import matplotlib
if not hasattr(matplotlib.RcParams, "_get"):
matplotlib.RcParams._get = dict.get
Modul 4: Teori#
Pyodide-bemærkning (Python direkte i Browseren)#
Koden i denne Notebook er beregnet til at blive kørt lokalt på din egen computer og ikke direkte i browseren via Pyodide. Årsagen er, at vi bruger scikit-learn til at downloade hele MNIST-datasættet. Dette er en data-tung operation, som involverer download af en stor fil, hvilket ikke umiddelbart understøttes eller er praktisk i et browser-baseret miljø som Pyodide.
Digitale billeder#
import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import fetch_openml
# Hent MNIST (784 = 28x28 pixels)
mnist = fetch_openml('mnist_784', version=1, as_frame=False)
X, y = mnist.data, mnist.target.astype('int64')
---------------------------------------------------------------------------
KeyboardInterrupt Traceback (most recent call last)
Cell In[1], line 5
1 import matplotlib.pyplot as plt
2 import numpy as np
3 from sklearn.datasets import fetch_openml
4 # Hent MNIST (784 = 28x28 pixels)
----> 5 mnist = fetch_openml('mnist_784', version=1, as_frame=False)
6 X, y = mnist.data, mnist.target.astype('int64')
File /usr/local/lib/python3.11/site-packages/sklearn/utils/_param_validation.py:218, in validate_params.<locals>.decorator.<locals>.wrapper(*args, **kwargs)
212 try:
213 with config_context(
214 skip_parameter_validation=(
215 prefer_skip_nested_validation or global_skip_validation
216 )
217 ):
--> 218 return func(*args, **kwargs)
219 except InvalidParameterError as e:
220 # When the function is just a wrapper around an estimator, we allow
221 # the function to delegate validation to the estimator, but we replace
222 # the name of the estimator by the name of the function in the error
223 # message to avoid confusion.
224 msg = re.sub(
225 r"parameter of \w+ must be",
226 f"parameter of {func.__qualname__} must be",
227 str(e),
228 )
File /usr/local/lib/python3.11/site-packages/sklearn/datasets/_openml.py:1135, in fetch_openml(name, version, data_id, data_home, target_column, cache, return_X_y, as_frame, n_retries, delay, parser, read_csv_kwargs)
1133 # obtain the data
1134 url = data_description["url"]
-> 1135 bunch = _download_data_to_bunch(
1136 url,
1137 return_sparse,
1138 data_home,
1139 as_frame=bool(as_frame),
1140 openml_columns_info=features_list,
1141 shape=shape,
1142 target_columns=target_columns,
1143 data_columns=data_columns,
1144 md5_checksum=data_description["md5_checksum"],
1145 n_retries=n_retries,
1146 delay=delay,
1147 parser=parser_,
1148 read_csv_kwargs=read_csv_kwargs,
1149 )
1151 if return_X_y:
1152 return bunch.data, bunch.target
File /usr/local/lib/python3.11/site-packages/sklearn/datasets/_openml.py:689, in _download_data_to_bunch(url, sparse, data_home, as_frame, openml_columns_info, data_columns, target_columns, shape, md5_checksum, n_retries, delay, parser, read_csv_kwargs)
685 from pandas.errors import ParserError
687 no_retry_exception = ParserError
--> 689 X, y, frame, categories = _retry_with_clean_cache(
690 url, data_home, no_retry_exception
691 )(_load_arff_response)(
692 url,
693 data_home,
694 parser=parser,
695 output_type=output_type,
696 openml_columns_info=features_dict,
697 feature_names_to_select=data_columns,
698 target_names_to_select=target_columns,
699 shape=shape,
700 md5_checksum=md5_checksum,
701 n_retries=n_retries,
702 delay=delay,
703 read_csv_kwargs=read_csv_kwargs,
704 )
706 return Bunch(
707 data=X,
708 target=y,
(...) 712 target_names=target_columns,
713 )
File /usr/local/lib/python3.11/site-packages/sklearn/datasets/_openml.py:66, in _retry_with_clean_cache.<locals>.decorator.<locals>.wrapper(*args, **kw)
64 return f(*args, **kw)
65 try:
---> 66 return f(*args, **kw)
67 except URLError:
68 raise
File /usr/local/lib/python3.11/site-packages/sklearn/datasets/_openml.py:554, in _load_arff_response(url, data_home, parser, output_type, openml_columns_info, feature_names_to_select, target_names_to_select, shape, md5_checksum, n_retries, delay, read_csv_kwargs)
