SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The sampling efficiency of  $p_{\lambda}^{\beta}$  is again defined as the ratio of effective sample size and sample size:  $\mathrm{eff}_{\lambda}^{\beta} = (\mathbb{E}_{\theta \sim p}[v_{\beta}(w_{\lambda}(\theta))])^{2} / \mathbb{E}_{\theta \sim p}[v_{\beta}(w_{\lambda}(\theta))^{2}]$  and enjoys a relation similar to Theorem 2:
$$
D _ {\mathrm {K L}} \left(p _ {\lambda} ^ {\beta} \| q\right) \leq \log \left(1 + D _ {\chi^ {2}} \left(p _ {\lambda} ^ {\beta} \| q\right)\right) = - \log \operatorname {e f f} _ {\lambda} ^ {\beta}.
$$
6.1 Trade-off between Generalization and Sampling Efficiency
According to Eq. (5), we can see that the generalization is highly affected by extremely small  $w_{\lambda}(\theta)$ . A small weight indicates that the optimization produces certain parameters too often. This issue could be addressed by assigning a small weight for these parameters in importance sampling. On the other hand, the sampling efficiency of Bayesian learning is very sensitive to large value of  $w_{\lambda}(\theta)$ , according to Eq. (3). The high weight appears on the regime where output probability is low but posterior density is high. That essentially means the optimization oracle is not effective in exploring certain region, and the only way to bypass this bottleneck of Bayesian learning is changing the optimization oracle to produce a distribution with heavier tail than posterior. Due to the exponential dependence on energy change in weight, the distribution of weight is typically heavy-tailed, therefore the efficiency is low. Fortunately, this issue could be largely alleviated with interpolation. Since generalization bound is insensitive to large value of weight, it is possible to apply appropriate modification to weights such that the sampling efficiency is improved and generalization doesn't deteriorate too much.
Searching for the Pareto optimality among various trade-off methods is not a concern in this paper, although obviously this is an important issue and is a subject for future work. In this paper, we only
consider a simple weight clipping method:  $v_{\beta}(w) = \begin{cases} w & w \leq \beta \\ \beta & \text{otherwise} \end{cases}$ .
At the limit of  $\beta \to 0$ ,  $p_{\lambda}^{\beta}$  degenerates to output distribution  $q$  of optimization. At the limit of  $\beta \to \infty$ ,  $p_{\lambda}^{\beta}$  converges to posterior  $p_{\lambda}$ . A lower bound of sampling efficiency  $\mathrm{eff}_{\lambda}^{\beta} \geq (Z_{\lambda}^{\beta} / \beta)^{2}$  could be established. More importantly, weight clipping always achieves smaller generalization bounds than optimization, and higher sampling efficiency than Bayesian learning.
Theorem 3. For parameters  $\beta \in (0,\infty)$ ,  $D_{\mathrm{KL}}(p_\lambda^\beta \| p_\lambda)$  and  $\mathrm{eff}_{\lambda}^{\beta}$  are both monotonically decreasing functions of  $\beta$ .
7 Overparameterized Neural Network
The energy change  $\Delta_f V(\theta)$  and entropy change  $\Delta_f S(\theta)$  play a pivot role in previous sections. All quantities of interest, including KL divergence in the PAC-Bayesian bounds, weights in TransBL and sampling efficiency, depend on these two terms. Therefore, it is essential to understand the dynamics of these quantities. In this section, we will explore some simple infinitely wide neural network. In particular, this analysis will provide us insight for further developments in the deep Bayesian learning and PAC-Bayesian bounds.
