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I saw your tweets about this and was hoping that dask.array would get a PR :) cc @jcrist |
dask/array/linalg.py
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Is np.linalg.inv efficient in the triangular case? I did a quick benchmark and I think that scipy.linalg.solve_triangular might be faster here.
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In [1]: import numpy as np
In [2]: import scipy.linalg
In [3]: x = np.random.random((2000, 2000))
In [4]: y = np.random.random((2000, 2000))
In [5]: p, l, u = scipy.linalg.lu(x)
In [6]: %time np.linalg.inv(l).dot(y)
CPU times: user 7.05 s, sys: 295 ms, total: 7.35 s
Wall time: 1.86 s
Out[6]:
array([[ 0.47500179, 0.49931231, 0.98423146, ..., 0.32664585,
0.6498253 , 0.36445447],
[ 0.20188943, 0.42700747, 0.60685892, ..., 0.0924809 ,
0.6496587 , 0.09958131],
[ 0.88167014, 0.66419102, 0.75093161, ..., 0.21974697,
0.19297681, 0.59164822],
...,
[ 7.13030456, -0.52782879, 0.53777843, ..., 4.07304968,
1.4036643 , 4.20918249],
[ 5.25830631, 1.38309588, -3.68227833, ..., 11.42730652,
1.06180677, -0.80925082],
[ 11.57086192, 9.53704602, -3.79445665, ..., 8.64013982,
-2.69850912, 10.15152519]])
In [7]: %time scipy.linalg.solve_triangular(l, y, lower=True)
CPU times: user 743 ms, sys: 7.88 ms, total: 751 ms
Wall time: 755 ms
Out[7]:
array([[ 0.47500179, 0.49931231, 0.98423146, ..., 0.32664585,
0.6498253 , 0.36445447],
[ 0.20188943, 0.42700747, 0.60685892, ..., 0.0924809 ,
0.6496587 , 0.09958131],
[ 0.88167014, 0.66419102, 0.75093161, ..., 0.21974697,
0.19297681, 0.59164822],
...,
[ 7.13030456, -0.52782879, 0.53777843, ..., 4.07304968,
1.4036643 , 4.20918249],
[ 5.25830631, 1.38309588, -3.68227833, ..., 11.42730652,
1.06180677, -0.80925082],
[ 11.57086192, 9.53704602, -3.79445665, ..., 8.64013982,
-2.69850912, 10.15152519]])|
I'm curious about the permutation matrix part of this computation. How out-of-core is this algorithm? How does the size of the input array affect the memory requirements of the algorithm? |
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I may misunderstand what you're asking, but I think it's difficult to perform permutation (pivotting) over the dask chunks and results in error if permutation is needed. If we do not permute between blocks, the algorithm perform each blocks from left-top to right-bottom in out-of-core. Assuming 3x3 chunks:
My main interest is to use LU as intermediate results to get inverse matrix. |
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What is the maximum number of chunks that we must have in memory at once? For example, in the following computation What is maximum amount of memory used over time? At most this number is |
dask/array/linalg.py
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You should be able to replace this with just (name_lu, i, j). The scheduler accepts direct name aliases.
In [1]: import dask
In [2]: inc = lambda x: x + 1
In [3]: dsk = {'x': (inc, 1), 'y': 'x'}
In [4]: dask.get(dsk, 'y')
Out[4]: 2There was a problem hiding this comment.
Also, you don't need the parentheses when adding an element with a tuple key to a dict -- Python adds that automatically.
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@mrocklin Yes your example should work, and #886 intends to the case like below. Because calculated L and U is referred by intermediate calculation.
dsk = {('x', 0): np.array([1, 2]),
('x', 1): ('x', 0)}
darr = da.Array(dsk, 'x', chunks=(2, ), shape=(4, ))
darr.compute()
# KeyError: ('x', 0)
@shoyer Thanks, will fix that.
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This looks pretty cool so far. Image from current state (after removing In [1]: import dask.array as da
In [2]: x = da.random.random((16, 16), chunks=(4, 4))
In [3]: p, l, u = da.linalg.lu(x)
In [4]: from dask import visualize
In [6]: visualize(l, u, filename='dask.png')
Out[6]: <IPython.core.display.Image object>The image corroborates your reasoning about out-of-core. The algorithm looks like it will be easy to compute with little memory. @jcrist was working on a Cholesky decomposition a while ago. This might interest him. |
dask/array/linalg.py
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Is this necessary? @mrocklin any thoughts on if it's dangerous to share a task dict between multiple arrays?