544 arff_params: Dict = dict(
545 parser=parser,
546 output_type=output_type,
(...) 551 read_csv_kwargs=read_csv_kwargs or {},
552 )
553 try:
--> 554 X, y, frame, categories = _open_url_and_load_gzip_file(
555 url, data_home, n_retries, delay, arff_params
556 )
557 except Exception as exc:
558 if parser != "pandas":
File /usr/local/lib/python3.11/site-packages/sklearn/datasets/_openml.py:542, in _load_arff_response.<locals>._open_url_and_load_gzip_file(url, data_home, n_retries, delay, arff_params)
540 gzip_file = _open_openml_url(url, data_home, n_retries=n_retries, delay=delay)
541 with closing(gzip_file):
--> 542 return load_arff_from_gzip_file(gzip_file, **arff_params)
File /usr/local/lib/python3.11/site-packages/sklearn/datasets/_arff_parser.py:533, in load_arff_from_gzip_file(gzip_file, parser, output_type, openml_columns_info, feature_names_to_select, target_names_to_select, shape, read_csv_kwargs)
524 return _liac_arff_parser(
525 gzip_file,
526 output_type,
(...) 530 shape,
531 )
532 elif parser == "pandas":
--> 533 return _pandas_arff_parser(
534 gzip_file,
535 output_type,
536 openml_columns_info,
537 feature_names_to_select,
538 target_names_to_select,
539 read_csv_kwargs,
540 )
541 else:
542 raise ValueError(
543 f"Unknown parser: '{parser}'. Should be 'liac-arff' or 'pandas'."
544 )
File /usr/local/lib/python3.11/site-packages/sklearn/datasets/_arff_parser.py:404, in _pandas_arff_parser(gzip_file, output_arrays_type, openml_columns_info, feature_names_to_select, target_names_to_select, read_csv_kwargs)
392 default_read_csv_kwargs = {
393 "header": None,
394 "index_col": False, # always force pandas to not use the first column as index
(...) 401 "dtype": dtypes_positional,
402 }
403 read_csv_kwargs = {**default_read_csv_kwargs, **(read_csv_kwargs or {})}
--> 404 frame = pd.read_csv(gzip_file, **read_csv_kwargs)
405 try:
406 # Setting the columns while reading the file will select the N first columns
407 # and not raise a ParserError. Instead, we set the columns after reading the
408 # file and raise a ParserError if the number of columns does not match the
409 # number of columns in the metadata given by OpenML.
410 frame.columns = [name for name in openml_columns_info]
File /usr/local/lib/python3.11/site-packages/pandas/io/parsers/readers.py:873, in read_csv(filepath_or_buffer, sep, delimiter, header, names, index_col, usecols, dtype, engine, converters, true_values, false_values, skipinitialspace, skiprows, skipfooter, nrows, na_values, keep_default_na, na_filter, skip_blank_lines, parse_dates, date_format, dayfirst, cache_dates, iterator, chunksize, compression, thousands, decimal, lineterminator, quotechar, quoting, doublequote, escapechar, comment, encoding, encoding_errors, dialect, on_bad_lines, low_memory, memory_map, float_precision, storage_options, dtype_backend)
861 kwds_defaults = _refine_defaults_read(
862 dialect,
863 delimiter,
(...) 869 dtype_backend=dtype_backend,
870 )
871 kwds.update(kwds_defaults)
--> 873 return _read(filepath_or_buffer, kwds)
File /usr/local/lib/python3.11/site-packages/pandas/io/parsers/readers.py:306, in _read(filepath_or_buffer, kwds)
303 return parser
305 with parser:
--> 306 return parser.read(nrows)
File /usr/local/lib/python3.11/site-packages/pandas/io/parsers/readers.py:1947, in TextFileReader.read(self, nrows)
1940 nrows = validate_integer("nrows", nrows)
1941 try:
1942 # error: "ParserBase" has no attribute "read"
1943 (
1944 index,
1945 columns,
1946 col_dict,
-> 1947 ) = self._engine.read( # type: ignore[attr-defined]
1948 nrows
1949 )
1950 except Exception:
1951 self.close()
File /usr/local/lib/python3.11/site-packages/pandas/io/parsers/c_parser_wrapper.py:215, in CParserWrapper.read(self, nrows)
213 try:
214 if self.low_memory:
--> 215 chunks = self._reader.read_low_memory(nrows)
216 # destructive to chunks
217 data = _concatenate_chunks(chunks, self.names)
File pandas/_libs/parsers.pyx:832, in pandas._libs.parsers.TextReader.read_low_memory()
--> 832 'Could not get source, probably due dynamically evaluated source code.'
