| Data set | q(z2|x) | q(z1|z2,x) | p(z1|z2) | p(x|z1) |
| SVHN/CIFAR-10 | Share:
+Conv(3, 32, 4, 2, 1),
+Conv(32, 32, 4, 2, 1),
+Conv(32, 32, 4, 2, 1) | For x:
+Conv(3, 32, 4, 2, 1),
+Conv(32, 32, 4, 2, 1) | Share:
+FC(In=32, Out=256) | ConvTransp(32, 32, 4, 2, 1),
+ConvTransp(32, 32, 4, 2, 1),
+ConvTransp(32, 3, 4, 2, 1) |
| For mean:
+FC(In=512, Out=32) | Share:
+ConvTransp(64, 32, 4, 2, 1) | For mean:
+FC(In=256, Out=512) |
| For variance:
+FC(In=512, Out=32) | For mean:
+Conv(32, 32, 3, 1, 1) | For variance:
+FC(In=256, Out=512) |
| For variance:
+Conv(32, 32, 3, 1, 1) | |
| z1 dims: 512 | z2 dims: 32 | σx=0.71 | Total params: 475811 |
| MNIST (Binary) | Share:
+Conv(1, 16, 4, 2, 1),
+Conv(16, 16, 4, 2, 1),
+Conv(16, 16, 4, 1, 0) | For x:
+Conv(1, 16, 4, 2, 1),
+Conv(16, 16, 4, 1, 0) | Share:
+FC(In=20, Out=128) | ConvTransp(16, 16, 4, 1, 0),
+ConvTransp(16, 16, 4, 2, 1),
+ConvTransp(16, 1, 4, 2, 1) |
| For mean:
+FC(In=256, Out=20) | For mean:
+ConvTransp(32, 16, 4, 1, 0) | For mean:
+FC(In=128, Out=256) |
| For variance:
+FC(In=256, Out=20) | For variance:
+Conv(16, 16, 3, 1, 1) | For variance:
+FC(In=128, Out=256) |
| For variance:
+Conv(16, 16, 3, 1, 1) | |
| z1 dims: 256 | z2 dims: 20 | σx: N/A | Total params: 122713 |
+
+Table 3: Model architecture details for LVAEs used in Section 5. Conv and ConvTransp denote the convolutional and transposed convolutional layer, which has the corresponding input: input channel, output channel, kernel size, stride, padding. FC represents fully connected layer.
+
+| Data set | q(z2|x) | q(z1|z2,x) | p(z1|z2) | p(x|z1) |
| SVHN/CIFAR-10 | Share:
+Conv(3, 32, 4, 2, 1),
+Conv(32, 32, 4, 2, 1),
+Conv(32, 32, 4, 2, 1) | Involve d:
+Conv(32, 32, 4, 2, 1) | Share:
+FC(In=32, Out=256) | ConvTransp(32, 32, 4, 2, 1),
+ConvTransp(32, 32, 4, 2, 1),
+ConvTransp(32, 3, 4, 2, 1) |
| For mean:
+Conv(32, 32, 3, 1, 1) | For mean:
+FC(In=256, Out=512) |
| For mean:
+FC(In=512, Out=32) | For variance:
+Conv(32, 32, 3, 1, 1) | For variance:
+FC(In=256, Out=512) |
| For variance:
+FC(In=512, Out=32) | | | |
| z1 dims: 512 | z2 dims: 32 | σx=0.71 | Total params: 408131 |
+
+# A.2 ADDITIONAL RESULTS
+
+Here we attached the results for MNIST, as well as the full results for LVAE on SVHN and generalized top-down HVAEs on CIFAR-10.
+
+# A.2.1 RESULTS FOR GENERALIZED TOP-DOWN HVAES ON MNIST
+
+We also evaluate our proposed framework using generalized top-down HVAEs trained on binary MNIST data (i.e., black and white images rather than grayscale).
+
+We note that the inception score (IS) behaves different in our MNIST models compared to SVHN (see Figure 5) in that optimal IS in MNIST occurs for high $R(z_{1}|z_{2})$ rather than high $R(z_{2})$ . This indicates that semantically low-level properties (hand-writing style) of MNIST might have more variation than high level properties (the digit), whereas SVHN images show variation in additional high-level properties such as the background color.
+
+
+
+
+
+
+
+
+Figure 8: Trade-offs between rates and all metrics we used in Section 5 from the generalized top-down HVAEs trained with MNIST. The results from the standard VAE (i.e. $\beta_{2} = \beta_{1} = 1$ ) and the $\beta$ -VAE (i.e. $\beta_{2} = \beta_{1}$ ) are marked with “o” and “o”. The markers “o” highlight the optimal models selected using convex hull (see Figure 6 for details). The diagonal grid lines are references for equivalent total rates, i.e. points on the same line have the same total rates.
+
+
+
+
+
+
+
+# A.2.2 RESULTS FOR LVAE ON SVHN
+
+
+
+
+
+
+
+
+Figure 9: Trade-offs between rates and all metrics we used in Section 5 from LVAE trained with SVHN. The results from the standard VAE (i.e. $\beta_{2} = \beta_{1} = 1$ ) and the $\beta$ -VAE (i.e. $\beta_{2} = \beta_{1}$ ) are marked with “o” and “o”. The markers “o” highlight the optimal models selected using convex hull (see Figure 6 for details). The diagonal grid lines are references for equivalent total rates, i.e. points on the same line have the same total rates.
+
+
+
+
+
+
+
+# A.2.3 RESULTS FOR GENERALIZED TOP-DOWN HVAES ON CIFAR-10
+
+
+
+
+
+
+
+
+Figure 10: Trade-offs between rates and all metrics we used in Section 5 from the generalized top-down HVAEs trained with CIFAR-10. The results from the standard VAE (i.e. $\beta_{2} = \beta_{1} = 1$ ) and the $\beta$ -VAE (i.e. $\beta_{2} = \beta_{1}$ ) are marked with “o” and “o”. The markers “o” highlight the optimal models selected using convex hull (see Figure 6 for details). The diagonal grid lines are references for equivalent total rates, i.e. points on the same line have the same total rates.
+
+
+
+
+
+
+
+# B PROOF OF THE BOUND ON CLASSIFICATION ACCURACY
+
+This section provides a proof of Eq. 11 by reformulating the proof of Proposition 5 in the thesis by Meyen (2016) into the notation used in the present paper. We stress that this section contains no original contribution and is provided only as a convenience to the reader, motivated by reviewer feedback. All credits for this section belong to Meyen (2016).
+
+We consider an (unknown) true data generative distribution $p_{\mathrm{data}}(y, \boldsymbol{x})$ for data $\boldsymbol{x}$ with (unobserved) true labels $y$ , and a hierarchical VAE with an inference model $q_{\phi}(\{\boldsymbol{z}_{\ell}\} \mid \boldsymbol{x})$ of the form of Eq. 4. Focusing on a single layer $\ell$ of latents, we denote the joint probability over $y$ , $\boldsymbol{x}$ , and $\boldsymbol{z}_{\ell}$ as
+
+$$
+q (y, \boldsymbol {x}, \boldsymbol {z} _ {\ell}) := p _ {\text {d a t a}} (y, \boldsymbol {x}) q _ {\phi} (\boldsymbol {z} _ {\ell} | \boldsymbol {x}) \tag {13}
+$$
+
+where the marginal $q_{\phi}(\pmb{z}_{\ell}|\pmb{x})$ of $q_{\phi}(\{\pmb{z}_{\ell}\}|\pmb{x})$ is defined as usual. We further consider a classifier $p_{\mathrm{cls.}}(y|\pmb{z}_{\ell})$ that operates on $\pmb{z}_{\ell}$ . Denoting its top prediction as $\hat{y} \coloneqq \arg \max_y p_{\mathrm{cls.}}(y|\pmb{z}_{\ell})$ , the classification accuracy is $\alpha \coloneqq \mathbb{E}_q[\delta_{y,\hat{y}}]$ , where $\delta$ is the Kronecker delta.
+
+Theorem 1. The mutual information $I_{q}(y;z_{\ell})$ between the latent representation $z_{\ell}$ and the true label $y$ under the distribution $q$ defined in Eq. 13 is lower bounded as follows,
+
+$$
+I _ {q} (y; \boldsymbol {z} _ {\ell}) \geq f (\alpha) \quad \text {w i t h} \quad f (\alpha) = H _ {p _ {\text {d a t a}}} [ y ] - H _ {2} (\alpha) - (1 - \alpha) \log (M - 1) \tag {14}
+$$
+
+where $H_{2}(\alpha) = -\alpha \log \alpha - (1 - \alpha)\log(1 - \alpha)$ is the entropy of a Bernoulli distribution, $H_{p_{data}}[y] \leq \log M$ is the marginal entropy of the true labels, and $M$ denotes the number of classes.
+
+Before we prove Theorem 1, we note that the function $f$ is strictly monotonically increasing on the relevant interval $[\max_y p_{\mathrm{data}}(y), 1]$ . Thus, $f$ is invertible and we obtain the following corollary:
+
+Corollary 1. The classification accuracy $\alpha$ is upper bounded as in Eq. 11 of the main text, i.e.,
+
+$$
+\alpha \leq f ^ {- 1} \left(I _ {q} (y; \boldsymbol {z} _ {\ell})\right) \leq f ^ {- 1} \left(\mathbb {E} _ {p _ {\text {d a t a}} (\boldsymbol {x})} [ R (\boldsymbol {z} _ {\geq \ell}) ]\right). \tag {15}
+$$
+
+The second inequality in Eq. 15 results from the bound $I_{q}(y;z_{\ell}) \leq \mathbb{E}_{p_{\mathrm{data}}(\boldsymbol{x})}[R(\boldsymbol{z}_{\geq \ell})]$ derived in Eq. 10, using the fact that $f^{-1}$ is monotonically increasing (since $f$ is).
+
+Proof of Theorem 1. We split the mutual information into two contributions,
+
+$$
+I _ {q} (y; \boldsymbol {z} _ {\ell}) = H _ {p _ {\mathrm {d a t a}}} [ y ] - H _ {q} [ y | \boldsymbol {z} _ {\ell} ] = H _ {p _ {\mathrm {d a t a}}} [ y ] - \mathbb {E} _ {\boldsymbol {z} _ {\ell} \sim q (\boldsymbol {z} _ {\ell})} \left[ \mathbb {E} _ {y \sim q (y | \boldsymbol {z} _ {\ell})} [ - \log q (y | \boldsymbol {z} _ {\ell}) ] \right] \tag {16}
+$$
+
+where, as clarified in the second equality, $H_{q}[y|\pmb{z}_{\ell}]$ is the expectation over $\pmb{z}_{\ell}$ of the conditional entropy of $y$ given $\pmb{z}_{\ell}$ , and $q(\pmb{z}_{\ell})$ and $q(y|\pmb{z}_{\ell})$ are marginals and conditionals of $q$ (Eq. 13) as usual.
+
+Since $H_{p_{\mathrm{data}}}[\boldsymbol {y}]$ is fixed by the problem at hand, finding a lower bound on $I_q(y;z_\ell)$ for a given classification accuracy $\alpha$ is equivalent to finding an upper bound on the second term on the right-hand side of Eq. 16, $H_{q}[y|\boldsymbol{z}_{\ell}] = \mathbb{E}_{\boldsymbol{z}_{\ell}\sim q(\boldsymbol{z}_{\ell})}[\mathbb{E}_{y\sim q(y|\boldsymbol{z}_{\ell})}[-\log q(y|\boldsymbol{z}_{\ell})]]$ , with the constraint $\mathbb{E}_q[\delta_{y,\hat{g}}] = \alpha$ . We do this by upper bounding the conditional entropy $\mathbb{E}_{y\sim q(y|\boldsymbol{z}_{\ell})}[-\log q(y|\boldsymbol{z}_{\ell})]$ of $y$ given $\boldsymbol{z}_{\ell}$ for all $\boldsymbol{z}_{\ell}$ independently, and then taking the expectation over $\boldsymbol{z}_{\ell}\sim q(\boldsymbol{z}_{\ell})$ .
+
+For a fixed latent representation $\mathbf{z}_{\ell}$ , we first split off the contribution to $\mathbb{E}_{y \sim q(y|\mathbf{z}_{\ell})}[-\log q(y|\mathbf{z}_{\ell})]$ from $y = \hat{y}$ , where $\hat{y} = \arg \max_{y} p_{\mathrm{cls.}}(y|\mathbf{z}_{\ell})$ is the label that our classifier would predict for $\mathbf{z}_{\ell}$ ,
+
+$$
+\mathbb {E} _ {y \sim q (y | \boldsymbol {z} _ {\ell})} [ - \log q (y | \boldsymbol {z} _ {\ell}) ] = - q (y = \hat {y} | \boldsymbol {z} _ {\ell}) \log q (y = \hat {y} | \boldsymbol {z} _ {\ell}) - \sum_ {y \neq \hat {y}} q (y | \boldsymbol {z} _ {\ell}) \log q (y | \boldsymbol {z} _ {\ell}). \tag {17}
+$$
+
+Here, the second term on the right-hand side resembles the entropy of a distribution over the remaining $(M - 1)$ labels $(y\neq \hat{y})$ , except that the probabilities sum to $(1 - q(y = \hat{y} |z_{\ell}))$ rather than one. Thus, regardless of the value of $q(y = \hat{y} |z_{\ell})$ , this term is maximized if $q(y|z_{\ell})$ distributes the remaining probability mass $(1 - q(y = \hat{y} |z_{\ell}))$ uniformly over the remaining $(M - 1)$ labels, i.e.,
+
+$$
+\begin{array}{l} \mathbb {E} _ {y \sim q (y | \boldsymbol {z} _ {\ell})} [ - \log q (y | \boldsymbol {z} _ {\ell}) ] \leq - q (y = \hat {y} | \boldsymbol {z} _ {\ell}) \log q (y = \hat {y} | \boldsymbol {z} _ {\ell}) - (1 - q (y = \hat {y} | \boldsymbol {z} _ {\ell})) \log \frac {1 - q (y = \hat {y} | \boldsymbol {z} _ {\ell})}{M - 1} \\ = H _ {2} \left(q \left(y = \hat {y} \mid \boldsymbol {z} _ {\ell}\right)\right) + \left(1 - q \left(y = \hat {y} \mid \boldsymbol {z} _ {\ell}\right)\right) \log (M - 1). \tag {18} \\ \end{array}
+$$
+
+Plugging Eq. 18 back into Eq. 16, we obtain the bound
+
+$$
+I _ {q} (y; \boldsymbol {z} _ {\ell}) \geq H _ {p _ {\text {d a t a}}} [ y ] - \mathbb {E} _ {\boldsymbol {z} _ {\ell} \sim q (\boldsymbol {z} _ {\ell})} \left[ H _ {2} \left(q \left(y = \hat {y} \mid \boldsymbol {z} _ {\ell}\right)\right) \right] - \mathbb {E} _ {\boldsymbol {z} _ {\ell} \sim q (\boldsymbol {z} _ {\ell})} \left[ 1 - q \left(y = \hat {y} \mid \boldsymbol {z} _ {\ell}\right) \right] \log (M - 1). \tag {19}
+$$
+
+We arrive at the proposition (Eq. 14) by pulling the concave function $H_{2}$ out of the expectation using Jensen's inequality, and by then identifying $\mathbb{E}_{\boldsymbol{z}_{\ell}\sim q(\boldsymbol{z}_{\ell})}[q(y = \hat{y} |\boldsymbol{z}_{\ell})] = q(y = \hat{y}) = \alpha$ .
