0, there exists k > 0 and δ > 0 *such that, for all initializations* w0 satisfying ∥w0 − w\n⋆∥ ⩽ δ, *the restarted MEG satisfies:*\n\n$$\\|w_{t}-w^{\\star}\\|=O((1-\\varphi+\\varepsilon)^{t})\\|w_{0}-w^{\\star}\\|.$$\n\n## 7 Experiments\n\n\n\nFigure 3: *Illustration of the game Jacobian spectra and the performance of different algorithms considered.*\nJacobian spectrum in the first plot matches S\n⋆\n2 in (14) precisely, while that in the third plot inexactly follows S\n⋆\n2\n. The second (fourth) plot shows the performance of different algorithms for solving quadratic games in\n(16) with the Jacobian spectrum following the first (third) plot.\n\nIn this section, we perform numerical experiments to optimize a game when the Jacobian has a cross-shaped spectrum in (14). We focus on this spectrum as it may be the most challenging case, involving both real and complex eigenvalues (c.f., Theorem 3). To test the robustness, we consider two cases where the Jacobian spectrum exactly follows S\n⋆\n2 in (14), as well as the inexact case. We illustrate them in Figure 3.\n\nWe focus on two-player quadratic games, where player 1 controls x ∈ R\nd1 and player 2 controls y ∈ R\nd2 with loss functions in (16). In our setting, the corresponding vector field in (17) satisfies M12 = −M⊤\n21, but S1 and S2 can be nonzero symmetric matrices. Further, the Jacobian ∇v = A has the cross-shaped eigenvalue structure in (14), with c =\nL−µ 2(c.f., Proposition 1, Case 2). For the problem constants, we use µ = 1, and L = 200. The optimum [x\n⋆ y\n⋆]\n⊤ = w\n⋆ ∈ R\n200 is generated using the standard normal distribution. For simplicity, we assume b = [b1 b2]\n⊤ = [0 0]⊤. For the algorithms, we compare GD in (22), GDM in (5), EG in\n(23), and MEG in (4). All algorithms are initialized with 0. We plot the experimental results in Figure 3.\n\nFor MEG (optimal), we set the hyperparameters using Theorem 5. For GD (theory) and EG (theory), we set the hyperparameters using Theorem 7, both for the exact and the inexact settings. For GDM (grid search), we perform a grid search of h GDM and mGDM, and choose the best-performing ones, as Theorem 8 does not give a specific form for hyperparameter setup. Specifically, we consider 0.005 ⩽ h GDM ⩽ 0.015 with 10−3 increment, and 0.01 ⩽ mGDM ⩽ 0.99 with 10−2increment. In addition, as Theorem 7 might be conservative, we conduct grid searches for GD and EG as well. For GD (grid search), we use the same setup as h GDM.\n\nFor EG (grid search), we use 0.001 ⩽ h EG ⩽ 0.05 with 10−4increment.\n\nThere are several remarks to make. First, although the third plot in Figure 3 does not exactly follow the spectrum model in (14), MEG still works well with the optimal hyperparameters from Theorem 5. As expected, MEG (optimal) required more iterations in the inexact case compared to the exact case. Second, compared to other algorithms, MEG (optimal) indeed exhibits a significantly faster rate of convergence, even when compared to other methods that use grid-search hyperparameter tuning, supporting our theoretical findings in Section 4. Third, while EG (theory) is slower than GD (theory), which confirms Corrolary 1, EG (grid search) can be tuned to converge faster via grid search. Lastly, even though the best performance of GDM (grid search) is obtained through grid search, one can see the GD (grid search) obtains a slightly faster convergence rate than GDM (grid search), confirming Corollay 2.\n\n## 8 Conclusion\n\nIn the study of differentiable games, finding stationary points efficiently is crucial. This work analyzes the momentum extragradient method, revealing three distinct convergence modes dependent on the Jacobian eigenvalue distribution. Through a polynomial-based analysis, we derive optimal hyperparameters for each mode, achieving accelerated asymptotic convergence rates. We compared the obtained rates with other firstorder methods and showed that the considered methods do not achieve the accelerated convergence rate.\n\nNotably, our initial analysis for affine vector fields extends to guarantee local convergence rates on twicedifferentiable vector fields. Numerical experiments on quadratic games validate our theoretical findings.\n\n## Acknowledgments\n\nThe authors would like to thank Fangshuo Liao, Baptiste Goujaud, Damien Scieur, Miri Son, and Giorgio Young for their useful discussions and feedback. 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On linear convergence of iterative methods for the variational inequality problem. *Journal of* Computational and Applied Mathematics, 1995.\n\nJian Zhang and Ioannis Mitliagkas. Yellowfin and the art of momentum tuning. Proceedings of Machine Learning and Systems, 1, 2019.\n\n## A Missing Proofs In Section 3 A.1 Proof Of Lemma 1\n\nProof of Lemma 1 can be found for example in Azizian et al. (2020b, Section B).\n\nTo obtain the residual polynomials of MEG, w1 has to be set slightly differently from the rest of the iterates, as we write in the pseudocode below:\nAlgorithm 1: Momentum extragradient (MEG) method Input: Initialization w0, hyperparameters *h, γ, m*.\n\nSet: w1 = w0 −h 1+m v(w0 − γv(w0))\nfor t = 1, 2*, . . .* do wt+1 = wt − hv(wt − γv(wt)) + m(wt − wt−1)\nend\n\n## Derivation Of The First Part\n\nProof. We want to find the residual polynomial P˜t(A) of the extragradient with momentum (MEG) in (4).\n\nThat is, we want to find\n\n$$w_{t}-w^{\\star}=\\tilde{P}_{t}(A)(w_{0}-w^{\\star}),$$\n$$(32)$$\n⋆), (32)\nwhere {wt}t=0 is the iterates generated by MEG, which is possible by Lemma 1, as MEG is a first-order method (Arjevani & Shamir, 2016; Azizian et al., 2020b). We now prove this is by induction. To do so, we will use the following properties. First, note that as we are looking for a stationary point, it holds that v(w\n⋆) = 0. Further, as v is linear by the assumption of Lemma 1, it holds that v(w) = A(w − w\n⋆).\n\nBase case. For t = 0, P˜0(A) is a degree-zero polynomial, and hence equals Id, which denotes the identity matrix. Thus, w0 − w\n⋆ = Id(w0 − w\n⋆) holds true.\n\nFor completeness, we also prove when t = 1. In that case, observe that MEG in proceeds as w1 = w0 −\nh 1+m v(w0 − γv(w0)). Subtracting w\n⋆ on both sides, we have:\n\nw1 − w ⋆ = w0 − w ⋆ −h 1 + m v(w0 − γv(w0)) = w0 − w ⋆ −h 1 + m v(w0 − γA(w0 − w ⋆)) = w0 − w ⋆ −h 1 + m A(w0 − γA(w0 − w ⋆) − w ⋆) = w0 − w ⋆ −h 1 + m A(w0 − w ⋆) + hγ 1 + m A 2(w0 − w ⋆) = Id −h 1 + m A +hγ 1 + m A 2 (w0 − w ⋆) = Id −h 1 + m A(Id − γA) (w0 − w ⋆) = P˜1(A)(w0 − w ⋆).\nInduction step. As the induction hypothesis, assume P˜t satisfies (32). We want to prove this holds for t + 1. We have:\n\n$$w_{t+1}=w_{t}-hv(w_{t}-\\gamma v(w_{t}))+m(w_{t}-w_{t-1})$$ $$=w_{t}-hv(w_{t}-\\gamma A(w_{t}-w^{\\star}))+m(w_{t}-w_{t-1})$$ $$=w_{t}-hA(w_{t}-\\gamma A(w_{t}-w^{\\star})-w^{\\star})+m(w_{t}-w_{t-1})$$ $$=w_{t}-hA(w_{t}-w^{\\star})+h\\gamma A^{2}(w_{t}-w^{\\star})+m(w_{t}-w_{t-1})$$ $$=w_{t}-hA(I_{d}-\\gamma A)(w_{t}-w^{\\star})+m(w_{t}-w_{t-1}).$$\n\nSubtracting w\n⋆ on both sides, we have:\n\nwt+1 − w ⋆ = wt − w ⋆ − hA(Id − γA)(wt − w ⋆) + m(wt − wt−1) = (Id − hA(Id − γA)) (wt − w ⋆) + m(wt − w ⋆ − (wt−1 − w ⋆)) (32) = (Id − hA(Id − γA)) P˜t(A)(w0 − w ⋆) + m(P˜t(A)(w0 − w ⋆) − P˜t−1(A)(w0 − w ⋆)) = (Id + mId − hA(Id − γA)) P˜t(A)(w0 − w ⋆) − mP˜t−1(A)(w0 − w ⋆) = P˜t+1(A)(w0 − w ⋆),\nwhere in the third equality, we used the induction hypothesis in (32).\n\n## Derivation Of The Second Part In (6)\n\nProof. We show Pt = P˜t for all t via induction.\n\nBase case. For t = 0, by the definition of Chebyshev polynomials of the first and the second kinds, we have T0(λ) = U0(λ) = 1. Thus,\n\n$$P_{0}(\\lambda)=m^{0}\\left(\\frac{2m}{1+m}T_{0}(\\sigma(\\lambda))+\\frac{1-m}{1+m}U_{0}(\\sigma(\\lambda))\\right)$$ $$=\\frac{2m}{1+m}+\\frac{1-m}{1+m}=1=\\tilde{P}_{0}(\\lambda).$$\n\nAgain, for completeness, we prove when t = 1 as well. In that case, by the definition of Chebyshev polynomials of the first and the second kinds, we have T1(λ) = λ, and U1(λ) = 2λ. Therefore,\n\n$$P_{1}(\\lambda)=m^{t/2}\\left(\\frac{2m}{1+m}T_{1}(\\sigma(\\lambda))+\\frac{1-m}{1+m}U_{1}(\\sigma(\\lambda))\\right)$$ $$=m^{t/2}\\left(\\frac{2m}{1+m}\\,\\sigma(\\lambda)+\\frac{1-m}{1+m}\\cdot2\\cdot\\sigma(\\lambda)\\right)$$ $$=m^{t/2}\\left(\\frac{2\\sigma(\\lambda)}{1+m}\\right)$$ $$=1-\\frac{h\\lambda(1-\\gamma\\lambda)}{1+m}=\\tilde{P}_{1}(\\lambda).$$\n\nInduction step. As the induction hypothesis, assume that Pt = P˜t for t. In this step, we show that the same holds for t + 1.\n\nPt+1 = m(t+1)/2 2m 1 + m Tt+1(σ(λ)) + 1 − m 1 + m Ut+1(σ(λ)) = m(t+1)/2 2m 1 + m 2σ(λ)Tt(σ(λ)) − Tt−1(σ(λ)) + 1 − m 1 + m 2σ(λ)Ut(σ(λ) − Ut−1(σ(λ))) = 2σ(λ) · m1/2· mt/2 2m 1 + m Tt(σ(λ)) + 1 − m 1 + m Ut(σ(λ)) | {z } Pt(λ) − m · m(t−1)/2 2m 1 + m Tt−1(σ(λ)) + 1 − m 1 + m Ut−1(σ(λ)) | {z } Pt−1(λ) = 2σ(λ) · √m · P˜t(λ) − m · P˜t−1(λ) = (1 + m − hλ(1 − γλ))P˜t(λ) − mP˜t−1(λ),\n$\\square$\nwhere in the second to last equality we use the induction hypothesis.\n\n## A.3 Proof Of Lemma 2\n\nProof of Lemma 2 can be found in Goujaud & Pedregosa (2022).\n\nProof. We first recall that using (3), we can upper bound the worst-case convergence rate as:\n\n$$\\sup_{\\lambda\\in\\mathcal{S}^{+}}|P_{t}(\\lambda)|\\stackrel{{\\eqref{eq:20}}}{{=}}\\sup_{\\lambda\\in\\mathcal{S}^{+}}\\left|m^{t/2}\\bigg{(}\\frac{2m}{1+m}T_{t}(\\sigma(\\lambda))+\\frac{1-m}{1+m}U_{t}(\\sigma(\\lambda))\\bigg{)}\\right|$$ $$\\leqslant m^{t/2}\\bigg{(}\\frac{2m}{1+m}\\sum_{\\lambda\\in\\mathcal{S}^{+}}|T_{t}(\\sigma(\\lambda))|+\\frac{1-m}{1+m}\\sum_{\\lambda\\in\\mathcal{S}^{+}}|U_{t}(\\sigma(\\lambda))|\\bigg{)}\\tag{33}$$ $\\lambda$ is $\\lambda$-independent. For $\\lambda$, $\\lambda$ is $\\lambda$-independent. The $\\mathcal{T}_{t}(\\lambda)$ and $\\mathcal{U}_{t}(\\lambda)$ has a unique \nNow, denote σ¯ := supλ∈S⋆ |σ(λ; *h, γ, m*)|. For the first case, if σ¯ ⩽ 1, both Tt(x) and Ut(x) behave nicely, per Lemma 2. Thus, we have\n\n$$(33)\\stackrel{{(8)}}{{\\leqslant}}m^{t/2}\\bigg{(}\\frac{2m}{1+m}+\\frac{1-m}{1+m}(t+1)\\bigg{)}\\leqslant m^{t/2}(t+1)\\implies\\limsup_{t\\to\\infty}\\left(m^{t/2}(t+1)\\right)^{\\frac{1}{2t}}=\\sqrt[4]{m}.\\tag{33}$$\n\nFor the second case, we use the following expressions of Chebyshev polynomials:\n\n$$T_{n}(x)={\\frac{\\left(x-{\\sqrt{x^{2}-1}}\\right)^{n}+\\left(x+{\\sqrt{x^{2}-1}}\\right)^{n}}{2}},\\quad{\\mathrm{and}}$$ $$U_{n}(x)={\\frac{\\left(x+{\\sqrt{x^{2}-1}}\\right)^{n+1}-\\left(x-{\\sqrt{x^{2}-1}}\\right)^{n+1}}{2{\\sqrt{x^{2}-1}}}}.$$\n\nTherefore, in the second case where σ >¯ 1, we have both Tn(x) and Un(x) growing at rate (x +\n√x 2 − 1)n.\n\nHence, we have:\n\n (33) $\\leqslant O\\left({m}^{t/2}\\left(\\bar{\\sigma}+\\sqrt{{\\bar{\\sigma}}^{2}-1}\\right)^{t}\\right)\\implies\\limsup\\limits_{t\\to\\infty}$\n$$\\left(m^{t/2}\\left(\\bar{\\sigma}+\\sqrt{\\bar{\\sigma}^{2}-1}\\right)^{t}\\right)^{\\frac{1}{2t}}=\\sqrt[4]{m}\\left(\\bar{\\sigma}+\\sqrt{\\bar{\\sigma}^{2}-1}\\right)^{1/2}.$$\nFinally, in order for MEG to converge in the second case, we need:\n\n$$\\sqrt[4]{m}\\left(\\bar{\\sigma}+\\sqrt{\\bar{\\sigma}^{2}-1}\\right)^{1/2}<1$$\n\nwhich is equivalent to\n\n$$\\bar{\\sigma}\\leqslant\\frac{\\sqrt{m}(m+1)}{2m}=\\frac{m+1}{2\\sqrt{m}}.$$\n\n## A.5 Derivation Of Extreme Points Of Robust Region In (11)\n\nWe first write a general formula for inverting a quadratic function. For f(x) = ax2 + bx + c, its inverse is given by:\n\n$$f(x)=a x^{2}+b x+c:=y$$ $$f^{-1}(y)={\\frac{-b\\pm{\\sqrt{b^{2}-4a(c-y)}}}{2a}},$$\n\nwith some abuse of notation (i.e., f\n−1 above is not a function).\n\nApplying the above to the link function of MEG in (6), we get\n\n$$\\sigma^{-1}(y)=\\frac{1}{2\\gamma}\\pm\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{1+m}{h\\gamma}+\\frac{2\\sqrt{m}}{h\\gamma}\\cdot y}.$$\n\nWith this formula, we can plug in 1 and −1 to get:\n\n$$\\sigma^{-1}(-1)=\\frac{1}{2\\gamma}\\pm\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1+\\sqrt{m})^{2}}{h\\gamma}}\\quad\\mathrm{and}\\quad\\sigma^{-1}(1)=\\frac{1}{2\\gamma}\\pm\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}}.$$\n\nProof. We analyze each case separately.\n\nCase 1: There are two square roots: q1\nanonymous uare roots: $\\sqrt{\\frac{1}{4\\gamma^2}-\\frac{(1-\\sqrt{m})^2}{h\\gamma}}$ and $\\sqrt{\\frac{1}{4\\gamma^2}-\\frac{(1+\\sqrt{m})^2}{h\\gamma}}.$ The second one is real if: $$\\frac{1}{4\\gamma^2}\\geqslant\\frac{(1+\\sqrt{m})^2}{h\\gamma}\\implies\\frac{h\\gamma}{4\\gamma^2}=\\frac{h}{4\\gamma}\\geqslant(1+\\sqrt{m})^2,$$\n\nwhich implies the first is real, as (1 + √m)\n2 ⩾ (1 −\n√m)\n2.\n\n **Case 3:** There are two square roots: $\\sqrt{\\frac{1}{k\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{k\\gamma}}$ and $\\sqrt{\\frac{1}{k\\gamma^{2}}-\\frac{(1+\\sqrt{m})^{2}}{k\\gamma}}$. The first one is complex if: $$\\frac{1}{4\\gamma^{2}}<\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}\\implies\\frac{h\\gamma}{4\\gamma^{2}}=\\frac{h}{4\\gamma}<(1-\\sqrt{m})^{2},$$\nwhich implies the second is complex, as (1 + √m)\n2 ⩾ (1 −\n√m)\n2.\n\nCase 2: This case follows automatically from the above two cases.\n\n## A.7 Proof Of Proposition 1\n\nProof. Define D3 = diag(*a, . . . , a*) of dimensions d × d. Let us prove that if there exists *U, V* orthonormal matrices and D1, D2 matrices with non-zeros coefficients only on the diagonal such that (with a slight abuse of notation)\n\n$$S_{1}=U\\mathrm{diag}(D_{3},D_{1})U^{\\top},S_{2}=V D_{3}V^{\\top},\\quad\\mathrm{and}\\quad B=U D_{2}V^{\\top},$$\n\nthen the spectrum of A is crossed shaped. In that case, we have\n\n$$\\begin{array}{r l}{A={\\left[\\begin{array}{l l l}{U[D_{3};D_{1}]U^{\\top}}&{U D_{2}V^{\\top},}\\\\ {-V D_{2}^{\\top}U^{\\top}}&{V D_{3}V^{\\top}}\\end{array}\\right]}}\\\\ {={\\left[\\begin{array}{l l l}{U}&{0}\\\\ {0}&{V}\\end{array}\\right]{\\left[\\begin{array}{l l l}{[D_{3};D_{1}]}&{D_{2}}\\\\ {-D_{2}^{\\top}}&{D_{3}}\\end{array}\\right]}{\\left[\\begin{array}{l l l}{U}&{0}\\\\ {0}&{V}\\end{array}\\right]}^{\\top}.}\\end{array}$$\n\nNow by considering the basis W = ((U1, 0),(0, V1)*, . . . ,*(Udv\n, 0),(0, Vdv\n),(Udv+1, 0)*, . . . ,*(Ud, 0)) we have that A can be block diagonalized in that basis as\n\n$$A=W\\mathrm{diag}\\left(\\left[\\begin{matrix}a&[D_{2}]_{11}\\\\ -[D_{2}]_{11}&a\\end{matrix}\\right],\\ldots,\\left[\\begin{matrix}a&[D_{2}]_{d_{u},d_{v}}\\\\ -[D_{2}]_{d_{u},d_{v}}&a\\end{matrix}\\right],[D_{1}]_{1},\\ldots,[D_{1}]_{d_{u}-d_{v},d_{u}}\\right)W^{\\top}.$$\nW⊤. (34)\nNow, notice that\n\n$$(34)$$\n$${\\rm Sp}\\left(\\begin{bmatrix}a&-b\\\\ b&a\\end{bmatrix}\\right)=\\{a\\pm bi\\},\\tag{1}$$\n$$(35)$$\n\nsince the associated characteristic polynomial of the above matrix is:\n\n$$(a-\\lambda)^{2}+b^{2}=0\\implies$$\n\n2 = 0 =⇒ a − λ = ±bi =⇒ λ = a ± bi.\n\nHence, using (35) in the formulation of A in (34), we have that the spectrum of A is cross-shaped.\n\n## B Missing Proofs In Section 4 B.1 Proof Of Theorem 4\n\nProof. We write the conditions required for Theorem 5 below:\n\n$$\\frac{1}{2\\gamma}-\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}}=\\mu_{1},$$ $$\\frac{1}{2\\gamma}-\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1+\\sqrt{m})^{2}}{h\\gamma}}=L_{1},$$ $$\\frac{1}{2\\gamma}+\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1+\\sqrt{m})^{2}}{h\\gamma}}=\\mu_{2},\\quad\\text{and}$$ $$\\frac{1}{2\\gamma}+\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}}=\\mu_{2}.$$ In both the solutions (10) and (20) are \n\nBy adding (37) and (38) (or equivalently by ading (36) and (39)), we get\n\n$$\\gamma={\\frac{1}{\\mu_{1}+L_{2}}}={\\frac{1}{\\mu_{2}+L_{1}}}.$$\n. (40)\n$$(39)$$\n$$(40)$$\n\nFrom (36), we have:\n\n$$\\frac{1}{2\\gamma}+\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}}=\\mu_{1}$$ $$\\frac{1}{4\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}=\\left(\\frac{1}{2\\gamma}-\\mu_{1}\\right)^{2}$$ $$\\frac{(1-\\sqrt{m})^{2}}{h}=\\mu_{1}(1-\\gamma\\mu_{1})$$ $$h=\\frac{(1-\\sqrt{m})^{2}}{\\mu_{1}(1-\\gamma\\mu_{1})}=\\frac{(1-\\sqrt{m})^{2}(\\mu_{1}+L_{2})}{\\mu_{1}L_{2}}\\tag{41}$$ we:\n$$(41)$$\n\nSimilarly, from (38), we have:\n\n$$\\frac{1}{2\\gamma}+\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1+\\sqrt{m})^{2}}{h\\gamma}}=\\mu_{2}$$ $$\\frac{1}{4\\gamma^{2}}-\\frac{(1+\\sqrt{m})^{2}}{h\\gamma}=\\left(\\frac{\\mu_{2}-L_{1}}{2}\\right)^{2}$$ $$\\left(\\frac{\\mu_{2}+L_{1}}{2}\\right)^{2}-\\left(\\frac{\\mu_{2}-L_{1}}{2}\\right)^{2}=\\mu_{2}L_{1}=\\frac{(1+\\sqrt{m})^{2}}{h\\gamma}\\tag{5.1}$$ which is for $\\mu_{2}$. \n$$\\left(42\\right)$$\n$$(43)$$\n\nCombining (41) and (42), and solving for m, we get µ2L1(1 −\n√m)\n2 = µ1L2(1 + √m)\n2\n\n$$m\\rangle^{2}=\\mu_{1}L_{2}(1+\\sqrt{m})$$ $$m=\\left(\\frac{\\sqrt{\\mu_{2}L_{1}}-\\sqrt{\\mu_{1}L_{2}}}{\\sqrt{\\mu_{2}L_{1}}+\\sqrt{\\mu_{1}L_{2}}}\\right)^{2}\\stackrel{(13)}{=}\\left(\\frac{\\sqrt{\\zeta^{2}-R^{2}}-\\sqrt{\\zeta^{2}-1}}{\\sqrt{\\zeta^{2}-R^{2}}+\\sqrt{\\zeta^{2}-1}}\\right)^{2}.$$ (11)\n. (43)\nFinally, plugging (43) back to (41), we get:\n\n1), we get: $ h=\\frac{(1-\\sqrt{m})^2(\\mu_1+L_2)}{\\mu_1L_2}$ $ =\\frac{4\\mu_1L_2}{(\\sqrt{\\mu_2+L_1}+\\sqrt{\\mu_1+L_2})^2}\\cdot\\frac{\\mu_1+L_2}{\\mu_1L_2}$ $ =\\frac{4(\\mu_1+L_2)}{(\\sqrt{\\mu_2+L_1}+\\sqrt{\\mu_1+L_2})^2}$. \nProof. We write the conditions required for Theorem 5 below:\nFirst, by adding (44) and (45), we get:\n\n$${\\frac{1}{\\gamma}}=\\mu+L\\implies\\gamma={\\frac{1}{\\mu+L}}.$$\n. (47)\n$$\\frac{1}{2\\gamma}-\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}}=\\mu,$$ $$\\frac{1}{2\\gamma}+\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}}=L,\\quad\\text{and}$$ $$\\sqrt{\\frac{(1+\\sqrt{m})^{2}}{h\\gamma}-\\frac{1}{4\\gamma^{2}}}=c.$$\n$$(444)$$\n$$(45)$$\n$$(46)$$\n$$(47)$$\n\nPlugging (47) back into (44), we have:\n\n$$\\frac{1}{2\\gamma}-\\sqrt{\\frac{1}{4\\gamma^{2}}-\\frac{(1-\\sqrt{m})^{2}}{h\\gamma}}=\\mu$$ $$\\frac{\\mu+L}{2}-\\mu=\\sqrt{\\left(\\frac{\\mu+L}{2}\\right)^{2}-\\frac{(1-\\sqrt{m})^{2}(\\mu+L)}{h}}$$ $$\\left(\\frac{L-\\mu}{2}\\right)^{2}=\\left(\\frac{\\mu+L}{2}\\right)^{2}-\\frac{(1-\\sqrt{m})^{2}(\\mu+L)}{h}$$ $$\\frac{(1-\\sqrt{m})^{2}(\\mu+L)}{h}=\\left(\\frac{\\mu+L}{2}\\right)^{2}-\\left(\\frac{L-\\mu}{2}\\right)^{2}=\\mu L$$ $$h=\\frac{(1-\\sqrt{m})^{2}(\\mu+L)}{\\mu L}.\\tag{48}$$\n\nPlugging (47) and (48) into (46), we have:\n\ns(1 + √m) 2 hγ − 1 4γ 2 = c s(1 + √m) 2 · µL (1 − √m) 2− µ + L 2 2= c (1 + √m) 2· µL (1 − √m) 2= c 2 + µ + L 2 2= 4c 2 + (µ + L) 2 4 (1 + √m) 2 (1 − √m) 2 = 4c 2 + (µ + L) 2 4µL (1 + √m)p4µL = (1 − √m)p4c 2 + (µ + L) 2 √m(p4c 2 + (µ + L) 2 +p4µL) = p4c 2 + (µ + L) 2 −p4µL √m = p4c 2 + (µ + L) 2 − √4µL p4c 2 + (µ + L) 2 + √4µL . (49)\nFinally, to simplify (48) further, from (49), we have:\n\n$$1-\\sqrt{m}=\\frac{4\\sqrt{\\mu L}}{\\sqrt{4c^{2}+(\\mu+L)^{2}}+\\sqrt{4\\mu L}}.$$\n\nHence, from (48),\n\n$$h={\\frac{(\\mu+L)(1-\\sqrt{m})^{2}}{\\mu L}}={\\frac{{\\frac{16\\mu L(\\mu+L)}{(\\sqrt{4c^{2}+(\\mu+L)^{2}}+\\sqrt{4\\mu L})^{2}}}}{\\mu L}}={\\frac{16(\\mu+L)}{(\\sqrt{4c^{2}+(\\mu+L)^{2}}+\\sqrt{4\\mu L})^{2}}}.$$\n. (50)\n$$(50)$$\n\nProof. We write the conditions required for (6) below:\n\n $\\dfrac{1}{2\\gamma}+\\sqrt{\\dfrac{1}{4\\gamma^2}}-1$ $\\dfrac{1}{2\\gamma}-\\sqrt{\\dfrac{1}{4\\gamma^2}}-1$ $\\dfrac{1}{2\\gamma}+\\sqrt{\\dfrac{1}{4\\gamma^2}}-1$ $\\dfrac{1}{2\\gamma}+\\sqrt{\\dfrac{1}{4\\gamma^2}}-1$ $\\gamma$ is not a good approximation. \n− (1 + √m) 2 hγ = c + bi, (51) − (1 + √m) 2 hγ = c + ai, (52) − (1 + √m) 2 hγ = c − ai, and (53) − (1 − √m) 2 hγ = c − bi. (54)\nFirst, we can see from all cases that the optimal γ is\n\n$$(51)$$\n$$(52)$$\n$$(53)$$\n$$(54)$$\n$$\\gamma={\\frac{1}{2c}}.$$\n$$(55)$$\n. (55)\n(51) and (54) equivalently imply\n\n$$\\sqrt{\\frac{(1+\\sqrt{m})^{2}}{h\\gamma}-\\frac{1}{4\\gamma^{2}}}=b$$ $$(1+\\sqrt{m})^{2}=h\\gamma b^{2}+\\frac{h}{4\\gamma}=\\frac{h(c^{2}+b^{2})}{2c}$$ $$h=\\frac{2c(1+\\sqrt{m})^{2}}{c^{2}+b^{2}}.$$\n$$(56)$$\n\nSimilarly, (52) and (53) imply\n\n and (50) imply $$\\sqrt{\\frac{(1-\\sqrt{m})^2}{h\\gamma}-\\frac{1}{4\\gamma^2}}=a$$ $$\\frac{(1-\\sqrt{m})^2}{h\\gamma}=a^2+\\frac{1}{4\\gamma^2}=a^2+c^2$$ $$\\frac{(1-\\sqrt{m})^2(c^2+b^2)}{(1+\\sqrt{m})^2}=a^2+c^2$$ $$(1-\\sqrt{m})\\sqrt{c^2+b^2}=(1+\\sqrt{m})\\sqrt{c^2+a^2}$$ $$\\sqrt{m}=\\frac{\\sqrt{c^2+b^2}-\\sqrt{c^2+a^2}}{\\sqrt{c^2+b^2}+\\sqrt{c^2+a^2}}=1-\\frac{2\\sqrt{c^2+a^2}}{\\sqrt{c^2+b^2}+\\sqrt{c^2+a^2}}\\tag{57}$$ In (56), we get... \nPlugging (57) to (56), we get\n\n$$h={\\frac{2c(1+{\\sqrt{m}})^{2}}{c^{2}+b^{2}}}={\\frac{8c}{({\\sqrt{c^{2}+b^{2}}}+{\\sqrt{c^{2}+a^{2}}})^{2}}}.$$\n\n## C Missing Proofs In Sections 5 C.1 Proof Of Corollary 1\n\nProof. To compute the convergence rates of GD and EG from Theorem 7 applied to each spectrum model in (12), (14), and (15), we need to compute minλ∈∆⋆ R(1/λ) and minλ∈∆⋆ R(λ) for GD. Similarly for EG,\nwe need to compute additionally minλ∈∆⋆ |λ|, minλ∈∆⋆ |λ| 2, and supλ∈∆⋆ |λ| 2.\n\n$\\square$\nCase 1: It's straightforward to compute\n\n$$\\operatorname*{min}_{\\lambda\\in{\\mathcal{S}}_{1}^{*}}\\Re(1/\\lambda)=1/L_{2},\\quad\\mathrm{and}\\quad\\operatorname*{min}_{\\lambda\\in{\\mathcal{S}}_{1}^{*}}\\Re(\\lambda)=\\mu_{1}$$\n\nThus, GD for Case 1 has the rate\n\n$$1-{\\frac{\\mu_{1}}{L_{2}}}=1-\\tau.$$\n\nFor EG, it's also simple to obtain\n\n$$\\operatorname*{min}_{\\lambda\\in{\\mathcal{S}}_{1}^{*}}|\\lambda|=L_{2},\\quad\\operatorname*{min}_{\\lambda\\in{\\mathcal{S}}_{1}^{*}}|\\lambda|^{2}=\\mu_{1}^{2},\\quad\\mathrm{and}\\quad\\operatorname*{sup}_{\\lambda\\in{\\mathcal{S}}_{1}^{*}}|\\lambda|^{2}=L_{2}^{2}.$$\n\nThus, EG for Case 1 has the rate\n\n$$1-\\frac{1}{4}\\left(\\frac{\\mu_{1}}{L_{2}}+\\frac{1}{16}\\left(\\frac{\\mu_{1}}{L_{2}}\\right)^{2}\\right).$$\n\nCase 2: For a complex number z = p + qi ∈ C, we can compute R(1/z) as:\n\n$${\\frac{1}{z}}={\\frac{1}{p+q i}}={\\frac{p-q i}{p^{2}+q^{2}}}={\\frac{p}{p^{2}+q^{2}}}-{\\frac{q}{p^{2}+q^{2}}}i\\implies\\Re\\left({\\frac{1}{z}}\\right)={\\frac{p}{p^{2}+q^{2}}}.$$\n\nThe four extreme points of the cross-shaped spectrum model in (14) are:\n\n$$\\mu=\\mu+0i,$$\nµ = µ + 0*i, L* = L + 0i, and L − µ\n$$L=L+0i,\\quad\\mathrm{and}\\quad\\frac{L-\\mu}{2}\\pm c i.$$\n\nHence, R(1/z) for each of the above points is:\n\n$$\\Re\\left(\\frac{1}{\\mu}\\right)=\\frac{\\mu}{\\mu^{2}}=\\frac{1}{\\mu},$$ $$\\Re\\left(\\frac{1}{L}\\right)=\\frac{L}{L^{2}}=\\frac{1}{L},\\quad\\text{and}$$ $$\\Re\\left(\\frac{1}{\\frac{L-\\mu}{2}\\pm ci}\\right)=\\frac{\\frac{L-\\mu}{2}}{\\left(\\frac{L-\\mu}{2}\\right)^{2}+c^{2}}$$ $$=\\frac{2(L-\\mu)}{4c^{2}+(L-\\mu)^{2}}.$$\n\nTherefore, minλ∈S⋆\n2 R1λ\n=\n1 L\n. As *µ < L,* we only need to compare the last two values. Observe that:\n\n$$\\begin{array}{c}{{c>\\sqrt{\\frac{L^{2}-\\mu^{2}}{4}}}}\\\\ {{4c^{2}>(L-\\mu)(L+\\mu)}}\\\\ {{4c^{2}>2L(L-\\mu)-(L-\\mu)^{2}}}\\\\ {{\\frac{1}{L}>\\frac{2(L-\\mu)}{4c^{2}+(L-\\mu)^{2}}.}}\\end{array}$$\n\nTherefore,\n\n$$\\operatorname*{min}_{\\lambda\\in S_{2}^{*}}\\Re\\left({\\frac{1}{\\lambda}}\\right)={\\begin{cases}{\\frac{2(L-\\mu)}{4c^{2}+(L-\\mu)^{2}}}&{{\\mathrm{if}}\\quad c>{\\sqrt{\\frac{L^{2}-\\mu^{2}}{4}}}}\\\\ {{\\frac{1}{L}}}&{{\\mathrm{otherwise.}}}\\end{cases}}$$\n\nFor minλ∈S⋆\n2 R(λ), it's straightforward from the definition that\n\n$$\\operatorname*{min}_{\\lambda\\in{\\mathcal{S}}_{2}^{*}}\\Re(\\lambda)=\\mu.$$\n\nThus, GD for Case 2 has the rate\n\n$$\\begin{cases}1-{\\frac{2\\mu(L-\\mu)}{4c^{2}+(L-\\mu)^{2}}}&\\quad{\\mathrm{if}}\\quad c>{\\sqrt{\\frac{L^{2}-\\mu^{2}}{4}}}\\\\ 1-{\\frac{\\mu}{L}}&\\quad{\\mathrm{otherwise.}}\\end{cases}$$\n\nSimilarly for EG, we need to compute minλ∈S⋆\n2 R(λ), which was computed above; additionally, we need to compute minλ∈S⋆\n2 |λ|, minλ∈S⋆\n2 |λ| 2, and supλ∈S⋆\n2 |λ| 2. For z = p + qi ∈ C, |z| =pp 2 + q 2. Hence, we have\n\n$$|\\mu+0i|=\\mu,\\quad|L+0i|=L,\\quad{\\mathrm{and}}\\quad\\left|{\\frac{L-\\mu}{2}}\\pm c i\\right|={\\sqrt{c^{2}+\\left({\\frac{L-\\mu}{2}}\\right)^{2}}}\\,.$$\n\nObserve that:\n\n$$c>\\sqrt{\\frac{3L^{2}+2L\\mu-\\mu^{2}}{4}}$$ $$c^{2}>L^{2}-\\frac{L^{2}-2L\\mu+\\mu^{2}}{4}$$ $$c^{2}+\\left(\\frac{L-\\mu}{2}\\right)^{2}>L^{2}$$\n\nThus, for supλ∈S⋆\n2 |λ|, we have\n\n$$\\operatorname*{sup}_{\\lambda\\in{\\mathcal{S}}_{2}^{*}}|\\lambda|={\\begin{cases}{\\sqrt{c^{2}+\\left({\\frac{L-\\mu}{2}}\\right)^{2}}}&{{\\mathrm{if}}\\quad c>{\\sqrt{\\frac{3L^{2}+2L\\mu-\\mu^{2}}{4}}}}\\\\ L&{{\\mathrm{otherwise,}}}\\end{cases}}$$\n\nfrom which supλ∈S⋆\n2 |λ| 2can also be obtained. Lastly, minλ∈S⋆\n2 |λ| 2 = µ 2, as we know *µ < L,* and (L − µ)/2 is the center of [*µ, L*].\n\nCombining all three, we get the rate of EG for Case 3 is\n\nwe get the rate of L6 for Case 3 is $$\\begin{cases}1-\\frac{1}{4}\\left(\\frac{\\mu}{\\sqrt{c^{2}+\\left(\\frac{L-\\mu}{2}\\right)^{2}}}+\\frac{\\mu^{2}}{16\\left(c^{2}+\\left(\\frac{L-\\mu}{2}\\right)^{2}\\right)}\\right)&\\text{if}c\\geqslant\\sqrt{\\frac{3L^{2}+2L\\mu-\\mu^{2}}{4}},\\\\ 1-\\frac{1}{4}\\Big{(}\\frac{\\mu}{L}+\\frac{\\mu^{2}}{16L^{2}}\\Big{)}&\\text{otherwise.}\\end{cases}$$\nCase 3: Since (15) has fixed real component, minλ∈S⋆\n3 R(λ) = c.\n\nFor minλ∈S⋆\n3 R(1/λ) we can compute compare\n\n$$\\Re\\left(\\frac{1}{c+ai}\\right)=\\frac{c}{c^{2}+a^{2}}>\\frac{c}{c^{2}+b^{2}}=\\Re\\left(\\frac{1}{c+bi}\\right),$$ since $a<b$ from (15). Thus, GD for Case 3 has the rate $$1-\\frac{c^{2}}{c^{2}+b^{2}}.$$\n\nFor EG, it's also simple to obtain\n\n$$\\operatorname*{lim}_{\\lambda\\in{\\mathcal{S}}_{3}^{*}}|\\lambda|={\\sqrt{c^{2}+b^{2}}},\\quad\\operatorname*{min}_{\\lambda\\in{\\mathcal{S}}_{3}^{*}}|\\lambda|^{2}=c^{2}+a^{2},\\quad{\\mathrm{and}}\\quad\\operatorname*{sup}_{\\lambda\\in{\\mathcal{S}}_{3}^{*}}|\\lambda|^{2}=c^{2}+b^{2}.$$\n\nThus, EG has the rate\n\n$$1-{\\frac{1}{4}}\\left({\\frac{c}{\\sqrt{c^{2}+b^{2}}}}+{\\frac{1}{16}}{\\frac{(c^{2}+a^{2})}{(c^{2}+b^{2})}}\\right).$$\n\n## C.2 Proof Of Corollary 2\n\nProof. Per Theorem 8, the largest ϵ that permits acceleration for GDM is ϵ =\n√µL. Therefore, in the special case of (14) we consider, i.e., when c =\nL−µ 2, GDM *cannot* achieve acceleration if L−µ 2 >\n√µL. Hence, we have:\n\n$$\\begin{array}{c}{{\\frac{L-\\mu}{2}>\\sqrt{\\mu L}}}\\\\ {{L-\\mu>2\\sqrt{\\mu L}}}\\\\ {{L^{2}+\\mu^{2}>6\\mu L>6\\mu^{2}\\ \\ (\\because L>\\mu)}}\\\\ {{L>\\sqrt{5}\\mu.}}\\end{array}$$\n\n## C.3 Proof Of Proposition 2\n\nProof. For an arbitrary complex number p + qi with p > 0, and using the link function of GDM from (7), we have\n\n$$|\\xi(p+qi)|=\\sqrt{\\left(\\frac{1+m-hp}{2\\sqrt{m}}\\right)^{2}+\\left(\\frac{hq}{2\\sqrt{m}}\\right)^{2}}\\leqslant1$$ $$\\frac{(1+m-hp)^{2}+h^{2}q^{2}}{4m}\\leqslant1$$ $$(1+m-hp)^{2}+h^{2}q^{2}\\leqslant4m$$ $$(1-m)^{2}+hp(hp-2(1+m))+h^{2}q^{2}\\leqslant0$$ $$\\frac{(1-m)^{2}+h^{2}q^{2}}{hp}\\leqslant2(1+m)-hp$$\n$$(58)$$\n\nNotice that the LHS is positive. Therefore, if the RHS is negative, the above inequality cannot hold. In other words, if 2(1+m)\nh < p, GDM cannot stay in the robust region. This is very hard to satisfy, even with a small p.\n\n## D Missing Proofs In Section 6\n\nLet us consider an affine vector field v(w) = Aw + b and its associated augmented MEG linear operator J:\n\n$$\\begin{array}{r l}{{\\left[w_{t+1}-w^{*}\\right]=J\\left[\\begin{array}{l l}{w_{t}-w^{*}}\\\\ {w_{t}-w^{*}}\\end{array}\\right]}}&{{\\mathrm{with}}}\\quad J={\\left[\\begin{array}{l l}{(1+\\beta)I_{d}-h A(I_{d}-\\gamma A)}&{-\\beta I_{d}}\\\\ {I_{d}}&{0_{d}}\\end{array}\\right],}}\\end{array}$$\n, (58)\nwhere Id and 0d respectively stand for the identity and the null matrices. To show the local convergence of (restarted) MEG for non-affine vector fields in Theorem 9, we first establish the following lemma, which connects the augmented state and the non-augmented one.\n\nLemma 3. Let P MEG\ntbe the residual polynomial associated with t updates of MEG (c.f., Theorem *1). Let* J *be defined as* (58)*. If* w1 = w0 −h 1+m v(w0 − γv(w0))*, we then have*\n\n$$J^{t}\\begin{bmatrix}w_{1}-w^{\\star}\\\\ w_{0}-w^{\\star}\\end{bmatrix}=\\begin{bmatrix}P_{t}^{M E G}(A)(w_{1}-w^{\\star})\\\\ P_{t}^{M E G}(A)(w_{0}-w^{\\star})\\end{bmatrix}.$$\n\nConsequently, if we denote zt+1 := [wt+1, wt] and z∗ := [w\n⋆, w⋆]*, we have*\n\n$$\\|z_{t+1}-z_{*}\\|\\leqslant C(t+1)(1-\\varphi)^{t}\\|z_{0}-z_{*}\\|.$$\nt∥z0 − z∗∥. (60)\n$$(59)$$\n$$(60)$$\nProof. Let us express J J t = P 11 t(A) P 12 t(A) P 21 t(A) P 22 t(A) , and J t w1 − w ⋆ w0 − w ⋆ = P 11 t(A)(w0 − w ⋆) + P 12 t(A)(w0 − w ⋆) P 21 t(A)(w0 − w ⋆) + P 22 t(A)(w0 − w ⋆) By writing J t+1 = JJt and using the block-matrix form of J in (58), we get that for any t ⩾ 0, P 11 t+1(A) = ((1 + β)Id − hA(Id − γA))P 11 t(A) − βP21 t(A) P 21 t+1(A) = P 11 t(A) (62) P 12 t+1(A) = ((1 + β)Id − hA(Id − γA))P 12 t(A) − βP22 t(A) P 22 t+1(A) = P 12 t(A). (63)\ntsuch that\n\n. (61)\n$$(61)$$\n$$(62)$$\nHence, we have that,\n\n$$\\begin{array}{l}{{P_{t+1}^{11}(A)\\stackrel{(62)}{=}((1+\\beta)I_{d}-h A(I_{d}-\\gamma A))P_{t}^{11}(A)-\\beta P_{t-1}^{11}(A)}}\\\\ {{P_{t+1}^{12}(A)\\stackrel{(63)}{=}((1+\\beta)I_{d}-h A(I_{d}-\\gamma A))P_{t}^{12}(A)-\\beta P_{t-1}^{12}(A).}}\\end{array}$$\n\nWe claim that\n\n$$(63)$$\n(64) $\\left(65\\right)$ (65)\n$$P_{t}^{11}(A)+P_{t}^{12}(A)=P_{t}^{\\mathrm{MEG}}(A)\\quad\\mathrm{for~all}\\quad t\\geqslant0.$$\n$$(66)$$\nt(A) for all t ⩾ 0. (66)\nWe prove this via induction.\n\nFor the base case, using the fact that w1 = w0 −h 1+m (w0 − γv(w0)), we have that\n\n$(P_{1}^{11}(A)+P_{1}^{12}(A))(w_{0}-w^{\\star})=w_{1}-w^{\\star}=(I_{d}-\\frac{1}{1+m}(I_{d}-\\gamma A))(w_{0}-w^{\\star})=P_{0}^{\\rm MEG}(A)(w_{0}-w^{\\star})\\,,$ $(P_{0}^{11}(A)+P_{0}^{22}(A))(w_{0}-w^{\\star})=(P_{1}^{21}(A)+P_{1}^{22}(A))(w_{0}-w^{\\star})=I_{d}(w_{0}-w^{\\star})=P_{0}^{\\rm MEG}(A)(w_{0}-w^{\\star})$. \nTo show the induction step, by adding (64) and (65), we get\n\n$P_{t+1}^{11}(A)+P_{t+1}^{12}(A)=((1+\\beta)I_{d}-hA(I_{d}-\\gamma A))(P_{t}^{11}(A)+P_{t}^{12}(A))-\\beta(P_{t-1}^{11}(A)+P_{t-1}^{12}(A))$ $\\stackrel{{\\rm()}}{{=}}((1+\\beta)I_{d}-hA(I_{d}-\\gamma A))P_{t}^{\\rm MECG}(A)-\\beta P_{t-1}^{\\rm MEG}(A),$\nwhere in the last step we used the induction hypothesis. Also notice P\n11 t+1(A) + P\n12 t+1(A) = P\nMEG\nt+1 on the left-hand side.\n\nHence we have for any t ⩾ 0,\n\n$$(P_{t}^{11}(A)+P_{t}^{12}(A))(w_{0}-w^{\\star})=P_{t}^{\\mathrm{MEG}}(A)(w_{0}-w^{\\star}).$$\n\nTherfore, going back to (61), we have:\n\nwt+1 − w ⋆ wt − w ⋆ = J t w1 − w ⋆ w0 − w ⋆ = (P 11 t(A) + P 12 t(A))(w0 − w ⋆) (P 21 t(A) + P 22 t(A))(w0 − w ⋆) (62),(63) =(P 11 t(A) + P 12 t(A))(w0 − w ⋆) (P 11 t−1 (A) + P 12 t−1 (A))(w0 − w ⋆) (66) = P MEG t(A)(w0 − w ⋆) P MEG t−1(A)(w0 − w ⋆) Thm. 1 = P MEG t(A)(w0 − w ⋆) P MEG t(A)(w−1 − w ⋆) = P MEG t(A) 0 0 P MEG t(A) · w0 − w ⋆ w−1 − w ⋆ = P MEG t(A) ⊗ I2 · w0 − w ⋆ w−1 − w ⋆ ,\n$$(67)$$\nwhere we use the convention that w0 = w−1. Finally, using the fact that ∥A ⊗ B∥ = ∥A∥∥B∥ for ℓ2-operator norm (Lancaster & Farahat, 1972), we have\n\n$$\\|z_{t+1}-z_{*}\\|\\leqslant\\|P_{t}^{\\mathrm{MEG}}(A)\\|\\|z_{0}-z_{*}\\|\\,\\stackrel{(10)}{\\leqslant}\\,C(t+1)(1-\\varphi)^{t}\\|z_{0}-z_{*}\\|.$$\n\nProof. We first recall the restarted MEG algorithm we consider in (31):\n\n$[w_{tk+i+1},w_{tk+i}]=G([w_{tk+i},w_{tk+i-1}])\\quad\\text{for}\\quad1\\leqslant i\\leqslant k-1,\\quad\\text{and then}\\quad1$ $w_{(t+1)k+1}=w_{(t+1)k}-\\frac{h}{1+m}v(w_{(t+1)k}-\\gamma v(w_{(t+1)k})).$\n$$\\square$$\nIn other words, we repeat MEG for k steps, and then re-start the momentum at [w(t+1)k+1, w(t+1)k].\n\nWe can analyze this method as follows, where we denote zt := [wt, wt−1] and z∗ = [w\n⋆, w⋆]:\n\n∥z(t+1)k − z∗∥ = ∥G (k)(ztk) − z∗∥ (68) = ∥∇G (k)(˜ztk)(ztk − z∗)∥ ⩽ ∥∇G (k)(z∗)(ztk − z∗)∥ + ∥(∇G (k)(˜ztk) − ∇G (k)(z∗))(ztk − z∗)∥ (60) ⩽ C(k + 1)(1 − φ) k∥ztk − z∗∥ + ∥∇G (k)(˜ztk) − ∇G (k)(z∗)∥∥(ztk − z∗)∥, where in the second line we use the Mean Value Theorem:\n$$\\exists\\xi_{tk}\\in[z_{tk},z_{*}]\\quad\\text{such that}\\quad G^{(k)}(z_{tk})=G^{(k)}(z_{*})+\\nabla G^{(k)}(\\xi_{tk})(z_{tk}-z_{*})$$ $$=z_{*}+\\nabla G^{(k)}(\\hat{z}_{tk})(z_{tk}-z_{*})\\quad\\text{(since$z_{*}$is the fixed point).}$$\nIn the fourth line we used the fact that ∇G(k)(z∗)(ztk − z∗) exactly correspond to k updates of MEG when the vector field is affine, as well as Lemma 3 to account for the augmented state.\n\nNow let us consider *φ > ε >* 0 and k large enough such that C(k + 1)(1 − φ)\nk ⩽ (1 − φ +\nε 2\n)\nk. Since ∇G is assumed to be continuous, ∇G(k)is continuous too. Therefore, there exists δ > 0 such that ∥ztk − z∗∥ ⩽ δ implies ∥∇G(k)(˜ztk) − ∇G(k)(z∗)∥ ⩽ ε\n′. In particular, choose ε\n′ = (1 − φ + ε)\nk − (1 − φ +\nε 2\n)\nk ∼kε 2(1−φ)\n.\n\nThen, we have\n\n$$\\|z_{(t+1)k}-z_{*}\\|\\leqslant C(k+1)(1-\\varphi)^{k}\\|z_{tk}-z_{*}\\|+\\|\\nabla G^{(k)}(\\tilde{z}_{tk})-\\nabla G^{(k)}(z_{*})\\|\\|z_{tk}-z_{*}\\|$$ $$\\leqslant(1-\\varphi+\\frac{\\pi}{2})^{k}\\|z_{tk}-z_{*}\\|+\\varepsilon^{\\prime}\\|z_{tk}-z_{*}\\|$$ $$\\leqslant(1-\\varphi+\\varepsilon)^{k}\\|z_{tk}-z_{*}\\|<\\|z_{tk}-z_{*}\\|<\\|z_{0}-z_{*}\\|.$$\n\nFrom the above, we can conclude that for all ε > 0, there exists k > 0 and δ > 0 such that, for all initialization satisfying ∥w0 − w\n⋆∥ ⩽ δ, the restarted MEG described above satisfies:\n\n$$\\|w_{t}-w^{\\star}\\|=O((1-\\varphi+\\varepsilon)^{t})\\|w_{0}-w^{\\star}\\|.$$\n"},"reviews":{"kind":"string","value":"Review 1:\nSummary: This paper analyzes the momentum extragradient (MEG) method for differentiable games using residual polynomial.\nIt is shown that the convergence rate of the MAG can be characterized by the residual polynomial (Theorem 1),\nunder the assumption that the vector field is given by an affine function.\nBased on this theorem,\nthe paper provides sufficient conditions for convergence and a guide to setting optimal parameters.\nThe usefulness of this analysis is confirmed by numerical experiments as well.\n\nStrengths and Weaknesses: Strengths:\n\n- Motivation and contribution are well explained.\n- The research topic of this paper, convergence analysis of dynamics for games, would be of interest to ML communities.\n- The paper contains practical results, such as guidelines for optimal parameter settings.\n\nWeaknesses:\n\n- The main results of this paper are natural extensions of the results by Azizian et al. (2020a, 2020b) and do not appear to be a major breakthrough.\n- It was not clear to me that the main theoretical results regarding convergence rates (Theorems 1 and 2) were applicable to the general games with non-affine vector fields.\n\nReference:\n- Waïss Azizian, Ioannis Mitliagkas, Simon Lacoste-Julien, and Gauthier Gidel. A tight and unified analysis\nof gradient-based methods for a whole spectrum of differentiable games. In International Conference on\nArtificial Intelligence and Statistics, pp. 2863–2873. PMLR, 2020a.\n- Waïss Azizian, Damien Scieur, Ioannis Mitliagkas, Simon Lacoste-Julien, and Gauthier Gidel. Accelerating\nsmooth games by manipulating spectral shapes. In International Conference on Artificial Intelligence and\nStatistics, pp. 1705–1715. PMLR, 2020b\n\nRequested Changes: - It would be better to mention to what extent the analytical results given for an affine problem (Theorems 1 and 2) can be extended to non-affine problems. Even if there is no clear theoretical argument and the application to non-affine settings is heuristics, that in itself is fine in my opinion. However, in that case, I think it is necessary to clarify that it is heuristics.\n- I would prefer to see a clear statement of how this paper defines \"optimal parameters\" before theorems on optimal parameters given in Section 4.\n- There are several places (e.g., (12), (14) and (15), ...) where the symbol of $\\in$ is used in the sense of inclusion, i.e., in situations where the symbols $\\subset$ or $\\subseteq$ would normally be used. I would appreciate it if you could check them.\n\nBroader Impact Concerns: NA\n\n==================================================\n\nReview 2:\nSummary: In this work, the authors demonstrate the benefit of momentum extragradient (MEG) in terms of convergence for bilinear games over gradient descent (GD), extragradient (EG) and GDM (Gradient descent with momentum) methods. The authors assume the game vector field to be an affine function and analyses the convergence of these algorithms in 3 sub-cases where the game Jacobian takes only 1) real eigenvalues, 2) cross spectrum and 3) shifted imaginary spectrum. They showcase the accelaration of MEG by considering the optimal hyperparameters (step-size, extragradient parameter and momentum parameter) that result in the best convergence.\n\nStrengths and Weaknesses: Strengths:\n1) The problem considered here is important and the authors have made a significant contribution in this work. \n2) The paper is easy to read and follow. The mathematical explanations are quite easy to follow as well. \n3) The 3 case convergence analysis based on the spectra of the game Jacobian is rigorous. I did not find any mistakes in the derivation. \n\nWeakness:\n1) In my opinion, the assumption of the game vector field to be an affine function is too restrictive which makes this analysis valid for only quadratic games. I did not see the authors commenting about this in a more general setting where the game vector field is not affine and the polynomial based analysis will no longer be valid. \n\n2) Is it possible to extend the setup of the experiments to explore beyond quadratic games and analyze the convergence of these 4 methods (GD,EG,MEG,GDM)?\n\n3) Can the authors comment on how the convergence is affected with stochasticity in the gradients with all these cases? Currently only the deterministic setting is considered. \n\n4) Although EG methods are known to perform better than GD, the use of finite small step-size has an implicit effect of discretization drift [1] , where the implicit regularization is exactly the norm of the gradient. Infact applying a first order taylor expansion on the extra-gradient term would also have this term ( the norm of the gradient). In this context, how important is having the explicit regularization (as extra gradient term) compare with the implcit regularization due to the finite step size? Morevoer, in this context, implicit regularization for momentum (due to finite step-size and momentum parameter) is already explored in [2] and whether this can be compared with MEG (momentum extragradient). \n\n\n[1] Rosca, Mihaela C., et al. \"Discretization drift in two-player games.\" International Conference on Machine Learning. PMLR, 2021.\n\n\n[2] Ghosh, Avrajit, et al. \"Implicit regularization in Heavy-ball momentum accelerated stochastic gradient descent.\" The Eleventh International Conference on Learning Representations. 2022.\n\nRequested Changes: I would like to see some comments on the 4 weakness points i mentioned. For point 4, the implicit regualrization won't affect convergence on a quadratic games but can have a different effect for more complex games. \n\nI feel addressing point number 1 and 2 are critical in securing my recommendation for acceptance.\n\nBroader Impact Concerns: No ethical concerns\n\n==================================================\n\nReview 3:\nSummary: The authors consider the extra gradient method with momentum (MEG). They investigate how the spectrum of a Jacobian has an effect on the method. A detailed analysis and the choice of hyperparameters to get acceleration are provided.\n\nStrengths and Weaknesses: I will be very brief because I do not have huge expertise in the extra gradient method. In general, I think that the work is solid and considered an interesting problem. The authors carefully explored different spectrum regimes when MEG converges and outperforms GD and EG. \n\nSome weaknesses:\n1. As far as I understand, the whole provided theory only works with *affine functions*. The authors should more clearly emphasize it in the abstract and in the introduction. \n2. Moreover, the authors show that MEG enjoys acceleration in Theorem 4 and compare this result to GDM. While I agree with this result, I think that this comparison is not fair because GDM can obtain acceleration even with *non-affine gradients*. I believe that this should be explained. The same comment should be addressed when the authors compare MEG with other previous methods.\n\nI think the paper is good and deserves acceptance, but the authors should more clearly explain that they consider the setup where the objective is an *affine function*.\n\nRequested Changes: -\n\nBroader Impact Concerns: -\n\n==================================================\n\nMetareview:\nRecommendation: Accept as is\n\nComment: The primary concerns raised by the reviewers were (i) the difference from the previous result [Azizian et al. (2020a, 2020b)] and (ii) the restriction of the problem that assumes the affine vector field. The authors have addressed these concerns well in the rebuttal and the revision. Specifically, the authors clearly described the originality of the work and extended the theory into the non-affine case with the local convergence analysis. Therefore, the paper meets the acceptance criteria.\n\n==================================================\n\n"}}},{"rowIdx":85,"cells":{"paper":{"kind":"string","value":"# How To Train Your Vit? Data, Augmentation, And Regularization In Vision Transformers\n\nAndreas Steiner∗*andstein@google.com* Alexander Kolesnikov∗*akolesnikov@google.com* Xiaohua Zhai∗*xzhai@google.com* Jakob Uszkoreit *usz@google.com* Google Research, Brain Team, Zürich\n∗ Equal technical contribution, †*independent researcher* Reviewed on OpenReview: *https: // openreview. net/ forum? id= 4nPswr1KcP*\n\n## Abstract\n\nVision Transformers (ViT) have been shown to attain highly competitive performance for a wide range of vision applications, such as image classification, object detection and semantic image segmentation. In comparison to convolutional neural networks, the Vision Transformer's weaker inductive bias is generally found to cause an increased reliance on model regularization or data augmentation (\"AugReg\" for short) when training on smaller training datasets. We conduct a systematic empirical study in order to better understand the interplay between the amount of training data, AugReg, model size and compute budget.1 As one result of this study we find that the combination of increased compute and AugReg can yield models with the same performance as models trained on an order of magnitude more training data: we train ViT models of various sizes on the public ImageNet-21k dataset which either match or outperform their counterparts trained on the larger, but not publicly available JFT-300M dataset.\n\n## 1 Introduction\n\nThe Vision Transformer (ViT) (13) has recently emerged as a competitive alternative to convolutional neural networks (CNNs) that are ubiquitous across the field of computer vision. Without the translational equivariance of CNNs, ViT models are generally found to perform best in settings with large amounts of training data (13) or to require strong AugReg schemes to avoid overfitting (39). However, so far there was no comprehensive study of the trade-offs between model regularization, data augmentation, training data size and compute budget in Vision Transformers.\n\nIn this work, we fill this knowledge gap by conducting a thorough empirical study. We pre-train a large collection of ViT models (different sizes and hybrids with ResNets (18)) on datasets of different sizes, while 1We release more than 50 000 ViT models trained under diverse settings on various datasets. We believe this to be a treasure trove for model analysis. Available at https://github.com/google-research/vision_transformer and https:\n//github.com/rwightman/pytorch-image-models. The code for full reproduction of model training is available at https: //github.com/google-research/big_vision.\nRoss Wightman†*rwightman@gmail.com* Lucas Beyer∗*lbeyer@google.com*\n\n\n\nFigure 1: Adding the right amount of regularization and image augmentation can lead to similar gains as increasing the dataset size by an order of magnitude.\nat the same time performing carefully designed comparisons across different amounts of regularization and data augmentation. We then proceed with extensive transfer learning experiments for the resulting models.\n\nWe focus mainly on gaining insights useful for a practitioner with limited compute and data budgets.\n\nThe homogeneity of the performed study constitutes one of the key contributions of this paper. For the vast majority of works involving Vision Transformers it is not practical to retrain all baselines and proposed methods on equal footing, in particular those trained on larger amounts of data. Furthermore, there are numerous subtle and implicit design choices that cannot be controlled for effectively, such as the precise implementation of complex augmentation schemes, hyper-parameters (e.g. learning rate schedule, weight decay), test-time preprocessing, dataset splits and so forth. Such inconsistencies can result in significant amounts of noise added to the results, quite possibly affecting the ability to draw any conclusions. Hence, all models on which this work reports have been trained and evaluated in a consistent setup.\n\nThe insights we draw from our study constitute another important contribution of this paper. In particular, we demonstrate that carefully selected regularization and augmentations roughly correspond (from the perspective of model accuracy) to a 10x increase in training data size. However, regardless of whether the models are trained with more data or better AugRegs, one has to spend roughly the same amount of compute to get models attaining similar performance. We further evaluate if there is a difference between adding data or better AugReg when fine-tuning the resulting models on datasets of various categories. Other findings, such as the overall beneficial effect of AugRegs for medium-sized datasets, simply confirm commonly held beliefs.\n\nFor those findings, the value of this study lies not in novelty, but rather in confirming these assumptions and quantifying their effect in a strictly controlled setting.\n\nIn addition, we aim to shed light on other aspects of using Vision Transformers in practice such as comparing transfer learning and training from scratch for mid-sized datasets. Finally, we evaluate various compute versus performance trade-offs. We discuss all of the aforementioned insights and more in detail in Section 4.\n\n## 2 Scope Of The Study\n\nWith the ubiquity of modern deep learning (25) in computer vision it has quickly become common practice to pre-train models on large datasets once and re-use their parameters as initialization or feature extraction part in models trained on a broad variety of other tasks (32; 45).\n\nIn this setup, there are multiple ways to characterize computational and sample efficiency. When simply considering the overall costs of pre-training and subsequent training or fine-tuning procedures together, the cost of pre-training usually dominates, often by orders of magnitude. From the vantage point of a researcher aiming to improve model architectures or pre-training schemes, the pre-training costs might therefore be most relevant. Most practitioners, however, rarely, if ever perform pre-training on today's largest datasets but instead use some of the many publicly available parameter sets. For them the costs of fine-tuning, adaptation or training a task-specific model from scratch would be of most interest.\n\nYet another valid perspective is that all training costs are effectively negligible since they are amortized over the course of the deployment of a model in applications requiring a very large number of invocations of inference.\n\nIn this setup there are different viewpoints on computational and data efficiency aspects. One approach is to look at the overall computational and sample cost of both pre-training and fine-tuning. Normally,\n\"pre-training cost\" will dominate overall costs. This interpretation is valid in specific scenarios, especially when pre-training needs to be done repeatedly or reproduced for academic/industrial purposes. However, in the majority of cases the pre-trained model can be downloaded or, in the worst case, trained once in a while.\n\nContrary, in these cases, the budget required for adapting this model may become the main bottleneck.\n\nThus, we pay extra attention to the scenario, where the cost of obtaining a pre-trained model is free or effectively amortized by future adaptation runs. Instead, we concentrate on time and compute spent on finding a good adaptation strategy (or on tuning from scratch training setup), which we call \"practitioner's cost\".\n\nA more extreme viewpoint is that the training cost is not crucial, and all that matters is eventual inference cost of the trained model, \"deployment cost\", which will amortize all other costs. This is especially true for large scale deployments, where a visual model is expected to be used a massive number of times. Overall, there are three major viewpoints on what is considered to be the central cost of training a vision model. In this study we touch on all three of them, but mostly concentrate on \"practitioner\" and \"deployment\" costs.\n\n## 3 Experimental Setup\n\nIn this section we describe our unified experimental setup, which is used throughout the paper. We use a single JAX/Flax (19; 3) codebase for pre-training and transfer learning using TPUs. Inference speed measurements, however, were obtained on V100 GPUs (16G) using the timm PyTorch library (42). All datasets are accessed through the *TensorFlow Datasets* library (15), which helps to ensure consistency and reproducibility. More details of our setup are provided below.\n\n## 3.1 Datasets And Metrics\n\nFor pre-training we use two large-scale image datasets: ILSVRC-2012 (ImageNet-1k) and ImageNet-21k.\n\nImageNet-21k dataset contains approximately 14 million images with about 21 000 distinct object categories (11; 22; 30). ImageNet-1k is a subset of ImageNet-21k consisting of about 1.3 million training images and 1000 object categories. We make sure to de-duplicate images in ImageNet-21k with respect to the test sets of the downstream tasks as described in (13; 22). Additionally, we used ImageNetV2 (29) for evaluation purposes. For transfer learning evaluation we use 4 popular computer vision datasets from the VTAB benchmark (45):\nCIFAR-100 (24), Oxford IIIT Pets (28) (or Pets37 for short), Resisc45 (6) and Kitti-distance (14). We selected these datasets to cover the standard setting of natural image classification (CIFAR-100 and Pets37), as well as classification of images captured by specialized equipment (Resisc45) and geometric tasks (Kitti-distance).\n\nIn some cases we also use the full VTAB benchmark (19 datasets) to additionally ensure robustness of our findings. For all datasets we report top-1 classification accuracy as our main metric. Hyper-parameters for fine-tuning are selected by the result from the *validation* split, and final numbers are reported from the *test* split. Note that for ImageNet-1k we follow common practice of reporting our main results on the validation set. Thus, we set aside 1% of the training data into a *minival* split that we use for model selection. Similarly, we use a minival split for CIFAR-100 (2% of training split) and Oxford IIIT Pets (10% of training split). For Resisc45, we use only 60% of the training split for training, and another 20% for validation, and 20% for computing test metrics. Kitti-distance finally comes with an official validation and test split that we use for the intended purpose. See (45) for details about the VTAB dataset splits.\n\n| Model | Layers | Width | MLP | Heads | Params | Model | Resblocks | Patch-size | Params |\n|-------------|----------|---------|-----------------------|---------|----------|---------|-------------|--------------|----------|\n| | | | R+Ti/16 | [] | 8 | 6.4M | | | |\n| | | | R26+S/32 [2, 2, 2, 2] | 1 | 36.6M | | | | |\n| | | | R50+L/32 [3, 4, 6, 3] | 1 | 330.0M | | | | |\n| ViT-Ti (39) | 12 | 192 | 768 | 3 | 5.8M | | | | |\n| ViT-S (39) | 12 | 384 | 1536 | 6 | 22.2M | | | | |\n| ViT-B (13) | 12 | 768 | 3072 | 12 | 86M | | | | |\n| ViT-L (13) | 24 | 1024 | 4096 | 16 | 307M | | | | |\n\nTable 1: Configurations of ViT models.\n\nTable 2: ResNet+ViT hybrid models.\n\n## 3.2 Models\n\nThis study focuses mainly on the Vision Transformer (ViT) (13). We use 4 different configurations from\n(13; 39): ViT-Ti, ViT-S, ViT-B and ViT-L, which span a wide range of different capacities. The details of each configuration are provided in Table 1. We use patch-size 16 for all models, and additionally patch-size 32 for the ViT-S and ViT-B variants. The only difference to the original ViT model (13) in our paper is that we drop the hidden layer in the head, as empirically it does not lead to more accurate models and often results in optimization instabilities: when pre-training on ImageNet-1k we include both models with and without hidden layer, when pre-training on ImageNet-21k we always drop the hidden layer.\n\nIn addition, we train hybrid models that first process images with a ResNet (18) backbone and then feed the spatial output to a ViT as the initial patch embeddings. We use a ResNet stem block (7 × 7 convolution +\nbatch normalization + ReLU + max pooling) followed by a variable number of bottleneck blocks (18). We use the notation Rn+{Ti,S,L}/p where n counts the number of convolutions, and p denotes the patch-size in the input image - for example R+Ti/16 reduces image dimensions by a factor of two in the ResNet stem and then forms patches of size 8 as an input to the ViT, which results in an effective patch-size of 16.\n\n## 3.3 Regularization And Data Augmentations\n\nTo regularize our models we use robust regularization techniques widely adopted in the computer vision community. We apply dropout to intermediate activations of ViT as in (13). Moreover, we use the stochastic depth regularization technique (20) with linearly increasing probability of dropping layers.\n\nFor data augmentation, we rely on the combination of two recent techniques, namely Mixup (47) and RandAugment (7). For Mixup, we vary its parameter α, where 0 corresponds to no Mixup. For RandAugment, we vary the magnitude parameter m, and the number of augmentation layers l. Note that we use the original RandAugment implementation in TensorFlow, which differs from re-implementations found, for example, in timm (42).\n\nWe also try two values for weight decay (27) which we found to work well, since increasing AugReg may need a decrease in weight decay (2).\n\nOverall, our sweep contains 28 configurations, which is a cross-product of the following hyper-parameter choices:\n- Either use no dropout and no stochastic depth (e.g. no regularization) or use dropout with probability 0.1 and stochastic depth with maximal layer dropping probability of 0.1, thus 2 configuration in total.\n\n- 7 data augmentation setups for (*l, m, α*): none (0, 0, 0), light1 (2, 0, 0), light2 (2, 10, 0.2), medium1\n(2, 15, 0.2), medium2 (2, 15, 0.5), strong1 (2, 20, 0.5), strong2 (2, 20, 0.8).\n\n- Weight decay: 0.1 or 0.03. The weight decay is decoupled following (27), but multiplied by the learning-rate which peaks at 0.001.\n\n## 3.4 Pre-Training\n\nWe pre-trained the models with Adam (21), using β1 = 0.9 and β2 = 0.999, with a batch size of 4096, and a cosine learning rate schedule with a linear warmup (10k steps). To stabilize training, gradients were clipped at global norm 1. The images are pre-processed by Inception-style cropping (36) and random horizontal\n\n\n\nFigure 2: **Left**: When training small and mid-sized datasets from scratch it is very hard to achieve a test error that can trivially be attained by fine-tuning a model pre-trained on a large dataset like ImageNet-21k. With our recommended models (Section 4.5), one can find a good solution with very few trials (bordered green dots, using recipe from B). Note that AugReg is not helpful when transferring pre-trained models (borderless green dots). **Right**: Same data as on the left side (ignoring the borderless green dots), but simulating the results of a random search. For a given compute budget (x-axis), choosing random configurations within that budget leads to varying final performance, depending on choice of hyper parameters (shaded area covers 90%\nfrom 1000 random samples, line corresponds to median).\nflipping. On the smaller ImageNet-1k dataset we trained for 300 epochs, and for 30 and 300 epochs on the ImageNet-21k dataset. Since ImageNet-21k is about 10x larger than ImageNet-1k, this allows us to examine the effects of the increased dataset size also with a roughly constant total compute used for pre-training.\n\n## 3.5 Fine-Tuning\n\nWe fine-tune with SGD with a momentum of 0.9 (storing internal state as bfloat16), sweeping over 2-3 learning rates and 1-2 training durations per dataset as detailed in Table 4 in the appendix. We used a fixed batch size of 512, gradient clipping at global norm 1 and a cosine decay learning rate schedule with linear warmup. Fine-tuning was done both at the original resolution (224), as well as at a higher resolution (384) as described in (40).\n\n## 4 Findings 4.1 Scaling Datasets With Augreg And Compute\n\nOne major finding of our study, which is depicted in Figure 1, is that by judicious use of image augmentations and model regularization, one can (pre-)train a model to similar accuracy as by increasing the dataset size by about an order of magnitude. More precisely, our best models trained on AugReg ImageNet-1k (31) perform about equal to the same models pre-trained on the 10x larger plain ImageNet-21k (11) dataset. Similarly, our best models trained on AugReg ImageNet-21k, when compute is also increased (e.g. training run longer),\nmatch or outperform those from (13) which were trained on the plain JFT-300M (35) dataset with 25x more images. Thus, it is possible to *match these private results with a publicly available dataset*, and it is imaginable that training longer and with AugReg on JFT-300M might further increase performance.\n\nOf course, these results cannot hold for arbitrarily small datasets. For instance, according to Table 5 of (44),\ntraining a ResNet50 on only 10% of ImageNet-1k with heavy data augmentation improves results, but does not recover training on the full dataset.\n\n## 4.2 Transfer Is The Better Option\n\nHere, we investigate whether, for reasonably-sized datasets a practitioner might encounter, it is advisable to try training from scratch with AugReg, or whether time and money is better spent transferring pre-trained\n\n\n\nFigure 3: Pretraining on more data yields more transferable models on average, tested on the VTAB suite (45)\nof 19 tasks across 3 categories.\nmodels that are freely available. The result is that, for most practical purposes, transferring a pre-trained model is both more cost-efficient and leads to better results.\n\nWe perform a thorough search for a good training recipe2for both the small ViT-B/32 and the larger ViT-B/16 models on two datasets of practical size: Pet37 contains only about 3000 training images and is relatively similar to the ImageNet-1k dataset. Resisc45 contains about 30 000 training images and consists of a very different modality of satellite images, which is not well covered by either ImageNet-1k or ImageNet-21k.\n\nFigure 2 shows the result of this search.\n\nThe most striking finding is that, no matter how much training time is spent, for the tiny Pet37 dataset, it does not seem possible to train ViT models from scratch to reach accuracy anywhere near that of transferred models. Furthermore, since pre-trained models are freely available for download, the pre-training cost for a practitioner is effectively zero, only the compute spent on transfer matters, and thus *transferring a pre-trained* model is simultaneously significantly cheaper and gives better results.\n\nFor the larger Resisc45 dataset, this result still holds, although spending two orders of magnitude more compute and performing a heavy search may come close (but not reach) to the accuracy of pre-trained models. Notably, this does not account for the \"exploration cost\", which is difficult to quantify. For the pre-trained models, we highlight those which performed best on the *pre-training* validation set and could be called recommended models (see Section 4.5). We can see that using a recommended model has a high likelihood of leading to good results in just a few attempts, while this is not the case for training from-scratch, as evidenced by the wide vertical spread of points.\n\n## 4.3 More Data Yields More Generic Models\n\nWe investigate the impact of pre-training dataset size by transferring pre-trained models to unseen downstream tasks. We evaluate the pre-trained models on VTAB, including 19 diverse tasks (45). Figure 3 shows the results on three VTAB categories: natural, specialized and structured. The models are sorted by the inference time per step, thus the larger model the slower inference speed. We first compare two models using the same compute budget, with the only difference being the dataset size of ImageNet-1k\n(1.3M images) and ImageNet-21k (13M images). We pre-train for 300 epochs on ImageNet-1k, and 30 epochs on ImageNet-21k. Interestingly, the model pre-trained on ImageNet-21k is significantly better than the ImageNet-1k one, across all the three VTAB categories. This is in contrast with the validation performance on ImageNet-1k (Figure 6), where this difference does not appear so clearly.\n\nAs the compute budget keeps growing, we observe consistent improvements on ImageNet-21k dataset with 10x longer schedule. On a few almost solved tasks, e.g. flowers, the gain is small in absolute numbers. For\n\n2Not only do we further increase available AugReg settings, but we also sweep over other generally important training hyperparameters: learning-rate, weight-decay, and training duration, as described in Appendix A.\n\n \n\nFigure 4: Validation accuracy (for ImageNet-1k: minival accuracy) when using various amounts of augmentation and regularization, highlighting differences to the unregularized, unaugmented setting. For relatively small amount of data, almost everything helps. However, when switching to ImageNet-21k while keeping the training budget fixed, almost everything hurts; only when also increasing compute, does AugReg help again.\n\nThe single column right of each plot show the difference between the best setting with regularization and the best setting without, highlighting that regularization typically hurts on ImageNet-21k.\n\nthe rest of the tasks, the improvements are significant compared to the model pre-trained for a short schedule.\n\nAll the detailed results on VTAB could be found from supplementary section C.\n\nOverall, we conclude that *more data yields more generic models*, the trend holds across very diverse tasks.\n\nWe recommend the design choice of using *more data* with a fixed compute budget.\n\n## 4.4 Prefer Augmentation To Regularization\n\nIt is not clear a priori what the trade-offs are between data augmentation such as RandAugment and Mixup, and model regularization such as Dropout and StochasticDepth. In this section, we aim to discover general patterns for these that can be used as rules of thumb when applying Vision Transformers to a new task. In Figure 4, we show the upstream validation score obtained for each individual setting, i.e. numbers are not comparable when changing dataset. The colour of a cell encodes its improvement or deterioration in score when compared to the unregularized, unaugmented setting, i.e. the leftmost column. Augmentation strength increases from left to right, and model \"capacity\" increases from top to bottom.\n\nThe first observation that becomes visible, is that for the mid-sized ImageNet-1k dataset, any kind of AugReg helps. However, when using the 10x larger ImageNet-21k dataset and keeping compute fixed, i.e. running for 30 epochs, any kind of AugReg *hurts* performance for all but the largest models. It is only when also increasing the computation budget to 300 epochs that AugReg helps more models, although even then, it continues hurting the smaller ones. Generally speaking, there are significantly more cases where adding augmentation helps, than where adding regularization helps. More specifically, the thin columns right of each map in Figure 4 shows, for any given model, its best regularized score minus its best unregularized score. This view, which is expanded in Figure 7 in the Appendix, tells us that when using ImageNet-21k, regularization almost always hurts.\n\n## 4.5 Choosing Which Pre-Trained Model To Transfer\n\nAs we show above, when pre-training ViT models, various regularization and data augmentation settings result in models with drastically different performance. Then, from the practitioner's point of view, a natural question emerges: how to select a model for further adaption for an end application? One way is to run downstream adaptation for all available pre-trained models and then select the best performing model, based on the validation score on the downstream task of interest. This could be quite expensive in practice.\n\nAlternatively, one can select a single pre-trained model based on the upstream validation accuracy and then only use this model for adaptation, which is much cheaper.\n\n\n\nvalidation score on pre-training data vs. validation score on fine-tuning data (negative values mean that selecting models by pre-training validation deteriorates fine-tuning test metrics). **Right**: Correlation between\n\"minival\" validation score vs. ImageNetV2 validation score and official ImageNet-1k validation score (that serves as a test score in this study). Red circles highlight the best models by validation score, see Section 4.5 for an explanation.\nIn this section we analyze the trade-off between these two strategies. We compare them for a large collection of our pre-trained models on 5 different datasets. Specifically, in Figure 5 (left) we highlight the performance difference between the cheaper strategy of adapting only the best pre-trained model and the more expensive strategy of adapting all pre-trained models (and then selecting the best).\n\nThe results are mixed, but generally reflect that the cheaper strategy works equally well as the more expensive strategy in the majority of scenarios. Nevertheless, there are a few notable outliers, when it is beneficial to adapt all models. Thus, we conclude that selecting a single pre-trained model based on the upstream score is a cost-effective practical strategy and also use it throughout our paper. However, we also stress that if extra compute resources are available, then in certain cases one can further improve adaptation performance by fine-tuning additional pre-trained models.\n\nA note on validation data for the ImageNet-1k dataset. While performing the above analysis, we observed a subtle, but severe issue with models pre-trained on ImageNet-21k and transferred to ImageNet-1k dataset. The validation score for these models (especially for large models) is not well correlated with observed test performance, see Figure 5 (right). This is due to the fact that ImageNet-21k data contains ImageNet-1k training data and we use a \"minival\" split from the training data for evaluation (see Section 3.1).\n\nAs a result, large models on long training schedules memorize the data from the training set, which biases the evaluation metric computed in the \"minival\" evaluation set. To address this issue and enable fair hyper-parameter selection, we instead use the independently collected ImageNetV2 data (29) as the validation split for transferring to ImageNet-1k. As shown in Figure 5 (right), this resolves the issue. We did not observe similar issues for the other datasets. We recommend that researchers transferring ImageNet-21k models to ImageNet-1k follow this strategy.\n\n## 4.6 Prefer Increasing Patch-Size To Shrinking Model-Size\n\nOne unexpected outcome of our study is that we trained several models that are roughly equal in terms of inference throughput, but vary widely in terms of their quality. Specifically, Figure 6 (right) shows that models containing the \"Tiny\" variants perform significantly worse than the similarly fast larger models with \"/32\" patch-size. For a given resolution, the patch-size influences the amount of tokens on which self-attention is\n\n\n\nFigure 6: ImageNet transfer. **Left**: For every architecture and upstream dataset, we selected the best model by upstream validation accuracy. Main ViT-S,B,L models are connected with a solid line to highlight the trend, with the exception of ViT-L models pre-trained on i1k, where the trend breaks down. The same data is also shown in Table 3. **Right**: Focusing on small models, it is evident that using a larger patch-size (/32)\nsignificantly outperforms making the model thinner (Ti).\n\nTable 3: ImageNet-1k transfer. Column i1kup evaluates best checkpoint without adaptation, columns i1k300, i21k30 and i21k300 (ImageNet-1k 300 epochs and ImageNet-21k 30 and 300 epochs) report numbers after fine-tuning, which are shown in Figure 6, the \"recommended checkpoints\" (see Section 4.5) were fine-tuned with two different learning rates (see Section B). For the column i21kv2(ImageNet-21k, 300 epochs), the upstream checkpoint was instead chosen by ImageNetV2 validation accuracy. The JFT-300M numbers are taken from (13) (bold numbers indicate our results that are on par or surpass the published JFT-300M results without AugReg for the same models). Inference speed measurements were computed on an NVIDIA V100 GPU using timm (42), sweeping the batch size for best throughput.\n\n| | 224px resolution | | | 384px resolution | | | | | | | |\n|----------|--------------------|-------|--------|--------------------|---------|---------|--------|--------|---------|--------|---------|\n| Model | img/sec | i1kup | i1k300 | i21k30 | i21k300 | img/sec | i1k300 | i21k30 | i21k300 | i21kv2 | JFT300M |\n| L/16 | 228 | 75.72 | 74.01 | 82.05 | 83.98 | 50 | 77.21 | 84.48 | 85.59 | 87.08 | 87.12 |\n| B/16 | 659 | 79.84 | 78.73 | 80.42 | 83.96 | 138 | 81.63 | 83.46 | 85.49 | 86.15 | 84.15 |\n| S/16 | 1508 | 79.00 | 77.51 | 76.04 | 80.46 | 300 | 80.70 | 80.22 | 83.73 | 83.15 | - |\n| R50+L/32 | 1047 | 76.84 | 74.17 | 80.26 | 82.74 | 327 | 76.71 | 83.19 | 85.99 | 86.21 | - |\n| R26+S/32 | 1814 | 79.61 | 78.20 | 77.42 | 80.81 | 560 | 81.55 | 81.11 | 83.85 | 83.80 | - |\n| Ti/16 | 3097 | 72.59 | 69.56 | 68.89 | 73.75 | 610 | 74.64 | 74.20 | 78.22 | 77.83 | - |\n| B/32 | 3597 | 74.42 | 71.38 | 72.24 | 79.13 | 955 | 76.60 | 78.65 | 83.59 | 83.59 | 80.73 |\n| S/32 | 8342 | 72.07 | 69.19 | 68.49 | 73.47 | 2154 | 75.65 | 75.74 | 79.58 | 80.01 | - |\n| R+Ti/16 | 9371 | 70.13 | 67.30 | 65.65 | 69.69 | 2426 | 73.48 | 71.97 | 75.40 | 75.33 | - |\n\nperformed and, thus, is a contributor to model capacity which is not reflected by parameter count. Parameter count is reflective neither of speed, nor of capacity (10).\n\n## 5 Related Work\n\nThe scope of this paper is limited to studying pre-training and transfer learning of Vision Transformer models and there already are a number of studies considering similar questions for convolutional neural networks\n(23; 22). Here we hence focus on related work involving ViT models.\n\nAs first proposed in (13), ViT achieved competitive performance only when trained on comparatively large amounts of training data, with state-of-the-art transfer results using the ImageNet-21k and JFT-300M datasets, with roughly 13M and 300M images, respectively. In stark contrast, (39) focused on tackling overfitting of ViT when training from scratch on ImageNet-1k by designing strong regularization and augmentation schemes. Yet neither work analyzed the effects of stronger augmentation of regularization and augmentation in the presence of larger amounts of training data. Ever since (22) first showed good results when pre-training BiT on ImageNet-21k, more architecture works have mentioned using it for select few experiments (13; 38; 37; 8), with (30) arguing more directly for the use of ImageNet-21k. However, none of these works thoroughly investigates the combined use of AugReg and ImageNet-21k and provides conclusions, as we do here.\n\nAn orthogonal line of work introduces cleverly designed inductive biases in ViT variants or retain some of the general architectural parameters of successful convolutional architectures while adding self-attention to them.\n\n(33) carefully combines a standard convolutional backbone with bottleneck blocks based on self-attention instead of convolutions. In (26; 17; 41; 43) the authors propose hierarchical versions of ViT. (9) suggests a very elegant idea of initializing Vision Transformer, such that it behaves similarly to convolutional neural network in the beginning of training.\n\nYet another way to address overfitting and improve transfer performance is to rely on self-supervised learning objectives. (1) pre-trains ViT to reconstruct perturbed image patches. Alternatively, (4) devises a selfsupervised training procedure based on the idea from (16), achieving impressive results. We leave the systematic comparison of self-supervised and supervised pre-training to future work.\n\n## 6 Discussion\n\nSocietal Impact. Our experimental study is relatively thorough and used a lot of compute. This could be taken as encouraging anyone who uses ViTs to perform such large studies. On the contrary, our aim is to provide good starting points and off-the-shelf checkpoints that remove the need for such extensive search in future work.\n\nLimitations. In order to be thorough, we restrict the study to the default ViT architecture and neither include ResNets, which have been well studied over the course of the past years, nor more recent ViT variants.\n\nWe anticipate though that many of our findings extend to other ViT-based architectures as well.\n\n## 7 Summary Of Recommendations\n\nBelow we summarize three main recommendations based on our study:\n- We recommend to use checkpoints that were pre-trained on more upstream data, and not relying only on ImageNet-1k as a proxy for model quality, since ImageNet-1k validation accuracy is inflated when pre-training on ImageNet-1k, and more varied upstream data yields more widely applicable models (Figure 3 and Section 4.3).\n\n- Judiciously applying data augmentation and model regularization makes it possible to train much better models on a dataset of a given size (Figure 1), and these improvements can be observed both with medium sized datasets like ImageNet-1k, and even with large datasets like ImageNet-21k. But there are no simple rules which AugReg settings to select. The best settings vary a lot depending on model capacity and training schedule, and one needs to be careful not to apply AugReg to a model that is too small, or when pre-training for too short - otherwise the model quality may deteriorate\n(see Figure 4 for an exhaustive quantitative evaluation and Section 4.4 for further comments on regularization vs augmentations).\n\n- How to select the best upstream model for transfer on your own task? Aside from always using ImageNet-21k checkpoints, we recommend to select the model with the best upstream validation performance (Section 4.5, table with paths in our Github repository3). As we show in Figure 5, this choice is generally optimal for a wide range of tasks. If the user has additional computational resources available to fine-tune all checkpoints, they may get slightly better results in some scenarios, but also need to be careful with respect to ImageNet-1k and ImageNet-21k data overlap when it comes to model selection (Figure 5, right).\n\n## 8 Conclusion\n\nWe conduct the first systematic, large scale study of the interplay between regularization, data augmentation, model size, and training data size when pre-training Vision Transformers, including their respective effects on the compute budget needed to achieve a certain level of performance. We also evaluate pre-trained models through the lens of transfer learning. As a result, we characterize a quite complex landscape of training settings for pre-training Vision Transformers across different model sizes. Our experiments yield a number of surprising insights around the impact of various techniques and the situations when augmentation and regularization are beneficial and when not.\n\nWe also perform an in-depth analysis of the transfer learning setting for Vision Transformers. We conclude that across a wide range of datasets, even if the downstream data of interest appears to only be weakly related to the data used for pre-training, transfer learning remains the best available option. Our analysis also suggests that among similarly performing pre-trained models, for transfer learning a model with more training data should likely be preferred over one with more data augmentation.\n\nWe hope that our study will help guide future research on Vision Transformers and will be a useful source of effective training settings for practitioners seeking to optimize their final model performance in the light of a given computational budget. 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A large-scale study of representation learning with the visual task adaptation benchmark, 2020. 2, 3, 6, 15\n[46] Xiaohua Zhai, Xiao Wang, Basil Mustafa, Andreas Steiner, Daniel Keysers, Alexander Kolesnikov, and Lucas Beyer. Lit: Zero-shot transfer with locked-image text tuning. *CVPR*, 2022. 16\n[47] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In *ICLR*, 2018. 4\n\n## A From-Scratch Training Details\n\nWe present from-scratch training details for B/32 and B/16 models, on both Resisc45 and Pets37 datasets. We perform a grid search over the following parameters:\n- B/32 on Pets37\n- Epochs: {1k, 3k, 10k, 30k, 300k} - Learning rates: {1e−4, 3e−4, 1e−3, 3e−3}\n- Weight decays4: {1e−5, 3e−5, 1e−4, 3e−4}\n- B/16 on Pets37\n- Epochs: {1k, 3k, 10k} - Learning rates: {3e−4, 1e−3}\n- Weight decays: {3e−5, 1e−4}\n- B/32 on Resisc45\n- Epochs: {75, 250, 750, 2.5k, 7.5k, 25k}\n- Learning rates: {1e−4, 3e−4, 1e−3, 3e−3}\n- Weight decays: {1e−5, 3e−5, 1e−4, 3e−4}\n- B/16 on Resisc45\n- Epochs: {75, 250, 750, 2.5k, 7.5k}\n- Learning rates: {1e−3}\n- Weight decays: {1e−4, 3e−4}\nAll these from-scratch runs sweep over dropout rate and stochastic depth in range\n{(0.0, 0.0),(0.1, 0.1),(0.2, 0.2)}, and data augmentation (*l, m, α*) in range { (0, 0, 0), (2, 10, 0.2), (2, 15, 0.2),\n(2, 15, 0.5), (2, 20, 0.5), (2, 20, 0.8), (4, 15, 0.5), (4, 20, 0.8) }.\n\nFor the definition of (*l, m, α*) refer to Section 3.3\n\n## B Finetune Details\n\nIn Table 4, we show the hyperparameter sweep range for finetune jobs. We use the same finetune sweep for all the pre-trained models in this paper.\n\n| Dataset | Learning rate | Total, warmup steps |\n|----------------|--------------------|---------------------------|\n| ImageNet-1k | {0.01, 0.03} | {(20k, 500)} |\n| Pets37 | {1e-3, 3e-3, 0.01} | {(500, 100), (2.5k, 200)} |\n| Kitti-distance | {1e-3, 3e-3, 0.01} | {(500, 100), (2.5k, 200)} |\n| CIFAR-100 | {1e-3, 3e-3, 0.01} | {(2.5k, 200), (10k, 500)} |\n| Resisc45 | {1e-3, 3e-3, 0.01} | {(2.5k, 200), (10k, 500)} |\n\nTable 4: Finetune details for the pre-trained models.\n4 As opposed to 3.3 where we specify weight decay values as typically defined in common frameworks, here the values are\n\"decoupled\" following (27) that is better suited for sweeps; multiplying weight decay by the base learning-rate recovers the\n\"coupled\" value as used elsewhere.\nTable 5: Detailed VTAB results, including the \"Mean\" accuracy shown in Figure 3. We show datasets under natural, specialized, structured groups, following (45). \n\n| Caltech101 | CIFAR-100 | DTD | | Flowers102 | Pets | Sun397 | SVHN | | Mean | Camelyon | EuroSAT | Resisc45 | Retinopathy | Mean | Clevr-Count | Clevr-Dist | DMLab | dSpr-Loc | dSpr-Ori | KITTI-Dist | sNORB-Azim | sNORB-Elev | Mean |\n|------------------------------|-------------|-----------|------|--------------|-----------|----------|-----------|------|-----------|------------|-----------|------------|--------------------|-----------|---------------|--------------|-----------|------------|------------|--------------|--------------|--------------|--------|\n| R+Ti/16 | 91.6 | 81.9 | 68.0 | 94.0 | 91.9 | 70.6 | 95.6 | 84.8 | 85.2 | 98.4 | 94.8 | 80.4 | 89.7 | 96.1 89.8 | 67.4 | 99.9 86.9 | 81.9 | 25.1 | 46.3 | 74.2 | | | |\n| S/32 | 92.7 | 86.4 | 70.7 | 93.6 | 91.2 | 72.9 | 95.8 | 86.2 | 83.6 | 98.6 | 95.5 | 79.6 | 89.3 | 94.2 88.4 | 65.8 | 99.9 86.1 | 80.7 | 24.9 | 68.2 | 76.0 | | | |\n| B/32 | 92.6 | 87.6 | 72.7 | 94.4 | 92.2 | 73.8 | 95.8 | 87.0 | 82.7 | 98.6 | 94.9 | 79.8 | 89.0 | 94.0 89.6 | 66.1 | 99.8 84.7 | 80.3 | 24.7 | 62.4 | 75.2 | | | |\n| Ti/16 | 92.7 | 84.0 | 68.9 | 93.8 | 92.5 | 72.0 | 96.1 | 85.7 | 83.7 | 98.7 | 95.6 | 81.6 | 89.9 | 98.0 91.9 | 68.5 | 99.7 83.2 | 82.0 | 26.5 | 65.9 | 77.0 | | | |\n| ImageNet-1k (300ep) R26+S/32 | 90.2 | 86.2 | 74.0 | 95.5 | 94.3 | 74.5 | 95.6 | 87.2 | 84.5 | 98.6 | 96.0 | 83.4 | 90.6 | 99.7 91.6 | 73.3 | 100 | 84.8 | 84.5 | 28.2 | 51.3 | 76.7 | | |\n| S/16 | 93.1 | 86.9 | 72.8 | 95.7 | 93.8 | 74.3 | 96.2 | 87.5 | 84.1 | 98.7 | 95.9 | 82.7 | 90.3 | 98.7 91.5 | 69.8 | 100 | 84.3 | 79.6 | 27.3 | 58.0 | 76.1 | | |\n| R50+L/32 90.7 | 88.1 | 73.7 | 95.4 | 93.5 | 75.6 | 95.9 | 87.6 | 85.8 | 98.4 | 95.4 | 83.1 | 90.7 | 99.8 90.4 | 71.1 | 100 | 87.5 | 82.4 | 23.5 | 53.0 | 76.0 | | | |\n| B/16 | 93.0 | 87.8 | 72.4 | 96.0 | 94.5 | 75.3 | 96.1 | 87.9 | 85.1 | 98.9 | 95.7 | 82.5 | 90.5 | 98.1 91.8 | 69.5 | 99.9 84.5 | 84.0 | 25.9 | 53.9 | 76.0 | | | |\n| L/16 | 91.0 | 86.2 | 69.5 | 91.4 | 93.0 | 75.3 | 94.9 | 85.9 | 81.0 | 98.7 | 93.8 | 81.6 | 88.8 | 94.3 88.3 | 63.9 | 98.5 85.1 | 81.3 | 25.3 | 51.2 | 73.5 | | | |\n| R+Ti/16 | 92.4 | 82.7 | 69.5 | 98.7 | 88.0 | 72.4 | 95.1 | 85.6 | 83.6 | 98.8 | 94.9 | 80.7 | 89.5 | 95.7 90.2 | 66.6 | 99.9 87.0 | 80.3 | 24.4 | 47.0 | 73.9 | | | |\n| S/32 | 92.7 | 88.5 | 72.4 | 98.9 | 90.5 | 75.4 | 95.4 | 87.7 | 83.5 | 98.7 | 95.0 | 79.5 | 89.2 | 94.5 89.8 | 64.4 | 99.8 87.9 | 81.2 | 24.9 | 57.7 | 75.0 | | | |\n| B/32 | 93.6 | 90.5 | 74.5 | 99.1 | 91.9 | 77.8 | 95.7 | 89.0 | 83.5 | 98.8 | 95.1 | 78.8 | 89.1 | 93.6 90.1 | 62.9 | 99.8 89.0 | 78.3 | 24.1 | 55.9 | 74.2 | | | |\n| Ti/16 | 93.3 | 85.5 | 72.6 | 99.0 | 90.0 | 74.3 | 95.1 | 87.1 | 85.5 | 98.8 | 95.5 | 81.6 | 90.4 | 97.7 91.7 | 67.4 | 99.9 83.8 | 81.2 | 26.3 | 55.1 | 75.4 | | | |\n| ImageNet-21k (30ep) R26+S/32 | 94.7 | 89.9 | 76.5 | 99.5 | 93.0 | 79.1 | 95.9 | 89.8 | 86.3 | 98.6 | 96.1 | 83.1 | 91.0 | 99.7 92.0 | 73.4 | 100 | 88.7 | 84.8 | 26.2 | 53.3 | 77.3 | | |\n| S/16 | 94.3 | 89.4 | 76.2 | 99.3 | 92.3 | 78.1 | 95.7 | 89.3 | 84.5 | 98.8 | 96.3 | 81.7 | 90.3 | 98.4 91.5 | 68.3 | 100 | 86.5 | 82.8 | 25.9 | 52.7 | 75.8 | | |\n| R50+L/32 95.4 | 92.0 | 79.1 | 99.6 | 94.3 | 81.7 | 96.0 | 91.1 | 85.9 | 98.7 | 95.9 | 82.9 | 90.9 | 99.9 90.9 | 72.9 | 100 | 86.3 | 82.6 | 25.4 | 57.4 | 76.9 | | | |\n| B/16 | 95.1 | 91.6 | 77.9 | 99.6 94.2 | 80.9 | 96.3 | 90.8 | 84.8 | 99.0 | 96.1 | 82.4 | 90.6 | 98.9 90.9 | 72.1 | 100 | 88.3 | 83.5 | 26.6 | 69.6 | 78.7 | | | |\n| L/16 | 95.7 | 93.4 | 79.5 | 99.6 | 94.6 | 82.3 | 96.7 | 91.7 | 88.4 98.9 | 96.5 | 81.8 | 91.4 | 99.3 91.8 | 72.1 | 100 | 88.5 | 83.7 | 25.0 | 62.9 | 77.9 | | | |\n| ImageNet-21k (300ep) R+Ti/16 | 93.2 | 85.3 | 71.5 | 99.0 | 90.3 | 74.7 | 95.2 | 87.0 | 85.2 | 98.3 | 95.3 | 81.3 | 90.0 | 95.5 90.5 | 67.4 | 99.9 87.4 | 78.2 | 24.5 | 45.2 | 73.6 | | | |\n| S/32 | 93.2 | 89.7 | 75.3 | 99.2 | 92.0 | 78.1 | 96.1 | 89.1 | 84.0 | 98.5 | 95.4 | 80.6 | 89.6 | 96.9 88.7 | 68.1 | 100 | 91.0 | 79.6 | 26.2 | 55.0 | 75.7 | | |\n| B/32 | 95.2 | 92.3 | 77.2 | 99.5 | 92.8 | 81.2 | 96.6 | 90.7 | 87.0 | 98.8 | 96.0 | 81.3 | 90.8 | 97.7 89.8 | 70.5 | 100 | 92.3 82.7 | 25.9 | 83.1 80.2 | | | | |\n| Ti/16 | 93.7 | 87.2 | 73.1 | 99.2 | 91.0 | 77.3 | 95.7 | 88.2 | 86.0 | 98.5 | 95.8 | 81.9 | 90.6 | 98.3 89.7 | 70.8 | 100 | 86.0 | 82.6 | 26.8 | 49.9 | 75.5 | | |\n| R26+S/32 | 94.8 | 90.9 | 78.9 | 99.5 | 94.1 | 81.3 | 96.7 | 90.9 | 87.5 | 98.7 | 96.4 | 84.2 91.7 | 99.9 92.4 77.0 100 | 87.1 | 83.4 | 28.6 56.0 | 78.1 | | | | | | |\n| S/16 | 95.2 | 90.8 | 77.8 | 99.6 | 93.2 | 80.6 | 96.6 | 90.5 | 86.7 | 98.8 | 96.4 | 82.9 | 91.2 | 99.1 89.8 | 73.9 | 100 | 87.6 | 85.1 | 26.8 | 61.1 | 77.9 | | |\n| R50+L/32 95.7 | 93.9 | 81.6 99.5 | 94.9 | 83.6 97.1 | 92.3 | 85.8 | 98.7 | 96.7 | 84.2 | 91.3 | 100 | 92.0 | 76.8 | 100 | 87.2 | 85.2 | 26.8 | 61.8 | 78.7 | | | | |\n| B/16 | 96.0 93.2 | 79.1 | 99.6 | 94.7 | 83.0 | 97.0 | 91.8 | 87.4 | 98.7 | 96.8 83.5 | 91.6 | 99.7 89.0 | 76.0 | 100 | 86.7 | 85.7 28.3 | 68.2 | 79.2 | | | | | |\n| L/16 | 95.5 | 94.1 80.3 | 99.6 | 95.0 83.4 | 97.4 92.2 | 86.4 | 99.0 96.6 | 83.3 | 91.3 | 99.8 91.7 | 75.6 | 100 | 90.4 | 84.7 | 27.5 | 76.5 | 80.8 | | | | | | |\n\n## C Vtab Results\n\nIn Table 5, we show all the results in percentage for all the models on the full VTAB. We report VTAB\nscore only for the best pre-trained models, selected by their upstream validation accuracy (\"recommended checkpoints\", see Section 4.5). For VTAB tasks, we sweep over 8 hyper parameters, include four learning rates {0.001, 0.003, 0.01, 0.03} and two schedules {500, 2500} steps. The best run was selected on VTAB\nvalidation split.\n\n\n\nFigure 7: The improvement or deterioration in validation accuracy when using or not using regularization\n(e.g. dropout and stochastic depth) - positive values when regularization improves accuracy for a given model/augmentation. For absolute values see Figure 4.\n\n## D The Benefit And Harm Of Regularization\n\nIn Figure 7, we show the gain (green, positive numbers) or loss (red, negative numbers) in accuracy when adding regularization to the model by means of dropout and stochastic depth. We did verify in earlier experiments that combining both with (peak) drop probability 0.1 is indeed the best setting. What this shows, is that model regularization mainly helps larger models, and only when trained for long. Specifically, for ImageNet-21 pre-training, it hurts all but the largest of models across the board.\n\n## E Using Recommended Checkpoints For Other Computer Vision Tasks\n\nOne limitation of our study is that it focuses mainly on the classification task. However, computer vision is a much broader field, and backbones need to excel at many tasks. While expanding the full study to many more tasks such as detection, segmentation, tracking, and others would be prohibitive, here we take a peek at one further task: multi-modal image-text retrieval.\n\nA detailed analysis of this question is beyond the scope of this study, but we evaluated our *recommended*\n(see Section 4.5) B/32 checkpoint pre-trained on ImageNet-21k in a contrastive training setup with a locked image tower (46). We initialize the text tower with a BERT-Base (12) checkpoint and train for 20 epochs on CC12M (5). The results in Table 6 indicate that the upstream validation accuracy is a good predictor for zero-shot classification. Moreover, the representations produced by such a model yield similarly better results for image-text retrieval, when compared to models that do not have the ideal amount of AugReg applied. We hope the community will adopt our backbones for other tasks, as already done by (34).\n\nTable 6: Comparing our *recommended* (see Section 4.5) B/32 checkpoint with models that apply too little or too much AugReg. The final validation accuracy from the ImageNet-21k pre-training is the same that is reported in Figure 4. The other columns are ImageNet-1K zero-shot accuracy, and image-text retrieval accuracy on different datasets, after contrastively training as described in (46).\n\n| AugReg | I21k Val | I1k 0shot | Coco I2T | Coco T2I | Flickr I2T | Flickr T2I |\n|-------------|------------|-------------|------------|------------|--------------|--------------|\n| none/0.0 | 41.6 | 54.9 | 33.4 | 20.1 | 58.1 | 39.9 |\n| heavy2/0.1 | 43.5 | 57.3 | 39.1 | 24.4 | 62.1 | 44.6 |\n| Recommended | 47.7 | 60.6 | 41.1 | 25.5 | 65.9 | 46.9 |\n\n16"},"reviews":{"kind":"string","value":"Review 1:\nSummary: The paper conducts extensive experiments, e.g., training 50,000 models under different settings, to investigate the effects of different training recipes and their combinations for training Vision Transformers. Specific aspects include regularization, augmentation, model size, and training data size. The authors draw a few conclusions from the aspects of (pre-)training and transfer learning in a strictly controlled setting.\nSome conclusions are:\n1) the gains from increasing dataset size can be alternatively obtained by carefully selecting image regularization and augmentations. \n2) transferring a pre-trained ViT is more efficient and effective compared to training from scratch.\n3) using more pre-training data leads to better results on unseen downstream tasks.\n\n\nStrengths and Weaknesses: Strengths\n- The authors provide a consistent setup and 50,000 ViT models with different training recipes, which could be a valuable model set for model analysis. \n- The paper is organized and well written. It’s not easy to organize and present so many experiments intuitively. The well-designed figures make it easy to comprehend the corresponding findings.\n- The ViTs have become more and more popular. Thus the findings in the paper could be useful to researchers and practitioners working on ViTs, and provide insights into future research.\n\nWeaknesses:\n- All the experiments are conducted in supervised image classification. However, the success and popularity of ViTs are beyond it. Do the findings still hold for other vision tasks, e.g., object detection, and other training paradigms, e.g., self-supervised learning.\n- I’m not surprised by the findings in this paper. Most of them are natural and straightforward, or commonly held beliefs. For example, either using more data or using appropriate image AugReg can improve accuracy, pre-training is better than training from scratch, and more pre-training data achieves better downstream results.\n\nRequested Changes: The authors are suggested to add some discussion on other vision tasks or training paradigms as described above. Overall I recommend accepting this paper according to the policy of TMLR, i.e., the claims are convincing and clearly supported by experiments, and the practitioners working on ViTs could be interested in the findings.\n\nBroader Impact Concerns: The broader impact concerns are well addressed in Section 6.\n\n==================================================\n\nReview 2:\nSummary: This paper investigates the training strategies of vision transformers systematically, especially the impact of training data, augmentation, and regularization. There are extensive experiments to get solid findings on vision transformers.\n\nStrengths and Weaknesses: - This paper presents some meaningful conclusions, which may inspire the community. These findings answer some important questions, whether transfer learning always helps, whether more data enhance the generalization ability, etc. These findings can help researchers/engineers implement a deep learning model to solve practical problems.\n\n- The authors conduct extensive experiments to obtain solid conclusions. For example, the authors investigate 28 configurations of training strategies with different weight decay, data augmentation, and regularization. All these models have been trained adequately with a large lot of computing resources, making the conclusions reliable.\n\n- The experimental results are impressive. With proper training strategy, the model trained on ImageNet-21k can similar performance to those trained on JFT-300M, though there is a large gap between the numbers of the training dataset.\n\nRequested Changes: - The authors conduct most of the experiments on the standard ViT [1]. Could the findings in this paper generalize to other variants of vision transformers?\n\n- Obtaining the conclusions in these papers usually consumes extremely many computing resources, which is usually unaffordable in most of the research institutions. Could toy experiments with fewer training epochs obtain consistent conclusions?\n\n[1] An image is worth 16x16 words: Transformers for image recognition at scale. ICLR 2021.\n\nOverall, I recommend accepting with minor revision.\n\nBroader Impact Concerns: N/A\n\n==================================================\n\nReview 3:\nSummary: This paper presents evaluations from a hyperparameter sweep on ViT and Resnet+ViT architectures, and evaluates the results to make some conclusions on what matters for downstream performance. \n\nThey consider training on ImageNet and ImageNet-21k data, with and without dropout and stochastic depth, with varying amounts of augmentation, and varying amounts of weight decay.\n\nThey arrive at a few interesting results. One that jumps out at me is: a model trained with data augmentation performs similar to a model trained without augmentations on 10x data. Another result I like is: there is clearly a huge benefit to transferring models learned on larger datasets (which is maybe obvious), but there is still substantial variance: many transferred models are worse than models trained from scratch. Another interesting result is that regularization hurts the smaller models, and improves the larger ones, whereas augmentation helps them all. \n\n\nStrengths and Weaknesses: The main strength of this work is that it provides a large set of apples-to-apples comparisons, from which people can either take the authors' conclusions or draw their own. The experimental setup appears to be clean and simple, and the results are presented clearly.\n\nA weakness here is that the paper does not use the conclusions to create something entirely new. For example, I might expect that the authors implementing and evaluating all this might walk away with a feeling that they can design a better transformer, a better stem, or a more general or safer strategy for transfer. There are helpful takeaways in the individual sections, but I think I would appreciate something more global. For example, even providing a summary of the optimal choices across the sweep, summarizing the analysis across the many experiments, would be useful. \n\n\nRequested Changes: I would like to see a new section, with a title like \"Summary of recommendations\". Even if these have some caveats, that's OK. I want to know what the authors learned. \n\nBroader Impact Concerns: All good\n\n==================================================\n\nReview 4:\nSummary: This paper presents a comprehensive empirical study on the pre-training and fine-tuning of Vision Transformers (ViT). The aim is to understand how the size of training data, augmentation/regularization techniques, and fine-tuning policies affect the performance of ViT in practical image recognition tasks. The major conclusions include:\n\n(1)\tStrong data augmentation and regularization techniques are beneficial, and can alleviate the problem of insufficient pre-training data.\n\n(2)\tFor many downstream tasks, transferring existing pre-trained models leads to better performance and less computational cost (for training).\n\n(3)\tAs an efficient model selection policy, one can simply adopt the best pre-trained model on upstream datasets.\n\n\n\nStrengths and Weaknesses: \nStrengths:\n\n(1)\tThe experiments are pretty comprehensive and well-designed.\n\n(2)\tThe authors release more than 50 000 ViT models trained under diverse settings on various datasets for further application or research.