File pandas/_libs/parsers.pyx:911, in pandas._libs.parsers.TextReader._read_rows()
--> 911 'Could not get source, probably due dynamically evaluated source code.'
File pandas/_libs/parsers.pyx:1009, in pandas._libs.parsers.TextReader._convert_column_data()
-> 1009 'Could not get source, probably due dynamically evaluated source code.'
File pandas/_libs/parsers.pyx:1048, in pandas._libs.parsers.TextReader._convert_tokens()
-> 1048 'Could not get source, probably due dynamically evaluated source code.'
File pandas/_libs/parsers.pyx:1145, in pandas._libs.parsers.TextReader._convert_with_dtype()
-> 1145 'Could not get source, probably due dynamically evaluated source code.'
File /usr/local/lib/python3.11/site-packages/pandas/core/dtypes/dtypes.py:531, in CategoricalDtype.construct_array_type(self)
528 combined_hashed = combine_hash_arrays(iter(cat_array), num_items=len(cat_array))
529 return np.bitwise_xor.reduce(combined_hashed)
--> 531 def construct_array_type(self) -> type_t[Categorical]:
532 """
533 Return the array type associated with this dtype.
534
(...) 537 type
538 """
539 from pandas import Categorical
KeyboardInterrupt:
plt.figure(figsize=(12, 12))
plt.subplot(221)
plt.imshow(X[0].reshape(28, 28), cmap='gray')
plt.title(f"Digit: {y[0]}")
plt.subplot(222)
plt.imshow(X[1].reshape(28, 28), cmap='gray')
plt.title(f"Digit: {y[1]}")
plt.subplot(223)
plt.imshow(X[2].reshape(28, 28), cmap='gray')
plt.title(f"Digit: {y[2]}")
plt.subplot(224)
plt.imshow(X[3].reshape(28, 28), cmap='gray')
plt.title(f"Digit: {y[3]}")
# vis grafen
plt.show()
Vektorfunktioner#
Lad \(d,k \in \mathbb{N}\). En vektorfunktion af flere variable er en funktion af formen
\[\begin{equation*}
\pmb{f} \colon \operatorname{dom}(\pmb{f}) \to \mathbb{R}^k, \text{\; hvor }\operatorname{dom}(\pmb{f}) \subseteq \mathbb{R}^d.
\end{equation*}\]
Altså har en vektorfunktion \(\pmb{f} = \pmb{x} \mapsto \pmb{f}(\pmb{x})\):
Input (domænet): vektorer \(\pmb{x}\) i \(\mathbb{R}^d\)
Output (kodomænet): vektorer \(\pmb{f}(\pmb{x})\) i \(\mathbb{R}^k\)
Input \(\pmb{x}_0\)#
# Vis det tredje billede fra træningssættet
x0 = X[2].reshape(28, 28)
plt.figure(figsize=(7, 7))
plt.imshow(x0, cmap='gray')
plt.axis('off')
plt.show()
Output \(\pmb{f}(\pmb{x}_0)\)#
Output:
"Dette er cifret 4"
Ideelt output:
\[\begin{equation*}
\pmb{f}(\pmb{x}_0) = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}
\end{equation*}\]
Dette betyder: "100% sikker på, at dette er cifret 4"
Mere realistisk output:
\[\begin{equation*}
\pmb{f}(\pmb{x}_0) = \begin{bmatrix} 0.01 \\ 0.03 \\ 0.01 \\ 0.01 \\ 0.87 \\ 0.01 \\ 0.01 \\ 0.03 \\ 0.01 \\ 0.01 \end{bmatrix}
\end{equation*}\]
Dette betyder: "87% sikker på, at dette er cifret 4. Lille sandsynlighed (3%) for, at det er 1 eller 7."
