションまたは (Parallel Computing Toolbox™ がある場合は) ローカルの並列プールが使用されます。関数 mapreducer を使用して実行環境を変更できます。

クラスラベルと予測子データの準備

フライトが遅れた場合に真になる論理変数を定義することにより、10 分以上遅れたフライトを判別します。この変数にクラスラベルを含めます。この変数のプレビューには、はじめの数行が含まれています。

Y = tt.DepDelay > 10 % Class labels

Y =

  M×1 tall logical array

   1
   0
   1
   1
   0
   1
   0
   0
   :
   :

予測子データの tall 配列を作成します。

X = tt{:,1:end-1} % Predictor data

X =

  M×6 tall double matrix

          10          21           3         642           8         308
          10          26           1        1021           8         296
          10          23           5        2055          21         480
          10          23           5        1332          13         296
          10          22           4         629           4         373
          10          28           3        1446          59         308
          10           8           4         928           3         447
          10          10           6         859          11         954
          :           :            :          :           :           :
          :           :            :          :           :           :

欠損データが含まれている X および Y の行を削除します。

R = rmmissing([X Y]); % Data with missing entries removed
X = R(:,1:end-1); 
Y = R(:,end);

`OptimizeHyperparameters` を使用したベイズ最適化の実行

名前と値のペアの引数 'OptimizeHyperparameters' を使用して、自動的にハイパーパラメーターを最適化します。

予測子変数を標準化します。

Z = zscore(X);

5 分割交差検証損失を最小化する、名前と値のペアの引数 'KernelScale' および 'Lambda' に最適な値を求めます。再現性を得るために、tall 配列の乱数シードを設定し、'expected-improvement-plus' の獲得関数を使用します。

rng('default') 
tallrng('default') % For reproducibility
Mdl = fitckernel(Z,Y,'Verbose',0,'OptimizeHyperparameters','auto',...
    'HyperparameterOptimizationOptions',struct('AcquisitionFunctionName','expected-improvement-plus'))

Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 3: Completed in 9.1 sec
- Pass 2 of 3: Completed in 2 sec
- Pass 3 of 3: Completed in 2.9 sec
Evaluation completed in 17 sec

Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 2.4 sec
Evaluation completed in 2.7 sec
|=====================================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  KernelScale |       Lambda |
|      | result |             | runtime     | (observed)  | (estim.)    |              |              |
|=====================================================================================================|
|    1 | Best   |     0.19684 |      142.25 |     0.19684 |     0.19684 |       1.2297 |    0.0080902 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|    2 | Accept |     0.19684 |      65.457 |     0.19684 |     0.19684 |     0.039643 |   2.5756e-05 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|    3 | Accept |     0.19684 |       64.04 |     0.19684 |     0.19684 |      0.02562 |   1.2555e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2 sec
|    4 | Accept |     0.19684 |      70.552 |     0.19684 |     0.19684 |       92.644 |   1.2056e-07 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|    5 | Best   |     0.11465 |      110.25 |     0.11465 |     0.12696 |       11.173 |   0.00024836 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|    6 | Accept |     0.11482 |      120.81 |     0.11465 |     0.11469 |         11.6 |   0.00023941 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 2 sec
Evaluation completed in 2.2 sec
|    7 | Accept |     0.19684 |      69.935 |     0.11465 |     0.11464 |       25.242 |   0.00039858 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 1.9 sec
|    8 | Accept |     0.11612 |      112.26 |     0.11465 |     0.11428 |       7.8238 |   0.00051928 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|    9 | Best   |     0.10315 |      56.865 |     0.10315 |      0.1034 |       9.5079 |   1.6228e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   10 | Accept |     0.10361 |      53.425 |     0.10315 |     0.10339 |       9.8089 |    8.544e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2 sec
|   11 | Best   |     0.10269 |      53.639 |     0.10269 |     0.10271 |       8.7043 |   8.8678e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   12 | Best   |     0.10264 |      57.069 |     0.10264 |     0.10262 |       5.0702 |   8.7446e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2 sec
|   13 | Accept |      0.1032 |      53.495 |     0.10264 |     0.10266 |       6.0627 |   1.0286e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   14 | Best   |     0.10247 |      52.683 |     0.10247 |      0.1025 |       3.6969 |    8.965e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|   15 | Accept |     0.19684 |      75.223 |     0.10247 |      0.1025 |       993.29 |   8.8815e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 2 sec
Evaluation completed in 2.2 sec
|   16 | Accept |      0.1075 |      101.64 |     0.10247 |     0.10249 |       3.7513 |   0.00052631 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|   17 | Best   |     0.10223 |      53.351 |     0.10223 |     0.10214 |       4.1228 |   8.7857e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   18 | Accept |     0.19684 |      63.202 |     0.10223 |     0.10214 |    0.0010043 |    0.0052836 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|   19 | Accept |     0.10301 |      52.799 |     0.10223 |     0.10251 |       4.1588 |   1.0109e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|   20 | Accept |     0.10291 |      54.454 |     0.10223 |     0.10251 |       7.3336 |   8.4868e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|=====================================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  KernelScale |       Lambda |
|      | result |             | runtime     | (observed)  | (estim.)    |              |              |
|=====================================================================================================|
|   21 | Best   |     0.10194 |      49.856 |     0.10194 |     0.10225 |       3.8113 |   8.8671e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|   22 | Best   |     0.10121 |      53.558 |     0.10121 |     0.10141 |       3.3132 |   8.9989e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   23 | Accept |     0.10294 |      53.957 |     0.10121 |     0.10137 |       5.4153 |   1.1478e-05 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   24 | Accept |     0.10297 |      49.455 |     0.10121 |     0.10136 |       4.5918 |   7.4273e-07 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   25 | Accept |     0.10226 |      49.383 |     0.10121 |     0.10134 |       2.9349 |    8.293e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   26 | Accept |     0.10188 |      53.047 |     0.10121 |     0.10128 |       3.2244 |    1.019e-07 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   27 | Best   |     0.10096 |      52.627 |     0.10096 |     0.10121 |       3.2763 |   1.4374e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.3 sec
Evaluation completed in 1.5 sec
|   28 | Accept |     0.19684 |      61.924 |     0.10096 |     0.10121 |    0.0048849 |    0.0039902 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   29 | Accept |      0.1018 |      49.423 |     0.10096 |     0.10138 |       3.2511 |   8.7094e-09 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
|   30 | Accept |     0.10164 |      49.933 |     0.10096 |     0.10143 |       3.2002 |   8.4371e-09 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 2031.7757 seconds.
Total objective function evaluation time: 2006.5588

