from manim import * import numpy as np import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt config.frame_width = 16 config.frame_height = 9 config.pixel_width = 1920 config.pixel_height = 1080 config.background_color = "#0B0E14" BLUE_C0 = "#4C8DFF" ORANGE_C1 = "#FF9F43" TEAL = "#5EEAD4" GOLD = "#F0C75E" CREAM = "#E8ECF1" MUTED = "#8B93A1" AXIS_COL = "#6B7380" HUD_FILL = "#161B26" HUD_STROKE = "#2A3140" TITLE_COL = "#F2F5F8" PURPLE_HI = "#23183A" TEAL_MID = "#2F6F88" def sigmoid(z): z = np.clip(z, -30.0, 30.0) return 1.0 / (1.0 + np.exp(-z)) def make_data(): rng = np.random.RandomState(42) n = 25 x0 = rng.randn(n, 2) * 0.9 + np.array([-0.7, -0.5]) x1 = rng.randn(n, 2) * 0.9 + np.array([0.7, 0.6]) X = np.vstack([x0, x1]) y = np.concatenate([np.zeros(n), np.ones(n)]) X = (X - X.mean(axis=0)) / X.std(axis=0) return X, y def batch_gd(X, y, w1=-1.0, w2=0.5, b=1.0, alpha=0.3, n_iter=200): hist = np.zeros((n_iter + 1, 8)) for i in range(n_iter + 1): z = w1 * X[:, 0] + w2 * X[:, 1] + b f = sigmoid(z) f_clip = np.clip(f, 1e-12, 1.0 - 1e-12) J = -np.mean(y * np.log(f_clip) + (1.0 - y) * np.log(1.0 - f_clip)) err = f - y dw1 = np.mean(err * X[:, 0]) dw2 = np.mean(err * X[:, 1]) db = np.mean(err) hist[i] = (i, w1, w2, b, J, dw1, dw2, db) w1 = w1 - alpha * dw1 w2 = w2 - alpha * dw2 b = b - alpha * db return hist def cost_wb(w1, w2, b, X, y): z = w1 * X[:, 0] + w2 * X[:, 1] + b f = np.clip(sigmoid(z), 1e-12, 1.0 - 1e-12) return -np.mean(y * np.log(f) + (1.0 - y) * np.log(1.0 - f)) def interp_row(hist, t): t = float(np.clip(t, 0.0, hist.shape[0] - 1)) i0 = int(np.floor(t)) i1 = min(i0 + 1, hist.shape[0] - 1) a = t - i0 return (1.0 - a) * hist[i0] + a * hist[i1] def halfplane_polys(w1, w2, b, xmin, xmax, ymin, ymax, eps=1e-9): corners = [(xmin, ymin), (xmax, ymin), (xmax, ymax), (xmin, ymax)] edges = [(0, 1), (1, 2), (2, 3), (3, 0)] def zval(p): return w1 * p[0] + w2 * p[1] + b def hit(p, q): zp, zq = zval(p), zval(q) if abs(zp) < eps: return p if abs(zq) < eps: return None if zp * zq < 0: t = zp / (zp - zq) return (p[0] + t * (q[0] - p[0]), p[1] + t * (q[1] - p[1])) return None pos, neg = [], [] for i, j in edges: p, q = corners[i], corners[j] zp = zval(p) if zp > eps: pos.append(p) elif zp < -eps: neg.append(p) else: pos.append(p) neg.append(p) h = hit(p, q) if h is not None and (abs(h[0] - p[0]) > 1e-8 or abs(h[1] - p[1]) > 1e-8): pos.append(h) neg.append(h) def dedupe(seq): out = [] for p in seq: if not out or abs(out[-1][0] - p[0]) > 1e-7 or abs(out[-1][1] - p[1]) > 1e-7: out.append(p) if ( len(out) > 1 and abs(out[0][0] - out[-1][0]) < 1e-7 and abs(out[0][1] - out[-1][1]) < 1e-7 ): out = out[:-1] return out return dedupe(pos), dedupe(neg) def clip_boundary(w1, w2, b, xmin, xmax, ymin, ymax): pts = [] if abs(w2) > 1e-9: for x in (xmin, xmax): yy = -(w1 * x + b) / w2 if ymin - 1e-8 <= yy <= ymax + 1e-8: pts.append((x, float(np.clip(yy, ymin, ymax)))) if abs(w1) > 1e-9: for