Under alle omstændigheder: Outputtet er en vektor af sandsynligheder i \(\mathbb{R}^{10}\). Derfor er \(\operatorname{co\text{-}dom}(\pmb{f})=\mathbb{R}^{10}\).
Hvad med inputtet?#
Det er faktisk en matrix på størrelse 28x28:
\[\begin{split}
%\tiny
\scriptsize
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 67 & 232 & 39 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 62 & 81 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 120 & 180 & 39 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 126 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 153 & 210 & 40 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 220 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27 & 254 & 162 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 222 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 183 & 254 & 125 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 46 & 245 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 198 & 254 & 56 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 120 & 254 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 23 & 231 & 254 & 29 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 159 & 254 & 120 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 163 & 254 & 216 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 159 & 254 & 67 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 86 & 178 & 248 & 254 & 91 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 159 & 254 & 85 & 0 & 0 & 0 & 47 & 49 & 116 & 144 & 150 & 241 & 243 & 234 & 179 & 241 & 252 & 40 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 150 & 253 & 237 & 207 & 207 & 207 & 253 & 254 & 250 & 240 & 198 & 143 & 91 & 28 & 5 & 233 & 250 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 119 & 177 & 177 & 177 & 177 & 177 & 98 & 56 & 0 & 0 & 0 & 0 & 0 & 102 & 254 & 220 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 137 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 57 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 57 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 255 & 94 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 96 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 153 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 255 & 153 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 96 & 254 & 153 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\end{split}\]
Men det er ikke umiddelbart en (søjle)vektor!
Vi kan gøre den om til en vektor ved at stakke billedets rækker oven på hinanden. Dette kaldes flattening (udfladning) af billedet i Python.
# Udskriv som søjle med 784 tal
print("\nSøjle-repræsentation af billedet (784 tal):")
x0.reshape(784,1)
Søjle-repræsentation af billedet (784 tal):
array([[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
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[232],
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[120],
[180],
[ 39],
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[126],
[163],
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[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
[ 0],
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[ 0],
[ 0],
[ 2],
[153],
[210],
[ 40],
[ 0],
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[ 0],
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[ 0],
[220],
[163],
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[ 27],
[254],
[162],
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[222],
[163],
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[125],
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[254],
[ 91],
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[159],
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[ 85],
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[234],
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[241],
[252],
[ 40],
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[ 0],
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[150],
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[237],
[207],
[207],
[207],
[253],
[254],
[250],
[240],
[198],
[143],
[ 91],
[ 28],
[ 5],
[233],
[250],
[ 0],
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Så vi kan tænke på inputtet som vektorer i \(\mathbb{R}^{784}\)! Det vil sige
En AI funktion#
AI’en er en vektorfunktion med \(d=784\), \(k=10\).
I maskinlæring kaldes funktionen for en model, og den specifikke AI-funktion afhænger typisk af tusindvis eller millioner af parametre (kaldet vægte). For hvert sæt af parametre/vægte får vi en ny AI-funktion. Sådanne funktioner (også dybe neurale netværk) er ikke særligt komplicerede, men ofte lange at skrive eksplicit. De er bygget af
Sammensætning af simple vektorfunktioner \(\pmb{g} \circ \pmb{h}\)
De simple funktioner er normalt kun: 1. Affine vektorfunktioner \(\pmb{x} \mapsto A \pmb{x} + \pmb{b}\) (elementerne i matrixen \(A\) og vektoren \(\pmb{b}\) er parametrene/vægtene) 2. En ikke-lineær aktiveringsfunktion, fx ReLU.
Antallet af sammensætninger \(\pmb{g}_1 \circ \pmb{g}_2 \circ \pmb{g}_3 \circ \cdots \circ \pmb{g}_N\) beskriver netværkets dybde.
