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

@@ -3,17 +3,74 @@ 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 uses_nth_neighbors(n):
+    """Decorator to specify how many neighboring intervals the loss function uses.
+
+    Wraps loss functions to indicate that they expect intervals together
+    with ``n`` nearest neighbors
+
+    The loss function is then guaranteed to receive the data of at least the
+    N nearest neighbors (``nth_neighbors``) in a dict that tells you what the
+    neighboring points of these are. And the `~adaptive.Learner1D` will
+    then make sure that the loss is updated whenever one of the
+    ``nth_neighbors`` changes.
+
+    Examples
+    --------
+
+    The next function is a part of the `get_curvature_loss` function.
+
+    >>> @uses_nth_neighbors(1)
+    ... def triangle_loss(interval, scale, data, neighbors):
+    ...     x_left, x_right = interval
+    ...     xs = [neighbors[x_left][0], x_left, x_right, neighbors[x_right][1]]
+    ...     # at the boundary, neighbors[<left boundary x>] is (None, <some other x>)
+    ...     xs = [x for x in xs if x is not None]
+    ...     if len(xs) <= 2:
+    ...         return (x_right - x_left) / scale[0]
+    ...
+    ...     y_scale = scale[1] or 1
+    ...     ys_scaled = [data[x] / y_scale for x in xs]
+    ...     xs_scaled = [x / scale[0] for x in xs]
+    ...     N = len(xs) - 2
+    ...     pts = [(x, y) for x, y in zip(xs_scaled, ys_scaled)]
+    ...     return sum(volume(pts[i:i+3]) for i in range(N)) / N
+
+    Or you may define a loss that favours the (local) minima of a function.
+
+    >>> @uses_nth_neighbors(1)
+    ... def local_minima_resolving_loss(interval, scale, data, neighbors):
+    ...     x_left, x_right = interval
+    ...     n_left = neighbors[x_left][0]
+    ...     n_right = neighbors[x_right][1]
+    ...     loss = (x_right - x_left) / scale[0]
+    ...
+    ...     if not ((n_left is not None and data[x_left] > data[n_left])
+    ...         or (n_right is not None and data[x_right] > data[n_right])):
+    ...         return loss * 100
+    ...
+    ...     return loss
+    """
+    def _wrapped(loss_per_interval):
+        loss_per_interval.nth_neighbors = n
+        return loss_per_interval
+    return _wrapped
+
+
+@uses_nth_neighbors(0)
+def uniform_loss(interval, scale, data, neighbors):
     """Loss function that samples the domain uniformly.
 
     Works with `~adaptive.Learner1D` only.
@@ -34,7 +91,8 @@ def uniform_loss(interval, scale, function_values):
     return dx
 
 
-def default_loss(interval, scale, function_values):
+@uses_nth_neighbors(0)
+def default_loss(interval, scale, data, neighbors):
     """Calculate loss on a single interval.
 
     Currently returns the rescaled length of the interval. If one of the
@@ -42,7 +100,7 @@ def default_loss(interval, scale, function_values):
     never touched. This behavior should be improved later.
     """
     x_left, x_right = interval
-    y_right, y_left = function_values[x_right], function_values[x_left]
+    y_right, y_left = data[x_right], data[x_left]
     x_scale, y_scale = scale
     dx = (x_right - x_left) / x_scale
     if y_scale == 0:
@@ -58,35 +116,39 @@ def default_loss(interval, scale, function_values):
 
 
 def _loss_of_multi_interval(xs, ys):
-    pts = list(zip(xs, ys))
-    N = len(pts) - 2
-    return sum(volume(pts[i:i+3]) for i in range(N)) / N
+    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, neighbours, scale, function_values):
+@uses_nth_neighbors(1)
+def triangle_loss(interval, scale, data, neighbors):
     x_left, x_right = interval
-    neighbour_left, neighbour_right = neighbours
-    xs = [neighbour_left, x_left, x_right, neighbour_right]
-    # The neighbours could be None if we are at the boundary, in that case we
-    # have to filter this out
+    xs = [neighbors[x_left][0], x_left, x_right, neighbors[x_right][1]]
     xs = [x for x in xs if x is not None]
-    y_scale = scale[1] or 1
-    ys = [function_values[x] / y_scale for x in xs]
 
     if len(xs) <= 2:
         return (x_right - x_left) / scale[0]
     else:
+        y_scale = scale[1] or 1
+        ys_scaled = [data[x] / y_scale for x in xs]
         xs_scaled = [x / scale[0] for x in xs]
-        return _loss_of_multi_interval(xs_scaled, ys)
+        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, neighbours, scale, function_values):
-        triangle_loss_ = triangle_loss(interval, neighbours, scale, function_values)
-        default_loss_ = default_loss(interval, scale, function_values)
-        dx = interval[1] - interval[0] / scale[0]
+    @uses_nth_neighbors(1)
+    def curvature_loss(interval, scale, data, neighbors):
+        triangle_loss_ = triangle_loss(interval, scale, data, neighbors)
+        default_loss_ = default_loss(interval, scale, data, neighbors)
+        dx = (interval[1] - interval[0]) / scale[0]
         return (area_factor * (triangle_loss_**0.5)
-                + euclid_factor * old_loss
+                + euclid_factor * default_loss_
                 + horizontal_factor * dx)
     return curvature_loss
 
@@ -114,6 +176,15 @@ def _get_neighbors_from_list(xs):
     return sortedcontainers.SortedDict(neighbors)
 
 
+def _get_intervals(x, neighbors, nth_neighbors):
+    nn = nth_neighbors
+    i = neighbors.index(x)
+    start = max(0, i - nn - 1)
+    end = min(len(neighbors), i + nn + 2)
+    points = neighbors.keys()[start:end]
+    return list(zip(points, points[1:]))
+
+
 class Learner1D(BaseLearner):
     """Learns and predicts a function 'f:ℝ → ℝ^N'.
 