\ No newline at end of file
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+# TRAINABILITY PRESERVING NEURAL PRUNING
+
+Huan Wang| Method | Model | Unpruned top-1 (%) | Pruned top-1 (%) | Top-1 drop (%) | Speedup |
| L1(pruned-B) (Li et al., 2017) | | 73.23 | 72.17 | 1.06 | 1.32× |
| L1(pruned-B, reimpl.) (Wang et al., 2023) | | 73.31 | 73.67 | -0.36 | 1.32× |
| Taylor-FO (Molchanov et al., 2019) | ResNet34 | 73.31 | 72.83 | 0.48 | 1.29× |
| GReg-2 (Wang et al., 2021b) | | 73.31 | 73.61 | -0.30 | 1.32× |
| TPP (ours) | | 73.31 | 73.77 | -0.46 | 1.32× |
| ProvableFP (Liebenwein et al., 2020) | | 76.13 | 75.21 | 0.92 | 1.43× |
| MetaPruning (Liu et al., 2019a) | ResNet50 | 76.6 | 76.2 | 0.4 | 1.37× |
| GReg-1 (Wang et al., 2021b) | 76.13 | 76.27 | -0.14 | 1.49× |
| TPP (ours) | | 76.13 | 76.44 | -0.31 | 1.49× |
| IncReg (Wang et al., 2019) | | 75.60 | 72.47 | 3.13 | 2.00× |
| SFP (He et al., 2018) | | 76.15 | 74.61 | 1.54 | 1.72× |
| HRank (Lin et al., 2020) | | 76.15 | 74.98 | 1.17 | 1.78× |
| Taylor-FO (Molchanov et al., 2019) | | 76.18 | 74.50 | 1.68 | 1.82× |
| Factorized (Li et al., 2019) | | 76.15 | 74.55 | 1.60 | 2.33× |
| DCP (Zhuang et al., 2018) | ResNet50 | 76.01 | 74.95 | 1.06 | 2.25× |
| CCP-AC (Peng et al., 2019) | | 76.15 | 75.32 | 0.83 | 2.18× |
| GReg-2 (Wang et al., 2021b) | | 76.13 | 75.36 | 0.77 | 2.31× |
| CC (Li et al., 2021) | | 76.15 | 75.59 | 0.56 | 2.12× |
| MetaPruning (Liu et al., 2019a) | | 76.6 | 75.4 | 1.2 | 2.00× |
| TPP (ours) | | 76.13 | 75.60 | 0.53 | 2.31× |
| LFPC (He et al., 2020) | | 76.15 | 74.46 | 1.69 | 2.55× |
| GReg-2 (Wang et al., 2021b) | ResNet50 | 76.13 | 74.93 | 1.20 | 2.56× |
| CC (Li et al., 2021) | | 76.15 | 74.54 | 1.61 | 2.68× |
| TPP (ours) | | 76.13 | 75.12 | 1.01 | 2.56× |
| IncReg (Wang et al., 2019) | | 75.60 | 71.07 | 4.53 | 3.00× |
| Taylor-FO (Molchanov et al., 2019) | ResNet50 | 76.18 | 71.69 | 4.49 | 3.05× |
| GReg-2 (Wang et al., 2021b) | | 76.13 | 73.90 | 2.23 | 3.06× |
| TPP (ours) | | 76.13 | 74.51 | 1.62 | 3.06× |
| Method | Network | Top-1 (%) | FLOPs (G) |
| CHEX* (Hou et al., 2022) | | 77.4 | 2 |
| CHEX* (Hou et al., 2022) | ResNet50 | 76.0 | 1 |
| TPP* (ours) | | 77.75 | 2 |
| TPP* (ours) | | 76.52 | 1 |
+
+compensating isometry after pruning can help recover trainability. In stark contrast, our TPP method outperforms $L_{1}$ pruning and scratch consistently against different pruning ratios (only one exception is pruning ratio 0.7 on ResNet56, but our method is still the second best and the gap to the best is only marginal: 93.46 vs. 93.51). Besides, note that the accuracy trend – in general, with a larger sparsity ratio, TPP beats $L_{1}$ or Scratch by a more pronounced margin. This is because, ar a larger pruning ratio, the trainability is impaired more, where our method can help more, thus harvesting more performance gains. We will see similar trends many times.
+
+(4) In Tabs. 1 and 10, we also present the results when the initial retraining LR is $1 \mathrm{e} - 3$ . Wang et al. (2021a) argue that if the broken dynamical isometry can be well maintained/recovered, the final performance gap between LR $1 \mathrm{e} - 2$ and $1 \mathrm{e} - 3$ should be diminished. Now that TPP is claimed to be able to maintain trainability, the performance gap should become smaller. This is empirically verified in the table. In general, the accuracy gap between LR $1 \mathrm{e} - 2$ and LR $1 \mathrm{e} - 3$ of TPP is smaller than that of $L_{1}$ pruning. Two exceptions are PR 0.9/0.95 on ResNet56: LR $1 \mathrm{e} - 3$ is unusually better than LR $1 \mathrm{e} - 2$ for $L_{1}$ pruning. Despite them, the general picture is that the accuracy gap between LR $1 \mathrm{e} - 3$ and $1 \mathrm{e} - 2$ turns smaller with TPP. This is a sign that trainability is effectively maintained.
+
+# 4.3 IMAGENET BENCHMARK
+
+We further evaluate TPP on ImageNet-1K (Deng et al., 2009) in comparison to many existing filter pruning algorithms. Results in Tab. 2 show that TPP is consistently better than the others across different speedup ratios. Moreover, under larger speedups, the advantage of our method is usually more evident. E.g., TPP outperforms Taylor-FO (Molchanov et al., 2019) by $1.15\%$ in terms of the top-1 acc. drop at the $2.31 \times$ speedup track; at $3.06 \times$ speedup, TPP leads Taylor-FO (Molchanov et al., 2019) by $2.87\%$ . This shows TPP is more robust to more aggressive pruning. The reason is easy to see - more aggressive pruning hurts trainability more (Lee et al., 2019; Wang et al., 2023), where our method can find more use, in line with the observations on CIFAR (Tabs. 1 and 10).
+
+Table 3: Test accuracy (without retraining) comparison between two plausible schemes diagonal vs. decorrelate in our TPP method.
+
+| Model | Speedup | Pruned top-1 acc. (%) | PR |
| ResNet34 | 1.32× | 73.77 | [0, 0.50, 0.60, 0.40, 0, 0] |
| ResNet50 | 1.49× | 76.44 | [0, 0.30, 0.30, 0.30, 0.14, 0] |
| ResNet50 | 2.31× | 75.60 | [0, 0.60, 0.60, 0.60, 0.21, 0] |
| ResNet50 | 2.56× | 75.12 | [0, 0.74, 0.74, 0.60, 0.21, 0] |
| ResNet50 | 3.06× | 74.51 | [0, 0.68, 0.68, 0.68, 0.50, 0] |
+
+# B SENSITIVITY ANALYSIS OF HYPER-PARAMETERS
+
+Among the three hyper-parameters in Tab. 6, regularization ceiling $\tau$ works as a termination condition. We only require it to be large enough to ensure the weights are compressed to a very small amount. It does not have to be 1. The final performance is also less sensitive to it. The pruned performance seems to be more sensitive to the other two hyper-parameters, so here we conduct hyper-parameter sensitivity analysis to check their robustness.
+
+Results are presented in Tab. 8 and Tab. 9. Pruning ratio 0.7 (for ResNet56) and 0.5 (for VGG19) are chosen here because the resulted sparsity is the most representative (i.e., not too large or small). (1) For $K_{u}$ , in general, a larger $K_{u}$ tends to deliver a better result. This is no surprise since a larger $K_{u}$ allows more iterations for the network to adapt and recover when undergoing the penalty. (2) For $\Delta$ , we do not see a clear pattern here: either a small or large $\Delta$ can achieve the best result (for different networks). On the whole, when varying the hyper-parameters within a reasonable range, the performance is pretty robust, no catastrophic failures. Moreover, note that the default setting is actually not the best for both $K_{u}$ and $\Delta$ . This is because we did not heavily search the best hyper-parameters; however, they still achieve encouraging performance compared to the counterpart methods, as we have shown in the main text.
+
+Table 8: Robustness analysis of ${K}_{u}$ on the CIFAR10 and 100 datasets with our TPP algorithm. In default, ${K}_{u} = {10}$ . Layerwise $\mathrm{{PR}} = {0.7}$ for ResNet56 and 0.5 for VGG19. The best is highlighted in red and the worst in blue.
+
+ | Constituency Parser | Scene Graph Parser |
| Example 0 | CC-500 Prompt: A white sheep and a red car |
| “A white sheep”, “a red car” | “A white sheep”, “a red car” |
| Example 1 | Prompt: A silver car with a black cat sleeping on top of it |
| “A silver car”, “a black cat”, “A silver car with a black cat” | “A silver car”, “a black cat”, “top of it”, “a black cat sleeping on top of it” |
| Example 2 | Prompt: A horse running in a white field next to a black and green pole |
| “A horse”, “a white field”, “a black and green pole”, “a white field next to a black and green pole” | “A horse”, “a white field”, “a black and green pole”, “A horse running in a white field” |
| Example 3 | Prompt: Rice with red sauce with eggs over the top and orange slices on the side |
| “red sauce”, “the side”, “the top and orange slices”, “the top and orange slices on the side” | “red sauce”, “the side”, “the top and orange slices”, “Rice with red sauce”, “red sauce with eggs”, “the top and orange slices on the side”, “red sauce with eggs over the top and orange slices” |
| Example 4 | Prompt: A pink scooter with a black seat next to a blue car |
| “A pink scooter”, “a black seat”, “a blue car” | “A pink scooter”, “a black seat”, “a blue car”, “a pink scooter with a black seat”, “a black seat next to a blue car” |
+
+Table 3: Comparison between the constituency parser and scene graph parser. For CC-500 prompts, both parsers end up with the same results. As for general prompts, scene graph parser tends to generate more text spans with middle concepts appearing multiple times across different spans.
+
+
+Figure 12: Synthesized images corresponding to prompts in Table 3. Yellow boxes annotate compositions that are improved using different parsers.
+
+style descriptions. The parsing mechanism may categorize a style description, e.g. "in Van Gogh style" as a separate noun phrase that cannot be grounded in the image space. In addition, we discover that StructureDiffusion tends to generate similar images as Stable Diffusion. Thus we filtered out $20\%$ of most similar image pairs in Table 1, considering the efficiency of human evaluation. Therefore, the improvement could be compromised when evaluated on the full set of generated images. Future work may focus on devising explicit methods to associate attributes to objects using spatial information as input. For example, how to make a text-to-image synthesis model interpret coordinate information with limited fine-tuning or prompt tuning steps would be an appealing direction.
+
+# F ADDITIONAL RESULTS
+
+# Stable Diffusion Ours
+
+A green apple and a red banana
+
+
+
+A gold clock and a green bench
+
+
+
+A red cake and a blue suitcase
+
+
+
+A brown dog and a blue suitcase
+
+
+
+A yellow bowl and a blue cat
+
+
+Figure 13: Qualitative results on CC-500
+
+# Stable Diffusion Ours
+
+A red bird and a green banana
+
+
+
+A blue bird and a brown bowl
+
+
+
+A blue backpack and a brown bear
+
+
+
+A blue backpack and a brown cow
+
+
+
+A brown dog and a blue horse
+
+
+
+# Stable Diffusion
+
+a purple cat with a orange hat on its head
+
+
+A red helmet is on a yellow toilet in the dirt
+
+
+
+
+
+
+
+
+A bathroom with red tile and a green shower curtain
+
+
+A white toilet bowl with a purple rug in front
+
+
+
+
+A spoon and bowl of red pea soup and green beans with onions
+
+
+Figure 14: Qualitative results on ABC-6K
+
+# Stable Diffusion
+
+A red cat sits on a rug with a black cord
+
+
+
+
+
+
+A red stop sign above a white walk across road sign
+
+
+A spacious kitchen has white walls , red countertops , and a large stove
+
+
+A pink towel stands out greatly in the white bathroom
+
+
+A cow standing outside of a white building with a blue entrance
+
+# Stable Diffusion
+
+# Ours
+
+A yellow cat is wearing a blue plastic baseball hat.
+
+
+
+
+Two elephants walking by a green wall with tan palm trees painted on it
+
+
+A large white bed sitting in a hotel room next to a red couch
+
+
+A large pizza on a white plate sitting on a blue table
+
+
+A black and white curtain hanging in a room that is decorated in black, white and red
+
+
+Stable Diffusion
+Structure Diffusion
+
+
+a skateboarder jumps over a silver dome with stone domes in front of it and bicyclist behind
+
+
+a stone building with a wooden bench placed near it
+
+
+a wooden building with a stone bench placed near it
+
+
+a red sphere and a blue cube
+Figure 15: Qualitative results characterizing attributes beyond colors, including shape, size and materials.