Neurale netværk#
Hvordan ser disse “AI-funktioner” ud?
Generel notation for et feedforward ReLU-netværk#
Et feedforward-netværk beregner outputtet ved sekventielt at føre input gennem en række lag. For hvert lag \(\ell\) beregnes først en pre-aktivering (også kaldet logits), \(z^{(\ell)}\), efterfulgt af en aktivering (eller hidden state), \(h^{(\ell)}\).
\[\begin{equation*}
\begin{aligned}
&h^{(0)} = x \in \mathbb{R}^{n_0} &&\text{(Input)}\\
&z^{(\ell)} = W_\ell h^{(\ell-1)} + b_\ell, && \ell = 1,2,\dots,L &&\text{(Logits)}\\
&h^{(\ell)} =
\begin{cases}
\sigma (z^{(\ell)}), & \ell < L, \\
z^{(\ell)}, & \ell = L,
\end{cases}
&&\text{(Activation)}
\end{aligned}
\end{equation*}\]
hvor
\(h^{(0)}\) er input-vektoren \(x\).
\(W_\ell \in \mathbb{R}^{n_\ell \times n_{\ell-1}}\) er vægtmatricen for lag \(\ell\).
\(b_\ell \in \mathbb{R}^{n_\ell}\) er bias-vektoren.
\(\sigma:\mathbb{R}\to\mathbb{R}\) er en ikke-lineær aktiveringsfunktion (anvendt koordinatvis) typisk ReLU:
\[\begin{equation*}
\sigma(z) = \max(0,z)
\quad\text{(ReLU)}
\end{equation*}\]
\(n_0=d\) er inputdimensionen, \(n_L=k\) outputdimensionen.
Vi siger kort at netværket er af formen \(n_0 \to n_1 \to \cdots \to n_L\).
Netværkets samlede funktion \(\Phi: \mathbb{R}^d \to \mathbb{R}^k\) giver det endelige output:
\[\begin{equation*}
\Phi(x) = z^{(L)} = W_L h^{(L-1)} + b_L,
\end{equation*}\]
idet sidste lag her er lineært (uden ReLU-aktivering).
Shallow netværk (ét skjult lag med L=2)#
For et shallow netværk \(\Phi:\mathbb{R}^2\to\mathbb{R}\) med ét skjult lag af størrelse \(n\):
\[\begin{equation*}
\Phi(x) = W_2 \, \sigma(W_1 x + b_1) + b_2,
\end{equation*}\]
hvor
\(x \in \mathbb{R}^2\),
\(W_1 \in \mathbb{R}^{n\times 2},\ b_1 \in \mathbb{R}^{n},\)
\(W_2 \in \mathbb{R}^{1\times n},\ b_2 \in \mathbb{R}.\)
Illustration af lagene:
\[\begin{equation*}
h^{(0)} \xrightarrow{W_1,b_1} z^{(1)} \xrightarrow{\sigma} h^{(1)}
\xrightarrow{W_2,b_2} z^{(2)} \xrightarrow{\sigma} h^{(2)} \xrightarrow{W_3,b_3} z^{(3)} = \Phi(x)
\end{equation*}\]
Hvert ReLU-lag opdeler rummet i lineære regioner bestemt af ligningerne \( (W_\ell h^{(\ell-1)} + b_\ell)_i = 0 \), så \(\Phi\) er en stykkevist lineær funktion på \(\mathbb{R}^d\).
Generel formel for et ReLU-netværk med \(L=3\)#
Vi betragter en funktion
\[\begin{equation*}
\Phi:\mathbb{R}^{n_0}\to\mathbb{R}^{n_3}
\end{equation*}\]
defineret som et fuldt forbundet netværk med to skjulte lag:
\[\begin{equation*}
\begin{aligned}
h^{(0)} &= x \in \mathbb{R}^{n_0}, \\[2pt]
z^{(1)} &= W_1 h^{(0)} + b_1,\\
h^{(1)} &= \sigma \bigl(z^{(1)}\bigr), \\[4pt]
z^{(2)} &= W_2 h^{(1)} + b_2,\\
h^{(2)} &= \sigma \bigl(z^{(2)}\bigr), \\[4pt]
z^{(3)} &= W_3 h^{(2)} + b_3,\\
h^{(3)} &= z^{(3)}.
