Resolve "(Learner1D) add possibility to use the direct neighbors in the loss"

Closes #119 (closed)

Currently works for one R^1 \to R^1

Still have to make it work for R^1 \to R^N

Also performance is actually quite good. As in: the learner slows down about 1.5 times. Going (on my laptop) from 3 seconds per 1000 points to 4.5 seconds per 1000 points.

Which, I believe, will be more than compensated by the fact that the chosen points are generally better

adaptive/learner/learner1D.py

@@ -3,16 +3,19 @@ from copy import deepcopy
 import heapq
 import itertools
 import math
+from collections import Iterable
 
 import numpy as np
 import sortedcontainers
 
 from .base_learner import BaseLearner
+from .learnerND import volume
+from .triangulation import simplex_volume_in_embedding
 from ..notebook_integration import ensure_holoviews
 from ..utils import cache_latest
 
 
-def uniform_loss(interval, scale, function_values):
+def uniform_loss(interval, scale, function_values, neighbors):
     """Loss function that samples the domain uniformly.
 
     Works with `~adaptive.Learner1D` only.
@@ -33,7 +36,7 @@ def uniform_loss(interval, scale, function_values):
     return dx
 
 
-def default_loss(interval, scale, function_values):
+def default_loss(interval, scale, function_values, neighbors):
     """Calculate loss on a single interval.
 
     Currently returns the rescaled length of the interval. If one of the
@@ -56,6 +59,45 @@ def default_loss(interval, scale, function_values):
     return loss
 
 
+def _loss_of_multi_interval(xs, ys):
+    N = len(xs) - 2
+    if isinstance(ys[0], Iterable):
+        pts = [(x, *y) for x, y in zip(xs, ys)]
+        vol = simplex_volume_in_embedding
+    else:
+        pts = [(x, y) for x, y in zip(xs, ys)]
+        vol = volume
+    return sum(vol(pts[i:i+3]) for i in range(N)) / N
+
+
+def triangle_loss(interval, scale, function_values, neighbors):
+    x_left, x_right = interval
+    xs = [x_left, x_right]
+    if x_left in neighbors:
+        xs.insert(0, neighbors[x_left][1])
+    if x_right in neighbors:
+        xs.append(neighbors[x_right][0])
+
+    if len(xs) <= 2:
+        return (x_right - x_left) / scale[0]
+    else:
+        y_scale = scale[1] or 1
+        ys_scaled = [function_values[x] / y_scale for x in xs]
+        xs_scaled = [x / scale[0] for x in xs]
+        return _loss_of_multi_interval(xs_scaled, ys_scaled)
+
+
+def get_curvature_loss(area_factor=1, euclid_factor=0.02, horizontal_factor=0.02):
+    def curvature_loss(interval, scale, function_values, neighbors):
+        triangle_loss_ = triangle_loss(interval, scale, function_values, neighbors)
+        default_loss_ = default_loss(interval, scale, function_values, neighbors)
+        dx = (interval[1] - interval[0]) / scale[0]
+        return (area_factor * (triangle_loss_**0.5)
+                + euclid_factor * default_loss_
+                + horizontal_factor * dx)
+    return curvature_loss
+
+
 def linspace(x_left, x_right, n):
     """This is equivalent to
     'np.linspace(x_left, x_right, n, endpoint=False)[1:]',
@@ -79,6 +121,13 @@ def _get_neighbors_from_list(xs):
     return sortedcontainers.SortedDict(neighbors)
 
 
+def _get_interval(x, neighbors, nn_neighbors):
+    n = nn_neighbors
+    index = neighbors.bisect_left(x)
+    return [neighbors.iloc[i] for i in range(index - n - 1, index + 2 + n)
+            if 0 <= i < len(neighbors)]
+
+
 class Learner1D(BaseLearner):
     """Learns and predicts a function 'f:ℝ → ℝ^N'.
 
@@ -93,6 +142,10 @@ class Learner1D(BaseLearner):
         A function that returns the loss for a single interval of the domain.
         If not provided, then a default is used, which uses the scaled distance
         in the x-y plane as the loss. See the notes for more details.
+    nn_neighbors : int, default: 0
+        The number of neighboring intervals that the loss function
+        takes into account. If ``loss_per_interval`` doesn't use the neighbors
+        at all, then it should be 0.
 