Best observed feasible point:
    KernelScale      Lambda  
    ___________    __________

      3.2763       1.4374e-08

Observed objective function value = 0.10096
Estimated objective function value = 0.10143
Function evaluation time = 52.6271

Best estimated feasible point (according to models):
    KernelScale      Lambda  
    ___________    __________

      3.2763       1.4374e-08

Estimated objective function value = 0.10143
Estimated function evaluation time = 51.861

Mdl = 
  ClassificationKernel
            PredictorNames: {'x1'  'x2'  'x3'  'x4'  'x5'  'x6'}
              ResponseName: 'Y'
                ClassNames: [0 1]
                   Learner: 'svm'
    NumExpansionDimensions: 256
               KernelScale: 3.2763
                    Lambda: 1.4374e-08
             BoxConstraint: 576.5789


  Properties, Methods

`bayesopt` の使用によるベイズ最適化の実行

あるいは、関数 bayesopt を使用して、ハイパーパラメーターに最適な値を求めることもできます。

データセットを学習セットとテストセットに分割します。テストセット用に 1/3 のホールドアウト標本を指定します。

rng('default') 
tallrng('default') % For reproducibility
Partition = cvpartition(Y,'Holdout',1/3);
trainingInds = training(Partition); % Indices for the training set
testInds = test(Partition);         % Indices for the test set

学習データとテストデータを抽出し、予測子データを標準化します。

Ytrain = Y(trainingInds); % Training class labels
Xtrain = X(trainingInds,:);
[Ztrain,mu,stddev] = zscore(Xtrain); % Standardized training data

Ytest = Y(testInds); % Testing class labels
Xtest = X(testInds,:);
Ztest = (Xtest-mu)./stddev; % Standardized test data

名前と値のペアの引数 'KernelScale' および 'Lambda' に最適な値を求めるため、変数 sigma および lambda を定義します。optimizableVariable を使用し、最適な値は不明なので、変数について広い範囲を指定します。最適な値を対数スケールで求めるため、対数変換を変数に適用します。

N = gather(numel(Ytrain)); % Evaluate the length of the tall training array in memory

Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1 sec
Evaluation completed in 1.3 sec

sigma = optimizableVariable('sigma',[1e-3,1e3],'Transform','log');
lambda = optimizableVariable('lambda',[(1e-3)/N, (1e3)/N],'Transform','log');