7.1 Network Definition
We consider a feedforward network with  $d$  fully connected layers. The parameter  $\theta = \operatorname{vec}(W_1, \ldots, W_d)$  is a vector of flattened weights. For a specific input  $s_a$ , the forward propagation of the network is defined as follows:
$$
x _ {0} (\theta , s _ {a}) = s _ {a}, \quad y _ {i} (\theta , s _ {a}) = c _ {i} W _ {i} x _ {i - 1} (\theta , s _ {a}), \quad x _ {i} (\theta , s _ {a}) = \sigma_ {i} (y _ {i} (\theta , s _ {a})).
$$
The output  $y_{d}(\theta, s_{a})$  is a scalar. The point-wise function  $\sigma_{i}(y)$  is an activation function with bounded first to fourth order derivative. The hidden layer width at  $i$ -th layer is  $n_{i}$  such that  $y_{i}(\theta, s_{a}), x_{i}(\theta, s_{a}) \in \mathbb{R}^{n_{i}}$ .  $W_{i} \in \mathbb{R}^{n_{i} \times n_{i-1}}$  are trainable weights. The coefficient  $c_{i} = 1 / \sqrt{n_{i-1}}$  is used to ensure the output lies in a proper scale at infinite width limit, and is widely adopted in previous works [43, 51, 70, 40, 39, 3, 18, 59]. The network is initialized with Gaussian prior, such that  $(W_{i})_{j,k} \sim \mathcal{N}(0,1)$ .
We introduce the following notation to simplify the result. For a multi-variable scalar function  $f(X)$ ,  $\nabla_X f(X)$  is a row vector when  $X$  is a vector. If  $X$  is a matrix, then  $(\nabla_X f(X))_{i,j} = \frac{\partial f(X)}{\partial X_{j,i}}$  and  $\nabla_X f(X)$  share the same shape as  $X^T$ . For a multi-variable vector-value function  $F(X)$ , we define  $(\nabla_X F(X))_{i,j} = \frac{\partial F_i(X)}{\partial X_j}$ .
The network is trained by gradient flow or infinitesimal step size gradient descent. Let output vector be  $(Y(\theta))_a = y_d(\theta, s_a)$  and total loss function as  $\mathcal{L}(Y) = \sum_{a=1}^{m} l^s(Y_a, z_a)$ , the gradient flow is defined as:
$$
\frac {\mathrm {d}}{\mathrm {d} t} \theta (t) = - (\nabla_ {Y} \mathcal {L} (Y (\theta (t))) \nabla_ {\theta} Y (\theta (t))) ^ {T}.
$$
7.2 Energy Term
The energy for the prior is defined as  $V(\theta) = \sum_{i=1}^{d} \frac{1}{2} \| W_i\|_{\mathrm{Fro}}^2$ . We define  $y_d^{(i)}(\theta, s_a) = \mathrm{Tr}(\nabla_{W_i}y_d(\theta, s_a)W_i)$  and the dynamics of  $V(\theta(t))$  is:
$$
\frac {\mathrm {d}}{\mathrm {d} t} V (\theta (t)) = - \nabla_ {Y} \mathcal {L} (Y (\theta (t))) \sum_ {i = 1} ^ {d} Y ^ {(i)} (\theta (t)). \tag {7}
$$
We find that  $y_{d}^{(i)}(\theta, s_{a})$  can be computed by a group of auxiliary vector as Eq. (27) detailed in the appendix, and could be regarded as the tangent vector at output space by propagating a tangent vector  $W_{i}$  from  $i$ -th layer.
For deriving the dynamics of  $(Y^{(i)}(\theta))_a = y_d^{(i)}(\theta ,s_a)$ , we define  $\Theta^{(i)}(\theta) = (\nabla_{\theta}Y^{(i)}(\theta))(\nabla_{\theta}Y(\theta))^T$  and it follows
$$
\frac {\mathrm {d}}{\mathrm {d} t} Y ^ {(i)} (\theta (t)) = - \Theta^ {(i)} (\theta (t)) \left(\nabla_ {Y} \mathcal {L} \left(Y (\theta (t)))\right) ^ {T}. \right. \tag {8}
$$
We note that the definition of  $Y^{(d)}$  follows  $Y = Y^{(d)}$ , therefore  $\Theta^{(d)}$ , as a gradient Gram matrix, is just neural tangent kernel (NTK) which has been studied in [39, 3, 18].