+
+
+Stable Diffusion
+small silver laptop sitting on top of a wooden computer desk
+
+
+a silver stove sits next to a counter on a wooden floor
+
+
+a wooden building with a stone bench placed near it
+
+
+a red sphere and a yellow cube
+
+
+a blue sphere and a red cube
+
+
+Stable Diffusion
+Structure Diffusion
+two back to back wooden benches next to a shrub near a stone walkway
+
+
+a shiny silver metal microwavenear wooden cabinets
+
+
+a cat is perched upon a stone bench that sits on a wooden patio
+
+
+a brown sphere and a cyan cube
+
+
+a green sphere and a red cube
+
+
+
+
+
+
+
+
+Figure 16: A prompt "an astronaut riding a horse" appended with different (combinations of) style descriptions. Our method has no negative effects on the image style. "base" refers to Stable Diffusion.
+
+
+
+
+
+
+
+
\ No newline at end of file
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+++ b/transferableunlearnableexamples/full.md
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+# TRANSFERABLE UNLEARNABLE EXAMPLES
+
+Jie Ren*
+
+Michigan State University
+
+renjie3@msu.edu
+
+Han Xu*
+
+Michigan State University
+
+xuhan1@msu.edu
+
+Yuxuan Wan
+
+Michigan State University
+
+wanyuxua@msu.edu
+
+Xingjun Ma
+
+Fudan University
+
+xingjunma@fudan.edu.cn
+
+Lichao Sun
+
+Lehigh University
+
+lis221@lehigh.edu
+
+Jiliang Tang
+
+Michigan State University
+
+tangjili@msu.edu
+
+# ABSTRACT
+
+With more people publishing their personal data online, unauthorized data usage has become a serious concern. The unlearnable examples strategies have been introduced to prevent third parties from training on the data without permission. They add perturbations to the users' data before publishing, so as to make the models trained on the perturbed published dataset invalidated. These perturbations have been generated for a specific training setting and a target dataset. However, their unlearnable effects significantly decrease when used in other training settings or datasets. To tackle this issue, we propose a novel unlearnable strategy based on Class-wise Separability Discriminant (CSD), which boosts the transferability of the unlearnable perturbations by enhancing the linear separability. Extensive experiments demonstrate the transferability of the unlearnable examples crafted by our proposed method across training settings and datasets. The implementation of our method is available at https://github.com/renjie3/TUE.
+
+# 1 INTRODUCTION
+
+With more people posting their personal data online (intentionally or unintentionally), it has raised the concern that the data might be utilized without the owner's consent to train commercial or malicious machine learning models. While large-scale datasets collected from the Internet like LFW(Huang et al., 2008), Freebase(Bollacker et al., 2008), and Ms-celeb-1m(Guo et al., 2016) have greatly advanced the development of deep learning, they may contain a certain amount of private data, which has the potential risk of privacy leakage. Thus, growing efforts (Huang et al., 2020; Fowl et al., 2021) have been made to protect data from unauthorized usage by making the data samples "unlearnable" (Huang et al., 2020; Fowl et al., 2021; He et al., 2022). These methods generate the unlearnable examples by injecting imperceptible "shortcut" perturbation. If the data is used by unauthorized training, the models will be tricked to extract such easy-to-learn shortcut features and ignore the real semantics in the original data (Geirhos et al., 2020). Consequently, the trained model fails to recognize the user's data during the test phase and the user's data gets protected.
+
+As shown in Figure 1, the practical usage procedure of unlearnable examples (UEs) consists of two stages, i.e., the generation stage and the evaluation stage. Before releasing data, in the generation stage, one can make the data unlearnable by adding perturbations into original data. According to whether the label information is used to generate the perturbations, we divide the UEs into Supervised UEs which is generated with the guidance of the label information, and Unsupervised UEs which is generated without the guidance of label information. In the evaluation stage, the unlearnable version of data is released to the public, and the unauthorized third parties might use different algorithms, like supervised training and unsupervised training, to learn from the data. The UEs aim to provide protection in this stage against unauthorized training and invalidate the models trained on them.
+
+However, existing UEs are in lack of training-wise transferability and have weaknesses in data-wise transferability in the evaluation stage. First, low training-wise transferability implies that the perturbed samples generated towards one target training setting cannot be transferred to other
+
+
+Figure 1: Two stages of Unlearnable Examples.
+
+training settings. As shown in Section 3.2.1, although Error-Minimizing Noise (EMN) (Huang et al., 2020) can protect data from supervised training, we can use unsupervised learning methods, such as Contrastive Learning (Chen et al., 2020a; Chen & He, 2021; Chen et al., 2020b), to extract useful representations from the EMN-protected dataset and obtain high accuracy. In this way, the perturbed dataset is still exposed to no protection. Second, insufficient data-wise transferability indicates that the unlearnable effect of perturbations generated for one target dataset will significantly decrease when transferred to other datasets. We have to generate perturbations for each dataset, reducing its flexibility in real-world scenarios. For example, data in various applications such as social media is dynamic or even streaming, which makes it extremely challenging to generate the entire perturbation set when new data is continuously emerging.
+
+In this work, we aim to enhance the training-wise and data-wise transferability of unlearnable examples. In detail, our method is motivated by Synthetic Noise (SN) (Yu et al., 2021), which devises a manually designed linear separable perturbation to generate UEs. Such perturbations do not target specific dataset, thus it has the potential to enhance data-wise transferability. However, SN is manually designed and it is not quantifiable or estimizable. As such, it is impossible to incorporate SN into other optimization processes. Meanwhile, SN lacks training-wise transferability. Therefore, in this work, we propose Class-wise Separability Discriminant (CSD) to generate estimizable linear-separable perturbations. Our framework Transferable Unlearnable Examples with enhanced linear separability can generate UEs with superior training-wise and data-wise transferability.
+
+# 2 RELATED WORK
+
+Unlearnable Examples. Unlearnable examples are close to availability attack which aims at making the data out of service for training the models by the unauthorized third parties (Muñoz-González et al., 2017). Several supervised UEs are proposed (Huang et al., 2020; Fowl et al., 2021; Shan et al., 2020). The vanilla unlearnable examples are crafted by generating and adding unlearnable perturbations into the clean samples (Huang et al., 2020). Being induced to trust that the perturbation can minimize the loss better than the original image features, the model will pay more attention to the perturbations. Unlearnable Contrastive Learning (UCL) is proposed (He et al., 2022) to generate UEs to protect data from unsupervised learning. Our studies show that both Supervised and Unsupervised UEs lack training-wise transferability as shown in Section 3.2.1 and Appendix A, respectively. Table 1 summarizes the settings of existing methods, where $X$ means ineffective protection.
+
+Table 1: Unlearnable Examples Settings of Existing Approaches
+
+| Num Samples | 4 | 10 | 20 | 100 | 200 |
| FT-Transformer + LH-E2E Tuned | 1.00 | 1.00 | 1.00 | 1.08 | 1.17 |
| FT-Transformer + LH-E2E Default | 1.25 | 1.25 | 1.17 | 1.42 | 1.17 |
| FT-Transformer + MLP-E2E Tuned | 1.00 | 1.00 | 1.08 | 1.00 | 1.08 |
| FT-Transformer + MLP-E2E Default | 1.08 | 1.00 | 1.25 | 1.33 | 1.58 |
| FT-Transformer + LH Tuned | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| FT-Transformer + LH Default | 1.17 | 1.25 | 1.33 | 1.58 | 1.67 |
| FT-Transformer + MLP Tuned | 1.08 | 1.00 | 1.00 | 1.00 | 1.00 |
| FT-Transformer + MLP Default | 1.17 | 1.08 | 1.33 | 1.58 | 1.75 |
| ResNet + LH-E2E Tuned | 1.00 | 1.17 | 1.33 | 1.25 | 1.33 |
| ResNet + LH-E2E Default | 1.08 | 1.00 | 1.00 | 1.17 | 1.25 |
| ResNet + MLP-E2E Tuned | 1.00 | 1.00 | 1.08 | 1.08 | 1.00 |
| ResNet + MLP-E2E Default | 1.17 | 1.00 | 1.08 | 1.58 | 1.42 |
| ResNet + LH Tuned | 1.00 | 1.00 | 1.08 | 1.00 | 1.00 |
| ResNet + LH Default | 1.17 | 1.25 | 1.25 | 1.50 | 1.50 |
| ResNet + MLP Tuned | 1.00 | 1.00 | 1.00 | 1.00 | 1.08 |
| ResNet + MLP Default | 1.17 | 1.17 | 1.33 | 1.58 | 1.75 |
| MLP + LH-E2E Tuned | 1.17 | 1.25 | 1.42 | 1.42 | 1.17 |
| MLP + LH-E2E Default | 1.08 | 1.08 | 1.08 | 1.33 | 1.50 |
| MLP + MLP-E2E Tuned | 1.00 | 1.08 | 1.33 | 1.25 | 1.42 |
| MLP + MLP-E2E Default | 1.08 | 1.00 | 1.25 | 1.25 | 1.33 |
| MLP + LH Tuned | 1.00 | 1.25 | 1.17 | 1.17 | 1.08 |
| MLP + LH Default | 1.08 | 1.08 | 1.17 | 1.17 | 1.25 |
| MLP + MLP Tuned | 1.08 | 1.08 | 1.08 | 1.00 | 1.00 |
| MLP + MLP Default | 1.00 | 1.00 | 1.17 | 1.17 | 1.42 |
| TabTransformer + LH-E2E Tuned | 1.00 | 1.00 | 1.08 | 1.08 | 1.00 |
| TabTransformer + LH-E2E Default | 1.17 | 1.25 | 1.25 | 1.50 | 1.25 |
| TabTransformer + MLP-E2E Tuned | 1.00 | 1.00 | 1.08 | 1.08 | 1.00 |
| TabTransformer + MLP-E2E Default | 1.17 | 1.25 | 1.25 | 1.42 | 1.25 |
| TabTransformer + LH Tuned | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| TabTransformer + LH Default | 1.25 | 1.17 | 1.25 | 1.25 | 1.17 |
| TabTransformer + MLP Tuned | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| TabTransformer + MLP Default | 1.25 | 1.33 | 1.25 | 1.25 | 1.25 |
+
+# D ADDITIONAL RESULTS
+
+# D.1 RESULTS FOR TABTRANSFORMER
+
+Figure 6 is equivalent to Figure 2, but also includes the results for TabTransformer model, a tabular neural network consisting of transformer block for categorical features and an MLP block on top for numerical features. We find that TabTransformer performs poorly compared to other deep tabular neural networks, which might be explained by the fact that the original paper Huang et al. (2020) only suggests to tune the hyperparameters of transformer block, but not the MLP part, while Meta-MIMIC data has only one categorical features and 171 numerical features.
+
+
+Figure 6: Average model ranks including TabTransformer. Deep tabular models and GBDT performance is presented on the corresponding panels. Within each panel, columns represent transfer learning setups, and rows correspond to the number of available downstream samples. Warmer colors indicate better performance. FS denotes training from scratch (without pre-training on upstream data), LH and MLP denote linear and MLP heads correspondingly, E2E denotes end-to-end training. Rank is averaged across all downstream tasks.
+
+# D.2 RANKS STANDARD ERROR
+
+Figure 7 presents standard errors for ranks in Figure 2 computed across tasks.
+
+
+Figure 7: Standard error for model ranks computed across all downstream tasks. This figure complements Figure 2. Darker colors indicate lower standard errors.
+
+# D.3 MODEL-WISE RANKS
+
+In Figure 8 we compare different transfer learning setups for each deep tabular model.
+
+# D.4 AVERAGE ROC-AUC IMPROVEMENT OVER CATBOOST
+
+In Figure 9 we display ROC-AUC improvements over Catboost baseline averaged across all downstream tasks. We observe trends similar to ones with ranks; MLP model performs best in low data regimes, while FT-Transformer offers consistent gains across all data levels.
+
+
+Figure 8: Model-wise ranks for GBDT and neural networks. Within each panel, columns represent transfer learning setups, and rows correspond to the number of available downstream samples. Warmer colors indicate better performance. FS denotes training from scratch (without pre-training on upstream data), LH and MLP denote linear and MLP heads correspondingly, E2E denotes end-to-end training. Rank is computed across training setups for each model and is averaged across all downstream tasks.
+
+
+Figure 9: Average ROC-AUC improvements over Catboost baseline. Within each panel, columns represent transfer learning setups, and rows correspond to the number of available downstream samples. Warmer colors indicate better performance. FS denotes training from scratch (without pre-training on upstream data), LH and MLP denote linear and MLP heads correspondingly, E2E denotes end-to-end training. ROC-AUC improvement is computed as difference in ROC-AUC with Catboost model and is averaged across all downstream tasks.
+
+# D.5 DATASET-LEVEL RESULTS
+
+In Tables 11, 12, 13, 14 we report ROC-AUC measurements for each model on each downstream task. We include the results for GBDT models, neural baselines and neural networks with transfer learning.
+
+# D.6 EXPERIMENTS ON ADDITIONAL DATASETS
+
+To expand the experimental evaluation of tabular transfer learning beyond the medical domain, we include experiments on two additional multi-label datasets: Yeast functional genomics data Elisseeff and Weston (2001) and Emotions data Trohidis et al. (2008). Yeast contains micro-array expression data and phylogenetic profiles of genes with labels corresponding to different gene functional classes. Each gene is associated with a set of functional classes, therefore each task aims to predict if a gene is associated with a particular gene functional group. The dataset contains 2417 samples (genes) with 103 numerical features and 14 labels for each sample. Emotions data aims to detect emotions in music, it contains rhythmic and timbre features of 593 songs and labels corresponding to 6 emotions. In our experimental setup, we treat each classification label as a separate task. Similarly to the experiments with MetaMIMIC, we split the tasks into downstream and upstream by reserving $n - 1$ labels for the upstream task and the $n$ -th label for the downstream task. We further limit the amount of downstream data to 4, 10, 20, 100 and 200 samples. This strategy results in 70 combinations of
+
+
+Figure 10: Average model ranks across downstream tasks in Yeast data. Deep tabular models and Catboost performance is presented on the corresponding panels. Within each panel, columns represent transfer learning setups, and rows correspond to the number of available downstream samples. Warmer colors indicate better performance. FS denotes training from scratch (without pre-training on upstream data), LH and MLP denote linear and MLP heads correspondingly, E2E denotes end-to-end training. Rank is averaged across all downstream tasks.