\n\nWeaknesses:\n\n(1)\tThe experimental conclusions of this paper seem to be trivial and widely-known, e.g., strong AugRegs helps, large-scale pre-training helps, and fine-tuning is beneficial. Even though in the context of ViT, many recent papers have pointed these out, at least implicitly. Systemically studying these important issues is an acceptable contribution for me. However, I do believe that the authors should comment on this weakness.\n\n(2)\tThe study is limited to the recognition task and the plain ViT networks. Considering more downstream tasks (e.g., objection detection and semantic segmentation) or the recently proposed pyramid-style ViTs (e.g. swin Transformer and PVT) may make the experiments more strong.\n\n(3)\tIn Figure 5, if I do not misunderstand, adapting all pre-trained models should always achieve the best performance, since it traverses every pre-trained model. Why there are both positive and negative values in the left Figure?\n\n\n\n\nRequested Changes: The authors may consider revising this paper according to the listed weaknesses.\n\nBroader Impact Concerns: The authors have summarized the societal impacts and limitations of their work. I do not have some particular concerns on the broader impacts.\n\n==================================================\n\nMetareview:\nRecommendation: Accept as is\n\nComment: This paper investigates the size of data, augmentation, and regularization techniques in training vision transformers comprehensively. All the reviewers agree that this paper provide useful knowledge and contributions. The main concerns raised by the reviewers is that more experiments and discussions should be provided. \n\nDuring the rebuttal period, the authors address these concerns by performing new experiments on multi-modal image-text retrieval and adding the \"Summary of recommendations\" section to summarize the findings across the experiments. All the reviewers give positive recommendations after reading the authors' response.\n\nTherefore, I recommend acceptance for this paper. \n\n==================================================\n\n"}}},{"rowIdx":86,"cells":{"paper":{"kind":"string","value":"# Understanding Convolution On Graphs Via Energies\n\nFrancesco Di Giovanni∗*fd405@cam.ac.uk* University of Cambridge James Rowbottom∗*jr908@cam.ac.uk* University of Cambridge Benjamin P. Chamberlain Charm Therapeutics Thomas Markovich Cash App Michael M. Bronstein University of Oxford Reviewed on OpenReview: *https: // openreview. net/ forum? id= v5ew3FPTgb*\n\n## Abstract\n\nGraph Neural Networks (GNNs) typically operate by message-passing, where the state of a node is updated based on the information received from its neighbours. Most message-passing models act as graph convolutions, where features are mixed by a shared, linear transformation before being propagated over the edges. On node-classification tasks, graph convolutions have been shown to suffer from two limitations: poor performance on heterophilic graphs, and over-smoothing. It is common belief that both phenomena occur because such models behave as low-pass filters, meaning that the Dirichlet energy of the features decreases along the layers incurring a smoothing effect that ultimately makes features no longer distinguishable. In this work, we rigorously prove that simple graph-convolutional models can actually enhance high frequencies and even lead to an asymptotic behaviour we refer to as *over-sharpening*,\nopposite to over-smoothing. We do so by showing that linear graph convolutions with symmetric weights minimize a multi-particle energy that generalizes the Dirichlet energy; in this setting, the weight matrices induce edge-wise attraction (repulsion) through their positive (negative) eigenvalues, thereby controlling whether the features are being smoothed or sharpened. We also extend the analysis to non-linear GNNs, and demonstrate that some existing time-continuous GNNs are instead always dominated by the low frequencies. Finally, we validate our theoretical findings through ablations and real-world experiments.\n\n## 1 Introduction\n\nGraphNeural Networks (GNNs) represent a popular class of neural networks operating on graphs (Sperduti, 1993; Goller & Kuchler, 1996; Gori et al., 2005; Scarselli et al., 2008; Bruna et al., 2014; Defferrard et al.,\n2016). Most GNNs follow the *message-passing* paradigm (Gilmer et al., 2017), where node embeddings are computed recursively after collecting information from the 1-hop neighbours. Typically, Message Passing Neural Networks (MPNNs) implement convolution on graphs, where messages are first acted upon by a linear transformation (referred to as *channel-mixing*), and are then propagated over the edges by a (normalized)\nadjacency matrix (Kipf & Welling, 2017; Hamilton et al., 2017; Xu et al., 2019; Bronstein et al., 2021).\n\nWhile issues such as bounded expressive power (Xu et al., 2019; Morris et al., 2019) and over-squashing (Alon\n& Yahav, 2021; Topping et al., 2022; Di Giovanni et al., 2023) pertain to general MPNNs, graph-convolutional MPNNs have also been shown to suffer from additional problems more peculiar to node-classification tasks:\nperformance on *heterophilic* graphs - i.e. those where adjacent nodes often have different labels - and over-smoothing of features in the limit of many layers.\n\nSeveral works have gathered evidence that graph convolutions seem to struggle on *some* heterophilic graphs\n(Pei et al., 2020; Zhu et al., 2020; Platonov et al., 2023). A common reasoning is that these MPNNs tend to enhance the low-frequency components of the features and discard the high-frequency ones, resulting in a smoothing effect which is detrimental on tasks where adjacent nodes have different labels (Nt & Maehara, 2019). Accordingly, a common fix for this problem is allowing signed-message passing to revert the smoothing effect (Bo et al., 2021; Yan et al., 2021). Nonetheless, the property that graph convolutions act as low-pass filters is actually only proven in very specialized scenarios, which leads to an important question:\nQ.1 Are simple graph convolutions *actually capable* of enhancing the high-frequency components of the features without explicitly modelling signed message-passing?\n\nA positive answer to Q.1 might imply that, in the limit of many layers, simple graph convolutions can avoid over-smoothing. *Over-smoothing* occurs when node features become indistinguishable in the limit of many layers (Nt & Maehara, 2019; Oono & Suzuki, 2020). In fact, Cai & Wang (2020); Bodnar et al. (2022); Rusch et al. (2022) argued that over-smoothing is defined by the Dirichlet energy E\nDir (Zhou & Schölkopf, 2005)\ndecaying to zero (exponentially) as the depth increases. However, these theoretical works focus on a classical instance of GCN (Kipf & Welling, 2017) with assumptions on the non-linear activation and the singular values of the weight matrices, thereby leaving the following questions open:\nQ.2 Can over-smoothing be avoided by simple graph convolutions? If so, do graph convolutions admit asymptotic behaviours other than over-smoothing, in the limit of many layers?\n\nContributions and outline. Both the poor performance on heterophilic graphs and the issue of over-smoothing, can be characterized in terms of a *fixed energy functional* (the Dirichlet energy E\nDir) decreasing along the features F(t) *computed by graph-convolutional* MPNNs.\n\nNamely, if E\nDir(F(t)) decreases w.r.t. the layer t, then the features become smoother and hence likely not useful for separating nodes on heterophilic graphs, ultimately incurring over-smoothing. But does actually E\nDir(F(t)) decrease for all graph-convolutional models? In this work we show that for many graph convolutions, a more general, parametric energy Eθ, which recovers E\nDir as a special case, decreases along the features as we increase the number of layers. Since this energy can also induce repulsion along the edges, we are able to provide affirmative answers to both Q.1 and Q.2 above, thereby showing that graph convolutions are not bound to act as low-pass filters and over-smooth in the limit. Namely, our contributions are:\n\n\n\nFigure 1: Actual gradient flow dynamics: repulsive forces separate heterophilic labels.\n- In Section 4 we prove that a large class of linear graph convolutions are **gradient flows**, meaning that the features are evolved in the direction of steepest descent of some energy Eθ, as long as the weight (channel-mixing) matrices are symmetric. We study Eθ and show that it provides a physical interpretation for graph convolutions as multi-particle dynamics where the weight matrices induce attraction (repulsion) on the edges, via their positive (negative) eigenvalues.\n\n- In light of this, we provide an affirmative answer to Q.1, proving that the class of gradient-flow MPNNs can enhance the high-frequency components of the features (see Theorem 4.3 for the time-continuous case and Theorem 5.1 for the discrete setting). Our analysis suggests that gradient-flow MPNNs may be able to deal with heterophilic graphs, differently from other time-continuous MPNNs which instead can only induce smoothing of the features (Theorem 5.4), provided that a residual connection is available (Theorem 5.3).\n\n- In Section 5 we also answer Q.2 positively by proving that in the limit of many layers, linear gradient-flow MPNNs admit two possible asymptotic behaviours. If the positive eigenvalues of the weight matrices are larger than the negative ones, then we incur over-smoothing; conversely, the phenomenon of *over-sharpening* occurs, where the features become dominated by the projection onto the eigenvector of the Laplacian associated with the highest frequency (Theorem 5.1). This result proves that asymptotic behaviours other than over-smoothing are possible, even for graph convolutions, depending on the interactions between the spectra of the graph Laplacian and of the weight matrices. To determine whether the underlying MPNN is smoothing/sharpening the features, we also introduce a measure based on the normalized Dirichlet energy, of independent interest.\n\n- In Section 6 we extend our theoretical analysis to graph convolutions with non-linear activations, showing that despite not being gradient-flows, the same parametric energy Eθ(F(t)) introduced above, is monotonically decreasing with respect to t, thereby preserving the interpretation of the weight matrices as edge-wise potentials inducing attraction (repulsion) via their eigenvalues (Theorem 6.1).\n\n- Finally, in Section 7 we validate our theoretical findings through ablations and real-world experiments.\n\nReproducibility. Source code can be found at: https://github.com/JRowbottomGit/graff.\n\n## 2 Graph Convolutions And The Dirichlet Energy\n\nIn this Section, we first review the message-passing paradigm, with emphasis on the class of convolutional-type MPNNs. We then introduce the *Dirichlet energy* on graphs, a known functional which can be used to study the performance of graph convolutions on heterophilic tasks and the over-smoothing phenomenon.\n\n## 2.1 Preliminaries On Graphs And The Mpnn **Class**\n\nNotations and conventions. Consider a graph G = (V, E), with n := |V| nodes and E ⊂ V×V representing the edges. We assume that G is simple, *undirected*, and connected. Its adjacency matrix A is defined as aij = 1 if (*i, j*) ∈ E and zero otherwise. We let D = diag(d0*, . . . , d*n−1) be the degree matrix. We typically use A to denote some (normalized) version of the adjacency matrix. We are interested in problems where the graph has node features {fi ∈ R\nd: i ∈ V} whose matrix representation is F ∈ R\nn×d. In particular, fi ∈ R\ndis the i-th row (transposed) of F, while f r ∈ R\nn is its r-th column. Later, we also rely on the vectorization of the feature matrix vec(F) ∈ R\nnd, simply obtained by stacking all columns of F.\n\nMPNN**s and the 'convolutional flavour'.** In this work we investigate the dynamics of an MPNN in terms of its smoothing or sharpening effects on the features, which are properties particularly relevant in the context of heterophily and over-smoothing. Accordingly, we focus on node-level tasks, where each node i ∈ V has a label yi ∈ {1*, . . . , K*} that we need to predict. Message Passing Neural Networks (MPNNs) (Gilmer et al.,\n2017) compute node embeddings fi(t) at each node i and layer t, using the following recursion:\n\n## Fi(T + 1) = Upt (Fi(T), Aggt ({{Fj (T) : (I, J) ∈ E}})),\n\nwhere AGGt is invariant to permutations. The over-smoothing phenomenon was studied for a class of MPNNs that we refer to as *graph convolutions* (Bronstein et al., 2021), which essentially consist of two operations:\napplying a shared linear transformation to the features (*'channel mixing'*) and propagating them along the edges (*'diffusion'*). Namely, we consider MPNNs whose layer update can be written in matricial form as:\nF(t + 1) = F(t) + σ−F(t)Ωt + AF(t)Wt − F(0)W˜t\n, (1)\nwhere Ωt,Wt, and W˜t are learnable matrices in R\nd×d performing channel mixing, while σ is a pointwise nonlinearity; the message-passing matrix A instead, leads to the aggregation of features from adjacent nodes.\n\nWe first note how this system of equations include common instances of the MPNN class with residual connections. If A = D−1/2AD−1/2 and Ωt = W˜t = 0, then we recover GCN (Kipf & Welling, 2017). The case of A = D−1A and W˜t = 0 results in GraphSAGE (Hamilton et al., 2017), while by choosing Ωt = 0 and Wt and W˜t as convex combinations with the identity we recover GCNII (Chen et al., 2020). Finally, if A = A, Ωt = −(1 + ϵ)Wt, and W˜t = 0, then (1) is a shallow variant of GIN (Xu et al., 2019).\n\n## 2.2 The Dirichlet Energy On Graphs\n\nTo assess whether graph convolutions induce a smoothing or sharpening behaviour over the features, we monitor their Dirichlet energy. Consider a (feature) signal F : V → R\nd, where we associate a vector fi to each node i ∈ V. The graph **Dirichlet energy** of F is defined by (Zhou & Schölkopf, 2005):\n\n$${\\mathcal{E}}^{\\mathrm{Dir}}(\\mathbf{F}):={\\frac{1}{2}}\\sum_{(i,j)\\in E}\\|(\\nabla\\mathbf{F})_{i j}\\|^{2},\\quad(\\nabla\\mathbf{F})_{i j}:={\\frac{\\mathbf{f}_{j}}{\\sqrt{d_{j}}}}-{\\frac{\\mathbf{f}_{i}}{\\sqrt{d_{i}}}},$$\n\nwhere ∇F is the *gradient* of F and is defined edge-wise. We see that E\nDir measures the smoothness of F,\nmeaning whether F has similar values across adjacent nodes.\n\nThe graph Laplacian. The (normalized) graph *Laplacian* is the operator ∆ = I − D−1/2AD−1/2. The Laplacian is symmetric and positive semidefinite, and its eigenvalues are referred to as (graph) *frequencies* and written in ascending order as λ0 ≤ λ1 ≤ *. . .* ≤ λn−1. It is known that λ0 = 0 while λn−1 ≤ 2 (Chung &\nGraham, 1997). The low frequencies are associated with macroscopic (coarse) information in the graph, while the high frequencies generally describe microscopic (fine-grained) behaviour. In fact, if we let {ϕℓ} be an orthonormal frame of eigenvectors for ∆ (i.e. ∆ϕℓ = λℓϕℓ), then we can rewrite the Dirichlet energy of the features F with vector representation vec(F) ∈ R\nnd obtained by stacking all columns, as\n\n$$\\mathcal{E}^{\\text{Dir}}(\\mathbf{F})=\\sum_{r=1}^{d}\\sum_{\\ell=0}^{n-1}\\lambda_{\\ell}(\\mathbf{\\phi}_{\\ell}^{\\top}\\mathbf{f}^{r})^{2}=\\text{trace}(\\mathbf{F}^{\\top}\\mathbf{\\Delta}\\mathbf{F})=(\\text{vec}(\\mathbf{F}))^{\\top}(\\mathbf{I}_{d}\\otimes\\mathbf{\\Delta})\\text{vec}(\\mathbf{F}),\\tag{2}$$\n\nwhere ⊗ denotes the Kronecker product - some properties are reviewed in Appendix A.2 - and Id is the identity matrix in R\nd×d. Features are smooth if the Dirichlet energy is small, which occurs when for each channel r, we have large projections onto the eigenvectors ϕℓ associated with lower frequencies.\n\n## 3 Heterophily And Over-Smoothing: Overview On Related Work\n\nOn a broad scale, our work is related to studying GNNs as filters (Defferrard et al., 2016; Bresson & Laurent, 2017; Hammond et al., 2019; Balcilar et al., 2020; He et al., 2021) and adopts techniques similar to Cai &\nWang (2020). Our approach is also inspired by Haber & Ruthotto (2018); Chen et al. (2018); Biloš et al.\n\n(2021), where layers of an architecture are regarded as Euler discretizations of a time-continuous dynamical system. This direction has been studied in the context of GNNs in several flavours (Xhonneux et al., 2020; Zang & Wang, 2020; Chamberlain et al., 2021a; Eliasof et al., 2021; Chamberlain et al., 2021b; Bodnar et al.,\n2022; Rusch et al., 2022); we share this perspective in our work and expand the connection, by studying the energy associated with GNNs and think of features as 'particles' in some generalized position space.\n\nUnderstanding heterophily via the Dirichlet energy. A node-classification task on a graph G is heterophilic when adjacent nodes often have different labels. In this case the label signal has high Dirichlet energy, meaning that enhancing the high-frequency components of the features could be useful to separate classes - the prototypical example being the eigenvector of ∆ with largest frequency λn−1, that can separate the clusters of a bipartite graph. While MPNNs as in (1) have shown strong empirical performance on node-classification tasks with low heterophily, it has been observed that simple convolutions on graphs could struggle in the case of high heterophily (Pei et al., 2020; Zhu et al., 2020). Such findings have spurred a plethora of methods that modify standard MPNNs to make them more effective on heterophilic graphs, by either ignoring or rearranging the graph structure, or by introducing some form of signed message-passing to\n'reverse' the dynamics and magnify the high frequencies (Bo et al., 2021; Luan et al., 2021; Lim et al., 2021; Maurya et al., 2021; Zhang et al., 2021; Wang et al., 2022; Luan et al., 2022; Eliasof et al., 2023). Understanding over-smoothing via the Dirichlet energy. Whether a given MPNN acts as a low-pass filter or not, is also at the heart of the *over-smoothing* phenomenon, regarded as the tendency of node features to approach the same value - up to degree rescaling - as the number of layers increase, independent of any input information (Li et al., 2018; Oono & Suzuki, 2020). Cai & Wang (2020) formalized over-smoothing as the Dirichlet energy of the features E\nDir(F(t)) decaying to zero as the number of layers t diverge. A similar approach has also been studied in Zhou et al. (2021); Bodnar et al. (2022); Rusch et al. (2022) - for a review see Rusch et al. (2023). Although over-smoothing is now regarded as a general plague for MPNNs, it has in fact only been proven to occur for special instances of graph convolutional equations that (i) have nonlinear activation ReLU, (ii) have small weight matrices (as measured by their singular values), and (iii) have no residual connections.\n\nThe general strategy. Both the problem of graph convolutions struggling on heterophilic tasks and of over-smoothing, arise when a **fixed** energy functional, in this case the Dirichlet energy, decreases along the features computed at each layer. Namely, both these phenomena occur when, in general, d dtE\nDir(F(t)) ≤ 0, with F(t) the features at layer t. To address the questions Q.1 and Q.2 in Section 1, *we study whether* there exist general energy functionals promoting dynamics more expressive than simple smoothing of the features, that decrease along graph convolutions as in (1). Analyzing complicated dynamical systems through monotonicity properties of energy functionals is common in physics and geometry. Indeed, in this work we show that the same approach sheds light on the role of the channel-mixing weight matrices for graph-convolutions, and on the dynamics that MPNNs can generate beyond (over-)smoothing.\n\n## 4 **Gradient-Flow** Mpnn**S: Understanding Convolution On Graphs Via Parametric Energies**\n\nIn this Section we study when graph-convolutions (1) admit an energy functional Eθ such that t *7→ E*θ(F(t)) is decreasing as t increases. This allows us to interpret the underlying MPNN-dynamics as forces that dissipate\n(or in fact, minimize) the energy Eθ. By analysing Eθ we can then derive qualitative properties of the MPNN,\nand determine if the features F(t) incur over-smoothing or not when t is large. Put differently, we investigate when MPNNs as in (1) are *gradient flows*, a special class of dynamical systems we briefly review next.\n\nWhat is a gradient flow? Consider an N-dimensional dynamical system governed by the differential equation F˙ (t) = F(F(t)) that evolves some input F(0) for time t ≥ 0. We say that the evolution equation is a *gradient flow* if there exists E : R\nN → R such that F(F(t)) = −∇E(F(t)). In this case, since E˙(F(t)) = *−∥∇E*(F(t))∥\n2, the energy E *decreases* along the solution F(t). Gradient flows are valuable since the existence of E and the knowledge of its functional expression allow for a better understanding of the underlying dynamics. To motivate our analysis of graph convolutions, we first review examples of gradient flows on graphs that do not involve learnable weights - we discuss variational methods for image processing\n(You et al., 1996; Kimmel et al., 1997) in Appendix A.1 and Appendix B.5.\n\n## 4.1 Gradient Flows On Graphs: The Non-Learnable Case\n\nA prototypical gradient flow: heat equation. Let F ∈ R\nn×d be the feature matrix. The most common form of diffusion process on G is the *heat equation*, which consists of the system ˙f r(t) = −∆f r(t), where f r ∈ R\nn is the r-th entry of the features for 1 ≤ r ≤ d. The heat equation is an example of gradient flow: if we stack the columns of F into vec(F) ∈ R\nnd, we can rewrite the heat equation as\n\n$$\\operatorname{vec}({\\dot{\\mathbf{F}}}(t))=-{\\frac{1}{2}}\\nabla\\mathcal{E}^{\\mathrm{Dir}}(\\operatorname{vec}(\\mathbf{F}(t))),$$\n\nwhere E\nDir : R\nnd → R is the (graph) Dirichlet energy defined in (2). The evolution of F(t) by the heat equation (3) decreases the Dirichlet energy E\nDir(F(t)) and is hence a smoothing process; in the limit, E\nDir(F(t)) → 0, which is attained by the projection of the initial state F(0) onto the null space of the Laplacian.\n\nLabel propagation. Given a graph G and labels {yi} on V1 ⊂ V, assume we want to predict the labels on V2 ⊂ V. Zhou & Schölkopf (2005) introduced *label propagation* (LP) to solve this task. First, the input labels are extended by setting yi(0) = 0 for each i ∈ V \\ V1. The labels are then updated recursively according to the following rule, that is derived as gradient flow of an energy E\nLP we want to be minimized:\n\n$${\\dot{\\mathbf{Y}}}(t)=-{\\frac{1}{2}}\\nabla\\mathcal{E}^{\\mathrm{LP}}(\\mathbf{Y}(t)),\\quad\\mathcal{E}^{\\mathrm{LP}}(\\mathbf{Y}):=\\mathcal{E}^{\\mathrm{Dir}}(\\mathbf{Y})+\\mu\\|\\mathbf{Y}-\\mathbf{Y}(0)\\|^{2}.$$\n2. (4)\n$\\eqref{eq:walpha}$\n$\\left(4\\right)$. \n\nThe gradient flow ensures that the prediction is attained by the minimizer of E\nLP, which both enforces smoothness via E\nDir and penalizes deviations from the available labels (soft boundary conditions).\n\nMotivations. Our goal amounts to extending the gradient flow formalism from the parameter-free case to a deep learning setting which includes features. We investigate when graph convolutions as in (1) admit\n(parametric) energy functionals that decrease along the features computed at each layer.\n\n## 4.2 Gradient Flows On Graphs: The Learnable Case\n\nIn the spirit of Haber & Ruthotto (2018); Chen et al. (2018), we regard the family of MPNNs introduced in\n(1) as the Euler discretization of a continuous dynamical system. More specifically, we can introduce a step size τ ∈ (0, 1] and rewrite the class of MPNNs as\n\n$$\\mathbf{F}(t+\\tau)-\\mathbf{F}(t)=\\tau\\sigma\\left(-\\mathbf{F}(t)\\mathbf{\\Omega}_{t}+\\mathbf{A}\\mathbf{F}(t)\\mathbf{W}_{t}-\\mathbf{F}(0)\\mathbf{\\tilde{W}}_{t}\\right),$$\n\nwhich then becomes the discretization of the system of differential equations\n\n$$\\dot{\\mathbf{F}}(t)=\\sigma\\left(-\\mathbf{F}(t)\\mathbf{\\Omega}_{t}+\\mathbf{A}\\mathbf{F}(t)\\mathbf{W}_{t}-\\mathbf{F}(0)\\mathbf{\\tilde{W}}_{t}\\right),$$\n\nwhere · denotes the time derivative. Understanding the mechanism underlying this dynamical system might shed light on the associated, discrete family of MPNNs. By this parallelism, we study both the time-continuous and time-discrete (i.e. layers) settings, and start by focusing on the former for the rest of the section.\n\n## 4.2.1 Which Energies Are Being Minimized Along Graph-Convolutional Models?\n\nAssume that A in (1) is **symmetric** and let F(0) be the input features. We introduce a parametric function Eθ : R\nnd → R, parameterised by d × d weight matrices Ω and W of the form:\n\nEθ(F) = X i,j ⟨fi, Ωfi⟩ − X + X i,j φ 0(F, F(0)) i,j Aij ⟨fi,Wfj ⟩ , (5) | {z } E ext Ω | {z } E pair W | {z } E source φ0\n$$\\left(5\\right)$$\n$$({\\mathfrak{h}})$$\n$$\\quad(7)$$\n$$({\\boldsymbol{\\delta}})$$\nNote that we can recover the energies in Section 4.1. If Ω = W = Id and φ 0 = 0, then Eθ = E\nDir as per (2),\nwhile if φ 0is an L2-penalty, then Eθ = E\nLP as per (4). We can also recover harmonic energies on graphs\n(see Appendix B.5). Importantly, if we choose φ 0(F, F(0)) = 2Pi\n⟨fi,W˜ fi(0)⟩, for W˜ ∈ R\nd×d, then we can rewrite (see Appendix B.1)\n\n$${\\mathcal{E}}_{\\theta}(\\mathbf{F})=\\langle\\mathrm{vec}(\\mathbf{F}),(\\mathbf{\\Omega}\\otimes\\mathbf{I}_{n}-\\mathbf{W}\\otimes\\mathbf{A})\\mathrm{vec}(\\mathbf{F})+2(\\tilde{\\mathbf{W}}\\otimes\\mathbf{I}_{n})\\mathrm{vec}(\\mathbf{F}(0))\\rangle.$$\n\nThe *gradient flow* of Eθ can then be simply derived as (once we divide the gradient by 2):\n\n$${\\dot{\\mathbf{F}}}(t)=-{\\frac{1}{2}}\\nabla_{\\mathbf{F}}\\mathcal{E}_{\\theta}(\\mathbf{F}(t))=-\\mathbf{F}(t)\\left({\\frac{\\mathbf{\\Omega}+\\mathbf{\\Omega}^{\\top}}{2}}\\right)+\\mathbf{A}\\mathbf{F}(t)\\left({\\frac{\\mathbf{W}+\\mathbf{W}^{\\top}}{2}}\\right)-\\mathbf{F}(0)\\dot{\\mathbf{W}}.$$\n\nSince Ω,W appear in (7) in a symmetrized way, without loss of generality we can assume Ω and W to be symmetric d × d channel mixing matrices. Therefore, (7) simplifies as\n\n$${\\dot{\\mathbf{F}}}(t)=-\\mathbf{F}(t)\\mathbf{\\Omega}+\\mathbf{A}\\mathbf{F}(t)\\mathbf{W}-\\mathbf{F}(0){\\ddot{\\mathbf{W}}}.$$\nF˙ (t) = −F(t)Ω + AF(t)W − F(0)W˜ . (8)\nProposition 4.1. Assume that G *has a non-trivial edge. The linear, time-continuous* MPNN*s of the form*\n\n$${\\dot{\\mathbf{F}}}(t)=-\\mathbf{F}(t)\\mathbf{\\Omega}+\\mathbf{A}\\mathbf{F}(t)\\mathbf{W}-\\mathbf{F}(0){\\tilde{\\mathbf{W}}}$$\n\nare gradient flows of the energy in (6) if and only if the weight matrices Ω and W *are symmetric.*\nTherefore, a large class of linear MPNNs evolve the features in the direction of steepest descent of an energy, provided that the weight matrices are symmetric. Note that despite reducing the degrees of freedom, the symmetry of the weight matrices does not diminish their power (Hu et al., 2019).\n\n## 4.2.2 Attraction And Repulsion: A Physics-Inspired Framework\n\nThanks to the existence of an energy Eθ, we can provide a simple explanation for the dynamics induced by gradient-flow MPNNs as pairwise forces acting among adjacent features and generating attraction and repulsion depending on the eigenvalues of the weight matrices.\n\nWhy gradient flows? A multi-particle point of view. Below, we think of node features as particles in R\n\nd with energy Eθ. In (5), the first term E\next Ω is *independent of the pairwise interactions* and hence represents an 'external' energy in the feature space. The second term E\npair W instead accounts for *pairwise interactions* along edges via the symmetric matrix W and hence represents an energy associated with the graph structure.\n\nFor simplicity, we set the source term φ 0to zero and write W = Θ⊤\n+Θ+ − Θ⊤−Θ−, by decomposing it into components with positive and negative eigenvalues. We can then rewrite Eθ in (5) as\n\n$${\\mathcal{E}}_{\\theta}(\\mathbf{F})=\\underbrace{\\sum_{i}\\langle\\mathbf{f}_{i},(\\mathbf{\\Omega}-\\mathbf{W})\\mathbf{f}_{i}\\rangle}_{g r m p h\\text{-}i n d e p e n d e n t}\\underbrace{+\\frac{1}{2}\\sum_{i,j}\\|\\mathbf{\\Theta}_{+}(\\nabla\\mathbf{F})_{i j}\\|^{2}}_{a t r n c t i o n}\\underbrace{-\\frac{1}{2}\\sum_{i,j}\\|\\mathbf{\\Theta}_{-}(\\nabla\\mathbf{F})_{i j}\\|^{2}}_{r e p u l s i o n},$$\n(9) $\\frac{1}{2}$\nwhich we have derived in Appendix B. Consider now the gradient flow (8) where features F(t) are evolved in the direction of steepest descent of Eθ. Recall that the edge gradient (∇F(t))ij is defined edge-wise and measures the difference between features fi(t) and fj (t) along (*i, j*) ∈ E. We note that:\n(i) The channel-mixing W encodes *attractive edge-wise interactions* via its positive eigenvalues since the gradient terms ∥Θ+(∇F(t))ij∥ are being minimized along (8), resulting in a smoothing effect where the edge-gradient is shrinking along the eigenvectors of W associated with positive eigenvalues;\n(ii) The channel-mixing W encodes *repulsive edge-wise interactions* via its negative eigenvalues since the gradient terms −∥Θ−(∇F(t))ij∥ are being minimized along (8), resulting in a sharpening effect where the edge-gradient is expanding along the eigenvectors of W associated with negative eigenvalues.\n\nNext, we formalize the smoothing vs sharpening effects discussed in (i) and (ii) in a more formal way.\n\n## 4.3 Low Vs High Frequency Dominant Dynamics: A New Measure\n\nAttractive forces reduce the edge gradients and are associated with smoothing effects which magnify low frequencies, while repulsive forces increase the edge gradients and hence afford a sharpening action enhancing the high frequencies. To determine which frequency is dominating the dynamics, we propose to monitor the *normalized* Dirichlet energy: E\nDir(F(t))/∥F(t)∥\n2. This is the *Rayleigh quotient* of Id ⊗ ∆ and so it satisfies 0 ≤ EDir(F)/∥F∥\n2 ≤ λn−1 (see Appendix A.2). If this Rayleigh quotient is approaching its minimum, then the lowest frequency component is dominating, whereas if it is approaching its maximum, then the dynamics is dominated by the highest frequencies. This allows us to introduce the following characterization, of independent interest, to *measure the frequency-response* of spatial MPNNs:\nDefinition 4.2. Given a time-continuous (discrete) MPNN computing features F(t) at each time (layer) t, we say that the MPNN is Low-*Frequency-Dominant* (LFD) if E\nDir(F(t))/∥F(t)∥\n2 → 0 for t → ∞. Conversely, we say that the MPNN is *High-Frequency-Dominant* (HFD) if E\nDir(F(t))/∥F(t)∥\n2 → λn−1 as t → ∞.\n\nNote that the LFD characterization differs from the notion of over-smoothing introduced in Cai & Wang\n(2020); Rusch et al. (2022), since it also accounts for the norm of the features. In fact, in Appendix B.2 we derive how our formulation also captures those cases where the Dirichlet energy is not converging to zero, yet the lowest frequency component is growing the fastest as time increases. Our notion of HFD-dynamics is also novel since previous works typically focused on showing when graph convolutions behave as low-pass filters. Similarly to previous works, we now focus on a class of graph-convolutions inspired by GCN and show that even if we remove the terms Ω and W˜ from (1), graph-convolutions can enhance the high frequencies and ultimately lead to a HFD dynamics opposite to over-smoothing.\n\nAssumption. We let A = D−1/2AD−1/2 and consider the simplified gradient flows with Ω = W˜ = 0.\n\nTheorem 4.3. *Given a continuous* MPNN *of the form* F˙ (t) = AF(t)W, let µ0 < µ1 ≤ . . . ≤ µd−1 be the eigenvalues of W. If |µ0|(λn−1 − 1) > µd−1*, then for almost every* F(0)*, the* MPNN is HFD*. Conversely, if* |µ0|(λn−1 − 1) < µd−1*, then for almost every input* F(0)*, the* MPNN is LFD.\n\nWe provide convergence rates in Theorem B.3 in the Appendix. In accordance with (9), Theorem 4.3 shows that the channel-mixing matrix W can generate both attractive and repulsive forces along edges, leading to either an LFD or a HFD dynamics. In fact, the MPNN is HFD when W has a negative eigenvalue µ0 sufficiently larger than the most positive one µd−1 - and viceversa for the LFD case. Therefore, Theorem 4.3 provides affirmative answers to Q.1, Q.2 in Section 1 in the time-continuous case, by showing that a simple class of gradient flow convolutions on graphs can learn to enhance the high frequencies and lead to a HFD dynamics where the highest frequency components dominate in the limit of many layers.\n\nRemark. We note that Eθ can generally be negative and unbounded. With slight abuse of nomenclature, we call them 'energy' since the associated MPNNs follow the direction of steepest descent of such functional.\n\n## 5 The Interactions Between The Graph And Channel-Mixing Spectra\n\nIn this Section we consider discretized gradient flows, so that we can extend the analysis to the case of layers of an architecture. We consider MPNNs that we derive by taking the Euler discretization of the gradient flow equations in Theorem 4.3; given a step size τ and W ∈ R\nd×d*symmetric*, we have\n\n$$\\mathbf{F}(t+\\tau)=\\mathbf{F}(t)+\\tau\\mathbf{A}\\mathbf{F}(t)\\mathbf{W},\\quad\\mathbf{F}(0)=\\psi_{\\mathrm{{EN}}}(\\mathbf{F}_{0}),$$\n$$\\left(10\\right)$$\n\nF(t + τ ) = F(t) + τAF(t)W, F(0) = ψEN(F0), (10)\nwhere an *encoder* ψEN : R\nn×p → R\nn×d processes input features F0 and the prediction ψDE(F(T)) is produced by a *decoder* ψDE : R\nn×d → R\nn×K. Here, K is the number of label classes, T = mτ is the *integration time*,\nand m is the number of *layers*. We note that (i) typically ψEN, ψDE are MLPs, making the entire framework in (10) non-linear; (ii) since we have a residual connection, this is not equivalent to collapsing the dynamics into a single layer with aggregation matrix A\nm as done in Wu et al. (2019) - see (31) in the Appendix.\n\n## 5.1 Discrete Gradient Flows And Spectral Analysis\n\nThe gradient flow in (10) is a linear, residual GCN with nonlinear encoder and decoder operations. Once we vectorize the features F(t) 7→ vec(F(t)) ∈ R\nnd, we can rewrite the update as vec(F(t + τ )) = vec(F(t)) +\nτ (W ⊗ A) vec(F(t)) (see Appendix A.2 for details). In particular, if we pick bases {ψr} ⊂ R\nd and {ϕℓ} ⊂ R\nn of orthonormal eigenvectors for W and ∆ respectively, we can write the features after m *layers explicitly*:\n\n$$\\operatorname{vec}(\\mathbf{F}(m\\tau))=\\sum_{r=0}^{d-1}\\sum_{\\ell=0}^{n-1}\\left(1+\\tau\\mu_{r}(1-\\lambda_{\\ell})\\right)^{m}c_{r,\\ell}(0)\\psi_{r}\\otimes\\phi_{\\ell},$$\n$$(11)$$\n\nwhere cr,ℓ(0) := ⟨vec(F(0)), ψr ⊗ ϕℓ⟩ and {µr} are the eigenvalues of W. We see that the interaction of the spectra {µr} and {λℓ} is the 'driving' factor for the dynamics, with positive (negative) eigenvalues of W magnifying the frequencies λℓ < 1 (> 1 respectively). In fact, note that the projection of the features onto the kernel of W *stay invariant*. In the following we let µ0 ≤ µ1 ≤ *. . .* ≤ µd−1 be in ascending order.\n\nNote that ϕn−1 is the Laplacian eigenvector associated with largest frequency λn−1. We formulate the result below in terms of the following:\n\n$$\\frac{\\mu_{d-1}}{\\lambda_{n-1}-1}<|\\mu_{0}|<\\frac{2}{\\tau(2-\\lambda_{n-1})}.\\tag{12}$$\n\nNote that if (12) holds, then µ0 < 0 since λn−1 − 1 < 1 whenever G is not bipartite. The first inequality means that the negative eigenvalues of W dominate the positive ones (once we factor in the graph spectrum contribution), while the second is a constraint on the step-size since if τ is too large, then we no longer approximate the gradient flow in (8). We restrict to the case where µ0, µd−1 are simple eigenvalues, but extending the results to the degenerate case is straightforward.\n\nTheorem 5.1. *Consider an* MPNN *with update rule* F(t + τ ) = F(t) + τAF(t)W, with W *symmetric. Given* m layers, if (12) holds, then there exists δ < 1 s.t. for all i ∈ V *we have:*\n\n$$\\mathbf{f}_{i}(m\\tau)=\\left(1+\\tau|\\mu_{0}|(\\lambda_{n-1}-1)\\right)^{m}\\left(c_{0,n-1}(0)\\,\\phi_{n-1}(i)\\cdot\\psi_{0}+\\mathcal{O}\\left(\\delta^{m}\\right)\\right).$$\nm)). (13)\nConversely, if µd−1 > |µ0|(λn−1 − 1)*, then*\n\n$$(13)$$\n\n$$n-1-1),\\;t h e n$$\n$$\\mathbf{f}_{i}(m\\tau)=\\left(1+\\tau\\mu_{d-1}\\right)^{m}\\left(c_{d-1,0}(0){\\sqrt{\\frac{d_{i}}{2|\\mathbf{E}|}}}\\cdot\\Psi_{d-1}+{\\mathcal{O}}\\left(\\delta^{m}\\right)\\right).$$\n$$(14)$$\n. (14)\nWe first note that δ is reported in (32) in Appendix C.1. If (13) holds, then repulsive forces (high frequencies)\ndominate since for all i ∈ V, we have fi(mτ ) ∼ ϕn−1(i) · ψ0, up to *lower order terms in the number of layers*.\n\nThus as we increase the depth, any feature fi(mτ ) becomes dominated by a multiple of ψ0 ∈ R\ndthat only changes based on the value of the Laplacian eigenvector ϕn−1 at node i. Conversely, if (14) holds, then fi(mτ ) ∼\n√di· ψd−1, meaning that the features become dominated by a multiple of ψd−1 only depending on the degree of i - which recovers the over-smoothing phenomenon (Oono & Suzuki, 2020).\n\nCorollary 5.2. *If (13) holds, then the* MPNN is HFD *for almost every* F(0) and F(mτ )/∥F(mτ )∥ converges to F∞ *s.t.* ∆f r∞ = λn−1f r∞ for each r*. Conversely, if (14) holds, then the* MPNN is LFD *for almost every* F(0) and F(mτ )/∥F(mτ )∥ converges to F∞ s.t. ∆f r∞ = 0 *for each* r.\n\nOver-smoothing and over-sharpening. Our analysis shows that (i) linear graph-convolutional equations can induce a sharpening effect (13) (which answers Q.1 in Section 1 affirmatively); (ii) graph convolutions can avoid over-smoothing through the negative eigenvalues of the channel-mixing by incurring the opposite behaviour of *over-sharpening* (i.e. HFD behaviour) in the limit of many layers, which provides an affirmative answer to Q.2. As per Corollary 5.2, over-sharpening entails that the features converge, up to rescaling, to the eigenvector of ∆ associated with the largest frequency. Similarly, Corollary 5.2 implies that LFD is a rigorous characterization of over-smoothing, given that features converge to a vector with zero Dirichlet energy in the limit of many layers. Accordingly, both over-smoothing and over-sharpening afford a loss of information, since the features converge to the Laplacian eigenvector with minimal or maximal Dirichlet energy, respectively. When the number of layers is small though, the features F(mτ ) also depend on the lower-order terms of the asymptotic expansion in Theorem 5.1; whether the dynamics is LFD or HFD will then affect if the lower or higher frequencies have a larger contribution to the prediction.\n\nThe role of the residual connection. The following result shows that the residual connection is crucial for the emergence of the over-sharpening (HFD) regime:\nTheorem 5.3. If G *is not bipartite, and we remove the residual connection, i.e.* F(t + τ ) = τAF(t)W*, with* W *symmetric, then the dynamics is* LFD *for almost every* F(0)*, independent of the spectrum of* W.\n\nWe see that the residual connection *enables the channel-mixing to steer the evolution* towards low or high frequencies depending on the task. If we drop the residual connection, W is less powerful and the only asymptotic regime is over-smoothing (LFD), independent of the spectrum of W.\n\n## 5.2 Frequency Response And Heterophily\n\nNegative eigenvalues flip the edge signs. Let W = Ψdiag(µ)Ψ⊤ be the eigendecomposition of W yielding the Fourier coefficients Z(t) = F(t)Ψ. We rewrite the discretized gradient flow F(t + τ ) =\nF(t) + τAF(t)W in the Fourier domain of W as Z(t + τ ) = Z(t) + τAZ(t)diag(µ) and note that along the eigenvectors of W, if µr < 0, then the dynamics is *equivalent* to flipping the sign of the edges. Therefore, negative edge-weight mechanisms as those proposed in Bo et al. (2021); Yan et al. (2021) to deal with heterophilic graphs, can in fact be simply achieved by a *residual* graph convolutional model where the channel-mixing matrix W has negative eigenvalues. We refer to (33) in the appendix for more details.\n\nCan other 'time-continuous' MPNN**s be** HFD? A framework that cannot enhance the high frequencies may struggle on heterophilic datasets. While by Theorem 5.1 we know that gradient-flow MPNNs can be HFD,\nwe prove that some existing time-continuous MPNNs - CGNN (Xhonneux et al., 2020), GRAND (Chamberlain et al., 2021a), PDE − GCND Eliasof et al. (2021) - are never HFD; in fact, in our experiments (Table 3) we validate that these models are more vulnerable to heterophily - for an explicit statement including convergence rates we refer to Theorem B.4.\n\nTheorem 5.4 (Informal). CGNN, GRAND and PDE − GCND *induce smoothing and are never* HFD.\n\nSummary of the theoretical results. We have analysed a class of linear MPNNs that represent discrete gradient flows of Eθ. This provides a 'multi-particle' interpretation for graph convolutions and sheds light onto the dynamics they generate. We have shown that the interaction between the eigenvectors and spectra of W and ∆ is what drives the dynamics. We have also proven how such interaction ultimately leads to two opposite asymptotic regimes referred to as over-smoothing (LFD) and over-sharpening (HFD). This has allowed us to draw connections with more recent frameworks, proving that simple convolutional models are not necessarily bound to induce smoothing among features, *provided that we have a residual connection*,\nand can indeed also magnify the high frequencies, a property that may be desirable on heterophilic tasks, ultimately providing affirmative answers to Q.1 and Q.2 in Section 1.\n\n## 6 **Extending The Analysis To Non-Linear Layers: The Family Of Energy-Dissipating** Mpnns\n\nAs discussed above, analysing energies along MPNNs is a valuable approach for investigating their dynamics.\n\nCai & Wang (2020); Bodnar et al. (2022) showed that, under some assumptions, E\nDir is decreasing (exponentially) along some classes of graph convolutions, implying *over-smoothing* - see also Rusch et al. (2022).\n\nConsider the general family of (time-continuous) graph convolutions as in (1), with σ a nonlinear activation:\n\n$${\\dot{\\mathbf{F}}}(t)=\\sigma\\left(-\\mathbf{F}(t)\\mathbf{\\Omega}+\\mathbf{A}\\mathbf{F}(t)\\mathbf{W}-\\mathbf{F}(0){\\ddot{\\mathbf{W}}}\\right).$$\n$$(15)^{\\frac{1}{2}}$$\nF˙ (t) = σ−F(t)Ω + AF(t)W − F(0)W˜. (15)\nAlthough this is no longer a gradient flow (unless σ is linear), we prove that if the weights are symmetric, then the energy Eθ in (6) still decreases along (15).\n\nTheorem 6.1. Consider σ : R → R satisfying x 7→ xσ(x) ≥ 0. If F solves (15) with Ω,W being symmetric, then t 7→ Eθ(F(t)) is decreasing. If we discretize the system with step size τ and let c *denote the most positive* eigenvalue of Ω ⊗ In − W ⊗ A *– if no positive eigenvalues exists take* c = 0 *– then*\n\n$${\\mathcal{E}}_{\\theta}(\\mathbf{F}(t+\\tau))-{\\mathcal{E}}_{\\theta}(\\mathbf{F}(t))\\leq c\\|\\mathbf{F}(t+\\tau)-\\mathbf{F}(t)\\|^{2}.$$\n\nAn important consequence of Theorem 6.1 is that for non-linear graph convolutions with symmetric weights, the physics interpretation is preserved since the same multi-particle energy Eθ in (9) dissipates along the features. Note that in the time-discrete setting, the inequality can be interpreted as a Lipschitz regularity result. In general, in the nonlinear case we are no longer able to derive exact results in the limit of many layers, yet the channel-mixing W still induces attraction/repulsion along edges via its positive/negative eigenvalues (see Lemma D.1). Theorem 6.1 differs from Cai & Wang (2020); Bodnar et al. (2022) in two ways:\n(i) It asserts monotonicity of an energy Eθ more general than E\nDir, since it is parametric and in fact also able to enhance the high frequencies; (ii) it holds for an infinite class of non-linear activations (beyond ReLU).\n\n## 7 Experimental Validation Of The Theoretical Results\n\nIn this Section we validate the theoretical results through ablations and real-world experiments on node classification tasks. First, we study a linear, gradient-flow MPNN of the form:\n\n$$\\mathbf{F}(t+1)=\\mathbf{F}(t)+\\mathbf{A}\\mathbf{F}(t)\\mathbf{W}_{S},\\quad\\mathbf{W}_{S}:=(\\mathbf{W}+\\mathbf{W}^{\\top})/2,$$\n\nwhere W ∈ R\nd×dis a learnable weight matrix. Thanks to Proposition 4.1 we know that this is a discretized gradient flow that minimizes Eθ in (6). In particular, from Theorem 5.1 we derive that the positive eigenvalues of WS induce attraction along the edges, while the negative eigenvalues generate repulsion. We note that while the MPNN-update is linear, the map associating a label to each node based on its input feature, is actually non-linear, because the encoder and decoder are, typically, non-linear MLPs. Since this family of\n\n$$(16)$$\n\n| Dataset | Texas | Wisconsin | Cornell | Film | Squirrel | Chameleon | Citeseer | Pubmed | Cora |\n|-----------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|\n| Homophily | 0.11 | 0.21 | 0.30 | 0.22 | 0.22 | 0.23 | 0.74 | 0.80 | 0.81 |\n| #Nodes | 183 | 251 | 183 | 7,600 | 5,201 | 2,277 | 3,327 | 18,717 | 2,708 |\n| #Edges | 295 | 466 | 280 | 26,752 | 198,493 | 31,421 | 4,676 | 44,327 | 5,278 |\n| GCN | 60.81 ± 4.4 | 51.57 ± 3.51 | 59.19 ± 4.75 | 29.96 ± 0.84 | 42.77 ± 2.38 | 62.63 ± 2.02 | 75.9 ± 1.4 | 87.31 ± 0.44 | 86.0 ± 1.0 |\n| SGCNgf | 80.54 ± 4.95 | 82.75 ± 4.54 | 74.32 ± 6.97 | 33.72 ± 0.76 | 51.02 ± 1.68 | 68.57 ± 1.73 | 76.4 ± 1.72 | 88.39 ± 0.39 | 87.12 ± 0.61 |\n| GCNgf | 84.86 ± 4.22 | 84.12 ± 2.97 | 77.3 ± 7.85 | 35.13 ± 0.61 | 50.84 ± 1.92 | 68.27 ± 1.45 | 76.82 ± 1.59 | 88.49 ± 0.42 | 87.79 ± 0.93 |\n\nTable 1: Comparison of GCN and the models in (16) and (17) over datasets of varying homophily.\n\ngraph convolutions amount to a gradient-flow, simplified (i.e. linear) GCN with a residual-connection, we adopt the notation SGCNgf (we recall that A = D−1/2AD−1/2).\n\nBy Theorem 6.1, if we 'activate' (16) as\n\n$${\\bf F}(t+1)={\\bf F}(t)+\\sigma\\Big{(}{\\bf AF}(t){\\bf W}_{S}\\Big{)},\\quad{\\bf W}_{S}:=({\\bf W}+{\\bf W}^{\\top})/2\\tag{1}$$\n$$\\left(17\\right)$$\n\nwith σ s.t. xσ(x) ≥ 0, then Eθ in (6) is decreasing, so that we can think of such equations as more general\n'approximate' gradient flows. With slight abuse of notations, we refer to (17) as GCNgf , since this is just a GCN-model with a residual connection and symmetric weights, which ensure the monotonicity of the energy.\n\nReal-world experiments: performance on node-classification tasks. By Theorem 5.1 and Theorem 6.1, both (16) and (17) can induce repulsion along the edges through the negative eigenvalues of WS. Therefore, these models should be more robust to the heterophily of the graph when compared to the classical implementation of GCN - in fact, note that a linear GCN with symmetric weights is always LFD as per Theorem 5.3, and that non-linear GCN is bound to over-smooth if the singular values are sufficiently small (Oono & Suzuki, 2020; Cai & Wang, 2020). To validate this point, we run all three models on node classification tasks defined over graphs with varying homophily (Sen et al., 2008; Rozemberczki et al., 2021; Pei et al., 2020) (details in Appendix E). Training, validation and test splits are taken from Pei et al.\n\n(2020) for all datasets for comparison. The results are reported in Table 1. For all the datasets shown, we performed a simple, grid search over the space m ∈ {2, 4, 8} - recall that m is the depth - learning rate ∈ {0.001, 0.005} and decay ∈ {0.0005, 0.005, 0.05}. We find that both (16) and (17) surpass the performance of GCN, often by a significant margin, across all heterophilic datasets. We emphasize that this is in support of our theoretical findings, since we proved that *thanks to a residual connection*, graph-convolutional models enable the channel-mixing to also generate repulsion along the edges through its negative eigenvalues hence compensating for the underlying heterophily of the graphs. In particular, the results also show that imposing a symmetrization over the weights has no negative impact on performance.\n\nRemark. It is possible to significantly improve the results by considering additional Ω and W˜ -terms as in (10) and **tuning***, achieving strong results that often surpass more complicated benchmarks specifically* designed for heterophilic tasks. Our results then further bring about questions about the quality of these tasks and whether graph-convolutional models really need to be augmented or replaced for heterophilic tasks. Since the main goal of the Section amounts to validating the theory, we report these results in *Appendix E*.\n\nSpectral analysis. To corroborate our analysis, we also investigate spectral properties of the weight matrix WS. We report statistics of the gradient-flow MPNN in (16) after training, over 4 datasets (two heterophilic ones, and two homophilic ones). We recall that µd−1 and µ0 are the most positive and most negative eigenvalues of WS, respectively.\n\nDataset Squirrel Chameleon Pubmed Cora Homophily 0.22 0.23 0.8 0.81\n−µd−1/µ0 0.68 0.71 2.43 9.58 E\nDir(F(m))/∥F(m)∥\n2 EDir(F(0))/∥F(0)∥2 0.83 0.65 0.35 0.19 Table 2: Spectral properties of the weights of SGCNgf and behaviour of the (normalized) Dirichlet energy.\n\n\n\n \n\nFigure 2: Rayleigh Quotient of Id ⊗ ∆ (i.e.\n\nE\nDir(F(t))/∥F(t)∥\n2) on real graphs for SGCNgf .\n\nFigure 3: Rayleigh quotient of Id ⊗ ∆ on a complete bipartite graph for SGCNgf and GCN.\nIn fact, by Theorem 5.1 we know that the ratio −µd−1/µ0 indicates whether the underlying dynamics is dominated by the high frequencies (ratio smaller than 1), or by the low frequencies (ratio larger than one).\n\nWe confirm this behaviour in Table 2, where we see that the quantity −µd−1/µ0 is larger on the homophilic graphs and is instead smaller on the heterophilic graphs, where the model generally learns to enhance the high-frequency components more. We also monitor the profile of the normalized Dirichlet energy. We see that on homophilic graphs the low frequencies are indeed dominant since the normalized Dirichlet energy at the final layer is much lower than the one evaluated over the input features. This validates that our characterization of over-smoothing (LFD dynamics) and over-sharpening (HFD dynamics) is appropriate to study the dynamics of GNNs. Over-smoothing and over-sharpening. We also validate our theoretical results that linear gradient-flows MPNNs as in (16) admit two, opposite, behaviours in the limit of many layers, depending on the eigenvalues of WS: (i) over-smoothing, i.e. LFD dynamics where the normalized Dirichlet energy approaches zero and (ii)\nover-sharpening, i.e. HFD dynamics where the normalized Dirichlet energy approaches the largest eigenvalue of the normalized Laplacian λn−1 ∈ [1, 2). Accordingly, we report the values of the normalized Dirichlet energy\n(i.e. the Rayleigh quotient of Id ⊗ ∆, simply denoted by RQ) over several real-world datasets; here we are interested in validating that both over-smoothing and over-sharpening occur and that *no other* intermediate case is possible. Figure 2 confirms our theoretical analysis, showing that after training, the normalized Dirichlet energy approaches zero in 7 out of 9 cases (LFD dynamics, i.e. *over-smoothing*) whereas it converges to its maximum, depending on the graph, on the remaining two cases (HFD setting, i.e. *over-sharpening*),\nand that no other intermediate case arises. We point out that in this experiment we study the limiting behaviour of the underlying MPNNs and not the actual performance, which will of course degrade as we increase the number of layers for both over-smoothing and over-sharpening. To further support our claims, we also consider the synthetic case of a complete, bipartite graph with maximal heterophily (i.e. each node is connected to all other nodes with different label). In this extreme case, over-sharpening does actually provide the optimal classifier since the eigenvector of the graph Laplacian associated with largest frequency separates the two communities perfectly; in Figure 3 we see that for (16) the dynamics is HFD (i.e. over-sharpening),\nwhile a standard GCN over-smooths due to the lack of a residual connection as per Theorem 5.3.\n\n## 8 Conclusions\n\nSummary and the messages of the work. In this work we have shown that graph-convolutional models can enhance the high-frequency components of the features via the negative eigenvalues of the weight matrices, provided that there is a residual connection. In particular, we have studied a simple class of graph convolutions that represent the gradient flow of a parametric functional Eθ. The dynamics of gradient-flow MPNNs is fully characterized in terms of the interactions between the eigenvalues of the (symmetric) weight matrices and those of the graph Laplacian, ultimately leading to either over-smoothing or over-sharpening, in the limit of many layers. We have also extended some of the conclusions to gradient flows activated by non-linear maps and have finally validated our analysis through ablation studies and real-world experiments.\n\nOur work suggests that: (i) It is possible to shed light on the dynamics of GNNs once we identify an energy function that is decreasing along the features, thereby providing a general framework for analyzing existing models as well as building more principled ones; (ii) graph convolutions are not simple low-pass filters but can also learn to generate edge-wise repulsion once we have a residual connection; (iii) the role of heterophily as a major issue for GNNs and/or the validity of benchmarks commonly adopted in node classification, are brought into question, given that simple GCN models with residual connections are already on par, or better than GNN models designed specifically to deal with heterophily; (iv) the idea that over-smoothing is the dominant or only asymptotic behaviour for very deep MPNNs is questioned, considering that graph convolutions can also incur over-sharpening if the negative eigenvalues of the weight matrices are sufficiently large.\n\nLimitations and future work. We limited our attention to a class of energy functionals whose gradient flows give rise to evolution equations of the graph-convolutional type. In future work, we plan to study other families of energies that generalize different GNN architectures and provide new models that are more 'physics'-inspired; we believe this to be a powerful paradigm to design new GNNs. To the best of our knowledge, our analysis is a first step into studying the interaction of the graph and 'channel-mixing' spectra; it will be interesting to explore more general dynamics (i.e., that are neither LFD nor HFD).\n\nAcknowledgements. We thank the anonymous reviewers for valuable suggestions and feedback.\n\n## References\n\nUri Alon and Eran Yahav. On the bottleneck of graph neural networks and its practical implications. In International Conference on Learning Representations, 2021. 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Advances in Neural Information Processing Systems, 33:7793–7804, 2020.\n\n## Overview Of The Appendix\n\nTo facilitate navigating the appendix, where we report several additional theoretical results, analysis of different cases along with further experiments and ablation studies, we provide the following detailed outline.\n\n- In Appendix A.1 we review properties of the classical Dirichlet energy on manifolds that inspired traditional PDE variational methods for image processing whose extension to graphs and GNNs more specifically partly constitutes one of the main motivations of our work. We also review important elementary properties of the Kronecker product of matrices that are used throughout our proofs in Appendix A.2. We finally comment on the choice of the normalization (and symmetrization) of the graph Laplacian, briefly mentioning the impact of different choices.\n\n- In Appendix B.1 we derive the energy decomposition reported in (9). In Appendix B.2 we derive additional rigorous results to justify our characterization of LFD and HFD dynamics in Definition 4.2 along with examples. We also formalize (and prove) more quantitatively Theorem 4.3 in Theorem B.3.\n\nIn Appendix B.3 we report Theorem 5.4 including convergence rates and over-smoothing results. In Appendix B.4 we explore the special case of Ω = W which is equivalent to choosing ∆ rather than A as message-passing matrix, hence *providing new arguments as to why propagating messages using* A rather than ∆ *is 'more robust'*. Finally in Appendix B.5 we formally derive an analogy between the continuous energy used for manifolds (images) and a subset of the parametric energies in (5).\n\n- In Appendix C we prove Theorem 5.1, Corollary 5.2, and Theorem 5.3.\n\n- In Appendix D we prove Theorem 6.1 and an extra result confirming that even in the non-linear case the channel-mixing W still induces attraction and repulsion via its spectrum, thus magnifying the low or high frequencies respectively.\n\n- In Appendix E we report additional details on the evaluation along with further real-world experiments and ablation studies.\n\nAdditional notations and conventions used throughout the appendix. Any graph G is taken to be connected. We order the eigenvalues of the graph Laplacian as 0 = λ0 ≤ λ1 ≤ *. . .* ≤ λn−1 = λn−1 ≤ 2 with associated orthonormal basis of eigenvectors {ϕℓ}\nn−1 ℓ=0 so that in particular we have ∆ϕ0 = 0. Moreover, given a symmetric matrix B, we generally denote the spectrum of B by spec(B) and if B is positive semi-definite\n(written as B ⪰ 0), then gap(B) denotes the *positive, smallest eigenvalue* of B. Finally, if we write F(t)/∥F(t)∥\nwe always take the norm to be the Frobenius one and tacitly assume that the dynamics is s.t. the solution is not zero. Without losing generality, below we always take A = D−1/2AD−1/2.\n\n## A Motivations And Important Preliminaries\n\nA.1 Discussion on continuous Dirichlet energy and harmonic maps Starting point: a geometric parallelism. To motivate a gradient-flow approach for GNNs, we start from the continuous case. Consider a smooth map f : R\nn → (R\nd, h) with h a constant metric represented by H ⪰ 0. The *Dirichlet energy* of f is defined by\n\n$${\\mathcal{E}}(f,h)={\\frac{1}{2}}\\int_{\\mathbb{R}^{n}}\\|\\nabla f\\|_{h}^{2}\\,d x={\\frac{1}{2}}\\sum_{q,r=1}^{d}\\sum_{j=1}^{n}\\int_{\\mathbb{R}^{n}}h_{q r}\\partial_{j}f^{q}\\partial_{j}f^{r}(x)d x$$\n$$(18)^{\\frac{1}{2}}$$\n\nand measures the 'smoothness' of f. A natural approach to find minimizers of E - called *harmonic maps*\n- was introduced in Eells & Sampson (1964) and consists in studying the **gradient flow** of E, wherein a given map f(0) = f0 is evolved according to ˙f(t) = −∇f E(f(t)). These type of evolution equations have historically been the core of *variational* and *PDE-based image processing*; in particular, gradient flows of the Dirichlet energy (Kimmel et al., 1997), were shown to recover the Perona-Malik nonlinear diffusion Perona &\nMalik (1990).\n\n## A.2 Review Of Kronecker Product And Properties Of Laplacian Kernel\n\nKronecker product. In this subsection we summarize a few relevant notions pertaining to the Kronecker product of matrices, that are going to be applied throughout our spectral analysis of gradient flow equations for GNNs in both the continuous and discrete time setting.\n\nGiven a matricial equation of the form\n\n$$\\mathbf{Y}=\\mathbf{A}\\mathbf{X}\\mathbf{B},$$\n\nwe can vectorize X and Y by stacking columns into vec(X) and vec(Y) respectively, and rewrite the previous\nsystem as\n$$\\operatorname{vec}(\\mathbf{Y})=\\left(\\mathbf{B}^{\\top}\\otimes\\mathbf{A}\\right)\\operatorname{vec}(\\mathbf{X}).$$\n⊤ ⊗ Avec(X). (19)\nIf A and B are symmetric with spectra spec(A) and spec(B) respectively, then the spectrum of B ⊗ A is given by spec(A) · spec(B). Namely, if Ax = λ Ax and By = λ By, for x and y non-zero vectors, then λ Bλ A\nis an eigenvalue of B ⊗ A with eigenvector y ⊗ x:\n\n$$(19)$$\n\n$$(\\mathbf{B}\\otimes\\mathbf{A})\\,\\mathbf{y}\\otimes\\mathbf{x}=(\\lambda^{\\mathbf{B}}\\lambda^{\\mathbf{A}})\\mathbf{y}\\otimes\\mathbf{x}.$$\nA)y ⊗ x. (20)\nOne can also define the *Kronecker sum* of matrices A ∈ R\nn×n and B ∈ R\nd×d as\n\n$$(20)$$\n\n$$(21)$$\n$$\\mathbf{A}\\oplus\\mathbf{B}:=\\mathbf{A}\\otimes\\mathbf{I}_{d}+\\mathbf{I}_{n}\\otimes\\mathbf{B},$$\nA ⊕ B := A ⊗ Id + In ⊗ B, (21)\nwith spectrum spec(A ⊕ B) = {λ A + λ B : λ A ∈ spec(A), λB ∈ spec(B)}.\n\nAdditional details on E\nDir **and the choice of Laplacian.** We recall that the classical graph Dirichlet energy E\nDir is defined by\n\n$${\\mathcal{E}}^{\\mathrm{Dir}}(\\mathbf{F})=\\operatorname{trace}\\left(\\mathbf{F}^{\\mathsf{T}}\\,\\mathbf{\\Delta}\\mathbf{F}\\right).$$\n\nWe can use the Kronecker product to rewrite the Dirichlet energy as\n\n$${\\mathcal{E}}^{\\mathrm{Dir}}(\\mathbf{F})=\\mathrm{vec}(\\mathbf{F})^{\\top}(\\mathbf{I}_{d}\\otimes\\mathbf{\\Delta})\\mathrm{vec}(\\mathbf{F}),$$\n$$(22)$$\n⊤(Id ⊗ ∆)vec(F), (22)\nfrom which we immediately derive that 12∇vec(F)E\nDir(F) = (Id ⊗ ∆)vec(F) - since ∆ is *symmetric* - and hence recover the gradient flow in (3) leading to the graph heat equation across each channel.\n\nBefore we further comment on the characterizations of LFD and HFD dynamics, we review the main choices of graph Laplacian and the associated harmonic signals (i.e. how we can characterize the kernel spaces of the given Laplacian operator). Recall that throughout the appendix we always assume that the underlying graph G is *connected*. The symmetrically normalized Laplacian ∆ = I − A is symmetric, positive semi-definite with harmonic space of the form (Chung & Graham, 1997)\n\n$$\\ker(\\mathbf{\\Delta}):=\\operatorname{span}(\\mathbf{D^{\\frac{1}{2}}1_{n}}:\\ \\mathbf{1}_{n}=(1,\\ldots,1)^{\\top}).$$\n\nTherefore, if the node features F(t) satisfy F˙ (t) = GNNθ(F(t), t) with initial condition F(0), and we assume that over-smoothing occurs, meaning that ∆f r(t) → 0 for t → ∞ for each column 1 ≤ r ≤ d, then the only information persisting in the asymptotic regime is the degree and any dependence on the input features is lost, as studied in Oono & Suzuki (2020); Cai & Wang (2020). A slightly different behaviour occurs if instead of ∆, we consider the unnormalized Laplacian L = D − A with kernel given by span(1n), meaning that if Lf r(t) → 0 as t → ∞ for each 1 ≤ r ≤ d, then any node would be embedded into a single point, hence making any separation task impossible. The same consequence applies to the random walk Laplacian\n∆RW = I − D−1A. In particular, we note that generally a row-stochastic matrix is not symmetric - if it was, then this would in fact be doubly-stochastic - and the same applies to the random-walk Laplacian (a special exception is given by the class of *regular* graphs). In fact, in general any dynamical system governed by ∆RW\n(or simply D−1A) is not the gradient flow of an energy due to the lack of symmetry, as further confirmed below in (24).\n\n$$(23)$$\n\n## B Proofs And Additional Details Of **Section 4** B.1 Attraction Vs Repulsion: A Physics-Inspired Framework\n\nFirst, we show that we can rewrite the energy in (5) using the Kronecker product formalism. It suffices to focus on the term Ω-term since the other derivations are identical. We have\n\n$$\\sum_{i}\\langle\\mathbf{f}_{i}.\\Omega\\mathbf{f}_{i}\\rangle=\\sum_{i}f_{i}^{\\alpha}\\Omega_{\\alpha\\beta}f_{i}^{\\beta}.$$\n\nOn the other hand, since (Ω ⊗ I)vec(F) = IFΩ⊤, we also find\n\n$$\\langle\\mathrm{vec}(\\mathbf{F}),(\\mathbf{\\Omega}\\otimes\\mathbf{I})\\mathrm{vec}(\\mathbf{F})=f_{i}^{\\alpha}(\\mathbf{I}\\mathbf{F}\\mathbf{\\Omega}^{\\top})_{i}^{\\alpha}=f_{i}^{\\alpha}\\Omega_{\\alpha\\beta}f_{i}^{\\beta}\\,,$$\n\nwhich shows the equivalence claimed.\n\nProof of Proposition *4.1.* We first note that the system in (8) can be written using the Kronecker product as\n\n$$\\operatorname{vec}({\\dot{\\mathbf{F}}}(t))=-(\\mathbf{\\Omega}\\otimes\\mathbf{I}_{n})\\mathbf{v}$$\n\nvec(F˙ (t)) = −(Ω ⊗ In)vec(F(t)) + (W ⊗ A)vec(F(t)) − (W˜ ⊗ In)vec(F(0)).\n\nIf this is the gradient flow of F *7→ E*θ(F), then we would have\n\n$$\\nabla_{\\mathrm{vec}(\\mathbf{F})}^{2}{\\mathcal{E}}_{\\theta}(\\mathbf{F})=\\Omega\\otimes\\mathbf{I}_{n}-\\mathbf{W}\\otimes\\mathbf{A},$$\n$$(24)$$\nEθ(F) = Ω ⊗ In − W ⊗ A, (24)\nwhich must be symmetric due to the Hessian of a function being symmetric. The latter means\n\n$$(\\mathbf{\\Omega}^{\\mathsf{T}}-\\mathbf{\\Omega})\\otimes\\mathbf{I}_{n}=(\\mathbf{W}^{\\mathsf{T}}-\\mathbf{W})\\otimes\\mathbf{A}.$$\n\nTherefore, for each components *α, β* we must have\n\n($\\omega_{\\alpha\\beta}-\\omega_{\\beta\\alpha}$)I ($w_{\\alpha\\beta}-w_{\\beta\\alpha}$)A. \nIf A is non-diagonal, which happens as soon as there is a non-trivial edge in G, the equations are satisfied only if both Ω and W are *symmetric*. This shows that (8) is the gradient flow of Eθ if and only if Ω and W\nare symmetric.\n\nWe now rely on the spectral decomposition of W to rewrite Eθ explicitly in terms of attractive and repulsive interactions. If we have a spectral decomposition W = Ψdiag(µ)(Ψ)\n⊤, we can separate the positive eigenvalues from the negative ones and write\n\n$\\mathbf{W}=\\mathbf{\\Psi}\\mathrm{diag}(\\mathbf{\\mu}_{+})\\mathbf{\\Psi}^{\\top}+\\mathbf{\\Psi}\\mathrm{diag}(\\mathbf{\\mu}_{-})\\mathbf{\\Psi}^{\\top}:=\\mathbf{W}_{+}-\\mathbf{W}_{-}$. \nSince W+ ⪰ 0,W− ⪰ 0, we can use the Choleski decomposition to write W+ = Θ⊤\n+Θ+ and W− =\nΘ⊤−Θ− with Θ+, Θ− ∈ R\nd×d. Equation (9) then simply follows by direct computation: namely, if we let\n(D−1/2AD−1/2)ij = ¯aij , we derive\n\nEθ(F) = X i ⟨fi, Ωfi⟩ −X i,j a¯ij ⟨fi,Wfj ⟩ = X i ⟨fi,(Ω − W)fi⟩ + X i ⟨fi,Wfi⟩ −X i,j a¯ij ⟨Θ+fi, Θ+fj ⟩ + X i,j a¯ij ⟨Θ−fi, Θ−fj ⟩ i ⟨fi,(Ω − W)fi⟩ + 1 2 X i,j ∥Θ+(∇F)ij∥ 2 − 1 2 X i,j ∥Θ−(∇F)ij∥ 2, = X where we have used that Pi,j 1 di ∥Θ+fi∥ 2 =Pi ∥Θ+fi∥ 2.\n\n## B.2 Additional Details On Lfd And Hfd **Characterizations**\n\nIn this subsection we provide further details and justifications for Definition 4.2. We first prove the following simple properties.\n\nLemma B.1. *Assume we have a (continuous) process* t 7→ F(t) ∈ R\nn×d, for t ≥ 0*. The following equivalent* characterizations hold:\n\n_(i) $\\mathcal{E}^{\\rm Dir}({\\bf F}(t))\\to0$ for $t\\to\\infty$ if and only if $\\mathbf{\\Delta}{\\bf f}^{r}(t)\\to{\\bf0}$, for $1\\leq r\\leq d$._\n* $\\mathcal{E}^{\\rm Dir}(\\mathbf{F}(t)/\\|\\mathbf{F}(t)\\|)\\to\\lambda_{n-1}$ _for_ $t\\to\\infty$ _if and_ $d$ $t_{j_{k}}\\to\\infty$ _and a unit limit_ $\\mathbf{F}_{\\infty}$ _- depending on_ $1\\leq r\\leq d$_. \nDir(F(t)/∥F(t)∥) → λn−1 for t → ∞ if and only if for any sequence tj → ∞ *there exist a subsequence* tjk → ∞ and a unit limit F∞ *– depending on the subsequence - such that* ∆f r∞ = λn−1f r∞*, for* Proof. (i) Given F(t) ∈ R\nn×d, we can vectorize it and decompose it in the orthonormal basis {er ⊗ ϕℓ : 1 ≤\nr ≤ d, 0 ≤ ℓ ≤ n − 1}, with {er}\nd r=1 canonical basis in R\nd, and write\n\n$$\\operatorname{vec}(\\mathbf{F}(t))=\\sum_{r,\\ell}c_{r,\\ell}(t)\\mathbf{e}_{r}\\otimes\\boldsymbol{\\phi}_{\\ell},\\quad c_{r,\\ell}(t):=\\langle\\operatorname{vec}(\\mathbf{F}(t)),\\mathbf{e}_{r}\\otimes\\boldsymbol{\\phi}_{\\ell}\\rangle.$$\n\nWe can then use (22) to compute the Dirichlet energy as\n\n$${\\mathcal{E}}^{\\mathrm{Dir}}(\\mathbf{F}(t))=\\sum_{r=1}^{d}\\sum_{\\ell=0}^{n-1}c_{r,\\ell}^{2}(t)\\lambda_{\\ell}\\equiv\\sum_{r=1}^{d}\\sum_{\\ell=1}^{n-1}c_{r,\\ell}^{2}(t)\\lambda_{\\ell}\\geq\\operatorname{gap}(\\mathbf{\\Delta})\\sum_{r=1}^{d}\\sum_{\\ell=1}^{n-1}c_{r,\\ell}^{2}(t),$$\n\nwhere we have used that λ0 = 0 and that gap(∆) = λ1 ≤ λℓ for all ℓ ≥ 1. Therefore\n\n$${\\mathcal{E}}^{\\mathrm{Dir}}(\\mathbf{F}(t))\\to0\\quad\\Longleftrightarrow\\quad\\sum_{r=1}^{d}\\sum_{\\ell=1}^{n-1}c_{r,\\ell}^{2}(t)\\to0,\\quad t\\to\\infty,$$\n\nwhich occurs if and only if\n\n$$(\\mathbf{I}_{d}\\otimes\\mathbf{\\Delta})\\mathrm{vec}(\\mathbf{F}(t))=\\sum_{r=1}^{d}\\sum_{\\ell=1}^{n-1}c_{r,\\ell}(t)\\lambda_{\\ell}\\mathbf{e}_{r}\\otimes\\phi_{\\ell}\\to0.$$\n\n(ii) The argument here is similar. Indeed we can write Q(t) = F(t)/∥F(t)∥ with Q(t) a unit-norm signal.\n\nNamely, we can vectorize it and write\n\n$$\\operatorname{vec}(\\mathbf{Q}(t))=\\sum_{r,\\ell}q_{r,\\ell}(t)\\mathbf{e}_{r}\\otimes\\phi_{\\ell},\\quad\\sum_{r,\\ell}q_{r,\\ell}^{2}(t)=1.$$\n\nThen E\nDir(Q(t)) → λn−1 if and only if\n\n$$\\sum_{r,\\ell}q_{r,\\ell}^{2}(t)\\lambda_{\\ell}\\rightarrow\\lambda_{n-1},\\quad t\\rightarrow\\infty,$$\n\nwhich holds if and only if\n\n$$\\sum_{r}q_{r,\\lambda_{n-1}}^{2}(t)\\to1$$ $$q_{r,\\ell}^{2}(t)\\to0,\\quad\\ell:\\lambda_{\\ell}<\\lambda_{n-1},$$\nr,ℓ(t) → 0, ℓ : λℓ < λn−1, (25)\ngiven the unit norm constraint. This is equivalent to the Rayleigh quotient of Id ⊗ ∆ converging to its maximal value λn−1. When this occurs, for any sequence tj → ∞ we have that q 2 r,ℓ(tj ) ≤ 1, meaning that we can extract a converging subsequence that due to (25) will converge to a unit eigenvector F∞ of Id ⊗ ∆\nsatisfying (Id ⊗ ∆)F∞ = λn−1Q∞. Conversely assume for a contradiction that there exists a sequence\n\n$$(25)$$\n\ntj → ∞ such that E\nDir(F(tj )/∥F(tj )∥) < λn−1 − ϵ, for some ϵ > 0. Then (25) fails to be satisfied along the sequence, meaning that no subsequence converges to a unit norm eigenvector F∞ of Id ⊗ ∆ with associated eigenvalue λn−1 which is a contradiction to our assumption.\n\nBefore we address the formulation of low(high)-frequency-dominated dynamics, we solve explicitly the system F˙ (t) = AF(t) in R\nn×d, with some initial condition F(0). We can vectorize the equation and solve\n˙vec(F(t)) = (Id ⊗ A)vec(F(t)), meaning that\n\n$$\\operatorname{vec}(\\mathbf{F}(t))=\\sum_{r=1}^{d}\\sum_{\\ell=0}^{n-1}e^{(1-\\lambda_{\\ell})t}c_{r,\\ell}(0)\\mathbf{e}_{r}\\otimes\\phi_{\\ell},\\qquad c_{r,\\ell}(0):=\\langle\\operatorname{vec}(\\mathbf{F}(0)),\\mathbf{e}_{r}\\otimes\\phi_{\\ell}\\rangle.$$\n\nConsider any initial condition F(0) such that\n\n$$\\sum_{r=1}^{d}\\vert c_{r,0}\\vert=\\sum_{r=1}^{d}\\vert\\langle\\mathrm{vec}(\\mathbf{F}(0)),\\mathbf{e}_{r}\\otimes\\phi_{0}\\rangle\\vert>0,$$\n$\\square$\nwhich is satisfied for each vec(F(0)) ∈ R\nnd \\ U⊥, where U\n⊥ is the orthogonal complement of R\nd ⊗ span(ϕ0).\n\nSince U\n⊥ is a lower-dimensional subspace, its complement is dense. Accordingly for a.e. F(0), we find that the solution satisfies\n\n$$\\|{\\rm vec}({\\bf F}(t))\\|^{2}=e^{2t}\\left(\\sum_{r=1}^{d}c_{r,0}^{2}+{\\cal O}(e^{-2{\\rm gap}(\\Delta)t})\\right)=e^{2t}\\left(\\|P_{\\rm vec}^{\\perp}(\\Delta){\\rm vec}({\\bf F}(0))\\|^{2}+{\\cal O}(e^{-2{\\rm gap}(\\Delta)t})\\right),$$\n\nwith P\n⊥\nker(∆)\nthe projection onto R\nd ⊗ ker(∆). We see that the norm of the solution increases exponentially, however *the dominant term is given by the projection onto the lowest frequency signal* and in fact\n\n$${\\frac{\\operatorname{vec}(\\mathbf{F}(t))}{\\|\\operatorname{vec}(\\mathbf{F}(t))\\|}}={\\frac{P_{\\mathrm{{ker}}(\\mathbf{\\Delta})}^{\\operatorname{vec}}(\\mathbf{F}(0))+{\\mathcal{O}}(e^{-\\operatorname{sgn}(\\mathbf{\\Delta})t})(\\mathbf{I}-P_{\\mathrm{{ker}}(\\mathbf{\\Delta})}^{\\perp})\\operatorname{vec}(\\mathbf{F}(0))}{\\left(\\|P_{\\mathrm{{ker}}(\\mathbf{\\Delta})}^{\\perp}\\operatorname{vec}(\\mathbf{F}(0))\\|^{2}+{\\mathcal{O}}(e^{-2\\operatorname{sgn}(\\mathbf{\\Delta})t})\\right)^{\\frac{1}{2}}}}\\to\\operatorname{vec}(\\mathbf{F}_{\\infty}),$$\n\nsuch that (Id ⊗ ∆)vec(F∞) = 0 which means ∆f r∞ = 0, for each column 1 ≤ r ≤ d. Equivalently, one can compute E\nDir(F(t)/∥F(t)∥) and conclude that the latter quantity converges to zero as t → ∞ by the very same argument.\n\nIn fact, this motivates further the nomenclature LFD and HFD. Without loss of generality we focus now on the high-frequency case. Assume that we have a HFD dynamics t 7→ F(t), i.e. E\nDir(F(t)/∥F(t)∥) → λn−1, then we can vectorize the solution and write vec(F(t)) = ∥F(t)∥vec(Q(t)), for some time-dependent unit vector vec(Q(t)) ∈ R\nnd:\n\n$$\\operatorname{vec}(\\mathbf{Q}(t))=\\sum_{r,\\ell}q_{r,\\ell}(t)\\mathbf{e}_{r}\\otimes\\phi_{\\ell},\\quad\\sum_{r,\\ell}q_{r,\\ell}^{2}(t)=1.$$\n\nBy Lemma B.1 and more explicitly (25), we derive that the coefficients {qr,λn−1\n} associated with the eigevenctors er ⊗ ϕλn−1 are dominant in the evolution hence justifying the name *high-frequency dominated* dynamics.\n\nThe next result provides a theoretical justification for the characterization of low (high) frequency dominated dynamics in Definition 4.2.\n\nLemma B.2. *Consider a dynamical system* F˙ (t) = GNNθ(F(t), t)*, with initial condition* F(0).\n\n(i) GNNθ is LFD *if and only if* (Id ⊗ ∆)\nvec(F(t))\n∥F(t)∥ → 0 if and only if for each sequence tj → ∞ there exist a subsequence tjk → ∞ and F∞ *(depending on the subsequence) s.t.* F(tjk\n)\n∥F(tjk\n)∥ → F∞ *satisfying*\n∆f r∞ = 0*, for each* 1 ≤ r ≤ d.\n\n(ii) GNNθ is HFD if and only if for each sequence tj → ∞ there exist a subsequence tjk → ∞ and F∞\n(depending on the subsequence) s.t. F(tjk\n)\n∥F(tjk\n)∥ → F∞ *satisfying* ∆f r∞ = λn−1f r∞*, for each* 1 ≤ r ≤ d.\n\nProof. (i) Since ∆f r(t) → 0 for each 1 ≤ r ≤ d if and only if (Id ⊗ ∆)vec(F(t)) → 0, we conclude that the dynamics is LFD if and only if (Id ⊗ ∆)\nvec(F(t))\n∥F(t)∥ → 0 due to (i) in Lemma B.1. Consider a sequence tj → ∞. Since vec(F(tj ))/∥F(tj )∥ is a bounded sequence we can extract a converging subsequence tjk\n:\nvec(F(tjk\n))/∥F(tjk\n)∥ → vec(F∞). If the dynamics is LFD, then (Id ⊗ ∆)\nvec(F(tjk\n))\n∥F(tjk\n)∥ → 0 and hence we conclude that vec(F∞) ∈ ker(Id ⊗ ∆). Conversely, assume that for any sequence tj → ∞ there exists a subsequence tjk and F∞ such that F(tjk\n)\n∥F(tjk\n)∥ → F∞ satisfying ∆f r∞ = 0, for each 1 ≤ r ≤ d. If for a contradiction we had ε > 0 and tj → ∞ such that E\nDir(F(tj )/∥F(tj )∥ ≥ ε - for j large enough - then by (i)\nin Lemma B.1 there exist 1 ≤ r ≤ d, ℓ > 0 and a subsequence tjk satisfying\n\n$$|\\langle\\left(\\frac{\\operatorname{vec}(\\mathbf{F}(t_{j_{k}}))}{\\|\\mathbf{F}(t_{j_{k}})\\|}\\right),\\mathbf{e}_{r}\\otimes\\phi_{\\ell}\\rangle|>\\delta(\\varepsilon)>0,$$\n$\\square$\nmeaning that there is no subsequence of {tjk\n} s.t. (Id⊗∆)vec(F(tjk\n))/∥F(tjk\n)∥ → 0, providing a contradiction.\n\n(ii) This is equivalent to (ii) in Lemma B.1.\n\nRemark. We note that in Lemma B.2 an LFD *dynamics does not necessarily mean that the normalized* solution converges to the kernel of Id ⊗ ∆ - i.e. one in general has always to pass to subsequences. Indeed, we can consider the simple example t 7→ vec(F(t)) := cos(t)er¯ ⊗ ϕ0, for some 1 ≤ r¯ ≤ d*, which satisfies*\n∆f r(t) = 0 for each r*, but it is not a convergent function due to its oscillatory nature. Same argument* applies to HFD.\n\nWe will now show that (8) can lead to a HFD dynamics. To this end, we assume that Ω = W˜ = 0 so that (8)\nbecomes F˙ (t) = AF(t)W. According to (9) the negative eigenvalues of W lead to repulsion. We show that the latter can induce HFD dynamics as per Definition 4.2. We let P\nρ−\nW be the orthogonal projection into the eigenspace of W ⊗ A associated with the eigenvalue ρ− := |µ0|(λn−1 − 1). We recall that µ0 is the most negative eigenvalue of W, while µd−1 is the most positive eigenvalue of W. We define ϵHFD by:\nϵHFD := min{ρ− − µd−1, |µ0|gap(λn−1I − ∆), gap(|µ0|I + W)(λn−1 − 1)}.\n\nTheorem B.3. If ρ− > µd−1*, then* F˙ (t) = AF(t)W is HFD *for a.e.* F(0) *and indeed we have*\n\n$${\\mathcal{E}}^{\\mathrm{Dir}}(\\mathbf{F}(t))=e^{2t\\rho_{-}}\\left(\\lambda_{n-1}\\|P_{\\mathbf{W}}^{\\rho_{-}}\\mathbf{F}(0)\\|^{2}+{\\mathcal{O}}(e^{-2t\\epsilon_{\\mathrm{HFD}}})\\right),\\quad t\\geq0,$$\n\nand F(t)/∥F(t)∥ *converges to* F∞ ∈ R\nn×d*such that* ∆f r∞ = λn−1f r∞, for 1 ≤ r ≤ d. If instead ρ− < µd−1 then the dynamics is LFD and F(t)/∥F(t)∥ *converges to* F∞ ∈ R\nn×dsuch that ∆f r∞ = 0*, for* 1 ≤ r ≤ d, exponentially fast.\n\nProof of *Theorem B.3.* Once we compute the spectrum of W ⊗ A via (20), we can write the solution as –\nrecall that A = In − ∆ so we can express the eigenvalues of A in terms of the eigenvalues of ∆:\n\n$$\\operatorname{vec}(\\mathbf{F}(t))=\\sum_{r,\\ell}e^{\\mu_{r}(1-\\lambda_{\\ell})t}c_{r,\\ell}(0)\\psi_{r}\\otimes\\phi_{\\ell},$$\n\nwith Wψr = µrψr, for 0 ≤ r ≤ d − 1, where {ψr}r is an orthonormal basis of eigenvectors in R\nd. We can then calculate the Dirichlet energy along the solution as\n\n$${\\mathcal{E}}^{\\mathrm{Dir}}(\\mathbf{F}(t))=\\langle\\mathrm{vec}(\\mathbf{F}(t)),(\\mathbf{I}_{d}\\otimes\\Delta)\\mathrm{vec}(\\mathbf{F}(t))\\rangle=\\sum_{r,\\ell}e^{2\\mu_{r}(1-\\lambda_{\\ell})t}c_{r,\\ell}^{2}(0)\\lambda_{\\ell}.$$\n\nWe now consider two cases:\n- If µr > 0, then µr(1 − λℓ) ≤ µd−1.\n\n- If µr < 0, then µr(1 − λℓ) ≤ |µ0|(λn−1 − 1) := ρ−, with eigenvectors ψr ⊗ ϕn−1 for each r s.t.\n\nWψr = µ0ψr - without loss of generality we can assume that λn−1 is a simple eigenvalue for ∆. In particular, if µr < 0 and µr(1 − λℓ) < ρ−, then\n\n$$\\mu_{r}(1-\\lambda_{\\ell})<\\operatorname*{max}\\{|\\mu_{0}|(\\lambda_{n-2}-1),|\\mu_{1}|(\\lambda_{n-1}-1)\\},$$\n\nwhere µ1 is the second most negative eigenvalue of W and λn−2 is the second largest eigenvalue of\n∆. In particular, we can write\n\n$$\\lambda_{n-2}=\\lambda_{n-1}-\\mbox{gap}(\\lambda_{n-1}{\\bf I}_{n}-\\mathbf{\\Delta}),\\quad|\\mu_{1}|=|\\mu_{0}|-\\mbox{gap}(|\\mu_{0}|{\\bf I}_{d}+{\\bf W}).\\tag{26}$$\n\nFrom (i) and (ii) we derive that if µr(1 − λℓ) ̸= ρ−, then\n\n$$\\mu_{r}(1-\\lambda_{\\ell})-\\rho_{-}<-\\min\\{\\rho_{-}-\\mu_{d-1},\\rho_{-}-|\\mu_{0}|(\\lambda_{n-2}-1),\\rho_{-}-|\\mu_{1}|(\\lambda_{n-1}-1)\\}\\tag{27}$$ $$=-\\min\\{\\rho_{-}-\\mu_{d-1},\\,|\\mu_{0}|\\text{sgn}(\\lambda_{n-1}\\mathbf{I}-\\mathbf{\\Delta}),\\,\\text{sgn}(|\\mu_{0}|\\mathbf{I}+\\mathbf{W})(\\lambda_{n-1}-1)\\}=-\\epsilon_{\\text{HFD}},$$\n\nwhere we have used (26). Accordingly, if ρ− > µd−1, then\n\n$$\\mathcal{E}^{\\mathrm{Dir}}(\\mathbf{F}(t))=e^{2t\\rho_{-}}\\left(\\lambda_{n-1}\\sum_{r:\\mu_{r}=\\mu_{0}}c_{r,\\lambda_{n-1}}^{2}(0)+\\sum_{r,\\ell:\\mu_{r}(1-\\lambda_{\\ell})\\neq\\rho_{-}}e^{2\\left(\\mu_{r}(1-\\lambda_{\\ell})-\\rho_{-}\\right)t}c_{r,\\ell}^{2}(0)\\right)$$ $$=e^{2t\\rho_{-}}\\left(\\lambda_{n-1}\\|P_{\\mathbf{W}}^{\\mathrm{ow}}\\mathbf{F}(0)\\|^{2}+\\mathcal{O}(e^{-2t_{\\mathrm{EFD}}})\\right).$$\n\nBy the same argument we can factor out the dominant term and derive the following limit for t → ∞ and for a.e. F(0) since P\nρ−\nW vec(F(0)) = 0 only if vec(F(0)) belongs to a lower dimensional subspace of R\nnd:\n\n$$\\frac{\\operatorname{vec}(\\mathbf{F}(t))}{\\operatorname{vec}(\\mathbf{F}(t))}=\\frac{P_{\\mathbf{W}}^{\\mathcal{O}}\\operatorname{vec}(\\mathbf{F}(0))+\\mathcal{O}(e^{-\\epsilon_{\\mathbf{W}\\mathbf{D}}t})((\\mathbf{I}-P_{\\mathbf{W}}^{\\mathcal{O}})\\operatorname{vec}(\\mathbf{F}(0)))}{\\left(\\|P_{\\mathbf{W}}^{\\mathcal{O}}\\operatorname{vec}(\\mathbf{F}(0))\\|^{2}+\\mathcal{O}(e^{-2\\epsilon_{\\mathbf{W}\\mathbf{D}}t})\\right)^{\\frac{1}{2}}}\\to\\frac{P_{\\mathbf{W}}^{\\mathcal{O}}\\operatorname{vec}(\\mathbf{F}(0))}{\\|P_{\\mathbf{W}}^{\\mathcal{O}}\\operatorname{vec}(\\mathbf{F}(0))\\|},$$\n\nwhere the latter is a unit vector vec(F∞) satisfying (Id ⊗ ∆)vec(F∞) = λn−1vec(F∞), which completes the proof for the HFD case.\n\nFor the opposite case the proof can be adapted without efforts as explicitly derived in the proof of Theorem 5.1.\n\nWe emphasize that the previous result includes Theorem 4.3 as a special case.\n\n## B.3 Comparison With Continuous Gnns: Details And Proofs\n\nComparison with some continuous GNN models. In contrast with Theorem 4.3, we show that three main *linearized* continuous GNN models are either *smoothing* or more generally LFD. The linearized PDE-GCND model Eliasof et al. (2021) corresponds to choosing W˜ = 0 and Ω = W = K(t)\n⊤K(t) in (8), for some time-dependent family t 7→ K(t) ∈ R\nd×d:\n\n$${\\dot{\\mathbf{F}}}_{\\mathrm{PDE-GCN_{D}}}(t)=-\\mathbf{\\Delta}\\mathbf{F}(t)\\mathbf{K}(t)^{\\top}\\mathbf{K}(t).$$\n\nThe CGNN model Xhonneux et al. (2020) can be derived from (8) by setting Ω = I − Ω˜ ,W = −W˜ = I:\n\n$$\\dot{\\mathbf{F}}_{\\mathrm{CGNN}}(t)=-\\Delta\\mathbf{F}(t)+\\mathbf{F}(t)\\tilde{\\mathbf{\\Omega}}+\\mathbf{F}(0).$$\n\nFinally, in linearized GRAND Chamberlain et al. (2021a) a row-stochastic matrix A(F(0)) is *learned* from the encoding via an attention mechanism and we have\n\n$${\\dot{\\mathbf{F}}}_{\\mathrm{GRAND}}(t)=-\\mathbf{\\Delta}_{\\mathrm{RW}}\\mathbf{F}(t)=-(\\mathbf{I}-\\mathbf{\\mathcal{A}}(\\mathbf{F}(0)))\\mathbf{F}(t).$$\n\nWe note that if A is not symmetric, then GRAND is not a gradient flow.\n\nTheorem B.4. PDE − GCND, CGNN and GRAND *satisfy the following:*\n(i) PDE − GCND *is a smoothing model:* E˙Dir(FPDE−GCND\n(t)) ≤ 0.\n\n(ii) *For a.e.* F(0) *it holds:* CGNN *is never* HFD *and if we remove the source term, then* E\nDir(FCGNN(t)/∥FCGNN(t)∥) ≤ e\n−gap(∆)t.\n\n(iii) If G *is connected with self-loops,* FGRAND(t) → µ as t → ∞*, with* µ r = mean(f r(0)), 1 ≤ r ≤ d.\n\nBy (ii) the source-free CGNN-evolution is LFD *independent of* Ω˜ . Moreover, by (iii), over-smoothing occurs for GRAND. On the other hand, Theorem 4.3 shows that the negative eigenvalues of W can make the source-free gradient flow in (8) HFD. Experiments in Appendix E confirm that the gradient flow model outperforms CGNN and GRAND on heterophilic graphs.\n\nWe prove the following result which covers Theorem 5.4.\n\nProof of *Theorem B.4.* We structure the proof by following the numeration in the statement.\n\n(i) From direct computation we find\n\n$$\\frac{d\\mathcal{E}^{\\text{Dir}}(\\mathbf{F}(t))}{dt}=\\frac{d}{dt}\\left(\\langle\\text{vec}(\\mathbf{F}(t)),(\\mathbf{I}_{d}\\otimes\\boldsymbol{\\Delta})\\text{vec}(\\mathbf{F}(t))\\rangle\\right)$$ $$=-\\langle\\text{vec}(\\mathbf{F}(t)),(\\mathbf{K}^{\\top}(t)\\mathbf{K}(t)\\otimes\\boldsymbol{\\Delta}^{2})\\text{vec}(\\mathbf{F}(t))\\rangle\\leq0,$$\n\nsince K⊤(t)K(t) ⊗ ∆2 ⪰ 0. Note that we have used that (A ⊗ B)(C ⊗ D) = AC ⊗ BD.\n\n(ii) We consider the dynamical system F˙ CGNN(t) = −∆F(t) + F(t)Ω˜ + F(0).\n\nWe can write vec(F(t)) = Pr,ℓ cr,ℓ(t)ϕ Ω˜\nr ⊗ ϕℓ, leading to the system\n\n$$\\dot{c}_{r,\\ell}(t)=(\\lambda_{r}^{\\tilde{\\Omega}}-\\lambda_{\\ell})c_{r,\\ell}(t)+c_{r,\\ell}(0),\\quad0\\leq\\ell\\leq n-1,\\ 1\\leq r\\leq d.$$\n\nWe can solve explicitly the system as\n\n$$\\begin{array}{l l}{{}}&{{c_{r,\\ell}(t)=c_{r,\\ell}(0)\\left(e^{(\\lambda_{r}^{\\Omega}-\\lambda_{\\ell})t}\\left(1+\\frac{1}{\\lambda_{r}^{\\Omega}-\\lambda_{\\ell}}\\right)-\\frac{1}{\\lambda_{r}^{\\Omega}-\\lambda_{\\ell}}\\right),\\quad\\mathrm{if~}\\lambda_{r}^{\\Omega}\\neq\\lambda_{\\ell}}}\\\\ {{}}&{{c_{r,\\ell}(t)=c_{r,\\ell}(0)(1+t),\\quad\\mathrm{otherwise.}}}\\end{array}$$\n\nWe see now that for a.e. F(0) the projection (Id ⊗ ϕλn−1\n(ϕλn−1\n)\n⊤)vec(F(t)) is never the dominant term. In fact, if there exists r s.t. λ Ω˜\nr ≥ λn−1, then λ Ω˜\nr − λℓ > λΩ˜\nr − λn−1, for any other non-maximal graph Laplacian eigenvalue. It follows that there is no Ω˜ s.t. the normalized solution maximizes the Rayleigh quotient of Id ⊗ ∆, proving that CGNN is never HFD.\n\nIf we have no source, then the CGNN equation becomes\n\n$$\\begin{array}{r l}{{\\dot{\\mathbf{F}}}(t)=-\\mathbf{\\Delta}\\mathbf{F}(t)+\\mathbf{F}(t){\\hat{\\mathbf{\\Omega}}}}&{{}\\Longleftrightarrow\\operatorname{vec}({\\dot{\\mathbf{F}}}(t))}\\end{array}$$\n\nF˙ (t) = −∆F(t) + F(t)Ω˜ ⇐⇒ vec(F˙ (t)) = (Ω˜ ⊕ (−∆))vec(F(t)),\nusing the Kronecker sum notation in (21). It follows that we can write the vectorized solution in the basis\n{ϕ Ω˜\nr ⊗ ϕℓ}r,ℓ as\n\n$$\\text{vec}(\\mathbf{F}(t))=e^{\\lambda_{t}^{\\mathbf{\\Omega}}t}\\left(\\sum_{r:\\lambda_{r}^{\\mathbf{\\Omega}}=\\lambda_{r}^{\\mathbf{\\Omega}}}c_{r,0}(0)\\phi_{r}^{\\mathbf{\\Omega}}\\otimes\\phi_{0}+\\mathcal{O}(e^{-\\text{sgn}(\\lambda_{r}^{\\mathbf{\\Omega}}\\mathbf{L}_{t}-\\Omega)t})\\sum_{r:\\lambda_{r}^{\\mathbf{\\Omega}}<\\lambda_{r}^{\\mathbf{\\Omega}}}c_{r,0}(0)\\phi_{r}^{\\mathbf{\\Omega}}\\otimes\\phi_{0}\\right)$$ $$\\quad+e^{\\lambda_{t}^{\\mathbf{\\Omega}}t}\\left(\\mathcal{O}(e^{-\\text{sgn}(\\mathbf{\\Delta})t})\\left(\\sum_{r,L>0}c_{r,\\ell}(0)\\phi_{r}^{\\mathbf{\\Omega}}\\otimes\\phi_{\\ell}\\right)\\right),$$\n$\\overline{\\phantom{\\rule{0.3em}{0ex}}}$\n$$\\left.t\\right)\\right)=$$\n\nmeaning that the dominant term is given by the lowest frequency component and in fact, if we normalize we find E\nDir(F(t)/∥F(t)∥) ≤ e\n−gap(∆)t.\n\n(iii) Finally, we consider the dynamical system induced by linear GRAND\n\n$$\\dot{\\mathbf{F}}_{\\mathrm{GRAND}}(t)=-\\mathbf{\\Delta}_{\\mathrm{RW}}\\mathbf{F}(t)=-(\\mathbf{I}-\\mathbf{\\mathcal{A}}(\\mathbf{F}(0)))\\mathbf{F}(t).$$\n\nSince we have no channel-mixing, without loss of generality we can assume that d = 1 - one can then extend the argument to any entry. We can use the Jordan form of A to write the solution of the GRAND dynamical system as\n\n$$\\mathbf{f}(t)=P\\mathrm{diag}(e^{J_{1}t},\\ldots,e^{J_{n}t})P^{-1}\\mathbf{f}(0),$$\n\nfor some invertible matrix P of eigenvectors, with\n\n$$e^{J_{k}t}=e^{-(1-\\lambda_{k}^{\\mathbf{A}})t}\\begin{pmatrix}1&t&\\cdots&{\\frac{t^{m_{k}-1}}{(m_{k}-1)!}}\\\\ &&&\\vdots\\\\ &&&\\vdots\\end{pmatrix}$$\n1\n ,\nwhere mk are the eigenvalue multiplicities. Since by assumption G is connected and augmented with self-loops, the row-stochastic attention matrix A computed in Chamberlain et al. (2021a) with softmax activation is regular, meaning that there exists m ∈ N such that (Am)ij > 0 for each entry (*i, j*). Accordingly, we can apply Perron Theorem to derive that any eigenvalue of A has real part smaller than one except the eigenvalue λA\n0 with multiplicity one, associated with the Perron eigenvector 1n. Accordingly, we find that each block e Jkt decays to zero as t → ∞ with the exception of the one e J0t associated with the Perron eigenvector. In particular, the projection of f0 over the Perron eigenvector is just µ1n, with µ the average of the feature initial condition. This completes the proof.\n\n## B.4 Propagating With The Laplacian\n\nIn this subsection we briefly review the special case of (8) where Ω = W, and comment on why we generally expect a framework where the propagation is governed by A to be more flexible than one with −∆. If Ω = W\nand we suppress the source term i.e. W˜ = 0, the gradient flow in (8) becomes\n\n$${\\dot{\\mathbf{F}}}(t)=-\\mathbf{\\Delta}\\mathbf{F}(t)\\mathbf{W}.$$\nF˙ (t) = −∆F(t)W. (28)\nWe note that once vectorized, the solution to the dynamical system can be written as\n\n$$\\operatorname{vec}(\\mathbf{F}(t))=\\sum_{r=0}^{d-1}\\sum_{\\ell=0}^{n-1}e^{-\\mu_{r}{\\lambda_{\\ell}}t}c_{r,\\ell}(0)\\psi_{r}\\otimes\\phi_{\\ell}.$$\n$$(28)^{\\frac{1}{2}}$$\n\nIn particular, we immediately deduce the following counterpart to Theorem 4.3:\nCorollary B.5. If spec(W) ∩ R− ̸= ∅*, then (28) is* HFD *for a.e.* F(0).\n\nDifferently from (8) the lowest frequency component is *always preserved independent of the spectrum of* W.\n\nThis means that the system cannot learn eigenvalues of W to either magnify or suppress the low-frequency projection. In contrast, this can be done if Ω = 0, or equivalently one replaces −∆ with A providing a further justification in terms of the interaction between graph spectrum and channel-mixing spectrum for why graph-convolutional models use the normalized adjacency rather than the Laplacian for propagating messages Kipf & Welling (2017).\n\n## B.5 Revisiting The Connection With The Manifold Case\n\nIn (18) a constant nontrivial metric h in R\ndleads to the mixing of the feature channels. We adapt this idea by considering a symmetric positive semi-definite H = W⊤W with W ∈ R\nd×d and using it to generalize E\nDir by suitably weighting the norm of the edge gradients as\n\n$$\\mathcal{E}_{\\mathbf{W}}^{\\text{Dir}}(\\mathbf{F}):=\\frac{1}{2}\\sum_{q,r=1}^{d}\\sum_{i}\\sum_{j:(i,j)\\in\\mathsf{E}}h_{qr}(\\nabla\\mathbf{f}^{q})_{ij}(\\nabla\\mathbf{f}^{r})_{ij}=\\frac{1}{2}\\sum_{(i,j)\\in\\mathsf{E}}\\|\\mathbf{W}(\\nabla\\mathbf{F})_{ij}\\|^{2}.\\tag{29}$$\n\nWe note the analogy with (18), where the sum over the nodes replaces the integration over the domain and the j-th derivative at some point i is replaced by the gradient along the edge (*i, j*) ∈ E. We generally treat W as *learnable weights* and study the gradient flow of E\nDir W :\n\n$$\\bar{\\bf F}(t)=-\\frac{1}{2}\\nabla_{\\bf F}{\\cal E}_{\\bf W}^{\\rm Dir}({\\bf F}(t))=-\\mathbf{\\Delta}{\\bf F}(t)\\,{\\bf W}^{\\top}\\,{\\bf W}.\\tag{30}$$\n\nWe see that (30) generalizes (3).\n\nProposition B.6. Let P\nker W *be the projection onto* ker(W⊤W)*. Equation (30) is smoothing since*\n\nE\nDir(F(t)) ≤ e\n−2tgap(W⊤W)gap(∆)∥F(0)∥\n$$\\operatorname{gap}(\\mathbf{\\Delta})\\left\\|\\mathbf{F}(0)\\right\\|^{2}+{\\mathcal{E}}^{1}$$\nDir((P\nker\nW ⊗ In)vec(F(0))), t ≥ 0.\n\nIn fact F(t) → F∞ *s.t.* ∃ ϕ∞ ∈ R\nd: for each i ∈ V *we have* (f∞)i =\nq di 2|E|ϕ∞ + P\nker W fi(0).\n\nProof of Proposition *B.6.* We can vectorize the gradient flow system in (30) and use the spectral characterization of W⊤W ⊗ ∆ in (20) to write the solution explicitly as\n\n$$\\operatorname{vec}(\\mathbf{F}(t))=\\sum_{r,\\ell}e^{-(\\mu_{r}\\lambda_{\\ell})t}c_{r,\\ell}(0)\\psi_{r}\\otimes\\phi_{\\ell},$$\n\nwhere {µr}r = spec(W⊤W) ⊂ R≥0 with associated basis of orthonormal eigenvectors given by {ψr}r. Then\n\nE Dir(F(t)) = ⟨vec(F(t)),(Id ⊗ ∆)vec(F(t))⟩ = X r,ℓ e −2t(µrλℓ)c 2 r,ℓ(0)λℓ =X r:µr=0,ℓ c 2 r,ℓ(0)λℓ +X r:µr>0,ℓ>0 c 2 r,ℓ(0)e −2t(µrλℓ)λℓ = E Dir((P ker W ⊗ In)vec(F(0))) + X r:µr>0,ℓ>0 c 2 r,ℓ(0)e −2t(µrλℓ)λℓ ≤ EDir((P ker W ⊗ In)vec(F(0))) + λn−1e −2tgap(W⊤W)gap(∆)∥F(0)∥ 2,\nwhere we recall that P\nker W is the projection onto ker(W⊤W) and that by convention the index ℓ = 0 is associated with the lowest graph frequency λ0 = 0 - by assumption G is connected. This proves that the dynamics is in fact smoothing. By the very same argument we find that\n\n${\\rm vec}({\\bf F}(t))\\rightarrow({\\bf I}_{d}\\otimes P^{\\rm ker}){\\rm vec}({\\bf F}(0))+(P^{\\rm ker}_{\\bf W}\\otimes{\\bf I}_{n}){\\rm vec}({\\bf F}(0)),\\quad t\\rightarrow\\infty$. \nwith P\nker the orthogonal projection onto ker∆ - the other terms decay exponentially to zero. We first focus on the first quantity, which we can write as\n\n$$(\\mathbf{I}_{d}\\otimes P^{\\mathrm{ker}})\\mathrm{vec}(\\mathbf{F}(0))=\\sum_{r}c_{r,0}(0)\\psi_{r}\\otimes\\phi_{0},$$\n\nwhich has matrix representation ϕ0ϕ\n⊤\n∞ ∈ R\nn×d with\n\n$$\\phi_{\\infty}:=\\sum_{r}c_{r,0}(0)\\psi_{r}.$$\n\nBy (23) we deduce that the i-th row of ϕ0ϕ\n⊤\n∞ ∈ R\nn×dis the d-dimensional vector q di 2|E|ϕ∞. We now focus\n\non the term\n$$(P_{\\mathbf{W}}^{\\mathrm{ker}}\\otimes\\mathbf{I}_{n})\\mathrm{vec}(\\mathbf{F}(0))=\\sum_{r:\\mu_{r}=0,j}c_{r,j}(0)\\psi_{r}\\otimes\\phi_{j}$$\nwhich has matrix representation Pr:µr=0,j cr,j (0)ϕj (ψr)\n⊤. In particular, the i-th row is given by\n\n$$\\sum_{r\\mathbf{w},r=0,i}c_{r,j}(0)(\\phi_{j})_{i}\\psi_{r}=P_{\\mathbf{w}}^{\\mathrm{ker}}\\mathbf{f}_{i}(0).$$\n$\\square$\nr:µr=0,j\nThis completes the proof of Proposition B.6.\n\nProposition B.6 implies that no weight matrix W in (30) can separate the limit embeddings F∞ of nodes with same degree and same input features. In particular, we have the following characterization:\n- Projections of the edge gradients (∇F)ij (0) ∈ R\ndinto the eigenvectors of W⊤W with positive eigenvalues *shrink* along the GNN and converge to zero exponentially fast as integration time (depth) increases.\n\n- Projections of the edge gradients (∇F)ij (0) ∈ R\ndinto the kernel of W⊤W stay *invariant*.\n\nIf W has a trivial kernel, then nodes with same degrees converge to the same representation hence incurring over-smoothing. Differently from Nt & Maehara (2019); Oono & Suzuki (2020); Cai & Wang (2020), oversmoothing occurs independently of the spectral radius of the 'channel-mixing' if its eigenvalues are *positive* –\neven for equations which lead to residual GNNs when discretized (Chen et al., 2018). According to Proposition B.6, we do not expect (30) to succeed on heterophilic graphs where *smoothing* processes are generally harmful. To deal with heterophily, one needs negative eigenvalues to generate repulsive forces among adjacent features.\n\n## C Proofs And Additional Details Of **Section 5**\n\nWe first explicitly report here the expansion of the discrete gradient flow in (10) after m layers to further highlight how this is not equivalent to a single linear layer with a message passing matrix A\nm as for SGCN\nWu et al. (2019). For simplicity we suppress the source term.\n\n$${\\bf F}(t+\\tau)={\\bf F}(t)+\\tau\\left(-{\\bf F}(t)\\mathbf{\\Omega}+{\\bf A}{\\bf F}(t){\\bf W}\\right)$$ vec(${\\bf F}(t+\\tau)$) $$=({\\bf I}_{nd}+\\tau\\left(-{\\bf\\Omega}\\otimes{\\bf I}_{n}+{\\bf W}\\otimes{\\bf A}\\right))\\,{\\rm vec(F}(t))$$ vec(${\\bf F}(m\\tau)$) $$=\\sum_{k=0}^{m}\\left(\\matrix{m\\cr k\\cr}\\tau^{k}\\left(-\\mathbf{\\Omega}\\otimes{\\bf I}_{n}+{\\bf W}\\otimes{\\bf A}\\right)^{k}{\\rm vec(F}(0)\\right)$$ (31)\nand we see how the message passing matrix A actually enters the expansion after m layers with each power 0 ≤ k ≤ m. This is not surprising, given that *we are discretizing a linear dynamical system, meaning that we* are approximating an exponential matrix.\n\n## C.1 From Energy To Evolution Equations: Exact Expansion Of The Gnn Solutions\n\nWe first address the proof of the main result.\n\nProof of *Theorem 5.1.* We consider a linear dynamical system\n\n$$\\mathbf{F}(t+\\tau)=\\mathbf{F}(t)+\\tau\\mathbf{A}\\mathbf{F}(t)\\mathbf{W}$$\n\nwith W symmetric. We vectorize the system and rewrite it as\n\n$$r))=(\\mathbf{I}_{n d}+\\tau\\mathbf{W}\\otimes\\mathbf{A})\\mathrm{vec}(\\mathbf{F}(t))$$\nwhich in particular leads to\n$$\\operatorname{vec}(\\mathbf{F}(m\\tau))=(\\mathbf{I}_{n d}+\\tau\\mathbf{W}\\otimes\\mathbf{A})^{m}\\operatorname{vec}(\\mathbf{F}(0)).$$\n\nWe can then write explicitly the solution as\n\n$$\\operatorname*{vec}(\\mathbf{F}(m\\tau))=\\sum_{r,\\ell}\\left(1+\\tau\\mu_{r}(1-\\lambda_{\\ell})\\right)^{m}c_{r,\\ell}(0)\\psi_{r}\\otimes\\phi_{\\ell}.$$\n\nWe now verify that by assumption in (12) the dominant term of the solution is the projection into the eigenspace associated with the eigenvalue ρ− = |µ0|(λn−1 − 1). The following argument follows the same structure in the proof of Theorem B.3 with the extra condition given by the step-size. First, we note that for any r such that µr > 0, we have\n\n$$|1+\\tau\\rho_{-}|>|1+\\tau\\mu_{d-1}|\\geq|1+\\tau\\mu_{r}(1-\\lambda_{\\ell})|$$\n\nsince we required ρ− > µd−1 in (12). Conversely, if µr < 0, then\n\n$$|1+\\tau\\mu_{r}(1-\\lambda_{\\ell})|\\leq\\operatorname*{max}\\{|1+\\tau\\rho_{-}|,|1+\\tau\\mu_{0}|\\}$$\n\nAssume that τ |µ0| > 1, otherwise there is nothing to prove. Then |1 + τ ρ−| > τ |µ0| − 1 if and only if\n\n$$n-1\\,)<2,$$\n$\\downarrow$ $\\uparrow$ . \n$\\mathbf{T}$ = $\\mathbf{T}$. \nτ |µ0|(2 − λn−1) < 2, which is precisely the right inequality in (12). We can then argue exactly as in the proof of Theorem B.3 to derive that for each index r such that µr < 0 and µr ̸= µ0, then\n\n$$|1+\\tau\\mu_{\\tau}(1-\\lambda_{\\ell})|\\leq\\operatorname*{max}\\{|1+\\tau|\\mu_{1}|(\\lambda_{n-1}-1)|,|1+\\tau|\\mu_{0}|(\\lambda_{n-2}-1)|\\}$$\n\nwith µ1 and λn−2 defined in (26). We can then introduce\n\n$$\\delta_{\\rm HFD}:=\\max\\{\\mu_{d-1},\\rho_{-}-|\\mu_{0}|{\\rm gap}(\\lambda_{n-1}{\\bf I}-\\mathbf{\\Delta}),\\rho_{-}-(\\lambda_{n-1}-1){\\rm gap}(|\\mu_{0}|{\\bf I}+{\\bf W}),|\\mu_{0}|-\\frac{2}{\\tau}\\}\\tag{32}$$\nand conclude that $$\\mathbf{f}_{i}(m\\tau)=\\sum_{r,\\ell}\\left(1+\\tau\\rho_{r}(1-\\lambda_{\\ell})\\right)^{m}c_{r,\\ell}(0)\\phi_{\\ell}(i)\\psi_{r}$$ $$=(1+\\tau\\rho_{-})^{m}\\left(c_{0,n-1}(0)\\phi_{n-1}(i)\\cdot\\psi_{0}+\\mathcal{O}\\Big{(}\\Big{(}\\frac{1+\\tau\\delta_{\\mathrm{HFP}}}{1+\\tau\\rho_{-}}\\Big{)}^{m}\\Big{)}\\sum_{\\ell,r,\\mu_{r},(1-\\lambda_{\\ell})\\neq\\rho_{-}}c_{r,\\ell}(0)\\phi_{\\ell}(i)\\psi_{r}\\Big{)}\\right)$$ $$=(1+\\tau\\rho_{-})^{m}\\left(c_{0,n-1}(0)\\phi_{n-1}(i)\\cdot\\psi_{0}+\\mathcal{O}\\left(\\delta^{m}\\right)\\right),$$\n$$=\\left(1+\\tau\\rho_{-}\\right)^{m}\\left(c_{0,n-1}(0)\\phi_{n-1}(i)\\cdot\\psi_{0}+\\mathcal{O}\\left(\\delta^{m}\\right)\\right),$$\n\nwhich completes the proof of (13).