From 4762a088562adcbbd3e6b1d9e78d1439c3515fff Mon Sep 17 00:00:00 2001 From: Felix Dangel Date: Wed, 8 Jan 2025 12:58:13 -0500 Subject: [PATCH] [ADD] Diagonal estimation to example --- changelog.md | 2 +- .../basic_usage/diagonal_estimation.pdf | Bin 0 -> 182714 bytes .../example_trace_diagonal_estimation.py | 364 +++++++++++------- .../examples/basic_usage/trace_estimation.pdf | Bin 212469 -> 215486 bytes 4 files changed, 235 insertions(+), 131 deletions(-) create mode 100644 docs/examples/basic_usage/diagonal_estimation.pdf diff --git a/changelog.md b/changelog.md index f4e5a6a..2d064ec 100644 --- a/changelog.md +++ b/changelog.md @@ -31,7 +31,7 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0 - Add a [use case example](https://curvlinops.readthedocs.io/en/latest/basic_usage/example_trace_diagonal_estimation.html) - on trace estimation + on trace and diagonal estimation ([PR](https://github.com/f-dangel/curvlinops/pull/165)) - **Backward-incompatible:** Make linear operators purely PyTorch with a SciPy diff --git a/docs/examples/basic_usage/diagonal_estimation.pdf b/docs/examples/basic_usage/diagonal_estimation.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b9601301b914cde967fbc912fbb88a022fe204c6 GIT binary patch literal 182714 zcmagEV|1k57A+jx=s2mMW81dTQOCAzcF?hHTOHfBZKIQpeXHO1obNkh+#h$;*tMUv z_F8+*HEY)>%;&43s5l)HJuBST$_+q88ypjW5nyL%3CGI|VEE=?ZvtQtF>p4pwlf1T zC>ofVI02Y}0m=Y=emE0b|^cLRTM|3&X;0z~&Od>D>4Ef z%q)!bOl(|i%m8LiR(ej3zgIdyE`SXJcX0x6{R>mb*47SaG5t;XUsC=q|CdEs6I(N9 za{$wSDvMiK1N#Orh+6~65H&HfGdB4P-^tn0#J~p5J#$ifJeokladW$-)Wj*zu!{() zBbN6|6le!_xIdzM2jDwC5~MVKLT23iBR@sHt%>oLY0!dV2qel0(jqAlDusMz!{md)wqADh_SLrD+XLxG=h_DF`Mv)666d3{jq~kc{;O#7>xNU0o!x|8 zUQws&#~qaV+V+Qc>)Yv{J1gEjzAF#c*5EF;6rJ#lX`A1959!awD_I*x6LyhY3$?@9 zT2ibz$Y~RQ>Oa@2KRFhxx6#^fGk*I#vK48IlxI9TC=@k?y!Jm_9X{BtPp{9{fQx(e zZUjyj-7sbJn+*G5c7IFJf@cct>h$_?zx$nUo!@iVq;=mh9Q&IstO!ie1#!y?k6Lf4 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a/docs/examples/basic_usage/example_trace_diagonal_estimation.py b/docs/examples/basic_usage/example_trace_diagonal_estimation.py index 5cda8a0..63a10c0 100644 --- a/docs/examples/basic_usage/example_trace_diagonal_estimation.py +++ b/docs/examples/basic_usage/example_trace_diagonal_estimation.py @@ -2,18 +2,15 @@ Trace and diagonal estimation ============================= -In this example we will use Hutchinson's method to estimate the trace of a matrix. -We will also compare Hutchinson's method with Hutch++ from -`Meyer et al., 2020 `_. -The tutorial mostly follows their experimental setup. - -TODO Add diagonal estimation +In this example we will explore estimators for the trace and diagonal of a matrix. +:code:`curvlinops` implements different methods, and we will reproduce the results +from their original papers, using toy matrices with a power law spectrum. Here are the imports: """ from os import getenv -from typing import Dict, Tuple +from typing import Dict import matplotlib.pyplot as plt from numpy import ( @@ -21,8 +18,7 @@ array, diag, float64, - logspace, - mean, + linspace, median, ndarray, percentile, @@ -34,8 +30,7 @@ from scipy.sparse.linalg import aslinearoperator from tueplots import bundles -from curvlinops.trace.hutchinson import HutchinsonTraceEstimator -from curvlinops.trace.meyer2020hutch import HutchPPTraceEstimator +from curvlinops import hutchinson_diag, hutchinson_trace, hutchpp_trace, xdiag, xtrace # LaTeX is not available on RTD and we also want to analyze smaller matrices # to reduce build time @@ -46,9 +41,9 @@ ) # Dimension of the matrices whose traces we will estimate -DIM = 800 if RTD else 2000 +DIM = 200 if RTD else 1000 # Number of repeats for the Hutchinson estimator to compute error bars -NUM_REPEATS = 15 if RTD else 100 +NUM_REPEATS = 50 if RTD else 200 seed(0) # make deterministic @@ -111,25 +106,19 @@ def create_power_law_matrix(dim: int = DIM, c: float = 1.0) -> ndarray: # %% # -# Now, let's apply Hutchinson's method to estimate the trace. +# The simplest method for trace estimation is Hutchinson's method. # -# This method generates estimates from matrix-vector products with random vectors. -# To obtain better estimates we can draw multiple samples and average them. -# We usually want to repeat this process multiple times to get error estimates. +# The idea is to estimate the trace from matrix-vector products with random vectors. +# To obtain better estimates, we can use more queries. +# It is common to repeat this process multiple times to get error estimates. # # Let's estimate the trace and see if the estimate is decent: -samples_per_estimate = 5 - -# Initialize the Hutchinson trace estimator -estimator = HutchinsonTraceEstimator(Y) +# matrix-vector queries for one trace estimate +num_matvecs = 5 -# Generate estimates by averaging samples, repeat this process multiple times -# so we have error bars. -estimates = [] -for _ in range(NUM_REPEATS): - samples = [estimator.sample() for _ in range(samples_per_estimate)] - estimates.append(mean(samples)) +# Generate estimates, repeat process multiple times so we have error bars. +estimates = [hutchinson_trace(Y, num_matvecs) for _ in range(NUM_REPEATS)] # Calculate the median and quartiles (error bars) of the estimates med = median(estimates) @@ -152,107 +141,82 @@ def create_power_law_matrix(dim: int = DIM, c: float = 1.0) -> ndarray: # # Good! The estimate lies within the error bars. # -# In the following, we will look at Hutchinson's method and Hutch++, a variance-reduced -# version of Hutchinson's method. This method deterministically computes the trace in a -# sub-space whose dimension we can specify, and uses Hutchinson's method to estimate the -# trace in the remaining space, which reduces the variance at the cost of storing the -# basis. +# Comparison +# ^^^^^^^^^^ +# +# In the following, we will look at Hutchinson's method and two other algorithms: +# Hutch++ and XTrace. Hutch++ combines vanilla Hutchinson with variance reduction, +# by deterministically computing the trace in a sub-space, and using Hutchinson's +# method in the remaining space. XTrace uses variance reduction from Hutch++, +# and the exchangeability principle (i.e. the estimate is identical when permuting +# the random test vectors). All methods are unbiased, but Hutch++ and XTrace require +# additional memory to store the basis in which the trace is computed exactly. # -# We will analyze the quality of both methods on power law matrices of different decay -# rates. +# For matrices whose trace is dominated by a few large eigenvalues, i.e. have fast +# spectral decay, Hutch++ and XTrace can converge faster than vanilla Hutchinson. +# For matrices with slow spectral decay, the benefits of Hutch++ and XTrace become +# less pronounced. # -# Hutchinson versus Hutch++ -# ^^^^^^^^^^^^^^^^^^^^^^^^^ +# Let's reproduce these results empirically. # -# Generally speaking, Hutch++ yields better results when a matrix's spectrum decays -# rapidly, i.e. the trace is dominated by a small number of large eigenvalues. To -# reproduce this behavior, we will first consider a power law matrix with high decay -# rate :math:`c=2.0`: +# We will first consider a power law matrix with high decay rate :math:`c=2.0`: Y_mat = create_power_law_matrix(c=2.0) # %% # -# We will compare three methods: +# As before, we will repeat the trace estimation to obtain error bars for each method, +# and investigate how their accuracy evolves as we increase the number of matrix-vector +# products. We use the relative error, which is the absolute value of the difference +# between the estimated and exact trace, divided by the exact trace's absolute value. # -# - Vanilla Hutchinson -# - Hutch++ with basis dimension 10 -# - Hutch++ with basis dimension 100 -# -# Our goal is