+
+
+
+
+
+
+Figure 11: Average model ranks across downstream tasks in Emotions data. Deep tabular models and Catboost performance is presented on the corresponding panels. Within each panel, columns represent transfer learning setups, and rows correspond to the number of available downstream samples. Warmer colors indicate better performance. FS denotes training from scratch (without pre-training on upstream data), LH and MLP denote linear and MLP heads correspondingly, E2E denotes end-to-end training. Rank is averaged across all downstream tasks.
+
+
+
+upstream and downstream datasets constructed from the Yeast database and 30 combinations from the Emotions database. We conduct experiments with FT-Transformer, MLP, and Catboost models which demonstrated the best results in our previous experiments. We tune the hyperparameters for deep models and Catboost as described in Appendix C. Figures 10, 11 present average model ranks of two tabular neural network and Catboost models on downstream tasks as a heatmap with warmer colors representing better results.
+
+We observe that deep tabular models pre-trained on upstream data and finetuned with MLP head outperform models trained from scratch and Catboost models leveraging stacking. Interestingly, MLP model performs better on the Yeast dataset, while FT-Transformer performs better on the Emotions dataset.
+
+# D.7 EXPERIMENTS WITH FEWER PRE-TRAINING TASKS
+
+To further validate the effectiveness of transfer learning in scenarios where fewer pre-training tasks are available we repeat our experiments with the first 5 targets from Meta-MIMIC dataset: Diabetes,
+
+
+Figure 12: Average model ranks for models pre-trained and finetuned on the first five MetaMIMIC targets. Deep tabular models and Catboost performance is presented on the corresponding panels. Within each panel, columns represent transfer learning setups, and rows correspond to the number of available downstream samples. Warmer colors indicate better performance. FS denotes training from scratch (without pre-training on upstream data), LH and MLP denote linear and MLP heads correspondingly, E2E denotes end-to-end training. Rank is averaged across 5 downstream tasks.
+
+Hypertensive, Ischematic, Heart, Overweight. In particular we pre-train on 4 available targets and finetune on the remaining task. We limit the number of downstream tasks to 5 for consistency of the number of pre-training targets across downstream tasks. Figure 12 presents the results of this experiment, we observe similar trends to the original experiments with all targets. In particular, we find that deep neural networks pre-trained even on fewer upstream targets definitively outperform Catboost with stacking with MLP model performing better in low-data regimes and FT-Transformer performing better when more downstream data is available.
+
+# D.8 WHEN IS TRANSFER LEARNING MOST EFFECTIVE? CASE STUDY
+
+One might wonder when deep neural networks are effective for transferring knowledge between tasks. We hypothesize that transfer learning is most effective in scenarios where upstream and downstream tasks are related. In the previous experiments with pre-training on fewer tasks (see Appendix D.7) we found that transfer learning is effective when transferring knowledge between the first five tasks, which include a few cardiovascular-related conditions (hypertensive, ischemic, heart), diabetes and overweight conditions. From the correlation matrix in Figure 5, we can see that these tasks are slightly more correlated with each other than with other conditions such as anemia, respiratory, hypotension, purpura, alcohol. We notice that the latter set of tasks is also less related to cardiovascular conditions. To evaluate our hypothesis that transfer learning is less effective when pre-training and downstream tasks are less related, we run experiments with pre-training the model on the first 5 related tasks and transferring knowledge to the set of "less related" tasks. The results for this experiment are presented in Figure 13, we observe that although pre-trained MLP model still provides performance boosts in small data regimes compared to Catboost, the trends are less pronounced compared to results for transferring knowledge to "more related tasks" in Figure 12.
+
+# D.9 EXPERIMENTS WITH HIGHER DATA REGIMES IN DOWNSSTREAM DATA
+
+To expand the experimental evaluation to higher data regimes in downstream data we repeat our experiments with 400, 1000 and 2000 samples available for finetuning in the downstream data. The results for this experiment are presented in Figure 14. We find that even in higher data regimes pre-training on upstream data provides significant boost to FT-Transformer model compared to Catboost with stacking. At the same time, MLP performs on par with Catboost when there is more downstream data available, which aligns with our previous results.
+
+
+Figure 13: Average model ranks for transferring knowledge to "less related" tasks. Deep tabular models and Catboost performance is presented on the corresponding panels. Within each panel, columns represent transfer learning setups, and rows correspond to the number of available downstream samples. Warmer colors indicate better performance. FS denotes training from scratch (without pre-training on upstream data), LH and MLP denote linear and MLP heads correspondingly, E2E denotes end-to-end training. Rank is averaged across 5 "less related tasks".
+
+
+
+
+
+
+Figure 14: Average model ranks in higher-data regimes for downstream tasks. Deep tabular models and Catboost performance is presented on the corresponding panels. Within each panel, columns represent transfer learning setups, and rows correspond to the number of available downstream samples. Warmer colors indicate better performance. FS denotes training from scratch (without pre-training on upstream data), LH and MLP denote linear and MLP heads correspondingly, E2E denotes end-to-end training. Rank is averaged across all downstream tasks.
+
+
+
+# D.10 ADDITIONAL EXPERIMENTS WITH PSEUDO-Feature METHOD
+
+We further verify the effectiveness of our pseudo-feature method in scenarios with multiple features misaligned between upstream and downstream data. We repeat the experiment from Section 6, but instead of a single feature we remove three features. In Figure 15, the top left heatmap represents the experiment where the downstream data has three additional features missing from the upstream data. The bottom left heatmap represents the opposite scenario of the upstream data having three additional features not available in the downstream data. To ensure that the features we experiment with are meaningful and contain useful information, we chose important features according to GBDT feature importances. In Figure 15, we include the average ranks of FT-Transformer models pre-trained with missing features, pre-trained with imputed features and with the original features, which we assume are unavailable. Additionally, in the right panel we present standard errors for model ranks computed across all downstream tasks. From these experiments, we see that pre-training on data with imputed features significantly outperforms pre-training with missing features and that our feature imputation method enables effective transfer learning in scenarios where upstream and downstream feature sets differ in multiple features.
+
+
+
+
+
+
+Figure 15: Pseudo-Feature method for aligning upstream and downstream feature sets with 3 misaligned features. Left: Comparison of ranks of FT-Transformer model trained on data with 3 missing features, with pseudo-features and with the original features. Right: Standard error for ranks computed across all tasks. Figures in the top row correspond to the experiments with missing features in the upstream data, while figures in the bottom row present experiments with missing features in the downstream data.
+
+
+
+# D.11 ADDITIONAL EXPERIMENTS WITH CONTRASTIVE SELF-SUPERVISED LEARNING
+
+We extend our comparison of supervised pre-training and contrastive pre-training strategies by including two additional corruption hyperparameters $\lambda = 0.7, 0.8$ for contrastive pre-training used in experiments in Section 5 (which used $\lambda = 0.9$ ). We also include SCARF contrastive pretraining strategy proposed in (Bahri et al., 2021). For SCARF method we generate a random mask over features, then for each masked feature we sample uniformly from the joint distribution of that feature (in the training set). Then we compute the InfoNCE loss using these corrupted inputs and the clean inputs for contrastive pretraining. We use the default corruption hyperparameter $\lambda = 0.6$ from the SCARF paper (Bahri et al., 2021). We compare all contrastive pre-training strategies and supervised pre-training in Figure 16. We find that increasing the level of corruption (i.e., decreasing $\lambda$ ) does not lead to improved performance on downstream tasks for contrastive pre-training. Also, while the SSL methods outperform training from scratch, especially in the lower data regimes, supervised pre-training is consistently better than all considered SSL methods.
+
+
+Figure 16: Comparison of different pre-training strategies for FT-Transformer. The left panel illustrates end-to-end fine-tuning with linear head and the right plot illustrates end-to-end fine-tuning with MLP head. Within each panel, columns represent pre-training strategies and rows correspond to the number of available downstream samples. FS stands for training from scratch, $\lambda$ denotes the corruption parameter in contrastive pre-training, SCARF corresponds to contrastive pre-training strategy from (Bahri et al., 2021) and Supervised denotes supervised pre-training on upstream data.
+
+
+
+
+
+| Notation | Meaning |
| Ω | an underlying spacial domain in R2 |
| D | a subdomain in Ω (not necessarily topologically-connected) |
| ∂D, ∂Ω | D's and Ω's boundary, 1-dimensional manifolds |
| ∇u | the gradient vector of a function, ∇u(x) = (∂x1u(x), ∂x2u(x)) |
| ||·||ω | the L2-norm on a region ω |
| ||·||= ||·||Ω | the L2-norm on whole domain Ω |
| δx | the delta function such that ∫Ωf(y)δx(y) dy = f(x), ∀f. |
| n | the unit outer normal vector on the boundary ∂Ω. |
| ∇u·n | normal derivative of u, measures the rate of change along the direction of n |
| Λσ | NtD map from Neumann data g := ∇u·n |
| Hs(Ω), s≥0 | (how fast the solution changes toward the outward normal direction) to Dirichlet data f := u|∂Ω (the solution's value along the tangential direction) |
| Hs(Ω), s<0 | the Sobolev space of functions |
| H0s(Ω) | all u ∈ Hs(Ω) such that u's integral on Ω vanishes |
| |·|Y | the seminorm defined for functions in Y |
+
+# B BACKGROUND OF EIT
+
+For EIT, an immediate question is whether $\mathcal{F}^{-1}$ and $\mathcal{F}_L^{-1}$ in (4) and (5) are well-defined, namely whether $\sigma$ can be uniquely determined. In fact, for the case of full measurements ( $L = \infty$ ), the uniqueness for $\mathcal{F}^{-1}$ has been well established, (Bruhl, 2001; Hanke & Bruhl, 2003; Astala & Paivärinta, 2006; Nachman, 1996; Kohn & Vogelius, 1984; Sylvester & Uhlmann, 1987). It is worthwhile to point out that, in this case, $\sigma$ is not necessarily a piecewise constant function. In (1), we present a simplified case for purposes of illustrating as well as benchmarking. In general, with the full spectral information of the NtD map, $\sigma$ can be uniquely determined as a general positive function.
+
+If infinitely many eigenpairs are known, then the operator itself can be precisely characterized using infinitely many feature channels by Reproducing Kernel Hilbert Space (RKHS) theory, e.g., Mercer (1909); Aronszajn (1950); Minh et al. (2006); Morris (2015); Kadri et al. (2016); Lu et al. (2022). In the context of EIT, this is known as the "full measurement". A more challenging and practical problem is to recover $\sigma$ from only finitely many boundary data pairs. A common practice for the theoretical study of reconstruction using finite measurements is the assumption of $\sigma$ being a piecewise constant function. The task is usually set to recover the shape and location of the inclusion $D$ . Otherwise, the problem is too ill-posed. With finite measurements, the uniqueness of the inclusion remains a long-standing theoretical open problem, and it can be only established for several special classes of the inclusion shape, such as the convex cylinders in Isakov & Powell (1990) or convex polyhedrons in Barceló et al. (1994). We refer readers to some counter-examples in Kang & Seo (2001) where a two- or three-dimensional ball may not be identified uniquely by one single measurement if the values of $\sigma_0$ and $\sigma_1$ are unknown.
+
+Furthermore, here we provide one example to illustrate the difficulty in the reconstruction procedure (Pidcock et al., 1995). Let $\Omega$ be a unit circle, let $D$ be a circle with the radius $\rho < 1$ , and define
+
+$$
+\sigma (x) = \left\{ \begin{array}{l l} 1 & \text {i f} \| x \| \geq \rho , \\ \sigma_ {1} & \text {i f} \| x \| < \rho , \end{array} \right. \tag {24}
+$$
+
+with $\sigma_{1} < 1$ being an arbitrary constant. In this case, the eigenpairs of $\Lambda_{\sigma}$ can be explicitly calculated
+
+$$
+\lambda_ {l} = \frac {1}{l} \frac {1 - \rho^ {2 l} \mu}{1 + \rho^ {2 l} \mu}, \quad \nu_ {m} = \frac {1}{\sqrt {2 \pi}} \cos (l \theta) \text {o r} \frac {1}{\sqrt {2 \pi}} \sin (l \theta), l = 1, 2, \dots , \tag {25}
+$$
+
+with $\mu = (1 - \sqrt{\sigma_1}) / (1 + \sqrt{\sigma_1})$ , which are exactly the Fourier modes in a unit circle. In this case, if the set of basis $\{g_l\}_{l=1}^{\infty}$ is just chosen as $\{\cos(\theta), \sin(\theta), \cos(2\theta), \sin(2\theta), \ldots\}$ , the matrix representation $\mathbf{A}_{\sigma}$ in (3) can be written as an infinite diagonal matrix
+
+$$
+\mathbf {A} _ {\sigma} \left[ \begin{array}{c c c c} (1 - \rho^ {2} \mu) / (1 + \rho^ {2} \mu) & 0 & 0 & 0 \\ 0 & (1 - \rho^ {4} \mu) / (1 + \rho^ {4} \mu) & 0 & 0 \\ 0 & 0 & (1 - \rho^ {4} \mu) / (1 + \rho^ {4} \mu) & 0 \\ 0 & 0 & 0 & \ddots \end{array} \right]. \tag {26}
+$$
+
+Thanks to this special geometry, eigenvalues in (25) can clearly determine $\rho$ and $\sigma_{1}$ as follows:
+
+$$
+\rho = \sqrt {\frac {\left(1 - \lambda_ {2}\right) \left(1 + \lambda_ {1}\right)}{\left(1 + \lambda_ {2}\right) \left(1 - \lambda_ {1}\right)}} \quad \text {a n d} \quad \mu = \frac {\left(1 - \lambda_ {1}\right) ^ {2} \left(1 + \lambda_ {2}\right)}{\left(1 + \lambda_ {1}\right) ^ {2} \left(1 - \lambda_ {2}\right)}. \tag {27}
+$$
+
+However, in practice, $\mathbf{A}_{\sigma}$ does not have such a simple structure. Approximating $\mathbf{A}_{\sigma}$ itself requires a large number of data pairs that are not available in the considered case. Besides, an accurate approximation of the eigenvalues of $\mathbf{A}_{\sigma}$ is also very expensive. Furthermore, for complex inclusion shapes, two eigenvalues are not sufficient to exactly recover the shape and conductivity values.