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It's probably not necessary, so far all operations within dask are pure. However, it's also very cheap; we copy large dicts all the time.
In [1]: import dask.array as da
In [2]: x = da.random.random((10000, 10000), chunks=(100, 100))
In [3]: %time y = ((x + 1)**2).sum(axis=0).mean()
CPU times: user 473 ms, sys: 4.31 ms, total: 477 ms
Wall time: 477 ms
In [4]: %time _ = y.dask.copy()
CPU times: user 6.56 ms, sys: 0 ns, total: 6.56 ms
Wall time: 6 ms
In [5]: len(y.dask)
Out[5]: 40605|
It's pretty awesome that this works (mostly) out of core -- I didn't realize that that was possible! I would definitely document the memory requirements in the doc string for this function. |
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dask/array/linalg.py
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@jcrist Is there an easy way to plug into your new tree-reduction code here? Or would that be pointless?
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Is there a reason why we're not using sum directly here regardless?
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e.g. prevs = (sum, prevs)
Although, it's nice to break apart large computations like this into multiple keys. That helps debugging and diagnostic tools to properly classify what's going on. I wonder if each element in prevs should be a separate key in the graph followed by a final key for (sum, prev_keys).
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@mrocklin I think the reason not to use sum directly is that it lets you handle the sum in a streaming fashion? Otherwise you'll need to load each of prevs into memory together.
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At the moment these are all in the same task, so they'll all be loaded in at once. If we want to do streaming work then each computation will need to be a separate key-value pair in the dictionary.
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Is there anything I can do to help with this PR? |
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Thanks, I still meet the |
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I think that I see the problem. |
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Resolved in #903 I think |
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Tests pass |
dask/array/linalg.py
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I think that this should either be prevs = (sum, prevs) or it should be many small tasks within the graph.
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Sorry to take a long. Once updated to cover more larger matrices/chunks and looks OK. Could you review? |
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@ahmadia this PR may interest you. |
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Profile plots: https://rawgit.com/mrocklin/f73b2967b224b41a62d2/raw/9057bb64edea67f66ecd50a305e614265d03df57/lu.html In [1]: import dask.array as da
In [2]: x = da.random.random((10000, 1000), chunks=(1000, 1000))
In [3]: y = x.dot(x.T)
In [4]: p, l, u = da.linalg.lu(y)
In [5]: from dask.diagnostics import ResourceProfiler, Profiler, visualize
In [6]: with ResourceProfiler() as rprof, Profiler() as prof:
l.sum().compute()
...:
In [7]: visualize([rprof, prof], 'lu.html')
Out[7]: <bokeh.models.plots.GridPlot at 0x7f3290076780> |
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Indeed. Thanks for the ping :)
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From a parallel linear algebra standpoint, this is probably most interesting as an algorithm for solving dense (input) LU on low-memory compute nodes (not actually that uncommon of a situation) with the out-of-core pieces of the matrix stored somewhere with relatively high access cost compared to memory. This has implications for Hadoop-style clusters where the matrix to be solved for is not necessarily predistributed across the entire cluster. It would be worth doing a performance comparison here to see how the algorithm performs against a completely in-memory solution using just On a single-core machine, the approach seems straightforward. On a multi-core machine, one may tweak both the number of BLAS/LAPACK threads as well as the number of parallel processes. My intuition is that it will be fastest to use the largest possible blocks that would still fit in memory. I also would like to see some indication of the performance overhead to solve LU's out of core, so to speak, and where the costs are, if that's possible. |
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This conference paper provides more details on out-of-core parallel solver frameworks and is relatively recent: http://conferences.computer.org/sc/2012/papers/1000a062.pdf |
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This looks good to me. Merging in 24 hours if no comments. |
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Ah sorry, I've removed it because For efficient calculation, we should use I've defined Could you take a look once again? |
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Ah, I incorrectly assumed that This is exciting work! |
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Nice work @sinhrks!
