XGBoost: A Scalable Tree Boosting System
Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2016 · paper · AI and machine learning
Describes XGBoost, a scalable tree boosting system featuring a sparsity-aware algorithm, weighted quantile sketch, and designs for cache access, compression and sharding. The authors state it scales beyond billions of examples using far fewer resources than existing systems.
Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
5 findings
Substance Sec. 3.3 rewrites Eq (3) as "Σ ½ h_i(f_t(x_i) − g_i/h_i)² + Ω(f_t) + constant ... weighted squared loss with labels g_i/h_i"; under the paper's definition g_i = ∂l/∂ŷ, the square should be (f_t(x_i) + g_i/h_i)², i.e. the labels are −g_i/h_i.
- p. 4 Sec. 3.3, unnumbered equation after Eq. (9)
which is exactly weighted squared loss with labels gi/hi
Only the authors can say what is right here.
Judge's note: Crop C016-a confirms the printed '(f_t(x_i) − g_i/h_i)²' and 'labels g_i/h_i'. The text layer (p.2) defines g_i = ∂_{ŷ}l and gives Eq (3) the linear term +g_i f_t(x_i). My own working: ½h(f − g/h)² = ½hf² − gf + g²/(2h), so the linear term is −gf, the opposite sign to Eq (3). Completing the square instead gives ½hf² + gf = ½h(f + g/h)² − g²/(2h). Numerical check with h=1, g=2: Eq (3) is minimised at f = −2, but the printed form is minimised at f = +2, which also contradicts w* = −G/(H+λ) in Eq (5). The printed display is therefore false. The point the passage is making, that h_i acts as the weight, still holds once the sign is fixed.
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Substance Sec. 6.2: "1.7 billion instances with 67 features (13 integer, 26 average CTR statistics and 26 counts)"; the listed components sum to 65, while Table 2 also gives m = 67 for Criteo.
- p. 8 Sec. 6.2, Criteo paragraph
67 features (13 integer, 26 average CTR statistics and 26 counts)
- p. 8 Table 2, Criteo row, column m
67
- p. 8 Table 2, Criteo row
1.7 B 67
- p. 8 Sec. 6.2, Criteo paragraph
1.7 billion instances with 67 features (13 integer, 26 average CTR statistics and 26 counts)
Only the authors can say what is right here.
Judge's note: Page 8 image confirms the sentence and the Table 2 Criteo row (1.7 B, 67). The same paragraph says the raw data has "13 integer features and 26 ID features", which matches the breakdown, and nothing in the text mentions two further features. Either the total (text and table) or the breakdown is off; only the authors can say which.
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Wording Proof of Lemma A.2 prints "r̃+D(y) − r̃−D(y) − ω̃D(y) = [r̃+D(xi+1) − ω̃D(xi+1)] − [r̃+D(xi) + ω̃D(xi)] − 0"; the second bracket should contain r̃−D(xi), not r̃+D(xi).
- p. 12 Proof of Lemma A.2
−[r̃+D(xi)+ω̃D(xi)]−0
Only the authors can say what is right here.
Judge's note: Read on the page 12 image, the enlarged crop of that area and the text layer: the subtracted bracket is printed with r̃+. Reworked from Eq. (18): for y in (xi, xi+1), r̃−D(y) = r̃−D(xi) + ω̃D(xi), r̃+D(y) = r̃+D(xi+1) − ω̃D(xi+1) and ω̃D(y) = 0, so the left side equals [r̃+D(xi+1) − ω̃D(xi+1)] − [r̃−D(xi) + ω̃D(xi)] − 0. That is the left side of Eq. (23); the printed r̃+ version is not, and differs whenever r̃+D(xi) ≠ r̃−D(xi). The lemma as stated stands, so the step is a superscript slip.
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Wording Proof of Lemma A.2 prints "r̃+_D(y)−r̃−_D(y)−ω̃_D(y) = [r̃+_D(x_{i+1})−ω̃_D(x_{i+1})]−[r̃+_D(x_i)+ω̃_D(x_i)]−0"; by Eq (18) the second bracket should be [r̃−_D(x_i)+ω̃_D(x_i)].