+
+# C EXPERIMENT SET-UP
+
+# C.1 DATA GENERATION AND TRAINING
+
+In the numerical examples, the data generation mainly follows standard practice in theoretical prototyping for solving EIT problems, see e.g., Chow et al. (2014); Michalikova et al. (2014); Hamilton & Hauptmann (2018); Guo & Jiang (2020); Fan & Ying (2020). For examples, please refer to Figure 2. The computational domain is set to be $\Omega \coloneqq (-1,1)^2$ , and the two media with the different conductivities are with $\sigma_{1} = 10$ (inclusion) and $\sigma_0 = 1$ (background). The inclusions are four random ellipses. The lengths of the semi-major axis and semi-minor axis of these ellipses are sampled from $\mathcal{U}(0.1,0.2)$ and $U(0.2,0.4)$ , respectively. The rotation angles are sampled from $\mathcal{U}(0,2\pi)$ . There are 10800 samples in the training set, from which $20\%$ are reserved as validation. There are 2000 in the testing set for evaluation.
+
+
+Figure 2: Randomly selected samples of elliptic inclusion to represent the coefficient $\sigma$ (left 1-4). A Cartesian mesh $\Omega_h$ with a grid point $\{z_j\}$ (right). In computation, discretization of $\mathcal{I}^D$ consists of values taken as 1 at the mesh points of $\Omega_h$ inside $D$ and 0 at others.
+
+
+
+
+
+
+
+
+
+The noise $\xi = \xi (x)$ below (10) is assumed to be
+
+$$
+\xi (x) = \left(f (x) - \Lambda_ {\sigma_ {0}} g (x)\right) \tau G (x) \tag {28}
+$$
+
+where $\tau$ specifies the relative strength of noise, and $G(x)$ is a normal Gaussian distribution independent with respect to $x$ . As $\xi (x)$ is merely pointwise imposed, the boundary data can be highly rough, even if the ground truth $f(\cdot) - \Lambda_{\sigma_0}g(\cdot)$ is chosen to be smooth. Nevertheless, the harmonic extension makes the noise from boundary data have a minimal impact on the overall reconstruction, thanks to the smoothing property of the inverse of the Laplacian operator $(- \Delta)^{-1}$ ; for example, please refer to Figure 3. In data generation, the harmonic extension feature map is approximated by finite element methods incorporating stencil modification near the inclusion interfaces (Guo & Lin, 2019; Guo et al., 2019). Similar data augmentation practices using internal data can be found in Nachman et al. (2007); Bal (2013); Jin et al. (2022a).
+
+Thanks to the position-wise binary nature of $\mathcal{I}^D$ , another choice of the loss function during training can be the binary cross entropy $\mathcal{L}(\cdot, \cdot)$ , applied for a function in $\mathbb{P}$ , to measure the distance between the ground truth and the network's prediction
+
+$$
+\mathcal {L} \left(\mathbf {p} _ {h}, \mathbf {u} _ {h}\right) := - \sum_ {z \in \mathcal {M}} \left(\mathbf {p} _ {h} (z) \ln \left(\mathbf {u} _ {h} (z)\right) + \left(1 - \mathbf {p} _ {h} (z)\right) \ln \left(1 - \mathbf {u} _ {h} (z)\right)\right). \tag {29}
+$$
+
+
+Figure 3: The harmonic extensions $\phi_l$ with zero noise on boundary (left); the harmonic extensions $\tilde{\phi}_l$ with $20\%$ Gaussian noise on boundary (middle). Their pointwise difference (right). $\| \phi_l - \tilde{\phi}_l\| /\| \phi_l\| = 0.0203$ .
+
+
+
+
+
+Thanks to the Pinsker inequality (e.g., see (Cover, 1999, Section 11.6)), $\mathcal{L}(\mathbf{p}_h,\mathbf{u}_h)$ serves as a good upper bound for the square of the total variation, which can be further bounded below by the $L^2$ -error given the boundedness of the position-wise value.
+
+The training uses 1cycle (Smith & Topin, 2019) learning rate strategy with a warm-up phase. A mini-batch ADAM iterations are run for 50 epochs with no extra regularization, such as weight decay. The evaluated model is taken from the epoch that has the best validation metric. The learning rate starts and ends with $10^{-3} \cdot lr_{\max}$ , and reaches the maximum of $lr_{\max}$ at the end of the 10-th epoch. The $lr_{\max} = 10^{-3}$ . The result demonstrated is obtained from fixing the random number generator seed. Figure 6 shows the testing results for a randomly chosen sample. All models are trained on an RTX 3090 or an A4000. The codes to replicate the experiments are open-source and publicly available. $^{1}$ .
+
+
+Figure 4: The left: the training-testing pixel-wise binary cross entropy convergence for the CNN-based U-Net with 31 million parameters, a clear overfitting pattern is shown. The right: the training-testing convergence for the attention-based U-Transformer with 11.4 million parameters.
+
+
+
+# C.2 NETWORK ARCHITECTURE
+
+The difference in architectural hyperparameters, together with training and evaluation costs comparison for all the models compared in the task of EIT reconstruction can be found in Table 3.
+
+# U-Integral Transformer
+
+- Overall architecture. The U-Integral-Transformer architecture is a drop-in replacement of the standard CNN-based U-Net baseline model (7.7m) in Table 1. The CNN-based U-Net is used in DL-based approaches for boundary value inverse problem in Guo & Jiang (2020); Guo et al. (2021); Le et al. (2022). One of the novelties is that the input is a tensor that concatenates different measurement matrices as different channels, and a similar practice can be found in Brandstetter et al. (2023). Same with the baseline U-Net, the UIT has three downsampling layers as the encoder (feature extractor). The downsampling layers map $m \times m$ latent representations to $m/2 \times m/2$ , and
+
+
+Figure 5: The harmonic extension feature map $\phi_l$ (left 1-3 as different channels' inputs to the neural network) corresponding to a randomly chosen sample's inclusion map (right). No visible relevance shown with the ground truth. The layered heatmap appearance is adopted in plotly contour only for aesthetics purposes, no edge-like nor layer-like features can be observed from the actual harmonic extension feature map input.
+
+
+
+
+
+
+
+
+(a)
+
+
+(b)
+
+
+(c)
+
+
+(d)
+
+
+(e)
+
+
+(f)
+
+
+(g)
+
+
+(h)
+
+
+(i)
+Figure 6: The neural network evaluation result for the inclusion in Figure 5 using various models. A model's input is the left most function in Figure 5 if this model uses a single channel, or all left three in Figure 5 if a model uses 3-channel input. (a) Ground truth inclusion in Figure 5; (b) U-Net baseline $(7.7\mathrm{m})$ prediction; (c) U-Net big $(31\mathrm{m})$ prediction with 3 channels; (d) Fourier Neural Operator $(10.4\mathrm{m})$ prediction with 1 channel; (e) Fourier Neural Operator big $(33\mathrm{m})$ prediction with 1 channel; (f) Adaptive Fourier Neural Operator $(10.7\mathrm{m})$ prediction with 1 channel; (g) Multiwavelet Neural Operator $(9.8\mathrm{m})$ prediction with 1 channel; (h) Hybrid UT with a linear attention $(10.13\mathrm{m})$ prediction with 1 channel; (i) UIT $(11.4\mathrm{m})$ prediction with 1 channel; (j) UIT $(11.4\mathrm{m})$ prediction with 3 channels.
+
+
+(j)
+
+expand the number of channels from $C$ to $2C$ . To leverage the "basis $\Leftrightarrow$ channel" interpretation and the basis update nature of the attention mechanism in Section 3.3, the proposed attention block is first added on the coarsest grid, which has the most number of channels. UIT has three upsampling layers as the decoder (feature selector), which map $m/2 \times m/2$ latent representations to $m \times m$ , and shrink the number of channels from $2C$ to $C$ . In these upsampling layers, attention blocks are applied on each cross-layer propagation to compute the interaction between the latent representations on both coarse and fine grids (see below). Please refer to Figure 8 for a high-level encoder-decoder schematic.
+
+- Double convolution block. The double convolution block is modified from that commonly seen in Computer Vision (CV) models, such as ResNet (He et al., 2016). We modify this block such that upon being used in an attention block, the batch normalization (Ioffe & Szegedy, 2015) can be replaced by the layer normalization (Ba et al., 2016), which can be understood as a learnable approximation to the Gram matrices' inverse by a diagonal matrix.
+- Positional embedding. At each resolution, the 2D Euclidean coordinates of an $m \times m$ regular Cartesian grid are the input of a channel expansion through a fixed learnable linear layer and are
+
+then added to each latent representation. This choice of positional embedding enables a bilinear interpolation between the coarse and fine grids or vice versa (see below).
+
+- Mesh-normalized attention. The scaled dot-product attention in the network is chosen to be the integral kernel attention in (11) with a mesh-based normalization. Please refer to Figure 7 for a diagram in a single attention head.
+
+
+Figure 7: Detailed flow of the modified 2D attention-based encoder layer using (11). $C$ : the number of channels in the input, $N$ : the number of expanded channels (for the basis expansion interpretation in Theorem 2).
+
+- Interpolation. Instead of max pooling used in the standard CNN-based U-Net, we opt for a bilinear interpolation on the Cartesian grid to map a latent representation from the fine grid to the coarse grid or vice versa. Note that, in the upsampling layers, the interpolation's outputs are directly inputted into an attention block that computes the interaction of latent representations between coarse and fine grids (see below).
+- Coarse-fine attention in up blocks. A modified attention in Section 3.3 with a pre-inner-product normalization replaces the convolution layer on the coarsest level. The skip connection from the encoder latent representations to the ones in the decoder are generated using an architecture similar to the cross attention used in Petit et al. (2021). $Q$ and $K$ are generated from the latent representation functions on the same coarser grids. As such, the attention kernel to measure the interaction between different channels is built from the coarse grid. $V$ is associated with a finer grid. Compared with the one in Petit et al. (2021), the modified attention in our method is inspired by the kernel integral for a PDE problem. Thus, it has (1) no softmax normalization or (2) no Hadamard product-type skip connection.
+
+# Other operator learners compared
+
+- Fourier Neural Operator (FNO) and variants. Fourier Neural Operator (FNO2d) learns convolutional filters in the frequency domain for some pre-selected modes, efficiently capturing globally-supported spatial interactions for these modes. The weight filter in the frequency domain multiplies with the lowest modes in the latent representations (four corners in the FFT). The Adaptive Fourier Neural Operator (AFNO2d) adds a token-mixing layer, as seen in Figure 2 in Guibas et al. (2022), appending every spectral convolution layer in the baseline FNO2d model.
+- Multi-Wavelet Neural Operator (MWO). The MultiWavelet Neural Operator (MWO) is proposed in Gupta et al. (2021), which introduces a multilevel structure into the FNO architecture. MWO still follows FNO's practice on each level by pre-selecting the lowest modes.
+
+# D FROM DSM TO TRANSFORMER
+
+This section gives a more detailed presentation of how the attention-like operator is derived from the DSM ansatz (15) with learnable kernels.
+
+We begin with recalling the original indicator function from (10):
+
+$$
+\mathcal {I} ^ {D} (x) := C _ {f, g} \frac {\mathrm {d} (x) \cdot \nabla \phi (x)}{\left| \eta_ {x} \right| _ {H ^ {s} (\partial \Omega)}}, \tag {30}
+$$
+
+where $C_{f,g} = \| f - \Lambda_{\sigma_0}g\|_{L^2 (\partial \Omega)}^{-1}$ is a constant, and the equations of the functions $\phi$ and $\eta_{x}$ are:
+
+$$
+- \Delta \phi = 0 \quad \text {i n} \quad \Omega , \quad \mathbf {n} \cdot \nabla \phi = (f - \Lambda_ {\sigma_ {0}} g) + \xi \quad \text {o n} \quad \partial \Omega , \quad \int_ {\partial \Omega} \phi d s = 0, \tag {31}
+$$
+
+Table 3: The detailed comparison of the networks used in this study. For U-Net-based neural networks, the channel/width is the number of the base channels on the finest grid after the initial channel expansion. A torch.cfloat type parameter entry counts as two parameters. GFLOPs: Giga FLOPs for 1 backpropagation (BP) performed for a batch of 8 samples recorded the PyTorch autograd profiler for 1 BP averaging from 100 BPs. Eval: number of instances per second.