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This is very impressive! Are there any (preliminary) GFlop/s tests before and after going out of core? I have always wondered what the performance would be for running a 100 GB LU off of a 1 TB external hard disk. |
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No numbers yet. I agree that this would be interesting. @jcrist any interest in benchmarking this? |
* Make column projections stricter (dask#881) * Simplify again after lowering (dask#884) * Visual EXPLAIN (dask#885) * Fix merge predicate pushdowns with weird predicates (dask#888) * Handle futures that are put into map_partitions (dask#892) * Remove eager divisions from indexing (dask#891) * Add shuffle if objects are not aligned and partitions are unknown in assign (dask#887) Co-authored-by: Hendrik Makait <hendrik@makait.com> * Add support for dd.Aggregation (dask#893) * Fix random_split for series (dask#894) * Update dask version * Use Aggregation from dask/dask (dask#895) * Fix meta calculation error in groupby (dask#897) * Revert "Use Aggregation from dask/dask" (dask#898) * Parquet reader using Pyarrow FileSystem (dask#882) Co-authored-by: Patrick Hoefler <61934744+phofl@users.noreply.github.com> * Fix assign for empty indexer (dask#901) * Add dask.dataframe import at start (dask#903) * Add indicator support to merge (dask#902) * Fix detection of parquet filter pushdown 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little bit (dask#999) * Nicer read_parquet prefix (dask#998) Co-authored-by: Patrick Hoefler <61934744+phofl@users.noreply.github.com> * Set divisions with divisions already known (dask#997) * Start building and publishing conda nightlies (dask#986) * Fix zero division error when reading index from parquet (dask#1000) * Rename overloaded `to/from_dask_dataframe` API (dask#987) * Register json and orc APIs for "pandas" dispatch (dask#1004) * Fix pyarrow fs reads for list of directories (dask#1006) * Release for dask 2024.4.0 * Fix meta caclulation in drop_duplicates (dask#1007) * Release 1.0.7 * Support named aggregations in groupby.aggregate (dask#1009) * Make release 1.0.9 * Adjust version number in changes * Make setattr work (dask#1011) * Release for dask 2024.4.1 * Fix head for npartitions=-1 and optimizer step (dask#1014) * Deprecate ``to/from_dask_dataframe`` API (dask#1001) * Fix projection for rename if projection isn't renamed (dask#1016) * Fix unique with numeric columns (dask#1017) * Add changes for new version * Fix column projections in merge when suffixes are relevant (dask#1019) * Simplify dtype casting logic for shuffle (dask#1012) * Use implicit knowledge about divisions for efficient grouping (dask#946) Co-authored-by: Patrick Hoefler <61934744+phofl@users.noreply.github.com> Co-authored-by: Hendrik Makait <hendrik@makait.com> * Fix assign after set index incorrect projections (dask#1020) * Fix read_parquet if directory is empty (dask#1023) * Rename uniuqe_partition_mapping property and add docs (dask#1022) * Add docs for usefule optimizer methods (dask#1025) * Fix doc build error (dask#1026) * Fix error in analyze for scalar (dask#1027) * Add nr of columns to explain output for projection (dask#1030) Co-authored-by: Hendrik Makait <hendrik@makait.com> * Fuse more aggressively if parquet files are tiny (dask#1029) * Move IO docstrings over (dask#1033) * Release for dask 2024.4.2 * Add cudf support to ``to_datetime`` and ``_maybe_from_pandas`` (dask#1035) * Fix backend dispatching for `read_csv` (dask#1028) * Fix loc accessing index for element wise op (dask#1037) * Fix loc slicing with Datetime Index (dask#1039) * Fix shuffle after set_index from 1 partition df (dask#1040) * Bugfix release * Fix bug in ``Series`` reductions (dask#1041) * Fix shape returning integer (dask#1043) * Fix xarray integration with scalar columns (dask#1046) * Fix None min/max statistics and missing statistics generally (dask#1045) * Fix drop with set (dask#1047) * Fix delayed in fusing with multipled dependencies (dask#1038) * Add bugfix release * Optimize when from-delayed is called (dask#1048) * Fix default name conversion in `ToFrame` (dask#1044) Co-authored-by: Patrick Hoefler <61934744+phofl@users.noreply.github.com> * Add support for ``DataFrame.melt`` (dask#1049) * Fixup failing test (dask#1052) * Generalize ``get_dummies`` (dask#1053) * reduce pickle size of parquet fragments (dask#1050) * Add a bunch of docs (dask#1051) Co-authored-by: Hendrik Makait <hendrik@makait.com> * Release for