yy in (ymin, ymax): x = -(w2 * yy + b) / w1 if xmin - 1e-8 <= x <= xmax + 1e-8: pts.append((float(np.clip(x, xmin, xmax)), yy)) uniq = [] for p in pts: if not any(abs(p[0] - q[0]) < 1e-6 and abs(p[1] - q[1]) < 1e-6 for q in uniq): uniq.append(p) if len(uniq) < 2: return None best = (uniq[0], uniq[1]) best_d = -1.0 for i in range(len(uniq)): for j in range(i + 1, len(uniq)): d = (uniq[i][0] - uniq[j][0]) ** 2 + (uniq[i][1] - uniq[j][1]) ** 2 if d > best_d: best_d = d best = (uniq[i], uniq[j]) return best def j_color(j, jmin, jmax): t = float(np.clip((j - jmin) / max(jmax - jmin, 1e-9), 0.0, 1.0)) if t < 0.45: u = t / 0.45 return interpolate_color(ManimColor(GOLD), ManimColor(TEAL_MID), u) u = (t - 0.45) / 0.55 return interpolate_color(ManimColor(TEAL_MID), ManimColor(PURPLE_HI), u) def extract_contours(W1, W2, Jg, levels): fig, ax = plt.subplots() cs = ax.contour(W1, W2, Jg, levels=levels) out = [] for k, segs in enumerate(cs.allsegs): for seg in segs: if len(seg) < 4: continue step = max(1, len(seg) // 48) pts = seg[::step] if not np.allclose(pts[-1], seg[-1]): pts = np.vstack([pts, seg[-1]]) out.append((levels[k], pts)) plt.close(fig) return out class LogisticRegressionGD(Scene): def construct(self): self.X, self.y = make_data() self.hist = batch_gd(self.X, self.y) self.n_iter = self.hist.shape[0] - 1 self.b_star = float(self.hist[-1, 3]) self.tracker = ValueTracker(0.0) self.play_intro() self.build_main_stage() self.play_descent() self.play_convergence() def play_intro(self): title = Text( "Logistic Regression: Gradient Descent", font_size=38, color=TITLE_COL, weight=BOLD, ) eq1 = MathTex(r"z = w_1 x_1 + w_2 x_2 + b", font_size=38, color=CREAM) eq2 = MathTex(r"g(z) = \frac{1}{1 + e^{-z}}", font_size=38, color=TEAL) eq3 = MathTex( r"L = -y\log f - (1-y)\log(1-f)", font_size=34, color=CREAM, ) eq4 = MathTex( r"J(w,b) = \frac{1}{m}\sum_{i=1}^{m} L^{(i)}", font_size=36, color=GOLD, ) eqs = VGroup(eq1, eq2, eq3, eq4).arrange(DOWN, buff=0.38) block = VGroup(title, eqs).arrange(DOWN, buff=0.52) block.move_to([0, 0.12, 0]) self.play(FadeIn(title, shift=UP * 0.15), run_time=0.8) self.play( LaggedStart(*[FadeIn(e, shift=UP * 0.08) for e in eqs], lag_ratio=0.18), run_time=2.2, ) self.wait(1.5) self.play(FadeOut(block), run_time=0.5) def build_main_stage(self): self.main_title = Text( "Logistic Regression: Gradient Descent", font_size=26, color=TITLE_COL, weight=BOLD, ) self.main_title.to_edge(UP, buff=0.42) self.cxs = [-5.15, 0.10, 5.25] names = ["Decision boundary", "Sigmoid g(z)", "Cost J(w1, w2)"] self.ptitles = VGroup() for cx, name in zip(self.cxs, names): pt = Text(name, font_size=20, color=CREAM) pt.move_to([cx, 3.18, 0]) self.ptitles.add(pt) self._build_left_axes() self._build_center_axes() self._build_right_axes() self._build_hud() self.play( FadeIn(self.main_title, shift=UP * 