\n\nConversely, if ρ− < µd−1, then the projection onto the eigenspace spanned by ψd−1 ⊗ ϕ0 is dominating the dynamics with exponential growth (1 + τµd−1(1 + 0))m. We can then adapt the very same argument above by factoring out the dominating term once we note that due to the choice of symmetric normalized Laplacian\n∆, we have ϕ0(i) = pdi/2|E|, which then yields (14).\n\nWe can now also address the proof of Corollary 5.2.\n\nProof of Corollary *5.2.* Once we have the node-wise expansion we can simply compute the Rayleigh quotient of Id⊗∆. We report the explicit details for the HFD case since the argument for LFD extends without relevant modifications. Using (11), we can compute the Dirichlet energy along a solution of F(t+τ ) = F(t)+τAF(t)W\nsatisfying (12) by\n\nE Dir(F(mτ )) = X r,ℓ (1 + τµr(1 − λℓ))2m c 2 r,ℓ(0)λℓ = (1 + τ ρ−) 2mλn−1X r:µr=µ0 c 2 r,λn−1 (0) + O 1 + τ δHFD 1 + τ ρ− 2m X ℓ,r:µr(1−λℓ)̸=ρ− c 2 r,ℓ(0)λℓ = (1 + τ ρ−) 2mλn−1∥P ρ− W F(0)∥ 2 + O 1 + τ δHFD 1 + τ ρ− 2m,\n$$\\square$$\nwhere P\nρ−\nW is the orthogonal projector onto the eigenspace associated with the eigenvalue ρ− = |µ0|(λn−1 −1).\n\nIn particular, since vec(F(mτ )) = (1 + τ ρ−)\nmP\nρ−\nW vec(F(0)) + O(δ\n\n$$(\\delta^{m})\\;,$$\n\nwe find that the dynamics is HFD with vec(F(t))/∥vec(F(t))∥ converging to the unit projection of the initial projection by P\nρ−\nW provided that such projection is not zero, which is satisfied for a.e. initial condition F(0).\n\nProof of *Theorem 5.3.* If we drop the residual connection and simply consider F(t + τ ) = τAF(t)W, then\n\n$$\\operatorname{vec}(\\mathbf{F}(m\\tau))=(\\tau\\mathbf{W}\\otimes\\mathbf{A})^{m}\\mathrm{vec}(\\mathbf{F}(0)).$$\n\nSince G is not bipartite, the Laplacian spectral radius satisfies λn−1 < 2. Therefore, for each pair of indices\n(*r, ℓ*) we have the following bound:\n\n$$|\\mu_{r}(1-\\lambda_{\\ell})|\\leq\\operatorname*{max}\\{\\mu_{d-1},|\\mu_{0}|\\},$$\n$$\\square$$\n\nand the inequality becomes strict if ℓ > 0, i.e. λℓ > 0. The eigenvalues µd−1 and µ0 are attained along the eigenvectors ψd−1 ⊗ ϕ0 and ψ0 ⊗ ϕ0 respectively. Accordingly, the dominant terms of the evolution lie in the kernel of Id ⊗ ∆, meaning that for any F0 with non-zero projection in ker(Id ⊗ ∆) - which is satisfied by all initial conditions except those belonging to a lower dimensional subspace - the dynamics is LFD. In fact, without loss of generality assume that |µ0| > µd−1, then\n\n$$\\mathrm{vec}(\\mathbf{F}(m\\tau))=|\\mu_{0}|^{m}\\sum_{r:\\mu_{r}=\\mu_{0}}(-1)^{m}c_{r,0}(0)\\psi_{0}\\otimes\\phi_{0}$$ $$\\quad+|\\mu_{0}|^{m}\\Big{(}\\mathcal{O}(\\varphi(m))\\big{(}\\mathbf{I}_{nd}-\\sum_{r:\\mu_{r}=\\mu_{0}}(\\psi_{0}\\otimes\\phi_{0})(\\psi_{0}\\otimes\\phi_{0})^{\\top}\\Big{)}\\mathrm{vec}(\\mathbf{F}(0))\\Big{)},$$\n\nwith φ(m) → 0 as m → ∞, which completes the proof.\n\nGradient flow as spectral GNNs. We finally discuss (10) from the perspective of spectral GNNs as in Balcilar et al. (2020). Let us assume that W˜ = 0, Ω = 0. If we let ∆ = Ψdiag(µ)Ψ⊤ be the eigendecomposition of the graph Laplacian and {µr} be the spectrum of W with associated orthonormal basis of eigenvectors given by {ψr}, and we introduce z r(t) : V → R defined by z r i\n(t) = ⟨fi(t), ψr⟩, then we can rewrite the discretized gradient flow as\n\n$$\\mathbf{x}^{r}(t+\\tau)=\\mathbf{\\Psi}({\\bf I}+\\tau\\mu_{r}({\\bf I}-{\\bf\\Lambda}^{\\mathbf{\\Delta}}))\\mathbf{\\Psi}^{\\top}\\mathbf{x}^{r}(t)=\\mathbf{x}^{r}(t)+\\tau\\mu_{r}{\\bf\\Lambda}\\mathbf{x}^{r}(t),\\quad0\\leq r\\leq d-1.\\tag{33}$$\n\nAccordingly, for each projection into the r-th eigenvector of W, we have a spectral function in the graph frequency domain given by λ 7→ 1 + τµr(1 − λ). If µr > 0 we have a *low-pass* filter while if µr < 0 we have a high-pass filter. Moreover, we see that along the eigenvectors of W, if µr < 0 then the dynamics is equivalent to flipping the sign of the edge weights, which offers a direct comparison with methods proposed in Bo et al.\n\n(2021); Yan et al. (2021) where some 'attentive' mechanism is proposed to learn negative edge weights based on feature information.\n\nThe previous equation simply follows from\n\n$$z_{i}^{\\tau}(t+\\tau)=\\langle\\mathbf{f}_{i}(t+\\tau),\\psi_{r}\\rangle=\\langle\\mathbf{f}_{i}(t)+\\mathbf{W}(\\mathbf{Af}(t))_{i},\\psi_{r}\\rangle$$ $$=z_{i}^{\\tau}(t)+\\mu_{r}\\sum_{j}\\bar{a}_{ij}z_{j}^{\\tau}(t),$$\n\nwhich concludes the derivation of (33).\n\n## D Proofs And Additional Details Of **Section 6**\n\nProof of *Theorem 6.1.* First we check that if time is continuous, then Eθ in (6) is decreasing. We use the Kronecker product formalism to rewrite the gradient ∇FEθ(F) as a vector in R\nnd: explicitly, we get\n\n$${\\frac{1}{2}}\\nabla_{\\mathbf{F}}{\\mathcal{E}}_{\\theta}(\\mathbf{F})=(\\mathbf{\\Omega}\\otimes\\mathbf{I}_{n}-\\mathbf{W}\\otimes\\mathbf{A})\\mathrm{vec}(\\mathbf{F})+({\\tilde{\\mathbf{W}}}\\otimes\\mathbf{I}_{n})\\mathrm{vec}(\\mathbf{F}(0)).$$\n\nIt follows then that\n\n$$\\frac{1}{2}\\frac{d\\mathcal{E}_{\\theta}(\\mathbf{F}(t))}{dt}=\\frac{1}{2}\\left(\\nabla_{\\mathbf{F}}\\mathcal{E}_{\\theta}(\\mathbf{F}(t))\\right)^{\\top}\\mathrm{vec}(\\dot{\\mathbf{F}}(t))=$$ $$=\\left(\\frac{1}{2}\\nabla_{\\mathbf{F}}\\mathcal{E}_{\\theta}(\\mathbf{F}(t))\\right)^{\\top}\\sigma\\bigg{(}-\\frac{1}{2}\\nabla_{\\mathbf{F}}\\mathcal{E}_{\\theta}(\\mathbf{F}(t))\\bigg{)}.$$\n\nIf we introduce the notation Z(t) = −\n1 2∇FEθ(F(t), then we can rewrite the derivative as\n\n$${\\frac{d{\\mathcal{E}}_{\\theta}(\\mathbf{F}(t))}{d t}}=-\\mathbf{Z}(t)^{\\top}\\sigma(\\mathbf{Z}(t))=-\\sum_{\\alpha}\\mathbf{Z}(t)^{\\alpha}\\sigma(\\mathbf{Z}(t)^{\\alpha})\\leq0$$\n\nby assumption on σ. The discrete case follows similarly. Let us use the same notation as above so we can write F(t + τ ) = F(t) + τσ(Z(t)), with Z(t) = −\n1 2∇FEθ(F(t)).\n\nEθ(F(t + τ )) = ⟨vec(F(t + τ )),(Ω ⊗ In − W ⊗ A)vec(F(t + τ )) + 2(W˜ ⊗ In)vec(F(0))⟩ = ⟨vec(F(t)) + τσ(Z(t)),(Ω ⊗ In − W ⊗ A)vec(F(t + τ )) + 2(W˜ ⊗ In)vec(F(0))⟩ = ⟨vec(F(t)) + τσ(Z(t)),(Ω ⊗ In − W ⊗ A) (vec(F(t) + τσ(Z(t))) + 2(W˜ ⊗ In)vec(F(0))⟩ = ⟨vec(F(t)),(Ω ⊗ In − W ⊗ A)vec(F(t)) + 2(W˜ ⊗ In)vec(F(0))⟩ + τ ⟨vec(F(t)),(Ω ⊗ In − W ⊗ A)σ(Z(t))⟩ + τ ⟨σ(Z(t)),(Ω ⊗ In − W ⊗ A)vec(F(t) + 2(W˜ ⊗ In)vec(F(0))⟩ + τ 2⟨σ(Z(t)),(Ω ⊗ In − W ⊗ A)σ (Z(t))⟩.\nBy using that Ω ⊗ In − W ⊗ A is symmetric, we find that\n\nEθ(F(t + τ )) = Eθ(F(t)) + 2τ ⟨σ(Z(t),(Ω ⊗ In − W ⊗ A)vec(F(t) + (W˜ ⊗ In)vec(F(0))⟩ + τ 2⟨ 1 τ (F(t + τ ) − F(t)),(Ω ⊗ In − W ⊗ A) 1 τ (F(t + τ ) − F(t))⟩ = Eθ(F(t)) − 2τ ⟨σ(Z(t)), Z(t)⟩ + ⟨F(t + τ ) − F(t),(Ω ⊗ In − W ⊗ A)(F(t + τ ) − F(t))⟩ ≤ Eθ(F(t)) + C+∥F(t + τ ) − F(t))∥ 2,\nwhere again we have used that Z\n⊤σ(Z) ≥ 0. This completes the proof.\n\nTo further support the principle that the effects induced by W are similar even in this non-linear setting, we consider a simplified scenario.\n\nLemma D.1. *If we choose* Ω = W = diag(ω) *with* ω r ≤ 0 for 0 ≤ r ≤ d − 1 and W˜ = 0 *i.e.* t 7→ F(t)\nsolves the dynamical system F˙ (t) = σ (−∆F(t)diag(ω)),\nwith xσ(x) ≥ 0*, then the standard graph Dirichlet energy satisfies*\n\n$\\overline{\\overline{\\phantom{\\rule{0.3em}{0ex}}}}$ 4. \n$$\\sigma\\left(-\\lambda\\right)$$\n$${\\frac{d{\\mathcal{E}}^{\\mathrm{Dir}}(\\mathbf{F}(t))}{d t}}\\geq0.$$\nProof. This again simply follows from directly computing the derivative:\n\n$$\\frac{d\\mathsf{E}^{\\text{Dir}}(\\mathbf{F}(t))}{dt}=\\frac{1}{2}\\frac{d}{dt}\\Big{(}\\sum_{r=1}^{d}\\sum_{(i,j)\\in\\mathsf{E}}\\Big{(}\\frac{f_{i}^{r}(t)}{\\sqrt{d_{i}}}-\\frac{f_{j}^{r}(t)}{\\sqrt{d_{j}}}\\Big{)}^{2}\\Big{)}$$ $$=\\sum_{r=1}^{d}\\sum_{i\\in\\mathsf{V}}(\\boldsymbol{\\Delta}\\mathbf{f}^{r})_{i}\\sigma(-\\omega^{r}(\\boldsymbol{\\Delta}\\mathbf{f}^{r})_{i})=\\sum_{r=1}^{d}\\sum_{i\\in\\mathsf{V}}(\\boldsymbol{\\Delta}\\mathbf{f}^{r})_{i}\\sigma(|\\omega^{r}|(\\boldsymbol{\\Delta}\\mathbf{f}^{r})_{i})\\geq0.$$\n$\\square$\n$$\\left(34\\right)^{2}$$. \nImportant consequence: The previous Lemma implies that even with non-linear activations, negative eigenvalues of the channel-mixing induce repulsion and indeed the solution becomes less smooth as measured by the classical Dirichlet Energy increasing along the solution. Generalising this result to more arbitrary choices is not immediate and we reserve this for future work.\n\n## E Additional Details On Experiments E.1 Additional Results On Real-World Tasks\n\nGeneral details. The graph-convolutional models in (16) and (17) are implemented in PyTorch (Paszke et al., 2019), using PyTorch geometric Fey & Lenssen (2019) and torchdiffeq (Chen et al., 2018). Hyperparameters were tuned using wandb (Biewald, 2020) and random grid search. Experiments were run on AWS\np2.8xlarge machines, each with 8 Tesla V100-SXM2 GPUs.\n\nWe further improve our results on the node-classification tasks of by considering the following, more general, parameterizations. We consider an instance of the gradient-flow framework, termed GRAFF, of the form:\n\n$$\\mathrm{GRAFF}:\\quad\\mathbf{F}(t+\\tau)=\\mathbf{F}(t)+\\tau\\left(-\\mathbf{F}(t)\\mathrm{diag}(\\pmb{\\omega})+\\mathbf{AF}(t)\\mathbf{W}-\\beta\\mathbf{F}(0)\\right),$$\n\nwhere ω ∈ R\nd and β ∈ R, W is a *symmetric* d × d-matrix *shared* across layers and (node-wise) encoder and decoder are MLPs. We descibe different implementations of W below. In accordance with Theorem 6.1, we also consider a non-linear variant (termed GRAFFNL) as\n\n$${\\rm GRAFF}_{\\rm NL}:\\quad{\\bf F}(t+\\tau)={\\bf F}(t)+\\tau\\sigma\\left(-{\\bf F}(t){\\rm diag}(\\mathbf{\\omega})+{\\bf AF}(t){\\bf W}-\\beta{\\bf F}(0)\\right),\\tag{35}$$\n\nwith σ s.t. xσ(x) ≥ 0. We note that differently from (16) and (17), these models also allow for a source term controlled by β and a diagonal term acting on the state of a node, independent of the graph structure, controlled by ω.\n\nMethodology. In the experiments we rely on the following parameterizations. We implement encoder ψEN and ψDE as single linear layers or MLPs. We consider *diagonally-dominant* (DD) and *diagonal* (D) choices for the structure of W that offer explicit control over its spectrum. In the (DD)-case, we consider a W0 ∈ R\nd×d symmetric with zero diagonal and w ∈ R\nd defined by wα = qαPβ |W0αβ| + rα, and set W = diag(w) + W0.\n\nDue to the Gershgorin Theorem the eigenvalues of W belong to [wα −Pβ |W0αβ|, wα +Pβ |W0αβ|], so the model 'can' easily re-distribute mass in the spectrum of W via qα, rα. This generalizes the decomposition of W in Chen et al. (2020) providing a justification in terms of its spectrum. For (D) we take W to be diagonal.\n\nReal world experiments. In Table 3 we test GRAFF and GRAFFNL on the same datasets with varying homophily adopted in Section 7 (Sen et al., 2008; Rozemberczki et al., 2021; Pei et al., 2020). We use results provided in Yan et al. (2021, Table 1), which include GCNs models, GAT (Veličković et al., 2018), PairNorm\n(Zhao & Akoglu, 2019) and models designed for heterophily (GGCN (Yan et al., 2021), Geom-GCN (Pei et al.,\n2020), H2GCN (Zhu et al., 2020) and GPRGNN (Chien et al., 2021)). For Sheaf (Bodnar et al., 2022), a recent strong baseline with heterophily, we took the best performing variant (out of six) for each dataset. We include continuous baselines CGNN (Xhonneux et al., 2020) and GRAND (Chamberlain et al., 2021a) to corroborate Theorem 5.4. Training, validation and test splits are taken from Pei et al. (2020) for all datasets\n\n| Texas | Wisconsin | Cornell | Film | Squirrel | Chameleon | Citeseer | Pubmed | Cora | |\n|------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|\n| Hom level | 0.11 | 0.21 | 0.30 | 0.22 | 0.22 | 0.23 | 0.74 | 0.80 | 0.81 |\n| #Nodes | 183 | 251 | 183 | 7,600 | 5,201 | 2,277 | 3,327 | 18,717 | 2,708 |\n| #Edges | 295 | 466 | 280 | 26,752 | 198,493 | 31,421 | 4,676 | 44,327 | 5,278 |\n| #Classes | 5 | 5 | 5 | 5 | 5 | 5 | 7 | 3 | 6 |\n| GGCN | 84.86 ± 4.55 | 86.86 ± 3.29 | 85.68 ± 6.63 | 37.54 ± 1.56 | 55.17 ± 1.58 | 71.14 ± 1.84 | 77.14 ± 1.45 | 89.15 ± 0.37 | 87.95 ± 1.05 |\n| GPRGNN | 78.38 ± 4.36 | 82.94 ± 4.21 | 80.27 ± 8.11 | 34.63 ± 1.22 | 31.61 ± 1.24 | 46.58 ± 1.71 | 77.13 ± 1.67 | 87.54 ± 0.38 | 87.95 ± 1.18 |\n| H2GCN | 84.86 ± 7.23 | 87.65 ± 4.98 | 82.70 ± 5.28 | 35.70 ± 1.00 | 36.48 ± 1.86 | 60.11 ± 2.15 | 77.11 ± 1.57 | 89.49 ± 0.38 | 87.87 ± 1.20 |\n| GCNII | 77.57 ± 3.83 | 80.39 ± 3.40 | 77.86 ± 3.79 | 37.44 ± 1.30 | 38.47 ± 1.58 | 63.86 ± 3.04 | 77.33 ± 1.48 | 90.15 ± 0.43 | 88.37 ± 1.25 |\n| Geom − GCN | 66.76 ± 2.72 | 64.51 ± 3.66 | 60.54 ± 3.67 | 31.59 ± 1.15 | 38.15 ± 0.92 | 60.00 ± 2.81 | 78.02 ± 1.15 | 89.95 ± 0.47 | 85.35 ± 1.57 |\n| PairNorm | 60.27 ± 4.34 | 48.43 ± 6.14 | 58.92 ± 3.15 | 27.40 ± 1.24 | 50.44 ± 2.04 | 62.74 ± 2.82 | 73.59 ± 1.47 | 87.53 ± 0.44 | 85.79 ± 1.01 |\n| GraphSAGE | 82.43 ± 6.14 | 81.18 ± 5.56 | 75.95 ± 5.01 | 34.23 ± 0.99 | 41.61 ± 0.74 | 58.73 ± 1.68 | 76.04 ± 1.30 | 88.45 ± 0.50 | 86.90 ± 1.04 |\n| GCN | 55.14 ± 5.16 | 51.76 ± 3.06 | 60.54 ± 5.30 | 27.32 ± 1.10 | 53.43 ± 2.01 | 64.82 ± 2.24 | 76.50 ± 1.36 | 88.42 ± 0.50 | 86.98 ± 1.27 |\n| GAT | 52.16 ± 6.63 | 49.41 ± 4.09 | 61.89 ± 5.05 | 27.44 ± 0.89 | 40.72 ± 1.55 | 60.26 ± 2.50 | 76.55 ± 1.23 | 87.30 ± 1.10 | 86.33 ± 0.48 |\n| MLP | 80.81 ± 4.75 | 85.29 ± 3.31 | 81.89 ± 6.40 | 36.53 ± 0.70 | 28.77 ± 1.56 | 46.21 ± 2.99 | 74.02 ± 1.90 | 75.69 ± 2.00 | 87.16 ± 0.37 |\n| CGNN | 71.35 ± 4.05 | 74.31 ± 7.26 | 66.22 ± 7.69 | 35.95 ± 0.86 | 29.24 ± 1.09 | 46.89 ± 1.66 | 76.91 ± 1.81 | 87.70 ± 0.49 | 87.10 ± 1.35 |\n| GRAND | 75.68 ± 7.25 | 79.41 ± 3.64 | 82.16 ± 7.09 | 35.62 ± 1.01 | 40.05 ± 1.50 | 54.67 ± 2.54 | 76.46 ± 1.77 | 89.02 ± 0.51 | 87.36 ± 0.96 |\n| Sheaf(max) | 85.95 ± 5.51 | 89.41 ± 4.74 | 84.86 ± 4.71 | 37.81 ± 1.15 | 56.34 ± 1.32 | 68.04 ± 1.58 | 76.70 ± 1.57 | 89.49 ± 0.40 | 86.90 ± 1.13 |\n| GRAFF | 88.38 ± 4.53 | 88.83 ± 3.29 | 84.05 ± 6.10 | 37.11 ± 1.08 | 58.72 ± 0.84 | 71.08 ± 1.75 | 77.30 ± 1.85 | 90.04 ± 0.41 | 88.01 ± 1.03 |\n| GRAFFNL | 86.49 ± 4.84 | 87.26 ± 2.52 | 77.30 ± 3.24 | 35.96 ± 0.95 | 59.01 ± 1.31 | 71.38 ± 1.47 | 76.81 ± 1.12 | 89.81 ± 0.50 | 87.81 ± 1.13 |\n\n| dataset | Texas | Wisconsin | Cornell | Film | Squirrel | Chameleon | Citeseer | Pubmed | Cora |\n|----------------------|---------|-------------|-----------|--------|------------|-------------|------------|----------|--------|\n| Homophily | 0.11 | 0.21 | 0.3 | 0.22 | 0.22 | 0.23 | 0.74 | 0.8 | 0.81 |\n| Depth | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 4 |\n| µd−1 | 2.07 | 1.33 | 1.37 | 3.33 | 2.48 | 2.66 | 4.1 | 9.47 | 2.49 |\n| µ0 | -3.07 | -0.82 | -0.76 | -1.45 | -3.61 | -3.72 | -0.96 | -3.89 | -0.26 |\n| mean({µr}) | -0.02 | 0.02 | 0.02 | 0.06 | -0.1 | -0.1 | 0.13 | 0.15 | 0.11 |\n| E Dir(F(0)) | 13171 | 5367 | 3687 | 7460 | 68811 | 25772 | 4749 | 5737 | 1226 |\n| E Dir(F(m)) | 42674 | 6767 | 4932 | 12224 | 828532 | 398302 | 37168 | 119518 | 33700 |\n| E Dir(F(0))/∥F(0)∥ 2 | 1.68 | 1.33 | 1.35 | 1.1 | 1.77 | 1.86 | 0.55 | 0.82 | 0.57 |\n| E Dir(F(m))/∥F(m)∥ 2 | 1.67 | 0.71 | 0.91 | 0.71 | 1.47 | 1.21 | 0.16 | 0.29 | 0.11 |\n| Test accuracy (mean) | 81.08 | 81.96 | 72.43 | 34.16 | 52.48 | 68.99 | 76.43 | 88.56 | 87.4 |\n| Test accuracy (std) | 4.52 | 4.45 | 7.13 | 0.9 | 1.8 | 1.7 | 1.53 | 0.51 | 0.97 |\n\nTable 3: Node-classification results. Top three models are coloured by First, Second, Third Table 4: SGCNgf learned spectrum\nfor comparison. For the Cornell dataset we take the adjacency file from before the July 2022 update of Pei et al. (2020).\n\nResults. GRAFF and GRAFFNL are both versions of graph convolutions with stronger 'inductive bias' given by the energy Eθ *decreasing along the solution*; in fact, we can recover them from graph convolutions by simply requiring that the channel-mixing is *symmetric and shared across layers*. Nonetheless they achieve competitive results on all datasets often outperforming slower and more complex models. As noted in Section 7, the improved performance is mainly due to the presence of a residual connection at each layer, which simply enables the channel-mixing matrices to induce either attraction or repulsion, via their eigenvalues.\n\nA word of caution though: these tasks are very sensitive to the hyperparameter tuning, making them generally not ideal candidates to probe different models. While they suffice for our purposes, given the theoretical nature of our work, we believe them to be at times very poor indicators of the 'heterophilic problem'.\n\n## E.2 Details On Hyperparameters\n\nUsing wandb Biewald (2020) we performed a random grid search with uniform sampling of the continuous variables. We provide the hyperparameters that achieved the best results from the random grid search in Table 5. Input dropout and dropout are the rates applied to the encoder/decoder respectively with no dropout applied in the ODE block. Further hyperparameters decide the use of non-linearities, batch normalisation, parameter vector ω and source term multiplier β which are specified in the code.\n\n| w_style | lr | decay | dropout | input_dropout | hidden | time | step_size | |\n|-----------|----------|---------|-----------|-----------------|----------|--------|-------------|------|\n| chameleon | diag_dom | 0.0050 | 0.0005 | 0.36 | 0.48 | 64 | 3.33 | 1 |\n| squirrel | diag_dom | 0.0065 | 0.0009 | 0.17 | 0.35 | 128 | 2.87 | 1 |\n| texas | diag_dom | 0.0041 | 0.0354 | 0.33 | 0.39 | 64 | 0.6 | 0.5 |\n| wisconsin | diag | 0.0029 | 0.0318 | 0.37 | 0.37 | 64 | 2.1 | 0.5 |\n| cornell | diag | 0.0021 | 0.0184 | 0.30 | 0.44 | 64 | 2.0 | 1 |\n| film | diag | 0.0026 | 0.0130 | 0.48 | 0.42 | 64 | 1.5 | 1 |\n| Cora | diag | 0.0026 | 0.0413 | 0.34 | 0.53 | 64 | 3.0 | 0.25 |\n| Citeseer | diag | 0.0001 | 0.0274 | 0.22 | 0.51 | 64 | 2.0 | 0.5 |\n| Pubmed | diag | 0.0039 | 0.0003 | 0.42 | 0.41 | 64 | 2.6 | 0.5 |\n\nTable 5: Selected hyperparameters for real-world datasets\n\n## E.3 Additional Details On The Spectral Experiments\n\nWe report additional details on the spectrum of WS in (16) and on the normalized Dirichlet energy in Table 4. We also report the distribution of the eigenvalues of the learned channel-mixing matrix WS in Figure 4.\n\n\n\nFigure 4: Histogram of SGCNgf learned spectrum"},"reviews":{"kind":"string","value":"Review 1:\nSummary: This paper proposes an analysis and re-evaluation of basic MPNNs such as GCN via an energetic perspective. The paper shows that by setting the appropriate weights in models like GCN, they can be viewed as a gradient flow that is either smoothing (a commonly known observation), but also over-sharpening (a surprising result shown in this paper). \n\nThus, the authors suggest that despite the common belief that simple MPNNs like GCN are bound to always over-smooth, such issues can be avoided using their proposed parameterization.\n\nThe authors present several experiments that show the effectiveness of viewing MPNNs as gradient flows, resulting in improved performance compared to simple, known models, like GCN.\n\nStrengths and Weaknesses: **Strengths**:\n* The paper is highly organized and very well written.\n* It seems that the general concept proposed here is novel and can also be applied to other MPNNs.\n* The paper is theoretically rigorous, as well as practically meaningful, because of the various experiments that show the benefit of the interpretation of MPNNs as gradient flows and the inspection of their energy.\n\n**Weaknesses / Questions**:\n* Right before the end of page 1, the authors say that \"Several works have gathered evidence that graph-convolutional models seem to struggle on some heterophilic\ngraphs\". Can you please expand on that point? Is it not true for all or most of heterophilic graphs? Are there examples of heterophilic graphs where basic MPNNs (e.g. GCN) do not struggle?\n* With respect to signed message passing architectures, the authors may consider adding a reference to \"Magnet: A neural network for directed graphs\"\n* With respect to repulsive and attractive forces, the authors may consider adding a reference to \"ACMP: Allen-Cahn Message Passing for Graph Neural Networks with Particle Phase Transition\"\n* With respect to learning low/high pass filters, the authors may consider adding references to \"Revisiting heterophily for graph neural networks\", \"Improving Graph Neural Networks with Learnable Propagation Operators\"\n* The authors call subsection 4.1 \"non-parametric\" and to 4.2 \"parametric\". I think that the terms used here may be confusing to some readers because usually the notion of \"parametric models\" is used for closed-form models (e.g., exp(-theta) where theta is a scalar parameter). Perhaps it would be useful to distinguish between the two?\n* In equation 5 the authors add a source term but I am not sure it is used later. Is it a beneficial term? Also, how is the source term here different than the residual connection made in \"Simple and Deep Graph Convolutional Networks\", and the source term in \"GRAND++: Graph Neural Diffusion with A Source Term\"?\n* If I understand correctly, the authors use the same weights W throughout all time steps. May I ask what is the influence on the theoretical results if W can change at each time step, and how may it influence the empirical results?\n* In figure 2, what is cornell_old ? is it just a typo?\n* Is it safe to assume there always exists some desired energy of an optimal state of an MPNN? If so, can such energy always be achieved using the proposed method?\n* In the appendix, the authors define the \"GRAFF\" model, but it is not shown later in table 3.\n\nRequested Changes: I have read the whole paper including the appendix. I enjoyed reading it, and I think that addressing my questions in my review will improve the paper.\n\nBroader Impact Concerns: None\n\n==================================================\n\nReview 2:\nSummary: This paper explores the expressivity power of GNNs. Specifically, it answers whether simple message-passing GNNs can show a behaviour different from that of oversmoothing, in which the node features start to homogenize as more layers are added to the GNN, irrespectively of the initial feature values. By interpreting time-continuous GNNs are gradient flows of an energy function, the authors show that certain simple GNNs can indeed show oversharpening, an effect contrary to oversmoothing. Moreover, the authors extend their results to more complex GNNs with non-linear activation functions, and empirically validate the theoretical findings, showing that a properly parametrized simple GNN can outperform tailored oversmoothing solutions, casting doubts on commonly-adopted benchmarks and solutions in the literature. \n\nStrengths and Weaknesses: **Strengths**\n- The paper's results are relevant for the community, and challenge widespread beliefs by the GNN community.\n- The story of the paper is rather intuitive, going from gradient flow for time-continuous GNNs, to discrete-time GNNs (layers) and non-linearities.\n- The paper does a good job at providing explanations and intuitions for the technical results. \n- Empirical results corroborate the efficacy of the parametrization studied.\n\n**Weaknesses**\n- I find confusing having Q1 and Q2 in the manuscript, as to my eyes they are intimate related and Q2 may imply Q1.\n- The paper tries to be really general, yet I feel some parts are not clear, and it would benefit a lot from simplifying the formulations and extending the main explanations of the paper. For example:\n - The formulation in Eq. (1) is quite general, yet all results in the main paper assume a simplified version of it, unless I missed something.\n - Point (i) at the beginning of page 7 is not immediately obvious to me.\n - Non-trivial derivations, e.g. Eq. (6), are not written in the appendix nor main paper.\n- I feel that the assumption of the adjacency matrix A being symmetric can be easily missed/forgotten, and more emphasis should be put into it. An easy way to fix it, would be to focus on _undirected_ graphs, whose directed version is a symmetric adjacency matrices.\n\n_Disclaimer:_ I have not checked the correctness of the proofs in the appendix.\n\nRequested Changes: I think this is good work, and that the results are of interest for the community. In its current state, I wouldn't require any major changes for recommending its acceptance.\n\nHowever, I do strongly believe that it would be extremely beneficial for the possible outreach of this work to remove extra content that is not fully utilized (like additional unused parametrizations), in pursuit of strengthening the message and focus of the manuscript. As it stands right now, I think it is easily accessible to only a small fraction of the community. \n\nBroader Impact Concerns: I don't think there are any broader impact concerns for this work.\n\n==================================================\n\nReview 3:\nSummary: This paper analyzes the behavior of node features of a GNN represented by a gradient flow or its discretized version. Continuous-time GNN is defined as a gradient flow by parametric energy determined by node features (Eq. (6)). Then, this paper introduces the concepts of low-frequency-dominant (LFD) and high-frequency-dominant (HFD), which quantitatively evaluate the smoothness of the node representation using the normalized energy. This paper shows that a GNN can be both LFD and HFD depending on the spectra of weight matrices. Furthermore, this paper introduces a residual GNN as a discretized version of the continuous-time GNN and shows similar results. Also, it is shown that the discretized GNN is always LFD when there is no residual connection. The proposed model was applied to the node classification task on various heterophily datasets to verify its consistency with the theoretical analysis.\n\nStrengths and Weaknesses: **Strengths**\n\n- Based on normalized Dirichlet energy, over-smoothing and its opposite state (called over-shaping in this paper) are defined in a unified manner (Definition 4.2.)\n- In the previous analysis of over-smoothing, only the scale of the GNN (specifically, the singular value) was considered. This paper analyzes more precisely the role of the weight matrix and finds that the negative eigenvalue is helpful for avoiding over-smoothing (Theorem 4.3)\n- The effect of residual connection on over-smoothing is further clarified (Theorem 5.3).\n\n\n**Weaknesses**\n\n- This paper first introduces the general form of the model as Equations (1) or (8). Equation (1) claims its usefulness because it includes GCN, GraphSAGE, GCNII, and GIN with residual connections as special cases. However, only its special cases are considered in most analyses. Specifically, Section 4.2.2 considers the case $\\boldsymbol{\\mathrm{\\tilde{W}}}_t=0$ and $\\boldsymbol{\\mathsf{A}}=D^{-1/2}AD^{-1/2}$. Also, Section 4.3, Theorem 5.1, and Corollary 5.2 further impose the constraint $\\boldsymbol{\\Omega}_t=0$. Among the above examples above, only GCN satisfies this parametrization. The exception is Proposition 4.1, which used the general form of (8). However, since it only provides the necessary conditions for defining a GNN as a gradient flow, I think it may not answer why we should consider the ODE of the form (1) or (8).\n- The energy $\\mathcal{E}_\\theta$ is the Diriclet energy when $\\boldsymbol{\\mathrm{\\Omega}}=\\boldsymbol{\\mathrm{W}}=\\boldsymbol{\\mathrm{I}}$ (and $\\boldsymbol{\\tilde{\\mathrm{W}}}=0$) (Equation (5)). Since the model analyzed in Theorem 4.3 assumes $\\boldsymbol{\\mathrm{\\Omega}}=\\boldsymbol{\\mathrm{W}}=0$, the model is not a gradient flow of Dirichlet energy (or at least, it is not shown explicitly). However, LFD and HFD are defined using normalized Dirichlet energy. So, the model and the LFD/HFD definitions use different energies. This fact raises questions about how the theoretical analysis (e.g., Theorem 4.3) uses the fact that the proposed model is defined as a gradient flow with $\\mathcal{E}_\\theta$.\n\nRequested Changes: P.5: As explained at the beginning of Section 4, the main characteristic of gradient flow is that the energy decreases monotonically along the solution. However, this paper does not use such a property of gradient flow. Instead, the normalized energy converges to the maximum possible value under certain conditions in Theorem 4.3. I wonder how the theoretical analysis utilizes the benefit of gradient flows, such as the monotonically decreasing property.\n\nP.6: The *only if* part of Proposition 4.1 does not hold mathematically strictly speaking (i.e., if Equation (8) is the gradient flow of energy Equation (6), then, $\\boldsymbol{\\Omega}$ and $\\boldsymbol{\\mathrm{W}}$ are symmetric.) For example, in the extreme case where the graph has no edges (i.e., $\\boldsymbol{\\mathsf{A}}=0$), no constraint is imposed on $\\boldsymbol{\\mathrm{W}}$, which does not have to be symmetric in particular. Even if the only-if part of this proposition is not satisfied, it does not cause critical problems in constructing the proposed model. However, a more elaborate discussion is needed to make it a mathematically correct statement.\n\nP.8:\n> [...] (ii) graph convolutions can avoid over-smoothing through the negative eigenvalues of the channel-mixing by incurring the opposite behaviour of over-shaping [...]\n\nThis sentence implicitly assumes that $\\boldsymbol{\\mathrm{W}}$ has at least one negative eigenvalue when (12) holds. It is certainly true (if $\\mu_0\\geq 0$, then $\\lambda_{n-1}\\leq 2$ implies $\\mu_{d-1}/(\\lambda_{n-1}-1)\\geq \\mu_{d-1} \\geq \\mu_0$, which contradicts to (12).) However since it is not obvious, I suggest writing the proof explicitly.\n\nP.10, Theorem 6.1: [...] with $\\boldsymbol{\\Omega}, \\boldsymbol{W}$ symmetric, [...] -> [...] with $\\boldsymbol{\\Omega}$ and $\\boldsymbol{W}$ being symmetric, [...]\n\nP.10, Theorem 6.1: [...] let $c$ denote the most positive eigenvalue of $\\boldsymbol{\\Omega} \\otimes \\boldsymbol{I}_n - \\boldsymbol{W} \\otimes \\boldsymbol{\\mathsf{A}}$ [...]: Can we assume that there always exists positive eigenvalues?\n\nP.11:\n> We emphasize that this is in support of our theoretical findings, since we proved that thanks to a residual connection, graph-convolutional models enable the channel-mixing to also generate repulsion along the edges through its negative eigenvalues hence compensating for the underlying heterophily of the graphs.\n\nThis sentence implicitly assumes that the learned weights have negative eigenvalues. However, I wonder whether it is empirically confirmed in numerical experiments.\n\nP.11: Table 2: I think it is easier to read if we reverse the denominator and numerator of $\\frac{\\mathcal{E}^{\\mathrm{Dir}}(\\boldsymbol{\\mathrm{F}}(0))/\\|\\boldsymbol{\\mathrm{F}}(0)\\|^2}{\\mathcal{E}^{\\mathrm{Dir}}(\\boldsymbol{\\mathrm{F}}(m))/\\|\\boldsymbol{\\mathrm{F}}(m)\\|^2}$\n\nP.12: enhanche -> enhance\n\nBroader Impact Concerns: There are no major concerns about broader impacts.\n\n==================================================\n\nMetareview:\nRecommendation: Accept as is\n\nComment: All 3 reviewers agree that the claims are supported by clear evidence, that the work is relevant to the TMLR community, and recommended to accept the manuscript. All the concerns and clarifications asked by the reviewers were addressed during the rebuttal and revisions. I thus recommend to accept the paper as is. \n\n==================================================\n\n"}}},{"rowIdx":87,"cells":{"paper":{"kind":"string","value":"# Bayesian Optimization With Derivatives Acceleration\n\nGuillaume Perrin *guillaume.perrin@univ-eiffel.fr* Université Gustave Eiffel, COSYS, 5 Bd Descartes 77454 Marne-La-Vallée, France Rodolphe Leriche **leriche@emse.fr**\nLIMOS (CNRS, Mines Saint-Etienne, UCA) Saint-Etienne, France Reviewed on OpenReview: *https: // openreview. net/ forum? id= XXXX*\n\n## Abstract\n\nBayesian optimization algorithms form an important class of methods to minimize functions that are costly to evaluate, which is a very common situation. These algorithms iteratively infer Gaussian processes from past observations of the function and decide where new observations should be made through the maximization of an acquisition criterion. Often, the objective function is defined on a compact set such as in a hyper-rectangle of the d-dimensional real space, and the bounds are chosen wide enough so that the optimum is inside the search domain. In this situation, this work **provides a way to integrate**\nin the acquisition criterion the *a priori* **information that these functions, once modeled as**\nGP trajectories, should be evaluated at their minima, and not at any point as usual acquisition criteria do. We propose an adaptation of the widely used Expected Improvement acquisition criterion that accounts only for GP trajectories where the first order partial derivatives are zero and the Hessian matrix is positive definite. The new acquisition criterion keeps an analytical, computationally efficient, expression. This new acquisition criterion is found to improve Bayesian optimization on a test bed of functions made of Gaussian process trajectories in low dimension problems. The addition of first and second order derivative information is particularly useful for multimodal functions.\n\n## 1 Introduction\n\nOver the last 20 years, Bayesian optimization (BO) methods have established themselves as one of the references for approximating the point(s) minimizing an expensive-to-evaluate black-box function, from as few calls to this function as possible. This is reflected in the existence of many reviews and tutorials on BO in the literature (see for instance Jones (2001); Sobester et al. (2008); Shahriari et al. (2015); Gramacy (2020); Garnett (2023); Frazier (2018), as well as many applications of BO in industrial applications, such as aeronautics Forrester et al. (2007); Lam et al. (2018) or agriculture Picheny et al. (2017)). Today, the machine learning community is a key contributor to BO advances, motivated by the need to optimize hyperparameters Bergstra et al. (2011); Snoek et al. (2012); Klein et al. (2017); Wu et al. (2019); Turner et al.\n\n(2021) or exploration strategies in reinforcement learning Wang et al. (2023). More specifically, BO is concerned with minimization problems that can be written in **the following form:**\n\n$$x^{*}\\in{\\underset{x\\in\\mathbb{X}}{\\operatorname{arg\\,min}}}\\,y(x),$$\nypxq, (1)\nwhere y **is a pointwise observable function defined over the compact set** X Ă R\nd, d ě **1. BO assumes that**\ny **can be usefully represented as a Gaussian process (GP), whose mean and covariance functions have been** identified from a limited number of calls to function y**. It then sequentially adds new observations of** y at points maximizing an *acquisition criterion* **whose objective, in the search for the global minimum, is**\n\n$$(1)$$\n\nto make a judicious trade-off between the exploration of X **and the exploitation of past observations. In**\nthe theory of decision under uncertainty, acquisition criteria are the expectation of a *utility* **of the possible**\nfunction observations according to the stochastic model of **the objective function Žilinskas & Calvin (2019).** Several acquisition criteria have been proposed. The earliest, one-dimensional, version of BO Kushner (1962) involved the probability of improvement and an upper **confidence bound. The upper confidence** bound was later theoretically studied in many dimensions in **Srinivas et al. (2010). Another early BO**\nacquisition criterion was described in Močkus (1972) which **is, since Frazier & Powell (2007), called the** knowledge gradient. It is a one-step-ahead expected progress in GP mean. The Expected Improvement beyond the current best observation (EI) is the most classical acquisition criterion. The EI has a simple interpretation and an analytical expression deprived of parameters to tune, two features which have contributed to its popularity. It was first proposed in Saltenis **(1971) and popularized in Schonlau (1997);**\nJones et al. (1998); Močkus (2012). More recently, acquisition criteria based on information theory have been suggested which target entropy reductions in the GP model extrema Hernández-Lobato et al. (2014) or locations of extrema Villemonteix et al. (2009); Hennig & **Schuler (2012).**\nBO is particularly efficient when the dimension of the search space remains limited (d ď **5 to 10) and**\nwhen the function is multimodal with some structure Le Riche **& Picheny (2021). Several adaptations of** this formalism have been proposed to extend the efficiency of these approaches to larger input spaces, by playing directly on the acquisition criterion Siivola et **al. (2018), on the identification of latent spaces**\nof reduced dimensions Bouhlel et al. (2016); Gaudrie et al. (2020), on the introduction of trust regions Diouane et al. (2022), or by replacing the GP by a Bayesian neural network Kim et al. (2022). It sometimes happens that the derivatives of the true function at the observed points is available (e.g.,\nthrough automatic differentiation, or adjoint codes in partial differential equations solvers). It is then possible to add these derivatives as part of the observations of a vectorized Gaussian process composed of the function prediction and its derivatives Laurent et al. (2019). All the above acquisition criteria could then be calculated with such a gradient-enriched underlying Gaussian Process. This has been done with the gradient-knowledge criterion in Wu et al. (2017). It is nevertheless interesting to note that all of these methods only exploit a limited part of the information conveyed by the GP (once conditioned by the observations of the true function and potentially by the observations of its gradient). In particular, they do not take into account the information that the GP\nderivatives could bring, whether the function y is convex or not, and even if the derivatives of y **are not** observed. Indeed, when y is twice differentiable, it is well known that the first derivatives of y **become**\nzero and that its Hessian matrix is positive definite at its minimum (unless the minimum lies at an edge of the domain). It is reasonable to believe that the minimization strategy can only benefit from this supplementary knowledge on derivatives. Figure 1 provides **a graphical illustration to support this** intuition. It shows three plots with GP trajectories conditioned by three observations (also known as a kriging model). The two bottom plots further condition the trajectories on their derivatives so that they have local minima at the wrong (left) or right location. Because it is easier to force local minima where the true function really has an optimum, the information on local minima helps better discriminating the optimal from the non-optimal region. Note also that in Figure 1, while the first and second order derivatives of the GP trajectories are constrained, no information about the derivatives of the true function is used. This is a key difference with other work on BO with gradient knowledge such as Wu et al. (2017). With this in mind, the main contribution of this paper is to propose an adaptation of the famous EI criterion so that it integrates the information of zero derivative and positive definite Hessian matrix of the GP trajectories. In other terms, this new criterion only accounts for possible minima of the GP trajectories, as opposed to the traditional EI that can confer a utility to any part of a trajectory. We emphasize that the proposed criterion does not imply that derivatives of the true function ypxq **be calculated. The derivatives**\nonly concern the GP. The new criterion is meaningful if the minimum is located inside the search domain, which is a reasonable assumption in most applications where, precisely, the bounds are chosen as extremes that should not be reached. A complementary idea for cases where the bounds might be active is nevertheless given as a\n\n\n\nFigure 1: An illustration of the motivation for conditioning kriging trajectories with derivatives. Top: kriging trajectories in red, true function as a black solid line. Bottom: kriging trajectories forcing, by Gaussian conditioning, a zero derivative and a positive second derivative at the vertical dotted bar, i.e. at the global maximum of the true function in the figure on the left, and at the global minimum in the figure on the right. The difference between trajectories at the maximum and minimum of the true function is more apparent when forcing local minima at the right location.\nperspective: a method is proposed to estimate the likelihood that the minimum of y **is on the edge of the**\ndomain. Empirically investigating the effect of a new idea - here adding derivatives acceleration - on an optimization algorithm is difficult because the performance of an algorithm depends on both the function it is applied to and the tuning of its hyperparameters. The empirical tests we provide are designed to exclusively show the effects of the derivatives acceleration while avoiding all such experimental side effects. This is achieved firstly by testing on Gaussian processes whose hyper-parameters are known, therefore guaranteeing the compatibility of the model and the test function. Secondly, **the maximization of the acquisition criteria is**\ndone with extreme care, which is feasible up to dimension 5 and needs to be relaxed beyond. The outline of this paper is as follows. Section 2 recalls the **theoretical bases of the Gaussian process**\nregression (GPR) and its use for the minimization of black-box functions. Section 3 introduces the acquisition criterion we propose for taking into account information on the derivatives of y**. Section 4 then**\nillustrates the benefits of this new acquisition criterion on simulated test functions that can be modeled as realizations of Gaussian processes. Section 5 describes **how optima on the bounds can be handled and**\nconcludes the paper.\n\n## 2 The Bo General Framework\n\nFor d ě 1, let X **be a compact subset of** R\nd**. In this work, we are interested in finding the solution(s)**\nx\n‹ of the optimization problem defined by Eq. (1) using as few pointwise observations of y **as possible.**\nAnticipating the developments in the following sections exploiting the gradient of y**, we assume that** y is an element of C\n2pX, Rq, the set of real-valued twice continuously differentiable functions defined on X**. In**\naddition, we treat X as explicit, which means that the function y **cannot be evaluated outside the search**\nregion (it is defined as a product of intervals in the applications).\n\nTo solve this problem, we consider Bayesian Optimization guided by the Expected Improvement (EI) acquisition criterion. Such methods are sometimes called Efficient Global Optimization algorithms in reference to Jones et al. (1998), although implementations **(of the GP and of the EI maximization) vary.** The choice of the EI acquisition criterion is guided by simplicity: it is the most standard criterion and most importantly, it does not require GP simulations to be evaluated. Others criteria could benefit from derivatives acceleration as discussed in the perspectives **of this article (Section 5.2).** BO relies on the evaluation of the objective function at a sequence of well-chosen points as summarized hereunder and in Algorithm 1.\n\n## Initialization\n\nTo begin, the function y is evaluated at N0 points uniformly chosen in X **(typically according to a spacefilling design of experiments (DoE) Fang et al. (2006); Perrin & Cannamela (2017)). We note** px pnq, yn :\"\nypx pnqqqN0 n\"1 the obtained pairs. Given this available data, a GP-based surrogate model is trained for y. To obtain convergence results, a common theoretical assumption is that y **is a particular realization**\nof a Gaussian process Y \" GPp*µ, C*q**, whose prior mean and prior covariance functions are noted** µ and C **respectively (see Santner et al. (2003); Rasmussen (2003) for more details about Gaussian process**\nregression). In practice, it is only required that y **can be observed at a finite number of points and the** assumption of y being a sample of Y may not hold. The algorithm then conditions Y **to interpolate the**\nN0 **available input-output pairs, resulting in a new** YN0 \" GPpµN0\n, CN0 q, **where:**\n\n$$\\mu_{N_{0}}(\\mathbf{x})=\\mu(\\mathbf{x})+C(\\mathbf{x},\\mathbf{X})C(\\mathbf{X},\\mathbf{X})^{-1}(y(\\mathbf{X})-\\mu(\\mathbf{X})),\\ \\ \\mathbf{x}\\in\\mathbb{X},$$\n´1pypXq ´ µpXqq, x P X, (2)\n$$C(x,x^{\\prime})-C(x)$$\n\nCN0\npx, x\n1q \" Cpx, x\n1q ´ Cpx, XqCpX, Xq\n´1CpX, x\n$$\\mathbf{\\tau},\\ \\ \\mathbf{x},\\mathbf{x}^{\\prime}\\in\\mathbb{X}.$$\n1P X. (3)\n$$\\left(2\\right)$$\nIn the former expressions, X :\" rx p1q**¨ ¨ ¨** x pN0qs Tis the pN0 ˆ dq**-dimensional matrix that gathers the**\navailable input points, and for any function f and g defined on X and X ˆ X **respectively, the following**\nnotation is adopted:\n\n$$(f(\\mathbf{X}))_{n}=f(\\mathbf{x}^{(n)}),\\ \\ (g(\\mathbf{X},\\mathbf{X}))_{n m}=g(\\mathbf{x}_{n},\\mathbf{x}_{m}),\\ \\ 1\\leqslant n,m\\leqslant N_{0}.$$\n$$\\left({4}\\right)$$\n\n## Iteration\n\nGiven N ě N0 evaluations of y**, an acquisition criterion is introduced to choose at which point to carry out** the pN ` 1q th evaluation of y**. In the noise-free setting, the classical acquisition criterion is the Expected**\nImprovement (EI). It is the expectation of a utility at x **defined as the progress below the current best** observation:\n\n$$\\mathrm{EI}_{N}(\\mathbf{x}):=\\mathbb{E}\\left[\\max(0,y_{\\min}-Y_{N}(\\mathbf{x}))\\right]=\\int_{\\mathbb{R}}\\max(0,y_{\\min}-y)f_{Y_{N}(\\mathbf{x})}(y)dy$$ $$=\\sigma_{N}(\\mathbf{x})\\left(U_{N}(\\mathbf{x})\\Phi\\left(U_{N}(\\mathbf{x})\\right)+\\phi\\left(U_{N}(\\mathbf{x})\\right)\\right).$$\n\n$$\\left({\\overline{{5}}}\\right)$$\n\nHere, UN pxq :\" pymin ´ µN pxqq{σN pxq, σN pxq :\"\naCN px, xq, ymin **is the current minimum of the** N\nobservations of y**, noted** ypx p1qq*, . . . , y*px pNqq, Φ and φ **denote the probability density function (PDF)**\nand cumulative density function (CDF) of the standard Gaussian variables, and fYN pxqpyq \" φppy ´\nµN pxqq{σN pxqq{σN pxq is the PDF of the Gaussian random variable YN pxq \" N pµN pxq, σN pxq 2q**, where**\n\n$$Y_{N}:=Y\\mid Y(\\mathbf{x}^{(1)})=y(\\mathbf{x}^{(1)}),\\ldots,Y(\\mathbf{x}^{(N)})=y(\\mathbf{x}^{(N)}).$$\npNqq. (6)\nBy construction, this acquisition criterion seeks a compromise between exploitation (first term) and exploration (second term) for the global search of the minimum, and the next evaluation point is chosen such that\n\n$$({\\mathfrak{h}})$$\n$$\\mathbf{x}^{(N+1)}\\in\\arg\\operatorname*{max}_{\\mathbf{x}\\in\\mathbb{X}}\\operatorname{E}\\!\\mathrm{I}_{N}(\\mathbf{x}).$$\n$$\\left(7\\right)$$\nEIN pxq. (7)\n\n## Stopping Criterion\n\nFor most existing implementations of BO, the stopping criterion is a maximum number of evaluations of y. Indeed, unlike gradient-based approaches for minimizing **convex functions, once a local minimum of** y has been found, there is no theoretical guarantee that it corresponds to the global minimum of y**. While**\nit may be tempting, stopping the search when the expected improvement drops below a lower bound is unstable in practice as the EI changes a lot with the GP length **scales.**\n\n## Degrees Of Freedom\n\nThe performance of the BO method depends on several degrees of freedom that vary between implementations. The choice for µ and C, the way the parameters on which µ and C **depend are optimized, the**\nratio N0{**budget, the way the initial DoE is constructed, the way the acquisition criterion is maximized**\nare all important (see Le Riche & Picheny (2021) for an investigation of the influence of these choices).\n\nHowever, as the paper studies an adaptation of the acquisition criterion, it is clearer to fix these parameters to standard values of the literature. To this end, the function µ **is taken as a constant, and the function** C\nis chosen in the class of tensorized Matérn kernels with smoothing parameter ν \" 5{**2 (see Santner et al.**\n(2003) for alternative classes of functions):\n\n$$\\mu(\\mathbf{x}):=\\beta,\\ \\ C(\\mathbf{x},\\mathbf{x}^{\\prime}):=\\sigma^{2}\\prod_{i=1}^{d}\\kappa\\left({\\frac{|x_{i}-x_{i}^{\\prime}|}{\\ell_{i}}}\\right),\\ \\ \\mathbf{x},\\mathbf{x}^{\\prime}\\in\\mathbb{X},$$\n1P X, (8)\n$$({\\boldsymbol{\\delta}})$$\n\nAlgorithm 1: **Standard BO algorithm.**\nChoose N0, budget, Y \" GPp*µ, C*q ; Ñ **Initialization**\nDraw at random N0 **points** x p1q*, . . . ,* x pN0qin X ;\nCompute ypx pnqq, 1 ď n ď N0, estimate the parameters on which µ and C **depend ;**\nDefine YN0\n\n:\" Y |Y px p1qq \" ypx p1qq*, . . . , Y* px pN0qq \" ypx pN0qq ;\nSet k \" 0 ; Ñ Iteration while k ă *budget* do Search for x pN0`k`1q:\" **arg max**xPX EIN0`kpxq ;\nEvaluate y at x pN0`k`1q(and potentially adjust the expressions of µ and C) ;\nDefine YN0`k`1 :\" Y |Y px p1qq \" ypx p1qq*, . . . , Y* px pN0`k`1qq \" ypx pN0`k`1qq ;\nUpdate k Ð k ` 1 end Return min1ďiďN0`**budget** ypx piqq.\n\n$$\\kappa(u):=\\left(1+{\\sqrt{5}}u+{\\frac{5}{3}}u^{2}\\right)\\exp\\left(-{\\sqrt{5}}u\\right),\\ \\ u\\geqslant0.$$\n$\\left(\\mathfrak{g}\\right)$. \n¯, u ě 0. (9)\nThe Matérn 5/2 kernel is selected here because it is well-performing and common in the literature (Le Riche & Picheny (2021)). Furthermore, we will soon introduce an acquisition criterion that needs the fourth order derivatives of the kernel (to have information about the curvatures of the trajectories).\n\nThe Matérn 5/2 kernel is precisely four times differentiable, yielding trajectories that are two times continuously differentiable. The hyperparameter vector θ :\" pβ, σ, ℓ1*, . . . , ℓ*dq **will either be considered known**\n(via the definition of test functions to be minimized in the form of a particular realization of a Gaussian process of chosen parameters), or estimated by its maximum likelihood estimator (see Williams & Rasmussen\n(2006) for further details). As we focus on costly functions, we will set the maximal budget between 10 and 20 times the dimension d of the problem, while N0 **will be chosen small (most of the time we will**\nhave N0 \" **3). The initial DoE will always be a random space-filling Latin Hypercube Sample (LHS)**\nDamblin et al. (2013); Perrin & Cannamela (2017). For objective numerical comparisons, the maximization of the acquisition criteria, whether it is the EI in Equation (7) or one of the new criteria of Section 3, is always carried out in the same way. At each iteration, the acquisition criterion is first evaluated at a very large number of points randomly chosen in X (typically of the order of 10d`1**). The Nelder-Mead algorithm**\nNelder & Mead (1965) then maximizes the acquisition criterion starting from the 10 most promising points among the random points.\n\n## 3 Extending The Expected Improvement With Derivatives\n\nWe now show how to extend the Expected Improvement acquisition criterion so that it accounts for gradient and Hessian information. The principles underlying the calculations are that GP derivatives are GPs, and that local optima away from the bounds coincide with **canceling derivatives and positive definite** Hessians. These principles have already been used in the context of BO in Hernández-Lobato et al. (2014) for approximating the entropy of local optima. An independent and differing version, adapted to EI, is described hereafter.\n\n## 3.1 Reminders On Gaussian Process Derivation\n\nThe acquisition criteria reviewed in the Introduction, in particular the EI, are only based on the distribution of YN pxq **and do not include information related to higher derivatives. Yet, when the functions** x ÞÑ µpxq and px, x 1**q ÞÑ** Cpx, x 1q are sufficiently regular, the statistical properties of the derivatives of Y can be deduced by simple derivations of µ and C**. Indeed, as the Gaussian distribution is stable by linear**\noperations, for any linear operator L such that Ly **is a function from** R\ndto R\ndL , LY **is also a Gaussian**\nprocess, with:\n\n$$\\mathbb{E}\\left[{\\mathcal{L}}Y(\\mathbf{x})\\right]={\\mathcal{L}}\\mu(\\mathbf{x}),\\ \\ \\operatorname{Cov}({\\mathcal{L}}Y(\\mathbf{x}),{\\mathcal{L}}Y(\\mathbf{x}^{\\prime}))={\\mathcal{L}}C(\\mathbf{x},\\mathbf{x}^{\\prime}){\\mathcal{L}}^{T}.$$\nT. **(10)**\nHere, the notations LCpx, x 1q and Cpx, x 1qL\nTindicate that operator L **is applied as a function of** x and x 1**respectively, so that Cov**pLY pxq,LY px 1qq is a pdL ˆ dLq**-dimensional matrix. In particular, for**\ndL \" 1 ` dpd ` 3q{**2, if we choose**\n\n$${\\mathcal{L}}:Y\\;\\mapsto{\\mathcal{L}}Y:=\\left(Y,{\\frac{\\partial Y}{\\partial x_{1}}},\\ldots,{\\frac{\\partial Y}{\\partial x_{d}}},{\\frac{\\partial^{2}Y}{\\partial x_{1}^{2}}},\\ldots,{\\frac{\\partial^{2}Y}{\\partial x_{1}\\partial x_{2}}},\\ldots,{\\frac{\\partial^{2}Y}{\\partial x_{d}^{2}}}\\right),$$\n\nwe obtain the joint distribution of Y **and its first and second order derivatives. For each twice-differentiable**\nfunction z**, we introduce the following notations,**\n\n$$\\partial z:=\\left[\\begin{array}{c}\\frac{\\partial z}{\\partial x_{1}}\\\\ \\vdots\\\\ \\frac{\\partial z}{\\partial x_{d}}\\end{array}\\right],\\quad\\partial^{2}z:=\\left[\\begin{array}{cccc}\\frac{\\partial^{2}z}{\\partial x_{1}^{2}}&\\cdots&\\frac{\\partial^{2}z}{\\partial x_{1}\\partial x_{d}}\\\\ \\vdots&\\ddots&\\vdots\\\\ \\frac{\\partial z}{\\partial x_{1}\\partial x_{d}}&\\cdots&\\frac{\\partial z}{\\partial x_{d}^{2}}\\end{array}\\right],\\quad D^{2}z:=\\operatorname{diag}(\\partial^{2}z)=\\left[\\begin{array}{c}\\frac{\\partial^{2}z}{\\partial x_{1}}\\\\ \\vdots\\\\ \\frac{\\partial^{2}z}{\\partial x_{d}}\\end{array}\\right],\\tag{11}$$\n\n$$(10)$$\n\n$$(12)$$\n$$(13)$$\n\nand we denote by M`pdq the set of pd ˆ dq**-dimensional positive definite matrices.**\n\n## 3.2 An Acquisition Criterion Accounting For The Derivatives\n\nFor any x in X, it is well known that if Bzpxq \" 0 and B\n2zpxq P M`pdq, x is a local minimum of z**. As**\nthe input space X is bounded, the reciprocal is however not true, since a local **minimum can be found at**\nthe boundary of X **with a non-zero gradient and/or** B\n2zpxq R M`pdq**. The case when the optima are on**\nthe bounds will be discussed in Section 5.1. For now we focus on the interior of X**, to integrate as prior**\nknowledge that the gradient will be zero and the matrix of curvatures positive definite at the local minima of y**, the EI criterion defined by Eq. (5) can be replaced by:**\n\n$$\\operatorname{deriv-}\\operatorname{EI}_{N}(\\mathbf{x}):=\\mathbb{E}\\left[1_{R_{N}(\\mathbf{x})\\oplus Y_{N}(\\mathbf{x})\\oplus\\mathcal{B}(\\epsilon),\\epsilon^{2}Y_{N}(\\mathbf{x})\\in\\mathcal{M}^{+}(d)}\\operatorname*{max}(0,y_{\\operatorname*{min}}-Y_{N}(\\mathbf{x}))\\right],$$\n\nwhere for any ε ą 0, Bpεq :\"\n y9 P R\nd, }y9 } ď ε\n(is the d-dimensional hypersphere of radius ε, RN pxq **is a**\nmatrix such that\n\n$$\\mathbf{R}_{N}(\\mathbf{x})\\mathrm{Cov}(\\partial Y_{N}(\\mathbf{x}))\\mathbf{R}_{N}(\\mathbf{x})^{T}=\\mathbf{I}_{d},$$\nT \" Id, **(13)**\nand for any event a, 1a is equal to 1 if a is true and to 0 otherwise. BYN pxq **has a covariance matrix** (made of the second derivatives of CN px, x 1q**) with correlations and its density has ellipsoidal level sets.**\nThe normalized vector RN pxqBYN pxq is uncorrelated, its covariance is the identity Id**, and its density**\nhas spherical level sets that can be compared to the sphere Bpǫq**. The dependency of this scaling on** x disappears for ε **small (see Appendix A). Equivalently, we can write the criterion deriv-EI**N as\n\n$$\\operatorname{deriv-}\\operatorname{EI}_{N}(\\mathbf{x}):=\\mathbb{E}\\left[1_{\\partial Y_{N}(\\mathbf{x})\\in{\\mathcal{E}}(\\mathbf{x},\\varepsilon),\\,\\partial^{2}Y_{N}(\\mathbf{x})\\in{\\mathcal{M}}^{+}(d)}\\operatorname*{max}(0,y_{\\operatorname*{min}}-Y_{N}(\\mathbf{x}))\\right],$$\n\nwith Epx, εq :\"\n!y9 P R\nd, y9 T RN pxq T RN pxqy9 ď ε 2)a d-dimensional ellipsoid. In connection with theoretical decision under uncertainty Žilinskas & Calvin (2019), deriv-EIN pxq **is the expectation of a utility of**\nthe function model (the GP trajectories) at x**. Here, the utility is defined as the progress of the stochastic** model below the best observation knowing that the function model has a minimum at x **i.e., it has null**\nfirst order derivatives and positive curvatures. The key idea of deriv-EI is to account only for minima of\n\n$$(14)$$\n\nthe possible functions. On the contrary, EI accounts for any **value of the possible functions which is below** the best observation, which is less consistent with the goal **of minimization. Because it characterizes the**\nbehavior of the minima of the GP realizations, deriv-EI can be seen as a criterion between EI and information theoretic criteria based on the expected reduction in entropy of the optima Hernández-Lobato et al.\n\n(2014).\n\nBy considering deriv-EIN rather than EIN **as a new acquisition criterion in Algorithm 1, we expect to** improve its exploitation capabilities, without degrading **its exploration capabilities too much. Like EI**N ,\nderiv-EIN **needs only evaluations of the true function,** ypx p1qq*, . . . , y*px pNqq through YN , BYN and B\n2YN **. It**\ndoes not need derivatives of the true function, y**. Only the GP is differentiated. However, this acquisition** criterion can no longer be calculated simply, and in the general case it will require the use of sampling techniques for its evaluation, which may complicate its use. Nevertheless, if we choose ε **small, if we neglect**\nthe off-diagonal terms of the Hessian (as it was already proposed in Hernández-Lobato et al. (2014)) while assuming a well-chosen conditional independence of its diagonal terms, we obtain the following relaxed acquisition criterion (see Appendix A for a detailed derivation):\n\n$$\\operatorname{deriv-EI}_{N}(\\mathbf{x})\\approx\\operatorname{LikelyMin}_{N}(\\mathbf{x})\\times\\operatorname{cond-EI}_{N}(\\mathbf{x}),$$\nderiv-EIN pxq « LikelyMinN pxq ˆ cond-EIN pxq, **(15)**\n$$\\operatorname{L}_{\\mathrm{L}}^{\\mathrm{L}}(\\mathbf{x}):=v\\times\\varepsilon^{d}\\times\\exp\\left(-{\\frac{\\dot{\\mathbf{m}}^{T}{\\hat{\\mathbf{S}}}^{-1}\\dot{\\mathbf{m}}}{2}}\\right)\\times\\prod_{i=1}^{d}\\Phi\\left({\\frac{{\\bar{r}}_{i}}{\\sqrt{1-r_{i}^{2}}}}\\right),$$\n$$(15)$$\n$$(16)$$\n$$\\mathrm{cond-EI}_{N}(\\mathbf{x}):=s\\left((z_{\\mathrm{min}}-a)\\Phi(z_{\\mathrm{min}})+\\phi(z_{\\mathrm{min}})\\right),$$\ncond-EIN pxq :\" s ppzmin ´ aqΦpzminq ` φpzminqq, **(17)**\nwhere v is a constant that does not depend on x and ε d**, and where the following notations have been**\nintroduced to simplify the expressions:\n\n$$\\partial Y_{N}(\\mathbf{x})\\ \\sim\\ {\\mathcal{N}}\\left(\\dot{\\mathbf{m}},\\dot{\\mathbf{S}}\\right),\\quad D^{2}Y_{N}:=\\left((\\partial^{2}Y_{N})_{1,1},\\ldots,(\\partial^{2}Y_{N})_{d,d}\\right),$$\n$$(Y_{N}(\\mathbf{x}),D^{2}Y_{N}(\\mathbf{x})))|\\partial Y_{N}(\\mathbf{x})=\\mathbf{0}~\\sim~{\\mathcal{N}}\\left(\\left(\\begin{array}{c}{{m}}\\\\ {{\\hat{m}_{1}}}\\\\ {{\\vdots}}\\\\ {{\\hat{m}_{d}}}\\end{array}\\right),\\left[\\begin{array}{c c c c c}{{s^{2}}}&{{\\rho_{1,1}}}&{{\\cdots}}&{{\\rho_{1,d}}}\\\\ {{\\rho_{1,1}}}&{{\\hat{s}_{1}}}&{{\\ddots}}&{{\\vdots}}\\\\ {{\\vdots}}&{{\\ddots}}&{{\\ddots}}&{{\\rho_{d-1,d}}}\\\\ {{\\rho_{d,1}}}&{{\\cdots}}&{{\\rho_{d,d-1}}}&{{\\hat{s}_{d}}}\\end{array}\\right]\\right),$$\n$$(17)$$\n$$(18)$$\n$$\\quad(19)$$\n$$(20)$$\n, **(19)**\n$$z_{\\min}:=\\frac{y_{\\min}-m}{s},\\ r_{i}:=\\frac{\\rho_{1i}}{s\\overline{s}_{i}},\\ \\overline{r}_{i}=\\frac{\\tilde{m}_{i}}{\\overline{s}_{i}},\\ a=\\sum_{i=1}^{d}\\frac{r_{i}}{\\sqrt{1-r_{i}^{2}}}\\ \\Phi\\left(\\frac{\\tilde{z}_{i}}{\\sqrt{1-r_{i}^{2}}}\\right).\\tag{2}$$\n\nThe precise choice of ε **has thus no impact.**\n\n## 3.3 Comments On The Proposed Acquisition Criterion\n\nAnalysis of the terms in deriv-EI\nComparing the criteria EIN pxq and deriv-EIN pxq**, we first notice the presence of the function** x ÞÑ LikelyMinN pxq, whose role is to concentrate the search of the new point to be **evaluated around the points**\nx that are likely to lead to a zero gradient of y **(small values of** m9 TS9\n´1m9 **), while favouring the areas of**\npositive second derivatives (high values of Φpτ:i{\na1 ´ r 2 iq for all i). The second function x ÞÑ **cond-EI**N pxq estimates the expected improvement assuming that the function has a minimum at x, and looks particularly like the expression given by Eq. (5). The more the second derivatives of YN **will be positive in**\nprobability, which translates into large values of τ:i**, the more this similarity will be important because, in**\nthis case, the constant a tends towards 0. In addition, as the statistical properties of YN pxq, BYN pxq and D2YN pxq are known explicitly, it is important to notice that the evaluation cost of deriv-EIN pxq **is of the**\nsame order of magnitude as that of the classical EIN pxq**. Importantly, there is no need to use sampling**\nmethods to estimate it.\n\nIn addition, if µ is chosen to be constant and C **is a stationary covariance kernel (which remains the most**\ncommon configuration in BO), then BY pxq is statistically independent of Y pxq and D2Y pxq **for any** x in X. In particular, if we focus on the first iteration of the BO procedure (N \" **0), and put aside the**\nconstraint on the Hessian, it can be noted that for any ε ą **0 and any** x P X,\n\n$$\\mathbb{E}\\left[1_{R_{0}(\\mathbf{x})\\partial Y(\\mathbf{x})\\in\\mathcal{B}(\\varepsilon)}\\operatorname*{max}(0,y_{\\operatorname*{min}}-Y(\\mathbf{x}))\\right]=p_{\\varepsilon}\\times\\operatorname{EL}_{0}(\\mathbf{x}),$$\n$$(21)$$\n\nwhere pε :\" PpR0BY pxq P Bpεqq is a constant independent of x **as the statistical properties of** BY pxq do not depend on x (stationarity). In that case, EI0 is very close to deriv-EI0 **(up to the influence of the**\nsecond derivatives), and maximizing either of these criteria should give close results. Then, the more the process Y is conditioned by observations of y, the more Y , BY and B\n2Y **are correlated, and the more**\nchances there are for deriv-EIN pxq and EIN pxq **to propose different points. After many observations, it**\nis anticipated that the interesting areas from the EI point of view will have low gradients, so that the two criteria should again propose close new evaluation points. The *a priori* **interest of the deriv-EI**N pxq criterion thus lies in intermediate values of N, in the exploration of the various local minima of y**, and the**\nsearch for the smallest zone of X **likely to contain the global minimum of** y.\n\nAt last, when the dimension d increases, one of the classical difficulties of BO based on EIN **is to favor**\nexploration over exploitation, by placing a very large number of points on the edges of the domain, which effectively represent the majority of the volume of X when d **is large Siivola et al. (2018). This effect**\nshould be limited by substituting deriv-EIN for EIN **, i.e., by requiring that each partial derivative of** YN\nbe close to 0 and that each main curvature be positive through the factor LikelyMinpxq**, which becomes**\nmore restrictive as d **increases.**\n\n## A More Exploratory Deriv-Ei\n\nIn return, by trying to quickly visit potential high-performance local minima, it is possible that the deriv-EIN criterion explores fewer regions of X than EIN **, which could be penalizing for the minimization of functions with multiple local minima. If this were the case (this kind of phenomenon was not**\nobserved on the test cases studied in Section 4), several techniques could be proposed to rebalance the exploration/exploitation ratio. For instance, the control of the exploration-exploitation balance by changing target values has been studied in Jones (2001) for the probability of improvement and a likelihood criterion. Such a shift in target around ymin **was included in the EI criterion in Berk et al. (2019); Lizotte**\n(2008). Another way of reinforcing exploration with respect to exploitation consists in maximizing the expected improvement at a certain power p ě 1. Indeed, by taking p **greater than 1, we further encourage low-probability high improvements compared to more probable small improvements. This idea was**\npursued in Schonlau et al. (1998) where expressions for the generalized EIN criterion with p ě **2 can be**\nfound. The new EI with derivatives can also benefit from elevating the improvement at a given power. It becomes, deriv-EIppq N :\" E\n\"1BY pxqPEpx,εq,B\n2YN pxqPM`pdq maxp0, ymin ´ YN pxqqp‰.\n\nFor p \" **2 (see Appendix A for more details) and using the same notations as in Section 3.2, such a criterion**\ncan again be approximated under an analytical form close to the one of Eq. (15):\n\n$$\\operatorname{deriv-EI}_{N}^{(2)}(\\mathbf{x})\\approx\\operatorname{LikelyMin}(\\mathbf{x})\\times\\operatorname{cond-EI}^{(2)}(\\mathbf{x}),$$\nN pxq « LikelyMinpxq ˆ cond-EIp2qpxq, **(22)**\n$$\\mathrm{cond-EI}^{(2)}(\\mathbf{x}):=s^{2}\\left((1+z_{\\mathrm{min}}^{2}-2a z_{\\mathrm{min}})\\Phi(z_{\\mathrm{min}})+(z_{\\mathrm{min}}-2a)\\phi(z_{\\mathrm{min}})\\right).$$\n˘. **(23)**\n$$(22)$$\n$$(23)$$\n\n## Adaptation To Noisy Outputs\n\nThe criteria EIN and deriv-EIN are introduced in a context where the observations of Y **are assumed**\nnoise-free. If it turns out that these outputs are in fact noisy, and if this noise can be modeled as a centered Gaussian vector with covariance matrix Σ **(which can be assumed to be diagonal or not), only a** few adjustments are needed to calculate these two criteria (and these adjustments are the same in both cases). First, in order to integrate the noisy nature of the observations in the conditioning formulas, the distribution of YN for any value of N **is obtained by replacing** CpX, Xq\n´1 by pCpX, Xq ` Σq\n´1in equations (2) and (3). Then, as the observations are noisy, the notion of a current minimum value makes no longer sense, and we need to adapt the value of ymin **in the proposed expressions accordingly. This**\nvalue can, for example, be chosen as the value minimizing over X the predictive mean x ÞÑ ErYN pxqs**, as**\nit is done in the knowledge gradient approaches Frazier & Powell (2007). To limit the computational cost associated with the choice of ymin**, it is also common practice to minimize the predictive mean only at the** observed input points. Once these two adjustments have been made, the criteria EIN and deriv-EIN can be applied to noisy observations.\n\n## 4 Numerical Experiments\n\nIn this Section, we first illustrate the way the proposed criterion works, and the differences it implies with the classical EI criterion. In particular, it will be seen that iterates stemming from the maximization of deriv-EIN are more concentrated inside the search domain than EIN **iterates. Then, by considering**\nfunctions with minima inside the search domain, we show that deriv-EIN **allows faster average convergence** than EIN **does. This is particularly visible with highly multimodal functions. In the experiments, deriv-EI**N\nis calculated through the approximation of Eq. (15).\n\n## 4.1 Analysis Of The Deriv-Ein **Criterion In Dimension 1 And 2** Test Functions And Experimental Protocol\n\nWe analyze the behavior of the deriv-EIN **criterion through the study of an oscillating function in dimension**\nd \" **1, noted** y 1D, and of a modified Branin function in dimension d \" **2, noted** y 2D **(see Figure 2 for a**\ngraphical representation of these functions, and Appendix **B for their definitions). In order to focus** exclusively on the effects of the acquisition criterion, we fix the hyperparameters (length scales, variance, trend parameters) of the Gaussian predictor to their maximum likelihood estimate for a large number of points. It has been observed that a good optimization of the acquisition criterion is a condition for BO to be efficient Le Riche & Picheny (2021). For this reason, the maximization of the acquisition criteria is performed by an exhaustive search on a fine grid, which is possible in such low dimensions. In higher dimension, a careful choice of the initial points of the acquisition function maximization is required Zhao et al. (2024).\n\n## Visualizing The Terms Making The New Acquisition Criterion\n\nWe illustrate the roles of the LikelyMinN and cond-EIN **functions (which are defined in Section 3.2), by**\nevaluating y 1D at N0 \" **5 points and** y 2D at N0 \" 12 points randomly chosen in X**. The evolutions of**\nLikelyMinN and cond-EIN associated to these evaluations are given in Figures 2-a and **b. As expected,**\nthe function LikelyMinN **is large at the points the most likely to correspond to local minima, while the**\nfunction cond-EIN **highlights the areas the most likely to lead to GP trajectories that have a null gradient**\nwhile having values lower than the current minimum. For these particular examples, the product of the two functions, which yields the deriv-EIN **criterion, favors new points inside the input domain, when the EI**N\ncriterion encourages to evaluate y 1D **(resp.** y 2D) on an edge of X**. We also notice that by concentrating the**\nsearch at areas of low gradient for y 1D or y 2D, we limit the significant values of deriv-EIN **to sub-regions**\nof X that are smaller than what EIN **would give.**\n\n## Performance Of Deriv-Ein **Over One Step**\n\nThe performance of the deriv-EIN **criterion is now analyzed in terms of minimization of** y 1D and y 2D.\n\n\n\n(a) LikelyMinN ( ) and cond-EIN ( **) for** y 1D\n\n(b) LikelyMinN ( ) and cond-EIN ( **) for** y 2D\n\n\n\nFigure 2: The function y 1D **is shown in black thick solid lines in plots (a) and (c) where the grey areas**\ncorrespond to 95% confidence intervals of the Gaussian predictor. Identically, the black solid lines in plots\n(b) and (d) are the contours of function y 2D**. In each plot, the global minimum is indicated by a cyan**\ntriangle, while the green dots show the points where the function has been evaluated. Plots (a) and (b)\nshow, for the two considered functions, the evolution of x ÞÑ LikelyMinN pxq **in orange solid line, and of** x ÞÑ cond-EIN pxq in magenta dotted line. Plots (c) and (d) compare the evolution of x ÞÑ EIN pxq **(in blue** solid line) to that of x ÞÑ deriv-EIN pxq **(in red dotted line). For ease of reading, the functions LikelyMin**N ,\ncond-EIN , EIN , and deriv-EIN **are normalized in such a way that their maximum value is fixed to 1.**\nWe start with a single step. For each j P t1, 2u, and each k ě 2d ´ **1, we generate 500 space-filling LHS**\nmade of k points in X **Perrin & Cannamela (2017), which are written** !X\npjq k,i )500 i\"1\n. For each 1 ď i ď **500, we**\nthen construct a Gaussian predictor of y jD **based on its evaluations at each point in** X\npjq k,i **, and we denote**\nby x pjq,**deriv-EI**\nk,i and x pjq,EI\nk,i the points of X maximizing the criteria deriv-EIk and EIk**, respectively. Let**\nyp pjq,EI\nk,i and yp pjq,**deriv-EI**\nk,i **be the smallest value of** y jD **that we obtain:**\n\n$$\\widehat{y}^{(j),\\text{EL}}_{k,i}:=\\min_{\\mathbf{x}\\in\\mathcal{X}^{(j)}_{k,i}\\cup\\left\\{\\mathbf{x}^{(j),\\text{EL}}_{k,i}\\right\\}}y^{\\text{ID}}(\\mathbf{x}),\\quad\\widehat{y}^{(j),\\text{derive-EL}}_{k,i}:=\\min_{\\mathbf{x}\\in\\mathcal{X}^{(i)}_{k}\\cup\\left\\{\\mathbf{x}^{(j),\\text{demire-EL}}_{k,i}\\right\\}}y^{\\text{ID}}(\\mathbf{x}).\\tag{24}$$\n\nBy construction, the lower these values are, the better the acquisition criteria should be. In this prospect, for j P t1, 2u**, Figure 3 compares the evolution of the 25%, 50% and 75% empirical quantiles of** yp pjq,EI\nk,i and yp pjq,**deriv-EI**\nk,i as a function of k**. As announced in Section 3.3, the interest of the proposed criterion**\nlies in the intermediate (about r5, 18s ˆ d) values of k**. For too low values, as the Gaussian predictor and**\nits first-order derivatives are not very correlated, the criteria deriv-EIN and EIN **are very close, and lead**\nto similar results in terms of minimization of the objective function. For k **large, the Gaussian predictor**\napproaches the objective function with little uncertainty, and the criteria deriv-EIN and EIN **are equally**\ncapable of identifying the global minimum. For intermediate values of k**, this phenomenon can be clearly** seen in the evolution of the values of y 2D **(right plot). Because a one-dimensional space is rapidly explored,**\nthe advantage of deriv-EIN over EIN **is less clear in the evolution of the values of** y 1D **(left plot). There,**\nthe two criteria give almost the same results, with deriv-EIN **allowing only slight improvements.**\n\n## Performance Of Deriv-Ein **Over Many Steps**\n\nIn the above numerical experiments, one step was studied and **the new evaluation points were independent**\nof each other. Getting closer to a BO algorithm, we now quantify the effect of the acquisition criteria when defining a sequence of points where y jD is evaluated. To this end, for j P t1, 2u**, we generate 500**\nnew space-filling LHS in X **composed of 3 points each, which are written** !Xrpjq 3,i )500 i\"1\n. For each j P t1, 2u and each repetition of the experiment 1 ď i ď **500, the function** y jD **is evaluated at each point of** Xrpjq 3,i ,\nand Algorithm 1 presented in Section 2 is run twice, taking as **acquisition criterion deriv-EI first, then the**\nclassical criterion EI. At each iteration k ě **1 of the algorithm, we note** y pjq,**deriv-EI**\nk,i and y pjq,EI\nk,i **the obtained**\ncurrent minima of y jD**. The empirical estimates of the median and the mean of these current minima is**\nshown in Figure 4. The interest of the deriv-EI acquisition criterion is again underlined by these results.\n\nIndeed, for all iterations k**, the median and the mean of the current minima associated with the deriv-EI**\ncriterion are lower than those of the current minima associated with the EI criterion. Again, the advantage of deriv-EI over EI takes place in the middle of the iterations k**. Note that the median is well below the**\nmean for the minimization of y 1D. It comes from the fact that, for both EI and deriv-EI, some of **the runs**\nhave taken a significant number of iterations to identify the **area of the global minimum.**\n\n## 4.2 Performance Analysis In Dimensions 2, 3 And 5 Test Functions Construction\n\nThe EI and deriv-EI acquisition criteria are now compared on **a larger set of test functions. To define this** set of functions, we elaborate on the idea of using GPs Hennig **& Schuler (2012) which are by construction** compatible with the working assumptions. We start by noting Z\npdq θ**the Gaussian process defined on**\nX \" r0, 1s d**such that for any** x, x 1 P X **and any** θ ą 0\n\n$$\\mathbb{E}\\left[Z_{\\theta}^{(d)}(\\mathbf{x})\\right]=0,\\ \\ \\mathrm{Cov}(Z_{\\theta}^{(d)}(\\mathbf{x}),Z_{\\theta}^{(d)}(\\mathbf{x}^{\\prime}))=\\prod_{i=1}^{d}\\kappa\\left(\\sqrt{\\frac{2}{d}\\frac{|x_{i}-x_{i}^{\\prime}|}{\\theta}}\\right)\\ ,$$\n$$(25)$$\n, **(25)**\n\n\n\nFigure 3: Influence of the acquisition criterion deriv-EIN and EIN **when minimizing** y 1D and y 2D**. For**\nk ě 2d ´ **1, the lower and upper parts of the black rectangles correspond to the 25% and 75% quantiles**\nof yp pjq,EI\nk,i **, while the black circles show the median value. Similarly, the lower and upper parts of the red**\nrectangles correspond to the 25% and 75% quantiles of yp pjq,**deriv-EI**\nk,i **, while the red triangles show the median**\nvalue.\nEI **deriv-EI**\n\n\n\nFigure 4: Influence of the acquisition criterion deriv-EIN and EIN **when minimizing** y 1D and y 2D**. For**\nk ě **1, the filled black circles (resp. the filled red triangles) represent the empirical median of** !y pjq,EI\nk,i )500 i\"1\n(resp. of !y pjq,**deriv-EI**\nk,i )500 i\"1\n), and the empty black circles (resp. empty red triangles) indicate the empirical means.\nwhere κ **is the Matérn-5/2 covariance function of Equation (9), which is such that** Z\npdq θ**is twice differentiable**\nin the mean-square sense. Notice the normalization of the length scales in ad{**2, allowing to define**\nGaussian processes in any dimension d **with close dependence structures. This normalization can also be**\nunderstood by seeing that distances (between the two farthest points, or the expected distance of two points randomly drawn in X) grow in ?d, therefore the length scales have to grow in ?d**. We consider as**\ntest function class the set F\npdq θ**of realizations of** Z\npdq θthat admits a global minimum strictly inside X **(i.e.,**\nat a point of zero partial derivatives). The following numerical tests then focus on two particular values of θ: θ \" 0.**2 will characterize strongly oscillating functions admitting a large number of local minima, while** θ \" 0.5 will refer to more regular functions presenting a smaller number of local minima. For θ P t0.2, 0.5u and d P t2, 3, 5u**, we generate 100 functions from** F\npdq θ**in a random and independent way. These functions**\nare noted !y pdq i,θ )100 i\"1\n(see Appendix C for a detailed description of their construction). We finally subtract from each function its minimum value so that\n\n$$\\operatorname*{min}_{\\mathbf{x}\\in\\mathbb{X}}y_{i,\\theta}^{(d)}(\\mathbf{x})=0,$$\n$$(26)$$\n\ni,θ pxq \" 0, **(26)**\nand we proceed to the same shifting on the Y **process. Figure 5 shows four examples of such functions**\nbelonging to F\np2q 0.2 and F\np2q 0.5 in the case d \" 2.\n\n## Experimental Protocol\n\nThe global minimum of these functions is then searched twice **with Algorithm 1 by, first, taking derivEI and, then, EI as the acquisition criterion. The total number of calls to the objective function of**\neach optimization run is equal to budget \" **100. The two types of searches are initialized with the**\nevaluation of y pdq i,θ at the same space-filling LHS of dimension N0 \" **3 (a different design is generated for**\neach function minimization). The size of the design is small and does not depend on d**. As observed in** Le Riche & Picheny (2021); Hutter et al. (2013), small random **designs at the beginning of BO searches** are more efficient. Moreover, the effect of the acquisition criterion is more visible for small initial random designs. The growth of the length scales in ?d **(Equation 25) guarantees that the correlation between the**\nN0 points is the same, independently of d**. In order to investigate the influence of the acquisition criterion**\nonly on the optimization but not on the learning of the GP, the **properties of the Gaussian process** Y\nused to guide the search are chosen equal to those of Z\npdq θ**. The maximization of the acquisition criteria is**\nperformed in two steps: each acquisition criterion is first evaluated in 105 **points randomly chosen in** X,\nand 10 Nelder-Mead algorithms starting from the 10 most promising points among the random points are then launched in parallel to identify the new point at which to evaluate the objective function.\n\nTwo quantities of interest are then extracted from these Bayesian optimizations. For each 1 ď k ď **budget,**\neach d P t2, 3, 5u, and each θ P t0.2, 0.5u**, we first note** yp pdq,**deriv-EI**\nθpkq **(resp.** yp pdq,EI\nθpkq**) the empirical mean**\nof the current minimum (mean best-so-far performance) obtained at the k th **iteration on all the tested**\nfunctions when taking deriv-EI (resp. EI) as the acquisition criterion. Second, we define pk pjq,deriv-EIp*θ, s*q\n(resp. pk pjq,EIp*θ, s*q**), the mean time-to-target that is the average number of iterations necessary for the**\nbest-so-far observation to be lower than a threshold s ą **0 when using deriv-EI (resp. EI). Note that for**\nboth quantities of interest, the average is done on the different test-functions, which have the same kind of variations and the same minimum equal to 0, which makes them comparable although potentially very different.\n\n## Optimization Results\n\nThe evolution of these quantities of interest are shown in Figure 6 for θ \" 0.2 and Figure 7 for θ \" 0.**5. In**\nall of these figures, a substantial gain is brought by the deriv-EI criterion with respect to the EI criterion. The gain is visible both in terms of the mean best-so-far objective function (plots (a) to (c)) and the mean time-to-target (plots (d) to (f)). We notice, as we had hoped **in Section 3.3, that the observed improvements**\n\n \nFigure 5: Representation of 4 particular elements of F\np2q 0.2 and F\np2q 0.5\n, the set of test functions to be minimized.\n\nIn each function instance, the global minimum location is indicated with a cyan triangle.\nbrought by the deriv-EI criterion are greater as the dimension d **of the input space increases. As expected,**\nwe also observe that choosing deriv-EI rather than EI is of more interest for more multimodal functions, i.e., when the length scale θ **is small. Indeed, it is in these configurations with a large number of local**\nminima that adding information about null first order derivatives and positive definite Hessian matrices is useful.\n\nRemark **For all the test functions studied in this section, the condition numbers of the covariance**\nmatrices of the observation points were all between 103 **and 2** ˆ 106.\n\n## 5 Extensions And Conclusions 5.1 Management Of Minima On Bounds\n\nThe article has assumed until now that the minimum is inside the search space X**. If this is not the case,**\norienting the search towards areas with a zero gradient can actually be counterproductive, as the global minimum will typically be associated with a nonzero gradient. Denoting by BX the boundary of X, a first possibility to circumvent this problem is to penalize the objective function so that optima on the boundary are pushed inside the domain but arbitrarily close **to the boundary, and therefore are associated** to a null gradient and positive definite Hessian. This is the idea of the barrier functions of the interior point methods Wright & Nocedal (2006). With barrier functions, the objective x ÞÑ ypxq **is replaced** by x ÞÑ ypxq ` λcpxq, where λ is a positive constant and c **is a continuously twice-differentiable positive**\nfunction such that cpxq would be close to zero when x is far from the boundaries of X**, and would take**\npotentially infinite values when x P BX. For instance, if X \" r0, 1s, the function c **can be chosen as:**\n\n$$c(x)=\\frac{1}{\\min(x,1-x)}\\quad\\mbox{or}\\quad c(x)=-\\log(\\min(x,1-x))\\quad,\\ \\ 0\\leqslant x\\leqslant1.\\tag{27}$$\n\nThe larger λ is, the further from BX the global minimum of y ` λc **is, whether the global minimum of** y is on BX or not. And by making λ **progressively tend towards 0, we make this global minimum, which**\n\n\n\nEI **deriv-EI**\n\nFigure 6: Plots (a), (b), and (c) show the mean best performance, k ÞÑ yp pdq,EI\nθpkq **(in black solid line**\n) and k ÞÑ yp pdq,**deriv-EI**\nθpkq (in red dotted line ) for strongly multimodal functions (θ \" 0.**2) and**\nd P t2, 3, 5u**. Plots (d), (e), and (f) give the mean time-to-target** s ÞÑ pk pjq,EIp*θ, s*q **(in black solid line) and**\ns ÞÑ pk pjq,deriv-EIp*θ, s*q **(in red dotted line).**\n\n\n\nFigure 7: Plots (a), (b), and (c) compare the mean best-so-far performances for the two acquisition criteria, k ÞÑ yp pdq,EI\nθpkq (in black solid line **) and** k ÞÑ yp pdq,**deriv-EI**\nθpkq (in red dotted line **), for**\nmoderatly multimodal functions (θ \" 0.5) and and d P t2, 3, 5u**. Plots (d), (e), and (f) compare the mean**\ntime-to-target, s ÞÑ pk pjq,EIp*θ, s*q **(in black solid line) and** s ÞÑ pk pjq,deriv-EIp*θ, s*q **(in red dotted line).**\n\n## Ei **Deriv-Ei**\n\nwill be well associated to a zero gradient of y ` λc, tend to the global minimum of y**. Such an approach**\nis well-studied for convex optimization problems, with bounds linking the choice of λ **to the degradation**\nin the optimal value of the objective function Wright & Nocedal (2006). In general however, the effect of the λ **decay law is difficult to understand. And the penalty adds a steep function increase at the edge of**\nX **that GPR-based metamodels will have difficulty to learn.**\nAlternatively, we propose to evaluate *a priori* the likelihood, noted ℓN , that the minimum of y **lies on the**\nboundary of X.\n\nThe ℓN **likelihood can be evaluated through a three-step procedure. Starting from the GP-based surrogate** model of y, noted YN **, the first step would be to look for positions that may correspond to local minima of**\ny, by running in parallel M \" 1 regularized Newton descent algorithms on the trajectories of YN **. These**\nminimizations would be in that case initialized at randomly **chosen points** x m 0 P X, 1 ď m ď M**, and we**\nwould denote by x m\n˚ **the obtained minimum when starting from** x m 0\n. Then, we generate Q \" **1 random**\nsamples of the Gaussian random vector pYN px 1\n˚q*, . . . , Y*N pxM\n˚qq**, which we denote by**\n\n$$(Y_{N}^{\\omega_{q}}({\\mathbf{x}}_{*}^{1}),\\ldots,Y_{N}^{\\omega_{q}}({\\mathbf{x}}_{*}^{M})),\\ 1\\leqslant q\\leqslant Q.$$\n$$(28)$$\n˚qq, 1 ď q ď Q. **(28)**\nThe indicator ℓN **is finally estimated by counting how often the minima of the draws are on the bounds,**\n\n$$\\ell_{N}:=\\frac{1}{Q}\\sum_{q=1}^{Q}\\frac{1}{{\\mathbf{x_{*}^{q}}}^{*}\\in\\mathbb{Z}}\\quad,\\quad{\\mathbf{x_{*}^{m}}}^{*}\\in\\arg\\operatorname*{min}_{1\\leqslant m\\leqslant M}Y_{N}^{\\omega_{q}}({\\mathbf{x_{*}^{m}}}).$$\n$$(29)$$\n\nDepending on the value of ℓN **, the method described in the rest of the article can be complemented in** one of the two following fashions. If ℓN is too large, the traditional EIN **acquisition criterion should be** used instead of deriv-EIN . Alternatively, ℓN **can be calculated specifically for each bound and if it is likely**\nthat some specific bounds are hit, then the corresponding variables can be set to these bounds, the BO\niteration being carried out with the deriv-EIN **criterion in the lower dimensional space.**\nNonetheless, the objective of this work was to come up with an **acquisition criterion applicable when the**\nminimum of y is not on the boundary of X**. We leave the continuation of the above analysis, based on the**\nlikelihood to have the optimum at a bound, as a perspective to **this work.**\n\n## 5.2 Summary And Further Perspectives\n\nIn the context of Bayesian optimization, this work proposes **a novel acquisition criterion allowing to**\nintegrate as additional *a priori* **the fact that interior minima are associated to zero first order derivatives**\nand positive definite Hessians. With this addition, a classical acquisition criterion such as the expected improvement takes on a feature of information theoretic criteria by characterizing the distribution of potential optima when the plain expected improvement accounts for all improving values of the function model. The new expected improvement with derivatives, called deriv-EI, does not need the derivatives of the true function. A computationally efficient approximation to deriv-EI is proposed in the article. It has been observed through several test cases that the new criterion allows significant gains in terms of function minimization at intermediate budgets of function evaluations. This benefit is larger when the function to minimize presents several local minima or the dimension is high, since in these cases the classical expected improvement is too exploratory in particular in areas near the bounds Siivola et al. (2018).\n\nAll the consequences of the proposed acquisition criterion **could not be investigated in this paper. In order** to simplify the interpretation of the results, all the comparisons between the classical EI and the deriv-EI\ncriteria have been carried out in *ideal* **configurations in the sense that the test functions are realizations**\nof the Gaussian process guiding the minimization. The hyperparameters of the GP characterizing its statistical properties are always known by construction of **the test functions. Therefore, it will be interesting** to study the sensitivity of Bayesian optimization with deriv-EI to the iterative estimation of the GP hyperparameters, as it happens in practice. In the same manner, only problems in moderate dimensions are implemented (d ď **5), as it seems important to restrict ourselves to cases for which the maximization of**\nthe EI and deriv-EI criteria can be sufficiently well solved. During the analysis of configurations in higher dimensions (d ě 10), the maximization of these criteria becomes a problem in **itself and the performances**\nof the EI and deriv-EI criteria turned out to be too dependent **on our ability to correctly maximize them.** Working at the definition of efficient methods to maximize the deriv-EI criterion would therefore be an appropriate continuation to this work.\n\nFinally, this work has focused exclusively on the EI acquisition criterion because it is a standard in BO, but other acquisition criteria should also benefit from the predictor's derivatives. For example, the EI criterion of Equation (12) can be adapted to the related probability of **improvement Kushner (1964): instead of**\nmaximizing x ÞÑ PpYN pxq ă yminq \" E\n\"1YN pxqăymin ‰**, we could maximize**\n\n$$\\mathbf{x}\\mapsto\\mathbb{E}\\left[1_{Y_{N}(\\mathbf{x})<y_{\\mathrm{min}}}\\,\\times\\,1_{\\mathbf{R}_{N}(\\mathbf{x})\\hat{c}Y_{N}(\\mathbf{x})\\in\\mathcal{B}(\\varepsilon),\\hat{c}^{2}Y_{N}(\\mathbf{x})\\in\\mathcal{M}+(d)}\\right].$$\n$$(30)$$\n\n‰. **(30)**\nIn the same manner, the well-known upper confidence bound acquisition criterion Srinivas et al. (2009) could be adapted by adding a penalty term on the derivatives, **which would make it possible to explicitly** play on the exploitation vs. exploration tradeoff in the same **way as the standard deviation. The new** point to be evaluated could for instance be sought as a solution to the following problem:\n\n$${\\mathbf{x}}^{(N+1)}\\in\\arg\\operatorname*{min}_{{\\mathbf{x}}\\in\\mathbb{X}}\\mu_{N}({\\mathbf{x}})-\\lambda_{1}\\sigma_{N}({\\mathbf{x}})+\\lambda_{2}\\left\\|\\mathbb{E}\\left[{\\hat{c}}Y_{N}({\\mathbf{x}})\\right]\\right\\|,$$\n$$(31)$$\nµN pxq ´ λ1σN pxq ` λ2 }ErBYN pxqs} , **(31)**\nwith λ1 and λ2 **two positive constants.**\n\n## Acknowledgments\n\nThis action benefited from the support of the Chair Stress Test, Risk Management and Financial Steering, led by the French Ecole polytechnique and its **foundation and sponsored by BNP**\nParibas. This research also benefited from the consortium in **Applied Mathematics CIROQUO**\n(https://doi.org/10.5281/zenodo.6581217**), gathering partners in technological and academia in the**\ndevelopment of advanced methods for Computer Experiments.\n\n## References\n\nJames Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl. 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For each x P X**, the means and**\ncovariance matrices of random vectors BYN pxq and pYN pxq, D2YN pxqq|BYN pxq \" 0 **are written**\n\n$$\\partial Y_{N}(\\mathbf{x})\\ \\sim\\ {\\mathcal{N}}\\left({\\dot{\\mathbf{m}}},{\\dot{\\mathbf{S}}}\\right),$$\n$$(Y_{N}(\\mathbf{x}),D^{2}Y_{N}(\\mathbf{x}))|\\partial Y_{N}(\\mathbf{x})=\\mathbf{0}\\ \\sim\\ {\\cal N}\\left(\\left(\\begin{array}{c}{{m}}\\\\ {{\\tilde{m}_{1}}}\\\\ {{\\vdots}}\\\\ {{\\tilde{m}_{d}}}\\end{array}\\right),\\left[\\begin{array}{c c c c}{{s^{2}}}&{{\\rho_{1,1}}}&{{\\cdots}}&{{\\rho_{1,d}}}\\\\ {{\\rho_{1,1}}}&{{\\tilde{s}_{1}}}&{{\\ddots}}&{{\\vdots}}\\\\ {{\\vdots}}&{{\\ddots}}&{{\\ddots}}&{{\\rho_{d-1,d}}}\\\\ {{\\rho_{d,1}}}&{{\\cdots}}&{{\\rho_{d,d-1}}}&{{\\tilde{s}_{d}}}\\end{array}\\right]\\right)$$\n,\nwhere the notation D2YN :\"\n`pB2YN q1,1, . . . ,pB2YN qd,d˘**refers to the diagonal terms of the matrix** B\n2YN .\n\nUsing the notations and the general expressions introduced in Section 3.1, the expressions of m and s 2 come from the conditioning of the pd ` 1q-dimensional Gaussian vector pYN pxq, BYN pxqq**, whose mean is** pµN pxq, BµN pxqq**, and whose covariance matrix is**\n\n$$\\left[\\begin{array}{l l}{C_{N}(x,x)}\\\\ {\\partial C_{N}(x,x)}\\end{array}\\right.$$\n\"CN px, xq BCN px, xq\n$$\\left.\\begin{array}{l}{{\\partial C_{N}(x,x)^{T}}}\\\\ {{\\partial^{2}C_{N}(x,x)}}\\end{array}\\right].$$\nBCN px, xq B2CN px, xq\nApplying the conditioning formula yields, m :\" ErYN pxq|BYN pxq \" 0s \" µN px**q ` B**CN px, xq JB\n2CN px, xq\n´1p0 ´ BµN pxqq ,\ns 2:\" VarpYN pxq|BYN pxq \" 0q \" CN px, x**q ´ B**CN px, xq JB\n2CN px, xq\n´1BCN px, xq .\n\nThe expressions for the mean and covariance matrix of the Gaussian vector YN pxq, D2YN pxq|BYN pxq \" 0, i.e., the symbols m: i, s:i and ρi,j **in Equation (19), are obtained following the same conditioning principle, but**\napplied to the Gaussian vector pYN pxq, D2YN pxq, BYN pxqq, whose mean is pµN pxq, D2µN pxq, BµN pxqq**, and**\nwhose covariance matrix is made of properly ordered terms such as Cov´B\n2YN\nBx 2 i pxq, BYN\nBxjpxq\n¯\"\nB\n3CN\nBx 2 i Bxjpx, xq, Cov ´B\n2YN\nBx 2 i pxq, B\n2YN\nBxjpxq\n¯\"B\n4C\nBx 2 i B2xjpx, xq, *. . .*.\n\nThe conditioning can be applied one more time to pYN pxq, D2YN pxq|BYN pxq \" 0q **to account for an**\nobservation of YN pxq**, leading to**\n\n$$(D^{2}Y_{N}(\\mathbf{x}))_{i}|\\partial Y_{N}(\\mathbf{x})=\\mathbf{0},Y_{N}(\\mathbf{x})=y\\sim{\\cal N}(\\bar{m}_{i}+\\rho_{1i}(y-m)/s^{2},\\bar{s}_{i}^{2}-\\rho_{1i}^{2}/s^{2}).\\tag{32}$$\n\nFor p P t1, 2u**, let us first assume that the off-diagonal terms of** B\n2Y:N **can be neglected. In that case,**\nensuring that B\n2YN is in M`pdq comes down to ensuring that its diagonal terms are positive, **i.e. ensuring**\nthat D2YN is in r0, `8rd**, and we can write (using the former notations):**\n\nderiv-EIN pxq :\" ż ymin y\"´8 ż y9 PEpx,εq ż Y: PM`pdq pymin ´ yq pdPpy, y9 ,Y: q y:Pr0,`8rd pymin ´ yq pfBYN pxqpy9qfYN pxq,D2YN pxq|BYN pxq\"y9py, y:qdydy9 dy:. « ż ymin y\"´8 ż y9 PEpx,εq ż\n$$(33)$$\nAs it is necessary for a matrix to have positive terms on its diagonal to be positive definite, this approximation is an overestimation of the number of trajectories that are actually positive definite. In addition, if ε, the size of the ellipsoid centered at 0 to which BYN pxq **belongs, is sufficiently small, it is possible to**\napproximate fBYN pxqpy9q by fBYN pxqp0q for any y9 in Epx, εq**, which leads to:**\nderiv-EIN pxq\n\n$$\\operatorname{EI}_{N}(x)$$\n$\\approx\\text{Vol}(\\mathcal{E}(\\mathbf{x},\\varepsilon))f_{\\mathcal{D}Y_{N}(\\mathbf{x})}(0)\\int_{y=-\\infty}^{y_{\\text{min}}}\\int_{\\mathbf{y}\\in[0,+\\infty[^{d}}(y_{\\text{min}}-y)^{p}f_{Y_{N}(\\mathbf{x}),D^{2}Y_{N}(\\mathbf{x})]\\mathcal{D}Y_{N}(\\mathbf{x})=\\mathbf{0}}(y,\\tilde{\\mathbf{y}})dyd\\tilde{\\mathbf{y}}.$\n$$(34)$$\n\nwhere VolpEpx, εqq is the volume of Epx, εq.