- p. 12 Proof of Lemma A.2
The key is again consider y ∈ (xi, xi+1)
Only the authors can say what is right here.
Judge's note: Crop C058-a and page 12 show r̃+_D(x_i) in the second bracket. Page 11, Eq (18), defines r̃−_D(y) = r̃−_D(x_i) + ω̃_D(x_i) for y ∈ (x_i, x_{i+1}), r̃+_D(y) = r̃+_D(x_{i+1}) − ω̃_D(x_{i+1}) and ω̃_D(y) = 0. Substituting these gives [r̃+(x_{i+1}) − ω̃(x_{i+1})] − [r̃−(x_i) + ω̃(x_i)], which is the left side of Eq (23), the bound the proof then invokes. Counterexample, the exact summary of D = {(1,1),(2,1)} at y = 1.5: the true value is (2−1) − (0+1) = 0, and the printed right-hand side is (2−1) − (1+1) = −1. With r̃− in place of r̃+ the proof goes through, so the lemma stands.
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Wording Proof of Lemma A.4, case x* = x_{i+1}: the line "≥ 2[r̃+_D(x_{i+1}) + ω̃_D(x_{i+1})] − εω(D)" should have −ω̃_D(x_{i+1}), matching the preceding equality and Eq (33).
- p. 13 Proof of Lemma A.4, case x* = x_{i+1}, third line
≥ 2[r̃+D(xi+1) + ω̃D(xi+1)] − εω(D)
- p. 13 Proof of Lemma A.4, fourth case, first chain
≥ 2[r̃+D(xi+1) + ˜ωD(xi+1)] −ϵω(D)
Only the authors can say what is right here.
Judge's note: Page 13 image: the preceding line is 2[r̃+(x_{i+1}) − ω̃(x_{i+1})] − [r̃+(x_{i+1}) − ω̃(x_{i+1}) − r̃−(x_i) − ω̃(x_i)]. By Eq (23) the second bracket is at most εω(D), which gives ≥ 2[r̃+(x_{i+1}) − ω̃(x_{i+1})] − εω(D). The printed '+' makes a stronger claim that does not follow. Counterexample, the exact summary of D = {(1,1),(2,1)} with S = {1,2}: this case (i=1) is reached for 1 ≤ d < 1.5. The first line gives 2d ≥ 0 + 1 + 2 − 1 = 2, but the printed bound would require 2d ≥ 6, which is impossible since d ≤ ω(D) = 2. With the minus sign the bound is 2, which is consistent. Eq (33) needs the minus sign, so Lemma A.4 stands.
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31 minor (typos and the like; most checked by the judge only)
Wording Introduction says "The system runs more than ten times faster than existing popular solutions on a single machine", but Tables 3 and 4 show a >10x per-tree speed-up only against scikit-learn (28.51 vs 0.6841 s); R.gbm is 1.032 s and pGBRT 2.576 s vs 0.826 s. Sec. 6.3 also says R's GBM's one-branch approach "makes it faster" without saying faster than what, while Table 3 shows it slower than XGBoost.
- p. 1 Introduction, scalability paragraph
ten times faster than existing popular solutions on a single
- p. 8 Table 3
1.032
- p. 8 Table 4
2.576
- p. 1 Sec. 1 Introduction, right column
The system runs more than ten times faster than existing popular solutions on a single machine
Only the authors can say what is right here.
Judge's note: Viewed pages 1 and 8. Table 3: XGBoost 0.6841, scikit-learn 28.51, R.gbm 1.032. Table 4: XGBoost 0.826, pGBRT 2.576. Sec. 6.3 itself limits the claim to "more than 10x faster than scikit-learn". The plural, unqualified introduction claim is supported for one of three single-machine baselines; this is overstatement in wording, not a wrong number. The "makes it faster" sentence reads as a statement about the mechanism (one-branch expansion versus a full tree), so it is ambiguous rather than contradicted. No other single-machine comparison in the text gives 10x.