to look at the trace estimator's quality as a function of matrix-vector -# products. We will use the relative trace error, i.e. the absolute difference between -# the estimate and the true trace, normalized by the true trace. Here is a function that -# computes these relative trace errors for a given matrix: +# Here is a function that computes these relative trace errors for a given matrix: -NUM_MATVECS = unique(logspace(0, 3, 40, dtype=int)) +NUM_MATVECS_HUTCH = unique(linspace(1, 100, 50, dtype=int)) +# Hutch++ requires matrix-vector products divisible by 3 +NUM_MATVECS_HUTCHPP = unique(NUM_MATVECS_HUTCH + (3 - divmod(NUM_MATVECS_HUTCH, 3)[1])) +# XTrace requires matrix-vector products divisible by 2 +NUM_MATVECS_XTRACE = unique(NUM_MATVECS_HUTCH + (2 - divmod(NUM_MATVECS_HUTCH, 2)[1])) -def compute_relative_trace_errors( - Y_mat: ndarray, - basis_dims: Tuple[int, ...] = (10, 100), - num_matvecs: ndarray = NUM_MATVECS, -) -> Dict[str, Dict[str, ndarray]]: - """Compute the relative trace errors for Hutchinson's method and Hutch++. +def compute_relative_trace_errors(Y_mat: ndarray) -> Dict[str, Dict[str, ndarray]]: + """Compute the relative trace errors for Hutchinson's method, Hutch++, and XTrace. Args: Y_mat: Matrix to estimate the trace of. - basis_dims: Dimensions of the basis for Hutch++. - num_matvecs: Number of matrix-vector products to use. Returns: - Dictionary with the relative trace errors for Hutchinson's method and Hutch++. + Dictionary with the relative trace errors. """ Y = aslinearoperator(Y_mat) exact_trace = trace(Y_mat) # compute median and quartiles for Hutchinson's method - num_samples = list(num_matvecs) - errors = [] - - for _ in range(NUM_REPEATS): - estimator = HutchinsonTraceEstimator(Y) - samples = [estimator.sample() for _ in range(max(num_samples))] - estimates = [mean(samples[:n]) for n in num_samples] - errors.append([abs(est - exact_trace) / abs(exact_trace) for est in estimates]) - - errors = array(errors) - med = median(errors, axis=0) - quartile1 = percentile(errors, 25, axis=0) - quartile3 = percentile(errors, 75, axis=0) - - results = { - "hutch": { - "med": med, - "quartile1": quartile1, - "quartile3": quartile3, - "num_matvecs": num_matvecs, - } + estimators = { + "Hutchinson": hutchinson_trace, + "Hutch++": hutchpp_trace, + "XTrace": xtrace, } - # compute median and quartiles for Hutch++ with different basis dimensions - for basis_dim in basis_dims: - # Hutch++ spends basis_dim matvecs to build the basis, and another - # basis_dim matvecs to compute the exact trace in that sub-space. - n_matvecs = [n for n in num_matvecs if n > 2 * basis_dim] - num_samples = [n - 2 * basis_dim for n in n_matvecs] - errors = [] - - for _ in range(NUM_REPEATS): - estimator = HutchPPTraceEstimator(Y, basis_dim=basis_dim) - samples = [estimator.sample() for _ in range(max(num_samples))] - estimates = [mean(samples[:n]) for n in num_samples] - errors.append( - [abs(est - exact_trace) / abs(exact_trace) for est in estimates] - ) - - errors = array(errors) - med = median(errors, axis=0) - quartile1 = percentile(errors, 25, axis=0) - quartile3 = percentile(errors, 75, axis=0) - results[f"hutch++_{basis_dim}"] = { - "med": med, - "quartile1": quartile1, - "quartile3": quartile3, - "num_matvecs": n_matvecs, + num_matvecs = [NUM_MATVECS_HUTCH, NUM_MATVECS_HUTCHPP, NUM_MATVECS_XTRACE] + + results = {} + for (name, method), num_matvecs_method in zip(estimators.items(), num_matvecs): + med = [] + quartile1 = [] + quartile3 = [] + + for n in num_matvecs_method: + estimates = [method(Y, n) for _ in range(NUM_REPEATS)] + errors = [abs(e - exact_trace) / abs(exact_trace) for e in estimates] + med.append(median(errors)) + quartile1.append(percentile(errors, 25)) + quartile3.append(percentile(errors, 75)) + + results[name] = { + "med": array(med), + "quartile1": array(quartile1), + "quartile3": array(quartile3), + "num_matvecs": num_matvecs_method, } return