dask 2024.5.0 * Fix to_parquet in append mode (dask#1057) * Fix sort_values for unordered categories (dask#1058) * Fix dropna before merge (dask#1062) * Fix non-integer divisions in FusedIO (dask#1063) * Add cache argument to ``lower_once`` (dask#1059) * Use ensure_deterministic kwarg instead of config (dask#1064) * Fix isin with strings (dask#1067) * Fix isin for head computation (dask#1068) * Fix read_csv with positional usecols (dask#1069) * Release for dask 2024.5.1 * Use `is_categorical_dtype` dispatch for `sort_values` (dask#1070) * Fix meta for string accessors (dask#1071) * Fix projection to empty from_pandas (dask#1072) * Release for dask 2024.5.2 * Fix categorize if columns are dropped (dask#1074) * Fix resample divisions propagation (dask#1075) * Release for dask 2024.6.0 * Fix get_group for multiple keys (dask#1080) * Skip distributed tests (dask#1081) * Fix cumulative aggregations for empty partitions (dask#1082) * Move another test to distributed folder (dask#1085) * Release 1.1.4 * Release for dask 2024.6.2 * Add minimal subset of interchange protocol (dask#1087) * Add from_map docstring (dask#1088) * Ensure 1 task group per from_delayed (dask#1084) * Advise against using from_delayed (dask#1089) * Refactor shuffle method to handle invalid columns (dask#1091) * Fix freq behavior on ci (dask#1092) * Add first array draft (dask#1090) * Fix array import stuff (dask#1094) * Add asarray (dask#1095) * Implement arange (dask#1097) * Implement linspace (dask#1098) * Implement zeros and ones (dask#1099) * Remvoe pandas 2 checks (dask#1100) * Add unify-chunks draft to arrays (dask#1101) Co-authored-by: Patrick Hoefler <61934744+phofl@users.noreply.github.com> * Release for dask 2024.7.0 * Skip test if optional xarray cannot be imported (dask#1104) * Fix deepcopying FromPandas class (dask#1105) * Fix from_pandas with chunksize and empty df (dask#1106) * Link fix in readme (dask#1107) * Fix shuffle blowing up the task graph (dask#1108) Co-authored-by: Hendrik Makait <hendrik@makait.com> * Release for dask 2024.7.1 * Fix some things for pandas 3 (dask#1110) * Fixup remaining upstream failures (dask#1111) * Release for dask 2024.8.0 * Drop support for Python 3.9 (dask#1109) Co-authored-by: James Bourbeau <jrbourbeau@gmail.com> * Fix tuples as on argument in merge (dask#1117) * Fix merging when index name in meta missmatches actual name (dask#1119) Co-authored-by: Hendrik Makait <hendrik@makait.com> * Register `read_parquet` and `read_csv` as "dispatchable" (dask#1114) * Fix projection for Index class in read_parquet (dask#1120) * Fix result index of merge (dask#1121) * Introduce `ToBackend` expression (dask#1115) * Avoid calling ``array`` attribute on ``cudf.Series`` (dask#1122) * Make split_out for categorical default smarter (dask#1124) * Release for dask 2024.8.1 * Fix scalar detection of columns coming from sql (dask#1125) * Bump `pyarrow>=14.0.1` minimum versions (dask#1127) 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Remove recursion in task spec (dask#1158) * Fix value_counts with split_out != 1 (dask#1170) * Release 2024.12.0 * Use new blockwise unpack collection in array (dask#1173) * Propagate group_keys in DataFrameGroupBy (dask#1174) * Fix assign optimization when overwriting columns (dask#1176) * Remove custom read-csv stuff (dask#1178) * Fixup install paths (dask#1179) * Version 1.1.21 * Remove legacy conversion functions (dask#1177) * Remove duplicated files * Move repository * Clean up docs and imports * Clean up docs and imports --------- Co-authored-by: Hendrik Makait <hendrik@makait.com> Co-authored-by: Florian Jetter <fjetter@users.noreply.github.com> Co-authored-by: Miles <miles59923@gmail.com> Co-authored-by: Joris Van den Bossche <jorisvandenbossche@gmail.com> Co-authored-by: Richard (Rick) Zamora <rzamora217@gmail.com> Co-authored-by: Charles Blackmon-Luca <20627856+charlesbluca@users.noreply.github.com> Co-authored-by: James Bourbeau <jrbourbeau@gmail.com> Co-authored-by: alex-rakowski <alexrakowski90@gmail.com> Co-authored-by: Matthew Rocklin <mrocklin@gmail.com> Co-authored-by: Sandro <shfu29r4bu@liamekaens.com> Co-authored-by: Ben <55319792+benrutter@users.noreply.github.com> Co-authored-by: James Bourbeau <jrbourbeau@users.noreply.github.com> Co-authored-by: Guillaume Eynard-Bontemps <g.eynard.bontemps@gmail.com> Co-authored-by: Tom Augspurger <tom.augspurger88@gmail.com>


It is still under work. I'm going to built
scipy.linalg.lucompat API using blocked gaussian elimination.scipy results