From 34beb52478c088ed96c502e1d0ba520e66c192cd Mon Sep 17 00:00:00 2001 From: Dan Waxman Date: Thu, 26 Feb 2026 13:39:52 -0500 Subject: [PATCH 1/7] Fix differentiability of the PF implementation Previously, computing the gradient of the MLL would result in RunTime errors, as the default _inverse_cdf implementation on CPU used a `jax.pure_callback`. Since we don't expect the `inverse_cdf` function to be differentiable, anyways, we can safely wrap its inputs with a `lax.stop_gradient` call to ensure at least biased gradients are available. --- cuthbertlib/resampling/utils.py | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/cuthbertlib/resampling/utils.py b/cuthbertlib/resampling/utils.py index afbdea4..564fa37 100644 --- a/cuthbertlib/resampling/utils.py +++ b/cuthbertlib/resampling/utils.py @@ -4,7 +4,7 @@ import jax.numpy as jnp import numba as nb import numpy as np -from jax.lax import platform_dependent +from jax.lax import platform_dependent, stop_gradient from jax.scipy.special import logsumexp from jax.tree_util import tree_map @@ -32,7 +32,10 @@ def inverse_cdf(sorted_uniforms: ArrayLike, logits: ArrayLike) -> Array: """ weights = jnp.exp(logits - logsumexp(logits)) return platform_dependent( - sorted_uniforms, weights, cpu=_inverse_cdf_cpu, default=_inverse_cdf_default + sorted_uniforms, + stop_gradient(weights), + cpu=_inverse_cdf_cpu, + default=_inverse_cdf_default, ) From f7345d319e99797524ad34eb6026f3caeb187e63 Mon Sep 17 00:00:00 2001 From: Dan Waxman Date: Thu, 26 Feb 2026 13:58:07 -0500 Subject: [PATCH 2/7] Introduce a Stop Gradient Resampling Decorator This commit introduces the stop-gradient resampler of [Scibior and Wood (2021)](https://arxiv.org/abs/2106.10314) as a resample decorator. Wrapping a base resampling method in the stop_gradient_decorator results in automatic differentation estimates of the score matching the classical estimates obtained via Fisher's identity. --- cuthbertlib/resampling/__init__.py | 8 +++- cuthbertlib/resampling/stop_gradient.py | 63 +++++++++++++++++++++++++ docs/cuthbertlib_api/resampling.md | 4 ++ 3 files changed, 74 insertions(+), 1 deletion(-) create mode 100644 cuthbertlib/resampling/stop_gradient.py diff --git a/cuthbertlib/resampling/__init__.py b/cuthbertlib/resampling/__init__.py index b23d83a..a681b3b 100644 --- a/cuthbertlib/resampling/__init__.py +++ b/cuthbertlib/resampling/__init__.py @@ -1,4 +1,10 @@ -from cuthbertlib.resampling import adaptive, killing, multinomial, systematic +from cuthbertlib.resampling import ( + adaptive, + killing, + multinomial, + stop_gradient, + systematic, +) from cuthbertlib.resampling.adaptive import ess_decorator from cuthbertlib.resampling.protocols import ConditionalResampling, Resampling from cuthbertlib.resampling.utils import inverse_cdf diff --git a/cuthbertlib/resampling/stop_gradient.py b/cuthbertlib/resampling/stop_gradient.py new file mode 100644 index 0000000..8615bc9 --- /dev/null +++ b/cuthbertlib/resampling/stop_gradient.py @@ -0,0 +1,63 @@ +"""Stop gradient resampling decorator. + +The stop_gradient resampling scheme provides the classical Fisher estimates for +the score function via automatic differentiation. This can be wrapped around a +resampling scheme such as multinomial or systematic resampling. + +See [Scibior and Wood (2021)](https://arxiv.org/abs/2106.10314) for more details. +""" + +from functools import wraps + +import jax +import jax.numpy as jnp + +from cuthbertlib.resampling.protocols import Resampling +from cuthbertlib.resampling.utils import apply_resampling_indices +from cuthbertlib.smc.ess import log_ess +from cuthbertlib.types import Array, ArrayLike, ArrayTree, ArrayTreeLike + + +def stop_gradient_decorator(func: Resampling) -> Resampling: + """Wrap a Resampling function to use stop gradient resampling. + + Args: + func: A resampling function with signature + (key, logits, positions, n) -> (indices, logits_out, positions_out). + + Returns: + A Resampling function implementing stop gradient resampling. + """ + # Build a descriptive docstring that includes the wrapped function doc + wrapped_doc = func.__doc__ or "" + doc = f""" + Stop gradient resampling decorator. + + This wrapper will call the provided resampling function, and then apply + the stop gradient trick of [Scibior and Wood (2021)](https://arxiv.org/abs/2106.10314). + Resulting estimates of the score function (i.e., the gradient of the + log-likelihood with respect to model parameters) are unbiased, + corresponding to the classical Fisher estimate. + + Wrapped resampler documentation: + {wrapped_doc} + """ + + @wraps(func) + def _wrapped( + key: Array, logits: ArrayLike, positions: ArrayTreeLike, n: int + ) -> tuple[Array, Array, ArrayTree]: + idx_base, logits_base, positions_base = func( + key, jax.lax.stop_gradient(logits), positions, n + ) + + logits = jnp.asarray( + logits_base + + apply_resampling_indices(logits, idx_base) + - jax.lax.stop_gradient(apply_resampling_indices(logits, idx_base)) + ) + return idx_base, logits, positions_base + + # Attach the composed docstring and return a jitted version + _wrapped.