@@ -138,27 +209,40 @@ 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)
         The x and y scale over all the intervals, useful for rescaling the
         interval loss.
-    function_values : dict(float → float)
+    data : 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.
+        At the left ``x_left`` and right ``x_left`` most boundary it has
+        ``x_left: (None, float)`` and ``x_right: (float, None)``.
+
+    The `loss_per_interval` function should also have
+    an attribute `nth_neighbors` that indicates how many of the neighboring
+    intervals to `interval` are used. If `loss_per_interval` doesn't
+    have such an attribute, it's assumed that is uses **no** neighboring
+    intervals. Also see the `uses_nth_neighbors` decorator.
+    **WARNING**: When modifying the `data` and `neighbors` datastructures
+    the learner will behave in an undefined way.
     """
 
-    def __init__(self, function, bounds, loss_per_interval=None, loss_depends_on_neighbours=False):
+    def __init__(self, function, bounds, loss_per_interval=None):
         self.function = function
-        self._loss_depends_on_neighbours = loss_depends_on_neighbours
 
-        if loss_depends_on_neighbours:
-            self.loss_per_interval = loss_per_interval or curvature_loss
+        if hasattr(loss_per_interval, 'nth_neighbors'):
+            self.nth_neighbors = loss_per_interval.nth_neighbors
         else:
-            self.loss_per_interval = loss_per_interval or default_loss
+            self.nth_neighbors = 0
+
+        self.loss_per_interval = loss_per_interval or default_loss
 
         # A dict storing the loss function for each interval x_n.
         self.losses = {}
@@ -217,36 +301,27 @@ class Learner1D(BaseLearner):
         return max(losses.values()) if len(losses) > 0 else float('inf')
 
     def _get_loss_in_interval(self, x_left, x_right):
-        if x_left is None or x_right is None:
-            return None
+        assert x_left is not None and x_right is not None
 
-        dx = x_right - x_left
-        if dx < self._dx_eps:
+        if x_right - x_left < self._dx_eps:
             return 0
 
         # we need to compute the loss for this interval
-        interval = (x_left, x_right)
-        if self._loss_depends_on_neighbours:
-            neighbour_left  = self._find_neighbors(x_left , self.neighbors)[0]
-            neighbour_right = self._find_neighbors(x_right, self.neighbors)[1]
-            neighbours = neighbour_left, neighbour_right
-            return self.loss_per_interval(interval, neighbours,
-                                          self._scale, self.data)
-        else:
-            return self.loss_per_interval(interval, self._scale, self.data)
+        return self.loss_per_interval(
+            (x_left, x_right), self._scale, self.data, self.neighbors)
 
 
     def _update_interpolated_loss_in_interval(self, x_left, x_right):
         if x_left is None or x_right is None:
-            return None
+            return
 
-        dx = x_right - x_left
         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
@@ -266,17 +341,11 @@ 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)
-
-            # if the loss depends on the neighbors we should also update those losses
-            if self._loss_depends_on_neighbours:
-                neighbour_left  = self._find_neighbors(x_left , self.neighbors)[0]
-                neighbour_right = self._find_neighbors(x_right, self.neighbors)[1]
-                self._update_interpolated_loss_in_interval(neighbour_left, x_left)
-                self._update_interpolated_loss_in_interval(x_right, neighbour_right)
+            # (x_left, x), (x, x_right) and the nth_neighbors nearest
+            # neighboring intervals. Since the addition of the
+            # point 'x' could change their loss.
+            for ival in _get_intervals(x, self.neighbors, self.nth_neighbors):
+                self._update_interpolated_loss_in_interval(*ival)
 
             # Since 'x' is in between (x_left, x_right),
             # we get rid of the interval.
@@ -305,9 +374,7 @@ class Learner1D(BaseLearner):
     @staticmethod
     def _find_neighbors(x, neighbors):
         if x in neighbors:
-            return neighbors
-        if x is None:
-            return None, None
+            return neighbors[x]
         pos = neighbors.bisect_left(x)
         keys = neighbors.keys()
         x_left = keys[pos - 1] if pos != 0 else None
@@ -424,10 +491,8 @@ class Learner1D(BaseLearner):
 
         # The the losses for the "real" intervals.
         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)
-                if x_right - x_left >= self._dx_eps else 0)
+        for ival in intervals:
+            self.losses[ival] = self._get_loss_in_interval(*ival)
 
         # List with "real" intervals that have interpolated intervals inside
         to_interpolate = []