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Wording Figure 2 prints the structure score as "Obj = −Σj Gj²/(Hj+λ) + 3γ" without the factor ½ that Eq. (6) has (−½ Σ … + γT), yet the text says "Fig. 2 illustrates how this score can be calculated."
- p. 3 Figure 2, formula inside the figure
Obj = −Σj Gj²/(Hj+λ) + 3γ
- p. 3 Eq. (6)
Fig. 2 illustrates how this score can be calculated.
- p. 3 Eq (6)
˜L(t)(q) = −1/2 ΣT j=1
Judge's note: Page 3 image: the formula inside Figure 2 reads Obj = −Σ_j G_j²/(H_j+λ) + 3γ, and no ½ is visible. Eq. (6) on the same page prints −½ Σ_{j=1}^T (Σg)²/(Σh+λ) + γT. The γ term is kept unscaled, so this is not a uniform rescaling of the score: with one leaf, G=2, H=1, λ=0, γ=1, Eq. (6) gives −1 and the figure's formula gives −3. Eq. (6) and Eq. (7) in the text are correct, so I rate this an illustrative-figure inconsistency.
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Wording Figure 11 caption: "Adding compression gives 3x speedup, and sharding into two disks gives another 2x speedup." (Sec. 6.5: "compression helps to speed up computation by factor of three"); at the only plotted basic-algorithm point the gap to block compression reads as roughly 5–6x, not 3x.
- p. 9 Figure 11 caption
Adding compression gives 3x speedup, and sharding into two disks gives another 2x speedup.
- p. 9 Sec. 6.5
compression helps to speed up computation by factor of three
Only the authors can say what is right here.
Judge's note: Read by eye from the log-log Figure 11 on page 9: basic algorithm about 1,700 s per tree at the smallest subset, block compression about 310 s, compression+shard about 210 s at the same subset. That gives roughly 5.5x for compression and 1.5x for sharding at that point; the 2x sharding gain does hold at the larger subsets. Chart readings are approximate and the 3x could come from an unplotted measurement, so this is a question for the authors.
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Wording Proof of Lemma A.4, first case, ends with "= r̃−_D(x1) + ω̃+_D(x1)"; the symbol ω̃+ is defined nowhere, and the intended term is ω̃_D(x1) (equal to r̃+_D(x1)).
- p. 12 Proof of Lemma A.4, first bullet, last line
= r̃−D(x1) + ω̃+D(x1)
- p. 12 Proof of Lemma A.4, first case
= ˜r−D(x1) + ˜ω+D(x1)
Only the authors can say what is right here.
Judge's note: Page 12 image: the last line of the first bullet does show ω̃ with a superscript '+'. The parenthetical just above states ω̃_D(x1) = r̃+_D(x1) = ω(x1), so the step r̃−(x1) + r̃+(x1) = r̃−(x1) + ω̃(x1) is valid once the stray '+' is dropped. As printed, the step uses an undefined symbol. Lemma A.4 is unaffected.
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Wording Proof of Theorem A.1: "Here the first inequality is due to Lemma A.3." Lemma A.3 justifies the first relation in the chain, which is an equality, not the first inequality.
- p. 12 Proof of Theorem A.1
Here the first inequality is due to Lemma A.3.
Judge's note: The crop shows the chain: r̃+D(y) − r̃−D(y) − ω̃D(y) = [sums over D1 and D2] ≤ ε1ω(D1) + ε2ω(D2) ≤ max(ε1, ε2)ω(D1 ∪ D2). Lemma A.3 (Eqs. 30-32, additivity of the extended functions) gives the "=" step. The first "≤" follows from Definition A.3 applied to each summary. The proof is sound; the sentence should say "first equality".
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Typo Stray "is": "These innovations include: a novel tree learning algorithm is for handling sparse data" should read "a novel tree learning algorithm for handling sparse data".