7.3 Entropy Term
We write down the gradient flow in an abstract form  $\frac{\mathrm{d}}{\mathrm{d}t}\theta (t) = -g(\theta (t))$  and denote the solution as  $\theta (t) = f(t,\theta (0)) = \theta (0) - \int_0^t g(\theta (t))dt$ . This gradient flow is a special example of optimization flow in Section 3. Here the entropy change is  $\Delta_tS(\theta) = \log J_{f(t,)}(\theta)$ . According to Liouville formula, we have  $\frac{\mathrm{d}}{\mathrm{d}t} (\Delta_tS(\theta (0))) = -\mathbf{div}g(\theta (t)) = -\operatorname {Tr}(\nabla g(\theta (t)))$ . Given that the gradient has form  $g(\theta) = -(\nabla_Y\mathcal{L}(Y(\theta))\nabla_\theta Y(\theta))^T$ , the entropy change could be formulated as follows:
$$
\frac {\mathrm {d}}{\mathrm {d} t} \left(\Delta_ {t} S (\theta (0))\right) = - \operatorname {T r} \left(\nabla_ {Y} \nabla_ {Y} \mathcal {L} (Y (\theta (t))) \Theta^ {(d)} (\theta (t))\right) - \sum_ {a = 1} ^ {m} \nabla_ {Y _ {a}} \mathcal {L} (Y (\theta (t))) \operatorname {T r} \left(H _ {a} (\theta (t))\right). \tag {9}
$$
The content in the first trace term in Eq. (9) is called Gauss-Newton matrix, which only represents curvature information of loss  $l^s$ . The matrix  $H_{a} = \nabla_{\theta}\nabla_{\theta}Y_{a}$  is the Hessian of  $y_{d}(\theta ,s_{a})$ . It is known that spectral norm of Hessian vanishes for wide neural network [47]. However, we find that the trace is not constant and depends on initialization and training. For all  $1\leq i\leq j < k\leq d$ ,  $\alpha = 1,\ldots ,n_j$ , we introduce auxiliary vectors  $(\xi^{(i,j)})_{\alpha} = c_i^2\| x_{i - 1}\| ^2\| \nabla_{y_i}(y_j)_\alpha \| ^2$  and  $\gamma_k^{(j)} = (\nabla_{x_j}y_k)\sigma_j''(y_j)$ . Notice that we left out parameters  $(\theta ,s_a)$  for concision.
Theorem 4. Let  $\odot$  be element-wise product, we have
$$
\operatorname {T r} (H _ {a} (\theta (t))) = \sum_ {j = 1} ^ {d - 1} (\nabla_ {x _ {j}} y _ {d}) \left(\sigma_ {j} ^ {\prime \prime} (y _ {j}) \odot \left(\sum_ {i = 1} ^ {j} \xi^ {(i, j)}\right)\right).
$$
The above formula is a multi-variable version of  $(f_{d - 1} \circ \dots \circ f_1)'' = \sum_{j = 1}^{d - 1}(f_{d - 1} \circ \dots \circ f_{j + 1})'f_j''((f_{j - 1} \circ \dots \circ f_1)')^2$ .  $\xi^{(i,j)}$  plays the role of squared gradient and is a kind of diagonal NTK in middle layers.
We define  $(\Gamma^{(i)}(\theta))_a = \gamma_d^{(i)}(\theta, s_a)$  and matrices  $\Phi^{(i)}(\theta) = (\nabla_\theta \Gamma^{(i)}(\theta))(\nabla_\theta Y(\theta))^T$ . Then we have for all  $1 \leq i < d$ ,
$$
\frac {\mathrm {d}}{\mathrm {d} t} \Gamma^ {(i)} (\theta (t)) = - \Phi^ {(i)} (\theta (t)) (\nabla_ {Y} \mathcal {L} (Y (\theta (t)))) ^ {T}. \tag {10}
$$
7.4 Infinite Width Limit
The dynamics in previous two sections can be dramatically simplified in infinite width limit.