0.08), FadeIn(self.ptitles), run_time=0.6, ) self.play( FadeIn(self.left_static), FadeIn(self.center_static), FadeIn(self.right_static), FadeIn(self.hud_group), run_time=0.6, ) self.add(self.left_live, self.center_live, self.right_live, self.hud_live) def _build_left_axes(self): cx = self.cxs[0] ax_top = self.ptitles[0].get_bottom()[1] - 0.22 self.axL = Axes( x_range=[-2.4, 2.4, 1.2], y_range=[-2.4, 2.4, 1.2], x_length=3.25, y_length=3.25, tips=False, axis_config={ "include_numbers": False, "stroke_width": 1.6, "color": AXIS_COL, }, ) self.axL.move_to([cx + 0.12, ax_top - self.axL.height / 2.0, 0]) xl = Text("x1", font_size=18, color=MUTED).next_to(self.axL, DOWN, buff=0.12) yl = Text("x2", font_size=18, color=MUTED).next_to(self.axL, LEFT, buff=0.12) reason = ( "Scatter, half-plane shading and the z = 0 line share one pair of " "x1–x2 axes so the rotating boundary can be read against the data." ) dots = VGroup() for pt, yi in zip(self.X, self.y): col = ORANGE_C1 if yi > 0.5 else BLUE_C0 dots.add(Dot(self.axL.c2p(pt[0], pt[1]), radius=0.050, color=col, z_index=3)) dots def live_boundary(): row = interp_row(self.hist, self.tracker.get_value()) w1, w2, b = float(row[1]), float(row[2]), float(row[3]) pos, neg = halfplane_polys(w1, w2, b, -2.4, 2.4, -2.4, 2.4) parts = VGroup() if len(neg) >= 3: parts.add( Polygon( *[self.axL.c2p(x, y) for x, y in neg], fill_color=BLUE_C0, fill_opacity=0.16, stroke_width=0, z_index=0, ) ) if len(pos) >= 3: parts.add( Polygon( *[self.axL.c2p(x, y) for x, y in pos], fill_color=ORANGE_C1, fill_opacity=0.16, stroke_width=0, z_index=0, ) ) ends = clip_boundary(w1, w2, b, -2.4, 2.4, -2.4, 2.4) if ends is not None: (x_a, y_a), (x_b, y_b) = ends parts.add( Line( self.axL.c2p(x_a, y_a), self.axL.c2p(x_b, y_b), color=GOLD, stroke_width=3.2, z_index=2, ) ) return parts live = always_redraw(live_boundary) self.left_static = VGroup(self.axL, xl, yl, dots) self.left_live = live def _build_center_axes(self): cx = self.cxs[1] ax_top = self.ptitles[1].get_bottom()[1] - 0.22 self.axC = Axes( x_range=[-7.0, 7.0, 3.5], y_range=[0.0, 1.15, 0.5], x_length=3.40, y_length=3.05, tips=False, axis_config={ "include_numbers": False, "stroke_width": 1.6, "color": AXIS_COL, }, ) self.axC.move_to([cx + 0.14, ax_top - self.axC.height / 2.0 - 0.06, 0]) xl = Text("z", font_size=18, color=MUTED).next_to(self.axC, DOWN, buff=0.12) yl = Text("g(z)", font_size=18, color=MUTED).next_to(self.axC, LEFT, buff=0.10) reason = ( "The fixed sigmoid, its dashed guides, and the sliding examples " "share one (z, g(z)) plot so class-1 points can be seen climbing " "toward 1 and class-0 points falling toward 0." ) curve = self.axC.plot( lambda z: 1.0 / (1.0 + np.exp(-z)), x_range=[-7, 7], color=TEAL, stroke_width=3.0, ) vguide = DashedLine( self.axC.c2p(0, 0), self.axC.c2p(0, 1.15), color=MUTED, stroke_width=1.4, dash_length=0.10, ) hguide = DashedLine( self.axC.c2p(-7, 0.5), self.axC.c2p(7, 0.5), color=MUTED, stroke_width=1.4, dash_length=0.10, ) half_lbl = Text("0.5", font_size=16, color=MUTED) half_lbl.next_to(self.axC.c2p(7, 0.5), LEFT, buff=0.10) half_lbl.shift(UP * 0.16) self.sig_dots = VGroup() row0 = self.hist[0] z0 = np.clip(row0[1] * self.X[:, 0] + row0[2] * self.X[:, 1] + row0[3], -6.85, 6.85) g0 = sigmoid(z0) for i, yi in enumerate(self.y): col = ORANGE_C1 if yi > 0.5 else BLUE_C0 self.sig_dots.add(Dot(self.axC.c2p(z0[i], g0[i]), radius=0.046, color=col, z_index=3)) self.sig_dots def slide_dots(mob): row = interp_row(self.hist, self.tracker.get_value()) z = np.clip(row[1] * self.X[:, 0] + row[2] * self.X[:, 1] + row[3], -6.85, 6.85) gc = sigmoid(z) for i, d in enumerate(mob): d.move_to(self.axC.c2p(z[i], gc[i])) self.sig_dots.add_updater(slide_dots) self.center_static = VGroup(self.axC, xl, yl, curve, vguide, hguide, half_lbl) self.center_live = self.sig_dots def _build_right_axes(self): cx = self.cxs[2] ax_top = self.ptitles[2].get_bottom()[1] - 0.22 self.axR = Axes( x_range=[-1.8, 3.2, 1.0], y_range=[-0.4, 2.8, 0.8], x_length=3.30, y_length=3.15, tips=False, axis_config={ "include_numbers": False, "stroke_width": 1.6, "color": AXIS_COL, }, ) self.axR.move_to([cx + 0.10, ax_top - self.axR.height / 2.0, 0]) xl = Text("w1", font_size=18, color=MUTED).next_to(self.axR, DOWN, buff=0.12) yl = Text("w2", font_size=18, color=MUTED).next_to(self.axR, LEFT, buff=0.10) reason = ( "Heat cells, isolines, the descent trail and the riding marker " "share one (w1, w2) plane so the path can be read against J." ) nw = 10 w1s = np.linspace(-1.8, 3.2, nw + 1) w2s = np.linspace(-0.4, 2.8, nw + 1) heat = np.zeros((nw, nw)) for i in range(nw): for j in range(nw): wm1 = 0.5 * (w1s[j] + w1s[j + 1]) wm2 = 0.5 * (w2s[i] + w2s[i + 1]) heat[i, j] = cost_wb(wm1, wm2, self.b_star, self.X, self.y) jmin, jmax = float(heat.min()), float(heat.max()) cells = VGroup() for i in range(nw): for j in range(nw): bl = self.axR.c2p(w1s[j], w2s[i]) tr = self.axR.c2p(w1s[j + 1], w2s[i + 1]) rec = Rectangle( width=tr[0] - bl[0], height=tr[1] - bl[1], stroke_width=0, fill_color=j_color(heat[i, j], jmin, jmax), fill_opacity=0.92, z_index=0, ) rec.move_to([(bl[0] + tr[0]) / 2.0, (bl[1] + tr[1]) / 2.0, 0]) cells.add(rec) cells fine = 56 gg1 = np.linspace(-1.8, 3.2, fine) gg2 = np.linspace(-0.4, 2.8, fine) WW1, WW2 = np.meshgrid(gg1, gg2) Jg = np.zeros_like(WW1) for i in range(fine): for j in range(fine): Jg[i, j] = cost_wb(WW1[i, j], WW2[i, j], self.b_star, self.X, self.y) levels = [0.27, 0.32, 0.40, 0.55, 0.75, 1.05, 1.45] segs = extract_contours(WW1, WW2, Jg, levels) isolines = VGroup() for lev, pts in segs: corners = [self.axR.c2p(p[0], p[1]) for p in pts] iso = VMobject( stroke_color="#D7DEE8", stroke_width=1.3, stroke_opacity=0.55, z_index=1, ) iso.set_points_as_corners(corners) isolines.add(iso) isolines def live_path(): t = self.tracker.get_value() i_hi = int(np.floor(t)) pts = [self.axR.c2p(self.hist[k, 1], self.hist[k, 2]) for k in range(i_hi + 1)] row = interp_row(self.hist, t) here = self.axR.c2p(row[1], row[2]) if len(pts) == 0 or np.linalg.norm(np.array(pts[-1]) - np.array(here)) > 1e-4: pts.append(here) grp = VGroup() if len(pts) >= 2: trail = VMobject(color=GOLD, stroke_width=2.6, z_index=2) trail.set_points_as_corners(pts) grp.add(trail) if t < 16: i = min(int(round(t)), self.n_iter) w1, w2 = self.hist[i, 1], self.hist[i, 2] dw1, dw2 = self.hist[i, 5], self.hist[i, 6] start = self.axR.c2p(w1, w2) end = self.axR.c2p(w1 - 0.85 * dw1, w2 - 0.85 * dw2) fade = 1.0 if t < 9 else max(0.0, 1.0 - (t - 9) / 7.0) if fade > 0.05: grp.add( Arrow( start, end, buff=0.02, color=CREAM, stroke_width=3.0, max_tip_length_to_length_ratio=0.22, stroke_opacity=fade, fill_opacity=fade, z_index=3, ) ) glow = Dot(here, radius=0.11, color=GOLD, fill_opacity=0.22, z_index=3) mark = Dot(here, radius=0.065, color=GOLD, z_index=4) grp.add(glow, mark) return grp live = always_redraw(live_path) self.right_static = VGroup(cells, isolines, self.axR, xl, yl) self.right_live = live def _build_hud(self): hud_bg = RoundedRectangle( width=10.4, height=0.56, corner_radius=0.10, fill_color=HUD_FILL, fill_opacity=1.0, stroke_color=HUD_STROKE, stroke_width=1.4, ) hud_bg.move_to([0, -3.38, 0]) hud_bg def make_pair(label, seed, decimals, signed=False, color=CREAM): lab = Text(label, font_size=20, color=MUTED) if decimals == 0: val = Integer(int(seed), font_size=22, color=color) else: val = DecimalNumber( seed, num_decimal_places=decimals, font_size=22, color=color, include_sign=signed, ) return VGroup(lab, val).arrange(RIGHT, buff=0.10), val p_iter, self.v_iter = make_pair("iter", 0, 0) p_w1, self.v_w1 = make_pair("w1", -1.00, 2, signed=True) p_w2, self.v_w2 = make_pair("w2", 0.50, 2, signed=True) p_b, self.v_b = make_pair("b", 1.00, 2, signed=True) p_j, self.v_j = make_pair("J", 1.075, 3, signed=False, color=GOLD) self.hud_row = VGroup(p_iter, p_w1, p_w2, p_b, p_j).arrange(RIGHT, buff=0.42) self.hud_row.move_to(hud_bg.get_center()) self.hud_row def sync_hud(_dt=0): row = interp_row(self.hist, self.tracker.get_value()) self.v_iter.set_value(int(round(row[0]))) self.v_w1.set_value(row[1]) self.v_w2.set_value(row[2]) self.v_b.set_value(row[3]) self.v_j.set_value(row[4]) self.hud_row.add_updater(lambda m, dt: sync_hud(dt)) self.hud_group = VGroup(hud_bg) self.hud_live = self.hud_row def play_descent(self): self.play( self.tracker.animate.set_value(float(self.n_iter)), run_time=26.8, rate_func=linear, ) def play_convergence(self): end_title = Text( "Converged global minimum", font_size=28, color=GOLD, weight=BOLD, ) end_title.to_edge(UP, buff=0.42) row = self.hist[-1] ring = Circle(radius=0.16, color=GOLD, stroke_width=2.4) ring.move_to(self.axR.c2p(row[1], row[2])) ring self.play( ReplacementTransform(self.main_title, end_title), FadeIn(ring, scale=0.6), run_time=1.2, ) self.play(Indicate(ring, color=GOLD, scale_factor=1.25), run_time=0.9) self.wait(4.9)

Logistic Regression Gradient