- p. 1 Introduction, last paragraph
a novel tree learning algorithm
- p. 1 Sec. 1 Introduction, right column
a novel tree learning algorithm is for handling sparse data
Judge's note: Visible in the page 1 image, right column, last paragraph. Not an extraction artefact.
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Typo Footnote 3: "Solutions come from of top-3 teams of each competitions." has a stray "of" and "each competitions" should be "each competition".
- p. 1 footnote 3
Solutions come from of top-3 teams of each competitions.
Judge's note: Visible in the page 1 image, footnote 3.
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Typo Scattered typos: "This this resembles previous work" (Sec. 5); "Solutions come from of top-3 teams of each competitions" (footnote 3); "Specicially" and "reguralized objective" (Sec. 2); "Lettting I" (Sec. 2.2); "in tochastic optimization" (Sec. 2.3); "best previously pubished system" (Sec. 6.4).
- p. 7 Sec. 5, first paragraph
This this resembles previous work
- p. 1 Footnote 3
Solutions come from of top-3 teams of each competitions.
- p. 2 Sec. 2
Specicially the second order
- p. 2 Sec. 2
reguralized objective
Judge's note: "This this" seen on the page 7 image; "Specicially", "reguralized", "tochastic" and "pubished" seen on crops; "come from of ... each competitions" and "Lettting" found in the text layer only (none of these words involve ligatures). The "spase" item in this cluster is the same defect as C025 and is not counted here.
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Typo Loss arguments are ordered "l(ŷi, yi)" in Eq. (2) but l(yi, ŷi^(t−1) + ft(xi)) in the step-t objective and l(yi, ŷ^(t−1)) in the second-order expansion, which also drops the instance subscript i on ŷ^(t−1).
- p. 2 Eq. (2)
l(ŷi, yi)
- p. 2 Sec. 2.2, objective at step t
l(yi, ŷi(t−1) + ft(xi))
Judge's note: Read on the page 2 image. Eq. (2) has prediction first; the Sec. 2.2 objective and expansion have target first, and the expansion and the definitions of g_i and h_i print ŷ^(t−1) without the subscript. A notation inconsistency only; the meaning is clear.
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Typo Sec. 2 opening paragraph misspells two words: "Specicially the second order method is originated from Friedman et al. [12]" (Specifically) and "minor improvements in the reguralized objective" (regularized).
- p. 2 Sec. 2, intro paragraph
Specicially the second order
- p. 2 Sec. 2, intro paragraph
reguralized objective
- p. 2
Specicially the second order method is originated from Friedman et al. [12].
Judge's note: Both misspellings are visible in the page 2 image, left column, Sec. 2.
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Typo Subject–verb disagreement: "The major contributions of this paper is listed as follows" should be "are listed".
- p. 2 Sec. 1 Introduction, before the bullet list
The major contributions of this paper is listed as follows
Judge's note: Visible at the top of the left column on the page 2 image.
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Typo Algorithms 1 and 3 declare "Input: d, feature dimension" but loop "for k = 1 to m do"; Sec. 2.1 defines m as the number of features, and Sec. 4.1 reuses d as "Let d be the maximum depth of the tree".
- p. 3 Algorithm 1
Input: d, feature dimension
- p. 5 Algorithm 3
Input: d, feature dimension
- p. 6 Sec. 4.1, Time Complexity Analysis
Let d be the maximum depth
- p. 3 Algorithm 1 loop
for k = 1 to m do
Judge's note: Confirmed on the page 3 and page 5 images (Algorithms 1 and 3) and in the page 6 text for Sec. 4.1. d is never used inside either algorithm. Notation inconsistency with no effect on results.
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Typo Algorithms 1 and 3 initialise "gain ← 0" but then update "score ← max(score, …)", which is never initialised; Algorithm 1 outputs "Split with max score" while Algorithm 3 outputs "Split and default directions with max gain".
- p. 3 Algorithm 1
gain ← 0
- p. 5 Algorithm 3, output line
Output: Split and default directions with max gain
Judge's note: Seen on the page 3 (Algorithm 1) and page 5 (Algorithm 3) images. The same variable is evidently intended; a naming inconsistency in pseudocode.