__doc__ = doc + return jax.jit(_wrapped, static_argnames=("n",)) diff --git a/docs/cuthbertlib_api/resampling.md b/docs/cuthbertlib_api/resampling.md index 808fb90..4492e09 100644 --- a/docs/cuthbertlib_api/resampling.md +++ b/docs/cuthbertlib_api/resampling.md @@ -20,3 +20,7 @@ - conditional_resampling ::: cuthbertlib.resampling.killing + +::: cuthbertlib.resampling.adaptive + +::: cuthbertlib.resampling.stop_gradient From 96be48a0985ace3963dec9b7b1d06c01d4075af7 Mon Sep 17 00:00:00 2001 From: Dan Waxman Date: Thu, 26 Feb 2026 14:36:45 -0500 Subject: [PATCH 3/7] Add a DPF example to the documentation Adds an example to the documentatio showing the score estimates of a SG-DPF vs. a non-differentiable PF on a linear-Gaussian task. --- cuthbertlib/resampling/README.md | 18 +- docs/assets/pf_grad_bias_final_particles.png | Bin 0 -> 34935 bytes docs/assets/pf_grad_bias_mll_scores.png | Bin 0 -> 364092 bytes docs/assets/pf_grad_bias_score_boxplots.png | Bin 0 -> 72547 bytes docs/assets/pf_grad_bias_simulated_data.png | Bin 0 -> 96410 bytes docs/examples/diff_pf_resampling.md | 579 +++++++++++++++++++ docs/examples/index.md | 1 + mkdocs.yml | 1 + 8 files changed, 598 insertions(+), 1 deletion(-) create mode 100644 docs/assets/pf_grad_bias_final_particles.png create mode 100644 docs/assets/pf_grad_bias_mll_scores.png create mode 100644 docs/assets/pf_grad_bias_score_boxplots.png create mode 100644 docs/assets/pf_grad_bias_simulated_data.png create mode 100644 docs/examples/diff_pf_resampling.md diff --git a/cuthbertlib/resampling/README.md b/cuthbertlib/resampling/README.md index b532744..439d7c9 100644 --- a/cuthbertlib/resampling/README.md +++ b/cuthbertlib/resampling/README.md @@ -34,4 +34,20 @@ adaptive_resampling = resampling.adaptive.ess_decorator( adaptive_resampled_indices, _, adaptive_resampled_particles = adaptive_resampling( resampling_key, logits, particles, 100 ) -``` \ No newline at end of file +``` + +For consistent gradient estimates with respect to model parameters, the [stop-gradient particle filter](https://arxiv.org/abs/2106.10314) is also implemented as a decorator. + +```python +differentiable_resampling = resampling.stop_gradient.stop_gradient_decorator( + resampling.multinomial.resampling +) +# can be combined with adaptive resampling +adaptive_and_differentiable_resampling = resampling.adaptive.ess_decorator( + differentiable_resampling, + threshold=0.5, +) +resampled_indices, _, resampled_particles = adaptive_and_differentiable_resampling( + resampling_key, logits, particles, 100 +) +``` diff --git a/docs/assets/pf_grad_bias_final_particles.png b/docs/assets/pf_grad_bias_final_particles.png new file mode 100644 index 0000000000000000000000000000000000000000..92906351ce978ef5cc9b421664cae26c906a84f0 GIT binary patch literal 34935 zcmeFa2UJwswk=xd7L}w(lxRW`5JgFn1qD%2Qjvpz1j#u{FaQDqCKSoRKqzv?0+gsI zLCF*u0VM}XlKz-TxzGFUzW@HY@4Vao{q3yQcH3^DR;@MXm?QMw$JhaiSLG@#VgEnz|7In)C&Ok5dz#ev*biMf%3i3YN;U`JSkD1ta!yg`AEc05+zE<9*wz4o$u{;vX 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z!Br^C#f}#*S~;zf7LB(lKlA0>1o2y*2gIgM*LxlVdd}~l-*a)#vL!%S>+0Ezct>c5 z`=kUY*gYJ!$4hc`SoV!<47iW3C7u;=c%tavK^r+30ro91?YZT(+eNODHeN;@eT%$I zdQ7Q`rU?d+VpG+0;_$*QELG}CFDm(7-8UcZuW*#wrkS6sxwm&`Yi$CF7M;~y9#g!7 z?R+TrI10)`O=O}0`I@O71LYYG%?}IC)-@|w4&@R(Hy68&K8=cb+XL>->@UH?*Yco= z>%NZgPfP1VXdd~+YG~l4G@Kdo94$7M({&m&E&>fdWZ!=85IJTIaW3ch{N13lFR83i zPR-L0XI=1vraDYlSZb>jF%*lEK9~_hsrEGCa)O`emURX#$7+M;Oq%Z z+H|;(44Ut_G-db&fb#73xHS<%f^*bszCUjz6qpCmc1xrcYRaG%CGfW&K0fGOHYXlH z4Jo+2-i9Po*c2C`p-+z}q#H}OfNKf_X_m=f(LnN)7}l4Hi&x4}TcVCGwy-9(kN!HE zFtixlTm-VeN14ts}v9WEYEr#k=vVKWR;p zZ0}@H1D_$=YT1MoPLTY?`Tc5swxU}- zt5|ZrJwa5^G@vofov<_*PG(JTQ29Od>}M-PwF#+(QK+JHEaw;4up_e)X1>vUcf79h zbZZ_xl2OcR<{^K+0Zy63FPd4BV9ugcTMs7l583UwK(w`Q;OdZpJmU6v#&SuTozmfR zPo+2jS{yqpfZLr_@)B`>c)T$4W(>A_op{^#=azg-_gIg-4pL6Ro7xXJRg8JM2e#^G ztlF_AdD!8vrx)?fo4ft;>cHTvzL1YOgWS!;^*16o8cK&hlI6!C757P&@6p;-5pG}K zIWHcFkZ1z={3lp#1fP6Jwmc^a>5Wb=5)RqyCZ3^a9}p>cYW#$ zD}pF{87XdT=kC2jY{wv>H%Id2L5?HAENN-p?cn@h5JYFs+q>O^%9;aTR)+i0xAM`~ z&ztt=16g8{O?qK#KEkiZbxSHVk9E%%JZP{6XQWkGWxMGH#D(ZV zr*ct9uuHa0&(kNT+8P`Sj|E+1Gce!fZ6uiP8sx}Baj`5X!lec1)sIt1VdkKxDZ*U6 zZ{VKgteb);eKGElNT0P$3u6n)J&CAYQkCO=C_ zJ0MbZ$eW5k81#g(DtLLw+X!Z!Vtw}en?zYc8p5p7-IaM8r6)7d`B9oBA^2vNgKo8= z#iC|-Ya7y|SxdMoAtF=GhMWA&PcjIJowC%FOp%ZpYSyhr6g>WY>VZ0M2Xo0aGvE64 