\n\nWe further assume that for any y P R, the components of D2YN pxq conditioned by the event pBYN pxq \"\n0, YN pxq \" yq are statistically independent. In other words, a trajectory which passes through px, yq and which is flat is assumed to have independent curvatures. In this case, the density of D2YN pxq|pBYN pxq \"\n0, YN pxq \" yq **is a product of univariate densities. This leads to,**\n\nż ymin y\"´8 ż y:Pr0,`8rd pymin ´ yq pfYN pxq,D2YN pxq|BYN pxq\"0py, y:qdydy: \" ż ymin y\"´8 ż y:Pr0,`8rd pymin ´ yq pfYN pxq|BYN pxq\"0pyqfD2YN pxq|BYN pxq\"0,YN pxq\"ypy:qdydy: p2πq d`1 2 exp ˆ´ py ´ mq 2 2s 2 ˙¨ ˝ź d s:i b1 ´ ρ1i ss:i ˛ « ż ymin i\"1 ż `8 y:i\"0 exp ˆ´ py:i ´ pm: i ` ρ1ipy ´ mq{s 2qq2 2s: 2 ip1 ´ ρ 2 1i{ps 2s: 2 iqq ˙dy:i y\"´8 pymin ´ yq p ' dy s . (35)\n\nIf we now perform the following variable changes: z :\" py ´ mq{s, zmin :\" pymin ´ mq{s, ri:\" ρ1i{pss:iq and τ:i \" m: i{s:i**, it comes:**\n\np2πq d`1 2 exp ˆ´ py ´ mq 2 2s 2 ˙¨ ˝ź d s:i b1 ´ ρ1i ss:i ˛ ż ymin y\"´8 pymin ´ yq p i\"1 ż `8 y:i\"0 exp ˆ´ py:i ´ pm: i ` ρ1ipy ´ mq{s 2qq2 2s: 2 ip1 ´ ρ 2 1i{ps 2s: 2 iqq ˙dy:i ' dy s p2πq d`1 2 exp ˆ´ z 2 2 ˙ ˜ź d z:i\"´τ:i exp ˆ´ pz:i ´ rizq 2 2p1 ´ r 2 iq ˙ a dz:i 1 ´ r 2 i ¸ dz \" s p ż zmin i\"1 ż `8 z\"´8 pzmin ´ zq p i\"1 Φ ˜τ:i ` riz a1 ´ r 2 i ¸ dz. \" s p ż zmin z\"´8 pzmin ´ zq p ?2πexp ˆ´ z 2 2 ˙ź d (36)\n\nRecalling that Φ and φ are respectively the CDF and the PDF of the standard Gaussian **variables, the**\nformer expression can be further simplified by introducing the following first order Taylor expansion of the function Φ,\n\n$$\\Phi\\left(\\frac{\\tilde{r}_{i}+r_{i}z}{\\sqrt{1-r_{i}^{2}}}\\right)\\approx\\Phi\\left(\\frac{\\tilde{r}_{i}}{\\sqrt{1-r_{i}^{2}}}\\right)+\\frac{r_{i}z}{\\sqrt{1-r_{i}^{2}}}\\phi\\left(\\frac{\\tilde{r}_{i}}{\\sqrt{1-r_{i}^{2}}}\\right),\\tag{37}$$\n\nand by truncating to the first polynomial orders, so that:\n\ni\"1 Φ ˜τ:i ` riz a1 ´ r 2 i ¸ dz z\"´8 pzmin ´ zq p ?2πexp ˆ´ z 2 2 ˙ź d ż zmin i\"1 ˜ Φ ˜ a τ:i 1 ´ r 2 i ¸ ` riz a1 ´ r 2 i φ ˜ a τ:i 1 ´ r 2 i ¸¸ dz « ż zmin z\"´8 pzmin ´ zq p ?2πexp ˆ´ z 2 2 ˙ź d (38) i\"1 Φ ˜ a τ:i 1 ´ r 2 i ¸ ż zmin « ź d z\"´8 pzmin ´ zq pp1 ` zaq ?2πexp ˆ´ z 2 2 ˙dz ,\n$$\\quad(38)$$\n$$(39)$$\nwhere\n\n$$a=\\sum_{i=1}^{d}\\frac{r_{i}}{\\sqrt{1-r_{i}^{2}}}\\,\\frac{\\phi\\left(\\frac{\\vec{r}_{i}}{\\sqrt{1-r_{i}^{2}}}\\right)}{\\Phi\\left(\\frac{\\vec{r}_{i}}{\\sqrt{1-r_{i}^{2}}}\\right)}\\,\\,.$$\n\nIt finally comes deriv-EIN pxq « LikelyMinpxq ˆ cond-EIppqpxq, **(40)**\nwith\n\n$$\\text{LikelyMin}(\\mathbf{x}):=\\text{Vol}(\\mathcal{E}(\\mathbf{x},\\varepsilon))f_{\\widehat{\\varepsilon}\\mathcal{V}_{N}(\\mathbf{x})}(\\mathbf{0})\\prod_{i=1}^{d}\\Phi\\left(\\frac{\\tilde{r}_{i}}{\\sqrt{1-r_{i}^{2}}}\\right)$$ $$=v\\varepsilon^{d}\\times\\exp\\left(-\\frac{\\tilde{\\mathbf{m}}^{T}\\tilde{S}^{-1}\\tilde{\\mathbf{m}}}{2}\\right)\\times\\prod_{i=1}^{d}\\Phi\\left(\\frac{\\tilde{r}_{i}}{\\sqrt{1-r_{i}^{2}}}\\right),$$\n$$(40)$$\n$$(41)$$\n\nwhere v is a constant independent of x and ε**, and with**\n\n$$\\operatorname{cond-EL}(^{p})\\left(\\boldsymbol{x}\\right):={\\begin{cases}s\\left((z_{\\mathrm{min}}-a)\\Phi(z_{\\mathrm{min}})+\\phi(z_{\\mathrm{min}})\\right)&{\\mathrm{if~}}p=1,\\\\ s^{2}\\left((1+z_{\\mathrm{min}}^{2}-2a z_{\\mathrm{min}})\\Phi(z_{\\mathrm{min}})+(z_{\\mathrm{min}}-2a)\\phi(z_{\\mathrm{min}})\\right)&{\\mathrm{if~}}p=2.\\end{cases}}$$\n$$\\left(42\\right)$$\n\nThe expression when the constraint on the second order derivatives is not considered can be recovered by making τ:i tend to infinity. In this case, a **tends to 0 and** śd i\"1 Φ\nˆ\n?τ:i 1´r 2 i\n˙**tends to 1.**\nIn summary, the approximation provided by Eq. (40) is based on the following four assumptions:\n- the off-diagonal terms of B\n2Y:N **can be neglected,**\n- the value of ε is sufficiently small for fBYN pxqpy9q to be approximated by fBYN pxqp0q **for any** y9 in Epx, εq,\n- the components of D2YN pxq conditioned by the event pBYN pxq \" 0, YN pxq \" yq **are statistically**\nindependent,\n- the first order Taylor expansion of the function Φ provided in Eq. (37) holds.\n\n## Numerical Assessment Of The Proposed Approximation\n\nIn order to numerically evaluate the quality of the approximation of deriv-EIN **, we implement a second**\nMonte-Carlo based approximation :\n\n$$\\text{deriv-El}_{N}(\\mathbf{x})\\approx\\widehat{d}(\\mathbf{x}):=\\text{Vol}(\\mathcal{E}(\\mathbf{x},\\varepsilon))f_{\\mathcal{U}_{N}(\\mathbf{x})}(\\mathbf{0})\\frac{1}{M}\\sum_{m=1}^{M}1_{Y_{m}\\leqslant y_{\\text{min}}}1_{\\widehat{\\mathbf{Y}}_{m}\\in\\mathcal{M}^{+}(\\widehat{\\mathbf{x}})}(y_{\\text{min}}-Y_{m}),\\tag{43}$$\n\nwhere !Ym,Y: m\n)M\nm\"1 gathers M **independent realizations of** pYN pxq, B\n2YN pxqq|BYN pxq \" 0**. The term**\nVolpEpx, εqqfBYN pxqp0q **is explicitly calculated in the same way in both approximations.**\nFocusing on the test functions listed in Section 4.2 in dimensions d P t2, 3, 5u**, and limiting ourselves to** N P t2d, 5d, 10du, we uniformly sample 1000 points x1*, . . . ,* x**1000** in r0, 1s d**. We compute the coefficient**\nof determination R2 between the values of !LikelyMinpxj q ˆ **cond-EI**ppqpxj q\n)M\nj\"1\n(the proposed and fast approximation of deriv-EIN **) and the values of** !dppxj q\n)M\nj\"1\n(the time consuming Monte-Carlo approximation of deriv-EIN ). The mean value and standard deviation of these coefficients **of determination obtained**\nwhen repeating 10 times the whole process are finally summarized in Table 1. In each case, we observe high coefficients of determination, which justifies the use of **the fast approximation.**\n\n## B Expression Of The Analytical Test Functions\n\nThe functions y 1D and y 2D **which were considered in Section 4.1 have the following expressions (notice the**\noffset made such that the minimum of the functions is 0):\n\n$$y_{0}^{\\mathrm{1D}}:\\left\\{\\begin{array}{c c c}{{[0,1]}}&{{\\rightarrow}}&{{\\mathbb{R}}}\\\\ {{x}}&{{\\mapsto}}&{{\\cos(6\\pi x+0.4)+(x-0.5)^{2}}}\\end{array}\\right.$$ $$y^{\\mathrm{1D}}(x)=y_{0}^{\\mathrm{1D}}(x)-\\operatorname*{min}_{z\\in[0,1]}y_{0}^{\\mathrm{1D}}(z),\\ x\\in[0,1],$$\n$$y_{0}^{20}:\\left\\{\\begin{array}{r c l}{{[0,1]^{2}}}&{{\\to}}&{{\\mathbb{R}}}\\\\ {{(x_{1},x_{2})}}&{{\\to}}&{{10+x_{1}+\\left(15x_{2}-{\\frac{5(15x_{1}-5)^{2}}{(4\\pi)^{2}}}+{\\frac{5(15x_{1}-5)}{\\pi}}-6\\right)^{2}+10\\cos(15x_{1}-5)\\left(1-{\\frac{1}{5\\pi}}\\right)^{2}}\\end{array}\\right.$$\n\n| d | θ | N | mean of R2 | standard deviation of R2 |\n|-----|-----|-----|--------------|----------------------------|\n| 2 | 0.2 | 4 | 0.94 | 0.04 |\n| 2 | 0.5 | 4 | 0.96 | 0.03 |\n| 2 | 0.2 | 10 | 0.94 | 0.02 |\n| 2 | 0.5 | 10 | 0.95 | 0.02 |\n| 2 | 0.2 | 20 | 0.95 | 0.02 |\n| 2 | 0.5 | 20 | 0.98 | 0.02 |\n| 3 | 0.2 | 6 | 0.96 | 0.02 |\n| 3 | 0.5 | 6 | 0.96 | 0.06 |\n| 3 | 0.2 | 15 | 0.95 | 0.01 |\n| 3 | 0.5 | 15 | 0.98 | 0.02 |\n| 3 | 0.2 | 30 | 0.96 | 0.02 |\n| 3 | 0.5 | 30 | 0.98 | 0.01 |\n| 5 | 0.2 | 10 | 0.93 | 0.04 |\n| 5 | 0.5 | 10 | 0.97 | 0.03 |\n| 5 | 0.2 | 25 | 0.92 | 0.02 |\n| 5 | 0.5 | 25 | 0.96 | 0.03 |\n| 5 | 0.2 | 50 | 0.94 | 0.01 |\n| 5 | 0.5 | 50 | 0.95 | 0.06 |\n\nTable 1: Coefficients of determination R2 between the proposed, fast, approximation of deriv-EIN **and a**\nMonte-Carlo approximation.\n\n$$y^{\\mathrm{2D}}(\\mathbf{x})=y_{0}^{\\mathrm{2D}}(\\mathbf{x})-\\operatorname*{min}_{\\mathbf{z}\\in[0,1]^{2}}y_{0}^{\\mathrm{2D}}(\\mathbf{z}),\\ \\mathbf{x}\\in[0,1]^{2}.$$\n\n## C Generation Of Test Functions As Realizations Of A Gaussian **Process**\n\nWe describe here how deterministic functions are defined as particular realizations of a centered Gaussian process Z indexed by x P X, where X **is a compact subset of** R\nd and the covariance function of Z **is denoted**\nby C**. The process consists of the following steps.**\n1. We start by generating a design of experiments X :\" tx1*, . . . ,* xN u of large size N **covering as**\nmuch as possible X.\n\n2. We then project Z on this design of experiments, and we note z \" pz1*, . . . , z*N q the vector containing the values of Z realized at x1*, . . . ,* xN .\n\n3. By Gaussian conditioning, the function x ÞÑ rpxq T R\n´1z **can thus be seen as the continuous**\nextension of a realization of Z whose projection in X is equal to z**, where for all** x P X,\n\n$$\\mathbf{r}(\\mathbf{x})={\\left(\\begin{array}{l}{C(\\mathbf{x},\\mathbf{x}_{1})}\\\\ {\\vdots}\\\\ {C(\\mathbf{x},\\mathbf{x}_{N})}\\end{array}\\right)},\\ \\ \\mathbf{R}:={\\left[\\begin{array}{l l l}{C(\\mathbf{x}_{1},\\mathbf{x}_{1})}&{\\cdots}&{C(\\mathbf{x}_{1},\\mathbf{x}_{N})}\\\\ {\\vdots}&{\\cdots}&{\\vdots}\\\\ {C(\\mathbf{x}_{N},\\mathbf{x}_{1})}&{\\cdots}&{C(\\mathbf{x}_{N},\\mathbf{x}_{N})}\\end{array}\\right]}$$\n\nIt is clear that this type of construction strongly depends on the dimension of X**. Indeed, the larger the**\nsize of X, the closer the constructed function will look like a particular realization of Z**. In addition, the** larger the input space dimension, d, the more points will be needed in X **for the continuous extension to be**\nrelevant. For the examples treated in Section 4.2, X **is defined as the concatenation of a two-level factorial**\ndesign (in order to cover all the vertices of X \" r0, 1s d**) and a space filling LHS design of size 100** ˆ d."},"reviews":{"kind":"string","value":"Review 1:\nSummary: This work proposes the use of derivative & Hessian estimate of the target function in its acquisition criterion for BO. It provides an approximation of such criterion, and show that it indeed helps finding better estimate quicker than the one without those information on toy examples.\n\nStrengths and Weaknesses: Overall, this is a good and solid paper I think, if some of the concerns are addressed.\n\nStrength: \n1. The idea is very simple yet very general and practically useful.\n2. The paper is overall very well written and the idea is clear.\n3. It talks about some limitations too (the case the minima are on the edge etc.)\n\nWeakness:\n1. The major weakness is the lack of detailed step-by-step explanations on the derivation of approximation (especially Appendix A)\n2. Why focusing on certain kernels, EI etc. are not very well motivated\n\nMinor weakness:\n1. Some typos\n2. Clarity of figures\n\nRequested Changes: 1. Please add more detailed step-by-step explanations on Appendix A. Especially, although I might be missing something here, please define $m$, $\\dot{x}$, etc. etc. and please elaborate the approximate equality on (34) and (35) more so that I can easily verify the correctness.\n\n2. Please motivate the focus on the class of kernels used in this work and on EI; also please confirm that the experimental results are not very specific but that it should be the case this advantage is seen in wider domains of problems.\n\n3. Figure 1 is a bit hard to parse; please provide a bit more detailed explanation \n\nMinor points:\n1. In abstract, it says \"dimension 2, 3, and 5\" and it is a bit confusing why these specific numbers are important; otherwise, should say \"low dimension problems\" or so.\n2. a d-dimensional real space -> the?\n3. 4th row introduction: the parenthesis seems not to be closed\n4. What do you mean by \"BO is difficult to beat\"? it sounds a bit strange\n5. Please confirm that you use $C^2$ as twice differentiable function class, not twice continuously differentiable.\n6. What is \"MatÃľrn kernels\"? Matern?\n7. in page 6 \"A\" -> \"Appendix A\"\n8. in 12 what do you mean by \"These functions are call\"?\n9. in 12 \"Section ??\" please fix this\n10. in page 13 \"will typicallybe\" -> \"will typically be\"\nfix other minor typos as well.\n\nBroader Impact Concerns: N/A\n\n==================================================\n\nReview 2:\nSummary: The authors propose a variant of the well-known expected improvement (EI) acquisition function for (Gaussian-process-based) Bayesian Optimization. The proposed acquisition function restricts the original EI acquisition function only to consider function values of the GP posterior near local minima, i.e., having an almost vanishing gradient and a positive-definite Hessian matrix. The authors then introduce an approximation of said objective that avoids the necessity of Monte-Carlo approximations for evaluating the acquisition function. Subsequently, they compare the resulting BO method to the EI baseline, showing accelerated minimization of the chosen evaluation functions.\n\nStrengths and Weaknesses: Strengths:\n* The paper is well organized and written. As someone being familiar with the topic of BO but being rather an \"end-user\" of the methods, I had no issues following the paper.\n* I enjoyed the open discussion about the assumptions and limitations of the proposed method as well as the emphasis on it realizing a different tradeoff between exploration and exploitation than EI.\n* From a technical side, I enjoyed the use of the joint distribution over function values and first-and second-order derivatives.\n\nWeaknesses:\n* The proposed approach's motivation mostly stems from intuition, with no formal or statistical reason for why incorporating derivatives of GPs should improve performance (I'll detail that below).\n* The paper also lacks additional evaluations on other functions (detailed below) and an extended discussion of how and when the derived approximation to the deriv-EI objective fails.\n\nRequested Changes: * Minor: I would welcome legends in all Figures such that a reader does not need to search for the meaning of a line in the text caption.\n* Minor: There is a broken reference on Page 12\n* Minor: The statement that deriv-EI leads to lower, better values of $y^{1D}$ in Figure 2 for intermediate values of k seems overstated given the plot in Figure 2. Also why is Figure 2(a) in log-space compared to the others?\n* Minor: Could the authors clarify why they normalize the gradient with the square root of the covariance in Eq (12)?\n\n* The paper would benefit from a more extended discussion on why only looking at the minima of the \"imagined\" functions should improve optimization behavior. Looking at Figure 1(a), the number/location of the GP minima does not need to correlate with the minima of the true function and consequently, focusing only on the local minima of the GP may improve or degrade performance.\n* I would like to see more evaluations of the approximation of the deriv-EI objective. Table 1 shows that the objective works reasonably well for a low number of function samples but does not show how it evolves with an increasing number of samples. Given that, e.g., Figure 2(d) shows high levels of improvement of deriv-EI over EI for around N=13 data points on the corresponding two-dimensional function, it would be interesting to see how good the approximation is for this number of data points. I recommend that the authors add a hindsight analysis of the conducted experiments by comparing the approximated deriv-EI values with MC approximations for the generated optimization traces, thereby showing the approximation quality deriv-EI for the conducted optimizations. Additionally, it would be insightful to have an \"adversarial scenario\", in which the approximation clearly fails. I imagine that this happens with strong off-diagonal hessian terms, given that these are ignored in the approximation of deriv-EI.\n* Finally, I believe that adding the following experiments to Section 5 would strengthen the overall paper:\n * An evaluation on functions for which the maxima are on the border of the search space. Although the authors clearly state that the method is not well-suited for these scenarios, it would be interesting to see whether and how much worse it is compared to EI.\n * An evaluation on a real-world task, e.g., the [(surrogate) SVM task](https://github.com/automl/HPOBench/blob/master/hpobench/benchmarks/surrogates/svm_benchmark.py) used in [1]\n\n\n[1] Falkner, S., Klein, A., & Hutter, F. (2018, July). BOHB: Robust and efficient hyperparameter optimization at scale. In International conference on machine learning (pp. 1437-1446). PMLR.\n\nBroader Impact Concerns: No broader impact concerns\n\n==================================================\n\nReview 3:\nSummary: The paper proposes deriv-EI, a BO acquisition function that is an extension of expected improvement (EI). deriv-EI selects the next input to sample based not only on the improvement score, but on the likelihood that an input is a local minima, which is the case if the sample path gradient is 0 and the Hessian is positive semidefinite. This is possible because the derivatives of a GP are also GPs. deriv-EI is a tractable approximation to these criteria. deriv-EI is shown to outperform EI on synthetic test functions of dimension 1, 2, 3 and 5.\n\nStrengths and Weaknesses: ### Strengths:\n1. The central claim of the paper is met: deriv-EI appears to empirically outperform EI on the provided test functions.\n2. The idea is interesting and makes sense. There seems to be potential for this technique to be more generally applicable to other acquisitions beyond EI.\n3. The exposition is clear.\n\n### Weaknesses:\n1. For an algorithm with no theoretical guarantees, the empirical evaluation is somewhat narrow. All the test functions are synthetic and of low dimension. The empirical evaluation would be stronger if it included real world experiments, for example, hyperparameter tuning.\n2. While EI was originally proposed as for the noiseless evaluations setting, the modern BO research landscape almost always considers the case of noisy evaluations in which the observed values $y = f(x) + \\epsilon$ where $\\epsilon$ is a noise random variable. The proposed algorithm's practical usefulness is less proven when the experiments are run with noiseless observations.\n3. deriv-EI assumes that the global minima is within the interior of the search space, as opposed to on the boundary, an assumption that EI does not make and has to be handled as discussed in Sec 5.1. Furthermore, deriv-EI depends on an additional hyperparameter $\\epsilon$ which controls the size of the ellipsoid under which a gradient is taken to be sufficiently small. These additional complications provide a less compelling case for deriv-EI to be a drop-in replacement for EI. Can the authors explain the statement \"The precise choice of ε has thus no impact\"? Surely if we take $\\epsilon$ to be very large, the gradient condition disappears?\n4. A more complete discussion on related works also using GP derivatives would be helpful, e.g., explaining if this paper is related to \"Bayesian Optimization with Gradients\" (Wu et al., 2017) and other related papers.\n\nRequested Changes: ### Would strengthen work:\n1. Include real world experiments.\n2. Run experiments with noise.\n3. A more complete discussion on related works also using GP derivatives.\n\nBroader Impact Concerns: None.\n\n==================================================\n\nMetareview:\nRecommendation: Accept with minor revision\n\nComment: All the reviewers found that the claims were justified by evidence provided. The ratings provided by reviewers, however, were somewhat mixed with one Accept, one Leaning Accept and one Leaning Reject. Multiple reviewers leaned towards reject before the discussion period. One of them seems to have been satisfied by the author response. The other remaining reviewer cited the lack of response as a reason for their Leaning Reject rating, though they noted that they wouldn't be strongly opposed to acceptance. The authors have since responded, but after the reviewer posted their decision (and they haven't checked the response since - nor are they required to given the timeline). It seems that the authors addressed most of the reviewers' concerns within the response period and have made substantive changes to their manuscript. Having checked the authors' response, it seems that they have satisfied the concerns of the remaining negative reviewer. I appreciate that the authors updated the manuscript and highlighted the changes since the submission. Therefore, and given the two accepts, I am recommending accept with minor revisions. The minor revisions include the proposed changes from the rebuttal. Most of these are already worked into the manuscript, but obviously they should no longer be highlighted for the camera ready version. Given the one reviewers' concerns, I would like to verify that these changes have made it into the camera ready.\n\n==================================================\n\n"}}},{"rowIdx":88,"cells":{"paper":{"kind":"string","value":"# Large Language Models As Your Personal Data Scientist\n\nAnonymous authors Paper under double-blind review\n\n## Abstract\n\nLarge Language Models (LLMs) have contributed to massive performance improvements for various language understanding and generation tasks; however, their limits are yet to be fully explored for \"ill-defined\" complex tasks. One such task is conversational data science, where a user can talk to an intelligent agent to explain their data science needs, and the agent will serve the user by engaging in a conversation with them like any human data science would do and, accordingly, formulate and execute precise Machine Learning tasks.\n\nAlthough this is a very ambitious goal, given the recent developments in LLMs, a fully functional conversational data science system seems quite achievable in the near future.\n\nThrough an in-depth case study in this paper, we delved into the potential of employing LLMs as a solution to conversational data science. We hope that our findings will not only broaden the horizons of NLP research but also bring transformative changes in future AI technology.\n\n## 1 Introduction\n\nThe advent of Large Language Models (LLMs) is heralding a new era in the field of Natural Language Processing (NLP) research, opening the door to a plethora of untapped opportunities. These groundbreaking models have showcased their versatility in various tasks, including news summarization, paraphrase comprehension, inference generation, scientific writing improvement, code bug detection, and machine translation Fan et al. (2023). In addition, innovative prompt engineering techniques have significantly extended these capabilities Zhao et al. (2023). Despite these strides, the upper limit of LLMs' potential remains unknown. Testing LLMs in more complex and dynamic scenarios is essential to grasp this boundary, revealing a significant research opportunity.\n\nOne such complex and dynamic scenario is conversational data science, where a user can perform data science tasks on their own data sets merely by conversing with an intelligent agent. While such a data science task assistant does not exist yet, given the recent developments in LLMs, a fully functional conversational data science solution seems not only achievable but also quite feasible in the near future. At present, the closest endeavor toward this direction is \"Automated Machine Learning\" (AutoML), which is mostly inaccessible to the general public. Indeed, as illustrated in Figure 1, the existing AutoML process is fraught with challenges for the end users (e.g., domain experts), who are required to possess a profound understanding of the underlying machine learning tasks, objective functions, optimization techniques, training and validation process, etc. However, this level of understanding often eludes these end users, leading them to rely on the expertise of data scientists to perform these tasks Sarkar et al. (2023). This reliance manifests as time-consuming and inefficient dialogues with data scientists, thus obstructing AutoML from actualizing its potential for true automation Karmaker et al. (2021). To address these limitations, we envision a novel conversational AI agent called **VIDS** (Virtual Interactive Data Scientist), capable of assisting users in conducting basic Data Science tasks through intuitive, natural conversations without requiring in-depth knowledge of the underlying core Machine learning (ML) processes.\n\nThis agent's key functionalities are as follows: 1) accurately comprehend the user's prediction goals and, consequently, 2) formulate a precise ML task, 3) curate data sets and assign model hyper-parameters accordingly, 4) execute an autoML pipeline, and 5) communicate results effectively. To realize VIDS, we utilized\n\n\n\nFigure 1: A flowchart showing the end-to-end machine learning process. This figure highlights points of interaction between domain experts and data scientists, along with bottlenecks Karmaker et al. (2021). In this paper, we focus on automating three steps in the chat cycle with the largest communication bottleneck: Task Formulation (TF), Prediction Engineering (PE), and Result Summarization and Recommendation\n(RSR).\nLLMs to build a natural interface between the users and existing AutoML tools (like Scikit-Learn), which in turn, allowed us to approach this ambitious goal with a viable solution.\n\nWe designed the architecture of VIDS with four dialogue states: Data Visualization, Task Formulation, Prediction Engineering, and Result Summary and Recommendation. Each state marks a unique conversation phase, impacting the overall user-system interaction. Multiple LLM instances, serving as \"micro-agents\",\nensures a cohesive conversation flow, granting us granular control over the conversation's progression. In summary, we designed and developed an end-to-end system that demonstrates the viability of *Conversational* Data Science by evaluating the potency of LLMs in solving such an ill-defined complex task. Our main contribution in this paper is to make this connection between two seemingly independent yet complimentary research areas, i.e., LLMs and AutoML, and thus, build a novel solution for realizing Conversational Data Science. In terms of research questions, we explored the following: 1) Is conversational AI\na feasible way to perform data science tasks? 2) How accurate are LLMs in framing and solving ill-defined complex data science tasks? 3) What are the common challenges involved with systems like VIDS?\n\n## 2 Related Works\n\nLLMs like ChatGPT 1, BLOOM Workshop et al. (2023), LLama Touvron et al. (2023), PaLM Chowdhery et al. (2022), etc. (Chowdhery et al. (2022); Brown et al. (2020); Zeng et al. (2022); Zhang et al.\n\n(2022); Ouyang et al. (2022)), have reshaped the field of NLP research with their broad applicability and impressive performance across diverse tasks. For instance, they have been utilized in automatic news summarization Zhang et al. (2023), efficient healthcare research Sallam (2023), software bug fixing Surameery\n& Shakor (2023), and machine translation Jiao et al. (2023), showcasing their ability to handle diverse tasks and deliver practical real-world solutions. In benchmarking studies, LLMs have outperformed traditional models like BERT in many tasks and achieved comparable performance in others Zhong et al. (2023). These models' performances have been further enhanced through various prompting techniques such as chain of thought prompting Wei et al. (2023), self-consistency prompting Wang et al. (2022), synthetic prompting Shao et al. (2023), generated knowledge prompting Liu et al. (2022), and input-output prompting Ma et al. (2023). These techniques improve the models' reasoning, adaptability, and coherence over multiple turns, highlighting their flexible nature and inherent knowledge. Moreover, research like FrugalGPT Chen et al. (2023) has focused on using LLMs more sustainably and efficiently, matching the performance of top LLMs at a significantly reduced cost.\n\nDespite these advancements, the true upper limit of LLMs remains a tantalizing unknown, calling for further exploration of more complex, ill-defined tasks. One such complex task is to provide conversational data science support for end users, which are often ill-defined unseen tasks that could truly test the potential of LLMs. While the machine learning and systems communities have made substantial progress over the past decade in automating various aspects of data science pipelines, some areas still remain underdeveloped. For example, *Data Cleaning and visualization* Ilyas et al. (2015); Chu et al. (2016), *Feature Engineering* Katz et al. (2016); Kanter & Veeramachaneni (2015); Mountantonakis & Tzitzikas (2017); van den Bosch (2017);\nKhurana et al. (2017); Kaul et al. (2017), *Learning and Parameter Tuning* Bergstra & Bengio (2012); Snoek et al. (2012); Hutter et al. (2011); Bergstra et al. (2011); Bengio (2012); Bergstra et al. (2013); Swersky et al. (2013); Maclaurin et al. (2015), *Alternative Models Exploration, Testing and Validation* Thornton et al. (2013); Feurer et al. (2015); Swearingen et al. (2017); Zoph & Le (2016); Zoph et al. (2017); Liu et al. (2017b;a); Real et al. (2018); Pham et al. (2018); Baker et al. (2017) have all seen significant advancements. Yet, the inherently complex and human-centric parts like Prediction Task Formulation and Result Summarization and Recommendation have not seen the same level of advancement in terms of automation.\n\nThe limited progress in these areas not only presents a unique research opportunity but also highlights the potential impact of automation in data-reliant industries.\n\nWhile LLMs have demonstrated their ability to understand intricate language patterns and generate coherent, contextually appropriate responses, they are yet to be fully leveraged for complex human-centric tasks. This underexplored area presents both a challenge and an opportunity. On one hand, it highlights the hurdles such as managing the complexity of handling \"ill-defined\" tasks, ensuring consistency in responses, and mitigating the tendency of LLMs to generate implausible or 'hallucinated' information Xu et al. (2022); Mündler et al.\n\n(2023); Ji et al. (2023). On the other hand, it underscores the untapped potential of LLMs in bridging the automation gap in data science. This work aims to shed light on the potential of this interdisciplinary research area and contribute to the growing body of knowledge on the applications of LLMs in conversational data science.\n\n## 3 Model Architecture\n\nThis section delves into the methodology and technical details of VIDS, articulating the intricate interplay between overarching structures and localized nuances. Central to our model are four distinct dialogue states -\nData Visualization, Task Formulation, Prediction Engineering, and Result Summary and Recommendation, with each representing a unique phase in the conversation and contributing significantly to the overall usersystem interaction. VIDS employs multiple stateless global micro-agents, functioning independently of any state-related data or history, to create an overarching structure that enables fluid transitions throughout the 1https://openai.com/blog/chatgpt\n\n\n\nC\n\no n v e r s a tio n M\n\na n a g e r S\n\nD\n\nFigure 2: The state diagram of the VIDS dialogue system. The gray boxes are different states of the conversation history, the dark yellow boxes are global micro-agents, and the purple boxes are the local micro-agents.\ndialogue, irrespective of the specific state. This stateless design ensures a smooth narrative flow and avoids complications of state-dependent biases or entanglements, thus bolstering the versatility and adaptability of our dialogue system. Alongside these global agents, local micro-agents, each tailored to a specific dialogue state, proficiently handle the nuances of user utterances and conversation contexts, facilitating smooth transitions between states in line with the evolving dialogue. VIDS' strength lies in this symbiotic relationship between the global and local micro-agents across the different dialogue states.\n\n## 3.1 Global Micro-Agents\n\nVIDS deploys three global micro-agents across all dialogue states, which are briefly discussed below.\n\n## 3.1.1 State Detector\n\nAs a cornerstone of conversation management, VIDS deftly incorporates a number of distinct states, each one reflecting the various phases of interaction with the end user, as illustrated in Figure 2. The system is initialized with the \"*Data Visualization*\" state, simplifying the data exploration phase. It then progresses to the\n\"*Task Formulation*\" state, interpreting the user's intended task. The subsequent \"*Prediction Engineering*\"\nstate revolves around crafting training/testing data partitions and initializing model hyper-parameters based on the already defined tasks. The interaction culminates in the \"*Result Summarization and Recommendation*\" state, summarizing results and offering pertinent recommendations. Leveraging the latest context, current dialogue state, and the user's utterance, the system dynamically infers the next conversation state. It facilitates a seamless dialogue flow, aligning with user needs while enriching their interactive experience.\n\nTable 1 (appendix) showcases the unified prompt design guiding LLMs to identify the conversation state and the user intent accurately.\n\n## 3.1.2 Dialogue Summarizer\n\nThis global micro-agent distills ongoing conversations into concise summaries, ensuring efficient communication among various micro-agents. It crafts these summaries by integrating the latest user utterance, previous dialogue history, and current responses, maintaining a coherent context throughout the conversation. The unified prompt design for guiding LLMs in summarizing user-VIDS interactions is presented in Table 2\n(appendix).\n\n## 3.1.3 Conversation Manager\n\nThis micro-agent generates accurate responses by integrating inputs from other micro-agents, maintaining the conversation's context for an effective and seamless dialogue experience for the user. The promoting approach used to steer LLM is demonstrated in Table 3 (appendix).\n\n## 3.2 Local Micro-Agents 3.2.1 Micro-Agents For Data Visualization\n\nFirst, the user provides a dataset of their choice for customized exploration. Then, the visualization process begins by generating a condensed dataset version, followed by an LLM-driven extraction of insights using the Dataset Summarizer micro-agent. Subsequently, the **Task Suggestor** micro-agent, informed by these insights, proposes a suitable Machine Learning task. These cooperative micro-agents ensure efficient dataset exploration and readiness for the Task Formulation phase.\n\nIn summary, the Dataset Summarizer and Task Suggestor micro-agents, crucial to this stage, delve into the dataset and propose ML tasks, respectively. These agents, guided by prompting strategies presented in Table 4 and Table 5, pave the way for the Task Formulation stage.\n\n## 3.3 Micro-Agents For Task Formulation\n\nFollowing Data Visualization, VIDS proceeds to the Task Formulation stage. This state is broken down into two interconnected components: *Task Selection* and *PeTEL Representation Construction*, each managed by specialized micro-agents to ensure a precise formulation of the goal ML task.\n\nTask Selection: The Task Selection phase, managed by the *Task Selector* micro-agent, defines the machine learning task from the pool of suggestions based on the dataset summary and user's data science goals. It provides options like classification, regression, clustering, or user-suggested tasks. This iterative dialogue refines user requirements and aligns tasks with their dataset and goals, as directed by the prompting strategy in Table 6. This micro-agent leverages the conversation summary to properly guide the LLM in selecting a suitable ML model that meets both the dataset's characteristics and the user's needs. This process continues until the user is satisfied and confident with their task choice, enabling personalized, hands-on problem-solving within the micro-agent framework.\n\nPeTEL Representation Construction: Upon task selection, VIDS leverages the Prediction Task Expression Language (PeTEL) Karmaker et al. (2021), a concise, slot-value style structured language that encapsulates the core aspects of the chosen machine learning task. The slot-value format of the PeTEL expression provides an unambiguous task description, delineating desired outcomes and key search parameters.\n\nFor shaping a more user-tailored experience, VIDS uses the PeTEL Representation Construction group—a team of micro-agents working in concert, emphasizing precision and user satisfaction. Key components among them are the *Seeker* and *Feeder* micro-agents, which facilitate an iterative dialogue with the user to populate PeTEL slots. The *Seeker* drives the conversation toward Task formulation, ensuring no aspect of the task is missed (refer Table 7 in appendix). In parallel, the *Feeder* assimilates user responses into PeTEL\nslots, interpreting user inputs for precise task specification (prompts detailed in Table 8 of the appendix).\n\nCollectively, the *Seeker* and *Feeder* steer a dynamic dialogue until PeTEL accurately mirrors the user's intent. Finally, the PeTEL Descriptor micro-agent articulates the populated PeTEL in layman's terms, reinforcing user understanding (see Table 9) and active participation. Thus, the PeTEL Construction process yields an effective, user-specific task representation, paving the way for subsequent machine learning stages.\n\nRefer to Listing 1 in the appendix for a filled PeTEL example.\n\n## 3.4 Micro-Agents For Prediction Engineering\n\nThe Prediction Engineering phase is the bridge between abstract problem representation and a practical prediction model, composed of three steps: converting PeTEL to features, executing data preparation, and interfacing with AutoML systems. First, the *PeTEL to Feature Converter* micro-agent transforms the task representation into tangible features, readying the problem description for computational processing. Next, the *Data Preparer* microagent refines the dataset, handling missing data, outliers, and categorical variable encoding to facilitate the application of machine learning algorithms. Finally, the *AutoML Interfacer* micro-agent uses the prepared training and testing data sets with AutoML systems. Utilizing these platforms' automation, it selects, optimizes, and trains an appropriate machine-learning algorithm.\n\n## 3.5 Micro-Agents For Result Summary\n\nData scientists traditionally present findings and advise domain experts on strategies, but this stage is mostly manual. Therefore, automation of Result Summarization and Recommendation is indeed very challenging.\n\nNevertheless, we propose a hypothetical micro-agent *Result Summarizer* that will generate a comprehensive summary of findings by employing AutoML libraries like Auto-SKLearn to train multiple models.\n\nThis summary will enable users to compare and identify the most effective solution, rapidly understanding the core findings. Future versions of VIDS will implement such a *Result Summarizer* micro-agent to streamline the Result Summary and Recommendation phase.\n\nAdditionally, As part of its future vision, VIDS aims to introduce the *Result Visualizer* micro-agent, designed to elevate users' comprehension through visual representations of outcomes. This advanced mechanism will generate suitable visualizations, such as performance metrics and feature importance, providing users with an intuitive and enhanced understanding of the findings. Furthermore, our shoot-the-moon objective is to optimize decision-making through interactive dialogue, where the system recommends the optimal model based on the ongoing conversation. This personalized approach is set to simplify the process, empowering users to make well-informed decisions.\n\n## 4 Case Study And Qualitative Evaluation\n\nThis section reports a qualitative evaluation of LLMs in performing complex tasks through an in-depth case study. We exclusively focus on ChatGPT for this case study and utilize the Student Performance dataset2 for our analysis. Our evaluation revolves around three main criteria. Firstly, we examine the *usability* and *efficacy* of LLMs in facilitating seamless conversations for data science tasks by analyzing the chat cycle between users and the VIDS system. Secondly, we examine the interaction among the micro-agents themselves, each equipped with its unique LLM instance. This analysis reveals the intricacies of microagent dynamics in performing complex \"ill-defined\" tasks. Lastly, we assess the versatility and resilience of the TeLER Prompt Taxonomy Santu & Feng (2023), which we followed to design prompts that were fed 2https://www.kaggle.com/datasets/larsen0966/student-performance-data-set to the LLMs serving as various micro-agents, demonstrating the practicality and applicability of LLMs in demanding scenarios. Through this in-depth qualitative examination, we aim to provide a panoramic view of the essential role and effectiveness of LLMs in navigating complex tasks.\n\n## 4.1 Usability And Efficacy Of Llms\n\nTo evaluate the usability and efficacy of LLMs as a solution to conversational data science, we focused on the essential dynamics of the user-system chat cycle, which is pivotal for effective communication within VIDS. A deep dive into this cyclical process has elucidated how LLMs orchestrate fluid, efficient communication. This scrutiny has further shined a light on the user experience, underscoring the prowess and utility of the LLMs in conversational data science. Table 10 (in the appendix) presents a tangible snapshot of user-system interactions, exemplifying the successful execution of the user's goal task via natural conversation.\n\n## 4.2 Micro-Agents' Roles And Interactions\n\nIn this subsection, we explore the interplay among the micro-agents within the four primary states of the system. Each micro-agent, armed with an LLM in the background, contributes to the system's overall execution of complex tasks. Through an analysis of these internal dynamics, we sought to deepen our comprehension of LLM functionality.\n\nData visualization, the initial state, presents data in a visually intuitive format, unveiling inherent complexities and patterns. The effectiveness of this representation is hinged on the interaction of *Data Summarizer* and *Task Suggestor* micro-agents, as illustrated in Table 11. Next, during the Task Formulation state, the task definition and specifics are established. The interplay among the Task Selector, Seeker, *Feeder*, and *PeTel Descriptor* micro-agents was pivotal in shaping the task, significantly influencing the eventual success of execution. The specifics of these interactions are presented in Table 12.\n\nMoving forward, in the Prediction Engineering state, the dataset is tailored according to the formulated task. The collaboration among micro-agents (PeTEL-to-Attribute-Converter, *Data Preparer* and *AutoML* Interfacer) in this phase directly impacts the dataset preparation, influencing the accuracy of predictions.\n\nFurther details are provided in Table 13 (appendix), fostering a comprehensive understanding of micro-agents' roles within this critical AutoML phase.\n\nFinally, VIDS interfaces with AutoML tools like AutoSKLearn to train selected models. From these training performances, VIDS generates result summaries and recommendations aligned with user preferences, as depicted in Table 14. Future work involves customizing such recommendations based on the user's business goals.\n\n## 4.3 Versatility And Resilience Of Teler\n\nUnderstanding the interaction between humans and AI at different levels of detail, as defined by Santu &\nFeng (2023) as the TeLER taxonomy, is vital for effective collaboration, especially in complex tasks. Below, we discuss the role of different levels of prompt details (as defined by the TeLER taxonomy) for three micro-agents— State Detector, Dialogue Summarizer, and Conversation Manager—which work together to facilitate efficient user-AI dialogue.\n\nState Detector micro-agent: Table 15 displays LLM's responses to each TeLER level (0-5). On analyzing this table, it becomes clear that as task specificity in the prompt increases, the results become more targeted. Low-detail prompts (Levels 0 and 1), due to their lack of precision, fail to generate the desired outputs, emphasizing the effectiveness of detailed prompt articulations.\n\nDialogue Summarizer micro-agent: Table 16 displays the LLM's responses to each TeLER level. A detailed review of this table offers several key findings. Firstly, LLM appears to bypass embedded instructions or subtasks when provided with high-complexity prompts. For instance, while a level 4 prompt necessitates an explanation of the response from LLM, a level 5 prompt also demands the provision of evaluation criteria.\n\nNevertheless, as evidenced in Table 16, LLM often neglects these subtasks, indicating a potential limitation with processing long prompts. Secondly, despite the micro agent's primary goal of dialogue summarization and the expectation to emphasize the most recent utterance, LLM often generates a more generic summary, consistently incorporating dataset-related information throughout the dialogue. This behavior suggests a bias towards longer dialogue turns.\n\nConversation Manager micro-agent: As presented in Table 17, LLM responses tend to become verbose with highly detailed prompts, implying an optimal length for prompts that can avoid overwhelming users in terms of cognitive load.\n\n## 5 Discussion\n\nThis section discusses the strengths and shortcomings of LLMs for realizing conversational data science based on the observations from our case study.\n\n## 5.1 The Strengths\n\nIn terms of strengths, our case study revealed that LLMs can- 1) achieve more automation within the endto-end ML pipeline, 2) enhance the precision of AI responses, 3) increase the efficiency in communication with users, 4) provide a more controlled dialogue flow, and 5) contribute to the democratization of Data Science technology. LLMs in Automation: In terms of automating an end-to-end AutoML pipeline, especially during the *Task* Formulation and *Prediction Engineering* stages, the utility of LLMs cannot be overstated. With their robust capabilities, LLMs can significantly streamline these highly interactive complex processes (see Tables 1-17), leading to broader accessibility of Data Science to the general public.\n\nPrecision Control: We found that decomposing larger tasks into manageable, specific micro-agents helped to ensure precise LLM responses. Our findings also suggest that using multiple targeted prompts, as opposed to a single comprehensive one, leads to better control over the LLMs' responses, making them more focused and reducing the likelihood of errors or misunderstandings.\n\nEfficiency: The precision garnered from micro-agent decomposition also increases the efficiency of LLMs. As an example, the *Seeker* micro-agent, as highlighted in the 7th row of Table 12, demonstrates high efficiency in understanding the current dialogue state and crafting appropriate questions. We believe such efficiency emerged from each connected micro-agent (e.g., Feeder, *PeTEL Descriptor* etc.) performing their assigned tasks precisely and coordinating with each other properly.\n\nDialogue Flow Control: LLM (ChatGPT) performed commendably in terms of overall dialogue flow control by carefully balancing multiple factors, such as the utterance, context, responses from various microagents, user intent, and the current state of the conversation. The dialogue flow was best controlled when Level 3 prompts (defined by the TeLER taxonomy) were used, as shown in Table 17 (appendix). Indeed, LLM skillfully extracted relevant information and steered the conversation in the right direction, which is very promising. LLM and Democratization of Data Science: Based on our case study, we believe a product level realization of *VIDS* will enable the general public/domain experts to directly engage with the fascinating world of predictive analytics without worrying about the underlying detailed knowledge of machine learning. VIDS will also automate many of the routine tasks performed by a *human* Data Scientist. Although this is a highly ambitious goal, if reached, it will expedite the implementation of ML systems allowing Data Science to appeal to a far broader audience across various domains.\n\n## 5.2 The Shortcomings\n\nIn this section, we scrutinize and discuss the challenges that LLMs face with \"ill-defined\" complex tasks and prompts, viewing these complexities as a means to test their upper limits. Shortcomings in State Detection: The State Detector micro-agent demonstrated lapses, struggling to interpret complex prompts, despite distinct delineations between states (Table 1). For instance, in one case, the model erroneously identified \"Model Training\" as the current state instead of the intended \"Task Selection\" state where the user utterance clearly conveyed the goal of performing a classification task (Table 11).\n\nTo enhance accuracy, we introduced a refined prompt design listing potential subsequent states, resulting in an improvement of the state selector's precision.\n\nShortcomings in Dialogue Summarization: The *Dialogue Summarizer* micro-agent brought a set of hurdles, especially by \"hallucinating\" irrelevant content. To address this issue, we provided few-shot examples of conversations and summaries, aiming for LLM to adopt that summarizing pattern. However, during evaluation, LLM still failed in some cases, erroneously weaving the examples into the summary rather than focusing on the latest input. Shortcomings in Handling Complex Prompts: Our investigation into LLM's responses to varying prompt complexities, as illustrated in Table 17, revealed noteworthy patterns. Firstly, LLM tends to generate verbose responses when faced with detailed prompts, potentially inundating users with excessive information. Secondly, with complex prompts to the *Conversation Summarizer* micro-agent, LLM often overlooks embedded instructions or subtasks. For instance, a level 4 prompt asks for a self-explanatory LLM response, while a level 5 prompt additionally requires following some evaluation criteria. However, LLM frequently sidesteps these subtasks, highlighting a potential challenge with handling intricate prompts (see Table 16).\n\n## 5.2.1 Summary And Future Directions\n\nOur exploration into LLMs' role in Conversational Data Science reveals their promise and the challenges surfacing at their operational boundaries. While LLMs showcase the potential to enhance precision, efficiency, and dialogue flow management in automated systems, their full capacities are yet to be unearthed. However, we also encountered difficulties with LLMs in state detection, dialogue summarization, and intricate prompt handling. For example, the *Dialogue Summarizer* micro-agent, tasked with summarization, tends to generate generic summaries with a potential bias towards lengthy dialogues over the most recent utterances. On top of that, overly complex prompting techniques generated responses that were voluminous, turned out to be counterproductive. These findings emphasize the need for further exploration and refinement to optimize the balance between task complexity and LLM performance and to reveal the true capability of these models in automation. As we strive to realize the full potential of LLMs, we remain committed to pushing their boundaries, thereby aiding in the advancement and democratization of Data Science.\n\n## 6 Conclusion\n\nThis paper illuminates our journey into the uncharted potential of Large Language Models (LLMs) toward the ambitious goal of conversational data science, an arena characterized by complex, ill-defined tasks. We have successfully architected an innovative concept of personal data scientist, VIDS, capitalizing on the capabilities of LLMs to use natural language as the primary interface. VIDS' architecture encompasses four distinct dialogue states, each symbolizing a unique phase of the conversation and crucially impacting the user-system interaction. Further, the concept of global micro-agents, forming a holistic structure and ensuring a coherent dialogue throughout the session, was introduced. Alongside these, we've integrated state-specific local micro-agents, instrumental in delivering VIDS' functionality.\n\nAlthough VIDS exhibits advanced capabilities, it's also very important to acknowledge areas of challenge.\n\nEven as we've applied amendments to address these issues, experiencing certain improvements, these obstacles highlight the need for further exploration and development. Identifying and rectifying these failures is key to enhancing the model's robustness, efficiency, and overall performance, thereby bringing us closer to harnessing the full potential of LLMs.\n\nOur study serves as a crucial stepping-stone towards our ultimate vision of a conversational data science assistant that can effectively understand and engage with its users. The knowledge garnered from this research will guide our future efforts to optimize LLMs. We are steadfast in our belief that the continued refinement of these models will catalyze more intuitive and effective human-AI interactions, revolutionizing how we approach complex tasks and data analysis and unveiling the true potential of LLMs for automation.\n\n## 7 Limitations\n\nWhile our explorations into the landscape of Large Language Models (LLMs) and conversational data science have been pioneering, we recognize our work is in its early stages and is not without its limitations. We've highlighted several areas for improvement and potential paths forward, each presenting a unique challenge and opportunity to propel VIDS and the field of LLMs into a new dimension.\n\nSingle Model Dependence: Presently, our investigation leans heavily on ChatGPT, bypassing a comparative examination of various LLMs. A promising avenue for future inquiry involves contrasting performance of multiple LLMs tackling such intricate tasks.\n\nAutomated Result Summarization & Model Recommendation: Our system lacks automated generation of Result Summaries and Model Recommendations, both potential game-changers. Future enhancements could include dynamic visual adjustments responsive to conversation context and model suggestions tailored to individual business objectives and requirements.\n\nScalability & Performance: With the ongoing expansion of VIDS' features, ensuring robust scalability and prompt responsiveness emerges as a critical concern. Future efforts should focus on refining system optimization to balance between large-scale deployments and performance. VIDS Personalization: At present, VIDS misses personalization, a feature that can cater to individual user preferences. Future versions should incorporate adaptable user-specific customization for a more engaging experience. Multilingual Adaptation: VIDS' current English-only operation restricts its global outreach. Future updates should incorporate multilingual capabilities, broadening the appeal and accessibility of VIDS. Interpretability & Transparency: VIDS currently doesn't elucidate its underlying reasoning, possibly causing user apprehension. Future iterations should emphasize improved model interpretability and transparency to build user confidence.\n\nEthical Considerations & Bias Detection: VIDS lacks mechanisms to detect or counterbalance biases.\n\nFuture research should dedicate efforts to uncover potential biases and devise strategies for equitable, ethical interactions.\n\nIn the complex landscape of conversational data science, navigating these limitations offers unique opportunities to pressure test and expand the capabilities of Large Language Models (LLMs). These challenges, intrinsic to ill-defined tasks, aren't merely obstacles but pivotal springboards that propel our understanding of LLMs. Tackling them not only refines our approach towards task automation with VIDS, but also contributes substantially to the broader field of NLP. 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Vrabec, Imane Bello, Ishani Dash, Jihyun Kang, John Giorgi, Jonas Golde, Jose David Posada, Karthik Rangasai Sivaraman, Lokesh Bulchandani, Lu Liu, Luisa Shinzato, Madeleine Hahn de Bykhovetz, Maiko Takeuchi, Marc Pàmies, Maria A Castillo, Marianna Nezhurina, Mario Sänger, Matthias Samwald, Michael Cullan, Michael Weinberg, Michiel De Wolf, Mina Mihaljcic, Minna Liu, Moritz Freidank, Myungsun Kang, Natasha Seelam, Nathan Dahlberg, Nicholas Michio Broad, Nikolaus Muellner, Pascale Fung, Patrick Haller, Ramya Chandrasekhar, Renata Eisenberg, Robert Martin, Rodrigo Canalli, Rosaline Su, Ruisi Su, Samuel Cahyawijaya, Samuele Garda, Shlok S Deshmukh, Shubhanshu Mishra, Sid Kiblawi, Simon Ott, Sinee Sang-aroonsiri, Srishti Kumar, Stefan Schweter, Sushil Bharati, Tanmay Laud, Théo Gigant, Tomoya Kainuma, Wojciech Kusa, Yanis Labrak, Yash Shailesh Bajaj, Yash Venkatraman, Yifan Xu, Yingxin Xu, Yu Xu, Zhe Tan, Zhongli Xie, Zifan Ye, Mathilde Bras, Younes Belkada, and Thomas Wolf.\n\nBloom: A 176b-parameter open-access multilingual language model, 2023.\n\nJing Xu, Arthur Szlam, and Jason Weston. Beyond goldfish memory: Long-term open-domain conversation.\n\nIn Smaranda Muresan, Preslav Nakov, and Aline Villavicencio (eds.), *Proceedings of the 60th Annual* Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), ACL 2022, Dublin, Ireland, May 22-27, 2022, pp. 5180–5197. Association for Computational Linguistics, 2022. doi: 10.18653/\nv1/2022.acl-long.356. URL https://doi.org/10.18653/v1/2022.acl-long.356.\n\nAohan Zeng, Xiao Liu, Zhengxiao Du, Zihan Wang, Hanyu Lai, Ming Ding, Zhuoyi Yang, Yifan Xu, Wendi Zheng, Xiao Xia, Weng Lam Tam, Zixuan Ma, Yufei Xue, Jidong Zhai, Wenguang Chen, Peng Zhang, Yuxiao Dong, and Jie Tang. GLM-130B: an open bilingual pre-trained model. *CoRR*, abs/2210.02414, 2022. doi: 10.48550/arXiv.2210.02414. URL https://doi.org/10.48550/arXiv.2210.02414.\n\nSusan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher Dewan, Mona T. Diab, Xian Li, Xi Victoria Lin, Todor Mihaylov, Myle Ott, Sam Shleifer, Kurt Shuster, Daniel Simig, Punit Singh Koura, Anjali Sridhar, Tianlu Wang, and Luke Zettlemoyer. OPT: open pre-trained transformer language models. *CoRR*, abs/2205.01068, 2022. doi: 10.48550/arXiv.2205.01068. URL https:\n//doi.org/10.48550/arXiv.2205.01068.\n\nTianyi Zhang, Faisal Ladhak, Esin Durmus, Percy Liang, Kathleen McKeown, and Tatsunori B. Hashimoto.\n\nBenchmarking large language models for news summarization, 2023.\n\nWayne Xin Zhao, Kun Zhou, Junyi Li, Tianyi Tang, Xiaolei Wang, Yupeng Hou, Yingqian Min, Beichen Zhang, Junjie Zhang, Zican Dong, Yifan Du, Chen Yang, Yushuo Chen, Zhipeng Chen, Jinhao Jiang, Ruiyang Ren, Yifan Li, Xinyu Tang, Zikang Liu, Peiyu Liu, Jian-Yun Nie, and Ji-Rong Wen. A survey of large language models. *CoRR*, abs/2303.18223, 2023. doi: 10.48550/arXiv.2303.18223. URL https:\n//doi.org/10.48550/arXiv.2303.18223.\n\nQihuang Zhong, Liang Ding, Juhua Liu, Bo Du, and Dacheng Tao. Can chatgpt understand too? a comparative study on chatgpt and fine-tuned bert, 2023.\n\nBarret Zoph and Quoc V Le. Neural architecture search with reinforcement learning. *arXiv preprint* arXiv:1611.01578, 2016.\n\nBarret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning transferable architectures for scalable image recognition. *arXiv preprint arXiv:1707.07012*, 2(6), 2017.\n\n## A Global Micro-Agents\n\nIn this section we discuss about the global micro-agents and how our prompt design has helped to actualize the those agents.\n\n## A.1 State Detector:\n\nAs shown in Figure 2, VIDS integrates a variety of well-defined states, each corresponding to the different stages of a conversation. The initial state is \"data visualization\", which centers around the presentation of data in a comprehensible and approachable manner. This transitions into the \"task formulation\" state, wherein the focus shifts to defining and structuring the task or problem the user wishes to address. Following this, the system moves into the \"prediction engineering\" state. In this phase, the system focuses on constructing and implementing predictive models that are based on the tasks as defined in the previous stage.\n\nFinally, the conversation arrives at the \"result summarization and recommendation\" state. Here, the system offers a succinct summary of the results, coupled with relevant recommendations based on the outcomes. Our system employs a micro-agent designed to dynamically determine the user's intent by considering not only the immediate context and user's utterance, but also the current dialogue state. This micro-agent, taking as input the current context, conversation state, and user input, decides whether the user wishes to proceed to the next state of the dialogue. By accurately aligning with the user's needs and objectives, this approach ensures a smooth flow of conversation while also providing an engaging user experience. The design of our system, therefore, strikes a balance between addressing user needs and enriching their interaction with the system. The unified prompt design used to guide the LLM in correctly identifying the current state of the conversation and the user's intent is presented in Table 1.\n\n## A.2 Dialogue Summarizer:\n\nOur system incorporates a micro-agent responsible for generating concise summaries of the ongoing conversation. This crucial feature enhances effective communication among various micro-agents. Given the dialogue history as input, this micro-agent takes into account the latest user utterance, preceding conversation history, and the current response from another micro-agent. Consequently, it generates a new dialogue summary that preserves coherence and context throughout the conversation. The unified prompt design used to guide the LLM in summarizing the interactions between the user and VIDS is displayed in Table 2.\n\n## A.3 Conversation Manager:\n\nOur system includes a conversation management micro-agent that weaves together the outputs from relevant micro-agents, generating a coherent, overarching dialogue. It accepts a variety of inputs including the context, the conversation state, the utterance, current intent, and the response from the micro-agent. By effectively maintaining the dialogue's structure and context, it assures a seamless user experience and adept task execution. The unified prompt design that steers the LLM in this process is shown in Table 3.\n\n## B Local Micro-Agents\n\nIn this section we discuss about the micro-agents that works together to make sure there is seamless conversation flow at each stage.\n\n## B.1 Data Visualization\n\nUpon dataset selection, VIDS efficiently generates a summarized version via LLM and its Dataset Summarizer and Task Suggestor micro-agents, providing a thorough understanding of the dataset and suggesting potential Machine Learning tasks.\n\n## B.1.1 Dataset Summarizer Micro-Agent:\n\nThe Dataset Summarizer micro-agent, at the core of the Data Visualization stage, employs a purposefully designed prompt to extract insights from the provided dataset. This agent accepts the dataset as its input and delves into its structure and content, gleaning an array of insights that furnish users with a comprehensive understanding of the dataset's potential applications. This extraction process is guided by the unified prompt design, as shown in Table 4, ensuring a thorough, user-friendly data analysis experience.\n\n## B.1.2 Task Suggestor Micro-Agent:\n\nThe Task Suggestor micro-agent, taking as its input the dataset summary, dovetails with the Dataset Summarizer to propose pertinent Machine Learning tasks. Leveraging a unified prompt design, detailed in Table 5, it guides the LLM to yield effective task suggestions. This capability, integral to the Data Visualization stage, enriches user interaction and effectively paves the way for the subsequent Task Formulation stage.\n\n## B.2 Task Formulation\n\nFollowing the Data Visualization stage, VIDS proceeds to Task Formulation. This section is broken down into two interconnected components: Task Selection and PeTEL Construction. , each managed by specialized micro-agents to ensure a thorough and user-oriented formulation of the machine learning task. Task Selection is managed by the Task Selector micro agent where PeTEL Construction component is handled with a group of micro-agents that consists of the three micro-agents: Feeder, Seeker, and PeTEL Descriptor. A sample populated PeTEL, demonstrating the iterative process of filling out the different components, is available in Listing 1.\n\n## B.2.1 Task Selector Micro-Agent:\n\nFunctioning within our system, the task selection micro-agent navigates through user conversations to pinpoint an apt machine learning problem from a variety of available options. It also provides valuable assistance in choosing a model best-suited to user needs. The micro-agent operates using inputs such as the context of the dialogue and user utterance, along with understanding user requirements and factoring in the dataset's characteristics. This enables it to select from diverse model types, including but not limited to, 'classification', 'regression', 'clustering', 'dimensionality reduction', and 'anomaly detection'. By ensuring alignment between the chosen problem, model, and user's objectives, it offers personalized recommendations, thereby enhancing user engagement. The unified prompt design guiding the LLM during this process is presented in Table 6.\n\n## B.2.2 Seeker Micro-Agent:\n\nAs an integral member of the PeTEL Construction micro-agent group, the Seeker micro-agent conducts user interactions to fill the next slot in the PeTEL representation. It takes two main inputs: the context of the dialogue and the current PeTEL. Through its systematic guidance, it ensures each unfilled slot is populated, facilitating a comprehensive and accurate task formulation. Table 7 demonstrates the unified prompt design utilized to direct the LLM effectively in querying about specific unfilled slots in the PeTEL expression.\n\n## B.2.3 Feeder Micro-Agent:\n\nOur PeTEL Construction micro-agent group includes the Feeder micro-agent, a crucial component that populates the PeTEL representation with user responses. This micro-agent accepts inputs such as the context of the dialogue, the user utterance, and the current PeTEL, ensuring accurate interpretation and integration of user's utterances into problem formulation. Table 8 presents the unified prompt design that guides the LLM to effectively fill unfilled slots in PeTEL expressions from user interactions.\n\n## B.2.4 Petel Descriptor Micro-Agent:\n\nRounding out the PeTEL Construction micro-agent group, the PeTEL Descriptor micro-agent communicates the fully populated PeTEL expression back to the user in natural language. It takes as input the fully populated PeTEL, and its function underlines VIDS' user-centric approach by ensuring that the user completely comprehends the formulated task. The unified prompt design that guides the LLM in this process is depicted in Table 9.\n\n{\nproblem_type : classification ,\ntarget_variable : delay_severity ,\nfeatures : [ departure_airport , arrival_airport , airline , scheduled_departure_time\n, scheduled_arrival_time , weather_conditions ] ,\ndataset_size : 10000/ Default ,\nperformance_metrics : [ accuracy , precision , recall , f1_score , confusion_matrix ] ,\nvalidation_method : cross_validation ,\nclassification_methods : [ logistic_regression , decision_tree_classifier ,\nrandom_forest_classifier , svm_classifier , knn_classifier , xgboost_classifier ,\nnaive_bayes ] ,\ndata_filters : [\n{ column : delay_duration , condition : greater_than , value : 15} ,\n{ column : departure_airport , condition : equals , value : JFK }\n] ,\nbusiness_goals : [ reduce customer complaints , optimize scheduling , improve airport operations ] ,\nadditional_requirements : [ robust to outliers , handle class imbalance ] ,\nmodel_preferences : interpretable\n}\nListing 1: Sample populated PeTEL for classification task based on FlightDelay dataset (one of our demo datasets).\n\n## B.3 Prediction Engineering\n\nFollowing Task Formulation, the journey progresses to Prediction Engineering, a fundamental stage where the system transforms the problem representation into a tangible prediction model. This phase is composed of three micro-agents: PeTEL to Feature, Data Prepper, and AutoML interfacer. Each agent is crucial in bridging the gap between the problem's conceptual representation and its practical implementation.\n\n## B.3.1 Petel To Attribute Converter Micro-Agent:\n\nThe PeTEL to Feature conversion is the first step in the Prediction Engineering process. Here, the PeTEL\nrepresentation, which succinctly describes the machine learning task, is translated into features that can be used by the prediction model. This process ensures that the machine learning algorithms can interpret and work with the problem definition, turning the abstract task representation into concrete, computable features.\n\n## B.3.2 Data Prepper Micro-Agent:\n\nOnce the features are defined, the next step is Data Cleaning and Preparation. This stage involves preprocessing the dataset to ensure it's suitable for the prediction model. Common procedures during this phase include handling missing data, dealing with outliers, and encoding categorical variables. The goal is to produce a clean, well-structured dataset that can be readily consumed by downstream machine learning algorithms, maximizing the potential for accurate and meaningful predictions.\n\n## B.3.3 Automl Interfacer Micro-Agent:\n\nThe final step in the Prediction Engineering phase is interfacing with AutoML systems. AutoML platforms automate the process of applying machine learning to real-world problems, making the technology accessible to non-experts and improving efficiency of experts. In this step, the prepared dataset is fed into an AutoML\nsystem, which automatically selects the most suitable machine learning algorithm, optimizes its parameters, and trains the model. The result is a robust prediction model that is ready to generate insights from new data, bringing the conceptual machine learning task to fruition.\n\n## B.4 Result Summary And Recommendation\n\nWe aim to automate and enhance the final phase of VIDS, the Result Summary and Recommendation, with systematic structuring and incorporation of Result Summarization and Visualization to facilitate informed decision-making in data science.\n\n## B.4.1 Result Summarizer Micro-Agent:\n\nCurrently, we have implemented the Result Summarization micro-agent, where the system produces a comprehensive summary of the findings once the machine learning tasks have been executed. Utilizing an AutoML library such as Auto-SKLearn, the system trains all specified models, equipping users with a broad comparison to discern the most effective solution. This process distills the results into an accessible format, enabling users to grasp the essence of the findings quickly.\n\n## B.4.2 Result Visualizer Micro-Agent (Future Work):\n\nLooking forward, we aim to implement the Result Visualization micro-agent. Visualizing the outcomes can significantly aid users' understanding and facilitate more informed decision-making. We plan to develop a process that generates suitable visualizations based on the results, such as performance metrics or feature importance, offering a more intuitive perspective of the findings.\n\nAdditionally, we aspire to further optimize the decision-making process, by having the system engage in an interactive dialogue with the user, recommending the most suitable model based on the conversation. This personalized approach would empower users to make informed decisions, streamline the process, and enhance the effectiveness of their machine learning endeavors.\n\n| next available state: task formulation, model training. current state: model training next available state: model training. The agent MUST response with the following json format: {\"intent\": \" \", \"current_state\": \" \", \"next_state\": \" \" } Demonstration User Taking into account the given context { In this dialogue, the AI assistant provided information on suitable machine learning tasks for three different datasets: airlines, flights, and airports. For the flights dataset, the assistant suggested that classification and regression would be suitable tasks. Classification could be used to predict flight delays or cancellations, while regression could be used to predict the amount of delay. The user expressed interest to know more about the dataset.}, the conversation state { dataset_understanding } the utterance { What details are included in the flight delay dataset? }, identify my current intent and next state of conversation. Please remember to only response in following format predefined json format without any additional information. Carefully examine the utterance and think about how the context might influence the current utterance, leading you to determine my present intent and next state. LLM {\"intent\": \"Get dataset info\", \"current_state\": \"dataset_understanding\", \"next_state\": \"dataset_understanding\"} User Taking into account the given context { In this dialogue, the AI assistant provided information on suitable machine learning tasks for three different datasets: airlines, flights, and airports. For the flights dataset, the assistant suggested that classification and regression would be suitable tasks. Classification could be used to predict flight delays or cancellations, while regression could be used to predict the amount of delay. The user expressed interest in the flights dataset and asked if it could be formulated as a time series problem, but the assistant did not provide a response to this question. }, the conversation state { dataset_understanding } the utterance { I want to predict if a flight will be delayed or not }, identify my current intent and next state of conversation. Please remember to only response in following format predefined json format without any additional information. Carefully examine the utterance and think about how the context might influence the current utterance, leading you to determine my present intent and next state. LLM {\"intent\": \"Select problem\", \"current_state\": \"dataset_understanding\", \"next_state\": \"problem_selection\"} Directive Taking into account the given context {context}, the conversation state {conversation state} the utterance {user input}, identify my current intent and next state of conversation. Please remember to only response in following format predefined json format without any additional information. Carefully examine the utterance and think about how the context might influence the current utterance, leading you to determine my present intent and next state. Table 1: The details of prompt design for the State Detector microagent. In the directive, the {context}, {conversation state}, and {user input} are placeholders which will be replaced dynamically in different stage of conversation |\n|---|\n\nPrompt Design\n\nSystem setup\n\nGiven the dialogue between user and assistant, the AI assistant summarizes the dialogue summary. The AI agent should not leave out any crucial information. The goal of this summary generation\n\nis not being precise, rather the goal should be to contain all crucial information. if the previous\n\ndialogue is empty then you should return the current user utterance.\n\nDirective\n\nGiven the context as {context}, latest user utterance as {utterance} and previous response as {response}, summarize the following dialogue. You should not exclude any important information.\n\n{history}\n\nTable 2: The details of prompt design for the Dialogue Summarizer micro-agent. In the directive, {history}, {context}, {utterance} and {response} are placeholders which will be replaced dynamically during the conversation\n\n## Prompt Design System Setup\n\nThe AI assistant serves as a virtual data scientist, designed to engage with users and comprehend their objectives. The purpose of this interaction is to develop a machine learning task tailored to the user's data.\n\nTo achieve this, the assistant will collaborate with various micro agents, each performing specialized tasks to support the primary agent. The assistant will receive context, utterances, dataset summaries, and micro agent responses as input, and should aim to steer the conversation towards the goal. The following micro agents will aid the assistant, providing their output as input to the AI agent for further processing and integration.\n\nDepending on the current conversation state, different micro agents will be activated to provide their respective responses:\nIntent Detector: Identifies the user's intent from a list including 'Get dataset info', 'Get dataset trend', 'Select problem', 'Formulate problem', 'Problem execution', and 'Chitchat'. The detected intent will be used to determine the direction of the conversation. State Selector: Determines the conversation state, choosing from \"data_visualization\", \"task_selection\",\n\"task_formulation\", or \"task_execution\". The chosen state helps the AI agent to adapt its responses and maintain a coherent discussion flow.\n\nTask Selector: Selects an appropriate ML task from options such as \"classification\", \"regression\", \"clustering\",\n\"dimensionality reduction\", \"anomaly detection\", and \"time series\". The selected task guides the AI agent in suggesting relevant solutions to the user.\n\nTask Formulator: Constructs the ML task by utilizing a slot-value filling process. The formulated problem, complete with specified parameters, is then provided to the AI agent, which can assist the user in refining or executing the task.\n\nDirective Taking into account the given context [context], the conversation state {state} the utterance {input}, current intent {intent} and the response from the {micro-agent} micro-agent {MA_resp}, provide appropriate response to the user to carry the conversation to its goal which is formulating a ML task based on user demands.\n\nTable 3: The details of prompt design for the Conversation Manager micro-agent. In the directive, {state}, {input}, {micro-agent}, and\n{MA_resp} are placeholders which will be replaced dynamically during the conversation.\n\nPrompt Design\n\nSystem setup\n\nContinued on next page\n\n## Table 4 - Continued From Previous Page Prompt Design\n\nYou are an AI agent who will provide a conprihensive summary of a given dataset. Your task is to provide a comprehensive summary of a given dataset in a strict \"JSON\" format.\n\nThe summary MUST include the following informations:\n1. dataset summary: the summary of the given dataset in natural language 2. column: it will list all columns and give a brief description about that column 3. Row: AI agent will select a row at random and describe what the row means in natural language 4. Trend: In natural language the AI agent will write the trends that can be found from the given dataset.\n\nThe response should be in a strict JSON format as follows: {\"summary\": \"...\", \"columns\": [\"name\": \"col1\",\n\"description\": \"...\", \"name\": \"col2\", \"description\": \"...\"], \"row\": \"description of a random row\", \"trend\", \"...\"}\nPlease make sure to provide clear and concise descriptions in natural language to facilitate understanding for non-technical users.\n\nDirective Please provide a comprehensive summary of the given dataset. The response MUST be in JSON format NOTHING ELSE. Use the following dataset: {dataset}.\n\nTable 4: The details of prompt design for the Dataset Summarizer microagent. In the directive, the {dataset} is a placeholders which will be replaced a miniature version of the user provided dataset.\n\n## Prompt Design System Setup\n\nThe AI agent must analyze the provided dataset summary and recommend appropriate machine learning\n(ML) tasks. Based on the summary, column descriptions, row information, and any observed trends, the agent should suggest at least two suitable ML task from the following task list: [\"classification\", \"regression\",\n\"clustering\", \"dimensionality reduction\", \"anomaly detection\", \"time series\"]. For each ML task the agent chooses a clear rationale must be provided which may include an explanation of why the chosen task aligns with the dataset, and a concrete example of how the task can be formulated. Directive Suggest ML tasks based on the following dataset summary: {summary}\nTable 5: The details of prompt design for the Task Suggestor microagent. In the directive, the {summary} is a placeholders which will be replaced by the dataset summary of the user provided dataset.\n\nPrompt Design\n\nSystem setup\n\nThe AI assistant is designed to comprehend the user's needs through conversation and assist them in selecting a suitable machine learning model for formulating a Machine Learning problem. The\n\nassistant must choose the appropriate model from the provided list: [\"classification\", \"regression\",\n\n\"clustering\", \"dimensionality reduction\", \"anomaly detection\"]. The assistant should consider the\n\nuser's problem, requirements, and dataset, which may be found in the dialogue context, to recommend the best model tailored to their specific needs.\n\nDemonstration\n\nUser I want to predict whether a flight will be delayed or not based on factors like weather conditions\n\nand previous delays.\n\nLLM {'model': 'classification', 'reason': 'A classification model can be used to categorize flights as delayed\n\nor not delayed based on the input features, such as weather conditions and previous delays.'}\n\nUser I need to find groups of flights with similar delay patterns, considering variables like departure\n\nairport, airline, and time of day.\n\nLLM {'model': 'clustering', 'reason': 'A clustering model can help identify groups of flights with similar\n\ndelay patterns by analyzing variables like departure airport, airline, and time of day, without\n\nrequiring labeled data.'}\n\nDirective\n\nPlease choose a most suitable model the conversation context: [context] and my latest utterance {input}. The output must be in a strict JSON format: {\"model\": \"model name\", \"reason\": \"your\n\ndetail reasons for the choice\"}.\n\nTable 6: The details of prompt design for the Task Selector micro-agent. In the directive, the [context], and {input} are placeholders which will be replaced dynamically during the conversation\n\n## Prompt Design System Setup\n\nThe AI assistant functions as a virtual data scientist, aiming to fill in any missing values in a given JSON object by asking the user about them. In order to not overwhelm the user, the assistant should focus on one unidentified slot at a time. With the JSON object, dataset summary, and the next empty slot provided, the agent should formulate a specific and clear question based on this information to accurately identify the missing value. The dataset summary is as follows:\n{dataset_summary}\nDemonstration Continued on next page\n\n| | Prompt Design |\n|----|-----------------|\n| System setup The AI assistant serves as a virtual data scientist, striving to complete a given JSON object by incorporating missing values through user input. Update the JSON object with the relevant unknown slot, considering that the user may also wish to modify existing filled slots. Ensure that the output is a strictly JSON format. If it is not possible to update the JSON object, return the original input JSON object. Demonstration Continued on next p | age |\n\n| | Table 7 - continued from previous page Prompt Design |\n|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| User | In order to obtain information about the next unidentified slot in the given JSON object { problem_type: time_series_forecasting, target_variable: null, forecast_horizon: null, business_goals: null, granularity: null, features: null, time_range: null, model_preferences: null, performance_metrics: null, validation_method: null, additional_requirements: null, time_series_methods: null, data_filters: [ column: null, condition: null, value: null, column: null, condition: null, value: null ] } The agent should ask the user a clear and specific question. Considering the provided context (User wants to formulate a time series forecasting problem by using the flight delay dataset. He is interacting with the agent and the agent is now asking questions to the user to formulate the problem), please formulate a suitable query to gather the necessary data for the missing slot. |\n| LLM | based on the provided context and the current state of the JSON object, the next unidentified slot is the target_variables slot. What specific variable are you trying to forecast using the flight delay dataset? This is also known as the target variable. For example, are you trying to forecast the number of flights that will be delayed, or the total duration of delays in minutes? |\n| Directive In order to obtain information about the next unidentified slot in the given JSON object ({petel}), the agent should ask the user a clear and specific question. Considering the provided context ({context}), please formulate a suitable query to gather the necessary data for the missing slot. Table 7: The details of prompt design for the Seeker micro-agent. In the system setup, {dataset_summary} is a placeholder for the current dataset summary user is working with. In the directive, the {petel}, and {context} are placeholders which will be replaced dynamically during the conversation | |\n\n| Table 8 - continued from previous page Prompt Design | | |\n|--------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|------------------------|\n| User | Using the provided information (I'd like to forecast the total number of flight delays.), please update the appropriate slot in the given JSON object: { problem_type: time_series_forecasting, target_variable: null, forecast_horizon: null, business_goals: null, granularity: null, features: null, time_range: null, model_preferences: null, performance_metrics: null, validation_method: null, additional_requirements: null, time_series_methods: null, data_filters: [ column: null, condition: null, value: null, column: null, condition: null, value: null ] } Ensure that you fill only the slot that is directly related to this information. Your response should consist solely of the updated JSON file. | |\n| LLM | { problem_type: time_series_forecasting, target_variables: [flight_delays], forecast_horizon: null, business_goals: null, granularity: null, features: null, time_range: null, model_preferences: null, performance_metrics: null, validation_method: null, additional_requirements: null, time_series_methods: null, data_filters: [ column: null, condition: null, value: null, column: null, condition: null, value: null ] } | Continued on next page |\n\n| Table 8 - continued from previous page Prompt Design | |\n|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| User | Using the provided information (for time horizon I would like to use one month I guess.), please update the appropriate slot in the given JSON object: { problem_type: time_series_forecasting, target_variables: [flight_delays], forecast_horizon: null, business_goals: null, granularity: null, features: null, time_range: null, model_preferences: null, performance_metrics: null, validation_method: null, additional_requirements: null, time_series_methods: null, data_filters: [ column: null, condition: null, value: null, column: null, condition: null, value: null ] } Ensure that you fill only the slot that is directly related to this information. Your response should consist solely of the updated JSON file. |\n| LLM | { problem_type: time_series_forecasting, target_variables: [flight_delays], forecast_horizon: 1 month, business_goals: null, granularity: null, features: null, time_range: null, model_preferences: null, performance_metrics: null, validation_method: null, additional_requirements: null, time_series_methods: null, data_filters: [ column: null, condition: null, value: null, column: null, condition: null, value: null ] } |\n| Directive Using the provided information ({utterance}, {context}), please update the appropriate slot in the given JSON object: {petel}. Ensure that you fill only the slot that is directly related to this information. Your response should consist solely of the updated JSON file. Table 8: The details of prompt design for the Feeder micro-agent. In the directive, the {utterance}, {context}, and {petel} are placeholders which will be replaced dynamically during the conversation Prompt Design System setup Continued on next page | |\n\n| Table 9 - continued from previous page Prompt Design |\n|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| As the AI agent, your primary objective is to carefully examine the problem illustrated within the json object and then develop a comprehensive, yet clear, natural language description. This explanation should effectively communicate the underlying issue to the reader, ensuring that they can grasp the key aspects and implications of the problem at hand. By providing an elaborative and accessible interpretation, the AI agent will enable a better understanding of the situation for individuals who may not be familiar with the json format or the specific context of the issue. Demonstration User Please provide a clear and concise natural language description of the problem formulation presented here: { problem_type: classification, target_variable: delay_severity, features: [departure_airport, arrival_airport, airline, scheduled_departure_time, scheduled_arrival_time, weather_conditions], dataset_size: 10000/Default, performance_metrics: [accuracy, precision, recall, f1_score, confusion_matrix], validation_method: cross_validation, classification_methods: [logistic_regression, decision_tree_classifier, random_forest_classifier, svm_classifier, knn_classifier, xgboost_classifier, naive_bayes], data_filters: [ column: delay_duration, condition: greater_than, value: 15, column: departure_airport, condition: equals, value: JFK ], business_goals: [reduce customer complaints, optimize scheduling, improve airport operations], additional_requirements: [robust to outliers, handle class imbalance], model_preferences: interpretable } LLM This is a time series forecasting problem that aims to predict the total delay duration of flights for the next month. The prediction will be based on various features such as departure and arrival airports, airline, weather conditions, and delay duration. The model should be interpretable and capable of handling seasonality, trends, and nonlinear relationships. The problem also aims to understand the impact of weather on flight delays and to forecast delay duration per airline. The dataset is at hourly granularity and covers the past three years. The performance metrics used to evaluate the model include mean squared error, mean absolute error, and R2 score. The model will be validated using time series cross-validation. The data filters used in this problem include only flights departing from JFK airport and operated by Delta airlines. Directive Please provide a clear and concise natural language description of the problem formulation presented |\n\n## C Qualitative Examples\n\nThis section explores the qualitative performance of Large Language Models (LLMs) in complex tasks.\n\nWe focus on ChatGPT as our chosen LLM and utilize the Student Performance dataset for our analysis.\n\nOur investigation revolves around three main areas.\n\nFirstly, we examine the chat cycle between users and the VIDS system, emphasizing the usability and efficacy of VIDS' LLMs in facilitating seamless conversations. Secondly, we explore the interaction among the micro-agents themselves, each equipped with its unique LLM instance. This analysis reveals the intricacies of micro-agent dynamics in orchestrating complex tasks. Lastly, we assess the versatility and resilience of the Prompt Taxonomy within these microagents, demonstrating the practicality and applicability of LLMs in demanding scenarios. Through this in-depth qualitative examination, we aim to provide a panoramic view of the essential role and effectiveness of LLMs in navigating complex tasks.\n\n## C.1 Dataset\n\nThe Student Performance (SP) dataset, publicly accessible on Kaggle3, includes 649 multilingual records with 33 diverse attributes. We use this dataset as a whole as if the user want to have some insight about this dataset (i.e. dataset not divided into training, testing, and validation splits also no data was excluded). Download the dataset directly from Kaggle, adhering to their usage terms and conditions. Since this dataset was pre-existing, there are no specific details pertaining to our data collection process or annotator instructions.\n\n## C.2 Overall Chat Cycle\n\nIn this subsection, we focus on the essential dynamics of the user-system chat cycle, pivotal for effective communication within VIDS. Facilitated by a sophisticated ensemble of LLM agents, this cycle forms the backbone of user-system interactions.\n\nA deep dive into this cyclical process will elucidate how LLMs orchestrate fluid, efficient communication.\n\nThis scrutiny will shine a light on the user experience, underscoring the prowess and utility of the LLMs in complex tasks. Table 10 presents a tangible snapshot of user-system interactions, exemplifying the successful execution of the user's goal task. The dialogue begins with the AI presenting a dataset summary and proposing potential ML tasks. The user selects clas-3https://www.kaggle.com/datasets/larsen0966/studentperformance-data-set sification as the most suitable task and identifies the\n'final grade' as the target variable. The AI then proceeds to query about the size of the dataset, to which the user responds with '10,000 samples.' Following several exchanges (elided for brevity), the AI finalizes the task as predicting 'final grade' based on various features, considering three classification methods:\nrandom forest, SVM, and logistic regression. The data is filtered to include only records with attendance over 75% and study hours over 1. The primary goal is to predict student performance for early interventions, with model interpretability and accuracy considered vital requirements. The user then approves this task setup, closing this dialogue cycle.\n\n## C.3 Interaction Between Micro-Agents\n\nIn this subsection, we delve into the interactions within the micro-agents themselves, each equipped with its own instance of a Large Language Model\n(LLM). These interactions occur in four distinct states within the system: data visualization, task formulation, prediction engineering, and result generation and summarization. By examining the interplay among these micro-agents in each state, we aim to enhance our understanding of the internal mechanics of the LLMs and their role in executing complex tasks.\n\n## C.3.1 Data Visualization State:\n\nThe first state, data visualization, involves the presentation of data in a visual format that aids in understanding the complexities and patterns within the data. The interaction between micro-agent agents in this state is crucial as it determines the effectiveness of the data representation. In Table 11, we show the specific roles and interactions of the micro-agents in this state. The table presents a step-by-step interaction of different micro-agents involved in the Data Visualization state, utilizing a dataset related to student performance. The Dataset Summarizer initially provides a comprehensive summary of the dataset, including detailed descriptions of dataset content, column names, a sample row, and observed trends. Key insights, such as the correlation between a student's academic performance and factors like gender, study time, and alcohol consumption, are highlighted. The Suggest ML task micro-agent then proposes two potential machine learning tasks—Classification and Regression—that can be derived from the dataset.\n\nClassification could predict a student's success or failure, whereas Regression could estimate a student's final grade based on numerous factors, offering critical insights for intervention and performance enhancement strategies.\n\nthe user in this stage and evaluate different models based on the user's business goals.