SlowGuess/NeurIPS_2023_OCR huggingface dataset:
An MCMC algorithm updates at each iteration its state according to  $x_{n + 1} = x_n + \mathbf{I}(u_n < \alpha(x_n, y_n))(y_n - x_n)$  where  $\alpha(x_n, y_n)$  is the M-H probability,  $u_n \sim U(0, 1)$  is an uniform random number and  $\mathbf{I}(\cdot)$  is the indicator function. This sets  $x_{n + 1}$  to either the proposal  $y_n$  or the previous state  $x_n$  based on the binary value  $\mathbf{I}(u_n < \alpha(x_n, y_n))$ . Similarly, we can consider the update of the score function  $s(x) = \nabla \log \pi(x)$  and conveniently re-arrange it as an increment,
$$
s _ {n} ^ {\delta} = s \left(x _ {n + 1}\right) - s \left(x _ {n}\right) = \mathbf {I} \left(u _ {n} <   \alpha \left(x _ {n}, y _ {n}\right)\right) \left(s \left(y _ {n}\right) - s \left(x _ {n}\right)\right). \tag {15}
$$
While both  $s_n$  and  $s_n^\delta$  have zero expectation when  $x_n$  is from stationarity, i.e.  $x_n \sim \pi$ , the increment  $s_n^\delta$  (unlike  $s_n$ ) tends in practice to be more centered and close to zero even when the chain is transient, e.g. note that  $s_n^\delta$  is zero when  $y_n$  is rejected. Further, since the difference  $s_n^\delta = s(x_{n+1}) - s(x_n)$  conveys information about the covariance of the score function we can use it in the recursion of
Algorithm 1 Fisher adaptive MALA (blue lines are omitted when not adapting  $(R,\sigma^2)$ )
Input: Log target  $\log \pi (x)$  ; gradient  $\nabla$  log  $\pi (x)$  .  $\lambda >0$  (default  $\lambda = 10$  );  $\alpha_{*} = 0.574$    
Initialize  $x_{1}$  and  $\sigma^2$  by running simple MALA (i.e. with  $\mathcal{N}(y|x + (\sigma^2 /2)\nabla \log \pi (x),\sigma^2 I))$  for  $n_0$    
(default 500) iterations where  $\sigma^2$  is adapted towards acceptance rate  $\alpha_{*}$    
Initialize square root matrix  $R = I_d$  and compute  $(\log \pi (x_{1}),\nabla \log \pi (x_{1}))$    
Initialize  $\sigma_R^2 = \sigma^2$  # placeholder for the normalized step size  $\sigma^2 /\frac{1}{d} tr(RR^\top)$    
for For  $n = 1,2,3,\ldots ,$  do : Propose  $y_{n} = x_{n} + (\sigma_{R}^{2} / 2)R(R^{\top}\nabla \log \pi (x_{n})) + \sigma_{R}R\eta$  n  $\eta \sim \mathcal{N}(0,I_d)$  : Compute  $(\log \pi (y_n),\nabla \log \pi (y_n))$  .. Compute  $\alpha (x_n,y_n) = \min \left(1,e^{\log \pi (y_n) + h(x_n,y_n) - \log \pi (x_n) - h(y_n,x_n)}\right)$  by using Proposition 1 : Compute adaptation signal  $s_n^\delta = \sqrt{\alpha(x_n,y_n)} (\nabla \log \pi (y_n) - \nabla \log \pi (x_n))$  : Use  $s_n^\delta$  to adapt  $R$  based on Proposition 4 (if  $n = 1$  use (12) and if  $n > 1$  use (13)) : Adapt step size  $\sigma^2\gets \sigma^2 [1 + \rho_n(\alpha (x_n,y_n) - \alpha_*)]$  # default const learning rate  $\rho_{n} = 0.015$  :Normalize step size  $\sigma_R^2 = \sigma^2 /\frac{1}{d}\mathrm{tr}(RR^\top)$  # tr(RRT)  $= sum(R\circ R)$  which is  $O(d^{2})$  :Accept/reject  $y_{n}$  with probability  $\alpha (x_n,y_n)$  to obtain  $(x_{n + 1},\log \pi (x_{n + 1}),\nabla \log \pi (x_{n + 1}))$    
end for