results @@ -277,8 +241,8 @@ def compute_relative_trace_errors( # print the first 3 values for i in range(3): print( - f"\t\t- {num_matvecs[i]} matvecs: {med[i]:.5f}" - + f" ({quartile1[i]:.3f} - {quartile3[i]:.3f})" + f"\t\t- {num_matvecs[i]} matvecs: median {med[i]:.5f}" + + f" (quartiles {quartile1[i]:.3f} - {quartile3[i]:.3f})" ) # %% @@ -290,38 +254,29 @@ def compute_relative_trace_errors( def plot_estimation_results( - results: Dict[str, Dict[str, ndarray]], - ax: plt.Axes, - target: str = "trace", + results: Dict[str, Dict[str, ndarray]], ax: plt.Axes, target: str = "trace" ) -> None: """Plot the trace estimation results on the given Axes. Args: - results: Dictionary with the relative trace errors for Hutchinson's method - and Hutch++. + results: Dictionary with the relative trace errors. ax: The matplotlib Axes to plot on. target: The property that is approximated (used in ylabel). Default is ``'trace'``. """ - ax.set_xscale("log") ax.set_yscale("log") - for method, data in results.items(): + for name, data in results.items(): num_matvecs = data["num_matvecs"] med = data["med"] quartile1 = data["quartile1"] quartile3 = data["quartile3"] - if method == "hutch": - label = "Hutchinson" - else: - label = f"Hutch++ ({method.split('_')[1]}d basis)" - - ax.plot(num_matvecs, med, label=label) + ax.plot(num_matvecs, med, label=name) ax.fill_between(num_matvecs, quartile1, quartile3, alpha=0.3) ax.set_xlabel("Matrix-vector products") - ax.set_ylabel("Relative trace error") + ax.set_ylabel(f"Relative {target} error") ax.legend() @@ -345,20 +300,169 @@ def plot_estimation_results( axes[0].set_title("Fast spectral decay ($c=2$)") axes[1].set_title("Slow spectral decay ($c=0.5$)") - # Remove xlabel and legend from the first plot + # Remove xlabel from the first, and legend from the second, plot axes[0].set_xlabel(None) - axes[0].legend().remove() + axes[1].legend().remove() plt.savefig("trace_estimation.pdf", bbox_inches="tight") # %% # -# For fast spectral decay, Hutch++ yields more accurate trace estimates compared to -# vanilla Hutchinson. Using a larger basis further improves the results, but at the -# cost of storing a larger basis. For slow spectral decay, the benefits of Hutch++ -# are much less pronounced. They will completely disappear if the matrix's spectrum -# is completely flat, i.e., :math:`c=0``. Thankfully, many curvature matrices in -# deep learning exhibit a decaying spectrum, which may allow Hutch++ to improve -# over Hutchinson. +# For fast spectral decay, Hutch++ and XTrace yield more accurate trace estimates than +# vanilla Hutchinson. For slow spectral decay, the benefits of Hutch++ and XTrace +# disappear. Thankfully, many curvature matrices in deep learning exhibit a decaying +# spectrum, which may allow Hutch++ and XTrace to improve over Hutchinson. +# +# Diagonal estimation +# ------------------- +# +# Basics +# ^^^^^^ +# +# Diagonal estimation is similar to trace estimation. +# +# To give a concrete example, let's create a power law matrix, turn it into a linear +# operator, and compute its diagonal for reference: + +Y_mat = create_power_law_matrix() +Y = aslinearoperator(Y_mat) +exact_diag = diag(Y_mat) + +# %% +# +# The diagonal is a vector, which makes comparing the estimates by printing their +# entries tedious. Therefore, we will use the relative :math:`L_\infty` error, which is +# the maximum entry of the absolute difference between the estimated and exact diagonal +# entries, divided by the maximum absolute entry of the exact diagonal. + + +def relative_l_inf_error(est: ndarray, exact: ndarray) -> float: + """Compute the relative L-infinity error between two vectors. + + Args: + est: Estimated vector. + exact: Exact vector. + + Returns: + Relative L-infinity error. + """ + return max(abs(est - exact)) / max(abs(exact)) + + +# %% +# +# The