\n\n## C.3.2 Task Formulation State:\n\nThe second state, task formulation, is the stage where the task to be performed is defined and structured.\n\nThe interaction between micro-agent agents in this state is critical as it shapes the formulation of the task and ultimately influences the success of the task's execution. In Table 12, we delve into the specifics of these interactions. The table illustrates a complex interaction of micro-agents within the Task Selection state, focusing on an example with a student performance dataset. The dialogue begins with a summary of the dataset and an initial classification task proposition by a user. As the conversation progresses, different agents come into play, summarizing the dialogue context, selecting states, feeding problem specifications, and asking questions to fill in missing data.\n\nThe process is iterative, updating a JSON object specifying the problem and model preferences. This iterative dialogue concludes once all the necessary slots in the JSON object are filled, resulting in a comprehensive task description. This task involves predicting student final grades with a dataset of 10,000 samples and specific classification methods, filtering data based on attendance and study hours, and emphasizing the importance of model interpretability.\n\n## C.3.3 Prediction Engineering State:\n\nThe third state, prediction engineering, is an integral part of the AutoML pipeline. As shown in Table 13, this state takes the formulated task and prepares the dataset accordingly. The interaction between microagent agents in this state is essential as it directly influences the preparation of the dataset, which in turn impacts the accuracy and reliability of the predictions made. In this state, the micro-agent agents work collaboratively to interpret the task requirements, adjust the dataset to align with these requirements, and set the stage for accurate prediction generation.\n\n## C.3.4 Model Training, Result Summary, And Recommendation State:\n\nAfter the task is formulated, VIDS interfaces with AutoML tools (e.g. AutoSKLearn) and trains downstream model(s) based on the task formulation determined beforehand. As shown in Table 14, from training performance of different models, VIDS generates summaries, including the results and recommendations based on user preferences defined in the task formulation. Our future work will be to interact with\n\n| # | Input | MicroAgent | Output | |\n|-----|---------|---|-----------------------------|------------------------|\n| 1 | Dataset | - | M in_ds : Miniature dataset | Continued on next page |\n\n| Turn | Conversation |\n|--------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| system1 | Welcome, I am your personal data scientist. I have processed the provided dataset for your convenience I present the summary here: {Summary}. I propose the following ML tasks for this dataset: {ML_task} |\n| utterance1 | Ok, from the description it seems like classification is a good choice. |\n| response1 | Great! Let's move forward with the classification task. Now that we have decided on the classification task, let's move on to formulating the problem. Can you provide me with more details on what you would like to achieve with this task? For example, what is the target variable you would like to predict? |\n| utterance2 | I think I will use final grade |\n| response2 | Based on the provided context, the next unidentified slot in the JSON object is 'dataset_size'. To obtain this information, the agent could ask the user: Can you please provide the size of your dataset for the Classification model you are building? |\n| utterance3 | Lets use 10000 samples for this task |\n| responsen | The task is to predict the final_grade of a student based on various features like study_hours, attendance, participation, homework_scores, and test_scores. The dataset size is hypothetical. Performance is measured using accuracy, F1 score, and the confusion matrix. K-fold cross-validation is used for validation. Three classification methods are considered: random forest, SVM, and logistic regression. The data is filtered to only include records where attendance is greater than 75 (assuming attendance is measured as a percentage) and study_hours is more than 1. The business goal is to predict student performance to implement early interventions for those at risk of failing. Model interpretability is highlighted as an important additional requirement, and there's a preference for models with higher accuracy, but interpretability is also considered important. |\n| utterancen+1 | That seems all right to me. go ahead with this task. Table 10: Interaction of user and system where utterance∗ are the user utterances and response∗ are the overall system response to the user. In system1, {summary} and {ML_task} are from Table 11. |\n\n| | | Table 11 - continued from previous page | | |\n|----|----------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------|-----------|-----------------------------------------------------------------------|\n| # | Input | Micro-Agent | Output | |\n| 2 | M in_ds | Dataset Summarizer | summary : { dataset_summary : This dataset contains information about students in a school. It includes their demographic information such as sex, age, address, family size, and parents' education level. It also includes information about their academic performance such as grades, study time, and number of failures. Additionally, it includes information about their social life such as alcohol consumption, going out with friends, and having a romantic relationship. columns : list of column names and description sample row : This row represents a female student who is 16 years old, lives in an urban area with a family size greater than 3, and her parents are living together. Her mother has a higher education level than her father and works in the health sector, while her father works in the services sector. She chose this school because of its reputation and her father is her guardian. It takes her less than 15 minutes to travel from home to school. She studies for 2 to 5 hours per week and has not failed any classes in the past. She receives no extra educational support from school or family members, and does not pay for extra classes. She participates in extracurricular activities and attended nursery school. She wants to take higher education and has internet access at home. She is not in a romantic relationship and has a good quality of family relationships. She has a moderate amount of free time and goes out with friends occasionally. She has a low workday alcohol consumption and a very low weekend alcohol consumption. Her current health status is good and she has had 6 school absences. Her grades are 12 in the first period, 12 in the second period, and 13 in the final period. trend : From the dataset, it can be observed that female students tend to have higher grades than male students. Additionally, students who receive extra educational support from school or family members tend to have higher grades than those who do not. Students who have a higher amount of study time per week also tend to have higher grades. On the other hand, students who have a higher amount of alcohol consumption tend to have lower grades. Finally, students who have a higher number of school absences tend to have lower grades. } | |\n| 3 | summary | Task | ML_task : | Based on the provided dataset summary, the following are the suitable |\n| | | ML tasks: | | |\n| | Suggestor | 1. Classification: Classification can be used to predict whether a student will pass or fail based on their demographic information, academic performance, and social life. This task can be formulated as a binary classification problem where the target variable is pass/fail. The features can include sex, age, address, family size, parents' education level, study time, number of failures, alcohol consumption, and school absences. 2. Regression: Regression can be used to predict the final grade of a student based on their demographic information, academic performance, and social life. This task can be formulated as a regression problem where the target variable is the final grade (G3). The features can include sex, age, address, family size, parents' education level, study time, number of failures, alcohol consumption, and school absences. The rationale for choosing classification is that it can help identify students who are at risk of failing and provide early intervention to improve their academic performance. The rationale for choosing regression is that it can help identify factors that contribute to a student's academic success and provide insights into how to improve their performance. | | |\n| | Table 11: Interaction of different micro-agent in the Data Visualization state. In this qualitative study we work with Flight Delay dataset. | | | |\n\n| # | Input | MicroAgent | Output | | | |\n|---------------|--------------------------------|---------------|-----------------------------------------------------------------------|------------------|-----------------------------|---------------|\n| 1 | utterance1, summary, ML_task | Dialogue | context1 : The dialogue provides a summary of a dataset containing | | | |\n| summarizer | information about students in a school, including their demographic information, academic performance, and social life. The dataset summary highlights trends such as female students having higher grades and students who receive extra educational support having higher grades. Based on this information, the user suggests that classification would be a suitable machine learning task to predict whether a student will pass or fail based on their demographic information, academic performance, and social life. The user's rationale for choosing classification is to identify students who are at risk of failing and provide early intervention to improve their academic performance. | | | | | |\n| 2 | context1, | State | Detec | | | |\n| tor | ds1 : Task Formulation | | | | | |\n| utterance1 | | | | | | |\n| 3 | utterance1, | Task Selector | ML_task | : { \"model\": | \"classification\", \"reason\": | \"Based on the |\n| context1, ds1 | dataset summary, the AI agent suggested two suitable machine learning tasks: classification to predict whether a student will pass or fail, and regression to predict a student's final grade. The user agreed that classification would be a good choice based on the dataset summary. Therefore, a classification model is a good choice for this task.\"} | | | | | |\n| 4 | utterance2, | Dialogue | context2 : In this dialogue, the AI assistant provides a summary of a | | | |\n| context1, ds1 | Summarizer | dataset containing information about students in a school, including their demographic information, academic performance, and social life. The assistant notes that female students tend to have higher grades than male students, and students who receive extra educational support from school or family members tend to have higher grades. The assistant suggests two suitable ML tasks for the dataset: classification and regression. The user decides to use classification to predict whether a student will pass or fail based on their demographic information, academic performance, and social life. The assistant asks for more details on the target variable, and the user decides to use the final grade as the target variable. | | | | |\n| 5 | context2, | State | Detec | | | |\n| tor | ds2 : Task Formulation | | | | | |\n| utterance2 | | | | | | |\n| 6 | utterance2, context2, P eT EL1 | Feeder | P eT EL2 : { problem_type : | classification , | | |\n| | target_variable : | final | grade , | | | |\n| | features : | None , | | | | |\n| | dataset_size : | None , | | | | |\n| | performance_metrics : | None , | | | | |\n| | validation_method : | None , | | | | |\n| | classification_methods : | None , | | | | |\n| | data_filters : | [ | | | | |\n| | { column : | final | grade , | condition : | less | than |\n| | , | value : | passing | grade } , | | |\n| | { column : | None , | condition : | None , | value : | |\n| | None } | | | | | |\n| | ] , | | | | | |\n| | business_goals : | None , | | | | |\n| | additional_requirements : | None , | | | | |\n| | model_preferences : | None | | | | |\n| } | Continued on next page | | | | | |\n\n| Table 12 - continued from previous page | | | | | | |\n|-------------------------------------------------------------------------------------|--------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|------------------|---------|------|\n| # | Input | Micro-Agent | Output | | | |\n| 7 | utterance2, context2, P eT EL2 | Seeker | Q1 : Based on the provided context, the next unidentified slot in the JSON object is 'dataset_size'. To obtain this information, the agent could ask the user: Can you please provide the size of your dataset for the Classification model you are building? | | | |\n| 8 | utterance3, | Dialogue | context3 : The AI assistant provided a summary of the dataset, including information about the columns and a sample row. The assistant | | | |\n| context2, Q1 | Summarizer | also mentioned trends observed in the dataset, such as the correlation between study time and grades, and the negative correlation between alcohol consumption and grades. The user asked for suggestions on machine learning tasks, and the assistant suggested classification and regression. The user chose classification and specified that the target attribute would be the final grade. The assistant then asked for more information about the problem to help define it more precisely. The user decided to use 10,000 samples for the task | | | | |\n| 9 | utterance3, context3, P eT EL3 | Feeder | P eT EL3 : { problem_type : | classification , | | |\n| | target_variable : | final | grade , | | | |\n| | features : | None , | | | | |\n| | dataset_size : | 10000, | | | | |\n| | performance_metrics : | None , | | | | |\n| | validation_method : | None , | | | | |\n| | classification_methods : | None , | | | | |\n| | data_filters : | [ | | | | |\n| | { column : | final | grade , | condition : | less | than |\n| | , | value : | passing | grade } , | | |\n| | { column : | None , | condition : | None , | value : | |\n| | None } | | | | | |\n| | ] , | | | | | |\n| | business_goals : | None , | | | | |\n| | additional_requirements : | None , | | | | |\n| | model_preferences : | None | | | | |\n| | } | | | | | |\n| 10 | P eT EL3 | PeTEL | P eT EL_is_complete | | | |\n| Checker | | | | | | |\n| Repeat step 6 - 10 until PeTEL is complete. Completed PeTEL is denoted by: P eT EL∗ | | | | | | |\n| n | P eT EL∗ , | PeTEL | De | | | |\n| scriptor | Description : | | | | | |\n| contextn | The task is to predict the final_grade of a student based on various features like study_hours, attendance, participation, homework_scores, and test_scores. The dataset size is hypothetical. Performance is measured using accuracy, F1 score, and the confusion matrix. K-fold cross-validation is used for validation. Three classification methods are considered: random forest, SVM, and logistic regression. The data is filtered to only include records where attendance is greater than 75 (assuming attendance is measured as a percentage) and study_hours is more than 1. The business goal is to predict student performance to implement early interventions for those at risk of failing. Model interpretability is highlighted as an important additional requirement, and there's a preference for models with higher accuracy, but interpretability is also considered important. Continued on next page | | | | | |\n\nTable 12 - continued from previous page\n\n# Input Micro-Agent **Output**\n\nn+1 *context*n+1,\n\nutterancen+1\n\nState Detectordsn+1 : Prediction Engineering\n\nTable 12: Interaction of different micro-agent in the Task Selection state.\n\nIn the table, utterance1, utterance2*, utterance*3 are from Table 10, and\n\nsummary, ML_*task* are from Table 11. At 10th row, Micro-Agent PeTEL Checker is a python function which check existance of null value in\n\nPeTEL.\n\n| Step | Input | Micro-Agent | Output | |\n|------------------------------------------------------------------------|--------------------------------------|-------------------|------------------------------------------------------|--------------------------------------------------------|\n| 1 | contextn, | P eT EL∗ , | Result Summarizer | Result : performance of each model based on evaluation |\n| AutoML response | criteria set in problem formulation. | | | |\n| 2 | contextn, Result | Result Visualizer | Output : Description of results in natural language. | |\n| Table 14: Interaction of different micro-agent in the Task Formulation | | | | |\n\n| # | Input | Micro-Agent | Output |\n|-----|----------|----------------------------|----------------------------|\n| 1 | P eT EL∗ | PeTEL to Feature Converter | List of attributes |\n| 2 | P eT EL∗ | Data Preparer | Prepares data with the conditions in PeTEL |\n| 3 | P eT EL∗ | AutoML Interfacer | Calls the AutoML interface |\n\n## C.4 Prompt Engineering Taxonomy\n\nThe successful collaboration between humans and artificial intelligence in complex tasks necessitates a comprehensive understanding of the various levels of interaction that occur between them. These levels span from Level 0, where AI is solely responsible for data processing, to Level 5, which involves the integration of evaluation criteria. Building upon the foundational work on taxonomy of prompt engineering (TELeR) by Santu & Feng (2023), we put forward the notion of considering the depth of information that the System Role discloses to the Large Language Model (LLM). To illustrate, if a system role is welldelineated, it precludes its prompt from being classified as Level 0. This study will specifically focus on three micro-agents: the Intent and State Detector, the Dialogue Summarizer, and the Conversation Manager. Each of these micro-agents plays a unique and integral role in fostering a dynamic and functional dialogue between the user and the AI, leading to a more streamlined and efficient system overall.\n\nThe revised taxonomy for these interaction levels is as follows: Level 0: No directive is given. The focus is solely on the exchange of data.\n\nLevel 1: A simple one-sentence directive is provided, expressing the high-level goal of the task.\n\nLevel 2: A multi-sentence (paragraph-style) directive is given, expressing the high-level goal and the sub-tasks needed to achieve this goal.\n\nLevel 3: A complex directive is provided, expressing the high-level goal along with a bulleted list of subtasks that need to be performed.\n\nLevel 4: This level includes a complex directive that encompasses the following: 1) A description of the high-level goal, 2) A detailed bulleted list of subtasks, and 3) An explicit statement asking the LLM to explain its response.\n\nLevel 5: This level includes a complex directive that encompasses the following: 1) A description of the high-level goal, 2) A detailed bulleted list of subtasks, 3) An explicit statement asking the LLM to explain its response, and 4) A guideline about how the response should be evaluated.\n\nBy understanding these levels of interaction, we can maximize the potential benefits of AI and guide future research into user experience, system performance, and ethical considerations in AI applications.\n\n## C.4.1 Intent And State Detector Micro-Agent:\n\nIn this case study, each level of the prompt will be constructed based on the following input:\n1. **context:** The user and the AI assistant discussed the summary of a dataset containing information about students in a school, including their demographic information, academic performance, and social life. The AI assistant suggested two suitable machine learning tasks based on the dataset: classification and regression. The user agreed that classification is a good choice to identify students who are at risk of failing and provide early intervention to improve their academic performance.\n\n2. **state:** dataset_understanding 3. **utterance:** Ok, from the description it seems like classification is a good choice.\n\nBy following the prompt taxonomy we discussed earlier, Table 15 represents the response from the LLM\nagent for each level of prompt.\n\n## C.4.2 Dialogue Summarizer Micro-Agent:\n\nIn this case study, each level of the prompt will be constructed based on the following history as its input:\n\n## - History:\n\n1. assistant: summary of Student Performance dataset 2. user: From the summary you provided can you suggest me any machine learning task?\n\n3. assistant: *context*1 from Table 12 4. user: Ok, from the description it seems like classification is a good choice.\n\nBy following the prompt taxonomy we discussed earlier, Table 16 represents the response from the LLM\nagent for each level of prompt.\n\n## C.4.3 Conversation Manager Micro-Agent\n\nIn this case study, each level of the prompt will be constructed based on the following input:\n1. **context:** The user and assistant discussed the dataset containing information about students in a school, including their demographic information, academic performance, and social life. The assistant suggested that the suitable machine learning tasks for this dataset are classification and regression, with classification being a good choice to identify students at risk of failing and provide early intervention to improve their academic performance. The user agreed that classification is a good choice.\n\n2. **state:** Dataset Visualization 3. **utterance:** Ok, from the description it seems like classification is a good choice.\n\n4. **intent:** Select problem 5. **mp_resp:** {\"model\": \"classification\", \"reason\": \"Classification is a good choice for identifying students at risk of failing and providing early intervention to improve their academic performance. The dataset contains demographic information, academic performance, and social life, which can be used as input features to predict the likelihood of a student failing or passing.\"}\nBy following the prompt taxonomy we discussed earlier, Table 17 represents the response from the LLM\nagent for each level of prompt.\n\n| Prompt Level | Response From LLM | | | | | | | | | |\n|----------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------|----------|----------------|------------------------|------------------------|-------------|------|------------|-----|\n| 0 | Can you tell me more about the dataset? What are the features and labels? | | | | | | | | | |\n| 1 | Intent: dataset_understanding State: The user and the AI assistant have discussed a dataset containing information about students in a school, and the AI assistant has suggested two suitable machine learning tasks: classification and regression. The user has agreed that classification is a good choice to identify students who are at risk of failing and provide early intervention to improve their academic performance. | | | | | | | | | |\n| 2 | { | intent: | Select | problem, | current_state: | Dataset_Understanding, | next_state: | | | |\n| | Task_Formulation } | | | | | | | | | |\n| 3 | { | intent: | Problem | Execution, | current_state: | Dataset_Understanding, | next_state: | | | |\n| | Task_Formulation } | | | | | | | | | |\n| 4 | {intent: | Select | problem, | current_state: | dataset_understanding, | next_state: | | | | |\n| | Task_Formulation, | explanation: | The | user | has | indicated | that | they | understand | the |\n| | dataset and have chosen classification as a suitable problem. | Therefore, the next state | | | | | | | | |\n| | should be problem_selection, where the user can select a problem related to classification.} | | | | | | | | | |\n| 5 | {intent: | Select | problem, | current_state: | dataset_understanding, | next_state: | prob | | | |\n| | lem_selection, explanation: The user has acknowledged the suggestion of classification as a suitable problem and is now ready to move to problem selection state. The dataset understanding state has been completed, and problem selection is the next logical step in the conversation flow., evaluation: The agent has correctly identified the userś intent and provided a clear explanation for the state transition. The evaluation criteria for the agentś response are: accuracy, clarity, and relevance to the conversation context.} Table 15: State Detector micro-agent response for different level of prompt engineering input. | | | | | | | | | |\n| Prompt Level | Response From LLM | | | | | | | | | |\n| 0 | Great! Classification is a suitable choice for predicting whether a student will pass or fail based on their demographic information, academic performance, and social life. You can use various classification algorithms such as logistic regression, decision trees, random forests, and support vector machines to build a model that can accurately predict the target variable. Additionally, you can use techniques such as feature selection and hyperparameter tuning to improve the performance of the model. | | | | | | | | | |\n| 1 | Great! Classification is a suitable choice for predicting whether a student will pass or fail based on their demographic information, academic performance, and social life. This can help identify students who are at risk of failing and provide early intervention to improve their academic performance. | | | | | | | | | |\n| 2 | The user and assistant discussed the summary of a dataset containing information about students in a school, including their demographic information, academic performance, and social life. The assistant suggested two suitable machine learning tasks based on the dataset: classification and regression. The user decided that classification would be a good choice to identify students at risk of failing and provide early intervention. | | | | | | | | | |\n| 3 | Based on the dataset summary, classification is indeed a suitable machine learning task to predict whether a student will pass or fail based on their demographic information, academic performance, and social life. This can be formulated as a binary classification problem where the target variable is pass/fail and the features can include sex, age, address, family size, parents' education level, study time, number of failures, alcohol consumption, and school absences. | | | | | | | | | |\n| 4 | Based on the dataset summary, classification is indeed a suitable choice for predicting whether a student will pass or fail based on their demographic information, academic performance, and social life. This task can help identify students who are at risk of failing and provide early intervention to improve their academic performance. Continued on next page | | | | | | | | | |\n\n| | | Table 16 - continued from previous page |\n|--------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------|\n| Prompt Level | | Response From LLM |\n| 5 | Based on the dataset summary, it does seem like classification is a suitable machine learning task. It can help identify students who are at risk of failing and provide early intervention to improve their academic performance. The features that can be used for classification include sex, age, address, family size, parents' education level, study time, number of failures, alcohol consumption, and school absences. Table 16: Dialogue Summarizer micro-agent response for different level of prompt engineering input. | |\n| Prompt Level | Response From LLM | |\n| 0 | The assistant then asked the user about the specific features in the dataset that could be used for classification. The user mentioned that features such as gender, age, ethnicity, parental education level, and previous academic performance could be used to predict whether a student is at risk of failing. The assistant agreed and suggested that the dataset could be split into training and testing sets to build and evaluate a classification model. The user agreed and asked the assistant for recommendations on which classification algorithms to use. The assistant suggested using logistic regression, decision trees, and random forests, as they are commonly used for classification tasks and can handle both categorical and numerical data. The user thanked the assistant for the suggestions and said they would explore these algorithms further. | |\n| 1 | Yes, based on the information you provided, classification seems like a suitable machine learning task for the dataset containing information about students in a school. It can help identify students who are at risk of failing and provide early intervention to improve their academic performance. Regression can also be a good choice to predict the academic performance of students based on their demographic and social information. | |\n| 2 | Great! Now that we have agreed that classification is a suitable task for your dataset, let's move on to formulating the problem. Can you please provide me with more details about the specific problem you want to solve using this dataset? For example, do you want to predict which students are at risk of failing a particular subject or identify factors that contribute to academic success? This information will help us formulate a more precise ML task that meets your needs. | |\n| 3 | Based on our conversation so far, it seems like classification is a suitable model for identifying students who are at risk of failing and providing early intervention to improve their academic performance. By analyzing various factors such as attendance, grades, and behavior, a classification model can predict which students are likely to fail and provide targeted support to improve their academic outcomes. Now that we have selected the ML task, let's move on to formulating the problem. Can you provide more information about the dataset you have and the specific problem you want to solve with this classification model? This will help us to formulate the problem and select appropriate parameters for the model. | |\n| 4 | Based on the given context, it seems like we have reached a consensus that classification is a suitable machine learning task for the dataset containing information about students in a school. Now, we need to formulate the problem to proceed further. To do that, I suggest we move to the next conversation state, which is \"problem_formulation\". In this state, we can discuss the specific details of the classification task, such as the target variable, the features to be used, and the algorithm to be employed. So, to move forward, can you please provide me with more information about the problem you want to solve using classification? This will help me formulate the problem and suggest relevant solutions to you. Continued on next page | |\n\n| | Table 17 - continued from previous page |\n|--------------|-------------------------------------------|\n| Prompt Level | Response From LLM |\n| 5 | Based on the current conversation state, it seems like we are in the Task_Formulation state, and the user has agreed that classification is a good choice for the given dataset. Therefore, we can proceed with formulating the classification problem. To do so, we need to determine the target variable that we want to predict. In this case, we want to identify students who are at risk of failing and provide early intervention to improve their academic performance. Therefore, we can use the \"final grade\" variable as our target variable, and classify students into two categories: those who are at risk of failing and those who are not. To achieve this, we can use a binary classification algorithm, such as logistic regression or decision tree. We can also use feature selection techniques to identify the most important features that contribute to the prediction of the target variable. Based on this, we can formulate the ML task as follows: Develop a binary classification model to predict whether a student is at risk of failing based on their demographic information, academic performance, and social life. The target variable is the final grade, and the model should be trained on a labeled dataset. Feature selection techniques can be used to identify the most important features. This task can be further refined based on the userś specific requirements and preferences. For example, the user may want to use a different classification algorithm or include additional features in the model. Evaluation criteria for the generated response: - The response provides a clear and concise formulation of the ML task based on the userś requirements and preferences. - The response takes into account the dataset description and the previous discussion between the user and the assistant. - The response suggests an appropriate classification algorithm and feature selection techniques based on the problem requirements. - The response is open-ended and allows for further refinement and customization based on the userś specific needs Table 17: Conversation Manager micro-agent response for different level of prompt engineering input. |\n\n## D Reproducibility Criteria\n\nFor all reported experimental results:\n- A clear description of the mathematical setting, algorithm, and/or model.\n\n- Model description is provided in Section 3.\n\n- Submission of a zip file containing source code, with specification of all dependencies, including external libraries, or a link to such resources (while still anonymized) Description of computing infrastructure used.\n\n- Zip file containing source code will be submitted in submission as supplementary material.\n\n- The average runtime for each model or algorithm\n(e.g., training, inference, etc.), or estimated energy cost\n- We used ChatGPT which provides API for getting response hence we are unable to estimate such performance.\n\n- Number of parameters in each model\n- We use ChatGPT as the LLM in our case study. As we do not get any such information from any document of ChatGPT, we can not report such number.\n\n- Corresponding validation performance for each reported test result\n- We utilize the LLM (ChatGPT) through API,\nwe do not train or finetune the model hence we can not report such performance metric.\n\n- Explanation of evaluation metrics used, with links to code\n- We provide qualitative study in this paper i.e.\n\nwe depend on human judgment for the evaluation.\n\nFor all experiments with hyperparameter search: - The exact number of training and evaluation runs\n- We utilize the LLM (ChatGPT) through API,\nwe do not train or finetune the model hence we can not report such number.\n\n- Bounds for each hyperparameter\n- We utilize the LLM (ChatGPT) through API,\nwe do not train or finetune the model hence we can not report such number.\n\n- Hyperparameter configurations for bestperforming models\n- We utilize the LLM (ChatGPT) through API,\nwe do not train or finetune the model hence we can not report such number.\n\n- Number of hyperparameter search trials\n- We utilize the LLM (ChatGPT) through API,\nwe do not train or finetune the model hence we can not report such number.\n\n- The method of choosing hyperparameter values\n(e.g., uniform sampling, manual tuning, etc.) and the criterion used to select among them (e.g., accuracy)\n- We utilize the LLM (ChatGPT) through API,\nwe do not train or finetune the model hence we can not report such number.\n\n- Summary statistics of the results (e.g., mean, variance, error bars, etc.)\n- We utilize the LLM (ChatGPT) through API,\nwe do not train or finetune the model hence we can not report such number.\n\nFor all datasets used:\n- Relevant details such as languages, and number of examples and label distributions\n- Refer to Section C.1.\n\n- Details of train/validation/test splits\n- Refer to Section C.1.\n\n- Explanation of any data that were excluded, and all pre-processing steps\n- Refer to Section C.1.\n\n- A zip file containing data or link to a downloadable version of the data\n- Refer to Section C.1.\n\n- For new data collected, a complete description of the data collection process, such as instructions to annotators and methods for quality control.\n\n- Refer to Section C.1."},"reviews":{"kind":"string","value":"Review 1:\nSummary: The paper presents a case study in developing an automatic conversational data science assistant through large language models. It presents a conversational AI agent called VIDS that helps users conduct data science tasks, describing the collection of micro-agents that constitute the full system. The authors claim with empirical evidence that VIDS powered by LLMs show strong potential for enabling conversational data science and thus making data science explorations more accessible to the general public. It shares details of the prompts used throughout the system and a single case study with the student performance dataset.\n\nStrengths and Weaknesses: Strengths \n- S1: Describes in interesting architecture for enabling conversational data science, a novel field of study. \n- S2: Details for the prompts used are shared and a case study with these prompts are shared. \n\nWeaknesses\n- W1: This paper presents a concept, rather than a scientific study. Although it is understandable that it is difficult to quantify the performance of each micro-agent, the contributions of this paper would be much more impactful if it proposed an evaluation framework for this task and established a baseline with the presented system. Without some measure of statistical reliability for each micro-agent, it is impossible for the readers to gauge whether the claims of this paper that VIDS is a promising system is supported. \n- W2: There are no ablation studies that motivate the design decisions. For example, how important is it to have a dialogue summarizer? Can we provide just the last few turns of the dialogue instead? \n- W3: Ethical considerations are only shallowly discussed despite the potential for severe negative outcomes if the conversational data science assistant provides an erroneous conclusion from the data.\n\nRequested Changes: - RC1: In general, better motivate each design decisions. Currently, the paper reads as \"we did this and this was the result\", rather than a scientific exploration that goes \"we tried XYZ and the best results were with X and our analysis shows that this is so because of ABC\".\n- RC2: Propose an evaluation framework, even if it involves human evaluation, for each micro-agent or state so that a concrete baseline is established for future work to build on. A lot of space can be made for this by removing redundancies and fancy language. \n- RC3: Citation format should be (X et al., 2023) instead of X et al. (2023) when citation is not used as the subject/object.\n- RC4: Contextualize the student performance dataset so that readers can understand why it is a good dataset for studying conversational data science. Even after reading the appendix, it is unclear why this is the chosen dataset when it seems like there can be many other alternatives. Unless there is a strong reason why this dataset is the most suitable, the author should consider replicating the study with at least one other dataset. \n- RC5: Briefly explain the TeLER taxonomy so the paper can be more self-contained.\n\nBroader Impact Concerns: The authors should expand on the limitations, especially for the ethical considerations. What are the broader ethical implications of an automatic conversational data sciences system? What are the risks that we should be especially careful of in this domain? How could LLMs malfunctioning in a certain dialogue state lead to negative outcomes, such as an incorrect result summary and thus an erroneous conclusion delivered to a naive data scientist? The current write-up in this section is extremely shallow and it suggests that the authors did not consider these risks.\n\n==================================================\n\nReview 2:\nSummary: The authors propose a pipeline for using large language models (LLMs) to automate the work currently done by data scientists. That is, the aim is to have the LLM help an end user determine what machine learning task should achieve the user’s goals for the dataset, and then determine how to set up and train the model, and summarize the results and provide recommendations to the end user. To pursue this goal, the authors propose a pipeline called VIDS (Virtual Interactive Data Scientist) that uses a sequence of prompts to get an LLM to act as an interface between the end user and a ML toolkit such as scikit-learn. The authors investigate their proposed approach with some case studies.\n\nStrengths and Weaknesses: Strengths:\n- S1: The goal is an interesting and ambitious one that aims to use the impressive power of LLMs in a way that could make data science work more efficient and widely accessible.\n- S2: The authors provide a careful breakdown of the data science analysis process. This breakdown may be useful to future authors investigating ways to make data science more efficient.\n- S3: The authors provide careful analysis for each part of the data science pipeline of how to make that part accessible to an LM. In particular, one aspect that I find useful is the prompting techniques for producing machine-interpretable content, such as PeTEL data, as a way to encapsulate the LLMs decisions in a way that can then be fed to an ML toolkit.\n- S4: The authors clearly lay out strengths and limitations of the approach.\n\nWeaknesses:\n- W1: The paper does not have quantitative results to support its claims. This makes it very challenging to evaluate how successful the proposed approach is. This means that unfortunately I don’t think the paper makes much headway toward answering its research questions on the bottom of page 2; we can’t tell if LLMs are feasible if we can’t get some reliable estimate of how well they perform the task; we can’t assess how accurate LLMs are without some quantitative analysis, and we can’t see what the common challenges are without a large enough sample size to determine what challenges are common. I recognize that, as the authors say, the task is ill-defined, making it difficult to evaluate; but it would still be possible to attain quantitative results by, e.g., using a sample of 100 data science scenarios and then computing statistics for how well VIDS performed on various dimensions of desired behavior (e.g., “92% accuracy at identifying the correct task.”)\n- W2: The paper does provide some anecdotal results, but it is difficult to assess the LLM’s performance here because most of the concrete material for these case studies is in the appendix and is formatted in a way that makes it hard to tell if the LLM performs well; we can see the statements that the LLMs make, but we can’t tell if its decisions are in fact the right way to analyze the dataset. \n- W3: The paper makes many unsubstantiated statements about the strength of the proposed approach. As a non-exhaustive list of examples, it states that the approach “**proficiently** handles the nuances of user utterances”, that “VIDS **deftly** incorporates a number of distinct states”, that “it facilitates a **seamless** dialogue flow, aligning with user needs while **enriching** their interactive experience”,that “this micro-agent generates **accurate** responses by integrating inputs from other micro-agents”, and that another component is “**ensuring** no aspect of the task is missed” (the emphases in the preceding quotes are mine). However, there is no evidence to support the claims that it is “proficient”, “deft”, “seamless”, “enriching”, or “accurate” or that it “ensures” strong performance. To make such statements, there should be concrete evidence. For instance, to state that the system is accurate, there should be an analysis of enough different use cases that you can quantitatively compute the accuracy; or to state that it enriches the user experience there should be some sort of analysis where users are asked about their experience. These statements are therefore risky because they could mislead readers into thinking that VIDS or more impressive or advanced than it is. The authors do carefully note limitations of their approach, but it is not enough to solely list limitations in a limitations section; the rest of the paper should also be written in a way that is consistent with the project’s strengths and limitations.\n- W4: One of the main motivations for the approach is that it can remove some of the need for expert data scientists in some phases of a project. However, I do not believe that the paper makes a convincing case that VIDS actually accomplishes this. Specifically, in order for this to be possible, we need to be able to trust the LLM so completely that we can be confident its recommendations are correct. Without this complete trust, we would still need a data scientist to verify that the recommendations are reasonable; the dialogues (e.g., in Table 10) do ask users for verification, but if the user doesn’t know data science, they don’t have the knowledge to really tell if the recommendations are reasonable. Therefore, in order to achieve the paper’s goal, it seems important to establish whether the LLMs perform well enough to be completely trusted, which seems unlikely given issues of hallucination in LLMs. Since the paper does not provide analysis of how reliable LLMs are, I remain unconvinced that LLMs can be a useful contribution in this area. An alternative framing could be that LLMs could be used by data scientists to speed up their work, but even this would still require some analysis to establish whether data scientists find LLMs useful for this purpose.\n\nIn sum, while I applaud the authors' ambition and their aim to explore the true extent of the usefulness of LLMs, I am concerned that the execution does not achieve the paper's stated goals.\n\nRequested Changes: Unfortunately, in order for me to recommend acceptance, the paper would need to be substantially rewritten and restructured, so I think it is unlikely that changes could be made that are in the scope of a revision and that would be sufficient for me to recommend acceptance. So below I will list the changes that would be necessary for me to recommend acceptance, but with the caveat that doing all of this would be quite an extensive amount of work that seems out of scope for a revision:\n- RC1: Add quantitative analysis of the pipeline’s performance. The analysis should cover a sufficiently large sample size of cases that we can obtain a reasonable estimate of how successful the approach is. At a minimum, it should include data on how accurate the model’s predictions were at each stage in the pipeline, and also how accurate the whole end-to-end system is (to address the potential issue that errors might accumulate as you go down the pipeline).\n- RC2: Add some experiments testing how useful VIDS is for users who know nothing about data science. (This could be done at the same time as the experiments in RC1, by having those experiments be run by non-data-scientists, or it could be done separately). This relates to W4 above.\n- RC3: Revise the paper to make sure that all of the claims it makes are backed up by evidence, removing any claims about the system’s strength that do not have evidence to back them up (see W3).\n\nBroader Impact Concerns: Given my concerns that the paper’s statements are not backed up by evidence, there is the possibility that this paper might mislead people into thinking that current LLMs can be trusted to perform data science tasks, when in fact we do not have enough evidence to determine whether this is the case.\n\n==================================================\n\nReview 3:\nSummary: The paper deals with applying LLMs to an interesting area: providing a natural language conversational interface to systems like AutoML used for common data science tasks. The paper use a LangChain-like approach to:\n1) define the various components needed to accomplish the data science task: Data visualization, Task Formulation, Prediction engineering, Results summarization and recommendation. \n2) connect various components required to accomplish a data science task by \n * iteratively invoking the LLM\n * refining the prompt to account for the state, task objective\n * performing slot-filling where necessary to augment the prompt with needed info.\n3) create separate global components for 2) above so that the whole framework can operate in the context of a conversational assistant.\n4) ultimately generate an execution plan that can be run on the user provided dataset in AutoML.\n\nThese components and their subcomponents are termed as micro-agents in the paper. \n\nThe paper details one possible design of such a system in it's entirety, named Virtual Interactive Data Scientist (VIDS). It provides the contents of the prompts used in each of the components. It also discusses the qualitative effect of adding more information in the prompt\n\nStrengths and Weaknesses: Strengths\n========\n1) Explores a very interesting, nascent and impactful area of research - using LLM advances for democratization of data science.\n2) References quite relevant literature for a detailed systems-level approach to solving a data science problem. \n3) The paper's main strength is clearly designing the VIDS sytem with various components - each with a capability granular enough that it can be accomplished by straightforward prompting techniques in LLMs (like LangChain)\n4) Able to demonstrate successfully completing an AutoML task via dialog (that might otherwise require complex UI navigation). \n\nWeaknesses\n==========\n1) I'm not sure if the paper clearly showed that conversational AI is feasible or preferable way to perform data science tasks in general. The paper would have to include examples of more query or analysis based tasks and leverage the reasoning capabilities of the LLM (vs leveraging it's tool-use capabilities as in the current paper).\n2) The paper doesn't answer the question about the accuracy of LLMs in framing and solving ill-defined\ncomplex data science tasks. The paper has qualitative analysis on a single dataset and a single task type. To ascertain accuracy, precision, or efficacy, it would need at least more than one example. It needs a benchmark to compare against, analysis of whether the output of the micro-agents themselves are accurate or not (and preferable an ablation analysis). Even if doing a qualitative analysis, it needs review by human raters to see if the outputs, summaries, solutions were usable and accurate, even if successfully completing the entire dialog workflow.\n3) The paper does not list details of why the specific dataset was chosen, how the mini version of that was selected so that the summarizer can extract trends.\n4) When discussing the strengths of the approach in Section 5.1, the paper does not list any metrics or criteria for evaluating precision control or efficiency. So it is hard to gauge whether the performance or the strength of the approach.\n\nRequested Changes: * The paper mentions the notion of ill-defined tasks, but presents examples where the tasks are from a relatively well-defined set. What is an example of this ill-defined task (since the example from Table 10 seems well-defined)? Is there a prior step missing where the user initiates the conversation with an overall goal (e.g. \"I want to find out what I can and cannot predict reliably from my dataset\")?\n* Performing Data Science on a given dataset is complex for practitioners not well-versed with the field. The paper postulates that a conversational AI or a dialog based interface is a feasible and preferred way to facilitate this. Can you please point to any prior evaluation-based study to support this? (i.e. If it can be shown that an end-to-end opinionated system - which only provides the users fixed levers to update the model - is inferior to a more free-form dialog based system (not necessarily LLM based))\n* In Section 2: \"...the true upper limit of LLMs remains a tantalizing unknown...\" which domains are the limits being mentioned here about?\n* In Section 3: \"...This stateless design ensures a smooth narrative flow and avoids complications of state-dependent biases or entanglements, thus bolstering the versatility and adaptability of our dialogue system...\" Can you please elaborate/provide example or reference on what would be a smooth vs non-smooth narrative flow? Also, can you please add some examples of entanglement-related complications? Additionally, can you please state what exactly is meant by stateless here (since based on the examples in Tables 1, 2, 3 the state and context of the conversation so far is passed into the prompt in every invocation).\n* In Section 3.2.1: \"...These cooperative micro-agents ensure efficient dataset exploration and readiness for the Task Formulation phase...\" Can you please explain what is meant by cooperative here? Based on Figure 2 and description in Section 3.2, the outputs from Dataset Summarizer and Task Suggestor micro-agents are then fed into Task formulation, but they don't affect each other.\n* Significant portions of Section 3.5 mentions about proposed future work for this component. It would probably be better to move this into the Future work or Discussions section.\n* Section 4: \"...demonstrating the practicality and applicability of LLMs in demanding scenarios...\" Can you please add some specificity to the scenarios? Are they demanding because of need to do CoT, or the length of the prompt is too long (hence complex)\n* Most of the information in Section 4.2 has previously been mentioned in the description of the micro-agents. It does not provide any information about the interaction, hence feels redundant. Consider updating?\n* In Section 5.1 Strengths of Dialogue Flow Control: \"LLM (ChatGPT) performed commendably in terms of overall dialogue flow control by carefully balancing multiple factors, such as the utterance, context, responses from various microagents, user intent, and the current state of the conversation.\" Based on the prior sections, the LLMs were individual global and local micro-agents within a (presumably) deterministic overall dialog workflow. Is there another root LLM agent that decides which of the global or local micro-agent to run? If yes, could you please add details about that? If not, could you please explain how it is the LLM that does the overall dialogue flow control? \n* The Appendix sections listing the micro-agents do not add any new information from what is presented in the main section of the paper. For instance, the Dataset Summarizer has no details on how the subset of data is selected by the user to embed in the prompt for that micro-agent to use for summarization. Please consider adding in more/new detail in these sections or merge the existing appendix sections into the main paper.\n* Typo on pg 20: \"ask selection\" -> \"task selection\"\n* Section C.4 mentions the paper used a revised version of the TELeR taxonomy. But the version in that section is the same as the version in the reference. Can you please update? Also, can you please provide a clear example of how those levels translates to different *input* prompts for the various micro-agents?\n* The temperature setting for the OpenAI calls have not been specified, nor the randomizer's seed. Can you please mention either of those for reproducibility?\n\nBroader Impact Concerns: N/A\n\n==================================================\n\nMetareview:\nRecommendation: Reject\n\nComment: All the reviewers commended the authors for working on a potentially impactful effort of leveraging LLMs to facilitate data science tasks. The authors also made a nice connection to AutoML. Reviewer suFq also noted that \"The authors provide a careful breakdown of the data science analysis process. This breakdown may be useful to future authors investigating ways to make data science more efficient.\" Reviewer tDKy mentioned that \"References quite relevant literature for a detailed systems-level approach to solving a data science problem.\" Reviewer MYpM concurred that \"While the concept is interesting, a lack of quantitative results make the claims of the effectiveness of the proposed system unconvincing.\"\n\nHowever, all the reviewers are concerned with the lack of quantitative results. In the recommendation, Reviewer suFq stated that \"The lack of quantitative results makes me unable to recommend acceptance.\" Reviewer tDKy is concerned with the \"rigor for metrics and evaluation of experimental results\". From the discussions between the authors and the reviewers, it seems that the authors have a concurrent paper under review at TMLR contains more quantitative evaluation of the work. All the reviewers suggest that the authors should submit a single expanded version of the work instead of having two separate submissions that each can't stand on their own.\n\n==================================================\n\n"}}},{"rowIdx":89,"cells":{"paper":{"kind":"string","value":"# Training-Free Graph Neural Networks And The Power Of Labels As Features\n\nRyoma Sato rsato@nii.ac.jp National Institute of Informatics Reviewed on OpenReview: *https: // openreview. net/ forum? id= 7DzU88VrNU*\n\n## Abstract\n\nWe propose training-free graph neural networks (TFGNNs), which can be used without training and can also be improved with optional training, for transductive node classification.\n\nWe first advocate labels as features (LaF), which is an admissible but not explored technique.\n\nWe show that LaF provably enhances the expressive power of graph neural networks. We design TFGNNs based on this analysis. In the experiments, we confirm that TFGNNs outperform existing GNNs in the training-free setting and converge with much fewer training iterations than traditional GNNs.\n\n## 1 Introduction\n\nGraph Neural Networks (GNNs) [12, 41] are popular machine learning models for processing graph data.\n\nGNNs show strong empirical performance in various machine learning and data mining tasks, including chemical modeling [10, 19], question answering [34, 44], and recommender systems [7, 15, 47, 49, 55].\n\nOne of the standard problem settings for GNNs is transductive node classification, where the goal is to predict the labels of the test nodes in a graph given the labels of other nodes. This setting has many applications, including document classification [20, 54], e-commerce [42, 56], and social analysis [13, 56]. Many GNNs, including Graph Convolutional Networks (GCNs) [20] and Graph Attention Networks (GATs) [45], tackled transductive node classification and showed excellent performance.\n\nOne of the challenges of GNNs is the computational cost. There are many huge graphs in practice, such as social networks and Web graphs, which contain billions of nodes. It is sometimes prohibitive to even scan the entire graph, e.g., the whole World Wide Web. Many methods to speed up GNNs have been proposed. The basic approach is sampling nodes and/or edges to reduce the graph size [4, 13, 56, 59]. In the extreme case, Sato et al. [40] proposed constant time graph neural networks by neighbor sampling. Although it drastically reduces the computational cost per iteration, it still requires many training iterations. PinSAGE [55] adopts parallel training with MapReduce as well as importance pooling to speed up the training. Although PinSAGE succeeded in training GNNs with Web-scale graphs, it requires massive computational resources. It is still challenging to instantly use GNNs with limited computational resources. In this paper, we propose training-free graph neural networks (TFGNNs).\n\nTo design TFGNNs, we first introduce the idea of labels as features (LaF). LaF uses the node labels as features, which is admissible in the framework of transductive node classification. GNNs with LaF can utilize the label information, such as the class distribution in the neighboring nodes, to compute the node embeddings, which contain much more information than the embeddings with only the node features. We show that LaF provably enhances the expressive power of GNNs. TFGNNs can be used without training and deployed instantly as soon as the model is initialized. This property reduces the burden of hyperparameter tuning as no training process is involved in this mode.\n\nTFGNNs can also be improved with optional training. Users can use TFGNNs with the training-free mode or train TFGNNs for few iterations when the computational resources for training are limited. This property is useful for online learning, where training data come in a streaming manner, and the model should be updated instantly. Users can also fully train TFGNNs when the computational resources are abundant or the accuracy is required. TFGNNs enjoy the best of both worlds of nonparametric models and GNNs.\n\nIn the experiments, we confirm that TFGNNs outperform existing GNNs in the training-free setting and converge with much fewer training iterations than traditional GNNs.\n\nThe contributions of this paper are as follows:\n- We advocate the use of LaF in transductive learning.\n\n- We formally show that LaF strengthens the expressive power of GNNs.\n\n- We proposed training-free graph neural networks (TFGNNs). - We confirm that TFGNNs outperform existing GNNs in the training-free setting.\n\nReproducibility: Our code is available at https://github.com/joisino/laf.\n\n## 2 Background 2.1 Notations\n\nFor every positive integer n ∈ Z+, [n] denotes the set {1, 2*, . . . , n*}. A graph is defined as a tuple of (i) the set V of nodes, (ii) the set E of edges, and (iii) the node features X = [x1, x2*, . . . ,* xn]\n⊤ ∈ R\nn×d. Without loss of generality, we assume that the nodes are numbered with 1, 2*, . . . , n*. Y denotes the set of labels. yv ∈ R\nY\ndenotes the one-hot encoding of the label of node v. For every node v ∈ V , N (v) denotes the set of neighbors of node v. We adopt the numpy-like notation for indexing. For example, X:,1 denotes the first column of X, X:,−1 denotes the last column of X, X:,−5: denotes the last five columns of X, and X:,:−5 denotes all the columns except for the last five columns of X.\n\n## 2.2 Transductive Node Classification\n\nProblem (Transductive Node Classification).\n\nInput: A graph G = (*V, E,* X), labelled nodes Vtrain ⊂ V , and node labels Ytrain ∈ YVtrain of Vtrain.\n\nOutput: Predicted labels Ytest ∈ YVtest of Vtest = V \\ Vtrain.\n\nThere are two settings for the node classification problem: transductive and inductive. In transductive node classification, one graph and the labels of some of its nodes are given, and we predict the labels of the remaining nodes. This setting uses the same graph for training and testing. This is in contrast to the inductive setting, which uses different graphs for training and testing. For example, in spam account detection, annotating spam accounts on Facebook and using the trained model on Facebook is an example of the transductive setting, while using the trained model on X (Twitter) is an example of the inductive setting.\n\nTransductive node classification is a popular setting in the GNN community; it has been employed in wellknown studies, such as GCNs [20] and GATs [45], and has been adopted in popular benchmarks such as Cora, PubMed, and CiteSeer. There are also many practical applications of transductive node classification, such as document classification [20, 54] and fraud detection[23, 24, 46].\n\n## 2.3 Graph Neural Networks\n\nGNNs are a popular solution for transductive node classification. We follow the message-passing framework of GNNs [10]. A message passing GNN is defined as follows:\n\n$\\mathbf{h}_{v}^{(0)}=\\mathbf{x}_{v}$ ($\\forall v\\in V$), $\\mathbf{h}_{v}^{(l)}=f_{\\text{agg}}^{(l)}(\\mathbf{h}_{v}^{(l-1)},\\{\\mathbf{h}_{u}^{(l-1)}\\mid u\\in\\mathcal{N}(v)\\}\\}$) ($\\forall l\\in[L],v\\in V$), $\\mathbf{\\hat{y}}_{v}=f_{\\text{pred}}(\\mathbf{h}_{v}^{(L)})$ ($\\forall v\\in V$),\nwhere f\n(l)\nagg is the aggregation function and fpred is the prediction head, which are typically modeled by neural networks.\n\n## 3 Laf Is Admissible, But Not Explored Well\n\nWe ask the readers to recall the problem setting of transductive node classification. We are given node labels yv of the training nodes. A typical approach for this problem is to feed node features xv for a training node v to the model, predict the labels of node v, compute the loss based on the ground truth label yv, and update the model parameters. However, how we use yv is not limited. We can use yv as features of node v as well as for the loss function. This is the idea of LaF.\n\nGNNs with LaF initialize the node embeddings in Eq. (1) as\n\n$$(1)$$\n$$\\mathbf{\\Sigma}^{(2)}$$\n$$\\left({\\mathrm{3}}\\right)$$\n$$\\mathbf{h}_{v}^{(0)}=[\\mathbf{x}_{v};{\\tilde{\\mathbf{y}}}_{v}]\\in\\mathbb{R}^{d+1+|{\\mathcal{Y}}|},$$\n$$\\left(4\\right)$$\nd+1+|Y|, (4)\nwhere [·; ·] denotes the concatenation of vectors, and\n\n$$\\tilde{\\mathbf{y}}_{v}=\\begin{cases}[1;\\mathbf{y}_{v}]&(v\\in V_{\\text{train}}),\\\\ \\mathbf{0}_{1+|\\mathcal{V}|}&(v\\in V_{\\text{test}}),\\end{cases}\\tag{1}$$\n$$\\left(5\\right)$$\n\nis the label vector of node v, and 0d is the zero vector of dimension d. LaF enables GNNs to utilize the label information, such as the class distribution in the neighboring nodes, to compute the node embeddings. Such embeddings are expected to be more informative than the embeddings without the label information. LaF\nis admissible in the sense that it uses only the information available in the transductive setting.\n\nWe emphasize that LaF has not been explored well in the literature on GNNs, regardless of its simplicity, with some notable exceptions [1, 50] (see Section 7 for detailed discussions). For example, GCNs [20] and GATs [45] adopt the transductive setting, and they are allowed to use the label information as features. However, they initialize the node embeddings as h\n(0)\nv = xv without using the label information. One of the contributions of this paper is that we affirm that LaF is allowed in the transductive setting.\n\nWe should be careful when training GNNs with LaF. LaF may harm the generalization performance by inducing a shortcut of copying the label feature h\n(0)\nv,d+1: to the prediction. To prevent this, we should remove the label of the center nodes in the minibatch and treat them as test nodes. Specifically, let B ⊂ Vtrain be the set of nodes in the minibatch and we set\n\n$$\\tilde{\\mathbf{y}}_{v}=\\begin{cases}[1;\\mathbf{y}_{v}]&(v\\in V_{\\text{train}}\\setminus B),\\\\ \\mathbf{0}_{1+|\\mathcal{Y}|}&(v\\in V_{\\text{test}}\\cup B),\\end{cases}\\tag{6}$$\n\nand predict the label yˆv for v ∈ B, and compute the loss based on yˆv and yv. This simulates the transductive setting where the label information of the test nodes is missing, and GNNs learn how to predict the labels of the test nodes based on the label information and node features of the surrounding nodes.\n\n## 4 Laf Strengthens The Expressive Power Of Gnns\n\nWe show that LaF provably strengthens the expressive power of GNNs. Specifically, we show that GNNs with LaF can represent label propagation [58], an important model for transductive node classification, while GNNs without LaF cannot. This result is interesting in its own right, and it also motivates the design of TFGNNs. Label propagation is a classic method for transductive node classification. It starts random walks from a test node and outputs the label distribution of the labeled nodes the random walks first hit. The following theorem shows that GNNs with LaF can represent label propagation. Theorem 4.1. *GNNs with LaF can approximate label propagation with any precision. Specifically, there* exists a series of GNNs {f\n(l)\nagg}l and fpred such that for any positive ε*, for any connected graph* G = (*V, E,* X),\nfor any labeled nodes Vtrain ⊂ V and node labels Ytrain ∈ YVtrain and test node v ∈ V \\ Vtrain*, there exists* L ∈\nZ+ such that l(≥ L)*-th GNN* (f\n(1)\nagg*, . . . , f*(l)\nagg, fpred) with LaF outputs the approximation of label propagation with the error at most ε*, i.e.,*\n\n$$\\left\\|{\\hat{y}}_{v}-{\\hat{y}}_{v}^{L P}\\right\\|_{1}\\leq\\varepsilon,$$\n$$\\left(7\\right)$$\n$$({\\mathfrak{g}})$$\n\n≤ ε, (7)\nwhere yˆ\nLP *is the output of label propagation for test node* v.\n\nProof. We prove the theorem by construction. Let p*l,v,i* def = Pr[The random walk from node v hits Vtrain within l steps and the first hit label is i]. (8)\nFor the labeled nodes, this is a constant, i.e.,\n\ns $V_{\\text{train}}$ within $l$ steps and. \n$$p_{l,v,i}=1[i=\\mathbf{y}_{v}]\\quad(\\forall l\\in\\mathbb{Z}_{\\geq0},v\\in V_{\\mathrm{train}},i\\in\\mathcal{Y}).$$\n$$(10)$$\np*l,v,i* = 1[i = yv] (∀l ∈ Z≥0, v ∈ Vtrain, i ∈ Y). (9)\nFor the other nodes, this can be recursively computed as follows: These equations can be represented by GNNs with LaF. Specifically, the base case\n\n$$p_{0,v,i}=\\begin{cases}1[i=\\mathbf{y}_{v}]&(v\\in V_{\\text{train}}),\\\\ 0&(v\\in V\\setminus V_{\\text{train}}),\\end{cases}\\tag{1}$$\n$$(14)$$\n\ncan be computed from y˜v in h\n(0)\nv . Let f\n(l)\nagg always concat the first argument (i.e., h\n(l−1)\nv in Eq. (2))\nto the output so that the GNN can keep the information of the input. f\n(l)\nagg handles two cases by y˜v,1 ∈ {0, 1}, i.e., whether v is in Vtrain or not. If v is in Vtrain, f\n(l)\nagg just outputs 1[i = yv], which can be computed by y˜v in h\n(l−1)\nv . If v is not in Vtrain, f\n(l)\nagg aggregates pl−1*,u,i* from u in N (v) and takes the average, i.e., Eq. (13), which can be realized by message passing in the second argument of f\n(l)\nagg.\n\n$$p_{0,v,i}=0\\quad(\\forall v\\in V\\setminus V_{\\mathrm{train}},i\\in{\\mathcal{Y}}),$$\n\np0*,v,i* = 0 (∀v ∈ V \\ Vtrain, i ∈ Y), (10)\n$p_{l,v,i}$ $$=\\sum_{u\\in\\mathcal{N}(v)}\\Pr[\\text{The first step is}v\\to u]\\cdot\\Pr[\\text{The random walk from node$v$bits$V_{\\text{train}}$within}]$$l\\text{steps and the first hit label is}i\\mid\\text{The first step is}v\\to u]$$ $$=\\sum_{u\\in\\mathcal{N}(v)}\\frac{1}{\\deg(v)}\\cdot\\Pr[\\text{The random walk from node$u$bits$V_{\\text{train}}$within}(l-1)\\text{steps}]$$ and the first hit label is $i]$ $$=\\frac{1}{\\deg(v)}\\sum_{u\\in\\mathcal{N}(v)}p_{l-1,u,i}.$$\n(11)\n(12)\n$$(13)$$\nThe final output of the GNN is p*l,v,i*. The output of label propagation can be decomposed as follows:\nyˆ\nLP\nv,i\n= Pr[The first hit label is i] (15)\n= Pr[The random walk from node v hits Vtrain within l steps and the first hit label is i]\n+ Pr[The random walk from node v does not hit Vtrain within l steps and the first hit label is i]\n(16)\n= p*l,v,i*\n+ Pr[The random walk from node v does not hit Vtrain within l steps and the first hit label is i]\n(17)\nAs the second term converges to zero as l increases, the GNNs approximate label propagation with\n\n any precision by increasing l.\n\nWe then show that GNNs without LaF cannot represent label propagation. Proposition 4.2. GNNs without LaF cannot approximate label propagation. Specifically, for any series of GNNs {f\n(l)\nagg}l and fpred, there exists positive ε*, a connected graph* G = (V, E, X), labelled nodes V*train* ⊂ V ,\nnode labels Ytrain ∈ YVtrain and test node v ∈ V \\Vtrain, such that for any l*, GNN* (f\n(1)\nagg*, . . . , f*(l)\nagg, fpred) *without* LaF has the error at least ε*, i.e.,*\n\n$$\\left\\|{\\hat{\\mathbf{y}}}_{v}-{\\hat{\\mathbf{y}}}_{v}^{L P}\\right\\|_{1}\\geq\\varepsilon,$$\n≥ ε, (18)\nwhere yˆ\nLP *is the output of label propagation for test node* v.\n\nProof. We construct a counterexample. Let G be a cycle of four nodes. The nodes are numbered as 1, 2, 3, 4 in the clockwise direction. All the nodes have the same node features x. Let Vtrain = {1, 2}\nand Ytrain = [1, 0]⊤. Label propagation classifies node 4 as class 1 and node 3 as class 0. However, GNNs without LaF always predict the same label for nodes 3 and 4 since they are isomorphic.\n\nTherefore, for any GNN without LaF, there is an irreducible error either for node 3 or 4.\n\nTheorem 4.1 and Proposition 4.2 show that LaF provably enhances the expressive power of GNNs. These results indicate that GNNs with LaF are more powerful than traditional message passing GNNs such as GCNs, GATs, and GINs without LaF. Note that GINs have been considered to be the most expressive message passing GNNs, but GINs cannot represent label propagation without LaF while message passing GNNs with LaF can. This does not lead to a contradiction since the original GINs do not take the label information as input. Put differently, the input domains of the functions differ. These results indicate that it is important to consider what to input to GNNs as well as the architecture of GNNs.\n\n## 5 Training-Free Graph Neural Networks\n\nWe propose training-free graph neural networks (TFGNNs) based on the analysis in the previous section. TFGNNs can be used without training and can also be improved with optional training. First, we define training-free models. Definition 5.1 (Training-free Model). We say a parametric model is training-free if it can be used without optimizing the parameters.\n\nIt should be noted that nonparametric models are training-free by definition. The real worth of TFGNNs is that it is training-free while it can be improved with optional training. Users can enjoy the best of both worlds of parametric and nonparametric models by choosing the trade-off based on the computational resources for training and the accuracy required.\n\n\n\n$$(19)$$\n\n\n\n$$(20)$$\n$$(\\forall v\\in V).$$\n$$(21)$$\n\nFigure 1: Initialization of TFGNNs. The parameters of the last (1 + |Y|) rows or |Y| rows are initialized by 0 or 1 in a special pattern\nThe core idea of TFGNNs is to embed label propagation in GNNs by Theorem 4.1. TFGNNs are defined as follows:\n\n$$h_{v}^{(0)}=[x_{v};{\\tilde{y}}_{v}]$$\nh (0) v = [xv; y˜v] (∀v ∈ V ), (19) h (l) v = ReLU S (l)h (l−1) v +1 |N(v)| Pu∈N(v) V (l)h (l−1) u(v ∈ Vtrain, l ∈ [L]), ReLU T (l)h (l−1) v +1 |N(v)| Pu∈N(v) W(l)h (l−1) yˆv = softmax(Uh(L) v) (∀v ∈ V ). (21)\nu(v ∈ Vtest, l ∈ [L]),(20)\nThe architecture of TFGNNs is standard, i.e., TFGNNs transform the center nodes and carry out mean aggregation from the neighboring nodes. The key to TFGNNs lies in initialization. The parameters are initialized as follows:\n\n$\\mathbf{S}^{(l)}_{-(1+|\\mathcal{Y}|)}$:$-(1+|\\mathcal{Y}|)=0$ $\\mathbf{S}^{(l)}_{-(1+|\\mathcal{Y}|)}$:$-(1+|\\mathcal{Y}|)$:$=\\mathbf{I}_{1+|\\mathcal{Y}|}$ $\\mathbf{V}^{(l)}_{-(1+|\\mathcal{Y}|)}$:$=0$ $\\mathbf{T}^{(l)}_{-(1+|\\mathcal{Y}|)}$:$=0$ $\\mathbf{W}^{(l)}_{-(1+|\\mathcal{Y}|)}$:$-(1+|\\mathcal{Y}|)=0$ $\\mathbf{W}^{(l)}_{-(1+|\\mathcal{Y}|)}$:$-(1+|\\mathcal{Y}|)$:$=\\mathbf{I}_{1+|\\mathcal{Y}|}$ $\\mathbf{U}_{-}$:$-|\\mathcal{Y}|=0$, $\\mathbf{U}_{-}$:$-|\\mathcal{Y}|$:$=\\mathbf{I}_{|\\mathcal{Y}|}$\n−(1+|Y|):,:−(1+|Y|) = 0 (∀l ∈ [L]), (22)\n−(1+|Y|):,−(1+|Y|): = I1+|Y| (∀l ∈ [L]), (23)\n−(1+|Y|):,:−(1+|Y|) = 0 (∀l ∈ [L]), (26)\n−(1+|Y|):,−(1+|Y|): = I1+|Y| (∀l ∈ [L]), (27)\nU:,:*−|Y|* = 0, (28)\nU:,−|Y|: = I|Y|, (29)\n\n−(1+|Y|): = 0 (∀l ∈ [L]), (24)\n\n−(1+|Y|): = 0 (∀l ∈ [L]), (25)\n\ni.e., the parameters of the last (1 + |Y|) rows or |Y| rows are initialized by 0 or 1 in a special pattern (Figure 1). Other parameters are initialized randomly, e.g., by Xavier initialization [11]. The following proposition shows that the initialized TFGNNs approximate label propagation.\n\nProposition 5.2. *The initialized TFGNNs approximate label propagation. Specifically,*\n\n$$\\mathbf{h}_{v,-(|{\\mathcal{Y}}|-i+1)}^{(L)}=p_{L,v,i}$$\n$$(30)$$\n$$(31)$$\nv,−(*|Y|−*i+1) = p*L,v,i* (30)\nholds, where pL,v,i *is defined in Eq. (8), and*\n\n$$a n d$$\n$$\\operatorname*{arg\\,max}_{i}{\\hat{\\mathbf{y}}}_{v i}=\\operatorname*{arg\\,max}_{i}p_{L,v,i}$$\ni\ni\np*L,v,i* (31)\nholds, and p*L,v,i* → yˆ\nLP\nv,i as L → ∞.\n\nProof. By the definitions of TFGNNs,\n\n$$\\mathbf{h}_{v_{i},-|\\mathcal{Y}|:}^{(0)}=\\begin{cases}\\mathbf{y}_{v}&(v\\in V_{\\text{train}}),\\\\ \\mathbf{0}_{|\\mathcal{Y}|}&(v\\in V_{\\text{test}}),\\end{cases}$$ $$\\mathbf{h}_{v_{i},-|\\mathcal{Y}|:}^{(l)}=\\begin{cases}\\mathbf{h}_{v_{i},-|\\mathcal{Y}|:}^{(l-1)}&(v\\in V_{\\text{train}},l\\in[L]),\\\\ \\frac{1}{|\\mathcal{N}(v)|}\\sum_{u\\in\\mathcal{N}(v)}\\mathbf{h}_{u_{i},-|\\mathcal{Y}|:}^{(l-1)}&(v\\in V_{\\text{test}},l\\in[L]).\\end{cases}$$\n$$(32)$$\n$$(33)$$\nThis recursion is the same: $\\newcommand{\\colv}[1]{\\left\\|}$ . \nThis recursion is the same as Eqs. (9) - (13). Therefore, $\\mathbf{\\hat{\\theta}}$\n$$\\mathbf{h}_{v,-(|{\\mathcal{Y}}|-i+1)}^{(L)}=p_{L,v,i}$$\n$$(34)$$\n$${\\mathrm{monotone}},$$\n$$(35)$$\n$$\\square$$\nv,−(*|Y|−*i+1) = p*L,v,i* (34)\nholds. As $U$ picks the last $|\\mathcal{Y}|$\nholds. As U picks the last |Y| dimensions, and softmax is monotone,\n\n$$\\operatorname*{arg\\,max}_{i}{\\hat{\\mathbf{y}}}_{v i}=\\operatorname*{arg\\,max}_{i}p_{L,v,i}$$\np*L,v,i* (35)\nholds. $p_{L,v,i}\\to\\widehat{\\boldsymbol{y}}_{v,i}^{\\mathrm{LP}}$ as $L\\to\\infty$ is shown in the proof of Theorem 4.1. \nTherefore, the initialized TFGNNs can be used for transductive node classification as are without training.\n\nThe approximation algorithm of label propagation is seamlessly embedded in the model parameters, and TFGNNs can also be trained as usual GNNs.\n\n## 6 Experiments 6.1 Experimental Setup\n\nWe use the Planetoid datasets (Cora, CiteSeer, PubMed) [54], Coauthor datasets, and Amazon datasets [42] in the experiments. We use 20 nodes per class for training, 500 nodes for validation, and the rest for testing in the Planetoid datasets following Kipf et al. [20], and use 20 nodes per class for training, 30 nodes per class for validation, and the rest for testing in the Coauthor and Amazon datasets following Shchur et al. [42]. We use GCNs [20] and GATs [45] for the baselines. We use three layered models with the hidden dimension 32 unless otherwise specified. We train all the models with AdamW [25] with learning rate 0.0001 and weight decay 0.01.\n\n## 6.2 Tfgnns Outperform Existing Gnns In Training-Free Setting\n\nWe compare the performance of TFGNNs with GCNs and GATs in the training-free setting by assessing the accuracy of the models when the parameters are initialized. The results are shown in Table 1. TFGNNs outperform GCNs and GATs in all the datasets. Specifically, both GCNs and GATs are almost random in the training-free setting, while TFGNNs achieve non-trivial accuracy. These results validate that TFGNNs\n\nTable 1: Node classification accuracy in the training-free setting. The best results are shown in **bold**.\n\nCS: Coauthor CS, Physics: Coauthor Physics, Computers: Amazon Computers, Photo: Amazon Photo.\n\nTFGNNs outperform GCNs and GATs in all the datasets. These results indicate that TFGNNs are trainingfree. Note that we use three-layered TFGNNs to make the comparison fair although deeper TFGNNs perform\n\nbetter in the training-free setting as we confirm in Section 6.3.\n\nCora CiteSeer PubMed CS Physics Computers Photo\n\nGCNs 0.163 0.167 0.180 0.079 0.101 0.023 0.119\n\nGCNs + LaF 0.119 0.159 0.407 0.080 0.146 0.061 0.142 GATs 0.177 0.229 0.180 0.040 0.163 0.058 0.122\n\nGATs + LaF 0.319 0.077 0.180 0.076 0.079 0.025 0.044\n\nTFGNNs + random initialization 0.149 0.177 0.180 0.023 0.166 0.158 0.090 TFGNNs (proposed) **0.600 0.362 0.413 0.601 0.717 0.730 0.637**\n\n\n\nFigure 2: Deep TFGNNs perform better in the training-free setting. The x-axis is the number of layers, and the y-axis is the accuracy of the models for the Cora dataset in the training-free setting. These results show that deeper TFGNNs perform better in the training-free setting.\nmeet the definition 5.1 of training-free models. We can also observe that GCNs, GATs, and TFGNNs do not benefit from LaF in the training-free settings if randomly initialized. These results indicate that both LaF\nand the initialization of TFGNNs are important for training-free performance.\n\n## 6.3 Deep Tfgnns Perform Better In Training-Free Setting\n\nWe confirm that deeper TFGNNs perform better in the training-free setting. We have used three-layered TFGNNs so far to make the comparison fair with existing GNNs. Proposition 5.2 shows that the initialized TFGNNs converge to label propagation as the depth goes to infinity, and we expect that deeper TFGNNs perform better in the training-free setting. Figure 2 shows the accuracy of TFGNNs with different depths for the Cora dataset. We can observe that deeper TFGNNs perform better in the training-free setting until the depth reaches around 10, where the performance saturates. It is noteworthy that GNNs have been known to suffer from the oversmoothing problem [22, 33], and the performance of GNNs degrades as the depth increases. It is interesting that TFGNNs do not suffer from the oversmoothing problem in the trainingfree setting. It should be noted that it does not necessarily mean that deeper models perform better in the optional training mode because the optional training may break the structure introduced by the initialization of TFGNNs and may lead to oversmoothing and/or overfitting. We leave it as a future work to overcome these problems by adopting countermeasures such as initial residual and identity mapping [5], MADReg [3], and DropEdge [35].\n\n\n\nFigure 3: TFGNNs converge fast. The x-axis is the number of training iterations, and the y-axis is the validation accuracy of the models for the Cora dataset. These results show that TFGNNs in the optional training mode converge much faster than GCNs. \n\n\n\nFigure 4: TFGNNs are robust to feature noise. The x-axis is the standard deviation of the Gaussian noise added to the node features, and the y-axis is the accuracy of the models for the Cora dataset. Both models are trained. These results show that TFGNNs are more robust to feature noise than GCNs.\n\n## 6.4 Tfgnns Converge Fast\n\nIn the following, we investigate the optional training mode of TFGNNs. We train the models with three random seeds and report the average accuracy and standard deviation. We use baseline GCNs without LaF (i.e., the original GCNs) as the baseline.\n\nFirst, we confirm that TFGNNs in the optional training mode converge faster than GCNs. We show the training curves of TFGNNs and GCNs for the Cora dataset in Figure 3. TFGNNs converge much faster than GCNs. We hypothesize that TFGNNs converge faster because the initialized TFGNNs are in a good starting point, while GCNs start from a completely random point and require many iterations to reach a good point. We can also observe that fully trained TFGNNs perform on par with GCNs. These results indicate that TFGNNs enjoy the best of both worlds: TFGNNs perform well without training and can be trained faster with optional training.\n\n## 6.5 Tfgnns Are Robust To Feature Noise\n\nAs TFGNNs use both node features and label information while traditional GNNs rely only on node features, we expect that TFGNNs are more robust to feature noise than traditional GNNs. We confirm this in this section. We add i.i.d. Gaussian noise with standard deviation σ to the node features and evaluate the accuracy of the models. We train TFGNNs and GCNs with the Cora dataset. The results are shown in Figure 4. TFGNNs are more robust to feature noise especially in high noise regimes where the performance of GCNs degrades significantly. These results indicate that TFGNNs are more robust to i.i.d. Gaussian noise to the node features than traditional GNNs.\n\n## 7 Related Work 7.1 Labels As Features And Training-Free Gnns\n\nThe most relevant work is Wang et al. [50], who proposed to use node labels in GNNs. This technique was also used in Addanki et al. [1] and analyzied in Wang et al. [51]. The underlying idea is common with LaF, i.e., use of label information as input to transductive GNNs. A similar result as Theorem 4.1 was also shown in [51]. However, the focus is different, and there are different points between this work and theirs.\n\nWe propose the training-free + optional training framework for the first time. The notable characteristics of GNNs are (i) TFGNNs receive both original features and LaF, (ii) TFGNNs can be deployed without training, and (iii) TFGNNs can be improved with optional training. Besides, we provide detailed analysis and experiments including the speed of convergence and noise robustness. Our results provide complementary insights to the existing works.\n\nAnother related topic is graph echo state networks [8, 9, 29], which lead to lightweight models for graph data.\n\nThe key idea is to use randomly initialized fixed weights for aggregation. The main difference is that graph echo state networks still require to train the output layer, while TFGNNs can be used without training. These methods are orthogonal, and it is an interesting direction to combine them to further improve the performance.\n\n## 7.2 Speeding Up Gnns\n\nVarious methods have been proposed to speed up GNNs to handle large graph data. GraphSAGE [13]\nis one of the earliest methods to speed up GNNs. GraphSAGE employs neighbor sampling to reduce the computational cost of training and inference. It samples a fixed number of neighbors for each node and aggregates the features of the sampled neighbors. An alternative sampling method is layer-wise sampling introduced in FastGCN [4]. Huang et al. [16] further improved FastGCN by using an adaptive node sampling technique to reduce the variance of estimators. LADIES [59] combined neighbor sampling and layer-wise sampling to take the best of both worlds. Another approach is to use smaller training graphs. ClusterGCN [6] uses a cluster of nodes as a mini-batch. GraphSAINT [56] samples subgraphs by random walks for each mini-batch.\n\nIt should also be noted that general techniques to speed up neural networks, such as mixed-precision training\n[30], quantization [17, 21, 43, 48, 52], and pruning [2, 14] can be applied to GNNs.\n\nThese methods mitigate the training cost of GNNs, but they still require many training iterations. In this paper, we propose training-free GNNs, which can be deployed instantly as soon as the model is initialized.\n\nBesides, our method can be improved with optional training. In the optional training mode, the speed up techniques mentioned above can be combined with our method to reduce the training time further.\n\n## 7.3 Expressive Power Of Gnns\n\nExpressive power (or representation power) means what kind of functional classes a model family can realize. The expressive power of GNNs is an important field of research in its own right. If GNNs cannot represent the true function, we cannot expect GNNs to work well however we train them. Therefore, it is important to elucidate the expressive power of GNNs. Originally, Morris et al. [31] and Xu et al. [53] showed that messagepassing GNNs are at most as powerful as the 1-WL test, and they proposed k-GNNs and GINs, which are as powerful as the k-(set)WL and 1-WL tests, respectively. GINs are the most powerful message-passing GNNs. Sato [38, 39] and Loukas [26] showed that message-passing GNNs are as powerful as a computational model of distributed local algorithms, and they proposed GNNs that are as powerful as port-numbering and randomized local algorithms. Loukas [26] showed that GNNs are Turing-complete under certain conditions\n(i.e., with unique node ids and infinitely increasing depths). Some other works showed that GNNs can solve or cannot solve some specific problems, e.g., GNNs can recover the underlying geometry [37], GNNs cannot recognize bridges and articulation points [57]. There are various efforts to improve the expressive power of GNNs by non-message-passing architectures [27, 28, 32]. We refer the readers to survey papers [18, 36] for more details on the expressive power of GNNs.\n\nWe contributed to the field of the expressive power of GNNs by showing that GNNs with LaF are more powerful than GNNs without LaF. Specifically, we showed that GNNs with LaF can represent an important model, label propagation, while GNNs without LaF cannot. It should be emphasized that GINs, the most powerful message-passing GNNs, and Turing-complete GNNs cannot represent label propagation without LaF because they do not have access to the label information label propagation uses, and also noted that GINs traditionally do not use LaF. This result indicates that it is important to consider what to input to the GNNs as well as the architecture of the GNNs for the expressive power of GNNs. This result provides a new insight into the field of the expressive power of GNNs.\n\n## 8 Limitations\n\nOur work has several limitations. First, LaF and TFGNNs cannot be applied to inductive settings while most GNNs can. We do not regard this as a negative point. Popular GNNs such as GCNs and GATs are applicable to both transductive and inductive settings and are often used for transductive settings.\n\nHowever, this also means that they do not take advantage of transductive-specific structures (those that are not present in inductive settings). We believe that it is important to exploit inductive-specific techniques for inductive settings and transductive-specific techniques (such as LaF) for transductive settings in order to pursue maximum performance.\n\nSecond, TFGNNs cannot be applied to heterophilious graphs, or its performance degrades as TFGNNs are based on label propagation. The same argument mentioned above applies. Relying on homophilious graphs is not a negative point in pursuing maximum performance. It should be noted that LaF may also be exploited in heterophilious settings as well. Developing training-free GNNs for heterophilious graphs based on LaF is an interesting future work.\n\nThird, we did not aim to achieve the state-of-the-art performance. Exploring the combination of LaF with fancy techniques to achieve state-of-the-art performance is left as future work.\n\nFinally, we did not explore applications of LaF other than TFGNNs. LaF can help other GNNs in nontraining-free settings as well. Exploring the application of LaF to other GNNs is left as future work.\n\n## 9 Conclusion\n\nIn this paper, we made the following contributions.\n\n- We advocated the use of LaF in transductive learning (Section 3).\n\n- We confirmed that LaF is admissible in transductive learning, but LaF has not been explored in the field of GNNs such as GCNs and GATs.\n\n- We formally showed that LaF strengthens the expressive power of GNNs (Section 4).\n\n- We showed that GNNs with LaF can represent label propagation (Theorem 4.1) while GNNs without LaF cannot (Proposition 4.2).\n\n- We proposed training-free graph neural networks, TFGNNs (Section 5).\n\n- We showed that TFGNNs defined by Eqs. (19) - (29) meet the requirements of training-free models (Definition 5.1) by showing that the initialized TFGNNs approximate label propagation in Proposition 5.2.\n\n- We confirmed that TFGNNs outperform existing GNNs in the training-free setting. (Section 6)\n- We showed that TFGNNs outperform GCNs and GATs in all of the seven datasets in the training-free setting (Table 1).\n\n- TFGNNs achieve non-trivial accuracy without training and can be deployed instantly as soon as the model is initialized. These results corroborate that TFGNNs are training-free (Definition 5.1).\n\nWe also note that our idea can be applied to other machine learning models than graph neural networks. We hope that this papers opens the door to a new research direction of training-free neural networks.\n\n## References\n\n[1] R. Addanki, P. W. Battaglia, D. Budden, A. Deac, J. Godwin, T. Keck, W. L. S. Li, A. SanchezGonzalez, J. Stott, S. Thakoor, and P. Velickovic. Large-scale graph representation learning with very deep gnns and self-supervision. *arXiv*, 2021. URL https://arxiv.org/abs/2107.09422.\n\n[2] D. W. Blalock, J. J. G. Ortiz, J. Frankle, and J. V. Guttag. What is the state of neural network pruning? 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In *Advances in Neural Information Processing Systems* 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS, pages 11247–11256, 2019."},"reviews":{"kind":"string","value":"Review 1:\nSummary: The authors propose the somewhat strange yet defensible idea of graph neural networks which work without any training. The idea is that in many situations, graph neural networks are used for transductive node classification, where the goal is to take a graph and impute information about neighboring nodes from some known nodes on the same graph. In particular, the authors point out that it is completely admissible in this setting to treat the label information on the known nodes as feature information (this is not “training on the test set”, since the labels of the target nodes are still unknown). The authors then show the somewhat straightforward fact that with labels as features, transductive GNNs can implement the classic algorithm of label propagation, while in general they cannot without labels as features. Finally, they describe their training-free graph neural networks (TFGNN). These are initialized to approximate label propagation and can therefore be immediately used, without any training, to perform transductive node classification. But they are also just regular graph neural networks which can still be trained for improved performance. In experiments on small standard datasets, the authors show that TF-GNNs outperform other GNNs in a number of interesting ways.\n\nStrengths and Weaknesses: Strengths:\t\n\nThe idea is simple and elegant. It is surprising that this obvious fact was not used before, so it's good for the authors to illustrate it.\n\nThe paper is admirably clear.\n\nThe approach does seem to work well, at least on the small examples tried.\n\nOverall I see why it has been submitted to TMLR, as it is clearly an interesting and useful contribution, yet one which might be rejected from other conferences for being \"too simple\" (in reality, a strength).\n\nWeaknesses:\n\n- The experimental section is a good demonstration of the approach, but it’s still relatively limited. I think the claims made aren't completely supported by the experiments. See requested changes, I think this is easily fixable.\n\n- I did not get any deep insight from the proof about label propagation (it seems to just point out the obvious fact that one can’t propagate labels if one doesn’t know the labels).\n\n- This is more a weakness of me as a reviewer, but I don't know whether or not there is some work out in the broader graph-machine-learning (not just graph neural network) literature that would make this approach less novel or less interesting.\n\nRequested Changes: Comments for authors: the discussion around equation 6 is a little unclear to me still. My sense is that doing things this way is necessary for good generalization, but that it would not actually be incorrect to simply train on the labels as features. If so, I think that should be made explicit, as “leaking information” sounds like something which would make any reported results invalid, rather than just being something that happens during training and hurts performance. [non-critical change]\n\nSubsections 6.5 and 6.6:\n\nThese are reasonable experiments, which are suggestive evidence for the benefits of TF-GNNs, but I think the claims made in the text are too strong given the experiments. I would ask the authors to simply add some small amount of hedging and qualifying language to acknowledge the more limited scope of the experiments, which should not be difficult. I have nonetheless answered \"Yes\" to the claims and evidence question on the review form, because I think these changes should be pretty small [critical change].\n\nBroader Impact Concerns: I don't see any broader impact concerns.