\end{aligned}
\end{equation*}\]
hvor \(n_0 = d\), \(n_3 = k\), og
\(W_1 \in \mathbb{R}^{n_1\times n_0},\ b_1\in\mathbb{R}^{n_1}\)
\(W_2 \in \mathbb{R}^{n_2\times n_1},\ b_2\in\mathbb{R}^{n_2}\)
\(W_3 \in \mathbb{R}^{n_3\times n_2},\ b_3\in\mathbb{R}^{n_3}\)
Det samlede funktionsudtryk bliver:
\[\begin{equation*}
\Phi(x)
= W_3 \, \sigma \bigl(W_2 \, \sigma(W_1 x + b_1) + b_2 \bigr) + b_3.
\end{equation*}\]
Eksempel: For et konkret netværk med to inputvariabler og ét output af formen \(2 \to n_1 \to n_2 \to 1\):
\[\begin{equation*}
\Phi(x_1,x_2)
= W_3 \,\sigma \Big(W_2 \, \sigma \big(W_1
\begin{bmatrix}x_1 \\ x_2\end{bmatrix} + b_1\big) + b_2\Big) + b_3,
\end{equation*}\]
hvor dimensionerne er
\[\begin{equation*}
W_1 \in \mathbb{R}^{n_1\times2},\quad
W_2 \in \mathbb{R}^{n_2\times n_1},\quad
W_3 \in \mathbb{R}^{1\times n_2}.
\end{equation*}\]
Netværk og træning direkte i SKLearn#
Vi skal finde en “AI-funktion”
\[\begin{equation*}
\Phi : \mathbb{R}^{784} \to \mathbb{R}^{10}
\end{equation*}\]
der “bedst”-muligt kan klassificere et billede af et håndskrevet ciffer.
I koden nedenfor opbygges denne som et ReLU-netværk med to skjulte lag, hvilket giver en samlet dybde på \(L=3\). Formen af netværket er \(784 \to 256 \to 128 \to 10\). Altså er vægt-matricerne af størrelse
\[\begin{equation*}
\begin{aligned}
&W_1 \in \mathbb{R}^{256 \times 784}, \quad b_1 \in \mathbb{R}^{256}, \\
&W_2 \in \mathbb{R}^{128 \times 256}, \quad b_2 \in \mathbb{R}^{128}, \\
&W_3 \in \mathbb{R}^{10 \times 128}, \quad b_3 \in \mathbb{R}^{10}.
\end{aligned}
\end{equation*}\]
from sklearn.model_selection import train_test_split
from sklearn.neural_network import MLPClassifier
# Normaliser til [0,1]
X = X / 255.0
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=1/7, random_state=42
)
# DNN med samme “størrelse” som PyTorch-eksemplet
clf = MLPClassifier(
hidden_layer_sizes=(256, 128), # to skjulte lag
activation='relu',
solver='adam',
batch_size=64,
learning_rate_init=1e-3,
max_iter=6,
verbose=True
)
Netværkets størrelse#
Hvor mange parametre er der?
Svar#
Det samlede antal parametre for netværket er:
Lag 1: (784 input × 256 neuroner) + 256 bias = 200.704 + 256 = 200.960
Lag 2: (256 input × 128 neuroner) + 128 bias = 32.768 + 128 = 32.896
Lag 3: (128 input × 10 neuroner) + 10 bias = 1.280 + 10 = 1.290
I alt: 200.960 + 32.896 + 1.290 = 235.146
Træning via SKLearn#
Vi finder de optimale værdier for alle parametrene i \(Phi-\)funktionen ved at træne modellen med fit-metoden:
# Træn
clf.fit(X_train, y_train)
Iteration 1, loss = 0.22757620
Iteration 2, loss = 0.08920455
Iteration 3, loss = 0.05955793
Iteration 4, loss = 0.04347159
Iteration 5, loss = 0.03397271
Iteration 6, loss = 0.02918372
/home/jakle/Python/envs/intermat/lib/python3.10/site-packages/sklearn/neural_network/_multilayer_perceptron.py:781: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (6) reached and the optimization hasn't converged yet.