+
+| Model | NFE ↓ | FID↓ | Recall ↑ |
| Improved DDPM (Nichol & Dhariwal, 2021) | 4000 | 2.90 | - |
| UDM (Kim et al., 2021) | 2000 | 2.33 | - |
| Likelihood SDE (Song et al., 2021a) | 2000 | 2.87 | - |
| Score SDE (VE) (Song et al., 2021b) | 2000 | 2.20 | 0.59 |
| Score SDE (VP) (Song et al., 2021b) | 2000 | 2.41 | 0.59 |
| NCSN (Song & Ermon, 2019) | 1000 | 25.3 | - |
| Adversarial DSM (Jolicoeur-Martineau et al., 2020) | 1000 | 6.10 | - |
| VDM (Kingma et al., 2021) | 1000 | 4.00 | |
| D3PMs (Austin et al., 2021) | 1000 | 7.34 | - |
| DiffuseVAE (Pandey et al., 2022), T=1000 | 1000 | 8.72 | - |
| DDPM (Ho et al., 2020) | 1000 | 3.21 | 0.57 |
| Recovery EBM (Gao et al., 2021) | 180 | 9.58 | - |
| Gotta Go Fast (Jolicoeur-Martineau et al., 2021) | 180 | 2.44 | - |
| LSGM (Vahdat et al., 2021) | 147 | 2.10 | 0.61 |
| Probability Flow (VP) (Song et al., 2021b) | 140 | 3.08 | 0.57 |
| DiffuseVAE (Pandey et al., 2022), T=100 | 100 | 11.71 | - |
| TDPM, Trunc=99 (ours) | 100 | 2.97 | 0.57 |
| TDPM+, Trunc=99 (ours) | 100 | 2.83 | 0.58 |
| FastDDPM, T=50 (Kong & Ping, 2021) | 50 | 3.41 | 0.56 |
| DDIM, T=50 (Song et al., 2020) | 50 | 4.67 | 0.53 |
| SNGAN+DGflow (Ansari et al., 2021) | 25 | 9.62 | 0.48 |
| TDPM, Trunc=49 (ours) | 50 | 3.11 | 0.57 |
| TDPM+, Trunc=49 (ours) | 50 | 2.96 | 0.58 |
| Progressive distillation (Salimans & Ho, 2022) | 8 | 2.57 | - |
| Denoising Diffusion GAN (Xiao et al., 2022), T=8 | 8 | 4.36 | 0.56 |
| Progressive distillation (Salimans & Ho, 2022) | 4 | 3.00 | - |
| Denoising Diffusion GAN (Xiao et al., 2022), T=4 | 4 | 3.75 | 0.57 |
| TDPM, Trunc=4 (ours) | 5 | 3.51 | 0.55 |
| TDPM+, Trunc=4 (ours) | 5 | 3.17 | 0.57 |
| Progressive distillation (Salimans & Ho, 2022) | 2 | 4.51 | - |
| Denoising Diffusion GAN (Xiao et al., 2022), T=2 | 2 | 4.08 | 0.54 |
| TDPM, Trunc=1 (ours) | 2 | 4.47 | 0.53 |
| TDPM+, Trunc=1 (ours) | 2 | 3.86 | 0.56 |
| DDPM Distillation (Luhman & Luhman, 2021) | 1 | 9.36 | 0.51 |
| SNGAN (Miyato et al., 2018) | 1 | 21.7 | 0.44 |
| AutoGAN (Gong et al., 2019) | 1 | 12.4 | 0.46 |
| TransGAN (Jiang et al., 2021) | 1 | 9.26 | - |
| StyleGAN2 w/o ADA (Karras et al., 2020a) | 1 | 8.32 | 0.41 |
| StyleGAN2 w/ ADA (Karras et al., 2020a) | 1 | 2.92 | 0.46 |
| StyleGAN2 w/ Diffaug (Zhao et al., 2020) | 1 | 5.79 | 0.42 |
| Progressive distillation (Salimans & Ho, 2022) | 1 | 9.12 | - |
| Denoising Diffusion GAN (Xiao et al., 2022), T=1 | 1 | 14.6 | 0.19 |
| TDPM, Trunc=0 (ours) | 1 | 7.34 | 0.46 |
+
+
+Figure 13: Qualitative results on CIFAR-10 and CelebA $(64\times 64)$
+
+
+
+
+
+
+
+
+Figure 14: Qualitative results of TDPM on LSUN-Church $(256\times 256)$ , with $T_{\mathrm{trunc}} = 99$ , 49, and 4. Note NFE $= T_{\mathrm{trunc}} + 1$ in TDPM. Each group presents generated samples from $p_{\theta}(\mathbf{x}_0)$ (left) and $p_{\theta}(\mathbf{x}_{T_{\mathrm{trunc}}})$ (right).
+
+
+
+
+
+
+Figure 15: Analogous qualitative results to Figure 14 on LSUN-Bedroom. Produced by TDPM.
+
+
+
+
+
+
+Figure 16: Analogous qualitative results to Figure 14 on CelebA-HQ. Produced by TDPM.
+
+
+
+
+
+
+Figure 17: Analogous qualitative results to Figure 14 on LSUN-Church. Produced by TDPM-CT.
+
+
+
+
+
+
+Figure 18: Analogous qualitative results to Figure 14 on LSUN-Bedroom. Produced by TDPM-CT.
+
+
+
+
+
+
+Figure 19: Analogous qualitative results to Figure 14 on CelebA-HQ. Produced by TDPM-CT.
+
+
+
+
+
+# D.8 ADDITIONAL RESULTS ON TEXT-TO-IMAGE GENERATION
+
+
+A white and gray bird with black wings.
+
+
+An airplan flying over a body of water.
+
+
+A sign reads "TDPM".
+
+
+Busy city street at dusk with sun setting.
+Figure 20: Additional text-to-image generation results with different text prompt, produced by TLDM with $T_{\mathrm{trunc}} = 49$ .
+
+
+A cluster of flower on the wooden table.
+
+
+The bathroom has a big mirror.
+
+
+The bagel is put in a square plate.
+
+
+A clock tower near the river.
+Figure 21: Additional text-to-image generation results with different text prompt, produced by TLDM with $T_{\mathrm{trunc}} = 4$ .
\ No newline at end of file
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+# TRUTHFUL SELF-PLAY
+
+# Shohei Ohsawa
+
+Founder & CEO
+
+Daisy AI
+
+6-13-9 Ginza, Chuo-ku, Tokyo, Japan
+
+o@daisy.inc
+
+# ABSTRACT
+
+We present a general framework for evolutionary learning to emergent unbiased state representation without any supervision. Evolutionary frameworks such as self-play converge to bad local optima in case of multi-agent reinforcement learning in non-cooperative partially observable environments with communication due to information asymmetry. Our proposed framework is a simple modification of self-play inspired by mechanism design, also known as reverse game theory, to elicit truthful signals and make the agents cooperative. The key idea is to add imaginary rewards using the peer prediction method, i.e., a mechanism for evaluating the validity of information exchanged between agents in a decentralized environment. Numerical experiments with predator prey, traffic junction and StarCraft tasks demonstrate that the state-of-the-art performance of our framework.
+
+# 1 INTRODUCTION
+
+Evolving culture prevents deep neural networks from falling into bad local optima (Bengio, 2012). Self-play (Samuel, 1967; Tesauro, 1995) has not only demonstrated the ability to abstract high-dimensional state spaces as typified by AlphaGo (Silver et al., 2017), but also improved exploration coverage in partially observable environments. Communication (Sukhbaatar et al., 2016; Singh et al., 2019) exchanges their internal representations such as explored observation and hidden state in RNNs. Evolutionary learning is expected to be a general framework for creating superhuman AIs as such learning can generate a high-level abstract representation without any bias in supervision.
+
+However, when applying evolutionary learning to a partially observable environment with non-cooperative agents, improper bias is injected into the state representation. This bias originates from the environment. A partially observable environment with non-cooperative agents induces actions that disable an agent from honestly sharing the correct internal state resulting in the agent taking actions such as concealing information and deceiving other agents at equilibrium (Singh et al., 2019). The problem arises because the agent cannot fully observe the state of the environment, and thus, it does not have sufficient knowledge to verify the information provided by other agents. Furthermore, neural networks are vulnerable to adversarial examples (Szegedy et al., 2014) and are likely to induce erroneous behavior with small perturbations. Many discriminative models for information accuracy are available; these include GANs (Goodfellow et al., 2014; Radford et al., 2016) and curriculum learning (Lowe et al., 2020). However, these models assume that accurate samples can be obtained by supervision. Because of this assumption, is it impossible to apply these models to a partially observable environment, where the distribution is not stable.
+
+We generalize self-play to non-cooperative partially observable environments via mechanism design (Myerson, 1983; Miller et al., 2005), which is also known as reverse game theory. The key idea is to add imaginary rewards by using the peer prediction method (Miller et al., 2005), that is, a mechanism for evaluating the validity of information exchanged between agents in a decentralized environment, which is calculated based on social influence on the signals. We formulate the non-cooperative partially observable environment as an extension of the partially observable stochastic games (POSG) (Hansen et al., 2004); introduce truthfulness (Vickrey, 1961), which is an indicator of the validity of state representation. We show that the imaginary reward enables us to reflect the bias of state representation on the gradient without oracles.
+
+As the first contribution, we propose truthful self-play (TSP) and analytically demonstrate convergence to the global optimum (Section 4). We propose the imaginary reward on the basis of the peer prediction method (Miller et al., 2005) and apply it to self-play. The mechanism affects the gradient of the local optima, but not the global optima. The trick is to use the actions taken by the agents as feedback to verify the received signal from the every other agent, instead of the true state, input, and intent, which the agents cannot fully observe. TSP only requires a modification of the baseline function for self-play; it drastically improves the convergence to the global optimum in Comm-POSG.
+
+As the second contribution, based on the results of numerical experiments, we report that the TSP achieved state-of-the-art performance for various multi-agent tasks made of up to 20 agents (Section 5). Using predator prey (Barrett et al., 2011), traffic junction (Sukhbaatar et al., 2016; Singh et al., 2019), and StarCraft (Synnaeve et al., 2016) environments, which are typically used in Comm-POSG research, we compared the performances of TSP with the current neural nets, including the state-of-the-art method, with LSTM, CommNet (Sukhbaatar et al., 2016), and IC3Net (Singh et al., 2019). We report that the model with IC3Net optimized by TSP has the best performance.
+
+This work is the first attempt to apply mechanism design to evolutionary learning. TSP is a general optimization algorithm whose convergence is theoretically guaranteed for arbitrary policies and environments. Since no supervision is required, TSP has a wide range of applications to not only game AIs (Silver et al., 2017), but also the robots (Jaderberg et al., 2018), chatbots (Gupta et al., 2019; Chevalier et al., 2019), and autonomous cars (Tang, 2019) employed in multiagent tasks.
+
+Notation: Vectors are columns. Let $\llbracket n\rrbracket \coloneqq \{1,\dots ,n\}$ . $\mathbb{R}$ is a set of real numbers. $\pmb{i}$ is the imaginary unit. Re $u$ and Im $u$ are a real and an imaginary part of complex number $u$ , respectively. $n$ -tuple are written as boldface of the original variables $\mathbf{a} \coloneqq \langle a_1, \ldots, a_n \rangle$ , and $\mathbf{a}_{-i}$ is a $(n - 1)$ -tuple obtained by removing the $i$ -th entry from $\mathbf{a}$ . Let $\mathbf{1} \coloneqq (1, \ldots, 1)^{\mathrm{T}}$ . Matrices are shown in uppercase letters $\mathcal{L} \coloneqq (\ell_{ij})$ . $E$ is the unit matrix. The set of probability distributions based on the support $\mathcal{X}$ is described as $\mathfrak{P}(\mathcal{X})$ .
+
+# 2 RELATED WORK
+
+Neural communication has gained attention in the field of multiagent reinforcement learning (MARL) for both discrete (Foerster et al., 2016) and continuous (Sukhbaatar et al., 2016; Singh et al., 2019) signals. Those networks are trained via self-play to exchange the internal state of the environment stored in the working memory of recurrent neural networks (RNNs) to learn the right policy in partially observable environments.
+
+The term self-play was coined by the game AI community in the latter half of the century. Samuel (Samuel, 1967) introduced self-play as a framework for sharing a state-action value among two opposing agents to efficiently search the state space at Checkers. TD-Gammon (Tesauro, 1995) introduced self-play as a framework to learn $\mathrm{TD}(\lambda)$ (Sutton & Barto, 1998) and achieve professional-grade levels in backgammon. AlphaGo (Silver et al., 2017) defeated the Go champion by combining supervised learning with professional game records and self-play. AlphaZero (Silver et al., 2018) successfully learnt beyond its own performance entirely based on self-play. All these studies explain that eliminating the bias of human knowledge in supervision is the advantage of self-play.
+
+Self-play is also known as evolutionary learning (Bengio, 2012) in the deep learning community mainly as an approach to emerging representations without supervision (Bansal et al., 2018; Balduzzi et al., 2019). Bansal et al. (2018) show that competitive environments contribute to emerging diversity and complexity. Rich generative models such as GANs (Goodfellow et al., 2014; Radford et al., 2016) are frameworks for acquiring an environmental model by employing competitive settings. RNNs such as world models (Ha & Schmidhuber, 2018; Eslami et al., 2018) are capable of more comprehensive ranges of exploration in partially observable environments and generation of symbols and languages (Bengio, 2017; Gupta et al., 2019; Chevalier et al., 2019). The difference between evolutionary learning and supervised learning is the absence of human knowledge and oracles.
+
+Several works have formalized those in which the agents exchange environmental information as a formal class of the games such as Dec-POMDP-Com (Goldman & Zilberstein, 2003) and COM-MTDP (Pynadath & Tambe, 2002), and several frameworks are proposed to aim to solve the problems. However, the limitation of the frameworks is that they assume a common reward. As there are yet no formal definition of non-cooperative communication game, we formalize such a game to
+
+Comm-POSG as a superset of POSGs (Hansen et al., 2004), a more general class of multi-agent games including the cases of non-cooperativity (Hansen et al., 2004).
+
+To the best of our knowledge, there are no studies that have introduced truthful mechanisms into the field of MARL, but it may be possible to introduce it by using agents that can learn flexibly, such as neural networks. A typical truthful mechanism is the VCG mechanism (Vickrey, 1961), which is a generalization of the pivot method used in auction theory, but whereas the subject of the report that must satisfy truthfulness must be a valuation (or a value function if interpreted from a RL perspective). In this study, the scope of application is different because the belief states of the environment are subject to reporting. Therefore, we introduce instead a peer prediction method (Miller et al., 2005) that guarantees truthfulness with respect to reporting beliefs about arbitrary probability distributions using proper scoring rules (Gneiting & Raftery, 2007).
+
+# 3 PROBLEM DEFINITION
+
+# 3.1 COMM-POSG
+
+A communicative partially-observable stochastic game (Comm-POSG) is a class of non-cooperative Bayesian games in which every agent does not fully observe the environment but interacts each other. We define Comm-POSG as an extension of POSG (Hansen et al., 2004) with a message protocol.