simplest method for diagonal estimation is Hutchinson's method. +# +# The idea is to estimate the diagonal from matrix-vector products with random vectors. +# To obtain better estimates, we can use more queries. +# It is common to repeat this process multiple times to get error estimates. +# +# Let's estimate the diagonal and see if the estimate is decent: + +# matrix-vector queries for one diagonal estimate +num_matvecs = 5 + +# Generate estimates, repeat process multiple times so we have error bars. +estimates = [hutchinson_diag(Y, num_matvecs) for _ in range(NUM_REPEATS)] +errors = [relative_l_inf_error(e, exact_diag) for e in estimates] + +# Calculate the median and quartiles (error bars) of the estimates +med = median(errors) +quartile1 = percentile(errors, 25) +quartile3 = percentile(errors, 75) + +# Print the exact trace and the statistical measures of the estimates +print("Relative errors:") +print(f"\t- Median: {med:.3f}") +print(f"\t- First quartile (25%): {quartile1:.3f}") +print(f"\t- Third quartile (75%): {quartile3:.3f}") + +# %% +# +# Comparison +# ^^^^^^^^^^ +# +# We will compare Hutchinson's method for diagonal estimation with the XDiag method +# on a matrices with fast and slow spectral decay. +# +# Here is a function that computes these relative diagonal errors for a given matrix: + +NUM_MATVECS_HUTCH = unique(linspace(1, 100, 50, dtype=int)) +# XTrace requires matrix-vector products divisible by 2 +NUM_MATVECS_XDIAG = unique(NUM_MATVECS_HUTCH + (2 - divmod(NUM_MATVECS_HUTCH, 2)[1])) + + +def compute_relative_diagonal_errors(Y_mat: ndarray) -> Dict[str, Dict[str, ndarray]]: + """Compute the relative diagonal errors for Hutchinson's method and XDiag. + + Args: + Y_mat: Matrix to estimate the diagonal of. + + Returns: + Dictionary with the relative diagonal errors. + """ + Y = aslinearoperator(Y_mat) + exact_diag = diag(Y_mat) + + # compute median and quartiles for Hutchinson's method + estimators = { + "Hutchinson": hutchinson_diag, + "XDiag": xdiag, + } + num_matvecs = [NUM_MATVECS_HUTCH, NUM_MATVECS_XDIAG] + + results = {} + for (name, method), num_matvecs_method in zip(estimators.items(), num_matvecs): + med = [] + quartile1 = [] + quartile3 = [] + + for n in num_matvecs_method: + estimates = [method(Y, n) for _ in range(NUM_REPEATS)] + errors = [relative_l_inf_error(e, exact_diag) for e in estimates] + med.append(median(errors)) + quartile1.append(percentile(errors, 25)) + quartile3.append(percentile(errors, 75)) + + results[name] = { + "med": array(med), + "quartile1": array(quartile1), + "quartile3": array(quartile3), + "num_matvecs": num_matvecs_method, + } + + return results + + +# %% +# +# For plotting, we can re-purpose the function we used earlier to visualize the trace +# estimation results: + +# Compute results for matrices with different spectral decay rates +Y_mat_fast = create_power_law_matrix() # Fast spectral decay with c=2 +results_fast = compute_relative_diagonal_errors(Y_mat_fast) + +Y_mat_slow = create_power_law_matrix(c=0.5) # Slow spectral decay with c=0.5 +results_slow = compute_relative_diagonal_errors(Y_mat_slow) + +# Plot the results for both fast and slow spectral decay +with plt.rc_context(PLOT_CONFIG): + fig, axes = plt.subplots(nrows=2, sharex=True) + plot_estimation_results(results_fast, axes[0], target="diagonal") + plot_estimation_results(results_slow, axes[1], target="diagonal") + axes[0].set_title("Fast spectral decay ($c=2$)") + axes[1].set_title("Slow spectral decay ($c=0.5$)") + + # Remove xlabel from the first, and legend from the second, plot + axes[0].set_xlabel(None) + axes[1].legend().remove() + + plt.savefig("diagonal_estimation.pdf", bbox_inches="tight") + +# %% +# +# For fast spectral decay, XDiag yields more accurate diagonal estimates than +# vanilla Hutchinson. 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