Theorem 5. When hidden layers width  $n = n_1, \ldots, n_{d-1}$  approaches infinity, the random vectors  $Y^{(i)}(\theta(0))$  and  $\Gamma^{(i)}(\theta(0))$  at initialization converge in law to a multivariate Gaussian distribution with zero mean and covariance matrix  $\Sigma$ . Moreover, the following quantities converge in probability to constant and don't change during finite training time:
$$
\Theta^ {(i)} (\theta (t)) \rightarrow \Theta^ {(i)}, \quad \Phi^ {(i)} (\theta (t)) \rightarrow \Phi^ {(i)}, \quad \left(\xi^ {(i, j)} \left(\theta (t), s _ {a}\right)\right) _ {\alpha} \rightarrow \left(\Xi^ {(i, j)}\right) _ {a}. \tag {11}
$$
The specific formulas of  $\Sigma$ ,  $\Theta^{(i)}$ ,  $\Phi^{(i)}$ , and  $\Xi^{(i,j)}$  are shown in Appendix D.
The last line of above limits is for all  $\alpha = 1,\dots ,n_j$ , therefore theorem 4 could be simplified to:
$$
\operatorname {T r} (H _ {a} (\theta (t))) = \sum_ {j = 1} ^ {d - 1} (\Gamma^ {(j)} (\theta (t))) _ {a} \left(\sum_ {i = 1} ^ {j} (\Xi^ {(i, j)}) _ {a}\right).
$$
Theorem 5 demonstrates a function space picture, where the dynamics of  $Y^{(i)}(\theta(t))$  and  $\Gamma^{(i)}(\theta(t))$  only depend on these values themselves but not on internal dynamics of the network. The auxiliary vectors  $Y^{(i)}(\theta(t))$  and  $\Gamma^{(i)}(\theta(t))$  could be obtained by solving ODEs (8) and (10),  $\frac{\mathrm{d}}{\mathrm{d}t} V$  and  $\frac{\mathrm{d}}{\mathrm{d}t} S$  can be integrated according to Eqs. (7) and (9). Therefore, the KL divergence is accessible as an expectation in Eq. (1). Based on this KL divergence, we can obtain the final result of the PAC Bayes bound formulated as a function of training time and training data. The complete formula of this PAC Bayes bound is shown in Appendix F.4.
In order to get a sense of the how the generalization bound and sampling efficiency are influenced by training time, we show the asymptotic behaviour under mild assumption on loss function.
Figure 2: Comparison of dynamics in parameter space and function space.
Corollary 6. If there exist  $C_1, C_2$  such that  $\left|\frac{\mathrm{d}}{\mathrm{d}y} l^s(y,t)\right| \leq C_1$ ,  $\left|\frac{\mathrm{d}^2}{\mathrm{d}y^2} l^s(y,t)\right| \leq C_2$ , then  $D_{\mathrm{KL}}(q_t\|p) \leq \mathcal{O}(1)t + \mathcal{O}(1)t^2$ ,  $\mathrm{eff}_{\lambda} \geq \mathcal{O}(1)\exp(-\mathcal{O}(1)t - \mathcal{O}(1)t^2)$  for any finite  $t$ , where  $\mathcal{O}(1)$  represents positive constant irrelevant to  $t$ ,  $q_t$  is the output distribution of an infinitely wide network after training for  $t$  length of time, and  $\mathrm{eff}_{\lambda}$  is the sampling efficiency of importance sampling with  $q_t$  as proposal distribution.
We first illustrate that Hessian trace doesn't vanish for overparameterized network and our analysis induces an efficient estimation of this value. Next, we verify our theoretical finding by comparing the dynamics of an overparameterized network in function space and parameter space. Finally, we demonstrate the interpolation of sampling and optimization.
8.1 Non-Diminishing Hessian Trace and Efficient Estimation
We consider a three hidden layers feedforward network with 2048 as hidden layer width and tanh as nonlinear function.