\n\n==================================================\n\nReview 2:\nSummary: 1. The authors advocate the labels as features for transductive learning. And they formally demonstrate the LaF can strengthen the expressive power of GNN. \n2. They propose the TFGNN based on the LaF technique, which is training-free and can be promoted by training as well.\n\nStrengths and Weaknesses: Strengths:\n1. The label as feature technique is simple but seems effective with experimental results.\n2. The effect of LaF are sufficiently proven.\n3. The TFGNN shows impressive performance compared with existing baselines in a training-free setting.\n\nWeakness:\n1. It seems the initialization of TFGNN is an important technique. But it lacks explanation of why such initialization works.\n2. It lacks sufficient comparison of TFGNN and other GNN baselines when conducting optimal training. Furthermore, in training-free setting, the baselines is not sufficient. For example, what will happen if you replace LaF to other features such as random feature or laplacian eigen-vectors?\n3. TFGNN is limited in transudative learning and in a semi-supervised learning setting. In my opinion, the training-free benefits come from the following aspects (not limited to): 1) annotation is expensive, 2) training large graph is expensive. However, the proposed TFGNN is relied on the annotations (they use labels as features). Furthermore, the scalability of TFGNN is not demonstrated when graph is very large. Although this is an interesting work, the value of \"training-free\" is limited.\n4. TFGNN is heavily relied on the labels. But in reality, there are many scenario that the labels are sparse. Is this method robust to these scenarios?\n5. The concerns of training and inference latency. This method needs to sample subgraphs for each target node, thus it has low efficiency.\n\nRequested Changes: 1. Please refer to Weakness 1. Please add explanations of why such initialization works.\n2. Please refer to Weakness 2. Please add suggested experimental results on both training-free and training setting to make it more convincing.\n3. Please refer to Weakness 3. Please demonstrate the value of \"training-free\" of your proposed method.\n4. Please refer to Weakness 4. Please add experiments and discussion on scenario with sparse labels.\n5. Please refer to Weakness 5. Please add experiments on training and inference latency.\n\nBroader Impact Concerns: No\n\n==================================================\n\nReview 3:\nSummary: In this artice, the authors propose a new methodology for performing transductive node classification within a graph, where the task is to classify some nodes of a given graph, given some other nodes (on the same graph) on which the labels are known. For this, the authors suggest to use labels as features (LaF), that is, to encode the ground-truth labels (of the training nodes) as explicit features for message-passing graph neural network (GNN) architectures, instead of only using external features for the GNNs and then inferring the labels from the GNNs outputs. On the theoretical side, the authors demonstrate that, upon careful initialization of the GNN weights, and even without any training, the GNNs fed with LaF are at least as powerful as label propagation, where labels are inferred from the first training nodes reached by random walks originating from the test node to classify. On the experimental side, the authors designed experiments where untrained message-passing GNNs with LaF and clever initialization achieved better accuracies than other untrained GNN architectures (even with LaF). They also empirically showed faster convergence to the saturating accuracies, as well as better robustness to external feature noise.\n\nStrengths and Weaknesses: This article is clearly written, tackles an important problem, and provide an interesting approach, which is both easy to deploy and backed up by theoretical guarantees in the context of transductive node classification. To me, the main weaknesses are about some imprecisions in the text, as well as some missing details and results in the experiments. Overall, I think it is well suited for TMLR after some clarifications have been made (see below).\n\nRequested Changes: p3, first paragraph in section 3: the vector y_v has not beed defined. I figured from the text that it must be some sort of one-hot encoding of the labels, but this should be made explicit.\n\np5, last line in the proof of theorem 4.1: the second term has to converge to zero (the result is not true if this term only diminishes, as it could decrease to a minimum other than zero). Furthermore, I think this convergence has to be uniform across all nodes for the result to hold in the case of graphs with infinitely many nodes (if the graph is assumed to be finite, simple convergence is enough).\n\np6: a complementary figure could definitely be useful to illustrate equations (22) to (28).\n\np6: the role of the matrix U is unclear to me. What are the coefficients of the columns 0 to -|Y|-1 (in numpy notations)? If they are nonzero, then I don't see why \"U picks the last |Y| dimensions\" (as claimed at the end of the page). If they are zero, then 1. it should be mentioned explicitly, 2. the external features (other than the labels) are not used at all when building the predictions.\n\np7, table 1: why are there no variances in the results? It would be more convincing to make several runs with different training nodes, and provide the average + standard deviations of the accuracies. Also, it would be nice to explain / comment on the fact that for other GNN architectures, adding LaF actually degrades the accuracy (such as, e.g., GAT in CiteSeer), which looks counter-intuitive.\n\np8, Figure 1: why is there no variance in the curve whereas there is variance in Figures 2 and 3?\n\np8, Sections 6.4 and 6.5: were the GCNs trained with or without LaF? Providing both (and thus making three curves in Figs 2 and 3; one for TFGNNs, one for GCNs without LaF, one for GCNs with LaF) would be better.\n\nBroader Impact Concerns: N/A\n\n==================================================\n\nMetareview:\nRecommendation: Accept as is\n\nComment: The paper is interesting to the GNN community, and proposes terminology and mapping of existing and new ideas that allow the training-free approach of Graph Neural Networks. There is a clear consensus among the reviewers that the paper should be accepted.\n\n==================================================\n\n"}}},{"rowIdx":90,"cells":{"paper":{"kind":"string","value":"# A Survey Of Lottery Ticket Hypothesis\n\nAnonymous authors Paper under double-blind review\n\n## Abstract\n\nThe Lottery Ticket Hypothesis (LTH) states that a dense neural network model contains a highly sparse subnetwork (i.e., winning tickets) that can achieve even better performance than the original model when trained in isolation. While LTH has been proved both empirically and theoretically in many works, there still are some open issues, such as efficiency and scalability, to be addressed. Also, the lack of open-source frameworks and consensual experimental setting poses a challenge to future research on LTH. For the first time, we examine previous research and studies on LTH from different perspectives. We also discuss issues in existing works and list potential directions for further exploration. This provides an in-depth look at the state of LTH.\n\n## 1 Introduction\n\nAlthough deep neural network models have shown impressive performance in many domains such as computer vision and natural language processing, recent models such as GPT with billions of parameters are notoriously slow for training and inference (Brown et al., 2020). Pruning techniques can reduce the parameter counts of trained networks, decrease storage requirements, and improve computational performance of inference without compromising accuracy (LeCun et al., 1989; Liang et al., 2021b). Existing pruning technologies can be classified into two categories: structured pruning and unstructured pruning. Specifically, structured pruning (He et al., 2017; Liu et al., 2017; Li et al., 2017; Hu et al., 2016; Wen et al., 2016; Hong et al., 2018; Nonnenmacher et al., 2022; Halabi et al., 2022; Yin et al., 2023b; Nova et al., 2023) aims to find structural sparse patterns such as layer-wise, channel-wise, block-wise, and column-wise. Unstructured pruning (Han et al., 2015; Liu et al., 2019; Cao et al., 2019; Ren et al., 2019; Zhang et al., 2018; Alizadeh et al., 2022; Liao et al., 2023) results in sparse individual parameters without pre-defining the positions of zeros. However, the sparse architectures produced by these pruning methods are difficult to train from the start, reaching lower accuracy than the original networks. The Lottery Ticket Hypothesis (LTH) (Frankle & Carbin, 2018) states that a dense neural network model contains a highly sparse subnetwork (i.e., winning tickets) that can achieve even better performance than the original model. The winning tickets can be identified by training a network and pruning its parameters with the smallest magnitude in an iterative way or one-shot way. Before it is trained in each iteration, the weights will be reset to the original initialization. Instead of resetting to the original initialization, some works (Mikler, 2021; Renda et al., 2020a; Gadhikar & Burkholz, 2024) rewind the parameters of the subnetwork to the early stage of training. The identified winning tickets are retrainable to reduce the high memory cost and long inference time of the original neural networks, and the existence of winning tickets has been verified in both experiments and theory (Frankle et al., 2019; Diffenderfer & Kailkhura, 2021; Savarese et al., 2020; Zhou et al., 2019; Malach et al., 2020; Zhang et al., 2021a; Ma et al., 2021a; Zhou et al., 2019; Chen et al., 2021d; Su et al., 2020; Chen et al., 2022a). Later works have extended LTH to find the winning tickets for different kinds of neural networks such as generative models (Chen et al., 2021h; Kalibhat et al., 2021; Jiang et al., 2023), Transformers (Brix et al., 2020; Prasanna et al., 2020; Chen et al., 2020a; Behnke\n& Heafield, 2020), and GNNs (Chen et al., 2021c; Hui et al., 2023; Zhang et al., 2024; Harn et al., 2022)\nDespite the success of LTH, the iterative train-prune-retrain process and even the subsequent fine-tuning are costly to compute (Liu et al., 2021). Many works have proposed algorithms to find lottery tickets efficiently\n(Yin et al., 2023a; Zhang et al., 2021c; Wang et al., 2020b; Tanaka et al., 2020; Ye et al., 2020; Evci et al.,\n\n\n\n2020; Hou et al., 2022; Ma et al., 2022; Jaiswal et al., 2023a). LTH is also related to many other domains including robustness, transfer learning, fairness, and federated learning (Yuan et al., 2021; Goli & Aamodt, 2020; Raihan & Aamodt, 2020; Chen et al., 2021e; Mehta, 2019; Morcos et al., 2019b; Itahara et al., 2020; Seo et al., 2021; Mugunthan et al., 2022; Shi et al., 2023; Hansen & Søgaard, 2021). This survey aims to provide a comprehensive overview of the state of LTH research, categorizing various approaches, highlighting open issues, and presenting a benchmark. In this paper, we first present a taxonomy of recent research that categorizes existing works into 8 topics:\ntheory, special models, experimental insight, algorithms, efficiency, relations with other topics, open issues of existing works, and applications. Figure 1 shows the overview of the taxonomy. We first explain the theoretical foundations of the lottery tickets hypothesis. LTH has been studied on three specific categories of models, i.e., Transformer, GNNs, and generative models. We show the difference regarding pruning between a general model and these specific models. While many novel algorithms have been developed, we discuss the effectiveness of different initialization methods and turning algorithms. In addition, we analyze the experimental insight in existing papers. Our survey also reviews efficient strategies that can reduce the high computational cost of iterative pruning. For each other topic (i.e., hardware, scalability, federated learning, and robustness), we show the connection with LTH. Lastly, we conclude our paper with open issues of LTH and potential research directions. All publications with available code are summarized in Table 1.\n\nWe hope this survey serves as a valuable resource for LTH researchers and those seeking pruning technologies for various models.\n\n## 2 Theory\n\nWe start with notations and functions related to LTH settings. Consider a network function f(·) that is initialized as f(*x, θ*0) where x denotes input training samples and θ0 denotes the weights.\n\nThe lottery ticket hypothesis - There exists an identically initialized subnetwork (i.e., winning tickets) that - when trained in isolation - can reach similar accuracy with the well-trained original network using the same or fewer iterations. The winning ticket can be identified by pruning its parameters with the smallest magnitude in an iterative way or one-shot way.\n\nBased on the definition of LTH, we provide the following notations:\n- Pretraining: train the network f(*x, θ*0) for T epochs, arriving at weights θT .\n\n- Pruning: generate a mask m by pruning s% of weights in θT .\n\n- Re-initialization: reset the remaining weights to their values in θ0.\n\nAccording to the definition, the combination of (θ0, m) is a winning ticket. The mechanism behind finding winning tickets using iterative magnitude-based pruning (IMP) is identified as three key inquiries (Paul et al., 2022): (i) Information Conveyed by IMP Masks: Masks obtained at the end of training carry crucial information about the identity of a specific subspace intersecting a desired linearly connected mode of a matching sublevel set; (ii) SGD's Exploitation of Information: The stochastic gradient descent (SGD)\noptimization method demonstrates robustness that enables it to return to a desired mode despite significant perturbations early in training. (iii) Role of Retraining in IMP: The retraining step in IMP is essential for finding a network with new, small weights to be pruned.\n\nFollowing the observation of LTH, an even more striking phenomenon (strong lottery tickets) is presented in (Ramanujan et al., 2020): not only do such sparse subnetworks exist at initialization but they already achieve impressive performance without any training. Specifically, this work asserts that proficient subnetworks can be derived from neural networks with randomly initialized weights, even without the necessity for weight training. Their proposed methodology entails the fixation of weights for each edge in the network, subsequently assigning a score to each edge. The scores are updated using the straight-through estimator and the top scores will be preserved. This approach does not require the utilization of pre-trained weight values, and the weights remain steadfast. The existence of a winning ticket has been proved theoretically in many works. The first theoretical evidence of the existence of winning tickets was given by Malach et al. (2020). By approximating ReLU networks, This work proves that optimizing the weights is equivalent to pruning entire neurons of a sufficiently large random network. For every bounded distribution and every target network with bounded weights, a sufficiently overparameterized neural network with random weights contains a subnetwork with roughly the same accuracy as the target network, without any further training. However, the polynomial over-parameterization requirement in the proof is at odds with recent experimental results with networks that are wider than the bound.\n\nAn exponential improvement to the over-parameterization requirement is offered in Pensia et al. (2020).\n\nInstead of providing significantly tighter bounds, (Orseau et al., 2020) points out that the overparameterized network only needs a logarithmic factor number of neurons per weight of the target subnetwork. Any network with a depth of L and width of W can be obtained by pruning a randomly initialized network with a depth of 2L and width increased by a logarithmic factor, or a network with a depth of L + 1 and width increased by another logarithmic factor (Burkholz, 2022b). This discovery implies that effective lottery tickets can be identified at frequently used depths, necessitating only logarithmic levels of overparameterization.\n\nThe proof is also extended to convolutional neural networks (CNN) by (da Cunha et al., 2022): with high probability, it is possible to approximate any CNN by pruning a random CNN whose size is larger by a logarithmic factor. Considering that most theoretical validations regarding the existence of winning tickets have focused on networks with the ReLU activation functions, (Burkholz, 2022a) demonstrate that winning tickets are likely to be present in modern architectures featuring convolutional and residual layers, which incorporate a wide range of activation functions. A unifying framework is also introduced to prove the strong lottery ticket hypothesis for MLPs, CNNs, and general equivariant networks in (Ferbach et al., 2022). The strong lottery ticket hypothesis is extended functions that maintain the operation of the group, known as G-equivariant networks. The study establishes, with a high likelihood, that it is possible to approximate\n\n| Table 1: LTH-related publications with code | | | | | |\n|-----------------------------------------------|---------------------------|---------------------------------------------------------------------------------------------------------|-------------------------------------------------------|-----------------------------------|-------------------------------|\n| Method | Conference Framework Code | Model | Datasets | | |\n| LTH (Frankle & Carbin, 2018) | ICLR | Tensorflow https://github.com/google-research/lottery-ticket-hypothesis VGG-19, ResNet-18 MNIST CIFAR10 | | | |\n| Robust Tickets (Zheng et al., 2022) | ACL | Pytorch | https://github.com/ruizheng20/robust_ticket | BERT | IMDB AGNEWS SST-2 |\n| Super Tickets (Liang et al., 2021a) | ACL | Pytorch | https://github.com/RICE-EIC/SuperTickets | BERT | GLUE MNLI |\n| Early BERT (Chen et al., 2020c) | ACL | Pytorch | https://github.com/VITA-Group/EarlyBERT | BERT | QNLI QQP SST-2 |\n| Eearly-GCN (You et al., 2022b) | AAAI | Pytorch | https://github.com/RICE-EIC/Early-Bird-GCN | GCN | Cora Citeseer Pubmed Cifar-10 |\n| Edge-popup (Ramanujan et al., 2020) | CVPR | Pytorch | https://github.com/allenai/hidden-networks | VGG | Cifar-100 |\n| | ResNet | Tiny-ImageNet | | | |\n| Life-long ticket s(Chen et al., 2020b) | ICLR | Pytorch | https://github.com/VITA-Group/Lifelong-Learning-LTH | ResNet18 | Cifar-10 ImageNet |\n| Elastic LTH (Chen et al., 2021f) | NeurIPS | Pytorch | https://github.com/VITA-Group/ElasticLTH | ResNet-20 ResNet-18 | Cifar-10 Cifar-100 MNIST |\n| DLRA (Schotthöfer et al., 2022) | NeurIPS | Tensorflow Pytorch https://github.com/ScSteffen/DLRT-Net https://github.com/COMPiLELab/DLRT-Net | LeNet5, ResNet-50 | Cifar-10 | |\n| | AlexNet, VGG16 | ImageNet1K | | | |\n| CS (Savarese et al., 2020) | NeurIPS | Pytorch | https://github.com/lolemacs/continuous-sparsification | VGG-16 ResNet-20 | Cifar-10 Cifar-10 |\n| DLTH (Bai et al., 2022) | ICLR | Pytorch | https://github.com/yueb17/DLTH | ResNet56 | Cifar-100 |\n| | ResNet18 | ImageNet | | | |\n| Early Bird (You et al., 2020) | ICLR | Pytorch | https://github.com/GATECH-EIC/Early-Bird-Tickets | PreResNet101 VGG16 | Cifar-10 Cifar-100 |\n| | ResNet-56 ResNet-34 | | | | |\n| Rewinding (Renda et al., 2020b) | ICLR | Tensorflow https://github.com/lottery-ticket/rewinding-iclr20-public | ResNet-50 GNMT | Cifar-10 ImageNet WMT6 EN-DE | |\n| ULTH (Burkholz et al., 2022) | ICLR | Pytorch | https://github.com/RelationalML/UniversalLT | Conv2 | Cifar-10 Imagenet |\n| SuperTickets (You et al., 2022a) | ECCV | Pytorch | https://github.com/GATECH-EIC/SuperTickets | Transformer | ImageNet Cityscapes ADE20K |\n| ULTH-GNN (Chen et al., 2021c) | ICML | Pytorch | https://github.com/VITA-Group/Unified-LTH-GNN | GNN | Cora Citeseer Pubmed Cifar-10 |\n| PrAC (Zhang et al., 2021c) | ICML | Pytorch | https://github.com/VITA-Group/PrAC-LTH | ResNet | Cifar-100 |\n| | VGG | Tiny-ImageNet ImageNet2021 | | | |\n| RigL (Evci et al., 2020) | ICML | Tensorflow Pytorch https://github.com/google-research/rigl https://github.com/nollied/rigl-torch | MobileNet-v1 ResNet-50 | Cifar-10 Mnist CIFAR-10 CIFAR-100 | |\n| Sanity Checks (Ma et al., 2021a) | NeurIPS | Pytorch | https://github.com/boone891214/sanity-check-LTH | VGG ResNet | Tiny-ImageNet ImageNet-1K |\n| Object Recognition (Girish et al., 2021) CVPR | Pytorch | https://github.com/boone891214/sanity-check-LTH | RCNN ResNet | COCO | |\n| LTH-BERT (Prasanna et al., 2020) | EMNLP | Pytorch | https://github.com/sai-prasanna/bert-experiments | BERT | GLUE |\n| SubsetSum (Pensia et al., 2020) | NeurIPS | Pytorch | https://github.com/acnagle/optimal-lottery-tickets | LeNet5 | MNIST |\n\nany G-equivariant network with fixed width and depth by pruning a randomly initialized over-parametrized G-equivariant network to obtain a G-equivariant subnetwork. Additionally, the research demonstrates that the proposed over-parametrization approach is optimal and sets a lower limit on the number of effective parameters relative to the error tolerance.\n\nIn an alternative way, Zhang et al. (2021b) apply the dynamical systems theory and inertial manifold theory to theoretically validate the effectiveness of the Lottery Ticket Hypothesis. Moreover, they demonstrate lossless pruning and one-shot pruning in theory. Gradient dynamical systems are employed to elucidate optimization problems in neural networks, and project high-dimensional systems onto inertial manifolds, yielding a low-dimensional system specific to pruned subnetworks. PAC-Bayesian theory has also been leveraged to elucidate the relationship between the Lottery Ticket Hypothesis (LTH) and its generalization ability (Sakamoto & Sato, 2022). It turns out that using an initial large learning rate during pruning leads to suboptimal performance of the final tickets. Also, the authors observe that an initial large learning rate causes the optimizer to converge to flatter minima. It is hypothesized that winning tickets are associated with relatively sharp minima, which is typically considered detrimental to generalization. Employing a spike-and-slab distribution in the PAC-Bayes bound analysis of winning tickets, the authors discover that fatness contributes to improved accuracy and robustness. Furthermore, the distance to the initial weights significantly impacts the winning tickets. A sparse deep neural network can be represented with a sparse graph and the connectivity in the pruned model can be characterized by the spectral expansion property in the graph. The validity of LTH has been studied based on Ramanujan graph properties in terms of certain spectral bounds (Pal et al., 2022). Accordingly, a modification of existing iterative pruning algorithms that preserves the Ramanujan graph property is proposed for more efficient winning ticket search.\n\n## 3 Special Models\n\nIt is non-trivial to extend the lottery tickets hypothesis to special models such as graph neural networks, transformers, and generative models since these models have distinctive architectures. In this section, we will elaborate on the variants of LTH for these models.\n\n## 3.1 Graph Neural Networks\n\nA distinctive characteristic of Graph Neural Networks (GNNs) Kipf & Welling (2017); Hamilton et al. (2017); Hui et al. (2020) is that a graph structure is fed into the model. Intuitively, the computational resource consumption in both training and inference of GNNs will increase substantially as the size and complexity of graphs grow. Also, redundant or noisy edges may exist in the graph structure. To tackle this issue, UGS (Chen et al., 2021c) introduced a unified GNN sparsification (UGS) framework. This framework concurrently prunes the graph adjacency matrix and model weights, thereby expediting GNN inference on large-scale graphs. Expanding on the UGS framework, they extended the lottery ticket hypothesis to GNNs, defining a graph lottery ticket (GLT) as a combination of a pruned graph and a sparse sub-network. GLTs can be discerned from the original GNNs and the densely connected graph through the iterative sparsification process based on mask weights.\n\nHowever, GLT requires a full pretraining process, which may not be universally applicable in real-world scenarios. Additionally, GLT's sparsification approach may lead to significant information loss. To address these limitations, (Wang et al., 2022b) propose the Dual Graph Lottery Ticket (DGLT) framework. This approach transforms a random ticket into a graph lottery ticket, allowing for a more comprehensive exploration of the relationships between the original network/graph and the sparse counterpart. Through regularization-based network pruning and hierarchical graph sparsification, DGLT achieves a triple-win scenario: high sparsity, commendable performance, and enhanced explainability. (Hui et al., 2023) also propose a novel approach to address the limitations of UGS. They observe that the performance of a sparsified GNN significantly declined as graph sparsity increased beyond a certain threshold. To mitigate this, they introduce two enhancements: First, they refine the adjacency matrix pruning by incorporating an auxiliary loss head to guide edge pruning more comprehensively. Second, they treat unfavorable graph sparsification as adversarial data perturbations, formulating the pruning process as a min-max optimization problem to enhance robustness.\n\nInstead of pruning based on magnitude, an algorithm is proposed in Harn et al. (2022) to prune weights and edges based on gradient. Similar to GLT, GEBT (Graph early-bird tickets) (You et al., 2022b) develops a training framework to boost the efficiency of GNNs training by enabling co-sparsification.\n\nPrevious GLT methods use the same topology structure across all layers. The strategy hinders the integration of GLT into deeper and larger-scale GNN contexts. To address this issue, an Adaptive, Dynamic, and Automated framework (Zhang et al., 2024) is introduced for identifying Graph Lottery Tickets (AdaGLT). The proposed method tailors layer-adaptive sparse structures for various datasets and GNNs and captures GLT across diverse sparsity levels. The theoretical proofs to mitigate over-smoothing issues and obtain improved sparse structures in deep GNNs are also provided.\n\n## 3.2 Transformer\n\nLarge pre-trained transformers have achieved great success in the past few years. However, due to the exponentially increasing parameter counts of these large pre-trained models, fine-tuning these models with non-industry standard hardware is becoming seemingly impossible. (Yu et al., 2019) conduct a comprehensive investigation into the applicability of the Lottery Ticket Hypothesis (LTH) within the domains of Natural Language Processing (NLP). They scrutinize recurrent LSTM models as well as large-scale Transformer models. Notably, the authors successfully identify initializations that qualified as winning tickets, facilitating Transformer models to achieve performance levels nearly equivalent to their full-size counterparts, despite a reduction in size by one-third. Two studies have addressed the feasibility of applying the LTH to BERT. (Prasanna et al., 2020) successfully identified subnetworks within fine-tuned BERT that exhibited performance on par with the complete model. They observed that similarly sized subnetworks sampled from other parts of the model demonstrated worse performance. However, they noted that even the least performing subnetworks exhibited notable trainability, indicating the substantial utility of the majority of pre-trained BERT weights. In a separate investigation, (Chen et al., 2020a) examined pre-trained BERT and unearthed the presence of trainable and transferable subnetworks within it. Notably, these subnetworks were discovered during the pre-training phase, distinguishing the earlier research, which obtained subnetworks during the fine-tuning phase, a process necessitating a certain degree of additional training. In addition to its validation in the field of NLP, (Gan et al., 2022) explore whether trainable subnetworks, as suggested by the LTH, exist in vision-and-language models. The findings indicate that while exact matches are challenging, \"relaxed\" winning tickets at 50%-70% sparsity maintain 99% accuracy. Task-specific pruning leads to reasonably transferable subnetworks, and pre-training tasks at 60%/70% sparsity achieve 98%/96% accuracy. (Behnke & Heafield, 2020) investigate the pruning of attention heads in Transformers, prompted by recent research indicating a lack of confidence in the decisions made by a significant portion of attention heads. Through experiments in machine translation, they demonstrate the potential to remove up to three-quarters of attention heads from transformer-big during early training.\n\nDuring the fine-tuning stage of Transformer models applied to NLP tasks, the integration of adapter modules has been found to expedite transfer learning. However, adapter fusion models, which incorporate knowledge from various tasks, can lead to redundancy and increased computational costs. To mitigate this, (Wu et al., 2022) propose a method to assess the impact of each adapter module and employ the Lottery Ticket Hypothesis to prune them effectively. (Gong et al., 2022) explore the applicability of the Lottery Ticket Hypothesis to pre-trained language models (PLMs) for fine-tuning. The authors introduce a regularization approach using L1 distance and identify a subnetwork structure termed the \"dominant winning ticket.\" They empirically demonstrate that this dominant winning ticket can achieve performance comparable to the fullparameter model, is transferable across different tasks, and exhibits a natural structure within each parameter matrix. To address the computational cost of repetitive training and pruning routine of the transformer, Instant Soup Pruning (ISP) (Jaiswal et al., 2023b) is introduced as an efficient alternative to the iterative magnitude pruning (IMP) method. ISP leverages the concept of model soups, merging fine-tuned weights from multiple models to achieve better minima. The proposed approach replaces the costly intermediate stages of IMP with a computationally efficient weak mask generation and aggregation process. Specifically, ISP generates multiple weak and noisy subnetworks through a few iterations with varying training protocols and data subsets, then combines them to create a high-quality denoised subnetwork.\n\n## 3.3 Generative Models\n\nWhile prior research mainly focused on supervised learning scenarios, (Kalibhat et al., 2021) extend the concept to deep generative models including GANs and VAEs. The study demonstrates that iterative magnitude pruning, combined with generative losses, effectively identifies winning tickets, achieving high sparsity levels. Moreover, the transferability of winning tickets across different generative models implies they have inherent biases beneficial for training various deep generative models. Furthermore, the paper introduces the concept of \"early-bird tickets,\" enabling substantial reductions in floating-point operations and training time, enhancing the feasibility of training large-scale generative models under resource constraints.\n\n(Chen et al., 2021h) validate the presence of trainable matching subnetworks in various deep GAN architectures. Additionally, the results suggest that pruning the discriminator has a minor impact on the existence and quality of matching subnetworks, while the initialization weights in the discriminator significantly influence the outcome. Furthermore, the study demonstrates the robust transferability of these subnetworks to unseen tasks. Drawing inspiration from previous findings on independently trainable and highly sparse subnetworks within GANs, (Chen et al., 2021a) introduce a novel approach to data-efficient GAN training.\n\nThe proposed method involves identifying a lottery ticket from the original GAN using a small training set of real images, and then focusing on training that sparse subnetwork using the same set. The coordinated framework demonstrates significant improvements over existing real image data augmentation methods and introduces a novel feature-level augmentation that can be used in conjunction with them.\n\nAn extensive exploration of the applicability of pruning techniques in generative models is conducted by Yeo et al. (2023) to discover subnetworks that perform similarly to or better than the trained dense model. They introduce the concepts of weak lottery tickets and strong lottery tickets. The former refers to tickets identified from networks with learned weights, while the latter denotes tickets identified without any weight updates. The experimental result shows that a subnetwork of GFMN with only 10% of the weights remaining can reach similar FID scores as GFMN. LTH has also been applied to diffusion models (Jiang et al., 2023).\n\nIt shows that we can find a winning ticket of DDPM (Ho et al., 2020) at sparsity 90% − 99% without compromising performance for denoising diffusion probabilistic models on benchmarks (Cifar-10, Cifar-100, MNIST). Different from existing LTH works which identify the subnetworks with a unified sparsity along different layers, this work proposes to find the winning ticket with varying sparsity along different layers in the model.\n\n## 4 Experimental Insight\n\nThe discrepancies regarding LTH have emerged due to different settings and several works have investigated the criteria to identify winning tickets. In this survey, we conclude several experimental insights.\n\n## 4.1 How Much Can Be Pruned?\n\nAlthough the existence of winning tickets has been verified in theory, how much weight can be pruned is still an open question in an experimental setting. The question is answered in (Ye et al., 2020) to find a subnetwork while tolerating an acceptable level of accuracy reduction. They propose a pruning method based on greedy optimization. This method results in an exponential decrease in the discrepancy between the pruned network and the original network as the number of winning tickets decreases. (Liang et al., 2021a)\nidentify a phase transition phenomenon, where the performance of winning tickets initially improves with increased compression, but then starts to deteriorate after a threshold. These threshold tickets are referred to as \"super tickets.\" The presence and characteristics of this phase transition are influenced by both the task and the model size, with larger models and smaller datasets intensifying the effect. Experimentation on the GLUE benchmark demonstrates that super tickets enhance single-task fine-tuning, and adaptively sharing them across tasks benefits multi-task learning. As an increasing number of researchers engage in the investigation of the LTH, discrepancies have emerged in the experimental design and criteria formulation for the identification of winning tickets. To resolve these issues, (Ma et al., 2021b) proposes a more rigorous definition of the lottery ticket hypothesis and provides concrete evidence of its applicability across various DNN architectures and applications. The study quantitatively explores the relationships between winning tickets and various experimental factors. The authors discover that crucial training hyperparameters, such as learning rate and training epochs, along with architectural features like capacities and residual connections, significantly influence the identification of winning tickets. Based on their findings, the study offers a guideline for parameter settings tailored to specific architecture characteristics.\n\n## 4.2 Pruning More In Deep Layers\n\nIn an investigation of applying LTH to computer vision models (Girish et al., 2021), two pruning methods are introduced: layer-wise pruning and global pruning. It has been observed that when sparsity levels are low, there is a marginal disparity in accuracy between the two pruning methods. However, as sparsity levels increase, global pruning exhibits a slight performance advantage. This is attributed to the uneven sparsity distribution across layers in the case of global pruning, leading to enhanced performance of subnetworks.\n\nThrough an analysis of sparsity rates in each layer, they also observe that deeper layers are pruned more.\n\nSimilar results are also observed in (Jiang et al., 2023) where an incremental rate is used to increase the pruning ratio in deeper layers.\n\n## 4.3 Key Factors\n\nThere are three critical components in LTH: Zeros, Signs, and the Supermask (Zhou et al., 2019). Three questions are answered by ablating these factors: why setting weights to zero is important, how signs are all you need to make the reinitialized network train, and why masking behaves like training. It is the sign in the original initialization, not the magnitude of the weights, that is crucial to the performance of LT networks.\n\nAlso, the masking procedure can be considered as a training operation. Supermasks are observed to produce partially working networks without training. (Evci et al., 2022) emphasize that training unstructured sparse Neural Networks (NNs) from random initialization often results in poor generalization, with Lottery Tickets and Dynamic Sparse Training being notable exceptions. They observe that in the initialization phase of sparse NNs, gradient flow tends to be weak. In comparison to other methods, Dynamic Sparse Training significantly bolsters gradient flow during the training phase, potentially contributing to its effectiveness. Conversely, Lottery Tickets does not exhibit a significant improvement in gradient flow. Therefore, the authors posit three key factors contributing to the success of Lottery Tickets: 1. Proximity to the pruning solution during initialization. 2. Residing within the basin of the pruning solution. 3. Learning functions akin to the pruning solution.\n\n## 5 Algorithms\n\nThere are two core parts in the pruning algorithm: initialization and pruning. Following the original IMPbased pruning, many works have proposed new methods regarding both initialization and pruning.\n\n## 5.1 Initialization\n\nTypically, pruning involves training, compressing, and then retraining to recover accuracy. The conventional method of fine-tuning uses a low fixed learning rate to train the unpruned weights from their final trained values. In contrast, (Renda et al., 2020b) introduce two alternative approaches: weight rewinding, which reverts unpruned weights to earlier training stages, and learning rate rewinding, which applies the original learning rate schedule to the unpruned weights. Both rewinding techniques surpass fine-tuning, forming the foundation of a network-agnostic pruning algorithm that achieves comparable accuracy and compression ratios to specialized state-of-the-art techniques. It is important to understand how learning rate rewinding (LRR) excels in pruning schemes and parameter learning because it can bring us closer to the design of more flexible deep learning algorithms. To this end, Gadhikar & Burkholz (2024) conducts experiments to investigate the ability of LRR to flip parameter signs early and stay robust to sign perturbations.\n\n## 5.2 Unstructured Pruning\n\nMost of the existing works in the literature focus on finding unstructured winning tickets where the positions of zeros are not predefined. These methods pruned individual elements with the smallest magnitude. (Shang et al., 2022) delve into the Magnitude-Based Pruning (MBP) scheme and provide a fresh perspective by employing Fourier analysis on the deep learning model to guide model selection. In addition to elucidating the generalization capability of MBP using the Fourier transform, a novel two-stage pruning approach is proposed. The first stage involves obtaining the topological structure of the pruned network, followed by a second stage of retraining to restore capacity using knowledge distillation from lower to higher frequencies. Different from these methods based on IMP, (You et al., 2022a) present an approach addressing Deep Neural Networks (DNNs) termed \"Two-in-One Training,\" which enables the direct identification of efficient DNNs and their associated lottery subnetworks, referred to as SuperTickets, from a supernet. Previous methods typically employ Neural Architecture Search (NAS) followed by pruning algorithms to obtain lottery subnetworks for efficient DNNs. This approach, known as \"First-search-then-prune (S+P),\" involves an initial search for an efficient network architecture and subsequent sparsity induction. In contrast, this paper proposes a unified approach that simultaneously conducts architecture search and pruning during the training of the supernet within NAS. In the search phase, the authors compute importance factors for search units and set a threshold ϵ to remove unnecessary units. In the pruning phase, they employ the Importance Masking Pruning (IMP) method to achieve the desired level of sparsity. Additionally, two techniques are introduced in the search and pruning phases. In the search phase, Iterative Reactivation is implemented. Unlike the conventional IMP method, the authors incorporate reactivation during the search process rather than after pruning. This choice is made to retain training opportunities for the sparsified network, which is believed to contribute to performance enhancement. Experimental results validate this hypothesis by comparing the performance of networks with reactivation during search versus pruning. Schotthöfer et al. (2022) introduce a novel approach to identify efficient low-rank subnetworks. These subnetworks are determined and adjusted during the training process, significantly reducing the time and memory resources required for both training and evaluation. The key concept involves restricting weight matrices to a low-rank manifold and updating the low-rank factors instead of the entire matrix during training. This method ensures that the training updates stay within the specified manifold, providing guarantees of approximation, stability, and descent.\n\nContinuous Sparsification (CS) is introduced as another alternative to the traditional IMP approach. Unlike IMP, CS does not require weight resetting between epochs, making it a more efficient method for subnetwork discovery. Additionally, CS allows for dynamic adjustment of sparsity during the training process, thus achieving a better balance between model accuracy and sparsity. In contrast to traditional methods, CS\nintroduces L1 regularization to encourage most elements in the weight vector to be zero:\n\n$$\\min_{\\theta\\in\\mathbb{R}^{d},m\\in\\{0,1\\}^{d}}L(f(\\cdot;m\\odot\\theta))+\\lambda\\cdot\\|m\\|_{1}\\tag{1}$$\n$$\\left(1\\right)$$\n\nSimilar to most methods that utilized regularization, to avoid the discrete space of m, they re-parameterize m by introducing a new variable s ∈ R\nd. To mitigate biased and/or noisy training caused by gradient estimators, they define the Heaviside step function such that m := H(s), with s ∈ R\nd\n̸=0 and H : R\nd\n̸=0 → {0, 1}, i.e.,\nH(s) = 1 if s > 0 and 0 otherwise. This leads to an equivalent form to solve equation (1):\n\n$$\\operatorname*{min}_{w\\in\\mathbb{R}^{d},s\\in\\mathbb{R}_{\\neq0}^{d}}L(f(\\cdot;H(s)\\odot\\theta))+\\lambda\\cdot\\|H(s)\\|_{1}$$\n$$\\left(2\\right)$$\nL(f(·; H(s) ⊙ θ)) + λ · ∥H(s)∥1 (2)\nDue to the discontinuity of the step function H at certain points, and its derivative being zero at all points, dealing with H in computer programs poses some challenges. The authors propose an approximation method, using an S-shaped function parameterized by β to approximate H. The form of this approximating function is s 7→ σ(βs), where σ is the sigmoid function. Thus, the final objective function is:\n\n$$L_{\\beta}(\\theta,s):=L(f(\\cdot;\\sigma(\\beta s)\\odot\\theta))+\\lambda\\cdot\\|\\sigma(\\beta s)\\|_{1}$$\n$$\\left({\\boldsymbol{3}}\\right)$$\nLβ(*θ, s*) := L(f(·; σ(βs) ⊙ θ)) + λ · ∥σ(βs)∥1 (3)\nDuring training, gradient descent is used to minimize the loss function Lβ(*θ, s*) to learn the sparse network.\n\nSimultaneously, β is jointly adjusted in the process of T parameter updates. After each update, corresponding parameter values θ\n(T), s\n(T), and β\n(T) are obtained. Thus, between each iteration, only s is adjusted based on the changes in parameter values, without modifying the network weights. This leads to higher efficiency in finding networks, and the continuous dynamic adjustments ensure the smoothness of sparsity and the accuracy of the network.\n\n## 5.3 Structured Pruning\n\nAlthough LTH can find element-wise sparse subnetworks and even outperform the original dense models, unstructured pruning is known to have an inability for real hardware acceleration. \"Generalized Lottery Ticket Hypothesis\" is investigated by Alabdulmohsin et al. (2021) to encompass both unstructured and structured pruning under a common framework. Specifically, structured pruning is achieved by carefully selecting the basis and using algorithms originally developed for unstructured pruning such as IMP. To address the limit of winning tickets in practice, (Chen et al., 2022a) propose a \"post-processing technique\" after each round of unstructured pruning to enforce the structural sparsity. Specifically, the pruned elements are re-filled back in some channels, and then non-zero elements are re-grouped to create flexible groupwise structural patterns. It turns out that the hardware acceleration roadblock of LTH can be removed.\n\n(Li et al., 2023) attempt to alleviate the constraints on network pruning by employing a mixed-precision quantization approach. They validate the existence of quantized lottery tickets, which are denoted as MPQtickets (Mixed-Precision Quantization). Additionally, the study demonstrates that MPQ lottery models exhibit greater flexibility compared to regular lottery models, and they derive more substantial benefits from pruning when compared to quantized neural networks (QNNs).\n\n## 6 Efficiency\n\nDespite the success of LTH, the iterative train-prune-retrain process is time-consuming. To address this issue, many works propose to find the winning tickets in the early state. Some other works reduce the computation cost by reducing the size of data or using efficient search algorithms. Remarkably, the possibility of re-using an identified winning ticket across different datasets or architectures has been investigated.\n\n## 6.1 Early Bird Tickets\n\nThe concept of Early-bird (EB) tickets is first introduced by You et al. (2020). It refers to winning tickets identified in the early stages of training using cost-effective approaches like early stopping and low-precision training at high learning rates. This discovery aligns with prior observations that critical network patterns emerge early. Additionally, they introduce a mask distance metric for efficiently identifying EB tickets with minimal computational overhead, without requiring knowledge of the final winning tickets. By capitalizing on the existence of EB tickets and employing the proposed mask distance, we develop efficient training methods that involve identifying EB tickets through low-cost techniques and then continuing training exclusively on these EB tickets to reach the target accuracy. Chen et al. (2020c) introduce EarlyBERT, drawing inspiration from the concept of Early-Bird Lottery Tickets proposed by (You et al., 2022b). The study put forth a computationally efficient training algorithm applicable to both pre-training and fine-tuning large-scale language models. By slimming down specific components of a transformer, the study identifies structured winning tickets in the early stages of BERT training, leading to efficient BERT training. In the application of the LTH to object recognition tasks, (Girish et al., 2021) also utilizes Early-Bird tickets.\n\nThey observe that after the middle stage, the mask stabilizes. This observation suggests that a substantial reduction in the number of training iterations in computer vision tasks can be achieved with minimal impact on network performance. (You et al., 2022b) is the first to identify early birds in Graph Convolutional Networks (GCN) and introduce a novel concept known as Graph Early-Bird (GEB) tickets. These emerge in the early stages of sparsifying GCN graphs. They also propose an effective detector to automatically identify the presence of GEB tickets. Additionally, they advocate for graph-model co-optimization and present a highly efficient training framework called GEBT. This framework enhances the efficiency of GCN\ntraining by jointly identifying early-bird tickets in both GCN graphs and models while enabling simultaneous sparsification of both components. (Shen et al., 2022) present an approach similar to early-bird tickets. They introduce an Early Pruning Indicator (EPI), which distinguishes itself from early-bird tickets by utilizing sub-network architectural similarity to initiate pruning once the architecture reaches stability.\n\nExpanding on the Early-Bird (EB) approach, (Kim et al., 2022) introduces a novel method termed EarlyTime (ET). They establish the stopping criterion for training by evaluating the Kullback–Leibler (KL)\ndivergence between class prediction distributions at distinct time intervals. Once the KL divergence reaches a predetermined threshold, it signifies the acquisition of winning tickets. The ET method can be synergistically employed with EB, resulting in a reduction of training time by up to 38 Instead of pruning the trained networks, (Wang et al., 2020a) endeavor to prune networks during the initialization phase to save computational costs during training. They introduce a novel criterion known as Gradient Signal Preservation (GraSP). This method involves the removal of weights that have the least impact on reducing the gradient norm after pruning. (Rachwan et al., 2022) further refine the Gradient Signal Preservation (GraSP) by introducing Early Compression via Gradient Flow Preservation (EarlyCroP). This approach prioritizes the preservation of Gradient Flow (GF) and leverages the close relationship between the Neural Tangent Kernel (NTK) and GF. Consequently, it allows for network pruning with minimal disruption to the NTK, ensuring high levels of sparsity while maintaining performance comparable to that of the dense network. EarlyCroP requires only a single model training session and can be implemented either before or early in the training process.\n\n## 6.2 Reduce Data Size\n\nAn alternative way to find lottery tickets efficiently is using a selected subset of data rather than using the full training set. With the small dataset, the cost of interactive pruning can be reduced significantly. (Zhang et al., 2021d) propose reducing the size of the training set to more efficiently acquire winning tickets. They specifically select a subset of data, termed the Pruning-Aware Critical set (PrAC set), instead of employing the entire training dataset. The PrAC set encompasses samples that are considered to be the most challenging and information-rich for dense models. Training and pruning networks using the PrAC set enables the discovery of high-quality winning tickets, while concurrently reducing the number of iterations needed for training. (Shen & Xing, 2022) introduce the \"dataset lottery ticket hypothesis,\" suggesting the presence of a subset within an extensive large-scale dataset. This subset accurately mirrors the performance disparities observed under various training frameworks, hyperparameters, and other conditions. Consequently, it facilitates a substantial reduction in resource demands and enhances the computational efficiency of experiments, all while maintaining accuracy on the complete dataset. They employ multiple approaches, including a uniform selection scheme commonly utilized in the literature, as well as the creation of subsets based on prior knowledge, to validate this hypothesis. (Shen et al., 2023) face difficulties in locating subnetworks within Vision Transformers (ViTs) at the weight level using conventional methods. Consequently, they attempt to sparsify the input data, particularly image patches. In essence, there exists a specific subset of input image patches. When a ViT is exclusively trained using this subset, it attains a level of accuracy similar to ViTs trained with the entire set of patches. They term this subset of input patches as \"em winning tickets,\"\nsignifying their substantial information content within the input data.\n\n## 6.3 Re-Use Winning Tickets\n\nSince the \"retrainability\" of a winning ticket is the most distinctive property of LTH, an intuitive method to reduce the cost is re-using the winning tickets across various tasks and models. (Morcos et al., 2019a)\nexplore the potential of reusing winning tickets across different datasets and optimizers. They observe that winning tickets generated from larger datasets tend to transfer better. Moreover, these initializations also show promise across different optimization methods. These results indicate that winning ticket initializations from larger datasets possess inductive biases that benefit neural networks more broadly, offering promise for improved initialization methods. (Van Soelen & Sheppard, 2019) lay the foundation for a transfer learning technique that distills the original network to its crucial connections, obviating the need to freeze entire layers. The authors show that in the case of CNNs, networks can not only be pruned down to a mere 10%\nof their original parameters but these sparsified networks can also be re-trained on similar datasets with only a minor decrease in accuracy. Rooted in the LTH, this approach presents a promising alternative to conventional transfer learning methods and has shown encouraging initial outcomes. Burkholz et al. (2022) also theoretically prove the existence of universal lottery tickets. These tickets can be directly reused across various tasks without the need for retraining. (Desai et al., 2019) investigate the feasibility of re-training a sparse subnetwork obtained from one domain to a dissimilar domain and explores the impact of different initialization strategies during transfer. The experiments reveal that subnetworks obtained through lottery ticket training possess an inductive bias that transcends specific domains, enabling effective application across diverse domains.\n\nHowever, not all subnetworks exhibit universal transferability. (Chen et al., 2020a) investigate subnetworks extracted in the context of masked language modeling tasks, which demonstrate broad transferability. In contrast, subnetworks identified in other NLP tasks may possess limited applicability or even lack transferability altogether when applied in a transfer learning context. In addition to specific studies on the knowledge transfer of LTH in particular tasks and domains, there are ongoing investigations into the model's capacity for continuous learning, known as lifelong learning. (Chen et al., 2020b) extend the LTH into the realm of lifelong learning, introducing the concept of \"lifelong tickets.\" This research showcases that lifelong tickets can enhance learning performance across continual tasks. However, pruning networks in the lifelong setting presents significant challenges. Firstly, conducting greedy weight pruning in sequential tasks is hindered by a bias towards earlier tasks. Secondly, the compact network capacity of tickets in lifelong learning exacerbates the issue of catastrophic forgetting. To address these, two pruning options are proposed: top-down and bottom-up. The top-down approach extends iterative pruning over sequential tasks.\n\nIn contrast, the bottom-up approach, allowing dynamic adjustment of model capacity, proves effective in mitigating early-stage excessive pruning. Additionally, the study introduces \"lottery teaching,\" a method leveraging knowledge distillation with external unlabeled data to further counteract forgetting.\n\n(Sun et al., 2020) present a novel parameter sharing mechanism, \"Sparse Sharing,\" for deep multi-task learning. Unlike conventional approaches relying on predefined sharing schemes, Sparse Sharing automatically identifies a sparse sharing structure tailored to the specific task relations. It starts with an over-parameterized base network, from which each task extracts a subnetwork. These subnetworks, associated with multiple tasks, exhibit partial overlap and undergo parallel training. (Chen et al., 2021b) investigate whether subnetworks extracted from pre-trained computer vision models maintain downstream transferability. Their experiments verify that in pre-trained models for various tasks, matching subnetworks of different sparsity levels can consistently be identified. These subnetworks demonstrate universal transferability to downstream tasks without performance degradation. However, even within the same domain of studying LTH in computer vision (CV), (Girish et al., 2021) indicate that the acquired tickets from ImageNet pre-training do not transfer effectively to downstream tasks, particularly in the context of object recognition, instance segmentation, and keypoint estimation.\n\n(Iofinova et al., 2022) investigate the transferability of winning tickets in the context of pruned convolutional neural networks (CNNs) trained on ImageNet. They consider various pruning methods and find that sparse models can match or surpass the performance of dense models in transfer tasks, even at high sparsity levels.\n\n(Liu et al., 2022b) discover that the preserved pre-training performance within the subnetworks is the reason behind the success of magnitude pruning, which is linked to downstream transferability. As a result, they propose a novel approach to adjust the model architecture during the pre-training phase, aiming to retain more transferability. By training binary masks over model weights on the pre-training tasks, the method aims to preserve the universal transferability of the subnetwork across various downstream tasks. Compared to conventional magnitude-based pruning, this technique significantly improves the overall performance of identified BERT subnetworks on downstream tasks. Additionally, it proves to be more efficient in subnetwork identification and particularly advantageous for fine-tuning with limited data availability. In addition, meta-learning, which focuses on training models to quickly adapt to new tasks with minimal data, is closely related to transferability. (Gao et al., 2022) explore the application of the lottery ticket hypothesis (LTH) in the context of meta-learning to achieve efficient learning with reduced computational cost. They demonstrate that there exist sparse sub-networks, referred to as \"meta winning tickets,\" which can be trained for few-shot classification tasks with performance comparable to the original network. The utilization of LTH in meta-learning facilitates the adaptation of networks on resource-constrained IoT devices.\n\nThe paper also introduces a novel approach for the early detection of meta-winning tickets, enabling efficient training on devices with limited resources.\n\n(Burkholz et al., 2022) further substantiates that winning tickets can be applied across various tasks, suggesting a level of universality. This notion is formally established and theoretically validated, demonstrating the existence of universal tickets that do not necessitate additional training. Leveraging these validation results, they introduce several novel pruning approaches for strong lottery tickets and explicitly outline the sparse structure of the universal function family. Subsequently, (Huang et al., 2022) apply the LTH to CNNs integrating plug-and-play self-attention modules (SAMs). Their investigation reveals that complete integration of SAMs with all blocks may not lead to maximal performance enhancement. Consequently, they introduce the concept of Lottery Ticket Hypothesis for Self-attention Networks, suggesting that a self-attention network encompasses a sparse subnetwork capable of expediting inference, reducing parameter overhead, and maintaining accuracy. This conjecture is substantiated by a combination of empirical observations and theoretical underpinnings. (Chen et al., 2021f) attempt to tackle a challenge: transferring winning tickets from one architecture to another network with a different structure. This endeavor aims to directly provide the latter with a winning ticket, thus eliminating the necessity for expensive iterative training. Therefore, they propose the Elastic Lottery Ticket Hypothesis (E-LTH). It suggests that through strategic replication, removal, and re-ordering of layers within a network, the corresponding winning ticket can be extended or compressed to fit into a deeper or shallower network from the same family. This subnetwork's performance closely rivals that of the latter network's winning ticket directly identified by IMP.\n\n## 6.4 Dual Lottery Ticket Hypothesis\n\nDual Lottery Ticket Hypothesis (DLTH) (Bai et al., 2022) posits that a randomly selected subnetwork from a randomly initialized dense network can be transformed into a trainable condition and achieve commendable performance through regular fine-tuning, as compared to the Lottery Ticket Hypothesis (LTH). DLTH employs the Random Sparse Network Transformation (RST) method to obtain sparse subnetworks. This process bypasses the need for pre-training the model and directly selects a random subnetwork from the randomly initialized network. Subsequently, it utilizes a regularization-based method to extrude information from other weights that are masked to target the sparse structure. Consequently, the training process of DLTH is more efficient than that of LTH.\n\nIn the process of information extraction, a loss function similar to that in Continuous Sparsification is employed, but with a switch from L1 regularization to L2 regularization:\n\n$$L_{R}=L(f(x;\\theta),D)+\\frac{1}{2}\\lambda\\|\\theta^{*}\\|_{2}^{2},$$\n$$\\left(4\\right)$$\n, (4)\nwhere LR is composed of the regular training loss L and an L2 regularization term, θ\n∗ denotes the masked weights.\n\nBuilding upon this, the authors continually increase the trade-off parameter λ during the optimization of LR:\n\n$$\\lambda_{p+1}=\\left\\{\\begin{array}{l l}{{\\lambda_{p}+\\eta,}}&{{\\lambda_{p}<\\lambda_{b},}}\\\\ {{\\lambda_{p},}}&{{\\lambda_{p}=\\lambda_{b},}}\\end{array}\\right.\\tag{1}$$\n$\\left(5\\right)^{\\frac{1}{2}}$\nwhere λp represents the regularization term at the p-th updating, η is the given mini-step for gradually increasing λ, and λb is the bound to limit the increase of λ. Through this process, the information of θ\n∗\nis gradually squeezed into the unmasked weights, denoted as ¯θ\n∗. The importance of weights dynamically adjusts, transitioning from a balanced state (where all weights are equal) to an imbalanced state (as the magnitude decreases, θ\n∗ becomes less significant). After enough extrusion, the influence of θ\n∗ on the network becomes extremely limited, and they are subsequently removed to obtain the final sparse network.\n\nFurthermore, LTH's sparsification relies on the mask obtained from the pre-trained network, resulting in a fixed sparse structure. In contrast, DLTH depends on the randomly selected network, offering greater flexibility in the selection of sparse subnetworks. When compared to Pruning at Initialization (PI), DLTH\nfocuses on transforming randomly selected subnetworks into trainable states, rather than starting from scratch to select a specific subnetwork. Thus, the sparsity structure in DLTH is controllable. In terms of accuracy, ResNet56 pruned using the RST method outperforms ResNet56 pruned using LTH at sparsity ratios of 50%, 70%, 90%, 95%, and 98% on the Cifar-10 and Cifar-100 datasets.\n\nIn summary, DLTH introduces a novel approach for training sparse networks that surpass existing methods\n(LTH and PI) in terms of generality, efficiency, accuracy, and flexibility in sparse structure selection. Inspired by the Lottery Ticket Hypothesis and Dual Lottery Ticket Hypothesis DLTH (Bai et al., 2022), (Wang et al., 2022a) propose two novel algorithms, UniLTH and UniDLTH, aimed at unifying LTH and DLTH.\n\nUniLTH modifies the LTH approach by employing progressively increasing regularization on all weights of the neural network to discern their respective importance. On the other hand, UniDLTH applies to grow L2 regularization exclusively to the masked weights for information extrusion. Both UniLTH and UniDLTH\nconsist of two stages. In the first stage, the neural network is trained without any regularization, and early stopping is employed with a patience parameter, denoted as ϕ. When the validation loss fails to decrease for ϕ consecutive epochs, training is halted, and parameters are reverted to their state from xi epochs earlier. In the second stage, iterative pruning is integrated with increased regularization to search or transform winning tickets. The network is alternatively trained with increased regularization and pruned until it attains the desired sparsity ratio.\n\nExperimental results demonstrate that after applying UniLTH and UniDLTH to prune ResNet50 at various sparsity ratios, the performance surpasses that of LTH and DLTH under equivalent experimental conditions.\n\n## 7 Relation With Other Topics\n\nExtensive work has been done to fill the gap between LTH and other topics including robustness, scalability, and fairness. In this section, we conclude these connections and summarize the experimental results.\n\n## 7.1 Robustness\n\nAlthough winning tickets can achieve remarkable performance, these subnetworks manifest vulnerability to adversarial attacks (Zheng et al., 2022). To tackle this issue, the authors in (Zheng et al., 2022) initially convert the binary mask into a hard concrete distribution. This renders the mask differentiable, allowing for continuous adjustments during training based on the objective. Furthermore, it incorporates an adversarial loss objective to guide the search for robust tickets, prioritizing both accuracy and robustness. In contrast to adversarial training, (Liu et al., 2022a) propose a novel approach involving randomly initialized binary networks with fixed parameters (+1 or -1) to identify a subnetwork structure resilient to attacks. By adaptively pruning various network layers, employing an effective binary initialization strategy, and integrating a final batch normalization layer for enhanced training stability, they achieve more compact networks with competitive performance compared to existing methods.\n\nThe limitations of adversarial learning are also highlighted in (Xi et al., 2022). This method requires the generation of adversarial samples through gradient descent, making it more computationally expensive compared to conventional fine-tuning. By investigating the optimization process of adversarial training, it was discovered that resilient connectivity patterns emerge early in training well before parameters stabilize.\n\nTherefore, the authors devise a two-step training approach to obtain robust early-bird tickets: (1) earlystage search for robust tickets with structured sparsity; (2) subsequent fine-tuning of these robust tickets.\n\n(Chen et al., 2022b) introduce a more stringent concept known as \"Double-Win Lottery Tickets.\" They employ the approach of robust pre-training to extract subnetworks from the network. These subnetworks can independently transfer to various downstream tasks while achieving both standard and robust generalization, mirroring the full pre-trained model's capabilities.\n\n## 7.2 Fairness\n\nThere is also a gap between fairness and LTH. Hansen & Søgaard (2021) argue that the use of lottery tickets may impact the fairness of machine learning models. They assess the fairness of extracting lottery tickets through layer-wise and global weight pruning in the field of natural language processing. Experimental results indicate that at high pruning rates, there is a slight increase in group disparities. Models trained with a distributed robust optimization objective are sometimes less sensitive to pruning, but the results are inconsistent. To alleviate unfairness with sparse networks, (Tang et al., 2023) propose to find appropriate binary masks for the weights to obtain fair sparse subnetworks. It turns out that such sparse subnetworks with inborn fairness exist in randomly initialized networks, achieving an accuracy-fairness trade-off. They theoretically provide fairness and accuracy guarantees for fair scratch tickets and empirically verify the existence of fair scratch tickets on various datasets.\n\n## 7.3 Federated Learning\n\nFederated learning (FL) is a collaborative decentralized learning paradigm to address privacy in machine learning. It suffers from heavy communication such as uploading and downloading large volumes of parameters in the model. LTH is introduced to reduce both the communication and computation costs through model compression (Itahara et al., 2020). The key idea is to obtain a winning ticket from the original model.\n\nLotteryFL (Li et al., 2020) further assign each client with a personalized winning ticket. Following LotteryFL, CELL (Seo et al., 2021) exploits downlink broadcast for communication efficiency and introduces a novel user grouping method to mitigate stragglers. FedLTN (Mugunthan et al., 2022) proposes post-pruning without rewinding to achieve faster and greater model sparsity in FL. Babakniya et al. (2023) identify a challenge in federated learning (FL) on resource-limited edge nodes due to constrained computation and communication capabilities. They explore the use of off-the-shelf sparse learning algorithms, which train a binary sparse mask on each client, aiming to achieve a consistent sparse server mask with sparse weight tensors. However, their investigation reveals that this approach leads to a significant drop in accuracy compared to FL with dense models, particularly for clients with limited resources. They observe a notable lack of consensus among the trained sparsity masks on clients, hindering convergence for the server mask and potentially causing a substantial performance decline. In response, the authors propose \"federated lottery aware sparsity hunting\" (FLASH), a unified sparse learning framework designed to enable the server to attain a sparse sub-model, maintaining classification performance under highly resource-limited client settings. Additionally, they introduce \"hetero-FLASH\" to accommodate FL on different devices with varying parameter density requirements based on their resource constraints.\n\n## 7.4 Reinforcement Learning\n\nDue to the distributional shift inherent in reinforcement learning (RL) problems, the performance of winning lottery tickets can be affected (Vischer et al., 2022). Specifically, the feed-forward networks trained with behavioral cloning compared to reinforcement learning can be pruned to higher levels of sparsity, suggesting that the RL agents require more degrees of freedom. The analysis shows that the effect in learning paradigms can be attributed to the identified mask rather than the weight initialization. Also, the input layer mask selectively prunes entire input dimensions that turn out to be irrelevant to the task. The existence of winning tickets has also been investigated in a number of discrete-action space tasks, including both classic control and pixel control (Yu et al., 2019). The result confirms that winning ticket initialization outperforms parameter-matched random initializations at extreme pruning rates in RL.\n\n## 8 Applications 8.1 Cv And Nlp\n\nVGG and ResNet are the two most widely used models to investigate LTH in the literature. The existence of winning tickets is verified on two benchmark datasets: MNIST and CIFAR10. Besides these classification models in computer vision, (Girish et al., 2021) apply LTH to object recognition tasks. The authors empirically investigated the applicability of LTH for model pruning in tasks related to object detection, instance segmentation, and keypoint estimation. Additionally, the research delves into the behavior of trained tickets in relation to attributes like object size, frequency, and detection difficulty.\n\nLarge-scale pre-trained language models such as transformers are usually over-parametrized (Liang et al.,\n2021a). LTH has been applied to the GLUE benchmark, a collection of tools for evaluating the performance of models across a diverse set of existing natural language understanding tasks. These tasks include question-answering, sentiment analysis, and textual entailment, and an associated online platform for model evaluation, comparison, and analysis. Recently, multimodal learning has been revolutionized by combining computer vision and natural language processing technologies. An empirical study has been performed to assess whether winning tickets also exist in pre-trained vision-and-language (VL) models Gan et al. (2022). The authors consolidate 7 representative VL tasks: visual question answering, visual commonsense reasoning, visual entailment, referring expression comprehension, image-text retrieval, GQA, and NLVR. It turns out that the sparse model can not strictly match the performance of the full model. However, a \"relaxed\" winning ticket at 50%-70% sparsity can reach 99% of the full accuracy.\n\n## 8.2 Deployment\n\nObtaining winning tickets for large models involves a significant computational cost. Consequently, these winning tickets are regarded as valuable assets, warranting the need to safeguard their copyright. (Chen et al., 2021g) present a distinctive approach for validating lottery tickets by leveraging sparse topological information. This is achieved through the development of various graph-based signatures that can be embedded as credentials. By integrating trigger-set-based methods, the proposal can be applied in both white-box and black-box validation scenarios. MSSR (Lin et al., 2023) introduces a single adaptable model that encompasses multiple SR models of varying sizes. The MSSR framework involves two stages: a forward stage for model compression and a backward stage for expansion. In the forward stage, LTH with rewinding weights is used to gradually reduce the size of the SR model, employing pruning-out masks that form nested sets.\n\nAdditionally, stochastic self-distillation (SSD) is utilized to enhance the performance of sub-networks. In the backward stage, the smaller SR model is expanded by reinstating and fine-tuning the pruned parameters based on the pruning-out masks acquired in the forward stage.\n\n## 9 Open Issues 9.1 Acceleration In Practice\n\nThe irregular sparse patterns in a winning ticket pose a challenge to accelerating on hardware because most of the accelerators are optimized for dense matrix operations. It will limit the practical use of unstructured pruned subnetworks even if the sparsity level is very high. Therefore, it is urgent to design effective algorithms that can find structured winning tickets or transform an unstructured subnetwork into a structured subnetwork without compromising performance. Another solution is to design hardware accelerators that can support sparse matrix operations. We remark that existing works use sparsity or FLOPs (floating point operations per second). A new performance metric is needed to evaluate the real running time.\n\n## 9.2 Theoretical Understanding And Design Better Network.\n\nThe identified winning tickets have the potential to improve our theoretical understanding of neural networks and help us design better networks. The original, overparameterized model has too much complexity and may result in overfitting. The lottery ticket hypothesis offers a complementary perspective on the relationship between compression and generalization: a larger network might explicitly contain simpler representations.\n\nBy pruning, we can find an approximated simple (e.g., linear) model. However, the potential implications for theoretical study of optimization and generalization have not been studied yet. Also, it is unclear how to design a better model based on the initialization of winning tickets and sub-structure.\n\n## 9.3 Lth In Diffusion Model.\n\nAlthough diffusion models have shown impressive performance in capturing distributions and sample quality, they are notoriously slow to generate data due to the long chain of reversing the diffusion process. It is intuitive to mitigate the computational cost by pruning the reverse model. An open question remains: Can we use sub-networks with different sparsity in the reverse process? Since a winning ticket can be considered an equivalent version of the original model, we can leverage a sparser sub-network in the late stage of denoising to further improve efficiency. Intuitively, the noise in the later reverse process has been reduced and it will be easier to optimize. The challenge lies in how to optimize a dense model and a winning ticket while guaranteeing the convergence of training. Another challenge is to decide at which step to use the winning ticket. We leave this open question for future investigation which we hope will stimulate the research on improving the efficiency of the diffusion model.\n\n## 10 Conclusion\n\nThis survey for the first time investigates the lottery ticket hypothesis in the literature. 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URL https://proceedings.neurips.cc/paper/2019/ hash/1113d7a76ffceca1bb350bfe145467c6-Abstract.html."},"reviews":{"kind":"string","value":""}}},{"rowIdx":91,"cells":{"paper":{"kind":"string","value":"# Coin++: Neural Compression Across Modalities\n\nEmilien Dupont* *dupont@stats.ox.ac.uk* Hrushikesh Loya* *loya@stats.ox.ac.uk* Milad Alizadeh *milad.alizadeh@cs.ox.ac.uk*\n\nAdam Goliński *adamg@robots.ox.ac.uk*\n\nYee Whye Teh *y.w.teh@stats.ox.ac.uk*\n\nArnaud Doucet doucet@stats.ox.ac.uk\n\nUniversity of Oxford\n\nReviewed on OpenReview: *https: // openreview. net/ forum? id= NXB0rEM2Tq*\n\n## Abstract\n\nNeural compression algorithms are typically based on autoencoders that require specialized encoder and decoder architectures for different data modalities. In this paper, we propose COIN++, a neural compression framework that seamlessly handles a wide range of data modalities. Our approach is based on converting data to implicit neural representations, i.e.\n\nneural functions that map coordinates (such as pixel locations) to features (such as RGB\nvalues). Then, instead of storing the weights of the implicit neural representation directly, we store modulations applied to a meta-learned base network as a compressed code for the data. We further quantize and entropy code these modulations, leading to large compression gains while reducing encoding time by two orders of magnitude compared to baselines. We empirically demonstrate the feasibility of our method by compressing various data modalities, from images and audio to medical and climate data.\n\n## 1 Introduction\n\nIt is estimated that several exabytes of data are created everyday (Domo, 2018). This data is comprised of a wide variety of data modalities, each of which could benefit from compression. However, the vast majority of work in neural compression has focused only on image and video data (Ma et al., 2019). In this paper, we introduce a new approach for neural compression, called COIN++, which is applicable to a wide range of data modalities, from images and audio to medical and climate data (see Figure 1).\n\nMost neural compression algorithms are based on autoencoders (Theis et al., 2017; Ballé et al., 2018; Minnen et al., 2018; Lee et al., 2019). An encoder maps an image to a latent representation which is quantized and entropy coded into a bitstream. The bitstream is then transmitted to a decoder that reconstructs the image.\n\nThe parameters of the encoder and decoder are trained to jointly minimize reconstruction error, or *distortion*,\nand the length of the compressed code, or *rate*. To achieve good performance, these algorithms heavily rely on encoder and decoder architectures that are specialized to images (Cheng et al., 2020b; Xie et al., 2021; Zou et al., 2022; Wang et al., 2022). Applying these models to new data modalities then requires designing new encoders and decoders which is usually challenging.\n\nFigure 1: COIN++ converts a wide range of data modalities to neural networks via optimization and then\n\n\n\nstores the parameters of these neural networks as compressed codes for the data. Different data modalities can be compressed by simply changing the input and output dimensions of the neural networks.\n\nRecently, a new framework for neural compression, called COIN (COmpression with Implicit Neural representations), was proposed which bypasses the need for specialized encoders and decoders (Dupont et al.,\n2021a). Instead of compressing images directly, COIN fits a neural network mapping pixel locations to RGB\nvalues to an image and stores the quantized weights of this network as a compressed code for the image.\n\nWhile Dupont et al. (2021a) only apply COIN to images, it holds promise for storing other data modalities.\n\nIndeed, neural networks mapping coordinates (such as pixel locations) to features (such as RGB values),\ntypically called *implicit neural representations* (INR), have been used to represent signed distance functions\n(Park et al., 2019), voxel grids (Mescheder et al., 2019), 3D scenes (Sitzmann et al., 2019; Mildenhall et al.,\n2020), temperature fields (Dupont et al., 2021b), videos (Li et al., 2021b), audio (Sitzmann et al., 2020b) and many more. COIN-like approaches that convert data to INRs and compress these are therefore promising for building flexible neural codecs applicable to a range of modalities.\n\nIn this paper, we identify and address several key problems with COIN and propose a compression algorithm applicable to multiple modalities, which we call COIN++. More specifically, we identify the following issues with COIN: 1. Encoding is slow: compressing a single image can take up to an hour, 2. Lack of shared structure: as each image is compressed independently, there is no shared information between networks, 3.\n\nPerformance is well below state of the art (SOTA) image codecs. We address these issues by: 1. Using meta-learning to reduce encoding time by more than two orders of magnitude to less than a second, compared to minutes or hours for COIN, 2. Learning a base network that encodes shared structure and applying modulations to this network to encode instance specific information, 3. Quantizing and entropy coding the modulations. While our method significantly exceeds COIN both in terms of compression and speed, it only partially closes the gap to SOTA codecs on well-studied modalities such as images. However, COIN++ is applicable to a wide range of data modalities where traditional methods are difficult to use, making it a promising tool for neural compression in non-standard domains.\n\n## 2 Method\n\nIn this paper, we consider compressing data that can be expressed in terms of sets of coordinates x ∈ X and features y ∈ Y. An image for example can be described by a set of pixel locations x = (*x, y*) in R\n2 and their corresponding RGB values y = (r, g, b) in {0, 1*, ...,* 255}\n3. Similarly, an MRI scan can be described by a set of positions in 3D space x = (*x, y, z*) and an intensity value y ∈ R\n+. Given a single datapoint as a collection of coordinate and feature pairs d = {(xi, yi)}\nn i=1 (for example an image as a collection of n pixel locations and RGB values), the COIN approach consists in fitting a neural network fθ : *X → Y* with parameters θ to the datapoint by minimizing\n\n$${\\mathcal{L}}(\\theta,\\mathbf{d})=\\sum_{i=1}^{n}\\|f_{\\theta}(\\mathbf{x}_{i})-\\mathbf{y}_{i}\\|_{2}.$$\n\n$$(1)$$\n∥fθ(xi) − yi∥2. (1)\nThe weights θ are then quantized and stored as a compressed representation of the datapoint d. The neural network fθ is parameterized by a SIREN (Sitzmann et al., 2020b), i.e. an MLP with sine activation functions, which is necessary to fit high frequency data such as natural images (Mildenhall et al., 2020; Tancik et al.,\n2020b; Sitzmann et al., 2020b). More specifically, a SIREN layer is defined by an elementwise sin applied to a hidden feature vector h ∈ R\nd as\n\n$$\\mathrm{SIIREN}(\\mathbf{h})=\\sin(\\omega_{0}(W\\mathbf{h}+\\mathbf{b}))$$\nSIREN(h) = sin(ω0(Wh + b)) (2)\nwhere W ∈ R\nd×dis a weight matrix, b ∈ R\nd a bias vector and ω0 ∈ R\n+ a positive scaling factor.\n\nWhile this approach is very general, there are several key issues. Firstly, as compression involves minimizing equation 1, encoding is extremely slow. For example, compressing a single image from the Kodak dataset\n(Kodak, 1991) takes nearly an hour on a 1080Ti GPU (Dupont et al., 2021a). Secondly, as each datapoint d is fitted with a separate neural network fθ, there is no information shared across datapoints. This is clearly suboptimal when several datapoints are available: natural images for example share a lot of common structure that does not need to be repeatedly stored for each individual image. In the following sections, we show how our proposed approach, COIN++, addresses these problems while maintaining the generality of COIN.\n\n$$(2)^{\\frac{1}{2}}$$\n\n## 2.1 Storing Modulations\n\nWhile COIN stores each image as a separate neural network, we instead train a base network shared across datapoints and apply *modulations* to this network to parameterize individual datapoints. Given a base network, such as a multi-layer perceptron (MLP), we use FiLM layers (Perez et al., 2018), to modulate the hidden features h ∈ R\nd of the network by applying elementwise scales γ ∈ R\nd and shifts β ∈ R\nd as\n\n## Film(H) = Γ ⊙ H + Β. (3)\n\n\n\nGiven a fixed base MLP, we can therefore parameterize families of neural networks by applying different scales and shifts at each layer. Each neural network function is therefore specified by a set of scales and shifts, which are collectively referred to as modulations (Perez et al., 2018). Recently, the FiLM approach has also been applied in the context of INRs. Chan et al. (2021) parameterize the generator in a generative adversarial network by a SIREN network and generate samples by applying modulations to this network as sin(γ ⊙ (Wh + b) + β). Similarly, Mehta et al. (2021) parameterize families of INRs using a scale factor via α⊙sin(Wh+b). Both of these approaches can be modified to use a low dimensional latent vector mapped to a set of modulations instead of directly applying modulations. Chan et al. (2021) map a latent vector to scales and shifts with an MLP, while Mehta et al.\n\n(2021) map the latent vector through an MLP of the same shape as the base network and use the hidden activations of this network as modulations. However, we found that both of these approaches performed poorly in terms of compressibility, requiring a large number of modulations to achieve satisfying reconstructions.\n\nInstead, we propose a new parameterization of modulations for INRs which, on top of yielding better compression rates, is also more stable to train. More specifically, given a base SIREN network, we only apply shifts β ∈ R\nd as modulations using\n\n$\\pm$ $\\beta$)). \nsin(ω0(Wh + b + β)) (4)\nat every layer of the MLP. To further reduce storage, we use a latent vector which is linearly mapped to the modulations as shown in Figure 2. In a slight overload of notation, we also refer to this vector as modulations or latent modulations. Indeed, we found empirically that using only shifts gave the same performance as using both shifts and scales while using only scales yielded considerably worse performance.\n\nIn addition, linearly mapping the latent vector to modulations worked better than using a deep MLP\nas in Chan et al. (2021). Given this parameterization, we then store a datapoint d (such as an image)\nas a set of (latent) modulations ϕ. To decode the datapoint, we simply evaluate the modulated base network fθ(·; ϕ) at every coordinate x,\n\n$$\\mathbf{y}=f_{\\theta}(\\mathbf{x};\\phi)$$\ny = fθ(x; ϕ) (5)\nas shown in Figure 3. To fit a set of modulations ϕ to a datapoint d, we keep the parameters θ of the Figure 2: COIN++ architecture. Latent modulations ϕ (in green) are mapped through a hypernetwork to modulations\n(in blue) which are added to activations of the base network fθ (in white) to parameterize a single function that can be evaluated at coordinates x to obtain features y.\n\n\n\nto a base network fθ, we obtain different functions that can be decoded into datapoints d\n(1), d\n(2), d\n(3) by evaluating the functions at various coordinates. While we show images in this figure, the same principle can be applied to a range of data modalities.\n3 base network fixed and minimize\n\n$${\\mathcal{L}}(\\theta,\\phi,\\mathbf{d})=\\sum_{i=1}^{n}\\|f_{\\theta}(\\mathbf{x}_{i};\\phi)-\\mathbf{y}_{i}\\|_{2}$$\n$\\dot{\\textbf{b}}$). \n\n∥fθ(xi; ϕ) − yi∥2 (6)\nover ϕ. In contrast to COIN, where each datapoint d is stored as a separate neural network fθ, COIN++\nonly requires storing O(k) modulations (or less when using latents) as opposed to O(k 2) weights, where k is the width of the MLP. In addition, this approach allows us to store shared information in the base network and instance specific information in the modulations. For natural images for example, the base network encodes structure that is common to natural images while the modulations store the information required to reconstruct individual images.\n\n## 2.2 Meta-Learning Modulations\n\nGiven a base network fθ, we can encode a datapoint d by minimizing equation 6. However, we are still faced with two problems: 1. We need to learn the weights θ of the base network, 2. Encoding a datapoint via equation 6 is slow, requiring thousands of iterations of gradient descent. COIN++ solves both of these problems with meta-learning.\n\nRecently, Sitzmann et al. (2020a); Tancik et al. (2020a) have shown that applying Model-Agnostic MetaLearning (MAML) (Finn et al., 2017) to INRs can reduce fitting at test time to just a few gradient steps.\nInstead of minimizing L(θ, d) directly via gradient descent from a random initialization, we can meta-learn an\ninitialization θ\n∗such that minimizing L(θ, d) can be done in a few gradient steps. More specifically, assume\nwe are given a dataset of N points {d\n(j)}\nN\nj=1. Starting from an initialization θ, a step of the MAML inner\nloop on a datapoint d\n(j)is given by\n$$\\theta^{(j)}=\\theta-\\alpha\\nabla_{\\theta}{\\mathcal{L}}(\\theta,\\mathbf{d}^{(j)}),$$\n(j)), (7)\nwhere α is the inner loop learning rate. We are then interested in learning a good initialization θ\n∗such that the loss L(θ, d\n(j)) is minimized after a few gradient steps across the entire set of datapoints {d\n(j)}\nN\nj=1. To update the initalization θ, we then perform a step of the outer loop, with an outer loop learning rate β, via\n\n$$\\left(7\\right)$$\n\n$$\\theta\\leftarrow\\theta-\\beta\\nabla_{\\theta}\\sum_{j=1}^{N}{\\mathcal{L}}(\\theta^{(j)},{\\mathbf{d}}^{(j)}).$$\n$$({\\boldsymbol{\\delta}})$$\n(j)). (8)\nIn our case, MAML cannot be used directly since at test time we only fit the modulations ϕ and not the shared parameters θ. We therefore need to meta-learn an initialization for θ and ϕ such that, given a new datapoint, the *modulations* ϕ can rapidly be computed while keeping θ constant. Indeed, we only store the modulations for each datapoint and share the parameters θ across all datapoints. For COIN++, a single step of the inner loop is then given by\n\n$$\\phi^{(j)}=\\phi-\\alpha\\nabla_{\\phi}{\\mathcal{L}}(\\theta,\\phi,{\\mathbf{d}}^{(j)}),$$\n$$({\\mathfrak{g}})$$\n\n(j)), (9)\nwhere θ is kept fixed. Performing the inner loop on a subset of parameters has previously been explored by Zintgraf et al. (2019) and is referred to as CAVIA. As observed in CAVIA, meta-learning the initialization for ϕ is redundant as it can be absorbed into a bias parameter of the base network weights θ. We therefore only need to meta-learn the shared parameter initialization θ. The update rule for the outer loop is then given by\n\n$$\\theta\\leftarrow\\theta-\\beta\\nabla_{\\theta}\\sum_{j=1}^{N}{\\mathcal{L}}(\\theta,\\phi^{(j)},{\\bf d}^{(j)}).$$\n(j)). (10)\nThe inner loop then updates the modulations ϕ while the outer loop updates the shared parameters θ. This algorithm allows us to meta-learn a base network such that each set of modulations can easily and rapidly be fitted (see Figure 4). In practice, we find that as few as 3 gradient steps gives us compelling results, compared with thousands for COIN.\n\n## 2.3 Patches, Quantization And Entropy Coding For Modulations\n\nPatches for large scale data. While meta-learning the base network allows us to rapidly encode new datapoints into modulations, the training procedure is expensive, as MAML must take gradients through the\n\n$$(10)^{\\frac{1}{2}}$$\n\n\n\n\n\nFigure 4: (Left) Starting from a random initialization θ, we meta-learn parameters θ\n∗ of the base network\n(with training progress shown as a solid line) such that modulations ϕ can easily be fit in a few gradient steps (with fitting shown in dashed lines). (Right) During training we sample patches randomly, while at test time we partition the datapoint into patches and fit modulations to each patch.\ninner loop (Finn et al., 2017). For large datapoints (such as high resolution images or MRI scans), this can become prohibitively expensive. While first-order approximations exist (Finn et al., 2017; Nichol et al., 2018; Rajeswaran et al., 2019), we found that they severely hindered performance. Instead, to reduce memory usage, we split datapoints into random patches during training. For large scale images for example, we train on 32×32 patches. At train time, we then learn a base network such that modulations can easily be fit to patches. At test time, we split a new image into patches and compute modulations for each of them. The image is then represented by the set of modulations for all patches (see Figure 4). We use a similar approach for other data modalities, e.g. MRI scans are split into 3D patches.\n\nQuantization. While COIN quantizes the neural network weights from 32 bits to 16 bits to reduce storage, quantizing beyond this severely hinders performance (Dupont et al., 2021a). In contrast, we find that modulations are surprisingly quantizable. During meta-learning, modulations are represented by 32 bit floats.\n\nTo quantize these to shorter bitwidths, we simply use uniform quantization. We first clip the modulations to lie within 3 standard deviations of their mean. We then split this interval into 2 bequally sized bins (where b is the number of bits). Remarkably, we found that reducing the number of bits from 32 to 5 (i.e. reducing the number of symbols from more than 109to only 32) resulted only in small decreases in reconstruction accuracy. Simply applying uniform quantization then improves compression by a factor of 6 at little cost in reconstruction quality.