z2JKj%E44NKu`k;9f|Ih_k>sW{7JAj%E~Nfo^DW&hzh~Gft6k-2A1b_rudzW@NjySf zAju?RQ2*3t@ao8!skBdihd*TW=W0P%Tlo!fSz1}hUXi_?KHtEHrzrK~oAhLG?yfbv zY8!)1d2k?0qd)wWRZJr$3u5j$@yu7&nxjn$^MQE zUBL3wi`t;bRt`?x%Ik3*+> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +``` +--> diff --git a/docs/examples/index.md b/docs/examples/index.md index 265afe4..c3c3a06 100644 --- a/docs/examples/index.md +++ b/docs/examples/index.md @@ -8,4 +8,5 @@ highlighting the flexibility of the library and the utility of the underlying me - [Parallel-in-time Kalman filter](temporal_parallelization_kalman.md) - [Online filtering and prediction for a stochastic volatility model](online_stoch_vol.md) - [Parameter estimation with expectation maximization](parameter_estimation_em.md) +- [Differentiable particle filtering](diff_pf_resampling.md) - [Using `cuthbert` with Dynamax](dynamax_integration.md) diff --git a/mkdocs.yml b/mkdocs.yml index 19ab09e..cf41d76 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -39,6 +39,7 @@ nav: - examples/temporal_parallelization_kalman.md - examples/online_stoch_vol.md - examples/parameter_estimation_em.md + - Differentiable Resampling in Particle Filters: examples/diff_pf_resampling.md - Dynamax Integration: examples/dynamax_integration.md - cuthbert API: - cuthbert_api/index.md From 819b2f84eb94c3c39a212f45d7d6314fbc9a5a09 Mon Sep 17 00:00:00 2001 From: Dan Waxman Date: Thu, 26 Feb 2026 16:49:04 -0500 Subject: [PATCH 4/7] Add unittesting for the stop gradient PF Unit testing in this setting is somewhat difficult, for a few reasons: 1. The estimator is still stochastic, and requires lots of trials or particles to make accurate. But this seems rather expensive for CI. 2. The existing utilities for generating test cases are time-varying, and provide somewhat ill-conditioned systems The resolution here is to choose an extremely simple system (random walk on x, identity Gaussian observation on y), and run with 10,000 particles, taking the median over 10 runs, with a 20% relative error tolerance. This seems to run fast enough, at least locally. But happy to explore other options here. --- cuthbertlib/resampling/__init__.py | 1 + tests/cuthbert/smc/test_particle_filters.py | 150 ++++++++++++++++++-- 2 files changed, 143 insertions(+), 8 deletions(-) diff --git a/cuthbertlib/resampling/__init__.py b/cuthbertlib/resampling/__init__.py index a681b3b..a3ae9ad 100644 --- a/cuthbertlib/resampling/__init__.py +++ b/cuthbertlib/resampling/__init__.py @@ -7,4 +7,5 @@ ) from cuthbertlib.resampling.adaptive import ess_decorator from cuthbertlib.resampling.protocols import ConditionalResampling, Resampling +from cuthbertlib.resampling.stop_gradient import stop_gradient_decorator from cuthbertlib.resampling.utils import inverse_cdf diff --git a/tests/cuthbert/smc/test_particle_filters.py b/tests/cuthbert/smc/test_particle_filters.py index f2dda63..417674a 100644 --- a/tests/cuthbert/smc/test_particle_filters.py +++ b/tests/cuthbert/smc/test_particle_filters.py @@ -11,7 +11,7 @@ from cuthbert.inference import Filter from cuthbert.smc import marginal_particle_filter, particle_filter from cuthbertlib.kalman.generate import generate_lgssm -from cuthbertlib.resampling import ess_decorator, systematic +from cuthbertlib.resampling import ess_decorator, stop_gradient_decorator, systematic from cuthbertlib.stats.multivariate_normal import logpdf from tests.cuthbert.gaussian.test_kalman import std_kalman_filter @@ -23,15 +23,74 @@ def config(): jax.config.update("jax_enable_x64", False) +def generate_random_walk_model(seed: int, x_dim: int, y_dim: int, num_time_steps: int): + r"""Generates data from the simple LTI SSM + x_t = x_{t-1} + \epsilon_t + y_t = x_t + \eta_t + where \epsilon_t ~ N(0, 0.05^2) and \eta_t ~ N(0, 0.5^2). + """ + key = random.key(seed) + + m0 = jnp.zeros((x_dim,)) + chol_P0 = jnp.eye(x_dim) + + F = jnp.eye(x_dim) + c = jnp.zeros((x_dim,)) + chol_Q = 0.5 * jnp.eye(x_dim) + + H = jnp.zeros((y_dim, x_dim)) + diag_len = min(x_dim, y_dim) + H = H.at[jnp.arange(diag_len), jnp.arange(diag_len)].set(1.0) + d = jnp.zeros((y_dim,)) + chol_R = 0.5 * jnp.eye(y_dim) + + key, x0_key = random.split(key) + x0 = m0 + chol_P0 @ random.normal(x0_key, (x_dim,)) + + def body(x_prev, k): + k_state, k_obs = random.split(k) + x = F @ x_prev + c + chol_Q @ random.normal(k_state, (x_dim,)) + y = H @ x + d + chol_R @ random.normal(k_obs, (y_dim,)) + return x, y + + scan_keys = random.split(key, num_time_steps) + _, ys = jax.lax.scan(body, x0, scan_keys) + + # Broadcast the parameters to (N_t, ...) for use with, e.g., std_kalman_filter. + Fs = jnp.broadcast_to(F, (num_time_steps, x_dim, x_dim)) + cs = jnp.broadcast_to(c, (num_time_steps, x_dim)) + chol_Qs = jnp.broadcast_to(chol_Q, (num_time_steps, x_dim, x_dim)) + Hs = jnp.broadcast_to(H, (num_time_steps, y_dim, x_dim)) + ds = jnp.broadcast_to(d, (num_time_steps, y_dim)) + chol_Rs = jnp.broadcast_to(chol_R, (num_time_steps, y_dim, y_dim)) + + return m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys + + def load_inference( - m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys, method, noop=False + m0, + chol_P0, + Fs, + cs, + chol_Qs, + Hs, + ds, + chol_Rs, + ys, + method, + use_differentiable_resampling=True, + noop=False, + n_filter_particles=None, + ess_threshold=0.7, ): # Maybe make this more flexible in the future if we want to support other methods. if method == "bootstrap": - n_filter_particles = 1_000_000 + if n_filter_particles is None: + n_filter_particles = 1_000_000 algo = particle_filter elif method == "marginal": - n_filter_particles = 3_000 + if n_filter_particles is None: + n_filter_particles = 3_000 algo = marginal_particle_filter else: @@ -61,9 +120,11 @@ def log_potential(state_prev, state, model_inputs: int): Hs[idx] @ state + ds[idx], ys[idx], chol_Rs[idx], nan_support=False ) - ess_threshold = 0.7 # Decorate the resampling with adaptive behaviour and pass that to the filter - adaptive_systematic = ess_decorator(systematic.resampling, ess_threshold) + resampling_fn = systematic.resampling + if use_differentiable_resampling: + resampling_fn = stop_gradient_decorator(resampling_fn) + adaptive_systematic = ess_decorator(resampling_fn, ess_threshold) inference = Filter( init_prepare=partial( @@ -97,15 +158,28 @@ class Test(chex.TestCase): y_dim=[2], num_time_steps=[20], method=["bootstrap", "marginal"], + use_differentiable_resampling=[True, False], ) - def test(self, seed, x_dim, y_dim, num_time_steps, method): + def test( + self, seed, x_dim, y_dim, num_time_steps, method, use_differentiable_resampling + ): m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys = generate_lgssm( seed, x_dim, y_dim, num_time_steps ) # Run the particle filter. inference, model_inputs = load_inference( - m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys, method + m0, + chol_P0, + Fs, + cs, + chol_Qs, + Hs, + ds, + chol_Rs, + ys, + method, + use_differentiable_resampling, ) key = random.key(seed + 1) states = self.variant(filter, static_argnames=("filter_obj", "parallel"))( @@ -208,6 +282,66 @@ def log_potential(state_prev, state, model_inputs): expected_shape = (num_time_steps + 1, n_filter_particles, 2) chex.assert_shape(particles, expected_shape) + @parameterized.product( + seed=[0, 41], x_dim=[2, 1], y_dim=[2, 1], num_time_steps=[10] + ) + def test_stop_gradient_resampling(self, seed, x_dim, y_dim, num_time_steps): + """Tests that the stop-gradient DPF provides estimates of the gradient that are + close (within 20% relative error) of the gradient provided by the Kalman filter when the median + of 10 trials is taken. + """ + y_dim = max(x_dim, y_dim) + m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys = generate_random_walk_model( + seed, x_dim, y_dim, num_time_steps + ) + + F = Fs[0] + n_particles = 10_000 + + P0 = chol_P0 @ chol_P0.T + Qs = chol_Qs @ chol_Qs.transpose(0, 2, 1) + Rs = chol_Rs @ chol_Rs.transpose(0, 2, 1) + + def kalman_mll(F_param): + Fs_param = jnp.tile(F_param[None, :, :], (num_time_steps, 1, 1)) + _, _, ells = std_kalman_filter(m0, P0, Fs_param, cs, Qs, Hs, ds, Rs, ys) + return ells[-1] + + def pf_mll(F_param, key): + Fs_param = jnp.tile(F_param[None, :, :], (num_time_steps, 1, 1)) + filt, model_inputs = load_inference( + m0, + chol_P0, + Fs_param, + cs, + chol_Qs, + Hs, + ds, + chol_Rs, + ys, + "bootstrap", + use_differentiable_resampling=True, + n_filter_particles=n_particles, + ess_threshold=1.01, # always resample + ) + states = filter(filt, model_inputs, parallel=False, key=key) + return states.log_normalizing_constant[-1] + + kalman_grad = jax.grad(kalman_mll) + pf_grad = jax.grad(pf_mll, argnums=0) + + grad_key = random.key(seed + 1) + pf_grads = [] + n_trials = 10 + for _ in range(n_trials): + grad_key, trial_key = random.split(grad_key) + pf_grads.append(pf_grad(F, trial_key)) + + pf_grad_median = jnp.median(jnp.array(pf_grads), axis=0) + kf_grad = kalman_grad(F) + rel_err = jnp.linalg.norm(pf_grad_median - kf_grad) / jnp.linalg.norm(kf_grad) + assert rel_err < 0.2 + @pytest.mark.parametrize("seed", [1, 43, 99, 123, 456]) @pytest.mark.parametrize("x_dim", [1, 10]) From 8e628c63b4c3e61bf056cadd8c0429408b0f4f5f Mon Sep 17 00:00:00 2001 From: Adrien Corenflos Date: Fri, 27 Feb 2026 10:53:35 +0000 Subject: [PATCH 5/7] Enhance explanation of differentiable resampling methods Clarified the impact of non-differentiable resampling on gradient estimates and provided math for the gradient it computes --- docs/examples/diff_pf_resampling.md | 14 +++++++++++--- 1 file changed, 11 insertions(+), 3 deletions(-) diff --git a/docs/examples/diff_pf_resampling.md b/docs/examples/diff_pf_resampling.md index 26a2964..f3f065c 100644 --- a/docs/examples/diff_pf_resampling.md +++ b/docs/examples/diff_pf_resampling.md @@ -1,8 +1,16 @@ # Differentiable Resampling -In the default implementation of a particle filter, particles are propagated through a Markov kernel $M(x_t \mid x_{t-1})$, reweighted according to a potential function $G_t(x_{t-1}, x_t)$, and potentially resampled. While [various resamplings strategies exist](../cuthbertlib_api/resampling.md), they are typically non-differentiable, leading to bias in estimates of the marginal log-likelihood's gradient, $\nabla_{\theta} \log p(y_{1:T})$. This is unfortunate, as unbiased gradients of the marginal log-likelihood are crucial for parameter estimation tasks. - -Because this task is so important, many different approaches have been proposed for differentiable resampling. A simple implementation is that of the stop-gradient differentiable particle filter, which results in classical estimates of the gradients of the marginal log-likelihood via automatic differentation, as proposed by [Scibior and Wood (2021)](https://arxiv.org/abs/2106.10314). This implementation is cheap, results in identical results in the forward pass of the filter, and is effective for estimating the parameters of the dynamics or observation model in a state-space model (but not parameters of the proposal!). Here, we illustrate the bias of non-differentiable resampling, and show how to use the `stop_gradient_decorator` from `cuthbertlib.resampling` to achieve better gradient estimates. +In the default implementation of a particle filter, particles are propagated through a Markov kernel $M(x_t \mid x_{t-1})$, reweighted according to a potential function $G_t(x_{t-1}, x_t)$, and potentially resampled. While [various resamplings strategies exist](../cuthbertlib_api/resampling.md), they are typically non-differentiable, leading to biased estimates of the marginal log-likelihood's gradient, $\nabla_{\theta} \log p(y_{1:T})$, if used directly within `jax.grad` autodifferentiation framework. This is unfortunate, as relying on these biased gradients of the marginal log-likelihood falsifies for parameter estimation tasks. + +Because of this, many different approaches have been proposed to make the resampling step differentiable. A simple implementation is that of [Scibior and Wood (2021)](https://arxiv.org/abs/2106.10314), which implements a stop-gradient trick recovering classical estimates of the gradients of the marginal log-likelihood via automatic differentation. +Formally, it is equivalent to computing +\begin{align} + \nabla_{\theta} \log p(y_{1:T}) + &= \int \nabla_{\theta} \log p(x_{0:T}, y_{1:T}) \, p(x_{0:T} \mid y_{1:T}) \mathrm{d}x_{0:T}\\ + &\approx \sum_{n=1}^N W_T^{n} \log p(X^{(n)}_{0:T}, y_{1:T}) +\end{align} +under the high variance *[backward tracing](https://github.com/state-space-models/cuthbert/blob/main/cuthbertlib/smc/smoothing/tracing.py)* version of the smoothing distribution. +However, this implementation is cheap, black-box, does not modify the forward pass of the filter when implement, and is therefore a reasonable start for estimating the parameters of the dynamics or observation model in a state-space model (but not parameters of the proposal!). Here, we illustrate the bias of non-differentiable resampling, and show how to use the `stop_gradient_decorator` from `cuthbertlib.resampling` to achieve better gradient estimates. This is partially based on the [PFJax gradient comparisons](https://pfjax.readthedocs.io/en/latest/notebooks/gradient_comparisons.html). From 41f858b33b36a25e0f43442975e39e3ab2f9e83e Mon Sep 17 00:00:00 2001 From: Dan Waxman Date: Mon, 2 Mar 2026 17:33:33 -0500 Subject: [PATCH 6/7] Move to a unit test on the SG resampling decorator Previously, a bootstrap PF was run in its entirety on a linear Gaussian system, and its parameter gradients checked against the Kalman filter. This was expensive and stochastic. Per @AdrienCorenflos's suggestion (https://github.com/state-space-models/cuthbert/pull/209#issuecomment-3969549371), this is tested much more directly. --- tests/cuthbert/smc/test_particle_filters.py | 150 +----------------- .../resampling/test_resamplings.py | 38 +++++ 2 files changed, 46 insertions(+), 142 deletions(-) diff --git a/tests/cuthbert/smc/test_particle_filters.py b/tests/cuthbert/smc/test_particle_filters.py index 417674a..f2dda63 100644 --- a/tests/cuthbert/smc/test_particle_filters.py +++ b/tests/cuthbert/smc/test_particle_filters.py @@ -11,7 +11,7 @@ from cuthbert.inference