     Attributes
     ----------
@@ -103,9 +156,9 @@ class Learner1D(BaseLearner):
 
     Notes
     -----
-    `loss_per_interval` takes 3 parameters: ``interval``,  ``scale``, and
-    ``function_values``, and returns a scalar; the loss over the interval.
-
+    `loss_per_interval` takes 4 parameters: ``interval``,  ``scale``,
+    ``data``, and ``neighbors``, and returns a scalar; the loss over
+    the interval.
     interval : (float, float)
         The bounds of the interval.
     scale : (float, float)
@@ -114,11 +167,18 @@ class Learner1D(BaseLearner):
     function_values : dict(float → float)
         A map containing evaluated function values. It is guaranteed
         to have values for both of the points in 'interval'.
+    neighbors : dict(float → (float, float))
+        A map containing points as keys to its neighbors as a tuple.
     """
 
-    def __init__(self, function, bounds, loss_per_interval=None):
+    def __init__(self, function, bounds, loss_per_interval=None, nn_neighbors=0):
         self.function = function
-        self.loss_per_interval = loss_per_interval or default_loss
+        self.nn_neighbors = nn_neighbors
+
+        if nn_neighbors == 0:
+            self.loss_per_interval = loss_per_interval or default_loss
+        else:
+            self.loss_per_interval = loss_per_interval or get_curvature_loss()
 
         # A dict storing the loss function for each interval x_n.
         self.losses = {}
@@ -176,25 +236,36 @@ class Learner1D(BaseLearner):
         losses = self.losses if real else self.losses_combined
         return max(losses.values()) if len(losses) > 0 else float('inf')
 
+    def _get_loss_in_interval(self, x_left, x_right):
+        assert x_left is not None and x_right is not None
+
+        if x_right - x_left < self._dx_eps:
+            return 0
+
+        # we need to compute the loss for this interval
+        interval = (x_left, x_right)
+        return self.loss_per_interval(
+            interval, self._scale, self.data, self.neighbors)
+
+
     def _update_interpolated_loss_in_interval(self, x_left, x_right):
-        if x_left is not None and x_right is not None:
-            dx = x_right - x_left
-            if dx < self._dx_eps:
-                loss = 0
-            else:
-                loss = self.loss_per_interval((x_left, x_right),
-                                              self._scale, self.data)
-            self.losses[x_left, x_right] = loss
-
-            # Iterate over all interpolated intervals in between
-            # x_left and x_right and set the newly interpolated loss.
-            a, b = x_left, None
-            while b != x_right:
-                b = self.neighbors_combined[a][1]
-                self.losses_combined[a, b] = (b - a) * loss / dx
-                a = b
+        if x_left is None or x_right is None:
+            return
+
+        loss = self._get_loss_in_interval(x_left, x_right)
+        self.losses[x_left, x_right] = loss
+
+        # Iterate over all interpolated intervals in between
+        # x_left and x_right and set the newly interpolated loss.
+        a, b = x_left, None
+        dx = x_right - x_left
+        while b != x_right:
+            b = self.neighbors_combined[a][1]
+            self.losses_combined[a, b] = (b - a) * loss / dx
+            a = b
 
     def _update_losses(self, x, real=True):
+        """Update all losses that depend on x"""
         # When we add a new point x, we should update the losses
         # (x_left, x_right) are the "real" neighbors of 'x'.
         x_left, x_right = self._find_neighbors(x, self.neighbors)
@@ -207,10 +278,12 @@ class Learner1D(BaseLearner):
 
         if real:
             # We need to update all interpolated losses in the interval
-            # (x_left, x) and (x, x_right). Since the addition of the point
-            # 'x' could change their loss.
-            self._update_interpolated_loss_in_interval(x_left, x)
-            self._update_interpolated_loss_in_interval(x, x_right)
+            # (x_left, x), (x, x_right) and the nn_neighbors nearest
+            # neighboring intervals. Since the addition of the
+            # point 'x' could change their loss.
+            points = _get_interval(x, self.neighbors, self.nn_neighbors)
+            for ival in zip(points, points[1:]):
+                self._update_interpolated_loss_in_interval(*ival)
 
             # Since 'x' is in between (x_left, x_right),
             # we get rid of the interval.
@@ -358,7 +431,7 @@ class Learner1D(BaseLearner):
         self.losses = {}
         for x_left, x_right in intervals:
             self.losses[x_left, x_right] = (
-                self.loss_per_interval((x_left, x_right), self._scale, self.data)
+                self._get_loss_in_interval(x_left, x_right)
                 if x_right - x_left >= self._dx_eps else 0)
 
         # List with "real" intervals that have interpolated intervals inside