The synthetic dataset is constructed on a unit circle. We choose 20 values  $\tau$  uniformly distributed between 0 to  $\pi$  and we let 2-dimensional input to the network be  $s(\tau) = [\cos(\tau)\sin(\tau)]$ .
Figure 1: Hessian trace for one randomly initialized network (left) and the probability density of Hessian trace at initialization (right).
For each input  $s(\tau)$ , we are concerned about Hessian trace  $\mathrm{Tr}(\nabla_{\theta}\nabla_{\theta}y_d(\theta ,s(\tau)))$ . Fast Hessian trace estimation by itself is a challenging problem, due to the high dimensional parameters and complex structure of the network. We
follow previous works [36, 6, 78] and adopt Hutchinson's method to estimate the ground truth value of Hessian trace.
On the other hand, our analysis on the dynamics of overparameterized network reveals that certain factors which Hessian trace depends on are insensitive to initialization and training, therefore could be estimated by a fixed value to reduce computation. More details are shown in Appendix F.3. We apply the Hutchinson's method with 1000 independent random Rademacher vectors. The  $3\sigma$  interval is plotted in the Figure 1. It is shown that our approximation aligns with ground truth well. Based on this approximation, the distribution of Hessian is also calculated in Figure 1.
8.2 Comparing the Dynamics in Parameter Space and Function Space
We use the same toy model as the previous section and set target values to be  $t(\tau) = \frac{1}{2}\sin (3\tau)$  and loss to be mean squared error (MSE). For dynamics in parameter space, we run SGD with finite step size 0.01 and mini-batch size 1. For dynamics in function space, we solve Eqs. (7) to (10) with fixed matrices  $\Theta^{(i)}$ ,  $\Phi^{(i)}$  and  $\Xi^{(i,j)}$ . The result is plotted in Figure 2, where for SGD, the training time is defined as step size times iteration number. The ground truth of Hessian trace on one input is again estimated by Hutchinson's method. We can see from all three kinds of outputs that the function space
dynamics produces similar output as SGD. The error between them is attributed to discretization error for finite step size and finite width fluctuation.
8.3 Interpolation of Optimization and Bayesian Learning
We consider one-shot learning on Fashion-MNIST [75]. We randomly select two classes for binary classification and select one sample for each class as training dataset. We use a single hidden layer network with width being 1024 and softmax activation. We use loss  $l(y,t) = 1 / (1 + \exp (yt))$  and surrogate loss  $l^{s}(y,t) = \log (1 + \exp (-yt))$  for gradient descent. For Gibbs measure, we fix  $\lambda = 180$ . The entropy change is approximately evaluated by integrating Eq. (9) with finite step size and fixed  $\Theta^{(d)}$ .
We train  $10^{5}$  independent network and the results are shown in Figure 3. The shadow region represents  $3\sigma$  interval. We first notice that during the training, the expectation of TransBL prediction is stable, yet the variance decreases, indicating the distribution of ensemble is becoming closer to Gibbs measure. This could be verified in middle figure in Figure 3, as the distribution of weights concentrates toward the partition function  $\log Z_{\lambda} \approx -43.6$ .
Figure 3: Test accuracy for ensemble trained by GD and TransBL with  $\beta = \exp(-40)$  for weight clipping (left). Distribution of weights in TransBL (middle). Trade-off between sampling efficiency and test accuracy (right).
We also show a clear trade-off between sampling efficiency and test accuracy for TransBL with weight clipping, where the parameter  $\beta$  plays a pivotal role in balancing these aspects. When  $\beta$  is set high, the model leans towards a Bayesian posterior framework. Although this setting noticeably reduces the number of effective samples obtained, we observed an enhancement in the model's generalization ability, as depicted in the right figure of Figure 3. Conversely, with a smaller  $\beta$ , the model's behavior tends to resemble that of a typical deep ensemble, indicating a high sampling efficiency but worse generalization.
