Overfitting — where it appears
Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.
The model is the compressor
One stream of 32,768 symbols has an entropy of 3.886 bits per symbol, and 2.243, and 1.186, and 0.991, and 0.909. All five numbers are correct, all five are floors, and nothing about the data changed between them. The only thing that changed is how many preceding symbols the model was allowed to look at — which makes the entropy of a file a property of a decision rather than a property of a file.
The matrix a corpus wrote
A substitution matrix is not a property of an alphabet. Fit one to four hundred pairs of sequences that rarely change and the dearest substitution costs five; fit the same model to four hundred pairs that often change and it costs two. Two hundred test pairs aligned under each matrix give different alignments in 115 cases — and a matrix fitted to eight pairs of the first kind moves 79 of them, from sampling alone.
The grain a corpus chooses for itself
A fitted substitution matrix can choose its own rounding lattice: the coarsest grain that keeps all six of its costs apart. On eight resamples of the same 400-pair corpus that grain comes out anywhere from an eighth of a bit to a forty-seventh, because it is set by whichever two costs happen to land closest, and those two differ by less than a tenth of their own standard error. Asked instead which costs it can actually tell apart, the corpus pools six into the generator's two and chooses an alignment the generator agrees with for every one of 200 test pairs on every resample.
The classes held-out pairs cannot rule out
A substitution matrix can choose how many distinct costs it has as part of the fit: try all 203 ways of grouping its six pairs into classes and keep the one that best predicts pairs it was not fitted to. On corpora from a generator with two costs, that choice finds the two on about a third of resamples, and no more often at 3,200 pairs than at 50. A spurious split gains the same bit or two of fit at every size, and held-out scoring charges it a fixed price. A penalty that grows with the corpus finds the two every time, and finds a third class half a bit away from 400 pairs.
What a planner pays to find out what to pay
Insurance against a row estimate is set from the error's median and spread, and a running system knows neither — it has to fit them from executed queries. Fitted from one query the divisor costs 1.263 times the better plan against 1.215 for a planner that never insures at all, so learning is worse than not learning until about sixteen queries have run. The tail, though, is bought immediately: one observation already holds the worst case to 24.7 against 36.6.
A cost built from two properties
A substitution matrix fitted as a partition gives each class of pairs one cost, and the best number of classes can be chosen from the data. When the true costs are a sum of two terms — a transversion penalty and a graded difference between the letters — every pair costs something different, and the chosen partition's number of classes becomes a statement about the corpus: two classes at 25 sequence pairs, four at 200, five at 1,600. A model with the two terms and three parameters predicts a fresh corpus better than any partition at every size, and gets 191 of 200 alignments right from 50 pairs where the chosen partition needs 3,200. Told the wrong property it is worse than the plain two-class matrix, and nothing in its own fit says so.
What the queries know that the map does not
A greedy rule that chooses landmark cells by rerunning a sample of past queries needs two hundred of them to draw level with a rule that reads only the map — and what it finally chooses, on map after map, is the four corners. Give the queries a destination instead of scattering them, and twenty are enough to beat the corners by 29% on eight maps out of eight. A query log is worth reading exactly to the extent that it is not uniform.
A line the corpus finds
Substitution costs built as a transversion penalty plus the distance between the two letters' values on a graded property beat every partition of the pairs, when the model is told the property, and fall below the plainest matrix when told the wrong one. Fitting each letter's position on a line instead was predicted to find the property's order from about four times the told model's data and then behave like it. It finds the order on every corpus of 100 sequence pairs or more but one, and every miss starts with the two letters whose values differ by a fifth of a bit. But knowing the order is not knowing the costs. Its alignments are no better than six free classes' until 800 pairs, sixteen times the data the told model needed for the same score, and on a generator with no graded property at all it finds a line anyway, a quarter as long as a real one at 200 pairs, and pays for it against the plain partition.
A line with a prior toward the classes
Substitution costs fitted as the letters' positions on a free line learn the letters' order quickly, and on a generator with no graded property they invent a line anyway. Charging the line for its length, at a rate growing with the corpus as the partitions' charge does, was predicted to collapse the invented line from 50 pairs and to cost the real one little. It does neither. On the generator with no property the charged line still has a median length of 0.12 bits at 3,200 pairs. On the generator with a property it takes the letters' order away at every size up to 400 pairs where the free line had it on sixteen seeds of sixteen. Charging the line for the number of gaps it keeps, the partitions' own rule, collapses the invented line on every seed at every size and matches the chosen partition to the bit — and then needs 1,600 pairs before it believes a real line is there.
A placement after the traffic moved
Four landmarks fitted to twenty queries that all end at one cell expand two thirds of the corners' cells there. Move the destination ten cells and the advantage is gone; move it thirty and the fitted placement expands nearly twice the corners' cells. Choosing again costs 317,000 cells of search — the saving of 8,600 queries — so on streams whose destination drifts, a policy that refits on a counter never beats fitting once and keeping the result, and past a drift of about ten cells nothing fitted beats the corners.
Named alongside it
The objects these essays reach for when they reach for this one.
Cost modelEstimatorFittingSubstitution matrixAlignmentHonest limitParameter choiceSamplingCorpusModel selectionBreak-evenHeuristic search