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Typo Misspellings on page 3: "Lettting I = IL ∪ IR" (Letting) before Eq. (7), and "in tochastic optimization" (stochastic) in Sec. 2.3.
- p. 3 Sec. 2.2, before Eq. (7)
Lettting I = IL ∪ IR
- p. 3 Sec. 2.3
in tochastic optimization
- p. 3
Lettting I = IL ∪IR, then
Judge's note: Both are visible in the page 3 image, left column.
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Typo Algorithm 2 uses a malformed assignment operator "←=" in both the G_kv and H_kv lines ("Gkv ←="), where the other algorithms use "←".
- p. 3 Algorithm 2
Gkv ←=
Judge's note: Both lines in Algorithm 2 on the page 3 image print an arrow followed by an equals sign.
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Typo Algorithm 3 prints "G ← Σ_{i∈I}, g_i, H ← Σ_{i∈I} h_i" with a stray comma between the summation and g_i.
- p. 5 Algorithm 3, G update
G ←P i∈I, gi,H ←P i∈I hi
Judge's note: Page 5 image, Algorithm 3: a comma follows the subscript i∈I before g_i. The comma separating the G and H assignments is legitimate; the first one is stray.
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Typo Sec. 4.2: "reduce the runtime overhead when number of rows in the is large" is missing a noun after "in the".
- p. 6 Sec. 4.2, second paragraph
reduce the runtime overhead when number of rows in the
- p. 6 Sec. 4.2
number of rows in the is large
Judge's note: Page 6 image shows "number of rows in the / is large" across the line break with no noun. The intended word (e.g. "dataset" or "block") is not determinable from the page, but a word is plainly missing.
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Typo Sec. 4.2: "non cache-aware algorithm on the the Higgs and the Allstate dataset" has a duplicated "the".
- p. 6 Sec. 4.2, right column
non cache-aware algorithm on the the Higgs
- p. 6
non cache-aware algorithm on the the Higgs and the All-
Judge's note: Seen on the page 6 image, right column.
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Typo Sec. 4.2: "maximum number of examples in contained in a block" has a stray "in".
- p. 6 Sec. 4.2, block size definition
imum number of examples in contained in a block
Judge's note: Seen on the page 6 image, right column.
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Typo Sec. 4.1: "time complexity of original spase aware algorithm"; "spase" should be "sparse".
- p. 6 Sec. 4.1, Time Complexity Analysis
original spase
- p. 6
original spase aware algorithm is O(Kd∥x∥0 log n)
Judge's note: Seen on the page 6 image, Time Complexity Analysis paragraph.
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Typo Figure 12 caption: "XGBoost runs more 10x than spark per iteration and 2.2x as H2O's optimized version" is missing words ("more than 10x faster than", "2.2x as fast as"); Sec. 4.2 also has "when number of rows in the is large" with a missing noun.
- p. 9 Figure 12 caption
XGBoost runs more 10x than spark per iteration and 2.2x as H2O
- p. 9 Figure 12 caption
XGBoost runs more 10x than spark per iteration and 2.2x as H2O’s optimized version
- p. 9 Figure 12 caption
XGBoost runs more 10x than
- p. 6 Sec. 4.2
when number of rows in the is large
Judge's note: Both seen on the images: the Figure 12 caption on page 9 and the Sec. 4.2 sentence in the page 6 crop. The meaning is recoverable from Fig. 12(b), so this is copy-editing only; the missing noun in Sec. 4.2 (block or dataset) is for the authors to supply.
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Typo Sec. 6.1: "The distributed XGBoost runs natively on Hadoop, MPI Sun Grid engine." appears to lack a separator between "MPI" and "Sun Grid engine".
- p. 7 Sec. 6.1
on Hadoop, MPI Sun Grid engine
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: MPI and Sun Grid Engine are separate platforms. The page 7 image shows the text as quoted, with no comma or "and" between them.
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Typo Sec. 5: "the techniques such as cache-aware pre-fecthing can be used"; "pre-fecthing" should be "prefetching" (line-broken as "pre-/fecthing").
