文献来源，然后复现之：
💥1 概述
SBP算法已成为用于训练多层感知器的标准算法，如图1所示。它是一种广义最小均方 （LMS） 算法，它最小化等于实际输出和所需输出之间误差平方和的标准。这个标准是：
详细文章下载链接：
📚2 运行结果
W1W1 = -0.1900 -0.7425 -2.9507 -0.3955 0.2059 -0.8937 -0.4751 0.5315 -1.2644 -2.5390 -1.8319 3.2741 -1.0816 -0.6534 0.7954 -1.4622 -0.0331 -0.4283 -0.3125 0.0840 -0.8470 -0.6496 -0.2922 -0.6272 b1b1 = 9.4243 1.5431 0.1860 2.4630 0.1097 0.5390 1.6335 -0.4229 W2W2 = -1.6282 0.5796 0.2008 1.0366 0.9238 -0.3099 0.6122 -0.0681 b2b2 = 0.3461 Mean Error Square at Iter = 2000eSq = 0.0016 eSq_v = 0.0022 eSq_t = 0.0017 Trained 时间已过 1540.739160 秒。 No. of Iterations = 2001 Final Mean Squared Error at Iter = 2001eSq = 0.0016 >>
部分代码：
%****Load the Input File****** load ./nnm_train.txt redData = nnm_train(:,2); nir1Data = [nnm_train(:,3) ./ redData]'; nir2Data = [nnm_train(:,4) ./ redData]'; nir3Data = [nnm_train(:,5) ./ redData]'; pg = [ nir1Data; nir2Data; nir3Data]; targetData = nnm_train(:,7) ; %*******Validate Data******* load ./nnm_validate.txt redData_v = nnm_validate(:,2); nir1Data_v = [nnm_validate(:,3) ./ redData_v]'; nir2Data_v = [nnm_validate(:,4) ./ redData_v]'; nir3Data_v = [nnm_validate(:,5) ./ redData_v]'; targetData_v = nnm_validate(:,7) ; pValidate = [nir1Data_v; nir2Data_v; nir3Data_v]; %*******Test Data******* load ./nnm_test.txt redData_t = nnm_test(:,2); nir1Data_t = [nnm_test(:,3) ./ redData_t]'; nir2Data_t = [nnm_test(:,4) ./ redData_t]'; nir3Data_t = [nnm_test(:,5) ./ redData_t]'; targetData_t = nnm_test(:,7) ; pTest = [nir1Data_t; nir2Data_t; nir3Data_t]; %---Plot the Original Function---- pa = 1 : length(targetData_t); actLine = 0:0.1:0.8; subplot(2,1,2), plot (actLine, actLine); legend('Actual');%scatter(targetData_t, targetData_t,'^b'); hold on %-----Randomized First Layer Weights & Bias------- fprintf( 'Initial Weights and Biases'); %****3-8-1****** W1 = [ -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand; -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand;... -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand]'; %Uniform distribution [-0.5 0.5] b1 = [ -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand]'; W2 = [ -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand -0.5+rand]; %-----Randomized Second Layer Bias------ b2 = [ -0.5+rand ]; if (lambda == 0) % Save the Weights and Bias on SBP W1_initial = W1; b1_initial = b1; W2_initial = W2; b2_initial = b2; else % Reuse the Weights and Bias on MBP W1 = W1_initial; b1 = b1_initial; W2 = W2_initial; b2 = b2_initial; end %-----RandPermutation of Input Training Set------- j = randperm(length(targetData)); j_v = randperm(length(targetData_v)); j_t = randperm(length(targetData_t)); %--Set Max. Iterations--- maxIter = 2000; tic for train = 1 : maxIter +1 eSq = 0; eSq_v = 0; eSq_t = 0; % **** Mean Square Error **** %if ( train <= maxIter ) for p = 1 : length(targetData) n1 = W1*pg(:,p)+ b1 ; a1 = logsig(n1); a2 = poslin( W2 * a1 + b2 ); e = targetData(p) - a2 ; eSq = eSq + (e^2); end eSq = eSq/length(targetData); %*******Validate Error********** for p = 1 : length(targetData_v) n1 = W1*pValidate(:,p)+ b1 ; a1 = logsig(n1); a2 = poslin( W2 * a1 + b2 ); e = targetData_v(p) - a2 ; eSq_v = eSq_v + (e^2); end eSq_v = eSq_v/length(targetData_v); %********Use Validate Error for Early Stopping******** if ( train > 200 ) earlyStopCount = earlyStopCount + 1; % fprintf('EarlyStop = %d', earlyStopCount); if (earlyStopCount == 50) if ( (prev_eSq_v - eSq_v) < 0 ) W2 = W2_25; b2 = b2_25; W1 = W1_25; b1 = b1_25; break; end prev_eSq_v = eSq_v; % Store previous validation error earlyStopCount = 0; % Reset Early Stopping %----Save the weights and biases------- disp('Saving'); eSq W2_25 = W2; b2_25 = b2; W1_25 = W1; b1_25 = b1; end else % ----Initialize the Weights---- if ( train == 200 ) W2_25 = W2; b2_25 = b2; W1_25 = W1; b1_25 = b1; end prev_eSq_v = eSq_v; % Store previous validation error end %*******Test Error********** for p = 1 : length(targetData_t) n1 = W1*pTest(:,p)+ b1 ; a1 = logsig(n1); a2 = poslin( W2 * a1 + b2 ); e = targetData_t(p) - a2 ; eSq_t = eSq_t + (e^2); end eSq_t = eSq_t/length(targetData_t); if (train == 1 || mod (train, 100) == 0 ) fprintf( 'Weights and Biases at Iter = %d\n',train); fprintf('W1'); (W1) fprintf('b1') b1 fprintf('W2') W2 fprintf('b2') b2 fprintf( 'Mean Error Square at Iter = %d',train); eSq eSq_v eSq_t end subplot(2,1,1), xlabel('No. of Iterations'); ylabel('Mean Square Error'); title('Convergence Characteristics '); loglog(train, eSq, '*r'); hold on loglog(train, eSq_v, '*g'); hold on loglog(train, eSq_t, '*c'); hold on legend('Training Error', 'Validation Error', 'Testing Error'); %************Train Data********************** % Update only when the error is decreasing % if ( earlyStopCount == 0 ) for p = 1 : length(targetData) %----Output of the 1st Layer----------- n1 = W1*pg(:,j(p))+ b1 ; a1 = logsig(n1) ; %-----Output of the 2nd Layer---------- n2 = W2 * a1 + b2; a2 = (poslin( n2 )); %a2 = (logsig( n2 )); %-----Error----- t = targetData(j(p)); e = t - a2; %******CALCULATE THE SENSITIVITIES************ %-----Derivative of logsig function---- %f1 = dlogsig(n1,a1) % f1 = [(1-a1(1))*a1(1) 0; 0 (1-a1(2))*a1(2)] ; f1 = diag((1-a1).*a1); %-----Derivative of purelin function--- f2 = 1; %f2 = diag((1-a2).*a2); %------Last Layer (2nd) Sensitivity---- S2 = -2 * f2 * e; S2mbp = ((t)-n2); %------First Layer Sensitivity--------- S1 = f1 *(W2' * S2); S1mbp = f1 * (W2' * S2mbp); %******UPDATE THE WEIGHTS********************** %-----Second Layer Weights & Bias------ W2 = W2 - (alpha * S2*(a1)') - (alpha * lambda * S2mbp *(a1)'); b2 = b2 - alpha * S2 - (alpha * lambda * S2mbp); %-----First Layer Weights & Bias------- W1 = W1 - alpha * S1*(pg(:,j(p)))' - (alpha * lambda * S1mbp *(pg(:,j(p)))'); b1 = b1 - alpha * S1 - (alpha * lambda * S1mbp ); % end end %end % End of 21 Input Training Sets % ********** Function Apporx. ***************** if (train == 1 || mod (train, 100) == 0 || train == maxIter ) disp('Trained'); subplot(2,1,2), xlabel('Actual Fraction of Weeds in 3 sq feet of grass area'); ylabel('Estimated Fraction of Weeds in 3 sq feet of grass area'); title('Correlation of Estimated Value with respect to the Actual Function using Standard Backpropagation'); legend('Estimated'); for p = 1 : length(targetData_t) n1 = W1*pTest(:,p)+ b1 ; % Test Data a1 = logsig(n1) ; a2(p) = (poslin( W2 * a1 + b2 )); end %end for %scatter(targetData_t, a2); hold on; end end toc %-------End of Iterations------------ %***Plot of Final Function****** subplot(2,1,2), for p = 1 : length(targetData_t) n1 = W1*pTest(:,p)+ b1 ; % Test Data a1 = logsig(n1) ; a2(p) = (poslin( W2 * a1 + b2 )); end %end for
🎉3 参考文献
[1]S. Abid, F. Fnaiech and M. Najim, "A fast feedforward training algorithm using a modified form of the standard backpropagation algorithm," in IEEE Transactions on Neural Networks, vol. 12, no. 2, pp. 424-430, March 2001, doi: 10.1109/72.914537.