Proposition 4 to learn the preconditioner  $A$ , where we simply replace  $s_n$  by  $s_n^\delta$ . As shown in the experiments this leads to a remarkably fast and effective adaptation of the inverse Fisher matrix  $\mathcal{I}^{-1}$  without observable bias, or at least no observable for Gaussian targets where the true  $\mathcal{I}^{-1}$  is known. We can further apply Rao-Blackwellization to reduce some variance of  $s_n^\delta$ . Since  $s_n^\delta$  enters into the estimation of the empirical Fisher, see Eq. (8) or (14), through the outer product  $s_n^\delta(s_n^\delta)^\top = \mathbf{I}(u_n < \alpha(x_n, y_n))(s(y_n) - s(x_n))(s(y_n) - s(x_n))^\top$  we can marginalize out the r.v.  $u_n$  which yields  $\mathbb{E}_{u_n}[s_n^\delta(s_n^\delta)^\top] = \alpha(x_n, y_n)(s(y_n) - s(x_n))(s(y_n) - s(x_n))^\top$ . After this Rao-Blackwellization an alternative vector to use for adaptation is
$$
s _ {n} ^ {\delta} = \sqrt {\alpha \left(x _ {n} , y _ {n}\right)} \left(s \left(y _ {n}\right) - s \left(x _ {n}\right)\right), \tag {16}
$$
which depends on the square root  $\sqrt{\alpha(x_n, y_n)}$  of the M-H probability. As long as  $\alpha(x_n, y_n) > 0$ , the learning signal in (16) depends on the proposed sample  $y_n$  even when it is rejected.
Finally, we can express the full algorithm for Fisher information adaptive MALA as outlined by Algorithm 1, which adapts by using the Rao-Blackwellized score function increments from Eq. (16). Note that, while Algorithm 1 uses  $s_n^\delta$  from Eq. (16), the initial signal from Eq. (15) works equally well; see Appendix E.3. Also, the algorithm includes an initialization phase where simple MALA runs for few iterations to move away from the initial state, as discussed further in Section 5.
5.1 Methods and experimental setup
We apply the Fisher information adaptive MALA algorithm (FisherMALA) to high dimensional problems and we compare it with the following other samplers. (i) The simple MALA sampler with proposal  $\mathcal{N}(y_n|x_n + (\sigma^2 /2)\nabla \log \pi (x_n),\sigma^2 I)$ , which adapts only a step size  $\sigma^2$  without having a preconditioner. (ii) A preconditioned adaptive MALA (AdaMALA) where the proposal follows exactly the from in (7) but where the preconditioning matrix is learned using standard adaptive MCMC based on the well-known recursion from [21]:
$$
\mu_ {n} = \frac {n - 1}{n} \mu_ {n - 1} + \frac {1}{n} x _ {n}, \Sigma_ {n} = \frac {n - 2}{n - 1} \Sigma_ {n - 1} + \frac {1}{n} \left(x _ {n} - \mu_ {n - 1}\right) \left(x _ {n} - \mu_ {n - 1}\right) ^ {\top}, \tag {17}
$$
where the recursion is initialized at  $\mu_{1} = x_{1}$  and  $\Sigma_{2} = \frac{1}{2} (x_{2} - \mu_{1})(x_{2} - \mu_{1})^{\top} + \lambda I$ , and  $\lambda > 0$  is the damping parameter that plays the same role as in FisherMALA. (iii) The Riemannian manifold MALA (mMALA) [20] which uses position-dependent preconditioning matrix  $A(x)$ . mMALA in high dimensions runs slower than other schemes since the computation of  $A(x)$  may involve second derivatives and requires matrix decomposition that costs  $O(d^{3})$  per iteration. (iv) Finally, we include in the comparison the Hamiltonian Monte Carlo (HMC) sampler with a fixed number of 10 leap frog steps and identity mass matrix. We leave the possibility to learn with our method a preconditioner in HMC for future work since this is more involved; see discussion at Section 7.