import Filter from cuthbert.smc import marginal_particle_filter, particle_filter from cuthbertlib.kalman.generate import generate_lgssm -from cuthbertlib.resampling import ess_decorator, stop_gradient_decorator, systematic +from cuthbertlib.resampling import ess_decorator, systematic from cuthbertlib.stats.multivariate_normal import logpdf from tests.cuthbert.gaussian.test_kalman import std_kalman_filter @@ -23,74 +23,15 @@ def config(): jax.config.update("jax_enable_x64", False) -def generate_random_walk_model(seed: int, x_dim: int, y_dim: int, num_time_steps: int): - r"""Generates data from the simple LTI SSM - x_t = x_{t-1} + \epsilon_t - y_t = x_t + \eta_t - where \epsilon_t ~ N(0, 0.05^2) and \eta_t ~ N(0, 0.5^2). - """ - key = random.key(seed) - - m0 = jnp.zeros((x_dim,)) - chol_P0 = jnp.eye(x_dim) - - F = jnp.eye(x_dim) - c = jnp.zeros((x_dim,)) - chol_Q = 0.5 * jnp.eye(x_dim) - - H = jnp.zeros((y_dim, x_dim)) - diag_len = min(x_dim, y_dim) - H = H.at[jnp.arange(diag_len), jnp.arange(diag_len)].set(1.0) - d = jnp.zeros((y_dim,)) - chol_R = 0.5 * jnp.eye(y_dim) - - key, x0_key = random.split(key) - x0 = m0 + chol_P0 @ random.normal(x0_key, (x_dim,)) - - def body(x_prev, k): - k_state, k_obs = random.split(k) - x = F @ x_prev + c + chol_Q @ random.normal(k_state, (x_dim,)) - y = H @ x + d + chol_R @ random.normal(k_obs, (y_dim,)) - return x, y - - scan_keys = random.split(key, num_time_steps) - _, ys = jax.lax.scan(body, x0, scan_keys) - - # Broadcast the parameters to (N_t, ...) for use with, e.g., std_kalman_filter. - Fs = jnp.broadcast_to(F, (num_time_steps, x_dim, x_dim)) - cs = jnp.broadcast_to(c, (num_time_steps, x_dim)) - chol_Qs = jnp.broadcast_to(chol_Q, (num_time_steps, x_dim, x_dim)) - Hs = jnp.broadcast_to(H, (num_time_steps, y_dim, x_dim)) - ds = jnp.broadcast_to(d, (num_time_steps, y_dim)) - chol_Rs = jnp.broadcast_to(chol_R, (num_time_steps, y_dim, y_dim)) - - return m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys - - def load_inference( - m0, - chol_P0, - Fs, - cs, - chol_Qs, - Hs, - ds, - chol_Rs, - ys, - method, - use_differentiable_resampling=True, - noop=False, - n_filter_particles=None, - ess_threshold=0.7, + m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys, method, noop=False ): # Maybe make this more flexible in the future if we want to support other methods. if method == "bootstrap": - if n_filter_particles is None: - n_filter_particles = 1_000_000 + n_filter_particles = 1_000_000 algo = particle_filter elif method == "marginal": - if n_filter_particles is None: - n_filter_particles = 3_000 + n_filter_particles = 3_000 algo = marginal_particle_filter else: @@ -120,11 +61,9 @@ def log_potential(state_prev, state, model_inputs: int): Hs[idx] @ state + ds[idx], ys[idx], chol_Rs[idx], nan_support=False ) + ess_threshold = 0.7 # Decorate the resampling with adaptive behaviour and pass that to the filter - resampling_fn = systematic.resampling - if use_differentiable_resampling: - resampling_fn = stop_gradient_decorator(resampling_fn) - adaptive_systematic = ess_decorator(resampling_fn, ess_threshold) + adaptive_systematic = ess_decorator(systematic.resampling, ess_threshold) inference = Filter( init_prepare=partial( @@ -158,28 +97,15 @@ class Test(chex.TestCase): y_dim=[2], num_time_steps=[20], method=["bootstrap", "marginal"], - use_differentiable_resampling=[True, False], ) - def test( - self, seed, x_dim, y_dim, num_time_steps, method, use_differentiable_resampling - ): + def test(self, seed, x_dim, y_dim, num_time_steps, method): m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys = generate_lgssm( seed, x_dim, y_dim, num_time_steps ) # Run the particle filter. inference, model_inputs = load_inference( - m0, - chol_P0, - Fs, - cs, - chol_Qs, - Hs, - ds, - chol_Rs, - ys, - method, - use_differentiable_resampling, + m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys, method ) key = random.key(seed + 1) states = self.variant(filter, static_argnames=("filter_obj", "parallel"))( @@ -282,66 +208,6 @@ def log_potential(state_prev, state, model_inputs): expected_shape = (num_time_steps + 1, n_filter_particles, 2) chex.assert_shape(particles, expected_shape) - @parameterized.product( - seed=[0, 41], x_dim=[2, 1], y_dim=[2, 1], num_time_steps=[10] - ) - def test_stop_gradient_resampling(self, seed, x_dim, y_dim, num_time_steps): - """Tests that the stop-gradient DPF provides estimates of the gradient that are - close (within 20% relative error) of the gradient provided by the Kalman filter when the median - of 10 trials is taken. - """ - y_dim = max(x_dim, y_dim) - m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys = generate_random_walk_model( - seed, x_dim, y_dim, num_time_steps - ) - - F = Fs[0] - n_particles = 10_000 - - P0 = chol_P0 @ chol_P0.T - Qs = chol_Qs @ chol_Qs.transpose(0, 2, 1) - Rs = chol_Rs @ chol_Rs.transpose(0, 2, 1) - - def kalman_mll(F_param): - Fs_param = jnp.tile(F_param[None, :, :], (num_time_steps, 1, 1)) - _, _, ells = std_kalman_filter(m0, P0, Fs_param, cs, Qs, Hs, ds, Rs, ys) - return ells[-1] - - def pf_mll(F_param, key): - Fs_param = jnp.tile(F_param[None, :, :], (num_time_steps, 1, 1)) - filt, model_inputs = load_inference( - m0, - chol_P0, - Fs_param, - cs, - chol_Qs, - Hs, - ds, - chol_Rs, - ys, - "bootstrap", - use_differentiable_resampling=True, - n_filter_particles=n_particles, - ess_threshold=1.01, # always resample - ) - states = filter(filt, model_inputs, parallel=False, key=key) - return states.log_normalizing_constant[-1] - - kalman_grad = jax.grad(kalman_mll) - pf_grad = jax.grad(pf_mll, argnums=0) - - grad_key = random.key(seed + 1) - pf_grads = [] - n_trials = 10 - for _ in range(n_trials): - grad_key, trial_key = random.split(grad_key) - pf_grads.append(pf_grad(F, trial_key)) - - pf_grad_median = jnp.median(jnp.array(pf_grads), axis=0) - kf_grad = kalman_grad(F) - rel_err = jnp.linalg.norm(pf_grad_median - kf_grad) / jnp.linalg.norm(kf_grad) - assert rel_err < 0.2 - @pytest.mark.parametrize("seed", [1, 43, 99, 123, 456]) @pytest.mark.parametrize("x_dim", [1, 10]) diff --git a/tests/cuthbertlib/resampling/test_resamplings.py b/tests/cuthbertlib/resampling/test_resamplings.py index 73651f4..190f14f 100644 --- a/tests/cuthbertlib/resampling/test_resamplings.py +++ b/tests/cuthbertlib/resampling/test_resamplings.py @@ -2,10 +2,14 @@ import chex import jax +import jax.numpy as jnp import pytest from absl.testing import parameterized +from jax import random +from jax.scipy.special import logsumexp from cuthbertlib.resampling import killing, multinomial, systematic +from cuthbertlib.resampling.stop_gradient import stop_gradient_decorator from tests.cuthbertlib.resampling.utils import ( conditional_resampling_tester, resampling_tester, @@ -122,3 +126,37 @@ def test_conditional_resampling(self, seed, test_case): conditional_resampling_tester( key_test, log_weights, conditional_resampling, M, self.K ) + + @pytest.mark.xdist_group(name="resampling") # Serialize to avoid OOM + @parameterized.parameters( + itertools.product([0, 42, 1337], ["systematic", "multinomial", "killing"]) + ) + def test_stop_gradient_resampling(self, seed, method): + """Tests that gradient estimates are approximately correct under the stop-gradient decorator.""" + xs = jnp.linspace(-2.0, 2.0, 2_000) + n = xs.shape[0] + key = random.key(seed) + true_sigma = 1.0 + resampling_fn = get_resampling(method) + + def _gaussian_log_weights(xs: jnp.ndarray, sigma: float) -> jnp.ndarray: + return -0.5 * (xs / sigma) ** 2 + + def base_lse(sigma): + return logsumexp(_gaussian_log_weights(xs, sigma)) + + def resampled_lse(sigma, resampling_fn): + logws = _gaussian_log_weights(xs, sigma) + _, logits_out, _ = resampling_fn(key, logws, xs, n) + return logsumexp(logits_out) + + grad_base = jax.grad(base_lse)(true_sigma) + grad_stop = jax.grad( + lambda sigma: resampled_lse(sigma, stop_gradient_decorator(resampling_fn)) + )(true_sigma) + grad_plain = jax.grad(lambda sigma: resampled_lse(sigma, resampling_fn))( + true_sigma + ) + + print(grad_base, grad_plain, grad_stop, method) + chex.assert_trees_all_close(grad_stop, grad_base, rtol=0.05, atol=0.05) From 81baec73ebfaed54a63c958dd9c73f724352d274 Mon Sep 17 00:00:00 2001 From: Dan Waxman Date: Tue, 3 Mar 2026 07:22:40 -0500 Subject: [PATCH 7/7] Migrate naming of stop_gradient -> autodiff submodule. --- cuthbertlib/resampling/__init__.py | 4 ++-- .../resampling/{stop_gradient.py => autodiff.py} | 9 +++++---- docs/cuthbertlib_api/resampling.md | 2 +- docs/examples/diff_pf_resampling.md | 12 +++++++----- tests/cuthbertlib/resampling/test_resamplings.py | 3 +-- 5 files changed, 16 insertions(+), 14 deletions(-) rename cuthbertlib/resampling/{stop_gradient.py => autodiff.py} (85%) diff --git a/cuthbertlib/resampling/__init__.py b/cuthbertlib/resampling/__init__.py index a3ae9ad..9b82f20 100644 --- a/cuthbertlib/resampling/__init__.py +++ b/cuthbertlib/resampling/__init__.py @@ -1,11 +1,11 @@ from cuthbertlib.resampling import ( adaptive, + autodiff, killing, multinomial, - stop_gradient, systematic, ) from cuthbertlib.resampling.adaptive import ess_decorator +from cuthbertlib.resampling.autodiff import stop_gradient_decorator from cuthbertlib.resampling.protocols import ConditionalResampling, Resampling -from cuthbertlib.resampling.stop_gradient import stop_gradient_decorator from cuthbertlib.resampling.utils import inverse_cdf diff --git a/cuthbertlib/resampling/stop_gradient.py