- p. 7 Sec. 5
cache-aware pre-fecthing
- p. 7
the techniques such as cache-aware pre-fecthing can be used
Judge's note: Seen on the page 7 image, right column: "cache-aware pre-" then "fecthing" on the next line, with the letters c and t transposed. The paper spells it "prefetching" elsewhere (Sec. 4.2).
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Typo Misspellings and a stray word: Sec. 4.3 "we substract the row index by the begining index of the block"; Sec. 5 "cache-aware pre-fecthing can be used"; Sec. 6.4 "the best previously pubished system" and "and but also gives a bit higher performance".
- p. 9 Sec. 6.4
the best previously pubished system
- p. 7 Sec. 5, second paragraph
fecthing can be used
- p. 7 Sec. 4.3, Block Compression
we substract the row index by the begining index
- p. 9 Sec. 6.4
and but also gives a bit higher performance
Judge's note: All verified on images: "substract" and "begining" in crop C038-c, "fecthing" in crop C038-b, "pubished" and "and but also" on page 9. Fixes: subtract, beginning, fetching, published, and delete the stray "and".
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Typo Sec. 6.3: "This could due to the fact that there are few important features" is missing "be"; the same slip recurs in Sec. 6.4: "This could due to the fact that the subsampling helps".
- p. 8 Sec. 6.3
This could due to the fact
Judge's note: Seen on the page 8 image (Sec. 6.3) and the page 9 image (Sec. 6.4). The fix is "could be due to".
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Typo Reference [16]: "Proceedings of the Twenty-Sixth Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI'10)" has a duplicated "Conference".
- p. 10 Reference [16]
Twenty-Sixth Conference Annual Conference on Uncertainty
Judge's note: Seen on the page 10 image. The duplication is evident from the printed text itself; delete the first "Conference".
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Typo Reference [4]: "L. Breiman. Random forests. Maching Learning, 45(1):5–32, Oct. 2001." misspells the journal name "Machine Learning".
- p. 10 Reference [4]
Maching Learning
Judge's note: Seen on the page 10 image and in crop C045-a. "Maching" is an evident misspelling of "Machine".
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Typo Reference [15] prints the last author as "J. Q. n. Candela", with a stray lowercase "n." between the initials and the surname.
- p. 10 Reference [15]
J. Q. n. Candela
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the intended author name is J. Quiñonero Candela (the stray "n." being the remains of the ñ). The crop and the page 10 image both show "S. Bowers, and J. Q. n. Candela." A lowercase initial is anomalous on its face, but the correct form of the name cannot be checked from the files.
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Typo Appendix A opening: "Approximate answer of quantile queries is for many real-world applications." A word is missing after "is".
- p. 11 Appendix A, first paragraph
Approximate answer of quantile queries is for many real-
Only the authors can say what is right here.
Judge's note: The crop shows the sentence exactly as quoted; it has no predicate adjective or participle (for example "important", "needed" or "sufficient"). The missing word is not determinable from the text, so the authors must supply it.
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Typo Sec. A.3 prints "Q(D2) = (S2, r̃+D1, r̃−D2, ω̃D2)"; the plus-rank function in the summary of D2 carries subscript D1 instead of D2.
- p. 12 Sec. A.3, first paragraph
Q(D2) = (S2, r̃+D1, r̃−D2, ω̃D2)
- p. 12 A.3
Q(D2) = (S2, ˜r+ D1
Judge's note: The crop clearly shows subscript D1 on r̃+ in the Q(D2) tuple, while the other two functions carry D2. Eq. (27) and (31) use r̃+D2, so the intended subscript is D2. Subscript typo with an evident fix.
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Typo "the constraints on the original function can lead to in more general constraints on the extended function" contains a stray "in".
- p. 12 Appendix A.1, 'Property of Extended Function'
original function can lead to in more general constraints
Judge's note: The crop shows "can lead to in more general constraints". Extra word; either "lead to" or "result in" was intended.
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