+
+Definition 3.1 (Hansen et al., 2004) POSG $\langle n, T, S, \mathcal{A}, \mathcal{X}, \mathcal{T}, \mathcal{P}, \mathcal{R} \rangle$ is a class for multi-agent decision making under uncertainty in which the state evolves over time $1 \leq t \leq T$ , where
+
+- $n$ is the number of agents,
+- $T$ is a horizon i.e., the episode length,
+- $S$ is a set of discrete/continuous state $s_t \in S$ with an initial probabilistic distribution $p(s_0)$ ,
+- $\mathcal{A}$ is a set of discrete/continuous action $a_{ti} \in \mathcal{A}$ ,
+- $\mathcal{X}$ is a set of discrete/continuous observation $x_{ti} \in \mathcal{X}$ ,
+- $\mathcal{T} \in \mathfrak{P}(\mathcal{S} \times \mathcal{A} \times \mathcal{S})$ is state transition probability,
+- $\mathcal{P} \in \mathfrak{P}\left(\mathcal{S} \times \mathcal{X}^n\right)$ is an observation probability, and
+- $\mathcal{R}: \mathcal{S} \times \mathcal{A}^n \to \mathbb{R}^n$ is a reward function that outputs an $n$ -dimensional vector. $\square$
+
+In Comm-POSGs, every agent further follows a message protocol $\mathcal{Z}^{n\times n}$ , where $\mathcal{Z}$ is the discrete/continuous signal space. The complete information exchanged among the agent in time is $\mathbf{Z}_t$ , where $\mathbf{Z}_t\coloneqq (z_{tij})_{i,j\in [[n]]}\in \mathcal{Z}^{n\times n}$ is a signal matrix in which $(i,j)$ -th entry $z_{tij}$ represents a signal from Agent $i$ to Agent $j$ at $t$ . The $i$ -th diagonal entry of $\mathbf{Z}_t$ , $h_{ti}\coloneqq z_{tii}$ represents the pre-state, an internal state of $i$ -th agent before receiving the singals from the others. A game in Comm-POSG is denoted as $\mathcal{G}\coloneqq \langle n,T,\mathcal{S},\mathcal{A},\mathcal{X},\mathcal{T},\mathcal{P},\mathcal{R},\mathcal{Z}\rangle$ .
+
+The objective of Comm-POSG is social welfare (Arrow, 1963) defined by the following,
+
+$$
+J := \sum_ {i = 1} ^ {n} V ^ {\pi_ {i}}; \quad V ^ {\pi_ {i}} := \mathbb {E} _ {\pi_ {i}} \left[ \sum_ {t = 1} ^ {T} \gamma^ {t - 1} r _ {t i} \right], \tag {1}
+$$
+
+where $\gamma \in [0,1]$ is discount rate, $r_{ti}$ is reward $\pi_i$ is a stochastic policy, and $V^{\pi_i}$ is the value function.
+
+In extensive-form games including Comm-POSG, in addition to the information in the environment, the information of other agents cannot be observed. In the optimization problem under these assumptions, a policy converges to a solution called the Bayesian Nash equilibrium (BNE) (Fudenberg, 1993). We denote the social welfare at the BNE is $J^{*}$ , and the global maximum $\hat{J}$ . In general, $J^{*} \neq \hat{J}$ holds, which is closely related to the information asymmetry.
+
+# 3.2 COMMUNICATIVE RECURRENT AGENTS
+
+In order to propose an optimization algorithm in this paper, we do not propose a concrete structure of the network, but we propose an abstract structure that can cover existing neural communication models (Sukhbaatar et al., 2016; Singh et al., 2019), namely communicative recurrent agents (CRAs) $\langle f_{\phi},\sigma_{\phi},q_{\phi},\pi_{\theta}\rangle$ , where
+
+- $f_{\phi}(\hat{h}_{t - 1,i},x_{ti}) \mapsto \mathcal{Z}$ is a deep RNN for the high-dimensional input $x_{ti} \in \mathcal{X}$ and the previous post-state $\hat{h}_{t - 1,i} \in \mathcal{Z}$ , with a parameter $\phi$ and an initial state $\hat{h}_0 \in \mathcal{Z}$ ,
+- $\sigma_{\phi}(\mathbf{z}_{ti}|h_{ti})$ is a stochastic messaging policy for a pre-state $h_{ti} \coloneqq f_{\phi}(\hat{h}_{t - 1,i},x_{ti})$
+- $q_{\phi}(\hat{h}_{ti}|\hat{\mathbf{z}}_{ti})$ is a stochastic model for a post-state $\hat{h}_{ti} \in \mathcal{Z}$ and the received messages $\hat{\mathbf{z}}_{ti} := \mathbf{Z}_{t:i}^{\mathrm{T}} = (z_{t1i}, \ldots, z_{t,i-1,i}, h_{ti}, z_{t,i+1,i}, \ldots, z_{tni})^{\mathrm{T}}$ , and
+- $\pi_{\theta}(a_{ti}|\hat{h}_{ti})$ is the stochastic action policy with a parameter $\theta$ .
+
+These agents are trained through self-play using on-policy learning such as REINFORCE (Williams, 1992). All $n$ -agents share the same weight per episode, and the weights are updated based on the cumulative reward after the episode.
+
+In addition to the recurrent agent's output of actions with the observation series as input, the CRA has signals for communication as input and output. CRAs estimate current state of the environment and current value of the agent herself based on the post-state model with the pre-state $h_{ti}$ in the hidden layer of the RNN and the received signals $\hat{\mathbf{z}}_{ti, -i}$ from other agents. Hence, the veracity of the signals $\mathbf{z}_{ti}$ is the point of contention.
+
+# 3.3 TRUTHFULNESS
+
+In mechanism design, a truthful game (Vickrey, 1961) is a game in which all agents make an honest reporting in the Bayesian Nash equilibrium. In Comm-POSGs, the truthfulness of the game is achieved if all the sent signal equals the pre-state $z_{tij} = h_{ti}$ i.e., all the agent share a complete information. In such case, every agent has the same information $\hat{\mathbf{z}}_{ti} = \mathbf{h}_{ti} \coloneqq (h_{t1},\dots,h_{tn})^{\mathrm{T}}$ for all $i$ 's and the same post-state model probability distribution, and hence the mean of the cross entropy between the distributions below will be minimized.
+
+$$
+D _ {\phi} \left(\mathbf {Z} _ {t}\right) := \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {H} \left[ q _ {\phi} \left(\hat {h} _ {t i} \mid \hat {\mathbf {z}} _ {t i}\right) \right] + \frac {1}{n ^ {2}} \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} \mathcal {D} _ {\mathrm {K L}} \left(q _ {\phi} \left(\hat {h} _ {t i} \mid \hat {\mathbf {z}} _ {t i}\right) \mid | q _ {\phi} \left(\hat {h} _ {t j} \mid \hat {\mathbf {z}} _ {t j}\right)\right). \tag {2}
+$$
+
+The first term represents the entropy of knowledge each agent has about the environment, and the second the information asymmetry between the agents. $D_{\phi}$ is a lower bound on the amount of true information the environment has $\mathbb{H}[p(s_t)]$ . Since achieving truthfulness is essentially the same problem as minimizing $D_{\phi}$ , it also maximizes $J^{*}$ simultaneously.
+
+Proposition 3.1. (global optimality) For any games $\mathcal{G}$ in Comm-POSG, if $D_{\phi}(\mathbf{Z}_t) = \mathbb{H}[p(s_t)]$ for $0 \leq t \leq T$ and $\mathcal{R}_i$ is symmetric for any permutation of $i \in [[n]]$ , $J^{*}(\mathcal{G}) = \hat{J}(\mathcal{G})$ holds.
+
+Proof. Let $\mathbf{w} \coloneqq \langle \theta, \phi \rangle$ on a parameter space $\mathcal{W}(\mathcal{G})$ , and $\mathcal{W}^*(\mathcal{G})$ the BNE. Since $J$ is obviously maximized if $\sigma_{\phi}$ is truthful, we prove $\sigma_{\phi}$ must be truthful under a given condition. To this end, we show the following Pareto-optimality.
+
+Lemma 3.1. For any $\mathcal{G}$ in Comm-POSG and given $\mathbf{w}$ , if $J(\mathbf{w}) \geq J(\mathbf{w}^{\prime})$ holds for any $\mathbf{w}^{\prime} \in \mathcal{W}(\mathcal{G})$ , then either of $U(\mathbf{w}) \geq U(\mathbf{w}^{\prime})$ or $\mathcal{D}_{\mathrm{KL}}(p||q_{\phi}) \leq \mathcal{D}_{\mathrm{KL}}(p||q_{\phi^{\prime}})$ holds, where
+
+$$
+U (\mathbf {w}) := \mathbb {E} _ {q _ {\phi}} \left[ V ^ {\pi_ {\theta}} \right] = \int_ {s _ {t} \in \mathcal {S}, \hat {\mathbf {z}} _ {t} \in \mathcal {Z} ^ {n}} V ^ {\pi_ {\theta}} \left(s _ {t}\right) \mathrm {d} q _ {\phi} \left(s _ {t} \mid h _ {t i}, \mathbf {z} _ {t, - i}\right) \mathrm {d} \sigma_ {\phi} (\hat {\mathbf {z}} _ {t}). \tag {3}
+$$
+
+Proof. The first inequality $U(\mathbf{w}) \geq U(\mathbf{w}^{\prime})$ indicates that $\mathbf{w}$ is on the BNE $\mathcal{W}^{*}(\mathcal{G})$ given $\phi$ , and of the second that the belief state $q_{\phi}$ is as close to the true state $p$ as possible. On a fully-observable environment, from the theorem of value iteration, there exists the solution $\pi^{\star}(s_t)$ w.r.t. $V^{\pi^{\star}}(s_t)$ for any $s_t \in S$ . We name $\pi^{\star}$ and $V^{\star} \coloneqq V^{\pi^{\star}}$ the unbiased policy and value, respectively.
+
+Since the unbiased policy solves the objective as $J(\mathbf{w}) = n\mathbb{E}_{p(s_t)}[V^{\pi_\theta}(s_t)]$ from Eq (1), the goal intrinsicity is to find the policy $\pi^{*}$ as close as $\pi^{\star}$ for $\langle \pi^{*},\phi^{*}\rangle \in \mathcal{W}(\mathcal{G})$ , that maximizes $U(\mathbf{w})$ . The $\pi^{*}$ further can be represented as a mixed policy made of a couple of the unbiased policy $\pi^{\star}$ and a biased policy $\pi^{\prime}\neq \pi^{\star}$ as follows,
+
+$$
+\begin{array}{l} \pi^ {*} (a _ {t i} | \mathbf {x} _ {t i}, \phi) = \mathbb {E} _ {q _ {\phi} (s _ {t i} | \mathbf {x} _ {t i})} [ \pi^ {\star} (a _ {t i} | s _ {t i}) ] \\ = q _ {\phi} \left(s _ {t 0} \mid \mathbf {x} _ {t i}\right) \pi^ {*} \left(a _ {t i} \mid s _ {t 0}\right) + \left(1 - q _ {\phi} \left(s _ {t 0} \mid \mathbf {x} _ {t i}\right)\right) \pi^ {\prime} \left(a _ {t i} \mid \phi , \mathbf {x} _ {t i}\right), \tag {4} \\ \end{array}
+$$
+
+for the observations $\mathbf{x}_{ti} \in \mathcal{X}^n$ , where $s_{t0} \in S$ is the true state. Hence
+
+$$
+\begin{array}{l} V ^ {\pi^ {*}} (s _ {t 0} | \phi) = \mathbb {E} _ {\mathcal {P} (\mathbf {x} _ {t i} | s _ {t 0})} [ q _ {\phi} (s _ {t 0} | \mathbf {x} _ {t i}) V ^ {\star} (s _ {t 0}) + (1 - q _ {\phi} (s _ {t 0} | \mathbf {x} _ {t i})) V ^ {\prime} (\mathbf {x} _ {t i} | \phi)) ] \\ = q _ {\phi} \left(s _ {t 0}\right) V ^ {\star} \left(s _ {t 0}\right) + \mathbb {E} _ {\mathcal {P} \left(\mathbf {x} _ {t i} \mid s _ {t 0}\right)} \left[ \left(1 - q _ {\phi} \left(s _ {t 0} \mid \mathbf {x} _ {t i}\right)\right) V ^ {\prime} \left(\mathbf {x} _ {t i} \mid \phi\right) \right] \\ = q _ {\phi} \left(s _ {t 0}\right) V ^ {\star} \left(s _ {t 0}\right) + \left(1 - q _ {\phi} \left(s _ {t 0}\right)\right) \bar {V} ^ {\prime} \left(s _ {t 0} \mid \phi\right), \tag {5} \\ \end{array}
+$$
+
+where
+
+$$
+V ^ {\prime} \left(\mathbf {x} _ {t i} \mid \phi\right) := \int_ {\mathbf {s} _ {t} \in \mathcal {S} ^ {n + 1}, \mathbf {a} _ {t} \in \mathcal {A} ^ {n}} \mathcal {R} _ {i} \left(s _ {t 0}, \mathbf {a} _ {t}\right) \prod_ {i = 1} ^ {n} \mathrm {d} \pi^ {\prime} \left(a _ {t i} \mid s _ {t i}\right) q _ {\phi} \left(s _ {t i} \mid \mathbf {x} _ {t i}\right), \tag {6}
+$$
+
+and
+
+$$
+\bar {V} ^ {\prime} \left(s _ {t 0} \mid \phi\right) := \mathbb {E} _ {\mathcal {P} \left(\mathbf {x} _ {t i} \mid s _ {t 0}\right)} \left[ V ^ {\prime} \left(\mathbf {x} _ {t i} \mid \phi\right) \frac {1 - q _ {\phi} \left(s _ {t 0} \mid \mathbf {x} _ {t i}\right)}{1 - q _ {\phi} \left(s _ {t 0}\right)} \right]. \tag {7}
+$$
+
+Thus, the error from the unbiased value function can be written as $V^{\star}(s_{t0}) - V^{\pi^{*}}(s_{t0}|\phi) = (1 - q_{\phi}(s_{t0}))(V^{\star}(s_{t0}) - \bar{V}'(s_{t0}|\phi))$ , which is minimized if $q_{\phi}(s_{t0}) = 1$ as $V^{\star}(s_{t0}) > \bar{V}'(s_{t0}|\phi)$ by the definition. From the Jensen's equation,
+
+$$
+\begin{array}{l} \log \mathbb {E} _ {p (s _ {t 0})} [ q _ {\phi} (s _ {t 0}) ] \geq \mathbb {E} _ {p (s _ {t 0})} [ \log q _ {\phi} (s _ {t 0}) ] \\ = - \mathcal {D} _ {\mathrm {K L}} (p | | q _ {\phi}) - \mathbb {H} [ p ]. \tag {8} \\ \end{array}
+$$
+
+The right-hand side of the equation corresponds to the negative cross-entropy to be maximized. Therefore, as the second term $\mathbb{H}[p]$ depends not on $\phi$ , the optimization is achieved by minimizing $\mathcal{D}_{\mathrm{KL}}(p||q_{\phi})$ .