\n\nEntropy coding. A core component of almost all codecs is entropy coding, which allows for lossless compression of the quantized code, using e.g. arithmetic coding (Rissanen & Langdon, 1979). This relies on a model of the distribution of the quantized codes. As with quantization, we use a very simple approach for modeling this distribution: we count the frequency of each quantized modulation value in the training set and use this distribution for arithmetic coding at test time. In our experiments, this reduced storage 8-15%\nat no cost in reconstruction quality. While this simple entropy coding scheme works well, we expect more sophisticated methods to significantly improve performance, which is an exciting direction for future work.\n\nFinally, we note that we only transmit the *modulations* and assume the receiver has access to the shared base network. As such, only the modulations are quantized, entropy coded and count towards the final compressed file size. This is similar to the typical neural compression setting where the receiver is assumed to have access to the autoencoder and only the quantized and entropy coded latent vector is transmitted.\n\n## 3 Related Work\n\nNeural compression. Learned compression approaches are typically based on autoencoders that jointly minimize rate and distortion, as initially introduced in Ballé et al. (2017); Theis et al. (2017). Ballé et al.\n\n(2018) extend this by adding a hyperprior, while Mentzer et al. (2018); Minnen et al. (2018); Lee et al. (2019)\nuse an autoregressive model to improve entropy coding. Cheng et al. (2020b) improve the accuracy of the entropy models by adding attention and Gaussian mixture models for the distribution of latent codes, while Xie et al. (2021) use invertible convolutional layers to further enhance performance. While most of these are optimized on traditional distortion metrics such as MSE or SSIM, other works have explored the use of\n\n\n\n\n\nFigure 5: (Left) Test PSNR on CIFAR10 during training using a fixed number of modulations (see appendix for experimental details). Our method outperforms both baselines, improving PSNR by 2dB for the same number of parameters. (Right) Comparison of using shifts, scales and scales & shifts for modulations on MNIST (note that shifts and scales & shifts overlap). As can be seen, shifts perform significantly better than scales.\ngenerative adversarial networks for optimizing perceptual metrics (Agustsson et al., 2019; Mentzer et al.,\n2020). Neural compression has also been applied to video (Lu et al., 2019; Goliński et al., 2020; Agustsson et al., 2020) and audio (Kleijn et al., 2018; Valin & Skoglund, 2019; Zeghidour et al., 2021).\n\nImplicit neural representations and compression. In addition to COIN, several recent works have explored the use of INRs for compression. Davies et al. (2020) encode 3D shapes with neural networks and show that this can reduce memory usage compared with traditional decimated meshes. Chen et al. (2021)\nrepresent videos by convolutional neural networks that take as input a time index and output a frame in the video. By pruning, quantizing and entropy coding the weights of this network, the authors achieve compression performance close to standard video codecs. Lee et al. (2021) meta-learn sparse and parameter efficient initializations for INRs and show that this can reduce the number of parameters required to store an image at a given reconstruction quality, although it is not yet competitive with image codecs such as JPEG. Lu et al. (2021); Isik et al. (2021) explore the use of INRs for volumetric compression. Zhang et al.\n\n(2021) compress frames in videos using INRs (which are quantized and entropy coded) while learning a flow warping to model differences between frames. Results on video benchmarks are promising although the performance still lags behind standard video codecs. In concurrent work, Strümpler et al. (2021) propose a method for image compression with INRs which is closely related to ours. The authors also meta-learn an MLP initialization and subsequently quantize and entropy code the weights of MLPs fitted to images, leading to large performance gains over COIN. In particular, for large scale images their method significantly outperforms both COIN and COIN++ as they do not use patches. However, their approach still requires tens of thousands of iterations at test time to fully converge, unlike ours which requires 10 iterations (three orders of magnitude faster). Further, the authors do not employ modulations but directly learn the weights of the MLPs at test time. Finally, unlike our work, their approach is not applied to a wide range of modalities, including audio, medical and climate data. Indeed, to the best of our knowledge, none of these works have considered INRs for building a unified compression framework across data modalities.\n\n## 4 Experiments\n\nWe evaluate COIN++ on four data modalities: images, audio, medical data and climate data. We implement all models in PyTorch (Paszke et al., 2019) and train on a single GPU. We use SGD for the inner loop with a learning rate of 1e-2 and Adam for the outer loop with a learning rate of 1e-6 or 3e-6. We normalize coordinates x to lie in [−1, 1] and features y to lie in [0, 1]. Full experimental details required to reproduce all the results can be found in the appendix. We train COIN++ using MSE between the compressed and ground truth data. As is standard, we measure reconstruction performance (or distortion) using PSNR\n(in dB), which is defined as PSNR = −10 log10(MSE). We measure the size of the compressed data (or\n\n\n\nFigure 6: (Left) Rate distortion plot on CIFAR10. COIN++ outperforms COIN, JPEG and JPEG2000 while partially closing the gap to state of the art codecs. (Right) Qualitative comparison of compression artifacts for models at similar reconstruction quality. COIN++ achieves 32.4dB at 3.29 bpp while BPG\nachieves 31.9dB at 1.88 bpp.\nrate) in terms of bits-per-pixel (bpp) which is given by number of bits number of pixels 1 and kilobits per second (kpbs) for audio. We benchmark COIN++ against a large number of baselines including standard image codecs -\nJPEG (Wallace, 1992), JPEG2000 (Skodras et al., 2001), BPG (Bellard, 2014) and VTM (Bross et al.,\n2021) - autoencoder based neural compression - BMS (Ballé et al., 2018), MBT (Minnen et al., 2018)\nand CST (Cheng et al., 2020b) - standard audio codecs - MP3 (MP3, 1993) - and COIN (Dupont et al.,\n2021a). For clarity, we use consistent colors for different codecs and plot learned codecs with solid lines and standard codecs with dashed lines. The code to reproduce all experiments in the paper can be found at https://github.com/EmilienDupont/coinpp.\n\n## 4.1 Comparisons To Other Inr Parameterizations\n\nWe first compare our parameterization of INRs with the methods proposed by Chan et al. (2021) and Mehta et al. (2021) as described in Section 2.1. As can be seen in Figure 5, our method significantly outperforms both in terms of compressibility, improving PSNR by 2dB with the same number of parameters2. Further, using shift modulations is more effective than using scales and shifts, and performs significantly better than scales alone. We also note that our method allows us to quickly fit INRs using only a few hundred parameters.\n\nThis is in contrast to existing works on meta-learning for INRs (Sitzmann et al., 2020a; Tancik et al., 2020a),\nwhich typically require fitting 3 orders of magnitude more parameters at test time.\n\n## 4.2 Images: Cifar10\n\nWe train COIN++ on CIFAR10 using 128, 256, 384, 512, 768 and 1024 latent modulations. As can be seen in Figure 6, COIN++ vastly outperforms COIN, JPEG and JPEG2000 while partially closing the gap to BPG, particularly at low bitrates. To the best of our knowledge, this is the first time compression with INRs has outperformed image codecs like JPEG2000. Part of the gap between COIN++ and SOTA codecs\n(BMS, CST) is likely due to entropy coding: we use the simple scheme described in Section 2.3, while BMS\nand CST use deep generative models. We hypothesize that using deep entropy coding for the modulations would significantly reduce this gap. Figure 6 shows qualitative comparisons between our model and BPG to highlight the types of compression artifacts obtained with COIN++. In order to thoroughly analyse and evaluate each component of COIN++, we perform a number of ablation studies.\n\nQuantization bitwidth. Quantizing the modulations to a lower bitwidth yields more compressed codes at the cost of reconstruction accuracy. To understand the tradeoff between these, we show rate distortion plots when quantizing from 3 to 8 bits in Figure 7a. As can be seen, the optimal bitwidths are surprisingly low:\n\n1For non image data a \"pixel\" corresponds to a single dimension of the data. 2Despite significant experimental effort, we were unable to achieve better performance using meta-learning with Mehta et al.\n\n(2021).\n7\n\n\n\n\n\nFigure 7: (a) Rate distortion plot on CIFAR10 when quantizing the modulations o to various bitwidths.\n\nAs can be seen, using 5 or 6 bits is optimal. (b) Drop in PSNR for COIN and COIN++ quantization.\n\nCOIN++ is significantly more robust to quantization. (c) Drop in PSNR when quantizing the modulations\n¢ to various bitwidths, for various latent dimensions. Larger latent dimensions have larger drops in PSNR.\n\n(d) Effect of of quantization (to 5 bits) and entropy coding on CIFAR10. As can be seen, both quantization and entropy coding improve rate distortion performance.\n5 bits is optimal at low bitrates while 6 is optimal at higher bitrates. Qualitative artifacts obtained from quantizing the modulations are shown in Figure 14 in the appendix.\n\nQuantization COINvs COIN++. We compare the drop in PSNR due to quantization for COIN and COIN++ in Figure 7b. As can be seen, modulations are remarkably quantizable: when quantizing the COIN weights directly, performance decreases significantly around 14 bits, whereas quantizing modulations yields small drops in PSNR, even when using 5 bits. However, as shown in Figure 7c, the drop in PSNR from quantization is larger for larger models.\n\nEntropy coding. Figure 7d shows rate distortion plots for full precision, quantized and entropy coded modulations. As can be seen, both quantization and entropy coding significantly improve performance. Encoding/decoding speed. Table 1 shows the average encoding and decoding time for BPG, COIN and COIN++ on CIFAR10. As BPG runs on CPU while COIN ++ run on GPU, these times are not directly comparable. However, we follow standard practice in the literature and run all neural codecs on GPU and standard codecs on CPU (see appendix B.1 for hardware details). In terms of encoding, COIN++\ncompresses images 300× faster than CQIN while achieving a 4× better compression rate. Note that these results are obtained from compressing each image separately. When using batches of images, we can compress the entire CIFAR10 test set (10k images) in 4mins when using 10 inner loop steps (and in just over a minute when using 3 steps). In addition, as shown in Figure 15 in the appendix, COIN++ requires only 3 gradient steps to reach the same performance as COIN does in 10,000 steps, while using 4x less storage. In terms of decoding, CQIN++ is slower than COIN as it uses a larger shared network and entropy coding. However, decoding with COIN++ remains fast (on the order of a millisecond).\n\n| | BPG | COIN | COIN++ |\n|---------------|-------|------------|----------|\n| Encoding (ms) | 5.19 | 2.97 × 104 | 94.9 |\n| Decoding (ms) | 1.25 | 0.46 | 1.29 |\n\nTable 1: Average encoding and decoding time on CIFAR10 for BPG, COIN and COIN++. As can be seen, COIN++ encodes images orders of magnitude faster than COIN, while only being marginally slower than BPG at decoding time.\n\n\n\nFigure 8: (Left) Rate distortion plot on ERA5. COIN++ vastly outperforms all baselines. (Right) COIN++\ncompression artifacts on ERA5. See appendix E.5 for more samples.\n\n## 4.3 Climate Data: Era5 Global Temperature Measurements\n\nTo demonstrate the flexibility of our approach, we also use COIN++ to compress data lying on a manifold.\n\nWe use global temperature measurements from the ERA5 dataset (Hersbach et al., 2019) with the processing and splits from Dupont et al. (2021b). The dataset contains 8510 train and 2420 test globes of size 46×90, with temperature measurements at equally spaced latitudes λ and longitudes φ on the Earth from 1979 to 2020. To model this data, we follow Dupont et al. (2021b) and use spherical coordinates x = (cos λ cos φ, cos λ sin φ,sin λ) for the inputs. As a baseline, we compare COIN++ against JPEG,\nJPEG2000 and BPG applied to flat map projections of the data. As can be seen in Figure 8, COIN++\nvastly outperforms all baselines. These strong results highlight the versatility of the COIN++ approach:\nunlike traditional codecs and autoencoder based methods (which would require spherical convolutions for the encoder), we can easily apply our method to a wide range of data modalities, including data lying on a manifold. Indeed, COIN++ achieves a 3000× compression rate while having an RMSE of 0.5◦C, highlighting the potential for compressing climate data. More generally, it is likely that many highly compressible data modalities are not compressed in practice, simply because an applicable codec does not exist. We hope the flexibility of COIN++ will help make neural compression more generally applicable to such modalities.\n\nWe also note that very few baselines exist for compressing data on manifolds. A notable exception is McEwen et al. (2011) which builds a codec analogous to JPEG2000 by using wavelet transforms on the sphere. However, their method is not likely to outperform ours as it is not a learned codec and so cannot take advantage of the low entropy of the climate data. Further, their method is only applicable to the sphere, while our method is applicable to any manifold where a coordinate system can be defined to pass as input to the INR. Finally, we note that, while autoencoder based methods may be able to outperform COIN++ on this data modality, building such an autoencoder would require a non-trivial amount of work (including the use of spherical convolutions both for the encoder/decoder and hyperprior). In contrast, with COIN++ we simply change the input coordinates.\n\n\n\nFigure 9: (Left) Rate distortion plot on Kodak. While COIN++ performs slightly better than COIN, the use of patches reduces compression performance. (Right) COIN++ compression artifacts on Kodak. See appendix E.5 for more samples.\n\n## 4.4 Compression With Patches\n\nTo evaluate the patching approach from Section 2.3 and to demonstrate that COIN++ can scale to large data (albeit at a cost in performance), we test our model on images, audio and MRI data.\n\nLarge scale images: Kodak. The Kodak dataset (Kodak, 1991) contains 24 large scale images of size 768×512. To train the model, we use random 32×32 patches from the Vimeo90k dataset (Xue et al., 2019),\ncontaining 154k images of size 448×256. At evaluation time, each Kodak image is then split into 384 32×32 patches which are compressed independently. As we do not model the global structure of the image, we therefore expect a significant drop in performance compared to the case when no patching is required. As can be seen in Figure 9, the performance of COIN++ indeed drops, but still outperforms COIN and JPEG at low bitrates. We expect that this can be massively improved by modeling the global structure of the image\n(e.g. two patches of blue sky are nearly identical, but that information redundancy is not exploited in the current setup) but leave this to future work.\n\nAudio: LibriSpeech. To evaluate COIN++ on audio, we use the LibriSpeech dataset (Panayotov et al., 2015) containing several hours of speech data recorded at 16kHz. As a baseline, we compare against the widely used MP3 codec (MP3, 1993). We split each audio sample into patches of varying size and compress each of these to obtain models at various bit-rates (we refer to appendix B.5 for full experimental details). As can be seen in Figure 10, even though audio is a very different modality from the rest considered in this paper, COIN++ can still be used for compression, highlighting the versatility of our approach. However, in terms of performance, COIN++ lags behind well-establised audio codecs such as MP3.\n\nFigure 10: Rate distortion plot on LibriSpeech. While\n\n\n\nCOIN++ does not outperform specialized codecs like MP3, it still performs fairly well on audio, even though this is a vastly different data modality.\nMedical data: brain MRI scans. Finally, we train our model on brain MRI scans from the FastMRI\ndataset (Zbontar et al., 2018). The dataset contains 565 train volumes and 212 test volumes with sizes ranging from 16×320×240 to 16×384×384 (see appendix A.2 for full dataset details). As a baseline, we compare our model against JPEG, JPEG2000 and BPG applied independently to each slice. Due to memory constraints, we train COIN++ on 16×16×16 patches. We therefore store roughly 400 independent patches at test time (as opposed to 16 slices for the image codecs). Even then COIN++ performs reasonably well, particularly at low bitrates (see Figure 11). As a large number of patches are nearly identical, especially close to the edges, we expect that large gains can be made from modeling the global structure of the data.\n\nQualitatively, our model also performs well although it has patch artifacts at low bitrates (see Figure 11).\n\n\n\nFigure 11: (Left) Rate distortion plot on FastMRI. As can be seen, even when using patches, COIN++ performs better than JPEG. However, our methods lags behind BPG and JPEG2000. (Right) COIN++\ncompression artifacts on FastMRI. See appendix E.5 for more samples.\n\n## 5 Conclusion, Limitations And Future Work\n\nConclusion. We introduce COIN++, the first (to the best of our knowledge) neural codec applied to a wide range of data modalities. Our framework significantly improves performance compared to COIN both in terms of compression and encoding time, while being competitive with well-established codecs such as JPEG.\n\nWhile COIN++ does not match the performance of SOTA codecs, we hope our work will help expand the range of domains where neural compression is applicable.\n\nLimitations. The main drawback of COIN++ is that, because of the second-order gradients required for MAML, training the model is memory intensive. This in turn limits scalability and requires us to use patches for large data. Devising effective first-order approximations or bypassing meta-learning altogether would mitigate these issues. In addition, training COIN++ can occasionally be unstable, although the model typically recovers from loss instabilities (see Figure 12 in the appendix). Further, there are several common modalities our framework cannot handle, such as text or tabular data, as these are not easily expressible as continuous functions. Finally, COIN++ lags behind SOTA codecs. Indeed, while we have demonstrated the ease with which COIN++ is *applicable* to different modalities compared to autoencoders (as they do not require specialized and potentially complicated modality specific encoders and decoders), it is not clear whether COIN++ can *outperform* autoencoders specialized to each modality. However, we believe there are several interesting directions for future work to improve the performance of INR based codecs.\n\nFuture work. In its current form, COIN++ employs very basic methods for both quantization and entropy coding - using more sophisticated techniques for these two steps could likely lead to large performance gains.\n\nIndeed, recent success in modeling distributions of functions (Schwarz et al., 2020; Anokhin et al., 2021; Skorokhodov et al., 2021; Dupont et al., 2021b) suggests that large gains could be made from using deep generative models to learn the distribution of modulations for entropy coding. Similarly, better post-training quantization (Nagel et al., 2019; Li et al., 2021a) or quantization-aware training (Krishnamoorthi, 2018; Esser et al., 2020) would also improve performance. More generally, there are a plethora of methods from the model compression literature that could be applied to COIN++ (Cheng et al., 2020a; Liang et al., 2021). For large scale data, it would be interesting to model the global structure of patches instead of encoding and entropy coding them independently. Further, the field of INRs is progressing rapidly and these advances are likely to improve COIN++ too. For example, Martel et al. (2021) use adaptive patches to scale INRs to gigapixel images - such a partition of the input is similar to the variable size blocks used in BPG (Bellard, 2014). In addition, using better activation functions (Ramasinghe & Lucey, 2021) to increase PSNR and equilibrium models (Huang et al., 2021) to reduce memory usage are exciting avenues for future research.\n\nFinally, as COIN++ replaces the encoder in traditional neural compression with a flexible optimization procedure and the decoder with a powerful functional representation, we believe compression with INRs has great potential. Advances in INRs, combined with more sophisticated entropy coding and quantization may allow COIN-like algorithms to equal or even surpass SOTA codecs, while potentially allowing for compression on currently unexplored modalities.\n\n## Acknowledgments\n\nWe would like to thank Hyunjik Kim, Danilo Rezende, Dan Rosenbaum and Ali Eslami for helpful discussions.\n\nWe thank Jean-Francois Ton for helpful discussions around meta-learning and for reviewing an early version of the paper. We also thank the anonymous reviewers for their valuable feedback which helped improve the paper. Emilien gratefully acknowledges his PhD funding from Google DeepMind.\n\n## References\n\nEirikur Agustsson, Michael Tschannen, Fabian Mentzer, Radu Timofte, and Luc Van Gool. Generative adversarial networks for extreme learned image compression. 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PMLR, 2019.\n\nRenjie Zou, Chunfeng Song, and Zhaoxiang Zhang. The devil is in the details: Window-based attention for image compression. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern* Recognition, pp. 17492–17501, 2022.\n\n## A Dataset Details A.1 Vimeo90K\n\nWe use the Vimeo90k triplet dataset Xue et al. (2019) containing 73,171 3-frame sequences from videos at a resolution of 448×256. We processed the dataset following Bégaint et al. (2020). The resulting dataset contains 153,939 training images and 11,346 test images.\n\n## A.2 Fastmri\n\nTo generate the dataset, we use the validation split from the FastMRI brain multicoil database (Zbontar et al., 2018). This contains 1378 fully sampled brain MRI images obtained through a variety of sources - T1, T1 post-contrast, T2 and FLAIR images. We then filter the dataset to only use scans from the T2 source. In addition, as the vast majority of volumes have 16 slices, we also filter by volumes with 16 slices. We then randomly split the filtered scans into a 565 training volumes and 212 testing volumes. The train dataset contains the following shapes (with their counts):\n(16, 384, 384): 329 (16, 320, 320): 229\n(16, 384, 312): 2\n(16, 320, 260): 2 (16, 320, 240): 1\n(16, 384, 342): 1\n(16, 320, 270): 1 While the test dataset contains the following shapes (with their counts):\n(16, 384, 384): 124\n(16, 320, 320): 86 (16, 320, 260): 2 We also normalize the data to lie in [0, 1] (while COIN++ can handle data in any range, we cannot apply the image compression baselines if the data is not in [0, 1]). As the data contains outliers, we first compute a histogram of the data distribution and choose the maximum value such that 99.99% of the data has value less than this. We then normalize by the minimum and maximum value and clip any value lying outside this range (<0.01% of the data).\n\nDisclaimer required when using the FastMRI dataset: *\"Data used in the preparation of this article were* obtained from the NYU fastMRI Initiative database (fastmri.med.nyu.edu) (Zbontar et al., 2018; Knoll et al.,\n2020). As such, NYU fastMRI investigators provided data but did not participate in analysis or writing of this report. A listing of NYU fastMRI investigators, subject to updates, can be found at:fastmri.med.nyu.edu. The primary goal of fastMRI is to test whether machine learning can aid in the reconstruction of medical images.\"\n\n## A.3 Era5\n\nThe climate dataset was extracted from the ERA5 database (Hersbach et al., 2019), using the processing and splits from Dupont et al. (2021b) (see this reference for details). The resulting dataset contains 12,096 grids of size 46×90, with 8510 training examples, 1166 validation examples and 2420 test examples.\n\n## A.4 Librispeech\n\nThe LibriSpeech dataset (Panayotov et al., 2015) contains several hours of read English Speech recorded at 16kHz. For training, we use the train-clean-100 split containing 28,539 examples and the test-clean split containing 2,620 examples. We train and evaluate on the first 3 seconds of every example, corresponding to 48,000 audio samples per example.\n\n## B Experimental Details B.1 Cifar10\n\nFor all models, we set ω0 = 50 and used an inner learning rate of 1e-2, an outer learning rate of 3e-6 and batch size 64. All models were trained for 500 epochs (400k iterations). We used the following architectures:\n- latent dim: 128, 10 layers of width 512\n- latent dim: 256, 10 layers of width 512\n- latent dim: 384, 10 layers of width 512\n- latent dim: 512, 15 layers of width 512\n- latent dim: 768, 15 layers of width 512\n- latent dim: 1024, 15 layers of width 512 We used 10 inner steps at test time for all models. While we only transmit modulations (which have between 128 and 1024 parameters), the receiver is assumed to have access to the based shared network, which has on the order of millions of parameters. For example, the latent dimension 384 model has 3.8 million parameters, corresponding to a model size of 15.5MB. We note that autoencoder based models typically have a similar size (Dupont et al., 2021a).\n\nINR baselines. We used a latent dimension of 384 for all models. For Chan et al. (2021), we used a ReLU\nMLP with a single hidden layer with 512 units to map the latent vector to the modulations (we found that more layers performed worse). For Mehta et al. (2021), we used the exact model proposed in their paper and set the dimension of the z vector to 384.\n\nCOINbaseline. We manually searched for the best architecture for each bpp level. We followed all other hyperparameters from COIN (Dupont et al., 2021a) and trained for 10k iterations (we found this was enough to converge on CIFAR10). Surprisingly, we found that for CIFAR10 depth did not improve performance and that increasing the width of the layers was better. This may be because the layers are already very small.\n\n- bpp: 3.6, 2 layers of width 12 - bpp: 4.6, 2 layers of width 14 - bpp: 5.8, 2 layers of width 16\n- bpp: 7.1, 2 layers of width 18 - bpp: 8.5, 2 layers of width 20\n- bpp: 10.0, 2 layers of width 22 For the COIN quantization experiments, we used uniform quantization for the weights and biases separately.\n\nWe chose the number of standard deviations k at which to define the quantization range using the formula k = 3 + 3 number of bits−1 15 . I.e. when using 1 bit, we use 3 standard deviations and when using 16 bits we use 6 standard deviations. Indeed, there is a tradeoff between how much data we are cutting off and how finely we can quantize the range. We found that this formula generally gave robust results across different bit values.\n\nAutoencoder baselines. All autoencoder baselines were trained using the CompressAI implementations\n(Bégaint et al., 2020). In order for these models to handle 32×32 images from the CIFAR10 dataset, we modified the architectures both for BMS and CST. Specifically, for BMS we changed the last two convolutional layers in the encoder from kernel size 5, stride 2 convolutions to kernel size 3 stride 1 convolutions, in order to preserve the spatial size (we made similar changes for the transposed convolutions in the decoder). For CST\nwe replaced the first three residual blocks in the image encoder with stride 1 convolutions instead of stride 2, hence preserving the size of the image. Similarly, we replaced the upsampling operations in the decoder with stride 1 upsampling (i.e. dimension preserving convolutions) instead of stride 2. Otherwise, we used the default parameters provided by CompressAI, i.e. for BMS, we used N=128 and M=192 and for CST N=128.\n\nWe trained all models for 500 epochs with a learning rate of 1e-4. We trained models for each of the following λ values: [0.0016, 0.0032, 0.0075, 0.015, 0.03, 0.05, 0.1, 0.15, 0.3, 0.5]. As particularly CST could be unstable to train, we trained two models for each value of λ and kept the best model for the rate distortion plot. Standard image codec baselines. We use three image codec baselines: JPEG (Wallace, 1992), JPEG2000\n(Skodras et al., 2001) and BPG (Bellard, 2014). For each of these, we perform a search over either the quality, quantization level or compression ratio to find the best quality image (in terms of PSNR) at a given bpp level.\n\nWe use the JPEG implementation from Pillow version 8.1.0. We use the OpenJPEG version 2.4.0 implementation of JPEG2000, calling the binary file with opj_compress -i <in filepath> -r <compression ratio> -o <out filepath>.\n\nWe use BPG version 0.9.8, calling the binary file with bpgenc -f 444 -q <quantization level> -o <out filepath> <in filepath>.\n\nEncoding and decoding time. We measure the encoding time of COIN and COIN++ on a 1080Ti GPU\nand the decoding time on a 2080Ti GPU. For COIN we fit a separate neural network for each image in the CIFAR10 test set and report the average encoding time. For COIN++ we similarly fit modulations for each image in the test set and report the average encoding time. For BPG, we measured encoding time on an AMD Ryzen 5 3600 (12) at 3.600GHz with 32GB of RAM.\n\n## B.2 Kodak And Vimeo90K\n\nFor all models, we set ω0 = 50 and used an inner learning rate of 1e-2, an outer learning rate of 1e-6 and batch size 64. All models were trained for 600 epochs (1.4 million iterations). We used the following architectures:\n- latent dim: 16, 10 layers of width 512 - latent dim: 32, 10 layers of width 512 - latent dim: 64, 10 layers of width 512\n- latent dim: 96, 10 layers of width 512\n- latent dim: 128, 10 layers of width 512 We used 32×32 patches from the Vimeo90k dataset to train the model and evaluated on the full Kodak images. We used 3 inner steps for the latent dim 32 and 64 models and 10 inner steps for the latent dim 16, 96 and 128 models as this gave the best results. We quantized all modulations to 5 bits.\n\nWe experimented with different patch sizes (8×8, 16×16, 32×32, 64×64) but found that 32×32 was optimal in our case. Indeed while using larger patches should help compression performance, it also requires more memory and hence a smaller batch size when memory is limited. As we use GPUs with only 11GB of RAM,\nthe batch size we had to use for 64×64 was prohibitively small, leading to unstable training and worse performance.\n\n## B.3 Fastmri\n\nFor all models, we set ω0 = 50 and used an inner learning rate of 1e-2, an outer learning rate of 3e-6 and batch size 16. All models were trained for 32,000 epochs (1.1 million iterations). We used the following architectures:\n- latent dim: 16, 10 layers of width 512\n- latent dim: 32, 10 layers of width 512 - latent dim: 64, 10 layers of width 512 - latent dim: 128, 10 layers of width 512 We trained on 16×16×16 patches and evaluated on the full volumes. We used 10 inner steps at encoding time as this gave the best results. On the rate distortion plot, the first two points are the latent dim 16 model, quantized to 5 and 6 bits, then the latent dim 32 model, quantized to 5 bits, then the latent dim 64 model quantized to 6 bits and finally the latent dim 128 model, quantized to 5 bits and 6 bits.\n\n## B.4 Era5\n\nFor all models, we set ω0 = 50 and used an inner learning rate of 1e-2, an outer learning rate of 3e-6 and batch size 32. All models were trained for 800 epochs (210k iterations). We used the following architectures:\n- latent dim: 4, 10 layers of width 384\n- latent dim: 8, 10 layers of width 384\n- latent dim: 12, 10 layers of width 384 We used 3 inner steps at encoding time as this gave the best results. On the rate distortion plot, the first two points are the latent dim 4 and 8 models quantized to 5 bits, then the latent dim 8 model quantized to 6 and 7 bits and finally the latent dim 12 model quantized to 7 and 8 bits.\n\n## B.5 Librispeech\n\nFor all models, we set ω0 = 50 and used an inner learning rate of 1e-2, an outer learning rate of 1e-6 and batch size 64. We further scaled the coordinates to lie in [−5, 5] as we found this improved performance\n(similar observations were made by Sitzmann et al. (2020b)). All models were trained for 1000 epochs (445k iterations), except the latent dim 256 model which was trained for 2000 epochs (890k iterations). We used the following architectures:\n- latent dim: 128, 10 layers of width 512, patch size 1600 - latent dim: 128, 10 layers of width 512, patch size 800 - latent dim: 128, 10 layers of width 512, patch size 400\n- latent dim: 128, 10 layers of width 512, patch size 200\n- latent dim: 256, 10 layers of width 512, patch size 200 We used 3 inner steps at encoding time. On the rate distortion plot, each point corresponds to one of the above models quantized to 5, 6, 6, 7 and 7 bits respectively.\n\nAudio codec baselines. We use the MP3 implementation from LAME version 3.100, calling the binary file with lame -b <bit rate> <in filepath> <out filepath>.\n\n## C Figure Details\n\nFigure 7b (COIN vs COIN++ quantization). The results in this figure are averaged across the entire CIFAR10 test set. We used COIN and COIN++ models that achieve roughly the same PSNR (30-31dB),\ncorresponding to the bpp 7.1 model for COIN and the latent dim 384 model for COIN++.\n\nTable 1 (Encoding time). The BPG model uses 1.25 bpp (PSNR: 28.7dB), the COIN++ model 1.14 bpp\n(PSNR: 28.9dB) and the COIN model 7.1 bpp (PSNR: 30.7dB).\n\nFigure 9 (Kodak qualitative samples). The COIN++ model used for this plot has a bpp of 0.537 (latent dim 128).\n\nFigure 11 (FastMRI qualitative samples). The COIN++ model used for this plot has a bpp of 0.168 (latent dim 128).\n\nFigure 8 (ERA5 qualitative samples). The COIN++ model used for this plot has a bpp of 0.012 (latent dim 8).\n\nFigure 14 (Qualitative quantization). This figure uses the COIN++ model with a latent dim of 768.\n\n## D Things We Tried That Didn'T Work\n\n- As MAML is very memory intensive, we experimented with first-order approximations. We ran firstorder MAML as described in Finn et al. (2017), but found that this severely hindered performance.\n\nFurther, methods such as REPTILE (Nichol et al., 2018) are not applicable to our problem, as the weights updated in the inner and outer loop are not the same.\n\n- Mehta et al. (2021) use a similar approach to us for fitting INRs (without meta-learning) by using overlapping patches in images. However, we found that using overlapping patches yielded a worse tradeoff between reconstruction accuracy and number of modulations and therefore used non-overlapping patches throughout.\n\n- We experimented with using a deep MLP (and the architecture from Mehta et al. (2021)) for mapping the latent vector to modulations but found that this decreased performance. As MLPs are strictly more expressive than linear mappings, we hypothesize that this is due to optimization issues arising from the meta-learning. If the base network is learned without meta-learning, it is likely a deep MLP\nwould improve performance over a linear mapping.\n\n## E Additional Results E.1 Meta-Learning Curves\n\n\n\nFigure 12: Validation PSNR (3 inner steps) during meta-learning on CIFAR10 (top left), Kodak (top right),\nFastMRI (middle left), ERA5 (middle right) and LibriSpeech (bottom). Note that for LibriSpeech, the legend corresponds to \"latent dimension - patch size\".\n\n## E.2 Cifar10 Ablations\n\n\n\nFigure 13: Effect of number of inner steps on CIFAR10 for a model that has been quantized to 5 bits, with entropy coding. While we use 3 inner steps for meta-learning, performing 10 steps at test time leads to an increase in reconstruction performance of 0.5-1.5dB, while fitting for more than 10 steps generally does not improve performance. Indeed, curves for 10 and 50 steps almost fully overlap.\n\n## E.3 Qualitative Quantization Results\n\n\n\nFigure 14: Qualitative effects of quantization. The top row shows ground truth data from MNIST and CIFAR10, the second row shows the reconstructions from full precision (32 bit) modulations. The subsequent rows show reconstructions when quantizing to various bitwidths. As can be seen, with only 5 bits, reconstructions are nearly perfect. Using as few as 1 or 2 bits, the class of the object is generally recognizable.\n\n## E.4 Encoding Curves\n\n\n\nFigure 15: Encoding curves for COIN and COIN++ on CIFAR10 (full curve on the left, zoomed in version on the right). The COIN model has a bpp of 7.1, while COIN++ has a bpp of 2.2.\nE'2 Additional qualitative results\n\n\n\nFigure 16: Qualitative compression artifacts on ERA5 using the latent dim 8 model with 0.012 bpp (original in first column, COIN++ in second column and residual in third column).\n\n\n\nFigure 17: Qualitative compression artifacts on Kodak (original in first column, COIN++ in second column and residual in third column). The first 4 rows correspond to the model with latent dim 128 (0.537 bpp),\nwhile the bottom two rows correspond to the model with latent dim 64 (0.398 bpp).\n\n\n\nFigure 18: Qualitative compression artifacts on FastMRI using the latent dim 128 model with 0.168 bpp\n(original in first column, COIN++ in second column and residual in third column)."},"reviews":{"kind":"string","value":"Review 1:\nSummary: The authors propose to tackle the problem of data compression with the goal of having one method that is works across moalidites. To do so, the authors build on the recent COIN method that shifts away from the traditional encoder/decoder for data compression. Instead, the idea is to train a network on a proxy task (such as predicting pixel values from pixel indexes) and to use the network's weights as compressed representation.\n\nMore specifically, the authors adress two shortcomings of COIN, namely slow compression time (requiring network training) and lack of shared structure (training network from scratch to compress two images from the same database). Namely, the authors propose to learn a common network that is particularized to each data point using modulations (data-specific learnt vectors) that are injected at specific layers (instance-adaptation like). They reduce training time by leveraging meta-learning (MAML) and quantize and entropy-code the modulations. \n\nThe authors present a wide set of experiments (image, climate and medical data) demonstrating that COIN++ performs well among multimodal methods while stil not on a par with data type-specific compression methods (such as image compression).\n\nStrengths and Weaknesses: Strengths:\n- The authors clearly state the tradeoff between specific methods (such as image compression) versus multimodal compression: the proposed COIN++ method performs well but does not bridge the gap with specific methods, trading off a wide range of applicability for some loss in performance.\n- The method is clearly exposed and well-executed. We thank the authors for providing experimental details in Section 4 and code in the appendix (the reviewer did not execute the code). A lot of experiments acorss modalities support the main claims of the paper.\n- The paper is well written and easy to follow. In particular, the authors leverage previous work and clearly expose ths underlying methods such as MAML.\n\nIn sumamry, I would say that the paper is well written and that the method is well executed. The claims are matched with convincing experimental results. Hence, this work seems to be a good candidate for acceptance.\n\nRequested Changes: No critical changes are required for acceptance. A minor change to strengthen the paper would be as follows:\n- The authors rightfully state that the modulations are easier to compress than the original models. This echoes some work in Federated Learning showing that it's generally easier to compress model deltas (that are client specific, ie the equivalent of the modulations) than models themselves. Adding some references on this could help substantiate this claim (for instance [1]).\n\n\"Reconciling Security and Communication Efficiency in Federated Learning\", Prasad et al.\n\nBroader Impact Concerns: Broader impact statement seems not to be present in the main paper. No particular concerns for this work. \n\n==================================================\n\nReview 2:\nSummary: The authors consider using implicit neural representations (INRs) for compressing data. INRs make use of the observation that data can be represented as a function that maps a location to some value, e.g. in the case of images mapping a pixel to its RGB value. One overfits a neural network (NN) to this function, which becomes a lossy compressed representation of the original data, called the INR. INRs are a promising approach for neural data compression, but encoding a single data point is extremely expensive, as it requires fitting a neural network.\n\nTo reduce the heavy computational burden, the authors suggest meta-learning a base INR, which is way cheaper to adapt to a given data point. The adaptation is performed by learning a shift for the pre-activations of the INR. The compressed representation is thus reduced to these learned shifts, which the authors call \"modulations\". The authors discuss how to further reduce the codelength of the representations by quantizing and entropy coding them.\n\nThe authors test their method by compressing small and large image data, climate data on a sphere, MRI scans and audio data and demonstrate large gains compared to previous INR-based methods. Their method shows promising performance compared to state-of-the-art (SOTA) methods for each data type.\n\nStrengths and Weaknesses: ## Strengths\nI think the core idea of using meta-learning to reduce the computational burden of using INRs for compression is sound, and demonstrating that it works reasonably well without much tuning is very promising and a good contribution. Moreover, the paper contains a lot of empirical verification of the idea and has a reasonable amount of ablation study, making it quite appealing.\n\n## Weaknesses\nUnfortunately, the presentation of the ideas is poor and confusing, with some dubious claims.\n - I don't think the authors' argument that INR-based methods are more easily applicable to different data types than other neural methods is backed-up in any way. E.g. the authors claim that: \"These strong results highlight the versatility of the COIN++ approach: unlike traditional codecs and autoencoder based methods (which would require spherical convolutions for the encoder), we can easily apply our method to a wide range of data modalities, including data lying on a manifold\". I appreciate that INRs can be applied more easily in the sense that they don't require specialized layers (e.g. spherical convolutions), but I don't see how this is an advantage. In every experiment, the authors show that without fine-tuning the INR (e.g. its architecture) for a particular data type, their method is outperformed by any of the more specialized compression methods. This surely demonstrates that at least there is good potential in using specialized layers for INRs. In case the authors would like to retain this claim in their paper, they need to show that their method outperforms at least some sort of simple neural baselines on less conventional data types (climate data, MRI). They only compare against traditional codecs that were not designed to compress these data types. Otherwise, this claim should be removed from the paper.\n - I found Figure 1 misleading. It makes it seem like the authors are proposing to simultaneously meta-learn across different data types, which I believe is not the case. If this is a misunderstanding on my part, then the authors need to clarify the writing because there is no further implication in the paper that they would be performing the meta-learning for all data types simultaneously.\n - I couldn't parse Figure 2. The component with the latent modulation (green) is not connected to the rest of the figure. Also, there are two blue components in the figure, so I'm not sure what is mapped to what. Please redesign this figure or remove it.\n - (✓) The left plot of Figure 4 is confusing. I believe that $\\theta^*$ is the only quantity that is ever used during test time, so it is not clear what the point of depicting $\\theta$ is. I believe the plane shown is the parameter space, but this is not mentioned anywhere. Furthermore, connecting the $\\phi$ points to $\\theta^*$ is confusing. In general, I don't see how this plot is supposed to aid in understanding the ideas in the paper. Could the authors clarify this for me please?\n - The label sizes in Figures 6-10 need to be increased as they are currently hard to read.\n - (✓) What are the plots in Figure 5 depicting? The performance during training or test time?\n - (✓) In Figure 7d) the comparison appears to be tenuous, as I believe the compression is done using a GPU for the neural methods, while BPG only requires a CPU. \n - (✓) The symbol $n$ appears to be overloaded: it represents both the input size (e.g. in Eq 1 & 3) and the width of the modulated MLP in the discussion following Eq 3.\n - In the paragraph following Eq 2, the authors claim: \"Secondly, as each datapoint $d$ is fitted with a separate neural network $f_\\theta$, there is no information shared across datapoints. This is clearly suboptimal: natural images for example share a lot of common structure that does not need to be repeatedly stored for each individual image\". I am not quite sure what the authors mean by suboptimal, but this claim is most likely false. For single-shot image compression, optimizing the INRs for individual data points is the optimal thing to do, at least in theory. Amortizing the representation across multiple images should only be optimal when considering compressing multiple images simultaneously.\n - (✓) The section describing how $\\theta$ is learned using meta-learning is confusing. It appears that the authors use the learning rules in Eqs 7 and 8, but I believe they only use them in Eqs 9 and 10.\n - (✓) The \"Entropy coding\" paragraph in Section 4.2 is entirely dedicated to describing a plot from the paper's appendix. This figure should be moved to the main body of the paper. Otherwise, the paragraph does not have a place.\n\n## Questions:\n - (✓) How much does patch size matter for compressing large-scale images? \n\nRequested Changes: For most necessary changes, see the \"Weaknesses\" section above. The authors would need to address all concerns I outline there for me to recommend acceptance.\n\nAdditionally:\n - MAML, as an abbreviation, is undefined.\n - Several citations are badly bracketed.\n\n\nBroader Impact Concerns: I don't believe that a Broader Impact Statement is required.\n\n==================================================\n\nReview 3:\nSummary: This paper introduces COIN++, an improvement of the methods introduced by COIN.\nCOIN++ is a implicit neural representation method, where individual data points are encoded in neural network weights (i.e., via overfitting the data).\nBoth COIN and COIN++ use sinusoidal layers (SIRENS).\nCOIN++ improves COIN’s encoding/training time by adding meta-learning, it improves COIN’s storage redundancy by factoring the COIN network into a shared “stem” and a set of modulations, and it improves COIN’s lack of quantization/entropy coding by implementing them.\nCOIN++ is evaluated on a range of data modalities, covering natural images, audio, MRI scans, and climate data (the latter two being 3D).\nCOIN++ is much faster at encoding than COIN and results in competitive (though not optimal) compression performance across experiments, demonstrating that a single neural compression method can have general and practical utility.\n\nStrengths and Weaknesses: **Strengths:**\n* Motivation is clear.\n* I found the paper to be easy to follow in terms of writing and content.\n* The paper outlines a general method for compressing many modalities of data.\n* Experiments are within scope of the claims.\n\n**Weaknesses:**\n* I found some of the captions on the figures to be lacking a conclusion or analysis.\n* References to encoding speed are somewhat vague. For example, the number of iterations is used as a proxy for speed, but the time per iteration or any constant-time preprocessing (e.g., an up-front meta-learning cost) are not directly addressed. Decoding time is not addressed.\n* Questions about the constants in space usage (e.g., the shared stem) are also not clear. It’s not clear what weights are considered part of the bit stream. For example, if the shared parts are not part of the bit stream due to having a single “fixed” shared model, then it’s not clear if this “fixed” model (along with the modulations) will degrade in performance once the overfitting capacity is stretched. In that case, either a new model is re-transmitted (incurring bitstream cost) or distortion increases.’\n* The primary metric is PSNR but there are other metrics, such as SSIM that could have also been used.\n\nRequested Changes: * [Critical] The information density of some of the captions seems unnecessarily low. Make the captions summarize key observations, such that the figure is more self-contained. I think, for example, mentions of the appendix are not necessary in the caption, but conclusions should be in the caption. Figure 5, in particular, has the conclusion of \"improving PSNR by 2dB with the same number of parameters\" in the text while the caption is mostly restating the labels on the plots. Similarly, the rate-distortion plots are captioned as rate-distortion plots without a conclusion.\n* [Critical] From a presentation point of view, it should be clear what the space usage calculation is. Maybe I missed something, but the assumptions about the setting under study are not apparent e.g., how shared model weights are considered.\n* [Would Strengthen] Directly measuring and ablating factors of encoding/decoding speeds (i.e., wallclock time and not iteration count) and calculations involving space usage. Figure 7d is a good start but it's not a complete characterization.\n* [Would Strengthen] Adding additional metrics e.g., SSIM to evaluations. PSNR is strongly related to the (MSE) objective function used and is not a typical end-goal of compression in itself.\n\nBroader Impact Concerns: None.\n\n==================================================\n\nMetareview:\nRecommendation: Accept with minor revision\n\nComment: All the reviewers and myself agree that the idea is sound: applying meta learning to solving the slow compression issue of COIN, also sharing part of the architectures across data can allow better representation learning for the base network.\n\nHowever there are also some suggestions in the reviews that I suggest the authors to include them in final revision:\n1. Clarify the claim better in terms of \"implicit neural representation based compression methods are easier to apply to multiple types of data\";\n2. Include a discussion of codecs on manifolds to better situate the climate experiments in the literature;\n\n==================================================\n\n"}}},{"rowIdx":92,"cells":{"paper":{"kind":"string","value":"# Multi-Dimensional Concept Discovery (Mcd): A Unifying Framework With Completeness Guarantees\n\nJohanna Vielhaben **johanna.vielhaben@hhi.fraunhofer.de**\nExplainable Artificial Intelligence Group Fraunhofer Heinrich-Hertz-Institute Stefan Blüecher **bluecher@tu-berlin.de**\nMachine Learning Group TU Berlin Nils Strodthoff *nils.strodthoff@uol.de* Division AI4Health Oldenburg University Reviewed on OpenReview: *https: // openreview. net/ forum? id= KxBQPz7HKh*\n\n## Abstract\n\nThe completeness axiom renders the explanation of a post-hoc eXplainable AI (XAI) method only locally faithful to the model, i.e. for a single decision. For the trustworthy application of XAI, in particular for high-stake decisions, a more global model understanding is required.\n\nTo this end, concept-based methods have been proposed, which are however not guaranteed to be bound to the actual model reasoning. To circumvent this problem, we propose Multi-dimensional Concept Discovery (MCD) as an extension of previous approaches that fulfills a completeness relation on the level of concepts. Our method starts from general linear subspaces as concepts and does neither require reinforcing concept interpretability nor re-training of model parts. We propose sparse subspace clustering to discover improved concepts and fully leverage the potential of multi-dimensional subspaces. MCD offers two complementary analysis tools for concepts in input space: (1) concept activation maps, that show where a concept is expressed within a sample, allowing for concept characterization through prototypical samples, and (2) concept relevance heatmaps, that decompose the model decision into concept contributions. Both tools together enable a detailed global understanding of the model reasoning, which is guaranteed to relate to the model via a completeness relation. Thus, MCD paves the way towards more trustworthy concept-based XAI. We empirically demonstrate the superiority of MCD against more constrained concept definitions.\n\n## 1 Introduction\n\nExplainable AI (XAI) allows to peek insight the black box of inherently complex deep learning models. *Local* interpretability methods are particular valuable, as they measure attributions for an individual instance, which are easily comprehensible for any kind of end-users, see (Covert et al., 2021; Lundberg & Lee, 2017; Montavon et al., 2018; Samek et al., 2021) for reviews. For example, local methods make a prediction interpretable on the level of single images or individual bank customers for an image or credit risk classifier, respectively. Importantly, the commonly employed *completeness axiom* **(attributions sum up to the model**\nprediction) ensures a meaningful interpretation of attributions (Lundberg & Lee, 2017; Sundararajan et al.,\n2017). However, to actually comprehend the model reasoning we require a *global* **model understanding,**\nwhich reliably explains the model behavior across multiple instances (e.g. a group of female vs. male bank customers). We stress that it is not viable to require an end-user to aggregate local attributions into common\n\n\n\nFigure 1: We strive for the most general decomposition of the hidden feature space, spanned by the neurons c1, c2, c3**, into linear structures that form the concepts** C\ni**. The most constrained approach is to identify**\nconcepts with single neurons (D1), i.e. directions in feature space aligned with canonical basis vectors. If one allows for arbitrary rotations of the concept directions, one arrives at D2. Leaving aside the orthogonality constraint, D3 allows concepts to form arbitrary directions in feature space. Finally, allowing concepts to form multi-dimensional subspaces, we arrive at the most general approach D4. Previous concept-based methods are based on D1, D2 and D3. We choose the most general approach D4, to discover concepts that are *faithful*.\n\nmodel features (concepts). Such a procedure is prone to human confirmation bias and it is not clear how the imagined concepts align with the actual model reasoning. This urges for novel *local* and *concept-based* interpretability methods, which allow to understand shared model structures (used across multiple samples)\nfor an individual instance. This idea was first formalized by Kim et al. (2018) and further developed by ACE (Ghorbani et al., 2019) and its successors (Yeh et al., 2020; Zhang et al., 2021). Crucially, our work re-introduces *completeness* **within the context of concept-based explanations. Thereby, concepts obtained**\nwithin our multi-dimensional concept discovery (MCD) scheme are *locally* and *globally* **interpretable in terms**\nof a well-defined completeness decomposition. We outline the benefits of MCD in the following paragraphs.\n\nConcepts as multi-dimensional subspaces **Indisputably, concept discovery in neural networks is inherently linked to structures in the activations of intermediate feature layers. In Figure 1, we illustrate different**\napproaches to decompose the hidden feature space (union of all possible activations) into meaningful concepts, which are mathematically formalized as linear structures. As an illustrative example, we consider the activations of the last convolutional layer just before average pooling and a linear classification head. In this case, the model part remaining after the intermediate feature layer can only exploit linearly separable concepts, hence justifying the linearity assumption here. The most constrained definition (left most panel, D1) is to directly identify concepts with canonical basis vectors of the feature space (Bau et al., 2017). In our example, D1 identifies each convolutional channel with a concept. A slightly more general definition is to allow concepts to lie on orthogonal directions other than the unit axes in feature space (D2). In our example, this means, concepts are formed by orthogonal linear combinations of convolutional channels. Such a concept decomposition can be obtained via a principal component analysis (PCA) of the feature space (Zhang et al., 2021). Going one step further, we disregard the orthogonality constraint and allow arbitrary directions in feature space (D3) (Ghorbani et al., 2019; Kim et al., 2018; Yeh et al., 2020; Zhang et al.,\n2021). Thereby, we can characterize related concepts which are linearly independent but not orthogonal.\n\nThis is sensible because in general, the model has no mechanism that enforces orthogonality of concepts (for example different parts of an animal). Allowing for arbitrary multi-dimensional subspaces unfolds the most general definition of a linear decomposition (D4). Coming back to the CNN example, D4 allows a concept to lie on a hyperplane spanned by multiple directions of the convolutional channels. We argue, that this general approach enables the most *faithful* **concepts among D1-D4, as it allows to capture any meaningful**\nlinear structure within the hidden feature layer (*benefit 1* ).\n\nMulti-dimensionality ensures concise explanations **Concepts strive to organize the information about**\nthe global model reasoning in a concise manner. To this end, we want to cover the relevant feature space with only a few concepts and avoid fragmentation into a large number of low/one-dimensional subspaces. As a first step, we propose a concept completeness score, which measures the fraction of model prediction jointly covered by all concepts. We find that it requires significantly fewer multi-dimensional MCD concepts to reach a specified level of completeness as compared to more constrained concept definitions (D1-D3), i.e.,\nMCD provides more concise explanations (*benefit 2* ). Re-establishing completeness for concepts **To define concept relevances, in Section 2.3, we uniquely**\ndecompose the hidden activations in conjunction with the model prediction into concept parts. To this end, we restrict to a high-level feature layer which is only succeeded by linear operations (e.g. a linear classification head with global pooling). Then, the concept relevances follow a completeness relation (*benefit* 3 **), i.e., summing all concept relevances equals the final prediction. Thus, we restore the often-desired**\ncompleteness property mentioned above for concept explanations. We stress, that our concept relevances follow directly from the decomposition into concept parts and do not invoke any additional XAI method nor retraining model parts. Phrased differently, MCD can completely capture the model reasoning solely in terms of linear operations on concept subspaces.\n\nIn summary, MCD is a consistent framework to discover *faithful* **concepts, which are guaranteed to rely**\non the actual model reasoning via the completeness relation. Our framework provides several possibilities to investigate the discovered reasoning structure in input space. Thus, we see the main utility of MCD\nin the domain of model understanding and certification. Concepts provide insights into model behavior that generalize across samples and are therefore a valuable tool for systematic investigations of spurious correlations (model biases) (Lapuschkin et al., 2019; Palatnik de Sousa et al., 2021; Weber et al., 2021), as well as for scientific discovery (Blücher et al., 2020; Hägele et al., 2020; McGrath et al., 2022; Šarčević et al.,\n2022), where the model serves as a proxy for the unknown relationships in the data.\n\n## 2 Multi-Dimensional Concept Discovery (Mcd)\n\nWe organize this methodological section into three parts: First, we introduce our novel concept definition in Section 2.1. Second, we describe practical concept discovery procedures that align with this definition in Section 2.2. Third, we introduce a concept decomposition and discuss how to construct local and global concept importance that fulfill a concept completeness relation in Section 2.3.\n\nFig. 2 presents a schematic summary of our MCD framework, which we shortly summarize here. During the training phase, the approach discovers multi-dimensional subspaces (concepts) in the hidden feature space of model, which we mathematically capture as concept bases. These concept bases are obtained from (i) clustering feature vectors (using some particular algorithm) and (ii) characterizing clusters by their dominant principal directions (lower left panel). During the testing phase, we can compare new feature vectors (e.g. from a new test sample) with these concept bases and thereby analyze/characterize all concepts\n(right panel). Here, we provide two complementary tools: *concept activation maps***, which highlight strongly** expressed regions of the concept in input space and *concept relevance maps***, which show how indicative a**\nparticular concept is for the class predictions.\n\n## 2.1 Concept Definition\n\nConcepts are inherently tied to the hidden representations of intermediate feature layers. For our concept definition, we split the model f into two parts, f = g ◦ h, where h **is the mapping to a hidden feature layer,**\nwhich is mapped to the prediction by g**. Our definition then relies on hidden representations** h(α) ∈ R\nH×W×F\nof input samples α (height H, width W and number of features F**, see upper left panel in Figure 2).**\n\n\n\n$$(1)$$\n\nFigure 2: Schematic illustration of the MCD framework for concept discovery. The **upper left panel**\nillustrates how the model is split into a representation and prediction mapping. Feature vectors are extracted from the representation mapping of a sample. The lower left panel **illustrates the concept discovery**\nmethodology of MCD (Section 2.2). First, randomly choose and cluster a set of feature vectors {φ} **from**\na selection of samples (using any clustering algorithm). Second, construct subspace bases for all clusters C\nl **via PCA (intrinsic dimension** d l). The upper right panel corresponds to the construction of *concept* activation maps and the lower right panel shows the construction of *concept relevance heatmaps***, both laid**\nout in Section 2.3.\n\nWe spatially deconstruct the feature maps h(α**) and obtain a feature vector**1 φ α xy ∈ R\nF **for each location**\n(*x, y*) ∈ {1*, . . . , H*} × {1*, . . . , W*}. We now strive to identify concepts as (linear) structures in this F**dimensional feature space and pose no additional restrictions (one-dimensionality and/or orthogonality) on**\nthe structure of the subspaces.\n\nDefinition 1. *We define a concept* C\nl *as a* d l dimensional linear subspace in the F**-dimensional feature**\nspace, spanned by the basis vectors c l j\n,\n\n$$C^{l}=s p a n\\left(\\{{\\mathbf{c}}_{j}^{l}|j=1,\\ldots,d^{l}\\}\\right)\\,.$$\nj|j = 1*, . . . , d*l\t . (1)\nIn particular, the dimensionality d lcan vary among the concepts l = 1*, . . . , n*c**. We denote the number**\nof concepts as nc **and present a constructive way to determine it in Section 2.3. assume that the concept**\nsubspaces are pairwise disjoint. In Figure 1, we illustrate the linear structures that concepts could possibly form in hidden feature space: from single directions (D1-D3) to linear subspaces (D4). The MCD concept definition above corresponds to D4, which is the most general linear structure, i.e., arbitrarily orientated multi-dimensional linear subspaces. Note, that exploring even more general, non-linear concept structures, such as sub-manifolds in feature space, is an interesting idea for layers where non-linear operations follow. For these, linear multi-dimensional subspaces represent an improvement over the previous, more constrained linear concept definitions in terms of faithfulness. However, for a last hidden layer that is only followed by linear classification head, which we specialize to in Section 2.3, linear subspaces are the most general structure that can be separated, and thus form a faithful model concept definition.\n\n1**Vectors are denoted lower-case bold (**φ ∈ RF ).\n\n## 2.2 Concept Discovery\n\nTypically, concept discovery, i.e., obtaining concepts as defined by Equation (1), can be subdivided into two steps: (i) cluster a user-defined set of feature vectors {φ α x,y} **into clusters** C\n1*, . . . ,* C\nnc **and (ii) identify a**\nrepresentative basis C\nl = {c l j |j = 1 *. . . d*l} **for each concept cluster** C\nl**(lower left panel in Figure 2).**\n(i) Clustering feature vectors **In principle, any clustering method can be considered to discover concept**\nclusters in feature space. This includes well-established baselines such as k-means clustering or PCA. Both have previously been proposed in (Zhang et al., 2021) to identify one-dimensional subspaces. However, kmeans does not incorporate any information about the final objective to identify linear subspaces as opposed to general clusters and PCA is restricted to orthogonal, one-dimensional subspaces. We, therefore, propose a dedicated approach for this particular purpose to discover multi-dimensional linear subspaces and draw on the rich body of literature on *sparse subspace clustering* **(SSC) (You et al., 2016a; Soltanolkotabi & Candes,**\n2012; You et al., 2016b; Elhamifar & Vidal, 2013). SSC clusters datapoints that lie on a union of separate low-dimensional subspaces embedded in a high-dimensional space. It is based on the idea, that a feature vector φi **can be expressed as a linear combination of other feature vectors from the same cluster/subspace** C\nl.\n\nAs nicely laid out in (Elhamifar & Vidal, 2013), SSC is ideally suited to identify clusters of linear subspaces and provides a number of advantages over standard clustering algorithms, which are directly applied to the data: SSC does not rely on the spatial proximity of the data, it can be implemented robustly against noise and outliers and does not require specifying the cluster dimensionalities in advance.\n\nWe start out with a user-specified set of samples S **for which we aim to discover concepts. The sample** selection S **is unrestricted: the user can decide on class-specific samples/concepts or use all training samples to**\nobtain completely class-unspecific concepts. Then, the first step of SSC is to find a sparse self-representation matrix that expresses each φi**in terms of a minimal number of other feature vectors** {φ α x,y}**. Second, spectral**\nclustering is applied to the self-representation to obtain clusters C\nl**. We provide technical details on the**\nparticular subspace algorithm in Appendix A.\n\n(ii) Constructing concept bases **We have now identified clusters** C\n1*, . . . ,* C\nnc\n**, which contain all feature**\nvectors φ α x,y **from the training set. Next, we want to obtain general concepts and become independent from**\nthe original specific cluster members. To this end, we aim to identify a basis C\nl**that robustly covers**\nthe cluster C\nl**. By construction, all** φi ∈ Cl**lie within a linear subspace, hence a linear tool like principal**\ncomponent analysis (PCA) constructs an accurate basis C\nl**for the cluster** C\nl**. We determine the intrinsic**\ndimension d l **of the subspace using a heuristic proposed by Fukunaga & Olsen (1971) and implemented by**\nBac et al. (2021). The PCA components up to the intrinsic dimension d l**then serve as basis vectors** c l j for the subspace C\nl**. Given two subspaces, we can quantify their relation in terms of their Grassmann distance,**\nsee Appendix B.\n\n## 2.3 Concept Decomposition\n\nPreviously, we have laid out how to discover an expressive set of concepts C\nl**. Next, we discuss how new**\nfeatures vectors {φ β x,y} (obtained from a test set sample β**) and the weights of the final linear classifier layer**\ncan be analyzed via a decomposition into concept contributions. To this end, we propose *concept activation* maps, *concept relevance heatmaps* and a *global concept relevance score***. These are complementary tools that**\nform the final concept explanation, informing about the overall meaning of a concept (*concept activation* maps) and its impact on the classification of a specific sample (*concept relevance heatmaps***) or on a global** level (*global concept relevance score*).\n\nTo ensure that the union of all concepts spans the entire feature space, we define C\n⊥ **to be the orthogonal**\ncomplement of the subspace spanned by all concepts, i.e., C\n⊥ **= span(**C\n1*, . . . , C*nc )\n⊥**. To simplify the**\nnotation, we identify C\nnc+1 ≡ C\n⊥**. In the following, we assume that the concept subspaces are pairwise**\ndisjoint. 2\n\n2**This assumption was never violated in our experiments. If necessary, this could be enforced by removing the intersection**\nbetween the subspaces from both and considering it as a separate concept.\nConcept activation maps **quantify the activation of a chosen concept at a certain spatial location in the**\ninput space of a sample β**. For this purpose, we decompose the feature vectors** {φ β x,y} **into its unique concept**\ncontributions. Since the union of all concepts (including the orthogonal complement) forms a basis for the entire feature space, we can uniquely decompose any feature vector φ as\n\n$$\\mathbf{\\phi}=\\sum_{l=1}^{n_{c}+1}\\sum_{i=1}^{d}\\varphi_{i}^{l}\\mathbf{c}_{i}^{l}\\equiv\\sum_{l=1}^{n_{c}+1}\\mathbf{\\phi}^{l}\\,,\\tag{2}$$\n\nwhere ϕ l i are the components of φ **in the given basis. Now, one can interpret** |φ l|2 **as a measure for the extent**\nto which a certain concept is expressed in the given feature vector. Performing this step for every feature vector within a sample β**, i.e.,** φ β x,y =Pnc+1 l=1 φ β,l x,y, leads to a *concept activation map* |φ β,l x,y|2 **whose spatial**\ndimensions match those of the feature layer. For a fixed sample β, we normalize φ **such that the maximum**\nlength across all elements of the feature layer is 1, i.e., we divide the vectors elementwise by maxx,y|φ β x,y|2.\n\nFor CNNs, we follow the example of Selvaraju et al. (2020) and compute the corresponding concept activation map in input space by bilinear upsampling in the spatial dimensions. Our concept activation maps extend the concept visualization of Zhang et al. (2021) to multi-dimensional concepts (upper right panel in Figure 2).\n\nFor the final explanation, we also use them to characterize a concept in terms of prototypical examples. To this end, for each concept l**, we sort test set samples by the maximum activation max**xy|φ β,l x,y| **and choose the**\ntop-k samples as *concept prototypes*.\n\nWe stress, that our methodology is applicable beyond CNNs. In particular, one can decompose feature representations of any model based on MCD. However, the prerequisite for showing concept (activation) maps in input space is the locality of the trained model, i.e., the ability to associate locations in feature and input space. Whereas this locality is built in as an inductive bias into convolutional architectures, it also emerges for vision transformer models during training, as manifested for example in localized attention maps (Caron et al., 2021). To substantiate these claims, we show the first concept-based explanations for a vision transformer model in Section 4. As a final remark, we emphasize that concept activation maps can be evaluated for any model, including self-supervised pretrained models before finetuning with a classifiaction head, and any layer in the model. They reveal the learned structures in feature space and are only constrained by the restriction to linear subspaces instead of more general non-linear structures.\n\nConcept relevance heatmaps and completeness relation **As a general requirement, any concept-based** XAI method should quantify the *relevance* **of a concept in terms of its impact on the classification decision.**\nTo this end, we specialize to the last hidden layer, which is only followed by linear operations (e.g., mean pooling and a linear classification head). We discuss the broad class of models to which this applies in the last paragraph of this section and empirically in Section 4.\n\nFor a given class, the weight vector w ∈ ❘F linearly connects the final F**-dimensional feature space with the**\nscalar class prediction. First, we consider the feature vector after pooling φ ≡ φ β =1 WH\nPx,y φ β xy (φ β ∈ ❘F )\nin this very layer (see Figure 2 lower right panel). Now, we are interested in a *local* **(per-sample) concept**\nrelevance. For this, we can decompose the class logit under consideration, φ ·w + b, up to the bias term b**, as**\n\n$$\\mathbf{\\phi}\\cdot\\mathbf{w}=\\sum_{l=1}^{n_{c}+1}\\mathbf{\\phi}^{l}\\cdot\\mathbf{w}\\equiv\\sum_{l=1}^{n_{c}+1}r^{l}\\,.\\tag{1}$$\n$$(3)$$\n\nThe decomposition above defines a *local concept relevance* r l = φ l· w**. Aggregating relevances** r l**from**\nall concepts recovers the class logit prediction (up to the bias term), and thus, Equation (3) defines a completeness relation.\n\n3 4 Second, we apply Equation (3) to the feature vectors φ β xy **before pooling. This leads to a relevance heatmap**\nr l xy = φ β,l xy · w **that has the same spatial dimension as the feature layer. Importantly,** r l xy **reduces to** r l **after**\n\n3\n**In the special case of one-dimensional concepts,** r l**reduces to the local concept relevance in (Zhang et al., 2021).**\n4**We briefly comment on the remaining commonly desired Shapley axioms Lundberg & Lee (2017). The local concept**\nrelevance trivially fulfills them since it is built on a linear additive model. Formally, the hidden activation φβ **of a given sample**\nβ **are segmented into concept contributions/unique features** φ l β\n. Thus, the value function corresponding to the underlying Shapley values is given by vβ(S) = Pl∈S\nφ l β\n· w **(linear in** φ l**) for** S ⊆ {1*, . . . , n*c + 1}.\nspatial pooling. As for the concept activation maps, we use spatial upsampling to map r l xy **back to the input**\nspace and obtain *concept relevance heatmaps***. Since upsampling preserves the completeness relation, these**\ndecompose the *local relevance maps***, commonly referred to as class activation maps (CAMs) (Zhou et al.,**\n2016), rx,y =1 WH φ β xy · w **into concept contributions.**\nGlobal relevance and completeness score Next, we establish a *global* **(model-wide) concept relevance**\nscore, which measures the extend to which the concepts explain/cover the overall prediction strategy of the model. Recall, that all c l j defined above represent a basis for the feature space ❘F **. Hence, we can directly**\ndecompose the weight vector w **into (analogously to Equation (2))**\n\n$$\\mathbf{w}=\\sum_{l=1}^{n_{c}+1}\\sum_{i=1}^{d^{l}}w_{i}^{l}\\mathbf{c}_{i}^{l}\\equiv\\sum_{l=1}^{n_{c}+1}\\mathbf{w}^{l}\\,,\\tag{1}$$\n$$\\quad(4)$$\n$$\\left(5\\right)$$\n$$(6)$$\n\nwhere w l =Pd l i=1 w l i c l i and by construction, w l· w\n⊥ = 0 for l = 1*, . . . n*c**. In this case, we have**\n\n$$|\\mathbf{w}|^{2}=|\\mathbf{w}^{\\perp}|^{2}+|\\sum_{l=1}^{n_{c}}\\mathbf{w}^{l}|^{2}=\\sum_{l=1}^{n_{c}+1}|\\mathbf{w}^{l}|^{2}+\\sum_{l,k=1,l\\neq k}^{n_{c}}|\\mathbf{w}^{l}||\\mathbf{w}^{k}|\\cos(\\angle(\\mathbf{w}^{l},\\mathbf{w}^{k}))$$\n\nThe first equality allows us to define\n\n$$\\eta(\\{C^{l}\\})=1-|\\mathbf{w}^{\\perp}|^{2}/|\\mathbf{w}|^{2}$$\n2(6)\nas a *completeness score* (fraction of w **which is explained by all concepts** {C\n1*, . . . , C*nc }**) with respect to a**\ngiven class. To the best of our knowledge, we are the first to introduce a concept completeness score directly based on model parameters. Previous work (Yeh et al., 2020) defined a related measure based on model accuracy. Later, we will fix η to define the number of concepts nc**. Note, that for an orthonormal basis (e.g.,**\nMCD-SSC-ortho, see below) the second term in Equation (5) (cosine) disappears. Then |w l|/|w| **can be**\ndirectly interpreted as (global) concept relevances, which sum up to the previous completeness score over all concepts. Further, the angles in Equation (5) are lower- and upper-bounded by the corresponding minimal or maximal principal angles5 **between the two corresponding subspaces, i.e.,** θ kl min ≡ minmθ kl m ≤ ∠(w k, wl) ≤\nmaxmθ kl m ≡ θ kl max**. This means we can lower- and upper-bound** |w| 2 by\n\n$$\\sum_{l}^{n_{c}+1}|\\mathbf{w}^{l}|^{2}+\\sum_{l,k=1,l\\neq k}^{n_{c}}|\\mathbf{w}^{l}||\\mathbf{w}^{k}|\\cos(\\theta_{\\max}^{lk})\\leq|\\mathbf{w}|^{2}\\leq\\sum_{l}^{n_{c}+1}|\\mathbf{w}^{l}|^{2}+\\sum_{l,k=1,l\\neq k}^{n_{c}}|\\mathbf{w}^{l}||\\mathbf{w}^{k}|\\cos(\\theta_{\\min}^{lk})\\,.\\tag{7}$$\n\nObviously, the lower and upper bound coincide in the case of orthogonal subspaces. This implies, that the |w l| **are also informative in the non-orthogonal case, provided the principal angles between the different**\nsubspaces are given. This highlights the intricate connection between (global) relevances and the geometry in feature space, i.e., the relative orientation of the concept spaces (specified via principal angles between pairs).\n\nFinally, we briefly comment on the applicability of our approach for local and global concept relevances via Equation (3) and Equation (4). In the form described above it can be used for any model with a linear layer as final layer, potentially preceded by a global pooling layer, if one aims to spatially resolve the relevances instead of considering only pooled feature vectors. This latter category covers a broad range of modern CNN\narchitectures such as ResNets, Inception-based models but also vision transformers, that do not base their prediction on a CLS token, such as Swin transformers (Liu et al., 2021). We envision, that our approach is even applicable, in approximate form, to other feature layers apart from the final hidden layer if one locally approximates the remainder of the model by a linear model, similarly as it is done by Ribeiro et al. (2016)\nor by Selvaraju et al. (2020) to generalize (Zhou et al., 2016).\n\n## 2.4 Alternative Mcd Variants\n\nIn Section 2.2, we describe our algorithmic choices for the two steps of concept discovery, namely SSC for clustering of feature vectors and PCA for basis construction. As a consequence of the modularity of the 5**A formal definition of principal angles is given in Appendix B.**\n7 MCD framework, we can easily define alternative variants of MCD, that also serve for an ablation study later. To differentiate the original MCD flavor described in Section 2.2, we name it *MCD-SSC***. Replacing**\nSSC with other clustering methods gives rise to the following two MCD variants:\n- *MCD-kmeans* **We consider k-means clustering directly applied to the features. Like SSC, it leads to**\nmulti-dimensional and in general non-orthogonal subspaces. However, the clustering algorithm does not include any information about the linear subspaces as desired clustering target.\n\n- *ICE/MCD-PCA* **We consider PCA applied to the feature vectors directly. This corresponds to**\nthe concept discovery algorithm considered by ICE (Zhang et al., 2021). Note, that this approach already encompasses the basis identification step and directly leads to one-dimensional, orthogonal subspaces by construction.\n\nMCD-SSC does not assume that two different subspaces C\nl and C\nm **are orthogonal, as there is no mechanism**\nthat promotes this during model training. Still, concept orthogonality could be enforced through the use of dedicated orthogonal subspace clustering methods (Rahmani & Atia, 2017a), however, at the potential cost of slightly sub-optimal subspace clusters (Rahmani & Atia, 2017b). Alternatively, this could be implemented by sequentially rotating each identified subspace into the orthogonal complement of its predecessors. The latter leads to the last MCD flavor:\n- *MCD-SSC-orth* **We construct orthogonal subspaces from those discovered by MCD-SSC in an iterative fashion. Starting with an empty set, we explore the effect of adding one of the subspaces**\non the completeness defined in Equation (6) and choose the one that leads to the largest increase.\n\nIteratively, we consider adding another subspace rotated into the orthogonal complement of the span of the subspaces in the set so far, again selecting the candidate that leads to the largest completeness increase.\n\nLater, we find evidence that orthogonal are less faithful to the model than those that allow for arbitrary rotation.\n\n## 3 Related Work\n\nACE (Ghorbani et al., 2019) uses a superpixel segmentation algorithm and k-means clustering to identify class-specific concept candidates for TCAV (Kim et al., 2018). The concept discovery scheme of ACE\nhas several shortcomings: The segmentation into candidate concept patches is model-independent and thus, segments are not necessarily meaningful as perceived by the model. To enable clustering of intermediate CNN\nactivations, segments are resized and mean padded to the original input shape. This leads to artificial, offmanifold samples with potentially distorted aspect ratios and discards the overall scale information. Finally, ACE relies on multiple heuristics to discard segments/clusters both before and after k-means clustering. In contrast, MCD is coherently based on hidden model representations without relying on additional pre- or post-processing. Similar limitations apply to methods that rely on ACE-discovered labeled concepts, like\n(Li et al., 2021), which uses Shapley values for concept importance, and (Wu et al., 2020), which occludes particular neurons for neuron-wise relevances and transforms them into concept importances via concept classification. Recently, Crabbé & van der Schaar (2022) proposed a generalization of TCAV by invoking the kernel trick, which generalizes the concept definition towards non-linear structures. However, unlike MCD,\nit does not allow quantifying the relevance of a concept towards the model prediction and can only verify predefined concepts instead of discovering them.\n\nICE (Zhang et al., 2021) defines concepts as directions in feature space. Technically, this is achieved via dimensionality reduction techniques applied to concatenated flattened feature maps. ICE measures the importance of its class-wise concepts using TCAV. Interestingly, ICE introduces the notion of a concept weight, which is analogous to our concept relevances on the logit layer. However, they do not consider spatially resolved concept relevance heatmaps and only address the special case of single-dimensional subspaces. Given these restrictions, ICE can be seen as a special realization of the MCD framework, which uses dimensionality reduction methods like PCA as clustering algorithms. Other methods learn concept vectors and a mapping to feature space either for all classes simultaneously (ConceptSHAP (Yeh et al., 2020)) or for each class separately (MACE (Kumar et al., 2021), PACE (Kamakshi et al., 2021)). ConceptShap, MACE and PACE\nall use additional regularizers to enforce concept dissimilarity. In contrast, MCD restricts the concept discovery process as little as possible. Importantly, each method above defines a custom measure for concept importance, which is based on approximations of the original model. In contrast, the local and global concept relevance within MCD is solely based on the original model parameters. Other approaches (Chormai et al., 2022) use a concept definition similar to ours but use information from external attribution methods as well as orthogonality constraints to restrict the discovered concepts, whereas MCD works without such restrictions.\n\nThere is a complementary line of work of frameworks that try to identify concepts associated with particular neurons in hidden CNN representations, in conjunction with (Bau et al., 2017) or without (Achtibat et al.,\n2022) special concept-annotated datasets. Network Dissection (Bau et al., 2017) investigates the alignment of human-understandable concepts and particular single hidden features (neurons). Net2vec (Fong & Vedaldi, 2018) extended this by allowing concepts to be represented by combinations of neurons.\n\nLastly, there is a line of research that constructs inherently interpretable concept models by design with (Koh et al., 2020; Radenovic et al., 2022; Chen et al., 2020; Marconato et al., 2022; Zarlenga et al., 2022) or without relying on concept annotations (Chen et al., 2019). The objective of the former ante-hoc concept models is to discover concepts (Koh et al., 2020; Chen et al., 2020; Marconato et al., 2022; Zarlenga et al., 2022) in feature space in order to identify them with known factors from the underlying data-generating process. In contrast, the objective of MCD is to recover the true concepts learned by an arbitrarily trained model, which do not have to align with concepts underlying the data-generating process. Even though technically feasible (Chen & Feng, 2012), we do not equip MCD with a mechanism that enforces identifiability with known concepts, e.g., through weak concept supervision. There is a crucial difference in enforcing concept interpretability in the sense of identifiability with known concepts between ante-hoc and post-hoc approaches.\n\nRegularizing concept interpretability of post-hoc explanations might obfuscate the explanation and make the model appear more interpretable than it actually is. Concerning the latter category of concept models, our approach is best comparable with (Chen et al., 2019), as both can be reduced to a linear model operating on concepts that can be characterized via prototypes. We stress the essential difference, that our approach does not require retraining (with special training objectives) but is an interpretable reformulation of the original model.\n\n## 4 Results\n\nWe carry out our experiments on ImageNet (Deng et al., 2009). As model architectures, we consider ResNet models (He et al., 2016) using original weights as provided by *torchvision* **and updated weights as provided by**\ntimm (Wightman, 2019) with an improved training procedure (Wightman et al., 2021). We also present results for a swin vision transformer (Liu et al., 2021), again using weights provided by *timm* **(SwinS3base224).**\nIn the following, we will refer to these models as ResNet50, ResNet50v2 and, Swin-T, respectively. We base all our experiments on images from a diverse selection of ten ImageNet classes, which roughly align with CIFAR10 classes6\n\n## 4.1 Completeness Arithmetic\n\nFirst, we provide a concrete example for an MCD explanation and showcase its completeness relation introduced in Section 2.3 (*benefit 3* **). To this end, Figure 3 shows an MCD-SSC explanation of a ResNet50v2**\nprediction for a sample of the police van class in ImageNet. The number of concepts was chosen such that the completeness measure in Equation (6) reaches η = 0.**5. The three information components of the explanation**\nall provide complementary information:\n\n6**namely (airliner, beach wagon, hummingbird, siamese cat, ox, golden retriever, tailed frog, zebra, container ship, police**\nvan)\n(1) *Concept relevance heatmaps* **show the alignment of a feature vector component** φ l **associated with concept**\nC\nl **and the weight vector of a specific class. Roughly speaking, this alignment indicates how typical the**\nnetwork perceives the particular instantiation of the concept for the class under consideration. Applying mean pooling leads to a corresponding decomposition of the class logit under consideration (up to the bias term) into contributions corresponding to different concepts. This demonstrates the completeness relation on the level of concept relevance heatmaps as well on the level of logits, which represents a unique feature of the MCD framework. Interestingly, for the explanation in Figure 3, only the orthogonal complement concept contributes negatively to the class logit. The contributions of the first two concepts clearly dominate the class logit.\n\n(2) *Concept activation maps* **have positive scores showing how much a particular feature vector aligns with**\na specific concept subspace. These maps identify input regions where the concept is highly expressed. We color-code concept activation maps as a transparent overlay over the image where transparent regions indicate high activation. To guide the eye, we also include a yellow contour line at a threshold value of 0.5 and a white one at a value of 0.4.\n\n(3) *Concept prototypes* **allow characterizing a concept subspace through examples. Here, we display the**\nconcept activation maps of three test set samples that show the highest activation with the given concept.\n\nIn many cases, an intuitive meaning of a concept can be inferred most easily from these samples and numerous previous approaches present concepts in this way (Zhang et al., 2021; Achtibat et al., 2022; Yeh et al., 2020).\n\nIn case of the explanation in Figure 3, this could be windows/livery, livery, blue lights, building, tires (and the orthogonal complement covering mainly the background). In addition, we also indicate the global concept relevances for the different concepts according to Equation (4).\n\nAt this point, we deem it worthwhile discussing the complementary nature of concept relevance maps and concept activation maps:\n- Concept activation maps facilitate concept identification as they do not entangle concept activation and model prediction. More precisely, concept activation maps provide insight into the structures present in the feature space, including those that may not directly contribute to the prediction.\n\nThis is particularly evident in the case of the orthogonal complement, which is slightly activated in Figure 3 but has no positive relevance.\n\n- Concept the positive parts of the concept relevance maps correlate with the activation maps: Among all test set samples, we find a mean Pearson correlation of 0.45 for the concepts of the CIFAR10 classes between the positive part of each concept relevance map and the corresponding concept activation map (for MCD-SSC and ResNet50v2). This result serves as a sanity check and confirms that concept relevance is high in sample areas where the respective concept is strongly activated.\n\nImportantly, the negative parts in the concept relevance maps represent additional information.\n\nIn summary, the sample in Figure 3, is classified as a police van mainly due to its windows/livery, which are perceived as typical for the class by the network and are also the most relevant concept for the class globally.\n\nFurther, all other concepts are expressed in the sample and contribute positively, except for the orthogonal complement. Thus, we can confirm that MCD-SSC concepts indeed capture and focus on the relevant model reasoning structure.\n\n## 4.2 Empirical Evaluation\n\nWe compare MCD with sparse subspace clustering (MCD-SSC), MCD with alternative clustering, and previous methods listed in Table 1 in terms of (1) faithfulness (*benefit 1* ) and (2) conciseness (*benefit 2* **) of the**\nexplanations.\n\n## 4.2.1 Comparing Faithfulness Via Concept Flipping\n\nIn order to compare the methods in Table 1 in terms of faithfulness, we invoke the Smallest Destroying Concepts (SDC) benchmark as proposed in (Ghorbani et al., 2019) and (Wu et al., 2020). For concepts that\n\n\n\nFigure 3: Completeness relation for the police van class in ImageNet. Concepts are discovered via MCD-SSC\nfor ResNet50v2. The number of concepts is chosen such that the completeness score reaches η = 0.**5. We**\ndistinguish between local (sample-specific) and global properties (characterizing a set of samples). Locally, we consider *concept relevance maps***, which quantify the spatially resolved contribution of a concept to the** prediction. These satisfy a *completeness relation*, as explicitly shown in the first line. **Concept activation** maps **provide complementary information and indicate how much a concept is activated depending on the**\nspatial location in the sample. Globally, the overall relevance of a particular concept is quantified by the global relevance **scores. Finally, we also present concept prototypes (concept activation maps of the most**\nstrongly activated samples) to characterize a particular concept.\n\n| Table 1: Summary of concept discovery methods considered in this | work. | |\n|--------------------------------------------------------------------|------------|-------------|\n| | Arbitrary | |\n| Method | Multi-dim. | orientation |\n| MCD-SSC | ✓ | ✓ |\n| MCD-SSC-ortho | ✓ | ✗ |\n| MCD-kmeans | ✓ | ✓ |\n| ICE/MCD-PCA (Zhang et al., 2021) | ✗ | ✗ |\n| ACE (Ghorbani et al., 2019) | ✗ | ✓ |\n\nreflect the model's actual reasoning structure in feature space and *faithful* **concept relevance scores, SDC**\nshould show a sharp decline of the model accuracy with the number of flipped concepts.\n\nTo evaluate SDC, we subsequently remove concepts, as represented by concept masks in input space, in order of their sample-wise (local) relevance starting from high to low. To inpaint the removed segments, we use a classical imputation algorithm (Bertalmio et al., 2001), which leads to comparably realistic imputed images. Thus, the model is evaluated on-manifold in contrast to imputing with gray patches as often done in the literature (Samek et al., 2017). For similar reasons, we avoid the Smallest Sufficient Concepts (SSC)\n\n\n\nFigure 4: Left: Concepts are flipped one at a time in descending order of local concept importance/TCAV score, respectively. We measure the decline in model accuracy and show the mean accuracy across CIFAR10 classes against the fraction of deleted pixels. Meaningful concept discovery and quantification methods are supposed to show a sharp decline in this figure, but the decline should not happen after flipping only a single concept (i.e. the whole object). Right: Qualitative comparison between hard concept assignments.\nbenchmark, which would require high-quality imputation algorithms to avoid evaluating the model far from the data manifold. We obtain concept masks, i.e., hard concept assignments, in input space by taking the argmax of the corresponding concept activation maps over all concepts including the orthogonal complement.