For all experiments and samplers we consider  $2 \times 10^{4}$  burn-in iterations and  $2 \times 10^{4}$  iterations for collecting samples. We set  $\lambda = 10$  in FisherMALA and AdaMALA. Adaptation of the proposal distributions, i.e. the parameter  $\sigma^2$ , the preconditioning or the step size of HMC, occurs only during burn-in and at collection of samples stage the proposal parameters are kept fixed. For all three MALA schemes the global step size  $\sigma^2$  is adapted to achieve an acceptance rate around 0.574 (see Algorithm 1) while the corresponding parameter for HMC is adapted towards 0.651 rate [9]. In FisherMALA from the  $2 \times 10^{4}$  burn-in iterations the first 500 iterations are used as the initialization phase in Algorithm 1 where samples are generated by just MALA with adaptable  $\sigma^2$ . Thus, only the last  $1.95 \times 10^{4}$  burn-in iterations are used to adapt the preconditioner. For AdaMALA this initialization scheme proved to be unstable and we used a more elaborate scheme, as described in Appendix D.
We compute effective sample size (ESS) scores for each method by using the  $2 \times 10^{4}$  samples from the collection phase. We estimate ESS across each dimension of the state vector  $x$ , and we report maximum, median and minimum values, by using the built-in method in TensorFlow Probability Python package. Also, we show visualizations that indicate sampling efficiency or effectiveness in estimating the preconditioner (when the ground truth preconditioner is known).
5.2 Gaussian targets
We consider three examples of multivariate Gaussian targets of the form  $\pi(x) = \mathcal{N}(x|\mu, \Sigma)$ , where the optimal preconditioner (up to any positive scaling) is the covariance matrix  $\Sigma$  since the inverse Fisher is  $\mathcal{I}^{-1} = \Sigma$ . For such case the Riemannian manifold sampler mMALA [20] is the optimal MALA sampler since it uses precisely  $\Sigma$  as the preconditioning. In contrast to mMALA which somehow has access to the ground-truth oracle, both FisherMALA and AdaMALA use adaptive recursive estimates of the preconditioner that should converge to the optimal  $\Sigma$ , and thus the question is which of them learns faster. To quantify this we compute the Frobenius norm  $||\widetilde{A}_n - \widetilde{\Sigma}||_F$  across adaptation iterations  $n$ , where  $\widetilde{B}$  denotes the matrix normalized by the average trace, i.e.  $\widetilde{B} = B / (\frac{\mathrm{tr}(B)}{d})$ , for either  $A_n$  given by FisherMALA or  $A_n := \Sigma_n$  given by AdaMALA and where  $\widetilde{\Sigma}$  is the optimal normalized preconditioner. The faster the Frobenius norm goes to zero the more effective is the corresponding adaptive scheme. For all three Gaussian targets the mean vector  $\mu$  was taken to be the vector of ones and samplers were initialized by drawing from standard normal. The first example is a two-dimensional Gaussian target with covariance matrix  $\Sigma = [10.995; 0.9951]$ . Both FisherMALA and AdaMALA perform almost the same (FisherMALA has faster convergence) in this low dimensional example as shown by Frobenius norm in Figure 1a; see also Figure 5 in the Appendix for visualizations of the adapted preconditioners. The following two examples involve 100-dimensional targets.
Gaussian process correlated target. We consider a Gaussian process to construct a 100-dimensional Gaussian where the covariance matrix is obtained by a non-stationary covariance function comprising the product of linear and squared exponential kernels plus small white noise, i.e.  $[\Sigma]_{i,j} = s_i s_j \exp\left\{-\frac{1}{2} \frac{(s_i - s_j)^2}{0.09}\right\} + 0.001 \delta_{i,j}$  where the scalar inputs  $s_i$  form a regular grid in the range [1, 2]. Figure 1b shows the evolution of the Frobenius norms and panels d,c depict as  $100 \times 100$  images the true covariance matrix and the preconditioner estimated by FisherMALA. For AdaMALA see Figure 6 in the Appendix. Clearly, FisherMALA learns much faster and achieves more accurate estimates of the optimal preconditioner. Further Table 1 shows that FisherMALA achieves significantly better ESS than AdaMALA and reaches the same performance with mMALA.