warnings.warn(
MLPClassifier(batch_size=64, hidden_layer_sizes=(256, 128), max_iter=6,
verbose=True)In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
Parameters
| hidden_layer_sizes | (256, ...) | |
| activation | 'relu' | |
| solver | 'adam' | |
| alpha | 0.0001 | |
| batch_size | 64 | |
| learning_rate | 'constant' | |
| learning_rate_init | 0.001 | |
| power_t | 0.5 | |
| max_iter | 6 | |
| shuffle | True | |
| random_state | None | |
| tol | 0.0001 | |
| verbose | True | |
| warm_start | False | |
| momentum | 0.9 | |
| nesterovs_momentum | True | |
| early_stopping | False | |
| validation_fraction | 0.1 | |
| beta_1 | 0.9 | |
| beta_2 | 0.999 | |
| epsilon | 1e-08 | |
| n_iter_no_change | 10 | |
| max_fun | 15000 |
Forudsigelse på et enkelt billede#
Denne funktion kan vi nu bruge på et enkelt input-billede for at få en forudsigelse. Lad os tage et enkelt billede fra vores testsæt og se, hvad modellen forudsiger.
image_index = 2 # Vælg et billede (input)
input_image = X_test[image_index:image_index+1] # Format (1, 784)
true_label = y_test[image_index]
probability_vector = clf.predict_proba(input_image) # Få sandsynlighedsvektoren
predicted_label = clf.predict(input_image) # Lav forudsigelse
print(input_image)
print(true_label)
print(probability_vector)
print(predicted_label)
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8
[[7.08580512e-05 2.61484819e-04 1.44964793e-05 1.17057312e-03
1.67150883e-05 5.34671916e-02 4.29566890e-01 8.57747096e-06
5.14443444e-01 9.79769433e-04]]
[8]
Total forudsigelse#
Den samlede procentandel af billeder i testsættet, som modellen klassificerede korrekt kan findes ved:
# Samlet evaluering
print("Test accuracy:", clf.score(X_test, y_test))
Test accuracy: 0.9778
Visualisering af forudsigelser#
For at få en bedre fornemmelse af, hvordan modellen opfører sig, kan vi visualisere dens forudsigelser på enkelte billeder fra testsættet. Nedenstående funktion plotter billedet, den korrekte label, den forudsagte label og et søjlediagram over de forudsagte sandsynligheder for hver klasse. Dette er nyttigt for at se, hvornår modellen er sikker, og hvornår den er i tvivl.
def show_images_with_mlp_probabilities(clf, X_test, y_test, X_test_orig,
num_images=5, only_incorrect=False,
image_shape=(8,8)):
# Predict full test set
pred_labels = clf.predict(X_test)
probas = clf.predict_proba(X_test)
# Select indices
all_indices = np.arange(len(X_test))
if only_incorrect:
indices = all_indices[pred_labels != y_test][:num_images]
else:
indices = all_indices[:num_images]
plt.figure(figsize=(12, 6))
for i, idx in enumerate(indices):
# --- Image plot ---
plt.subplot(2, num_images, i + 1)
img = X_test_orig[idx].reshape(image_shape)
plt.imshow(img, cmap='gray')
plt.title(f"Idx {idx}\nTrue {y_test[idx]}\nPred {pred_labels[idx]}")
plt.axis('off')
# --- Probability distribution ---
plt.subplot(2, num_images, num_images + i + 1)
p = probas[idx]
classes = np.arange(len(p))
colors = [
"red" if c == y_test[idx] else
("green" if c == pred_labels[idx] else "blue")
for c in classes
]
plt.bar(classes, p, color=colors)
plt.xticks(classes)
plt.ylim(0, 1)
plt.xlabel("Class")
plt.ylabel("Probability")
plt.tight_layout()
plt.show()
show_images_with_mlp_probabilities(
clf,
X_test,
y_test,
X_test_orig=X_test,
only_incorrect=False,
num_images=5,
image_shape=(28,28)
)
show_images_with_mlp_probabilities(
clf,
X_test,
y_test,
X_test_orig=X_test,
only_incorrect=True,
num_images=5,
image_shape=(28,28)
)