+
+Suppose that $\hat{J}(\mathcal{G}) = J(\mathbf{w}) > J^{*}(\mathcal{G})$ for a non-truthful reporting policy s.t. $\sigma_{\phi}(h|h) < 1$ . From lemma 3.1, $q_{\phi}(s_t|h_{ti})$ for an internal state $h_{ti} = f(x_{ti})$ of Agent $i$ with an encoder $f$ minimizes $\mathcal{D}_{\mathrm{KL}}(p||q_{\phi}(s_t|h_{ti}))$ . As that $q_{\phi}(s_t|z_{ti}) \neq q_{\phi}(s_t|h_{ti})$ and $\mathcal{D}_{\mathrm{KL}}(p||q_{\phi}(s_t|z_{ti})) > \mathcal{D}_{\mathrm{KL}}(p||q_{\phi}(s_t|h_{ti}))$ contradicts the Pareto-optimality, $\sigma_{\phi}$ must be truthful. $\square$
+
+# 4 PROPOSED FRAMEWORK
+
+An obvious way to achieve truthful learning is to add $D_{\phi}$ as a penalty term of the objective, but there are two obstacles to this approach. One is that the new regularization term also adds a bias to the social welfare $J$ , and the other is that $D_{\phi}$ contains the agent's internal state, post-state $\hat{h}_{ti}$ , so the exact amount cannot be measured by the designer of the learner. If post-states are reported correctly, then pre-states should also be reported honestly, and thus truthfulness is achieved. Therefore, it must be assumed that the post-states cannot be observed during optimization.
+
+Our framework, truthful self-play (TSP), consists of two elements: one is the introduction of imaginary rewards, a general framework for unbiased regularization in Comm-POSG, and the other is the introduction of peer prediction method (Miller et al., 2005), a truthful mechanism to encourage honest reporting based solely on observable variables. In the following, each of them is described separately and we clarify that the proposed framework converges to the global optimum in Comm-POSG. We show the whole procedure in Algorithm 1.
+
+# 4.1 IMAGINARY REWARD
+
+Imaginary rewards are virtual rewards passed from an agent and have a different basis $i$ than rewards passed from the environment, with the characteristic that they sum to zero. Since the most of RL environments, including Comm-POSG, are of no other entities than agent and environment, two-dimensional structure are sufficient to describe them comprehensively if we wish to distinguish the sender of the reward. To remain the social welfare of the system is real, the system must be designed so that the sum of the imaginary rewards, i.e., imaginary part of the social welfare, is zero. In other words, it is not observed macroscopically and affects only the relative expected rewards of agents. The real and imaginary parts of the complex rewards are ordered by the mass parameter $\beta$ during training, which allows the weights of the network to maintain a real structure.
+
+The whole imaginary reward is denoted as $i\mathbf{Y} = (iy_{ij})_{i,j \in [n]}$ where $iy_{ij}$ is the imaginary reward passed from Agent $i$ to Agent $j$ , and the complex reward for the whole game is $\mathbf{R}^{+} := \mathbf{R} + i\mathbf{Y}$ where $\mathbf{R}$ is a diagonal matrix with the environmental reward $r_i$ as $(i,i)$ -th entry. We write $\mathcal{G}[i\mathbf{Y}]$ as the structure in which this structure is introduced. In this case, the following proposition holds.
+
+Proposition 4.1. For any $\mathcal{G}$ in Comm-POSG, if $\mathcal{G}[i\mathbf{Y}]$ is truthful and $\mathbf{R}^{+}$ is an Hermitian matrix, $J^{*}(\mathcal{G}[i\mathbf{Y}]) = \hat{J} (\mathcal{G})$ holds.
+
+Proof. Since $\mathcal{G}[i\mathbf{Y}]$ is truthful, $J^{*}(\mathcal{G}[i\mathbf{Y}]) = \hat{J} (\mathcal{G}[i\mathbf{Y}])$ holds from Proposition 3.1. Further, since $\mathbf{R}^+$ is Hermitian, $iy_{ij} = -iy_{ji}$ , and hence $\operatorname {Im}\hat{J} (\mathcal{G}[i\mathbf{Y}]) = 0$ holds; $\hat{J} (\mathcal{G}[i\mathbf{Y}]) = \hat{J} (\mathcal{G})$ holds.
+
+This indicates that the BNE could be improved by introducing imaginary rewards: $J^{*}(\mathcal{G}[i\mathbf{Y}]) \geq J^{*}(\mathcal{G})$ . Also, since $\sum_{i=1}^{n} \sum_{j=1}^{n} iy_{ij} = 0$ from the condition that $\mathbf{R}^{+}$ is Hermitian, the imaginary rewards affect not the social welfare of the system, which is a macroscopic objective, but only the expected rewards of each agent. The baseline in policy gradient (Williams, 1992) is an example of a function that affects not the objective when the mean gets zero. However, the baseline function is a quantity that is determined based on the value function of a single agent, whereas the imaginary reward is different in that (1) it affects the value function of each agent and (2) it is a meaningful quantity only when $n \geq 2$ and is not observed when $n = 1$ .
+
+# 4.2 PEER PREDICTION MECHANISM
+
+The peer prediction mechanism (Miller et al., 2005) is derived from a mechanism design using proper scoring rules (Gneiting & Raftery, 2007), which aims to encourage verifiers to honestly report their beliefs by assigning a reward measure score to their responses when predicting probabilistic events. These mechanisms assume at least two agents, a reporter and a verifier.
+
+The general scoring rule can be expressed as $\mathcal{F}(p_s\| s)$ where $p_{s}$ is the probability of occurrence reported by the verifier for the event $s$ , and $\mathcal{F}(p_s\| s)$ is the score to be obtained if the reported event $s$ actually occurred. The scoring rule is proper if an honest declaration consistent with the beliefs of the verifier and the reported $p_{s}$ maximizes the expected value of the earned score, and it is strictly proper if it is the only option for maximizing the expected value. A representative example that is strictly proper is the logarithmic scoring rule $\mathcal{F}(p_s\| s) = \log p_s$ , where the expected value for a 1-bit signal is the cross-entropy $p_s^*\log p_s + (1 - p_s^*)\log (1 - p_s)$ for belief $p_s^*$ . One can find that $p_{s} = p_{s}^{*}$ is the only report that maximizes the score.
+
+Since the proper scoring rule assumes that events $s$ are observable, it is not applicable to problems such as partial observation environments where the true value is hidden. Miller et al. (2005), who first presented a peer prediction mechanism, posed scoring to a posterior of the verifiers that are updated by the signal, rather than the event. This concept is formulated by a model that assumes that an event $s$ emits a signal $z$ stochastically and infers the type of $s$ by the signal of the reporters who receive it. The peer prediction mechanism (Miller et al., 2005) is denoted as $\mathcal{F}(p(s|z)\|s)$ under the assumption that (1) the type of event $s$ and the signal $z$ emitted by each type follow a given prior, (2) the priors are shared knowledge among verifiers, and (3) the posterior is updated according to the reports.
+
+We apply the mechanism to RL, i.e., the problem of predicting the agent's optimal behavior $a_{ti} \sim \pi_{\theta}|s_t$ for the true state $s_t \in S$ . In self-play, the conditions of 1 and 2 can be satisfied because the prior $\pi_{\theta}$ is shared among the agents, and furthermore, the post-state in Comm-POSG corresponds to 3, so that the peer prediction mechanism can be applied to the problem of predicting agent behavior. To summarize the above discussion, we can see that we can allocate a score matrix $\mathcal{L}_t$ as follows,
+
+$$
+\mathcal {L} _ {t} := \left(\ell \left(a _ {t i} \mid z _ {t j i}\right)\right) _ {i, j \in [ n ]}; \quad \ell \left(a _ {t i} \mid z _ {t j i}\right) := \mathcal {F} \left(\pi_ {\theta} \left(a _ {t i} \mid z _ {t j i}\right) \| a _ {t i}\right) = \log \pi_ {\theta} \left(a _ {t i} \mid z _ {t j i}\right), \tag {9}
+$$
+
+which is an $n$ -order square matrix representing the score from Agent $i$ to Agent $j$ .
+
+# 4.3 THE TRUTHFUL SELF-PLAY
+
+In TSP, a truthful system is constructed by introducing a proper scoring rule into imaginary rewards. However, since the score matrix obtained from the proper scoring rule does not satisfy Hermitianity, we perform zero-averaging by subtracting the mean of the scores from each element of the matrix,
+
+Algorithm 1 The truthful self-play (TSP).
+
+Require: Comm-POSG $\mathcal{G} = \langle n,T,\mathcal{S},\mathcal{A}^{+},\mathcal{X},\mathcal{T},\mathcal{P},\mathcal{R}^{+}\rangle$ , recurrent neural network $\langle \sigma_{\phi},q_{\phi},\pi_{\theta}\rangle$ with initial weight $\mathbf{w}_0 = \langle \theta_0,\phi_0\rangle$ and initial state $h_0$ , learning rate $\alpha >0$ , and mass parameter $\beta \geq 0$
+
+Initialize $\mathbf{w}\gets \mathbf{w}_0$
+
+for each episode do
+
+Genesis: $s_1\sim p(s)$ $\hat{h}_{0i}\gets h_0\forall i\in [[n]]$
+
+for $t = 1$ to $T$ do
+
+1. Self-Play Observe $\mathbf{x}_t\sim \mathcal{P}^n (\cdot |s_t)$ Update pre-state $h_{ti}\gets f_{\phi}(\hat{h}_{t - 1,i},x_{ti}),\quad \forall i\in [[n]]$ Generate message $\mathbf{z}_{ti}\sim \sigma_{\phi}(\cdot |h_{ti})$ $\forall i\in [[n]]$ Send message $\mathbf{Z}_t\gets (\mathbf{z}_{t1},\dots ,\mathbf{z}_{tn})$ Receive the message $\hat{\mathbf{z}}_{ti}\gets \mathbf{Z}_{t:i}^{\mathrm{T}},\quad \forall i\in [[n]]$ Update post-state $\hat{h}_{ti}\sim q_{\phi}(\cdot |\hat{\mathbf{z}}_{ti}),\quad \forall i\in [[n]]$ Act $a_{ti}\sim \pi_{\theta}(\cdot |\hat{h}_{ti}),\quad \forall i\in [[n]]$ Get the real reward $\mathbf{r}_t\gets \mathcal{R}(s_t,\mathbf{a}_t)$
+
+2. Compute the score matrix with peer prediction mechanism (Miller et al., 2005),
+
+$$
+\ell_ {i j} \leftarrow \left\{ \begin{array}{l l} \log \pi_ {\theta} \left(a _ {j} \mid z _ {i}\right) & (i \neq j) \forall i, j \in [ n ]; \\ 0 & (i = j) \end{array} \right. \tag {12}
+$$
+
+3. Combine real and imaginary rewards and construct a complex reward.
+
+$$
+\mathbf {R} _ {t} ^ {+} \leftarrow \mathbf {R} _ {t} + 1 \Delta \mathcal {L}. \tag {13}
+$$
+
+4. Update the weights by policy gradient (Williams, 1992),
+
+$$
+\mathbf {g} _ {t} \leftarrow \sum_ {i = 1} ^ {n} r _ {t i} ^ {+} \nabla_ {\mathbf {w}} \left[ \log \pi_ {\theta} \left(a _ {t i} | \hat {h} _ {t i}\right) + \log q _ {\phi} \left(\hat {h} _ {t i} \mid \hat {\mathbf {z}} _ {t i}\right) + \log \sigma_ {\phi} \left(z _ {t i} \mid \hat {h} _ {t - 1, i}, x _ {t i}\right) \right] \tag {14}
+$$
+
+$$
+\mathbf {w} \leftarrow \mathbf {w} + \alpha \operatorname {R e} \mathbf {g} _ {t} + \alpha \beta \operatorname {I m} \mathbf {g} _ {t}
+$$
+
+5. Proceed to the next state $s_{t + 1}\sim \mathcal{T}(\cdot |s_t,\mathbf{a}_t)$
+
+end for
+
+end for
+
+return w
+
+thereby making the sum to zero. This can be expressed as follows using the graph Laplacian $\Delta := E - 11^{\mathrm{T}} / n$ .
+
+$$
+\mathbf {Y} = \Delta \mathcal {L} _ {\psi} = \frac {1}{n} \left[ \begin{array}{c c c c} n - 1 & - 1 & \dots & - 1 \\ - 1 & n - 1 & \dots & - 1 \\ \vdots & \vdots & \ddots & \vdots \\ - 1 & - 1 & \dots & n - 1 \end{array} \right] \left[ \begin{array}{c c c c} 0 & \ell_ {\psi} \left(a _ {t 2} \mid z _ {t 1}\right) & \dots & \ell_ {\psi} \left(a _ {t n} \mid z _ {t 1}\right) \\ \ell_ {\psi} \left(a _ {t 1} \mid z _ {t 2}\right) & 0 & \dots & \ell_ {\psi} \left(a _ {t n} \mid z _ {t 2}\right) \\ \vdots & \vdots & \ddots & \vdots \\ \ell_ {\psi} \left(a _ {t 1} \mid z _ {t n}\right) & \ell_ {\psi} \left(a _ {t 2} \mid z _ {t n}\right) & \dots & 0 \end{array} \right], \tag {10}
+$$
+
+to get
+
+$$
+\mathbf {R} ^ {+} = \mathbf {R} + i \Delta \mathcal {L} _ {\psi}, \tag {11}
+$$
+
+which is the formula that connects reinforcement learning and mechanism design.
+
+We show the truthful self-play (TSP) in Algorithm 1. The only modification required from self-play is the imaginary reward.
+
+Theorem 4.1. (global optimality) For any $\mathcal{G}$ in Comm-POSG, TSP converges to the global optimum $\hat{J}(\mathcal{G})$ if the following convergence condition are met,
+
+$$
+\sup _ {\phi} \left| \frac {\partial \operatorname {R e} V ^ {\pi_ {\theta}}}{\partial \operatorname {I m} V ^ {\pi_ {\theta}}} \right| < \beta , \tag {15}
+$$
+
+Table 1: Social welfare $J$ in five Comm-POSG tasks. PP- $n$ : Predator prey and TJ- $n$ : Traffic junction. All the scores are negatives. The experiment was repeated thrice. The average and standard deviation are listed. Bold is the highest score. The models listed in the top three rows show the models optimized by SP, and the bottom row shows IC3Net optimized by TSP.
+
+