We study the generalization and sampling efficiency of Bayesian learning with special attention to overparameterized network. We show that both KL divergence, which governs generalization in PAC-Bayesian theory, and  $\chi^2$  divergence, which determines sampling efficiency for importance sampling, are related to the change of energy and entropy loss during training. The dynamics of these two quantities in DNNs are studied, leading to a function space picture in infinite width limit. Our study also contributes to the understanding of DNNs Hessian trace, due to its involvement in entropy loss. By considering importance sampling with output distribution from gradient-based optimization as proposal distribution, we bridge the gap between optimization and Bayesian learning problems, and provide a flexible interpolation for accuracy-computation trade-off.
This work was partially supported by NSF IIS 1838627, 1837956, 1956002, 2211492, CNS 2213701, CCF 2217003, DBI 2225775.
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Generalization of Overparameterized Network Generalization for neural network has been studied based on stability [11, 12] norm [7, 57, 41, 27, 45], compression [4], and special function class [5, 2]. [21, 81] achieves non-vacuous PAC-Bayesian generalization bounds based compression and perturbation robustness. Clerico et al. [15] derives PAC-Bayes bound and learning method for training a Gaussian process. A realizable PAC-Bayesian bound is also derived in [8] with sampling realizable networks instead of training by gradient descent. Mean field theory [22, 60, 66] could also be applied to obtain generalization bound.
Early Stopping Restricting training time is a popular trick in deep learning. Related theoretical explanation includes a uniform stability approach [30], which related stability generalization bounds with training time, and a variational inference approach [19], which bound marginal likelihood with entropy. Compared to these works, our Corollary 6, although only applies for infinitely wide network, provides two new points of view. First we relates KL divergence with training time, thus provide a PAC-Bayes explanation for early stop. Second, we control the worst-case sampling efficiency with training time, thus provide a novel computational argument.
Bayesian Deep Learning A comprehensive introduction of Bayesian Deep Learning and its relation to generalization is provided in [74]. It is known that sampling from the Gibbs measure is costly [37]. Bulk of work on Bayesian deep learning based on variational approximation [49, 33, 28, 9] has been developed. Despite the theoretical difficulty of exact Bayesian deep learning, various heuristic methods including SGD [48, 54], deep ensemble [23], and cyclic step size [80] have been shown to be successful in practice.
Hessian of Overparameterized Network The convergence of various stochastic Hessian trace estimation methods are analyzed in [6]. There has been work focusing on empirical evaluation of Hessian of neural network [78]. Existing result in [47] shows that spectral norm of Hessian vanish in infinite width limit. Our paper contributes to this fields in two aspect. First, we show that Hessian trace doesn't vanish in infinite width limit. Second, we propose a novel Hessian trace estimation method in Appendix F.3, which is much more efficient than commonly used Hutchinson's method.
MCMC : Markov Chain Monte Carlo (MCMC) utilize a kernel that leaves Bayesian posterior as stationary distribution. One source of bias of MCMC is finite number of iterations. The mixing time of MCMC suffer from curse of dimension, making it impractical to run MCMC till converge. Several orbit MCMC [68, 56] has been explored, seeking to incorporate non-equilibrium map into the transition kernel to reduce mixing time. They typically use damped Hamiltonian system or momentum gradient descent to bypass the entropy loss due to network curvature, thus suffer from increased  $\chi^2$  divergence as discussed in Appendix F.2. Telescopic sum argument [26] with coupling technique [38, 32] have also been proposed to tackle this problem. Another source of bias is inaccurate implementation of transfer kernel. The discretization error of various Langevin dynamics method has been analyzed [1, 16, 64]. The bias from noisy loss function [62] and noisy gradient [72, 14] could be reduced but could not vanish. Papamarkou et al. [58] summarize more challenges of MCMC that leads to lack of convergence.
B Discussion on the strength of assumption
We required  $f$  to be a bijective and  $J_{f}$  exist everywhere. This assumption is not as restrictive as it seems.