ベイズ最適化の目的関数を作成します。目的関数は、変数 sigma および lambda が格納されている table を受け入れ、関数 fitckernel で学習をさせたバイナリガウスカーネル分類モデルの分類損失値を計算します。診断情報を反復表示しないようにするため、fitckernel で 'Verbose',0 を設定します。

minfn = @(z)gather(loss(fitckernel(Ztrain,Ytrain, ...
    'KernelScale',z.sigma,'Lambda',z.lambda,'Verbose',0), ...
    Ztest,Ytest));

bayesoptを使用して、カーネル分類モデルのパラメーター [sigma,lambda] を分類損失に関して最適化します。既定では、bayesopt は最適化に関する情報をコマンドラインに繰り返し表示します。再現性を得るために、AcquisitionFunctionName オプションを 'expected-improvement-plus' に設定します。既定の獲得関数は実行時に決定されるので、結果が異なる場合があります。

results = bayesopt(minfn,[sigma,lambda],'AcquisitionFunctionName','expected-improvement-plus')

Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 2: Completed in 0.9 sec
- Pass 2 of 2: Completed in 2.1 sec
Evaluation completed in 3.4 sec
|=====================================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |        sigma |       lambda |
|      | result |             | runtime     | (observed)  | (estim.)    |              |              |
|=====================================================================================================|
|    1 | Best   |     0.19649 |      157.57 |     0.19649 |     0.19649 |       1.2297 |     0.012135 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|    2 | Accept |     0.19649 |      195.79 |     0.19649 |     0.19649 |     0.039643 |   3.8633e-05 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|    3 | Accept |     0.19649 |      144.31 |     0.19649 |     0.19649 |      0.02562 |   1.8832e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.5 sec
Evaluation completed in 1.7 sec
|    4 | Accept |     0.19649 |       92.09 |     0.19649 |     0.19649 |       92.644 |   1.8084e-07 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.5 sec
Evaluation completed in 1.6 sec
|    5 | Accept |     0.19649 |      102.21 |     0.19649 |     0.19649 |       978.95 |   0.00015066 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|    6 | Accept |     0.19649 |      153.49 |     0.19649 |     0.19649 |    0.0089609 |    0.0059189 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.6 sec
Evaluation completed in 1.8 sec
|    7 | Accept |     0.19649 |      151.92 |     0.19649 |     0.19649 |    0.0010228 |    1.292e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.5 sec
Evaluation completed in 1.7 sec
|    8 | Accept |     0.19649 |      174.42 |     0.19649 |     0.19649 |      0.27475 |    0.0044831 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 2 sec
Evaluation completed in 2.2 sec
|    9 | Accept |     0.19649 |      145.09 |     0.19649 |     0.19649 |      0.81326 |   1.0753e-07 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 2.1 sec
Evaluation completed in 2.3 sec
|   10 | Accept |     0.19649 |      158.98 |     0.19649 |     0.19649 |    0.0040507 |   0.00011333 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|   11 | Accept |     0.19649 |      102.02 |     0.19649 |     0.19649 |       330.84 |   0.00010095 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.5 sec
|   12 | Accept |     0.19649 |       96.16 |     0.19649 |     0.19649 |        65.19 |   0.00010513 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   13 | Accept |     0.19649 |      99.916 |     0.19649 |     0.19649 |       46.548 |   1.2466e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   14 | Accept |     0.19649 |      100.04 |     0.19649 |     0.19649 |       721.08 |     0.012414 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.5 sec
Evaluation completed in 1.7 sec
|   15 | Accept |     0.19649 |      101.49 |     0.19649 |     0.19649 |       737.85 |   1.2515e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.5 sec
Evaluation completed in 1.7 sec
|   16 | Accept |     0.19649 |      173.23 |     0.19649 |     0.19649 |     0.010073 |   3.3049e-05 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   17 | Accept |     0.19649 |      157.35 |     0.19649 |     0.19649 |     0.025672 |   0.00019456 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   18 | Accept |     0.19649 |      103.59 |     0.19649 |     0.19649 |        710.7 |    0.0021827 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|   19 | Accept |     0.19649 |      170.31 |     0.19649 |     0.19649 |    0.0053051 |   0.00087841 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   20 | Accept |     0.19649 |      173.82 |     0.19649 |     0.19649 |      0.12625 |     0.011292 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.5 sec
Evaluation completed in 1.7 sec
|=====================================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |        sigma |       lambda |
|      | result |             | runtime     | (observed)  | (estim.)    |              |              |
|=====================================================================================================|
|   21 | Accept |     0.19649 |       149.3 |     0.19649 |     0.19649 |      0.00149 |   5.8754e-07 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   22 | Accept |     0.19649 |       98.03 |     0.19649 |     0.19649 |       52.749 |    1.265e-08 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 2 sec
Evaluation completed in 2.2 sec
|   23 | Best   |     0.10317 |       153.7 |     0.10317 |     0.10318 |       3.5534 |   0.00044046 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.6 sec
Evaluation completed in 1.8 sec
|   24 | Best   |     0.10282 |       140.2 |     0.10282 |     0.10271 |       3.6546 |   0.00024156 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   25 | Accept |     0.11286 |      133.23 |     0.10282 |     0.10292 |       4.3231 |    0.0013656 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   26 | Accept |     0.10409 |      156.29 |     0.10282 |     0.10304 |       3.4792 |   0.00021823 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   27 | Accept |     0.11883 |      128.77 |     0.10282 |      0.1029 |        3.672 |    0.0027929 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.6 sec
|   28 | Accept |     0.10451 |      144.39 |     0.10282 |     0.10339 |       3.8815 |   0.00031253 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.4 sec
Evaluation completed in 1.5 sec
|   29 | Accept |     0.10374 |       141.5 |     0.10282 |     0.10339 |       3.6237 |    0.0003024 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.9 sec
Evaluation completed in 2.1 sec
|   30 | Accept |     0.10362 |      165.36 |     0.10282 |     0.10336 |       3.6775 |   0.00024926 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 4343.3484 seconds.
Total objective function evaluation time: 4164.5815

