function gddemo% GDDEMO A little demonstration of gradient descent.  %% Click in the space above the error surface to drop a ball onto it.  Ball % descends gradient using local search (negative hill-climbing), after % which final weights are shown by intersecting lines on axes.  Click on % the axes below the error surface to show the error at the corresponding % weights.% clear out any stale graphic handles for callback functionclear all% arbitrary weight params to work with ACTFUNWMIN = -3;WMAX = 3;WCNT = 60;% get a grid of weight values[w1grid,w2grid] = meshgrid(linspace(WMIN, WMAX, WCNT)); % compute the error values at those weightsygrid = errfun(w1grid, w2grid);% show the error values as a meshmesh(w1grid, w2grid, ygrid)% use a black-and-white gridcolormap bone% make the grid half transparentalpha(0.5)% make the figure look niceaxis([WMIN WMAX WMIN WMAX 0 1])xlabel('Weight 1')ylabel('Weight 2')zlabel('Error')set(gcf, 'Name', 'Gradient Descent Demo')% set up callback function for mouse-button clicksset(gcf, 'WindowButtonDownFcn', @callback);% BEGIN CALLBACK FUNCTION ---------------------------------------------------function callback(src, eventData)% display paramsDROPSTEP = 0.05;    % vertical drop on ball from "sky" on each stepDROPPAUSE = 0.05;   % pause between drop steps, in secondsDESCPAUSE = 0.10;   % pause between gradient-descent stepsZGAP = 0.01;        % gap between bottom of ball and surfaceDERIVRAD = 0.1;     % radius for searching steepest gradientDERIVCNT = 100;     % how many points to sample in square enclosing radius% graphics handles, for erasing previous ball, linespersistent ballh    persistent xhpersistent yhpersistent zh% last X, Y axis values clickedpersistent xlastpersistent ylast% get foreground, background points where user clickedxyzfb = get(gca, 'CurrentPoint');% use foreground pointxyz = xyzfb(1,:); % parse point into individual coordinates x = xyz(1);y = xyz(2);z = xyz(3);% get axis limits[xmin,xmax] = getlim('XLim');[ymin,ymax] = getlim('YLim');[zmin,zmax] = getlim('ZLim');% if user clicked in region above surface, run gradient descentif z > zmax/2 & z <= zmax        % erase any previous ball, line images    ballh = trydel(ballh);    xh = trydel(xh);    yh = trydel(yh);    zh = trydel(zh);        % reset previous line coordinates    xlast = [];    ylast = [];        % avoid erasing previously plotted objects    hold on        % get Z coordinates for kinematic drop points    zdrop = z:-DROPSTEP:errfun(x, y)+ZGAP;        % animate ball dropping from clicked point onto error surface    for z = zdrop        ballh = drawball(x, y, z);        if z ~= zdrop(end)            pause(DROPPAUSE)            delete(ballh)        end    end        % lastz tracks previous error value for halting descent    lastz = Inf;        % "loop forever", but we'll break when error stops falling    while true                % get a bunch of weight pairs in a radius around the current pair        [nbrx,nbry] = discpoints(x, y, DERIVRAD, DERIVCNT);                % compute the errors at those weight pairs        nbrz = errfun(nbrx, nbry);                % choose weight pair where the error is lowest        j = find(nbrz == min(nbrz));        x = nbrx(j);        y = nbry(j);                % compute the error at that weight pair         z = errfun(x, y);                % if error has gone up, we're done        if z > lastz            xh = plotwgt(x, x, ymin, ymax); % draw line for final weight 1            yh = plotwgt(xmin, xmax, y, y); %  line for final weight 2            zh = plotvert(x, y, z);         %  line from weights to ball             break        end                % if we go off the edge of the surface, beep and halt        if x < xmin | x > xmax | y < ymin | y > ymax            beep            break        end                % otherwise, continue animating the ball        pause(DESCPAUSE)        delete(ballh)        ballh = drawball(x, y, z+ZGAP);        lastz = z;            end    % user clicked below surface or outside plot    else        % any click erases previous weight lines    if isempty(xlast) & isempty(ylast)        xh = trydel(xh);        yh = trydel(yh);        zh = trydel(zh);    end      % user clicked on bottom of plot (Z = 0)    if abs(z-zmin) < .1                % always remove previous ball image        ballh = trydel(ballh);                % draw line on X or Y axis if user clicked close enough to it        xh = tryline(x, xmin, xmin, xmax, y, y, xh);        yh = tryline(y, ymin, x, x, ymin, ymax, yh);                % clicking on X (Y) axis tracks value on Y (X) axis        if x == xmin            ylast = y;        elseif y == ymin            xlast = x;        end                % if we have X and Y axis values, plot error at that weight pair        if ~isempty(xlast) & ~isempty(ylast)            z = errfun(xlast, ylast);               % compute error            ballh = drawball(xlast, ylast, z+ZGAP); % draw ball            zh = trydel(zh);                        % erase old vert. bar            zh = plotvert(xlast, ylast, z);         % plot new vert. bar        end    endend% BEGIN LOCAL FUNCTIONS ---------------------------------------------------% draw a line if axis was clickedfunction h = tryline(xy, xymin, xlo, xhi, ylo, yhi, h)if xy == xymin    hold on    h = trydel(h);    h = plotwgt(xlo, xhi, ylo, yhi);end% get axis limitsfunction [lo,hi] = getlim(limstr)lim = get(gca, limstr);lo = lim(1);hi = lim(2);% delete a graphic object safelyfunction h = trydel(h)if ~isempty(h)    try % deal with invalid graphics handle        delete(h)    catch        h = [];    endend% plot a line corresponding to a weightfunction h = plotwgt(xlo, xhi, ylo, yhi)h = plot3([xlo xhi], [ylo yhi], [0 0], 'r');% plot a vertical line between a weight pair and the ball function h = plotvert(x, y, z)h = plot3([x x], [y y], [0 z], 'k');% draw the ball at a specified locationfunction h = drawball(x, y, z)h = plot3(x, y, z, '.r', 'MarkerSize',20);