b/cuthbertlib/resampling/autodiff.py similarity index 85% rename from cuthbertlib/resampling/stop_gradient.py rename to cuthbertlib/resampling/autodiff.py index 8615bc9..8e726b8 100644 --- a/cuthbertlib/resampling/stop_gradient.py +++ b/cuthbertlib/resampling/autodiff.py @@ -1,8 +1,9 @@ -"""Stop gradient resampling decorator. +"""Implements decorators for automatic differentiation of resampling schemes. -The stop_gradient resampling scheme provides the classical Fisher estimates for -the score function via automatic differentiation. This can be wrapped around a -resampling scheme such as multinomial or systematic resampling. +Current supported is the stop_gradient resampling scheme, which provides the +classical Fisher estimates for the score function via automatic differentiation. +This can be wrapped around a resampling scheme such as multinomial or systematic +resampling. See [Scibior and Wood (2021)](https://arxiv.org/abs/2106.10314) for more details. """ diff --git a/docs/cuthbertlib_api/resampling.md b/docs/cuthbertlib_api/resampling.md index 4492e09..a77d446 100644 --- a/docs/cuthbertlib_api/resampling.md +++ b/docs/cuthbertlib_api/resampling.md @@ -23,4 +23,4 @@ ::: cuthbertlib.resampling.adaptive -::: cuthbertlib.resampling.stop_gradient +::: cuthbertlib.resampling.autodiff diff --git a/docs/examples/diff_pf_resampling.md b/docs/examples/diff_pf_resampling.md index f3f065c..ce5c5bd 100644 --- a/docs/examples/diff_pf_resampling.md +++ b/docs/examples/diff_pf_resampling.md @@ -124,8 +124,10 @@ kalman_grad = jax.grad(kalman_mll) To select differentiable or non-differentiable resampling, we simply wrap the corresponding resampling function in the `stop_gradient_decorator`, i.e., ```python +from cuthbertlib.resampling import autodiff + resampling_fn = ( - stop_gradient.stop_gradient_decorator(systematic.resampling) + autodiff.stop_gradient_decorator(systematic.resampling) if differentiable_resampling else systematic.resampling ) @@ -137,7 +139,7 @@ Putting this together, we can define our PF MLL function. ```{.python #pf_grad_bias-pf_mll} from cuthbertlib.resampling import systematic from cuthbertlib.resampling import adaptive -from cuthbertlib.resampling import stop_gradient +from cuthbertlib.resampling import autodiff def pf_mll( theta, @@ -170,7 +172,7 @@ def pf_mll( return logpdf(H @ x + d, ys[idx], chol_R, nan_support=False) resampling_fn = ( - stop_gradient.stop_gradient_decorator(systematic.resampling) + autodiff.stop_gradient_decorator(systematic.resampling) if differentiable_resampling else systematic.resampling ) @@ -507,7 +509,7 @@ Finally, we should be sure that the forward pass is not actually being modified return logpdf(H @ x + d, ys[idx], chol_R, nan_support=False) resampling_fn = ( - stop_gradient.stop_gradient_decorator(systematic.resampling) + autodiff.stop_gradient_decorator(systematic.resampling) if diff else systematic.resampling ) @@ -561,7 +563,7 @@ Finally, we should be sure that the forward pass is not actually being modified ## Key Takeaways - **Gradient bias**: The default (non-differentiable) resampling in the particle filter yields biased score estimates when differentiated with autodiff; the bias is visible when compared to the exact Kalman score on a linear-Gaussian model. -- **Differentiable resampling**: Using `cuthbertlib.resampling.stop_gradient` with systematic resampling gives a differentiable PF whose score estimates are much less biased, at similar computational cost. +- **Differentiable resampling**: Using `cuthbertlib.resampling.autodiff.stop_gradient_decorator` with systematic resampling gives a differentiable PF whose score estimates are much less biased, at similar computational cost. - **Exact reference**: On linear-Gaussian SSMs, the Kalman filter provides exact log marginal likelihood and score; this example uses it as a ground truth to quantify PF gradient bias. ## Next Steps diff --git a/tests/cuthbertlib/resampling/test_resamplings.py b/tests/cuthbertlib/resampling/test_resamplings.py index 190f14f..2c6a684 100644 --- a/tests/cuthbertlib/resampling/test_resamplings.py +++ b/tests/cuthbertlib/resampling/test_resamplings.py @@ -9,7 +9,7 @@ from jax.scipy.special import logsumexp from cuthbertlib.resampling import killing, multinomial, systematic -from cuthbertlib.resampling.stop_gradient import stop_gradient_decorator +from cuthbertlib.resampling.autodiff import stop_gradient_decorator from tests.cuthbertlib.resampling.utils import ( conditional_resampling_tester, resampling_tester, @@ -158,5 +158,4 @@ def resampled_lse(sigma, resampling_fn): true_sigma ) - print(grad_base, grad_plain, grad_stop, method) chex.assert_trees_all_close(grad_stop, grad_base, rtol=0.05, atol=0.05)