Inhomogeneous Gaussian target. In the last example we follow [28, 39] and we consider a Gaussian target with diagonal covariance matrix  $\Sigma = \mathrm{diag}(\sigma_1^2,\dots,\sigma_{100}^2)$  where the standard deviation values  $\sigma_{i}$  take values in the grid  $\{0.01,0.02,\ldots ,1\}$ . This target is challenging because the different scaling across dimensions means that samplers with a single step size, i.e. without preconditioning, will adapt to the smallest dimension  $x_{1}$  of the state while the chain at the higher dimensions, such as  $x_{100}$ , will be moving slowly exhibiting high autocorrelation. Note that FisherMALA and AdaMALA run without knowing that the optimal preconditioner is a diagonal matrix, i.e. they learn a full covariance matrix. Figure 2a shows the ESS scores for all 100 dimensions of  $x$  for four samplers (except mMALA which has the same performance with FisherMALA), where we can observe that only FisherMALA is able to achieve high ESS uniformly well across all dimensions. In contrast, MALA and HMC that use a single step size cannot achieve high sampling efficiency and their
Figure 1: Panel (a) shows the Frobenius norm across burn-in iterations for the 2-D Gaussian and (b) for the GP target. The exact GP covariance matrix is shown in (c) and the estimated one by FisherMALA in (d).
(a)
Figure 2: Results in the inhomogeneous Gaussian target.
ESS for dimensions close to  $x_{100}$  drops significantly. The same holds for AdaMALA due to its inability to learn fast the preconditioner, as shown by the Frobenius norm values in Figure 2c and the estimated standard deviations in Figure 2b. AdaMALA can eventually get very close to the optimal preconditioner but it requires hundreds of thousands of adaptive steps, while FisherMALA learns it with only few thousand steps.
5.3 Bayesian logistic regression
We consider Bayesian logistic regression distributions of the form  $\pi (\theta |Y,Z)\propto p(Y|\theta ,Z)p(\theta)$  with data  $(Y,Z) = \{y_{i},z_{i}\}_{i = 1}^{m}$ , where  $z_{i}\in \mathbb{R}^{d}$  is the input vector and  $y_{i}$  the binary label. The likelihood is  $p(Y|\theta ,Z) = \prod_{i = 1}^{m}\sigma (\theta^{\top}z_{i})^{y_{i}}(1 - \sigma (\theta^{\top}z_{i}))^{1 - y_{i}}$ , where  $\theta \in \mathbb{R}^d$  are the random parameters assigned the prior  $p(\theta) = \mathcal{N}(\theta |0,I_d)$ . We consider six binary classification datasets (Australian Credit, Heart, Pima Indian, Ripley, German Credit and Caravan) with a number of data ranging from  $n = 250$  to  $n = 5822$  and dimensionality of the  $\theta$  ranging from 3 to 87. We also consider a much higher 785-dimensional example on MNIST for classifying the digits 5 and 6, that has 11339 training examples. To make the inference problems more challenging, in the first six examples we do not standardize the inputs  $z_{i}$  which creates very anisotropic posteriors over  $\theta$ . For the MNIST data, which initially are grey-scale images in [0, 255], we simply divide by the maximum pixel value, i.e. 255, to bring the images in [0, 1]. In Table 1 we report the ESS for the low 7-dimensional Pima Indians dataset, the medium 87-dimensional Caravan dataset and the higher 785-dimensional MNIST dataset, while the results for the remaining datasets are shown in Appendix E. Further, Figure 3 shows the evolution of the unnormalized log target density  $\log \{p(Y|\theta ,Z)p(\theta)\}$  for the best four samplers in
Figure 3: The evolution of the log-target across iterations for the best four algorithms in Caravan dataset.
Table 1: ESS scores are averages after repeating the simulations 10 times under different random initializations.
Max ESS	Median ESS	Min ESS
GP target (d=100)
MALA	15.235 ± 4.246	6.326 ± 1.934	3.619 ± 0.956
AdaMALA	845.100 ± 80.472	662.978 ± 81.127	552.377 ± 74.441
HMC	17.625 ± 6.706	6.680 ± 2.205	4.315 ± 0.889
mMALA	2109.441 ± 101.553	2007.640 ± 104.867	1841.978 ± 114.266
FisherMALA	2096.259 ± 94.751	1923.753 ± 95.820	1784.962 ± 104.440
Pima Indian (d=7)
MALA	106.668 ± 29.601	14.723 ± 3.821	4.061 ± 1.587
AdaMALA	211.948 ± 133.363	52.277 ± 26.566	6.401 ± 3.344
HMC	1624.773 ± 544.777	337.100 ± 212.158	6.052 ± 2.062
mMALA	6086.948 ± 117.241	5690.967 ± 118.401	5297.835 ± 160.084
FisherMALA	6437.419 ± 207.548	5981.960 ± 156.072	5628.541 ± 168.425
Caravan (d=87)
MALA	27.247 ± 7.554	5.890 ± 0.398	2.906 ± 0.150
AdaMALA	41.522 ± 9.343	7.144 ± 0.663	3.144 ± 0.135
HMC	787.901 ± 173.863	37.303 ± 6.808	4.238 ± 0.532
mMALA	179.899 ± 61.502	121.867 ± 41.801	51.414 ± 24.995
FisherMALA	2257.737 ± 45.289	1920.903 ± 55.821	498.016 ± 96.692
MNIST (d=785)
MALA	34.074 ± 4.977	7.589 ± 0.149	2.944 ± 0.066
AdaMALA	62.301 ± 9.203	8.188 ± 0.399	2.985 ± 0.089
HMC	889.386 ± 118.050	303.345 ± 10.976	114.439 ± 20.965
mMALA	51.589 ± 3.447	20.222 ± 1.259	5.240 ± 0.492
FisherMALA	1053.455 ± 35.680	811.522 ± 19.165	439.580 ± 52.800
Caravan dataset which visualizes chain autocorrelation. From all these results we can conclude that FisherMALA is better than all other samplers, and remarkably it outperforms significantly the position-dependent mMALA, especially in the high dimensional Caravan and MNIST datasets.