Best observed feasible point:
    sigma       lambda  
    ______    __________

    3.6546    0.00024156

Observed objective function value = 0.10282
Estimated objective function value = 0.10336
Function evaluation time = 140.2034

Best estimated feasible point (according to models):
    sigma      lambda  
    ______    _________

    3.6237    0.0003024

Estimated objective function value = 0.10336
Estimated function evaluation time = 143.2776

results = 
  BayesianOptimization with properties:

                      ObjectiveFcn: @(z)gather(loss(fitckernel(Ztrain,Ytrain,'KernelScale',z.sigma,'Lambda',z.lambda,'Verbose',0),Ztest,Ytest))
              VariableDescriptions: [1×2 optimizableVariable]
                           Options: [1×1 struct]
                      MinObjective: 0.1028
                   XAtMinObjective: [1×2 table]
             MinEstimatedObjective: 0.1034
          XAtMinEstimatedObjective: [1×2 table]
           NumObjectiveEvaluations: 30
                  TotalElapsedTime: 4.3433e+03
                         NextPoint: [1×2 table]
                            XTrace: [30×2 table]
                    ObjectiveTrace: [30×1 double]
                  ConstraintsTrace: []
                     UserDataTrace: {30×1 cell}
      ObjectiveEvaluationTimeTrace: [30×1 double]
                IterationTimeTrace: [30×1 double]
                        ErrorTrace: [30×1 double]
                  FeasibilityTrace: [30×1 logical]
       FeasibilityProbabilityTrace: [30×1 double]
               IndexOfMinimumTrace: [30×1 double]
             ObjectiveMinimumTrace: [30×1 double]
    EstimatedObjectiveMinimumTrace: [30×1 double]

関数 bestPoint を使用して、ベイズモデル results 内の最良の実行可能点を取得します。既定の基準 min-visited-upper-confidence-interval を使用します。この基準では、目的関数の値に対して信頼区間の上限を最小化する探索済みの点が最良の実行可能点になります。

zbest = bestPoint(results)

zbest=1×2 table
    sigma      lambda  
    ______    _________

    3.6237    0.0003024

テーブル zbest には、名前と値のペアの引数 'KernelScale' および 'Lambda' に最適な推定値が格納されます。これらの値は、次の構文を使用して新しい最適化済みカーネル分類器に学習をさせるときに指定できます。

Mdl = fitckernel(Ztrain,Ytrain,'KernelScale',zbest.sigma,'Lambda',zbest.lambda)

tall 配列の場合、最適化手順に時間がかかる可能性があります。最適化手順を実行するにはデータセットが大きすぎる場合、データの一部だけを使用してパラメーターの最適化を試すことができます。関数 datasample を使用し、'Replace','false' を指定して、データを非復元抽出します。

参考