There exist works that use some form of global preconditioning in gradient-based samplers for specialized targets such as latent Gaussian models [13, 44], which make use of the tractable Gaussian prior. Our method differs, since it is more agnostic to the target and learns a preconditioning from the history of gradients, analogously to how traditional adaptive MCMC learns from states [21, 38].
Several research works use position-dependent preconditioning  $A(x)$  within gradient-based samplers, such as MALA. This is for example the idea behind Riemannian manifold MALA [20] and extensions [45]. Similar to Riemannian manifold methods there are approaches inspired by second order optimization that use the Hessian matrix, or some estimate of the Hessian, for sampling in a MALA-style manner [32, 17, 27]. Recently, such samplers and their time-continuous diffusion limits have been theoretically analyzed by obtaining convergence guarantees [12, 26]. All such methods form a position-dependent preconditioning and not the preconditioning we use in this paper, e.g. note that  $\mathcal{I}^{-1}$  we consider here requires an expectation under the target and thus it is always a global preconditioner rather than a position-dependent one. Another difference is that our method has quadratic cost, while position-dependent preconditioning methods have cubic cost and they require computationally demanding quantities like the Hessian matrix. Therefore, in order for these methods to run faster some approximation may be needed, e.g. low rank [27] or quasi-Newton type [46, 24]. Furthermore, the Bayesian logistic results in Table 1 (see also Figure 3) show that the proposed FisherMALA method significantly outperforms manifold MALA [20] in Caravan and MNIST examples, despite the fact that manifold MALA preconditions with the exact negative inverse Hessian matrix of the log target. This could suggest that position-dependent preconditioning may be less effective in certain type of high-dimensional and log-concave problems.
Finally, there is recent work for learning flexible MCMC proposals by using neural networks [42, 25, 23, 41] and by adapting parameters using differentiable objectives [25, 29, 43, 14]. Our method differs, since it does not use objective functions (which have extra cost because they require an optimization to run in parallel with the MCMC chain), but instead it adapts similarly to traditionally MCMC methods by accumulating information from the observed history of the chain.
We derived an optimal preconditioning for the Langevin diffusion by optimizing the expected squared jumped distance, and subsequently we developed an adaptive MCMC algorithm that approximates the optimal preconditioning by applying an efficient quadratic cost recursion. Some possible topics for future research are: Firstly, it would be useful to investigate whether the score function differences that we use as the adaptation signal introduce any bias in the estimation of the inverse Fisher matrix. Secondly, it would be interesting to extend our method to learn the preconditioning for other gradient-based samplers such as Hamiltonian Monte Carlo (HMC), where such a matrix there is referred to as the inverse mass matrix. For HMC this is more complex since both the mass matrix and its inverse are needed in the iteration. Finally, it could be interesting to investigate adaptive schemes of the inverse Fisher matrix by using multiple parallel and interacting chains, similarly to ensemble covariance matrix estimation for Langevin diffusions [16].
We are grateful to the reviewers for their comments. Also, we wish to thank Arnaud Doucet, Sam Power, Francisco Ruiz, Jiaxin Shi, Yee Whye Teh, Siran Liu, Kazuki Osaawa and James Martens for useful discussions.
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