\n\nAfter the argmax operation, we disregard (do not remove) the orthogonal complement during the SDC experiments. For each concept mask we obtain local relevance scores by pooling the corresponding concept relevance heatmaps over the respective regions. This provides concept masks in input space which are ordered according to their importance. ACE does not provide a measure of per-sample concept relevance. Therefore, we revert to the order of their (global) TCAV scores after discarding concepts where statistical testing in comparison to random input samples fails to stay below p = 0.**05. In contrast to previous studies (Ghorbani**\net al., 2019; Wu et al., 2020), we report the model performance depending on the fraction of occluded pixels, which is essential for comparability since the segment size varies between different approaches. In order to show a meaningful average of the samples across all classes we flip only as many concepts as are present for the class with minimum number of concepts nc **for each method. We base our evaluation on the CIFAR10**\nclasses described above and work with the ResNet50v2 model, for which we extract concepts from the last hidden layer. For all methods within the MCD framework, we fix the number of concepts in a class-dependent way such that we reach a completeness score of η = 0.5.\n\nIn the left panel of Figure 4, we show the results of the SDC experiment. As mentioned above, a meaningful concept discovery and quantification method should show a sharp decline in Figure 4. However, in principle, the sharpest decline is achievable via assigning the complete object to a single concept. Such a concept would simply highlight the entire relevant region, i.e., this would not provide any insights beyond those that can be inferred by standard (non-concept-based) attribution methods such as LRP (Bach et al., 2015), PredDiff\n(Blücher et al., 2022) or Shapley values (Lundberg & Lee, 2017). The SCD results show, that ICE/MCDPCA and MCD-SSC-ortho consistently detect only a single relevant concept, as the accuracy curve stagnates after flipping the first concept. Thus, ICE/MCD-PCA and MCD-SCC-ortho, counteract the benefits of concept-based explanations. Since both approaches rely on orthogonal concepts, this constraint seems unfitted for a fine-grained analysis of related but distinct characteristics in feature space. Among the remaining algorithms, MCD-SSC shows the strongest decline as compared to MCD-kmeans and ACE, hence its discovered concepts are the most faithful.\n\nTo provide a qualitative impression of the concept relevance heatmaps across methods, we show them together with concept activation maps for a selected sample of the golden retriever class (one of the CIFAR10 classes)\n\n\n\n}\nFigure 5: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the golden retriever class in ImageNet. The number of concepts is chosen such that the completeness score reaches η = 0.**5. Concepts are ordered from left to right according to global concept relevance. Concept**\nheatmaps are titled by the pooled local concept relevance that sums to the prediction logit minus the bias.\n\nFor ICE, we only show the first six out of 142 and for ACE the first six out of 25 concepts. For ACE, no complement exists. For MCD-kmeans, relevance is distributed over all three concepts and also four of the five MCD-SSC concepts have notable relevance. In contrast, among the MCD flavors with orthogonal concepts (MCD-SSC-ortho and ICE), only one concept notably contributes to the prediction.\nin Figure 5, and more equivalent results for other classes in Figures 7 and 8.7**In Figure 4 (right panel),**\nwe also show hard concept assignments for an example image of the golden retriever class, which form the basis of the concept flipping experiment described above. These visually support the findings of the concept flipping experiment. Most approaches only discover a single concept for the dog (apart from a potential genuine background concept). In particular, consider the concept assignments in Figure 4 on the right: Here, both orthogonal approaches do not distinguish between the dog head and fur parts on the image. In contrast, the unconstrained MCD-SSC approach can successfully distinguish these two correlated regions in features space (incorporate two similar concepts) and shows the most fine-grained decomposition.\n\n7**Correspondong prototypes can be found in Figures 9 to 11.**\nTo summarize the results of the concept flipping experiment, our general MCD definition leads to the most faithful concepts, as the two unconstrained MCD flavors (MCD-SSC and MCD-kmeans), show the steepest descent among all methods without reverting to the non-informative solution of a single relevant concept.\n\n## 4.2.2 Conciseness Of Explanations\n\nFor an accessible explanation, it is desirable, to explain the model reasoning with as few meaningful concepts as completely as possible, i.e. to deliver concise concept explanations. To compare the conciseness of concept explanations we measure the number of concepts required to reach a certain completeness score η**, i.e. how**\nmany concepts are necessary to cover the whole relevant feature space. Again, there is a trivial solution, namely leveraging more and more dimensions to cover the majority of feature space via a single concept.\n\nTherefore, we additionally evaluate the average subspace dimension dl **and the mean (scaled) Grassmann**\ndistance ∆kl c\n, as defined in Equation (10), between all concept pairs (*k, l*) within one class c **to quantify**\nhow dissimilar two concepts are. 8In summary, we argue that concepts should be concise (small nc**), but**\ndissect the feature space into meaningful building blocks of model reasoning. While the latter is difficult to quantify, we argue that there is a trade-off between (1) covering feature space with very few concepts of high dimensionality and potentially small distance vs. (2) dissecting it into a high number of concepts with small dimensionality (extreme case: one-dimensional). To support this reasoning, we also inspect the visual impression of concepts for a selection of classes. Again, we base our evaluation on the CIFAR10 classes and concepts for the last hidden layer of the ResNet50v2 model. For all methods within the MCD framework, we fix the number of concepts in a class-dependent way such that we reach a completeness score of η = 0.5.\n\nWe list the number of concepts nc that is required to reach a completeness score of η = 0.**5 and** dlin Section 4.2.1. To provide a visual comparison of the concepts discovered by these methods, we show concept activation maps of prototypes for basketball, golden retriever and airliner class in ImageNet in Figures 9 to 11 and judge how broad they appear in input space. MCD-kmeans discovers the smallest number of concepts with the highest mean concept dimensionality of 74.7 and the smallest inter-concept distance\n(mean(∆kl c\n) = 0.**83) among all methods. This is reflected in the visual appearance of the concept prototypes,**\nwhich are visually broad and difficult to distinguish. MCD-SSC discovers on average 4.8 concepts with a 41% smaller mean concept dimensionality of 44.2. Visually, concepts are medium broad and are easier to distinguish in input space than for MCD-kmeans, which is reflected in a higher inter-concept distance of 1.19.\n\nWhen requiring orthogonality the Grassman angle is fixed to mean(∆kl c\n) = π/2 = 1.**57 (MCD-SSC-ortho**\nand ICE). For orthogonal concept one concept is medium broad in input space while all others are almost not activated. Most likely, the orthogonality constraint hinders the concepts to reflect a natural similarity between certain concepts. This aligns with the conclusions drawn from the SDC benchmark. Most notably, to achieve a comparable model faithfulness (completeness score of η = 0.**5) 30 times more one-dimensional**\nICE concepts than multi-dimensional MCD concepts are required, meaning this method delivers concept explanations that are not concise. Intuitively, a single concept is split up into several concepts, which is also reflected in their weak activation on test set samples. Lastly, the visual impression of ACE concepts is fixed by the choice of the superpixel algorithm. While ACE concepts are all one-dimensional, they do not provide a mechanism to quantify how complete they are, thus we cannot quantify nc **required to reach**\na completeness of 50%. As an overall summary, MCD-SSC is superior in dissecting the feature space into enclosed and meaningful concepts.\n\n## 4.3 Use Case: Mcd Concepts Reveal Differences In Classification Strategies Between Model Architectures And Training Procedures\n\nFinally, we showcase how MCD can unravel different classification strategies depending on the model architecture (ResNet50 vs. Swin-T) and the training strategies (ResNet50 vs. ResNet50v2), most notably the fact that the ResNet50v2 was trained using a multilabel loss. The test accuracies for the subset of CIFAR10-classes are 0.80 (ResNet50), 0.84 (ResNet50v2) and 0.86 (Swin-T). Here, we focus on MCD-SCC\nand, as before, restrict ourselves to concepts in the last hidden feature layer. First, we compare the discov-8**We use a scaled version of the original Grassmann distance that aggregates the principle angles (in radian) between two**\nsubspaces, for which 0 ≤ ∆kl c ≤ π**. Two special cases are ∆**kl c **= 0, meaning that subspace bases vectors are perfectly aligned,**\nand ∆kl c = π/**2, meaning that they are orthogonal.**\nTable 2: Summary of concept discovery methods considered in this work in comparison to prior work from Zhang et al. (2021) (ICE/MCD-PCA) and Ghorbani et al. (2019) (ACE). We measure average subspace dimension d l and the number of concepts nc that is required to reach a completeness score of η = 0.5 for ResNet50v2 on the CIFAR10 classes. A small number of relevant concepts nc **is desirable since this**\nsummarizes the complete model into an accessible and meaningful format. Here, multi-dimensional concepts have an advantage. Additionally, we evaluate the mean (scaled) Grassmann distance ∆kl c\n, see Equation (10),\nbetween all concept pairs (*k, l*) within one class c **to quantify the distinctness between concepts. The visual**\ninspection is based on prototypes of the basketball, golden retriever and airliner class concepts in Figures 9 to 11. Medium broad and distinct concepts are the most informative.\n\nMethod mean(d l) mean(nc**) mean(∆**kl c\n) Visual inspection\n\n| to 11. Medium broad and distinct concepts are the most informative. Method mean(d l ) mean(nc) mean(∆kl c ) | | Visual inspection | | |\n|---------------------------------------------------------------------------------------------------------------|------|---------------------|------|------------------------------|\n| MCD-SSC | 44.2 | 4.8 | 1.19 | medium broad |\n| MCD-SSC-ortho | 44.2 | 4.8 | 1.57 | only one broad (rest narrow) |\n| MCD-kmeans | 74.7 | 2.7 | 0.83 | very broad |\n| ICE/MCD-PCA | 1 | 146.7 | 1.57 | only one broad (rest narrow) |\n| ACE | 1 | n.a. | n.a. | medium broad |\n\nered concepts between the models by the activation maps of concept prototypes for the beach wagon class of ImageNet in Figure 6. We fix the number of concepts to nc **= 5. For Swin-T, we only apply a spatial**\nupsampling of the concept activation maps from the feature to the input space to 14×**14 in order to account**\nfor the 16×**16 patch tokenization. We find that ResNet50 concepts, which could roughly be identified as (car**\nbody, windows, car roof, wheels, street), are more narrow than the expression of Swin-T and ResNet50v2 concepts. The latter are related to broader views of the car, such as concepts (1, 2, 4) for ResNet50v2 and concepts (1, 3) for Swin-T. Interestingly, ResNet50v2 concepts reach a much lower completeness score of η = 0.49 than ResNet50 (η = 0.89) and Swin-T (η = 0.84) for fixed nc **= 5. In Figure 6 we show the** relation between the total concept space dimensionality, the number of concepts nc **and the completeness**\nscore η across the CIFAR10-classes. Even for nc = 30, the ResNet50v2 concepts have a lower η **than those**\nof the ResNet50 for nc **= 3, although the former covers already a much larger part of the concept space.**\nThese observations support the statement that feature space of the ResNet50v2 exhibits a comparably richer structure than ResNet50 . Thus, MCD-SSC concepts can reveal interesting differences in the character of the feature space as a consequence of two different training procedures for the same architecture. Interestingly, the dependence of η on nc **for the concepts between two models with different architectures, ResNet50 and**\nSwin-T, is quite similar. This also aligns with the visual appearance of the concepts.\n\nTo summarize, Swin-T and ResNet50 build on broader and more versatile concepts. In comparison, ResNet50v2 builds on more narrow and thus specific concepts for its classification strategy. These broad concepts are not unexpected for a transformer architecture like Swin-T with coarse self-attention windows, but a rather surprising finding for ResNet50 in comparison to ResNet50v2.\n\n## 5 Summary And Discussion\n\nIn this work, we put forward MCD, a general framework for concept discovery based on the hidden representation of a trained deep neural network. Unlike prior work in the field, we propose a general concept definition (incorporating previous approaches) as multi-dimensional linear subspaces without restricting to single directions or enforcing orthogonality between subspaces. We use concept activation maps to visualize concepts in input space. Considering the final hidden layer representation, we can reformulate the original model as a linear classifier acting on linear concept subspaces without the need to retrain with a special objective. This leads to a completeness relation, i.e., a natural decomposition of class logits into contributions corresponding to specific concepts and allows to resolve their spatial importance in terms of concept relevance heatmaps. As a particularly suited realization of our framework, we put forward MCD-SCC, which\n\n\n\nFigure 6: Left: Mean concept space completeness score ν **for the CIFAR10 classes across architectures**\nagainst the dimensionality of the union of all concept subspaces Pl d l**. The number of concepts can be**\ninferred from the points on the line where the first point on each line corresponds to nc **= 3 and the last**\none to nc = 30. ResNet50v2 shows a much lower completeness score at roughly the same nc and Pl d l as ResNet50. The feature space dimensionality is F = 2048 for ResNet50(v2) and F **= 768 for Swin-T. Right:**\nWe show MCD-SSC concept activation maps for concept prototypes for ResNet50, ResNet50v2 and Swin-T\nand the beach wagon class in ImageNet. We fixed the number of concepts to nc **= 5. In this way, ResNet50v2**\nreaches η = 0.49, ResNet50 η = 0.89 and Swin-T η = 0.**84. Each row shows a single concept and is titled by**\nits global concept importance. The last row shows the orthogonal complement of the concept space.\nrelies on sparse subspace clustering for concept discovery. Based on qualitative and quantitative insights, we show the superiority of MCD-SCC over other MCD flavors that build on traditional clustering algorithms.\n\nWe showcase the ability of MCD via discriminating between hidden representations obtained from different model architectures and training strategies. This paves the way towards further novel use-cases for MCD\nconcepts such as gaining insights in the natural sciences, e.g., identifying sub-classes of cancerous cells in histopathology or summarizing model behavior beyond single examples and thereby systematically discovering model biases. MCD prioritizes faithfulness of concepts over identifiability with known human concepts by not including any interpretability-enforcing regularizers that could obfuscate the original structures learned by the model. In this way, we can guarantee to cover the original model reasoning structure, which is crucial for auditing models or scientific discovery use cases. However, this might obfuscate non-expert human users at first sight, since deep learning models will most likely not rely entirely on human-like features.\n\nCode to reproduce our experiments is publicly available at **https://github.com/jvielhaben/MCD-XAI**.\n\n## Acknowledgments\n\nThis work was supported by the German Ministry for Education and Research (BMBF) through BIFOLD\n(refs. 01IS18025A and 01IS18037A).\n\n## References\n\nReduan Achtibat, Maximilian Dreyer, Ilona Eisenbraun, Sebastian Bosse, Thomas Wiegand, Wojciech Samek, and Sebastian Lapuschkin. 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(8)\nThe particular kind of sparsity constraints that are imposed on Equation (8) and how it is optimized depends on the chosen SSC algorithm. Here, we use elastic net subspace clustering (You et al., 2016a), which is robust against noise and scales well for large sample sizes. In all our experiments, we fix the hyperparameter γ, which balances sparsity vs. robustness, to γ **= 10. We confirmed that the results are not sensitive to**\nvariation of this parameter over a range of values from 5 to 50. As computation time for SSC is dependent on this parameter, we chose γ **such that this is minimized.**\nWe remove outliers based on the ℓ1**-norm as in (Soltanolkotabi & Candes, 2012), where we empirically fix**\nthe percentile threshold to 0.75 and re-fit the sparse self-representation for the remaining elements.\n\nAnother scalable alternative to the elastic net clustering is orthogonal matching pursuit (OMP)(You et al.,\n2016b), which is, however, not robust against noise and does not allow for outlier removal via thresholding.\n\nFinally, the original sparse subspace clustering method from (Elhamifar & Vidal, 2013) is robust against noise and outliers but does not scale to large datasets. The particular robustness and scalability properties make elastic net subspace clustering (with thresholding) an ideal choice for the first step of our concept discovery method.\n\nSpectral clustering **In a second step, we perform spectral clustering with the affinity matrix** W = |R|+|RT|, which encodes the similarity of two feature vectors according to their self-representations. We determine the number of clusters nc **either via the largest gap in the spectrum of the Laplacian (Von Luxburg, 2007) or**\nuse a predetermined value. This step assigns every input feature φi to a particular cluster C1*, . . . ,* Cnc or to the set of outliers.\n\n## B Characterizing Relations Between Subspaces By Principal Angles\n\nIn this section, we briefly review the definition of principal angles, which can be used to characterize the relation between two linear subspaces. The principal angles θ AB\ni(Jordan, 1875) (i = 1*, . . . ,* min(dim A, dim B))\nbetween two linear subspaces *A, B***, are defined recursively via**\n\n$$\\cos\\theta_{i}^{A B}=\\max_{\\mathbf{a}\\in A,\\mathbf{b}\\in B}\\frac{\\mathbf{a}^{T}\\mathbf{b}}{|\\mathbf{a}||\\mathbf{b}|}=:\\frac{\\mathbf{a}^{T}\\mathbf{b}_{i}}{|\\mathbf{a}||\\mathbf{b}_{i}|}\\,\\tag{1}$$\n\n, (9)\nwhere the maximum is taken subject to the orthogonality constraints a Taj **= 0 and** b Tbj **= 0 for** j =\n1*, . . . , i* − 1.\n\nTo quantify the similarity between two subspaces A and B**, we use a scaled version of their Grassmann**\ndistance Hamm (2008), which is defined as,\n\n$$\\Delta^{A B}=1/{\\sqrt{\\operatorname*{min}(\\dim A,\\dim B)}}{\\sqrt{(\\theta_{1}^{A B})^{2}+\\ldots+(\\theta_{\\operatorname*{min}(\\dim A,\\dim B)}^{A B})^{2}}}\\,.$$\n2 . **(10)**\nThis allows comparing the similarity of concepts within a given class or across classes regardless of the concept subspaces' dimensionality.\n\n$$({\\mathfrak{g}})$$\n\n$$(10)$$\n\n## C Qualitative Results\n\nFor a qualitative comparison between of the concept activation maps and relevance heatmaps between the methods in Section 4.2, we provide results for selected samples in Figures 5, 7 and 8. In Figures 9 to 11 we\n\n\n\n}\n \nFigure 7: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the basketball class in ImageNet. The number of concepts is chosen such that the completeness score reaches η = 0.**5. Concepts are ordered from left to right according to global concept relevance. Concept heatmaps**\nare titled by the pooled local concept relevance that sums to the prediction logit minus the bias. For ICE,\nwe only show the first six out of 105 and for ACE the first six out of 25 concepts.\n\n\n\n}\n \nFigure 8: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the airliner class in ImageNet. The number of concepts is chosen such that the completeness score reaches η = 0.**5. Concepts are ordered from left to right according to global concept relevance. Concept heatmaps**\nare titled by the pooled local concept relevance that sums to the prediction logit minus the bias. For ICE,\nwe only show the first seven out of 141 and for ACE the first seven out of 25 concepts.\n\n\n\n\n\nFigure 9: Concept activation maps for concept prototypes for basketball class of ImageNet. The last row shows prototype for the complement, except for ACE, where no complement exists. For ICE, we only show the first six out of 105 and for ACE the first six out of 25 concepts.\n\n\n\n\n\nFigure 10: Concept activation maps for concept prototypes for golden retriever class of ImageNet. The last row shows prototype for the complement, except for ACE, where no complement exists. For ICE, we only show the first six out of 142 and for ACE the first six out of 25 concepts.\n\n\n\nFigure 11: Concept activation maps for concept prototypes for airliner class of ImageNet. The last row shows prototype for the complement, except for ACE, where no complement exists. For ICE, we only show the first seven out of 141 and for ACE the first seven out of 25 concepts."},"reviews":{"kind":"string","value":"Review 1:\nSummary: This paper proposes a framework for concept discovery based on hidden representation of neural networks. Different from existing approaches, authors proposed a multi-dimensional linear subspaces without restricting to single directions or enforcing orthogonality between subspaces with complete relations. Empirical results show better performance.\n\nStrengths and Weaknesses: Strengths\n\n- Empirical results seem better\n\nWeakness\n\n- Presentation is not very clear. Many related concepts are not fully explained, and motivations are not clearly written. Hence, one cannot fully judge the motivation or understand the advantages of the proposed approach. \n- Overall, I find the paper lacks sound justification, and mostly are hand wavy explanations. (although I recognize that maybe this is standard in XAI field due to lack of theoretical understanding and very subjective.)\n\n\nComments:\n- XAI is not explained in abstract.\n- \"directly identify concepts with a neural directions\": is it one direction or 3?\n- losing orthogonality seems to lead to overlap of concepts. I'm not sure arbitrary orientation is a desired property for concise explanation, since they likely contain redundant information. More discussion (esp. theoretical justification) here would be helpful. \n\nSection 4.2.1 studies this aspect in more details. However, it is not fully clear to me why MCD-SSC is better than MCD-SSC-ortho. \"the greedy way these orthogonal subspaces are constructed\" - also very informal. If one purposely control more concepts to be desired (maybe via some hyperparameters tuning), would it result in more fine-grained concepts for MCD-SSC-ortho?\n\n- \"Multi-dimensionality ensures concise explanations\": although this is discussed in introduction, I don't find this argument clear. Later section also lacks a clear mathematical justifications on why higher dimension is better than one-dimension. In addition, what would a good way to determine the dimension size?\n- Some hyperparameter choices are not well motivated, via ablation study for example. For example, number of concepts.\n- How does the propose method differ from many disentangled representation learning or prototype learning?\n\nRequested Changes: clear and sound justification on why multi-dimension and non-orthogonality are desired. \n\nBroader Impact Concerns: n/a\n\n==================================================\n\nReview 2:\nSummary: This paper proposes a method to extract \"concepts\" in deep image classification models as subspaces in latent space.\nIn previous research, \"concepts\" are typically defined as vectors in latent space.\nThe idea in this paper can be thus interpreted as a generalization of existing \"concepts\".\nThe authors pointed out that existing sparse subspace clustering techniques can be used to estimate these \"concept\" subspaces.\nThe authors also proposed a method to quantify the contribution of each \"concept\" in image classification using the basis vectors of each subspace, and to extract prototypical data for each \"concept\".\nFurthermore, the authors enabled to visualize the contribution of each \"concept\" in the image through heatmaps.\nThe generalization of the notion of \"concepts\", the extraction of prototypical data, and the visualization through heatmaps are useful tools for users to understand deep image classification models.\nThe authors also conducted experiments using ResNet and Swtin Transformer, demonstrating that the proposed method is more effective in extracting \"concepts\", and that there are differences in the found \"concepts\" between models.\n\nStrengths and Weaknesses: ### Strong aspects\n\n#### Strength 1: Generalization of \"concepts\"\nThe major contribution of this study lies in the definition of \"concepts\" as subspaces in latent space.\nIn previous research, \"concepts\" are typically defined as vectors in latent space.\nHowever, it would not be natural to consider that complex \"concepts\" can be represented as a single vector in high-dimensional latent space.\nOn the other hand, subspaces can more naturally represent richer and more complex \"concepts\" by adjusting their dimensions appropriately.\nThis perspective would be an important step forward in the study of \"concepts\".\n\n#### Strength 2: Benefits of subspace\nThe strengths of this study is not only the generalization of the definition of \"concepts\", but also the extraction of prototypical data and the visualization through heatmaps, which will further help users to understand deep image classification models.\nFurthermore, the extraction of these prototypical data and the visualization through heatmaps are naturally realized through the basis expansion of the latent vectors as well as the weight coefficients of using the basis vectors of the subspaces.\nThis suggests that defining \"concepts\" as subspaces is also useful as a model interpretation tool.\n\n\n### Weak aspects\nI did not find any crucial weakness.\nAs minor suggestions, I recommend the authors to enlarge images in Figures 3 and 5 as thy are too small to inspect.\nI also recommend the authors to move some (not necessarily all) of Figures 6, 7, and 8 to the main body of the paper.\nThey are essential ingredients and should be presented in the main body.\n\nRequested Changes: #### Request 1.\nI recommend the authors to enlarge images in Figures 3 and 5 as thy are too small to inspect.\n\n#### Request 2.\nI recommend the authors to move some (not necessarily all) of Figures 6, 7, and 8 to the main body of the paper.\nThey are essential ingredients and should be presented in the main body.\n\nBroader Impact Concerns: There is no ethical concern.\n\n==================================================\n\nReview 3:\nSummary: The authors introduce a novel post-hoc, concept based explainer for neural networks. \n\nStrengths and Weaknesses: + English is good.\n+ Contribution generalizes existing approaches in a sensible manner.\n+ Contribution is of interest to the XAI/CBM community.\n+ Related work is well done.\n\n- The structure of the text should be improved.\n- Some comments are not entirely motivated.\n\nRequested Changes: *Writing* English is good, and I could not spot any major linguistic mistakes. Structure and clarity, however, should be improved.\n\n- The introduction is far too detailed, the description of the method is not detailed enough. I think the structure should be changed to improve clarity and flow. Specifically,\n\n 1) The technical paragraphs at the end of the introduction are very hard to follow at this stage (as the reader is not yet familiar with the intent of the various steps and with the techniques involves), and should be moved later, preferably between Sec 2.1 and 2.2.\n\n 2) The schematic overview of the algorithm should be moved between to the beginning of Sec 2, and it should briefly outline the clustering, basis construction, and relevance computation steps, clearly stating what are the expected inputs and outcomes of the various steps *formally*.\n\n 3) All algorithmic details should be kept in Sections 2.2+. Intuition about important steps of the procedure are currently missing. The authors should briefly describe:\n - What is subspace clustering?\n - How does SSC work?\n - Why is PCA enough to recover usable bases from the SSC clusters?\nShort, high-level explanations of all these build blocks would help tremendously. The intended inputs and outputs to all steps should be specified.\n\n 4) The MCD-kmeans and MCD-PCA approaches mentioned in p 5 should be moved to the experiments. This would keep the focus on the proposed approach.\n\n 5) The complement subspace $C^{n_c+1}$ defined in p 4 should be introduced later, when the need for it becomes apparent. Also, $C^{n_c+1}$ does not roll easily on the tongue. I suggest to introduce a more obvious mnemonic, say $C^\\bot$, $D$, or similar.\n\n 6) In the experiments, clearly state all reasearch questions and provide clear descriptions of all data sets and evaluation metrics.\n\n- Eq. 2: I suggest to use use a different symbol for $\\phi^l_i$, to avoid confusion with the feature vector $\\phi$.\n\n- p 6: Footnote 4 should be moved next to footnote 3.\n\n- p 6: In the first sentence on p 6, change \"i.e., $max_{x,y} ...$\" -> \"i.e., we divide the vectors elementwise by $max_{x,y} ...$\".\n\n- p 6: In the second sentence, what is $|\\phi^l|$? Is it the Euclidean norm? If so, please use $\\|\\phi\\|_2$ instead. Otherwise, please explain what $|\\cdot|$ means.\n\n- p 8: Sec 3. \"The segmentation into candidate concept patches is model-independent and thus, segments are not necessarily meaningful as perceived by the model\". But by locality - which holds in CNNs and (as the authors mention) empirically also for ViTs - superpixels generally map to subsets of latent features.\n\n- p 9: What does \"yellow(white)\" and \"0.5(0.4)\" mean?\n\n- p 9: \"the the\"\n\n- Everywhere: \"true-to-the-model\" - I suppose the authors refer to \"faithfulness\". If so, the latter term should be preferred as it is more common in the XAI literature.\n\n- Why are some of the equations boxed? The boxes are unnecessary and should be removed.\n\n- Fig 2: It would be useful to draw $\\phi^\\alpha_{xy}$ as a tube. Also, please rotate the $x$ and $y$ coordinates, for readability.\n\n\n*Quality* I have some issues with the message conveyed by the paper. Specifically:\n\n- p 2: \"Allowing for arbitrary multi-dimensional subspaces unfolds the most general definition of a linear decomposition. Thus, this general approach enables true-to-the-model concepts as it allows to capture any meaningful linear structure within the hidden feature layer (benefit 1).\" I agree with the first sentence - linear subspaces are the most complex linear substructure possible for a linear space - but not with the second. Why are linear subspaces \"true-to-the-model\"? Please, either remove the second sentence (as it doesn't add much), make it relative (e.g., \"truer-to-the-model\" or \"more faithful\"), or provide arguments in support for it. The reason why I am insisting on this is that the \"true-to-the-model\" slogan is repeated again in the text - but to the best of my knowledge it is not justified: the model may be carrying out non-linear operations during prediction, so faithfulness is not guaranteed. I think it's important to discuss this point thoroughly, as faithfulness is one of the main selling points of the technique.\n\n- p 4: Correlated to the above: the authors write \"Note, that exploring even more general structures, such as concepts as sub-manifolds in feature space, is an interesting idea. However, these do not allow for a decomposition of the feature vector and hence do not lead to a completeness property, which is central to the definition of concept relevance maps\". The meaning is unclear to me. I'd appreciate if the authors could elaborate on what they mean. A second observation is that this seems to be in conflict with the \"true-to-the-model\" motto: if the model does reason in terms of manifolds rather than linear subspaces, explanations given in terms of subspaces (that as such satisfy the completeness property) cannot possibly be faithful to the model. So I'm not sure what benefits completeness yields - especially from the perspective of faithfulness. My understanding is that they are both useful properties, but quite orthogonal to each other. They authors should discuss this briefly.\n\n- p 4: \"assume without loss of generality that their subspaces are pairwise disjoint.\" Why is this w.l.o.g.?\n\n- p 4: \"does not take advantage of the spatial proximity of the data\" -> \"does not rely on ...\" (the sentence, as it is, implies a negative connotation)\n\n\n*Experiments*: \n\n1) The SCD dataset is not described, so I don't know whether this evaluation is in any way useful.\n\n3) Ideally, MCD should be compared to ground-truth explanations. To this end, the authors could define data sets where the ground-truth explanations are known, for instance, using data sets with known concept annotations and defining synthetic labels based on simple logical combinations of concepts, then testing whether MCD can recover the concepts responsible for the label. Right now, it is very hard to know whether the MCD explanations are faithful to the concept's reasoning. This is optional, but it would help immensely.\n\n\n\n*Related Work* The related work is well done. Two aspects that may be worth discussing are:\n\n1) Automatically extracted concepts - even those featuring nice theoretical properties like completeness - are not necessarily human understandable. This is a very important aspect. In fact, assuming concept interpretability is viewed as concept identification - as done in:\n\n Marconato et al., \"GlanceNets: Interpretabile, Leak-proof Concept-based Models\", NeurIPS'22.\n\nand implicitly in other works on concept-based explanations - it is very hard to ensure concepts extracted automatically from networks are interpretable. This is exactly the reason why TCAV, Concept-Bottleneck Models, Concept Whitening, and GlanceNets require a modicum of concept supervision. The authors should explain what makes concepts extracted by MCD intepretable (regardless of the how the network under examination is structured and learned), and discuss possible limitations of their (unsupervised) approach.\n\n2) In the same vein, it would be worth briefly discussing whether concept supervision could be injected into the explanation procedure to guide the subspace decomposition.\n\n3) The idea of implementing concepts as subspaces has been considered in:\n\n Zarlenga et al., \"Concept embedding models\", NeurIPS'22.\n\nIt would make sense to discuss the relationship between their (ante-hoc) solution to the proposed (post-hoc) approach.\n\nBroader Impact Concerns: As I mentioned, explanations extracted by *unsupervised* concept extraction methods always risk being potentially misinterpreted by stakeholders. The authors should discuss this in the conclusion. See:\n\n Chen et al., \"Does the explanation satisfy your needs?: A unified view of properties of explanations\"\n\nfor a reference. This is a very recent overview of properties of explanations and mentions pitfalls in interpreting them.\n\n==================================================\n\nMetareview:\nRecommendation: Accept as is\n\nComment: The authors propose a new post-hoc explainable AI method, which leverages multi-dimensional linear subspaces without orthogonality restrictions to discover hidden concepts. Empirical results demonstrate the potential of the proposed method.\n\nThe reviewers agree about the contribution of the work to the problem of concept-based explanation. The presentation issues of the first draft were fixed during the revision. While some reviewers have concerns about the theoretical justifications of the proposed method, the empirical results are convincing enough to validate the proposed method and support the claim on the possible advantages of multi-dimensional concepts without orthogonality restrictions. All reviewers lean acceptance after the revision.\n\n\n\n==================================================\n\n"}}},{"rowIdx":93,"cells":{"paper":{"kind":"string","value":"# Taming Diffusion Times In Score-Based Generative Models: Trade-Offs And Solutions\n\nAnonymous authors Paper under double-blind review\n\n## Abstract\n\nScore-based diffusion models are a class of generative models whose dynamics is described by stochastic differential equations that map noise into data. While recent works have started to lay down a theoretical foundation for these models, an analytical understanding of the role of the diffusion time T is still lacking. Current best practice advocates for a large T\nto ensure that the forward dynamics brings the diffusion sufficiently close to a known and simple noise distribution; however, a smaller value of T should be preferred for a better approximation of the score-matching objective and higher computational efficiency. Starting from a variational interpretation of diffusion models, in this work we quantify this trade-off, and suggest a new method to improve quality and efficiency of both training and sampling, by adopting smaller diffusion times. Indeed, we show how an auxiliary model can be used to bridge the gap between the ideal and the simulated forward dynamics, followed by a standard reverse diffusion process. Empirical results support our analysis; for image data, our method is competitive w.r.t. the state-of-the-art, according to standard sample quality metrics and log-likelihood.\n\n## 1 Introduction\n\nDiffusion-based generative models (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Song et al., 2021c; Vahdat et al., 2021; Kingma et al., 2021; Ho et al., 2020; Song et al., 2021a) have recently gained popularity due to their ability to synthesize high-quality audio (Kong et al., 2021; Lee et al., 2022b), image Dhariwal\n& Nichol (2021); Nichol & Dhariwal (2021) and other data modalities Tashiro et al. (2021), outperforming known methods based on Generative Adversarial Networks (gans) (Goodfellow et al., 2014), normalizing flows (nfs) (Kingma et al., 2016) or Variational Autoencoders (vaes) and Bayesian autoencoders (baes)\n(Kingma & Welling, 2014; Tran et al., 2021).\n\nDiffusion models learn to generate samples from an unknown density p*data* by reversing a *diffusion process* which transforms the distribution of interest into noise. The forward dynamics injects noise into the data following a diffusion process that can be described by a Stochastic Differential Equation (sde) of the form, dxt = f(xt, t)dt + g(t)dwt with x0 ∼ p*data* , (1)\nwhere xt is a random variable at time t, f(·, t) is the *drift term*, g(·) is the *diffusion term* and wt is a Wiener process (or Brownian motion). We will also consider a special class of linear sdes, for which the drift term is decomposed as f(xt, t) = α(t)xt and the diffusion term is independent of xt. This class of parameterizations of sdes is known as *affine* and it admits analytic solutions. We denote the time-varying probability density by p(x, t), where by definition p(x, 0) = p*data* (x), and the conditional on the initial condition x0 by p(x, t| x0).\n\nThe forward sde is usually considered for a sufficiently long *diffusion time* T, leading to the density p(x, T).\n\nIn principle, when T → ∞, p(x, T) converges to Gaussian noise, regardless of initial conditions.\n\nFor generative modeling purposes, we are interested in the inverse dynamics of such process, i.e., transforming samples of the noisy distribution p(x, T) into p*data* (x). Formally, such dynamics can be obtained by considering the solutions of the inverse diffusion process (Anderson, 1982),\ndxt =-−f(xt, t′) + g 2(t\n′)∇ log p(xt, t′)dt + g(t\n′)dwt , (2)\n1 where t\n′\ndef = T − t, with the inverse dynamics involving a new Wiener process. Given p(x, T) as the initial condition, the solution of Eq. (2) after a *reverse diffusion time* T, will be distributed as p*data* (x). We refer to the density associated to the backward process as q(x, t′)q(x, t). The simulation of the backward process is referred to as *sampling* and, differently from the forward process, this process is not *affine* and a closed form solution is out of reach. PM: in Eq. (2), should we at least say that the Weiner process is in reverse time?\n\nPractical considerations on diffusion times. In practice, diffusion models are challenging to work with\n(Song et al., 2021c). Indeed, a direct access to the true *score* function ∇ log p(xt, t) required in the dynamics of the reverse diffusion is unavailable. This can be solved by approximating it with a parametric function sθ(xt, t), e.g., a neural network, which is trained using the following loss function,\n\n$${\\mathcal{L}}(\\mathbf{\\theta})=\\int_{0}^{T}\\mathbb{E}_{\\sim(\\perp)}\\lambda(t)\\|\\mathbf{s}_{\\mathbf{\\theta}}(\\mathbf{x}_{t},t)-\\mathbf{\\nabla}\\log p(\\mathbf{x}_{t},t\\,|\\,\\mathbf{x}_{0})\\|^{2}\\,,$$\n\n2, (3)\n$$\\mathcal{L}(\\mathbf{\\theta})=\\frac{1}{1+\\frac{\\mu_{0}}{p(t)}\\frac{\\mathbf{m}}{\\sim(1)^{\\Lambda(t)}||\\mathbf{s}\\mathbf{\\theta}(\\mathbf{x}_{t},t)-\\mathbf{v}_{\\mathrm{LOE}}(\\mathbf{x}_{t},t+\\mathbf{v}_{\\mathrm{LO}})||^{2}}},$$\n\nwhere λ(t) is a positive weighting factor and the notation E∼(1) means that the expectation is taken with respect to the random process xt in Eq. (1): for a generic function h, E∼(1)[h(xt, x0, t)] = Rh(x, z, t)p(x, t| z)p*data* (z)dxdz. p(t) = U(0, T). This loss, usually referred to as *score matching loss*,\nis the cost function considered in Song et al. (2021b) (Eq. (4) ). The condition λ(t) = g(t)\n2, adopted in this work, is referred to a *likelihood reweighting*. Due to the affine property of the drift, the term p(xt, t| x0)\nis analytically known and normally distributed for all t (expression available in Table 1, and in Särkkä &\nSolin (2019)). Note also that we will refer to λ as the *likelihood reweighting* factor when λ(t) = g(t)\n2(Song et al., 2021b). Intuitively, the estimation of the *score* is akin to a denoising objective, which operates in a challenging regime. Later we will quantify precisely the difficulty of learning the *score*, as a function of increasing diffusion times.\n\nWhile the forward and reverse diffusion processes are valid for all T, the noise distribution p(x, T) is analytically known only when the diffusion time is T → ∞. To overcome this problem, the common solution is to replace p(x, T) with a simple distribution p*noise* (x) which, for the classes of sdes we consider in this work, is a Gaussian distribution.\n\nIn the literature, the discrepancy between p(x, T) and p*noise* (x)\nhas been neglected, under the informal assumption of a sufficiently large diffusion time. Unfortunately, while this approximation seems a valid approach to simulate and generate samples, the reverse diffusion process starts from a different initial condition q(x, 0) and, as a consequence, it will converge to a solution q(x, T) that is different from the true p*data* (x). Later, we will expand on the error introduced by this approximation, but for illustration purposes Fig. 1 shows quantitatively this behavior for a simple 1D toy example p*data* (x) = πN (1, 0.1 2) + (1 − π)N (3, 0.5 2), with π = 0.3: when T is small, the distribution p*noise* (x) is very different from p(x, T) and samples from q(x, T) exhibit very low likelihood of being generated from p*data* (x).\n\n$$(3)$$\n\n\n\nFigure 1: Effect of T on a toy model: low diffusion times are detrimental for sample quality (likelihood of 1024 samples as median and 95 quantile, on 8 random seeds).\n\nCrucially, Fig. 1 (zoomed region) illustrates an unknown behavior of diffusion models, which we unveil in our analysis. The right balance between efficient *score* estimation and sampling quality can be achieved by diffusion times that are smaller than common best practices. This is a key observation we explore in our work.\n\nContributions. An appropriate choice of the diffusion time T is a key factor that impacts training convergence, sampling time and quality. On the one hand, the approximation error introduced by considering initial conditions for the reverse diffusion process drawn from a simple distribution pnoise (x) ̸= p(x, T) increases when T is small. This is why the current best practice is to choose a sufficiently long diffusion time. On the other hand, training convergence of the *score* model sθ(xt, t) becomes more challenging to achieve with a large T, which also imposes extremely high computational costs **both** for training and for sampling. This would suggest to choose a smaller diffusion time. Given the importance of this problem, in this work we set off to study—for the first time—the existence of suitable operating regimes to strike the right balance between computational efficiency and model quality. The main contributions of this work are the following.\n\nContribution 1: In § 2 we provide a new characterization of score-based diffusion models, which allows us to obtain a formal understanding of the impact of the diffusion time T. We do so by introducing a novel decomposition of the evidence lower bound (elbo), which emphasizes the roles of (i) the discrepancy between the \"ending\" distribution of the diffusion and the \"starting\" distribution of the reverse diffusion processes, and (ii) of the *score* matching objective. This allows us to claim the existence of an optimal diffusion time, and it provides, for the first time, a formal assessment of the current best practice for selecting T. we use an elbo decomposition which allows us to study the impact of the diffusion time T.\n\nThis elbo decomposition emphasizes the roles of (i) the discrepancy between the \"ending\" distribution of the diffusion and the \"starting\" distribution of the reverse diffusion processes, and (ii) of the *score* matching objective. Crucially, our analysis does not rely on assumptions on the quality of the score models. We explicitly study the existence of a trade-off and explore experimentally, for the first time, current approaches for selecting the diffusion time T.\n\nContribution 2: In § 3 we propose a novel method to improve *both* training and sampling efficiency of diffusion-based models, while maintaining high sample quality. Our method introduces an auxiliary distribution, allowing us to transform the simple \"starting\" distribution of the reverse process used in the literature so as to minimize the discrepancy to the \"ending\" distribution of the forward process. Then, a standard reverse diffusion can be used to closely match the data distribution. Intuitively, our method allows to build \"bridges\" across multiple distributions, and to set T toward the advantageous regime of small diffusion times. In addition to our methodological contributions, in § 4, we provide experimental evidence of the benefits of our method, in terms of sample quality and log likelihood. Finally, we conclude in § 5.\n\nRelated Work. A concurrent work by Zheng et al. (2022) presents an empirical study of a truncated diffusion process, but lacks a rigorous analysis, and a clear justification for the proposed approach. Recent attempts by Lee et al. (2022b) to optimize p*noise* , or the proposal to do so (Austin et al., 2021) have been studied in different contexts. Related work focus primarily on improving sampling efficiency, using a wide array of techniques. Sample generation times can be drastically reduced considering adaptive step-size integrators (Jolicoeur-Martineau et al., 2021). Other popular choices are based on merging multiple steps of a pretrained model through distillation techniques (Salimans & Ho, 2022) or by taking larger sampling steps with GANs (Xiao et al., 2022). Approaches closer to ours *modify* the sde, or the discrete time processes, to obtain inference efficiency gains. In particular, Song et al. (2021a) considers implicit non-Markovian diffusion processes, while Watson et al. (2021) changes the diffusion processes by optimal scheduling selection and Dockhorn et al. (2022) considers overdamped sdes. Finally, hybrid techniques combining VAEs and diffusion models (Vahdat et al., 2021) or simple auto encoders and diffusion models (Rombach et al., 2022), have positive effects on training and sampling times.\n\n## 2 A New Elbo Decomposition And A Tradeoff On Diffusion **Timeexploring A Tradeoff** On Diffusion Time\n\nThe dynamics of a diffusion model can be studied through the lens of variational inference, which allows us to bound the (log-)likelihood using an evidence lower bound (elbo) (Huang et al., 2021).\n\nOur interpretation The interpretation we consider in this work (see also Song et al. (2021b), Thm. 1)\nemphasizes the two main factors affecting the quality of sample generation: an imperfect *score*, and a mismatch, measured in terms of the Kullback-Leibler (kl) divergence, between the noise distribution p(x, T)\nof the forward process and the distribution p*noise* used to initialize the backward process.\n\n## 2.1 Preliminaries: **The Elbo Decomposition**\n\nOur goal is to study the quality of the generated data distribution as a function of the diffusion time T. Then, instead of focusing on the log-likelihood bounds for single datapoints log q(x, T), we consider the average over the data distribution, i.e. the cross-entropy Ep*data* (x)log q(x, T). By rewriting the Lelbo derived in Huang et al. (2021, Eq. (25)) (details of the steps in the Appendix), we have that\n\n$$\\mathbb{E}_{p_{\\sf{Min}}\\,(\\mathbf{\\omega})}\\log q(\\mathbf{x},T)\\geq\\mathcal{L}_{\\textsc{ELBO}}(\\mathbf{s_{\\theta}},T)=\\mathbb{E}_{\\sim(1)}\\log p_{\\sf{base}}\\left(\\mathbf{x}_{T}\\right)-I(\\mathbf{s_{\\theta}},T)+R(T).\\tag{4}$$\nwhere R(T) = 12 R T t=0 E∼(1) hg 2(t)∥∇ log p(xt, t| x0)∥ 2 − 2f ⊤(xt, t)∇ log p(xt, t| x0) idt, and I(sθ, T) = 1 2 R T t=0 g 2(t)E∼(1) h∥sθ(xt, t) − ∇ log p(xt, t| x0)∥ 2idt is equal to the loss term Eq. (3) when λ(t) = g 2(t).\nNote that R(T) depends neither on sθ nor on p*noise* , while I(sθ, T), or an equivalent reparameterization\n(Huang et al., 2021; Song et al., 2021b, Eq. (1)), is used to learn the approximated *score*, by optimization of the parameters θ. It is then possible to show that\n\n$$I(\\mathbf{s_{\\theta}},T)\\geq\\underbrace{I(\\mathbf{\\nabla}\\log p,T)}_{\\stackrel{{\\mathrm{def}}}{{=}}K(T)}=\\frac{1}{2}\\int\\limits_{t=0}^{T}g^{2}(t)\\mathbb{E}_{\\sim(1)}\\left[\\left\\|\\mathbf{\\nabla}\\log p(\\mathbf{x}_{t},t)-\\mathbf{\\nabla}\\log p(\\mathbf{x}_{t},t\\mid\\mathbf{x}_{0})\\right\\|\\right]^{2}\\mathrm{d}t.\\tag{5}$$\n\nNote that the term K(T) = I(∇ log *p, T*) does not depend on θ. Consequently, we can define G(sθ, T) =\nI(sθ, T) − K(T) (see Appendix for details), where G(sθ, T) is a positive term that we call the gap term, accounting for the practical case of an imperfect *score*, i.e. sθ(xt, t) ̸= ∇ log p(xt, t). It also holds that\n\n$$\\mathbb{E}_{\\sim(1)}\\log p_{\\mathsf{feas}}(\\mathbf{x}_{T})=\\int\\left[\\log p_{\\mathsf{feas}}(\\mathbf{x})-\\log p(\\mathbf{x},T)+\\log p(\\mathbf{x},T)\\right]p(\\mathbf{x},T)\\mathrm{d}\\mathbf{x}=$$ $$=\\mathbb{E}_{\\sim(1)}\\log p(\\mathbf{x}_{T},T)-\\mathrm{KL}\\left[\\log p(\\mathbf{x},T)\\parallel p_{\\mathsf{feas}}(\\mathbf{x})\\right].$$\n\nTherefore, we can substitute the cross-entropy term E∼(1)log p*noise* (xT ) of rewrite the elbo in Eq. (4) as to obtain\n\n$$\\mathbb{E}_{p_{\\sf{data}}\\left(\\mathbf{x}\\right)}\\log q(\\mathbf{x},T)\\geq-\\text{KL}\\left[p(\\mathbf{x},T)\\parallel p_{\\sf{data}}(\\mathbf{x})\\right]+\\mathbb{E}_{\\sim(1)}\\log p(\\mathbf{x}_{T},T)-K(T)+R(T)-\\mathcal{G}(\\mathbf{s_{\\theta}},T).\\tag{7}$$\n\nBefore concluding our derivation it is necessary to introduce an important Proposition observation (formal proof in Appendix), where we show how to combine different terms of Eq. (7) into the negative entropy term Ep*data* (x)log p*data* (x).\n\nProposition 1. Given the stochastic dynamics defined in Eq. (1)*, it holds that*\n\n$$\\mathbb{E}_{\\sim({\\LARGE1})}\\log p(\\mathbf{x}_{T},T)-K(T)+R(T)=\\mathbb{E}_{p_{\\sf{data}}(\\mathbf{x})}\\log p_{\\sf{data}}(\\mathbf{x}).$$\nFinally, we can now bound the value of Ep*data* (x)log q(x, T) as\n$$\\mathbb{E}_{p_{\\mathsf{data}}\\,(\\mathbf{x})}\\log q(\\mathbf{x},T)\\geq\\underbrace{\\mathbb{E}_{p_{\\mathsf{data}}\\,(\\mathbf{x})}\\log p_{\\mathsf{data}}\\,(\\mathbf{x})-\\mathcal{G}(\\mathbf{s}_{\\mathbf{\\theta}},T)-\\operatorname{KL}\\left[p(\\mathbf{x},T)\\parallel p_{\\mathsf{mean}}(\\mathbf{x})\\right]}_{C_{\\mathsf{train}}(\\mathbf{s}_{\\mathbf{\\theta}},T)}.$$\n$$\\mathbf{\\Sigma}$$\n$$\\mathbf{\\Sigma}$$\n\n$$\\mathbf{\\Sigma}$$\n\nEq. (9) clearly emphasizes the roles of an approximate score function, through the gap term G(·), and the discrepancy between the noise distribution of the forward process, and the initial distribution of the reverse\n\n| Diffusion process | p(xt, t | x0) = N(m, sI) | pnoise (x) | | | |\n|---------------------|----------------------------------|-------------------------------|----------------------------|---------|---------|\n| Variance Exploding | α(t) = 0, g(t) = pdσ2(t) dt | m = x0, s = σ 2 (t) − σ 2 (0) | N(0, (σ 2 (T) − σ 2 (0))I) | | |\n| − 1 2 R t | − R t | | | | |\n| Variance Preserving | α(t) = − 1 2 β(t), g(t) = p β(t) | m = e | 0 β(dτ) x0, s = 1 − e | 0 β(dτ) | N(0, I) |\n\nTable 1: Two main families of diffusion processes, where σ 2(t) = σ 2 max σ2 min tand β(t) = β0 + (β1 − β0)t\nprocess, through the kl term. The (negative) entropy term Ep*data* (x)log p*data* (x), which is constant w.r.t T\nand θ, is the best value achievable by the elbo. Indeed, by rearranging Eq. (9), kl [q(x, T) ∥ p*data* (x)] ≤\nG(sθ, T) + kl [p(x, T) ∥ p*noise* (x)]. Optimality is achieved when i) we have perfect *score* matching and ii)\nthe initial conditions for the reverse process are ideal, i.e. q(x, 0) = p(x, T) In the ideal case of perfect score matching, the elbo in Eq. (9) is attained with equality. If, in addition, the initial conditions for the reverse process are ideal, i.e. q(x, 0) = p(x, T), then the results in Anderson (1982) allow us to claim that q(x, T) = p*data* (x).\n\nNext, we show the existence of a tradeoff: the kl decreases with T, while the gap increases with T.\n\n## 2.2 The Tradeoff On Diffusion Time\n\nWe begin by showing that the kl term in Eq. (9) decreases with the diffusion time T, which induces to select large T to maximize the elbo. We consider the two main classes of sdes for the forward diffusion process defined in Eq. (1): sdes whose steady state distribution is the standard multivariate Gaussian, referred to as Variance Preserving (VP), and sdes without a stationary distribution, referred to as *Variance Exploding*\n(VE), which we summarize in Table 1. The standard approach to generate new samples relies on the backward process defined in Eq. (2), and consists in setting p*noise* in agreement with the form of the forward process sde. The following result bounds the discrepancy between the noise distribution p(x, T) and p*noise* .\n\nLemma 1. For the classes of sdes considered (Table 1), the discrepancy between p(x, T) and the p*noise*(x)\ncan be bounded as follows.\n\nFor Variance Preserving sdes, it holds that: kl [p(x, T) ∥ p*noise*(x)] ≤ C1 exp−R T\n0\n\n$$\\beta(t)d t)$$\n\nFor Variance Exploding sdes, it holds that: kl [p(x, T) ∥ p*noise*(x)] ≤ C21 σ2(T)−σ2(0) .\n\nOur proof uses results from Villani (2009), the logarithmic Sobolev Inequality and Gronwall inequality (see Appendix for details). The consequence of Lemma 1 is that to maximize the elbo, the diffusion time T\nshould be as large as possible (ideally, T → ∞), such that the kl term vanishes. This result is in line with current practices for training score-based diffusion processes, that argue for sufficiently long diffusion times\n(De Bortoli et al., 2021). Our analysis, on the other hand, highlights how this term is only one of the two contributions to the elbo.\n\nNow, we focus our attention on studying the behavior of the second component, G(·). Before that, we define a few quantities that allow us to write the next important result.\n\nDefinition 1. We define the optimal score bsθ for any diffusion time T*, as the score obtained using parameters* that minimize I(sθ, T). Similarly, we define the optimal score gap G(bsθ, T) for any diffusion time T*, as the* gap attained when using the optimal score.\n\nLemma 2. The optimal score gap term G(bsθ, T) is a non-decreasing function in T. That is, given T2 > T1, and θ1 = arg minθ I(sθ, T1), θ2 = arg minθ I(sθ, T2)*, then* G(sθ2\n, T2) ≥ G(sθ1\n, T1).\n\nThe proof (see Appendix) is a direct consequence of the definition of G and the optimality of the score. Note that Lemma 2 does not imply that G(sθa\n, T2) ≥ G(sθb\n, T1) holds for generic parameters θa, θb.\n\n## 2.3 Is There An Optimal Diffusion Time?\n\nWhile diffusion processes are generally studied for T → ∞, for practical reasons, diffusion times in score-based models have been arbitrarily set to be \"sufficiently large\" in the literature. Here we formally argue, for the first time, about the existence of an optimal diffusion time, which strikes the right balance between the gap G(·) and the kl terms of the elbo in Eq. (9).\n\nBefore proceeding any further, we clarify that our final objective is not to find and use an optimal diffusion time. Instead, our result on the existence of optimal diffusion times (which can be smaller than the ones set by than popular heuristics) serves the purpose of motivating the choice of small diffusion times, which however call for a method to overcome approximation errors.\n\nProposition 2. Consider the elbo decomposition in Eq. (9). We study it as a function of time T, and seek its optimal argument T\n⋆ = arg maxT Lelbo(bsθ, T)*. Then, the optimal diffusion time* T\n⋆ ∈ R\n+, and thus not necessarily T\n⋆ = ∞. Additional assumptions on the gap term G(·) *can be used to guarantee strict finiteness* of T\n⋆. There exists at least one optimal diffusion *time* T\n⋆in the interval [0, ∞], which maximizes the elbo, that is T\n⋆ = arg maxT Lelbo(bsθ, T).\n\nIt is trivial to verify that, since the optimal gap term G(bsθ, T) is a non decreasing function in T (Lemma 2),\nwe have ∂G\n∂T ≥ 0.Then, we study the sign of the kl derivative, which is always negative as shown in the Appendix. Moreover, we know that that lim T→∞\n∂kl\n∂T = 0. Consequently, the function ∂Lelbo\n∂T =\n∂G\n∂T +\n∂kl\n∂T has at least one zero in its domain R\n+. To guarantee a stricter bounding of T\n⋆, we could study asymptotically the growth rates of G and the kl terms for large T. The investigation is technically involved and outside the scope of this paper. Nevertheless, as discussed hereafter, the numerical investigation carried out in this work suggests finiteness of T\n⋆.\n\nWhile the proof for the general case is available in the Appendix, the analytic solution for the optimal diffusion time is elusive, as a full characterization of the gap term is particularly challenging. Additional assumptions would guarantee boundedness of T\n⋆.\n\n\n\nFigure 2: elbo decomposition, elbo and likelihood for a 1D toy model, as a function of diffusion time T. Tradeoff and optimality numerical results confirm our theory.\nEmpirically, we use Fig. 2 to illustrate the tradeoff and the optimality arguments through the lens of the same toy example we use in § 1. On first and third column, we show the elbo decomposition. We can verify that G(sθ, T) is an increasing function of T, whereas the kl term is a decreasing function of T. Even in the simple case of a toy example, the tension between small and large values of T is clear. On the second and fourth, we show the values of the elbo and of the likelihood as a function of T. We then verify the validity of our claims: the elbo is neither maximized by an infinite diffusion time, nor by a \"sufficiently large\" value.\n\nInstead, there exists an optimal diffusion time which, for this example, is smaller than what is typically used in practical implementations, i.e. T = 1.0 In the Appendix , we show that optimizing the elbo to obtain an optimal diffusion time T\n⋆is technically feasible, without resorting to exhaustive grid search. In § 3, we present a new method that admits much smaller diffusion times, and show that the elbo of our approach is at least as good as the one of a standard diffusion model, configured to use its optimal diffusion time T\n⋆.\n\n## 2.4 Relation With Diffusion Process Noise Schedule\n\nWe remark that a simple modification of the noise schedule to steer the the diffusion process toward a small diffusion time (Kingma et al., 2021; Bao et al., 2022) is not a viable solution. In the Appendix, we discuss how the optimal value of the elbo, in the case of affine sdes, is *invariant* to the choice of the noise schedule.\n\nIndeed, its value depends uniquely on the relative level of corruption of the initial data at the considered final diffusion time T, that is, the *Signal to Noise Ratio*. Naively, we could think that by selecting a twice as fast noise schedule, we would be able to obtain the same elbo of the original schedule by diffusing only for half the time. While true, this does not provide any practical benefit in terms of computational complexity. If the noise schedule is faster, the drift terms involved in the reverse process changes more rapidly. Consequently, to *simulate* the reverse sde with a numerical integration scheme, smaller step sizes are required to keep the same accuracy of the original noise schedule simulation. The net effect is that while the diffusion time for the continuous time dynamics is smaller, the number of integration steps is larger, with a zero net gain. The optimization of the noise schedule can however have important practical effects in terms of stability of the training and variance of the estimations, that we do not tackle in this work (Kingma et al., 2021).\n\n## 2.5 Relation With Literature On Bounds And Goodness Of Score Assumptions\n\nFew other works in the literature attempt to study the convergence properties of Diffusion models. In the work of De Bortoli et al. (2021) (Thm. 1), a total variation (TV) bound between the generated and data distribution is obtained in the form C1 exp(a1T) + C2 exp(−a2T), where the constant C1 depends on the maximum error over [0, T] between the true and approximated score, i.e. maxt∈[0,T] ∥sθ(x, t) − ∇ log p(x, t)∥. In the work of De Bortoli (2022) the requirement is relaxed by setting maxt∈[0,T]\nσ 2(t)\n1+∥x∥\n∥sθ(x, t) − ∇ log p(x, t)∥, where the 1-Wasserstein distance between generated and true data is bounded as C1 + C2 exp(−a2T) + C3 (Thm. 1).\n\nOther works consider the more realistic average square norm instead of the infinity norm, which is consistent with standard training of diffusion models. Moreover, Lee et al. (2022a) show how the TV bound can be expressed as a function of maxt∈[0,T] E\nh∥sθ(xt, t) − ∇ log p(xt, t)∥\n2i(Thms. 2.2,3.1,3.2). Related to our work, Lee et al. (2022a) find that the TV bound is optimized for a diffusion time that depends, among others, on the maximum score error. Finally, the work by Chen et al. (2022) (Thm. 2), which is concurrent to ours, shows that if maxt∈[0,T] E\nh∥sθ(xt, t) − ∇ log p(xt, t)∥\n2iis bounded, then the TV distance between true and generated data can be bound as C1 exp(−a1T) + √ϵT, plus a discretization error.\n\nAll prior approaches require assumptions on the maximum score error, which *implicitly* depends on: i) the maximum diffusion time T and ii) the class of parametric score networks considered. Hence, such methods allow for the study of convergence properties, but with the following limitations. It is not clear how the score error behaves as the fitting domain ([0, T]) is increased, for generic class of parametric functions and generic p*data* . Moreover, it is difficult to link the error assumptions with the actual training loss of diffusion models.\n\nIn this work, instead, we follow a more agnostic path, as we make no assumptions about the error behavior.\n\nWe notice that the optimal gap term is **always** a non decreasing function of T. First, we question whether current best practice for setting diffusion times are adequate: we find that in realistic implementations, diffusion times are larger than necessary. Second, we introduce an new approach, with provably the same performance of standard diffusion models but lower computational complexity, as highlighted in § 3.\n\n## 3 A New, Practical Method For Decreasing Diffusion Times\n\nThe elbo decomposition in Eq. (9) and the bounds in Lemma 1 and Lemma 2 highlight a dilemma. We thus propose a simple method that allows us to achieve **both** a small gap G(sθ, T), and a small discrepancy kl [p(x, T) ∥ p*noise* (x)]. Before that, let us use Fig. 3 to summarize all densities involved and the effects of the various approximations, which will be useful to visualize our proposal.\n\nThe data distribution p*data* (x) is transformed into the noise distribution p(x, T) through the forward diffusion process. Ideally, starting from p(x, T)\nwe can recover the data distribution by simulating using the exact score ∇ log p. Using the approximated score sθ and the same initial conditions, the backward process ends up in q\n(1)(x, T), whose discrepancy 1 to pdata (x) is G(sθ, T). However, the distribution p(x, T) is unknown and replaced with an easy distribution p*noise* (x), accounting for an error a measured as kl [p(x, T) ∥ p*noise* (x)]. With score and initial distribution approximated, the backward process ends up in q\n(3)(x, T), where the discrepancy 3 from p*data* is the sum of terms G(sθ, T) + kl [p(x, T) ∥ p*noise* ].\n\n\n\nFigure 3: Intuitive illustration of the forward and backward diffusion processes. Discrepancies between distributions are illustrated as distances. Color coding discussed in the text.\nMultiple bridges across densities. In a nutshell, our method allows us to reduce the gap term by selecting smaller diffusion times and by using a learned auxiliary model to transform the initial density p*noise* (x) into a density νϕ(x), which is as close as possible to p(x, T), thus avoiding the penalty of a large kl term. To implement this, we first *transform* the simple distribution p*noise* into the distribution νϕ(x), whose discrepancy b kl [p(x, T) ∥ νϕ(x)] is smaller than a . Then, starting from from the auxiliary model νϕ(x),\nwe use the approximate score sθ to simulate the backward process reaching q\n(2)(x, T). This solution has a discrepancy 2 from the data distribution of G(sθ, T) + kl [p(x, T) ∥ νϕ(x)], which we will quantify later in the section. Intuitively, we introduce two bridges. The first bridge connects the noise distribution p*noise* to an auxiliary distribution νϕ(x) that is as close as possible to that obtained by the forward diffusion process.\n\nThe second bridge—a standard reverse diffusion process—connects the smooth distribution νϕ(x) to the data distribution. Notably, our approach has important guarantees, which we discuss next.\n\n## 3.1 Auxiliary Model Fitting And Guarantees\n\nWe begin by stating the requirements we consider for the density νϕ(x). First, as it is the case for p*noise* , it should be easy to generate samples from νϕ(x) in order to initialize the reverse diffusion process. Second, the auxiliary model should allow us to compute the likelihood of the samples generated through the overall generative process, which begins in p*noise* , passes through νϕ(x), and arrives in q(x, T).\n\nThe fitting procedure of the auxiliary model is straightforward. First, we recognize that minimizing kl [p(x, T) ∥ νϕ(x)] w.r.t ϕ also minimizes Ep(x,T)[log νϕ(x)], that we can use as loss function. To obtain the set of optimal parameters ϕ\n⋆, we require samples from p(x, T), which can be easily obtained even if the density p(x, T) is not available. Indeed, by sampling from p*data* , and p(x, T | x0), we obtain an unbiased Monte Carlo estimate of Ep(x,T)[log νϕ(x)], and optimization of the loss can be performed. Note that due to the affine nature of the drift, the conditional distribution p(x, T | x0) is easy to sample from, as shown in Table 1. From a practical point of view, it is important to notice that the fitting of νϕ is independent from the training of the score-matching objective, i.e. the result of I(sθ) does not depend on the shape of the auxiliary distribution νϕ. This observation indicates that the two training procedures can be run concurrently, thus enabling considerable time savings.\n\nNext, we show that the first bridge in our model reduces the kl term, even for small diffusion times.\n\nProposition 3. Let's assume that pnoise(x) is in the family spanned by νϕ, i.e. there exists ϕe *such that*\nνϕe = pnoise*. Then we have that*\n$$KL\\left[p(\\mathbf{x},T)\\parallel\\nu_{\\mathbf{\\phi}^{*}}(\\mathbf{x})\\right]\\leq KL\\left[p(\\mathbf{x},T)\\parallel\\nu_{\\mathbf{\\phi}}^{*}(\\mathbf{x})\\right]=KL\\left[p(\\mathbf{x},T)\\parallel p_{\\sf meas}(\\mathbf{x})\\right].\\tag{10}$$\nSince we introduce the auxiliary distribution ν, we shall define a new elbo for our method:\n\n$T$) = $\\mathbb{K}_{\\text{pdelta}}\\left(x\\right)$ l0. \nL\nϕ elbo(sθ, T) = Ep*data* (x)log pdata (x) − G(sθ, T) − kl [p(x, T) ∥ νϕ(x)] (11)\n\n$$(11)$$\n\nRecalling that bsθ is the optimal score for a generic time T, Proposition 3 allows us to claim that L\nϕ\n⋆\nelbo(bsθ, T) ≥\nLelbo(bsθ, T). Then, we can state the following important result:\nProposition 4. *Given the existence of* T\n⋆, defined as the diffusion time such that the elbo *is maximized*\n(Proposition 2), there exists at least one diffusion time τ ≤ T\n⋆*, such that* L\nϕ\n⋆\nelbo(bsθ, τ ) ≥ Lelbo(bsθ, T ∗).\n\nProposition 4 has two interpretations. On the one hand, given two score models optimally trained for their respective diffusion times, our approach guarantees an elbo that is at least as good as that of a standard diffusion model configured with its optimal time T\n⋆. Our method achieves this with a smaller diffusion time τ , which offers sampling efficiency and generation quality. On the other hand, if we settle for an equivalent elbo for the standard diffusion model and our approach, with our method we can afford a sub-optimal score model, that requires a smaller computational budget to be trained, while guaranteeing shorter sampling times. We elaborate on this interpretation in § 4, where our approach obtains substantial savings in terms of training iterations.\n\nA final note is in order. The choice of the auxiliary model depends on the selected diffusion time. The larger the T, the \"simpler\" the auxiliary model can be. Indeed, the noise distribution p(x, T) approaches p*noise* , so that a simple auxiliary model is sufficient to transform p*noise* into a distribution νϕ. Instead, for a small T,\nthe distribution p(x, T) is closer to the data distribution. Then, the auxiliary model requires high flexibility and capacity. In § 4, we substantiate this discussion with numerical examples and experiments on real data.\n\n## 3.2 Comparison With Schrödinger Bridges\n\nIn this Section, we briefly compare our method with the Schrödinger bridges approach (Chen et al., 2021b;a; De Bortoli et al., 2021), which allows one to move from an arbitrary pnoise to p*data* in any finite amount of time T. This is achieved by simulating the sde\n\n$$\\mathrm{d}\\mathbf{x}_{t}=\\left[-\\mathbf{f}(\\mathbf{x}_{t},t^{\\prime})+g^{2}(t^{\\prime})\\mathbf{\\nabla}\\log\\hat{\\psi}(\\mathbf{x}_{t},t^{\\prime})\\right]\\mathrm{d}t+g(t^{\\prime})\\mathrm{d}\\mathbf{w}_{t},\\quad\\mathbf{x}_{0}\\sim\\mathbf{j}_{\\mathrm{max}}\\,,\\tag{12}$$\n\nwhere *ψ, ψ* ˆ solve the Partial Differential Equation (pde) system\n\n$$\\begin{cases}\\frac{\\partial\\psi(\\mathbf{x},t)}{\\partial t}=-\\mathbf{\\nabla}^{\\top}\\left(\\psi(\\mathbf{x},t)\\right)\\mathbf{f}(\\mathbf{x},t)-\\frac{g^{2}(t)}{2}\\Delta(\\psi(\\mathbf{x},t)),\\\\ \\frac{\\partial\\psi}{\\partial t}=-\\mathbf{\\nabla}^{\\top}\\left(\\hat{\\psi}(\\mathbf{x},t)\\mathbf{f}(\\mathbf{x},t)\\right)+\\frac{g^{2}(t)}{2}\\Delta(\\hat{\\psi}(\\mathbf{x},t)),\\end{cases}\\qquad\\quad\\psi(\\mathbf{x},0)\\hat{\\psi}(\\mathbf{x},0)=p_{\\texttt{data}}\\left(\\mathbf{x},t\\right),$$\n$\\psi(\\mathbf{x},0)\\hat{\\psi}(\\mathbf{x},0)=p_{\\texttt{data}}\\left(\\mathbf{x}\\right),\\psi(\\mathbf{x},T)\\hat{\\psi}(\\mathbf{x},T)=p_{\\texttt{data}}\\left(\\mathbf{x}\\right)$. \n$\\mathbf{a}\\cdot\\mathbf{a}=0$. \n(13)\nThis approach presents drawbacks compared to classical Diffusion models. First, the functions ψ, ψˆ are not known, and their parametric approximation is costly and complex. Second, it is much harder to obtain quantitative bounds between true and generated data as a function of the quality of such approximations.\n\nThe *ψ, ψ* ˆ estimation procedure simplifies considerably in the particular case where p*noise* (x) = p(x, T), for arbitrary T. The solution of Eq. (13) is indeed ψ(x, t) = 1, ψˆ(x, t) = p(x, t). The first pde of the system is satisfied when ψ is a constant. The second pde is the Fokker-Planck equation, satisfied by ψˆ(x, t) = p(x, t).\n\nBoundary conditions are also satisfied. In this scenario, a sensible objective is the score-matching, as getting\n∇ log ψˆ equal to the true score ∇ log p allows perfect generation.\n\nUnfortunately, it is difficult to generate samples from p(x, T), the starting conditions of Eq. (12). A trivial solution is to select T → ∞ in order to have p*noise* as the simple and analytically known steady state distribution of Eq. (1). This corresponds to the classical diffusion models approach, which we discussed in the previous Sections. An alternative solution is to keep T finite and *cover* the first part of the bridge from pnoise to p(x, T)\nwith an auxiliary model. This provides a different interpretation of our method, which allows for smaller diffusion times while keeping good generative quality.\n\n## 3.3 An Extension For Density Estimation\n\nDiffusion models can be also used for density estimation by transforming the diffusion sde into an equivalent Ordinary Differential Equation (ode) whose marginal distribution p(x, t) at each time instant coincide to that of the corresponding sde (Song et al., 2021c). The exact equivalent ode requires the score ∇ log p(xt, t),\nwhich in practice is replaced by the score model sθ, leading to the following ode\n\n$${\\rm d}\\mathbf{x}_{t}=\\left(\\mathbf{f}(\\mathbf{x}_{t},t)-\\frac{1}{2}g(t)^{2}\\mathbf{s}_{\\mathbf{\\theta}}(\\mathbf{x}_{t},t)\\right){\\rm d}t\\quad\\mbox{with}\\quad\\mathbf{x}_{0}\\sim p_{\\rm data}\\,\\tag{14}$$\n\nwhose time varying probability density is indicated with pe(x, t). Note that the density pe(x, t), is in general not equal to the density p(x, t) associated to Eq. (1), with the exception of perfect score matching (Song et al., 2021b). The reverse time process is modeled as a Continuous Normaxlizing Flow (cnf) (Chen et al.,\n2018; Grathwohl et al., 2019) initialized with distribution p*noise* (x); then, the likelihood of a given value x0 is\n\n$$\\log\\widetilde{p}(\\mathbf{x}_{0})=\\log p_{\\mathsf{train}}(\\mathbf{x}_{T})+\\int\\limits_{t=0}^{T}\\mathbf{\\nabla}\\mathbf{\\cdot}\\left(\\mathbf{f}(\\mathbf{x}_{t},t)-\\frac{1}{2}g(t)^{2}\\mathbf{s}_{\\mathbf{\\theta}}(\\mathbf{x}_{t},t)\\right)\\mathrm{d}t.\\tag{15}$$\n\nTo use our proposed model for density estimation, we also need to take into account the ode dynamics. We focus again on the term log p*noise* (xT ) to improve the expected log likelihood. For consistency, our auxiliary density νϕ should now maximize E∼(14)log νϕ(xT ) instead of E∼(1)log νϕ(xT ). However, the simulation of Eq. (14) requires access to sθ which, in the endeavor of density estimation, is available only once the score model has been trained. Consequently optimization w.r.t. ϕ can only be performed sequentially, whereas for generative purposes it could be done concurrently. While the sequential version is expected to perform better, experimental evidence indicates that improvements are marginal, justifying the adoption of the more efficient concurrent version.\n\n## 4 Experiments\n\nWe now present numerical results on the mnist and cifar10 datasets, to support our claims in §§ 2 and 3.\n\nWe follow a standard experimental setup (Song et al., 2021a;b; Huang et al., 2021; Kingma et al., 2021): we use a standard U-Net architecture with time embeddings (Ho et al., 2020) and we report the log-likelihood in terms of bit per dimension (bpd) and the Fréchet Inception Distance (fid) scores (uniquely for cifar10).\n\nAlthough the fid score is a standard metric for ranking generative models, caution should be used against over-interpreting fid improvements (Kynkäänniemi et al., 2022). Similarly, while the theoretical properties of the models we consider are obtained through the lens of elbo maximization, the log-likelihood measured in terms of bpd should be considered with care (Theis et al., 2016). Finally, we also report the number of neural function evaluations (nfe) for computing the relevant metrics. We compare our method to the standard score-based model (Song et al., 2021c). The full description on the experimental setup is presented in Appendix.\n\nOn the existence of T\n⋆. We look for further empirical evidence of the existence of a T\n⋆ < ∞, as stated in Proposition 2. For the moment, we shall focus on the baseline model (Song et al., 2021c),\nwhere no auxiliary models are introduced. Results are reported in Table 2. For mnist, we observe how times T = 0.6 and T = 1.0 have comparable performance in terms of bpd, implying that any T ≥ 1.0 is at best unnecessary and generally detrimental.\n\nSimilarly, for cifar10, it is possible to notice that the best value of bpd is achieved for T = 0.6, outperforming all other values.\n\nTable 2: Optimal T in (Song et al., 2021c)\n\n| Dataset | Time T | bpd (↓) |\n|-----------|----------|-----------|\n| mnist | 1.0 | 1.16 |\n| 0.6 | 1.16 | |\n| 0.4 | 1.25 | |\n| 0.2 | 1.75 | |\n| cifar10 | 1.0 | 3.09 |\n| 0.6 | 3.07 | |\n| 0.4 | 3.09 | |\n| 0.2 | 3.38 | |\n\nOur auxiliary models. In § 3 we introduced an auxiliary model to minimize the mismatch between initial distributions of the backward process. We now specify the family of parametric distributions we have considered. Clearly, the choice of an auxiliary model also depends on the data distribution, in addition to the choice of diffusion time T.\n\nFor our experiments, we consider two auxiliary models: (i)\na Dirichlet process Gaussian mixture model (dpgmm) (Rasmussen, 1999; Görür & Edward Rasmussen, 2010) for mnist and (ii) Glow Kingma & Dhariwal (2018), a flexible normalizing flow for cifar10. Both of them satisfy our requirements: they allow exact likelihood computation and they are equipped with a simple sampling procedure. As discussed in § 3, auxiliary model complexity should be adjusted as a function of T. This is confirmed experimentally in Fig. 4, where we use the number of mixture components of the dpgmm as a proxy to measure the complexity of the auxiliary model.\n\n\n\nFigure 4: Complexity of the auxiliary model as function of diffusion time (reported median and 95 quantiles on 4 random seeds).\nReducing T **with auxiliary models.** We now show how it is possible to obtain a comparable (or better)\nperformance than the baseline model for a wide range of diffusion times T. For mnist, setting τ = 0.4 produces good performance both in terms of bpd (Table 3) and visual sample quality (Fig. 5). We also consider the sequential extension (S) to compute the likelihood, but remark marginal improvements compared to a concurrent implementation. Similarly for the cifar10 dataset, in Table 4 we observe how our method achieves better bpd than the baseline diffusion for T = 1. Moreover, our approach outperforms the baselines for the corresponding diffusion time in terms of fid score (additional non-curated samples in the Appendix).\n\nIn Figure 10 we provide a non curated subset of qualitative results, showing that our method for a diffusion time equal to 0.4 still produces appealing images, while the vanilla approach fails. We finally notice how the proposed method has comparable performance w.r.t. several other competitors, while stressing that many orthogonal to our solutions (like diffusion in latent space (Vahdat et al., 2021), or the selection of higher order schemes (Jolicoeur-Martineau et al., 2021)) can actually be combined with our methodology.\n\nTraining and sampling efficiency In Fig. 7, the horizontal line corresponds to the best performance of a fully trained baseline model for T = 1.0 (Song et al., 2021c). To achieve the same performance of the baseline, variants of our method require fewer iterations, which translate in training efficiency. For the sake of fairness, the total training cost of our method should account for the auxiliary model training, which however can be done concurrently to the diffusion process. As an illustration for cifar10, using four GPUs, the baseline model requires ∼ 6.4 days of training. With our method we trained the auxiliary and diffusion models for ∼ 2.3 and 2 days respectively, leading to a total training time of max{2.3, 2} = 2.3 days. Similar training curves can be obtained for the mnist dataset, where the training time for dpgmms is negligible.\n\nCIFAR10 200 400 600 800 1000 1200 Iterations (thousands)\n3.1 3.2 3.3 ScoreSDE\nOur (T = 0.2) Our (T = 0.4)\nOur (T = 0.6)\nBPD\nFigure 7: Training curves of score models for different diffusion time T, recorded during the span of 1.3 millions iterations.\n\nTable 3: Experiment results on mnist. For our method, (S) is for the extension in § 3.3\n\n| | nfe(↓) | bpd (↓) |\n|--------------------|----------|---------------|\n| | Model | (ode) |\n| ScoreSDE | 300 | 1.16 |\n| ScoreSDE (T = 0.6) | 258 | 1.16 |\n| Our (T = 0.6) | 258 | 1.16 1.14 (S) |\n| ScoreSDE (T = 0.4) | 235 | 1.25 |\n| Our (T = 0.4) | 235 | 1.17 1.16 (S) |\n| ScoreSDE (T = 0.2) | 191 | 1.75 |\n| Our (T = 0.2) | 191 | 1.33 1.31 (S) |\n\nFigure 5: Visualization of some samples\n\n\n\n| fid(↓) | bpd (↓) | nfe (↓) | nfe (↓) | |\n|--------------------------------------------------|--------------------|---------------|-----------|-----|\n| Model | (sde) | (ode) | | |\n| ScoreSDE (Song et al., 2021c) | 3.64 | 3.09 | 1000 | 221 |\n| ScoreSDE (T = 0.6) | 5.74 | 3.07 | 600 | 200 |\n| ScoreSDE (T = 0.4) | 24.91 | 3.09 | 400 | 187 |\n| ScoreSDE (T = 0.2) | 339.72 | 3.38 | 200 | 176 |\n| Our (T = 0.6) | 3.72 | 3.07 | 600 | 200 |\n| Our (T = 0.4) | 5.44 | 3.06 | 400 | 187 |\n| Our (T = 0.2) | 14.38 | 3.06 | 200 | 176 |\n| ARDM (Hoogeboom et al., 2022) | − | 2.69 | 3072 | |\n| VDM(Kingma et al., 2021) | 4.0 | 2.49 | 1000 | |\n| D3PMs (Austin et al., 2021) | 7.34 | 3.43 | 1000 | |\n| DDPM (Ho et al., 2020) | 3.21 | 3.75 | 1000 | |\n| Gotta Go Fast (Jolicoeur-Martineau et al., 2021) | 2.44 | − | 180 | |\n| LSGM (Vahdat et al., 2021) | 2.10 | 2.87 | 120/138 | |\n| ARDM-P (Hoogeboom et al., 2022) | − | 2.68/2.74 | 200/50 | |\n| Real data | ScoreSDE (T = 0.4) | Our (T = 0.4) | | |\n\nTable 4: Experimental results on cifar10, including other relevant baselines and sampling efficiency enhancements from the literature.\n\n\n\nSampling speed benefits are evident from Tables 3 and 4. When considering the sde version of the methods the number of sampling steps can decrease linearly with T, in accordance with theory (Kloeden & Platen, 1995), while retaining good bpd and fid scores. Similarly, although not in a linear fashion, the number of steps of the ode samplers can be reduced by using a smaller diffusion time T. Finally, we test the proposed methodology on the more challenging celeba 64x64 dataset. In this case, we use a variance exploding diffusion and we consider again Glow as the auxiliary model. The results, presented in Table 6, report the log-likelihood performance of different methods (qualitative results are reported in Appendix). On the two extremes of the complexity we have the original diffusion (VE, T = 1.0) with the best bpd and the highest complexity, and Glow which provides a much simpler scheme with worse performance. In the table we report the bpd and the nfe metrics for smaller diffusion times, in three different configurations:\nnaively neglecting the mismatch (ScoreSDE) or using the auxiliary model (Our). Interestingly, we found that the best results are obtained by using a combination of diffusion models pretrained for T = 1.0. The summary of the content of this table is the following: by accepting a small degradation in terms of bpd we can reduce the computational cost by almost one order of magnitude. We think it would be interesting to study more performing auxiliary models to improve performance of our method on challenging datasets.\n\n## 5 Conclusion\n\nDiffusion-based generative models emerged as an extremely competitive approach for a wide range of application domains. In practice, however, these models are resource hungry, both for their training and for sampling new data points. In this work, we have introduced the key idea of considering diffusion times T\nas a free variable which should be chosen appropriately. We have shown that the choice of T introduces a trade-off, for which an optimal \"sweet spot\" exists. In standard diffusion-based models, smaller values of T\n\n| bpd (↓) | nfe (↓) | |\n|---------------------------------------|-----------|----|\n| Model | (ode) | |\n| ScoreSDE (Song et al., 2021c) | 2.13 | 68 |\n| ScoreSDE (T = 0.5) | 8.06 | 15 |\n| ScoreSDE (T = 0.2) | 12.1 | 9 |\n| Our (T = 0.5) | 2.48 | 16 |\n| Our (T = 0.2) | 2.58 | 9 |\n| Our with pretrain diffusion (T = 0.5) | 2.36 | 16 |\n| Our with pretrain diffusion (T = 0.2) | 2.32 | 9 |\n| Glow (Kingma & Dhariwal, 2018) | 3.74 | 1 |\n\nTable 5: Experimental results on CELEBA 64\nare preferable for efficiency reasons, but sufficiently large T are required to reduce approximation errors of the forward dynamics. Thus, we devised a novel method that allows for an arbitrary selection of diffusion times, where even small values are allowed. Our method closes the gap between practical and ideal diffusion dynamics, using an auxiliary model. Our empirical validation indicated that the performance of our approach was comparable and often superior to standard diffusion models, while being efficient both in training and sampling. Limitations. In this work, the experimental protocol has been defined to corroborate our methodological contribution, and not to achieve state-of-the-art performance. A more extensive empirical evaluation of model architectures, sampling methods, and additional datasets could benefit practitioners in selecting an appropriate configuration of our method. 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In *International Conference on Learning Representations*, 2022.\n\nHuangjie Zheng, Pengcheng He, Weizhu Chen, and Mingyuan Zhou. Truncated diffusion probabilistic models. *CoRR*,\nabs/2202.09671, 2022.\n\n## A Generic Definitions And Assumptions\n\nOur work builds upon the work in Song et al. (2021b), which should be considered as a basis for the\ndevelopments hereafter. In this supplementary material we use the following shortened notation for a generic\nω > 0:\n$${\\mathcal{N}}_{\\omega}(\\mathbf{x})\\ {\\stackrel{\\mathrm{def}}{=}}\\ {\\mathcal{N}}(\\mathbf{x};\\mathbf{0},\\omega\\mathbf{I}).$$\ndef = N (x; 0, ωI). (16)\nIt is useful to notice that ∇ log(Nω(x)) = −\n1 ω x.\n\nFor an arbitrary probability density p(x) we define the convolution (∗ operator) with Nω using notation\n\n$$(16)$$\n$$p_{\\omega}(\\mathbf{x})=p(\\mathbf{x})*{\\mathcal{N}}_{\\omega}(\\mathbf{x}).$$\n$$(17)$$\n$$(18)$$\npω(x) = p(x) ∗ Nω(x). (17)\nEquivalently, pω(x) = expω 2 ∆p(x), and consequently dpω(x)\ndω =\n1 2∆p(x), where ∆ = ∇⊤∇. Notice that by considering the Dirac delta function δ(x), we have the equality δω(x) = Nω(x).\n\nIn the following derivations, we make use of the Stam–Gross logarithmic Sobolev inequality result in (Villani, 2009, p. 562 Example 21.3):\n\n$$\\operatorname{KL}\\left[p(\\mathbf{x})\\parallel\\mathcal{N}_{\\omega}(\\mathbf{x})\\right]=\\int p(\\mathbf{x})\\log\\left({\\frac{p(\\mathbf{x})}{\\mathcal{N}_{\\omega}(\\mathbf{x})}}\\right)\\mathrm{d}\\mathbf{x}\\leq{\\frac{\\omega}{2}}\\int\\left\\|\\nabla\\left(\\log{\\frac{p(\\mathbf{x})}{\\mathcal{N}_{\\omega}(\\mathbf{x})}}\\right)\\right\\|^{2}p(\\mathbf{x})\\mathrm{d}\\mathbf{x}.$$\n\n## B Deriving Equation (4) From Huang Et Al. **(2021)**\n\nWe start with Eq. (25) of Huang et al. (2021) which, in our notation, reads\n\n$$\\log g(\\mathbf{x},T)\\geq\\mathbb{E}\\left[\\log p_{\\mathbf{x}_{0}}(\\mathbf{x}_{T})\\mid\\mathbf{x}_{0}=\\mathbf{x}\\right]-\\int_{0}^{T}\\mathbb{E}\\left[\\frac{1}{2}g^{2}(t)\\|\\mathbf{x}_{0}(\\mathbf{x}_{t})\\|^{2}+\\nabla^{\\top}\\left(g^{2}(t)\\mathbf{x}_{0}(\\mathbf{x}_{t})-\\mathbf{f}(\\mathbf{x}_{t},t)\\right)\\mid\\mathbf{x}_{0}=\\mathbf{x}\\right]\\mathrm{d}t.$$\n\nThe first step is to take the expected value w.r.t x0 ∼ p*data* on both sides of the above inequality\n\n$$\\mathbb{E}_{p_{\\text{data}}}\\left[\\log q(\\mathbf{x},T)\\right]\\geq\\mathbb{E}\\left[\\log p_{\\text{data}}(\\mathbf{x}_{T})\\right]-\\int_{0}^{T}\\mathbb{E}\\left[\\frac{1}{2}g^{2}(t)\\|\\mathbf{s_{\\theta}}(\\mathbf{x}_{t})\\|^{2}+\\nabla^{\\top}\\left(g^{2}(t)\\mathbf{s_{\\theta}}(\\mathbf{x}_{t})-\\mathbf{f}(\\mathbf{x}_{t},t)\\right)\\right]\\text{d}t.$$\n\nWe focus on rewriting the term\n\nZ T\n0\nE\n-∇⊤g\n2(t)sθ(xt) − f(xt, t) dt =\nZ T\n0\np(x, t|x0)p*data* (x0)∇⊤g\n2(t)sθ(x) − f(x, t)dxdx0dt =\n−\nZ T\n0\n∇⊤ (p(x, t|x0)p*data* (x0)) g\n2(t)sθ(x) − f(x, t)dxdx0dt =\n−\nZ T\n0\n(p(x, t|x0)pdata (x0))−1 ∇⊤ (p(x, t|x0)p*data* (x0)) g\n2(t)sθ(x) − f(x, t)(p(x, t|x0)p*data* (x0)) dxdx0dt =\n−\nZ T\n0\n∇⊤ (log(p(x, t|x0)) + log(p*data* (x0))) g\n2(t)sθ(x) − f(x, t)(p(x, t|x0)p*data* (x0)) dxdx0dt =\n−\nZ T\n0\n∇⊤ (log(p(x, t|x0))) g\n2(t)sθ(x) − f(x, t)(p(x, t|x0)p*data* (x0)) dxdx0dt =\n−\nZ T\n0\nE\n-∇⊤ (log(p(xt, t|x0))) g\n2(t)sθ(xt) − f(xt, t) dt.\n$$(19)$$\nConsequently, we can rewrite the r.h.s of Equation (19) as\n\nE [log pnoise (xT )] − Z T 0 E 1 2 g 2(t)∥sθ(xt)∥ 2 − g 2(t)∇⊤ (log(p(xt, t|x0))) sθ(x) + g 2(t)∇⊤ (log(p(xt, t|x0))) f(x, t) dt = E [log pnoise (xT )] − Z T 0 E 1 2 g 2(t)∥sθ(xt) − ∇ (log(p(xt, t|x0)))∥ 2dt− 1 2 Z T 0 E -g 2(t)∥∇ (log(p(xt, t|x0)))∥ + 2∇⊤ (log(p(xt, t|x0))) f(x, t)dt,\n\nthat is exactly Equation (4).\n\n## C Proof Of **Eq. (5)**\n\nWe prove the following result\n\n$$I(\\mathbf{s_{\\theta}},T)\\geq\\underbrace{I(\\mathbf{\\nabla}\\log p,T)}_{\\triangleq I_{K(T)}}=\\frac{1}{2}\\int\\limits_{t=0}^{T}g^{2}(t)\\mathbb{E}_{\\sim(1)}\\left[\\left\\|\\mathbf{\\nabla}\\log p(\\mathbf{x}_{t},t)-\\mathbf{\\nabla}\\log p(\\mathbf{x}_{t},t\\,|\\,\\mathbf{x}_{0})\\right\\|\\right]^{2}\\mathrm{d}t.$$\n\nProof. We prove that for generic positive λ(·), and T2 > T1 the following holds:\n\n$$\\int\\limits_{t=T_{1}}^{T_{2}}\\lambda(t)\\mathbb{E}_{\\sim(\\cdot)}\\left[||\\mathbf{s}(\\mathbf{x}_{t},t)-\\nabla\\log p(\\mathbf{x}_{t},t|\\mathbf{x}_{0})||^{2}\\right]\\mathrm{d}t\\geq\\int\\limits_{t=T_{1}}^{T_{2}}\\lambda(t)\\mathbb{E}_{\\sim(\\cdot)}\\left[||\\nabla\\log p(\\mathbf{x}_{t},t)-\\nabla\\log p(\\mathbf{x}_{t},t|\\mathbf{x}_{0})||^{2}\\right]\\mathrm{d}t.\\tag{20}$$\n\nFirst we compute the functional derivative (w.r.t s)\n\n$$\\frac{\\delta}{\\delta\\sigma}\\int\\limits_{t=T_{1}}^{T_{2}}\\lambda(t)\\mathbb{E}_{\\sim(1)}\\left[\\left|s(\\mathbf{x}_{t},t)-\\nabla\\log p(\\mathbf{x}_{t},t|\\mathbf{x}_{0})\\right|\\right|^{2}\\right]\\mathrm{d}t=2\\int\\limits_{t=T_{1}}^{T_{2}}\\lambda(t)\\mathbb{E}_{\\sim(1)}\\left[\\left(s(\\mathbf{x}_{t},t)-\\nabla\\log p(\\mathbf{x}_{t},t|\\mathbf{x}_{0})\\right)\\right]\\mathrm{d}t=2\\int\\limits_{t=T_{1}}^{T_{2}}\\lambda(t)\\mathbb{E}_{\\sim(1)}\\left[\\left(s(\\mathbf{x}_{t},t)-\\nabla\\log p(\\mathbf{x}_{t},t)\\right)\\right]\\mathrm{d}t,$$\n\nwhere we used\n\n$$\\begin{array}{l}{{\\operatorname{E}_{\\sim(1)}\\left[\\nabla\\log p(\\mathbf{x}_{t},t|\\mathbf{x}_{0})\\right]=\\int\\nabla\\log p(\\mathbf{x},t|\\mathbf{x}_{0})p(\\mathbf{x},t|\\mathbf{x}_{0})p_{\\mathrm{data}}\\left(\\mathbf{x}_{0}\\right)\\mathrm{d}\\mathbf{x}\\mathrm{d}\\mathbf{x}_{0}=}}\\\\ {{\\quad\\quad\\int\\nabla p(\\mathbf{x},t|\\mathbf{x}_{0})p_{\\mathrm{data}}\\left(\\mathbf{x}_{0}\\right)\\mathrm{d}\\mathbf{x}\\mathrm{d}\\mathbf{x}_{0}=\\int\\nabla p(\\mathbf{x},t)\\mathrm{d}\\mathbf{x}=\\operatorname{E}_{\\sim(1)}\\left[\\nabla\\log p(\\mathbf{x}_{t},t)\\right].}}\\end{array}$$\n\nConsequently we can obtain the optimal s through\n\n$$\\frac{\\delta}{\\delta\\mathbf{s}}\\int\\limits_{t=T_{1}}^{T_{2}}\\lambda(t)\\mathrm{E}_{\\sim(\\cdot)}\\left[\\left|\\mathbf{s}(\\mathbf{x}_{t},t)-\\nabla\\log p(\\mathbf{x}_{t},t|\\mathbf{x}_{0})\\right|\\right|^{2}\\right]\\mathrm{d}t=0\\to\\mathbf{s}(\\mathbf{x},t)=\\nabla\\log p(\\mathbf{x},t).\\tag{21}$$\n\nSubstitution of this result into Eq. (20) directly proves the desired inequality.\n\nAs a byproduct, we prove the correctness of Eq. (5), since it is a particular case of Eq. (20), with λ =\ng 2, T1 = 0, T2 = T. Since K(T) is a minimum, the decomposition I(sθ, T) = K(T) + G(sθ, T) implies K(T) + G(sθ, T) ≥ K(T) → G(sθ, T) ≥ 0.\n\n## D Proof Of **Proposition 1**\n\nProposition 1. Given the stochastic dynamics defined in Eq. (1)*, it holds that*\n\n$$\\mathbb{E}_{\\sim(\\mathbb{1})}\\log p(\\mathbf{x}_{T},T)-K(T)+R(T)=\\mathbb{E}_{p_{\\mathrm{data}}(\\mathbf{x})}\\log p_{\\mathrm{data}}(\\mathbf{x}).$$\nE∼(1)log p(xT , T) − K(T) + R(T) = Ep*data*(x)log p*data*(x). (8)\nProof. We consider the pair of equations\n\n$$\\begin{array}{l}{\\rm d}\\mathbf{x}_{t}=\\left[-\\mathbf{f}(\\mathbf{x}_{t},t^{\\prime})+g^{2}(t^{\\prime})\\mathbf{\\nabla}\\log q(\\mathbf{x}_{t},t)\\right]{\\rm d}t+g(t^{\\prime}){\\rm d}\\mathbf{w}(t),\\\\ {\\rm d}\\mathbf{x}_{t}=\\mathbf{f}(\\mathbf{x}_{t},t){\\rm d}t+g(t){\\rm d}\\mathbf{w}(t),\\end{array}\\tag{22}$$\n\nwhere t\n′ = T − t, q is the density of the backward process and p is the density of the forward process. These equations can be interpreted as a particular case of the following pair of sdes (corresponding to Huang et al.\n\n(2021) eqn (4) and (17)1).\n\n$$\\mathrm{d}\\mathbf{x}_{t}=\\underbrace{\\left[-\\mathbf{f}(\\mathbf{x}_{t},t^{\\prime})+g^{2}(t^{\\prime})\\mathbf{\\nabla}\\log q(\\mathbf{x}_{t},t)\\right]}_{\\mathbf{\\mu}(\\mathbf{x}_{t},t)}\\mathrm{d}t+\\underbrace{g(t^{\\prime})}_{\\sigma(t)}\\mathrm{d}\\mathbf{w}(t),$$ $$\\mathrm{d}\\mathbf{x}_{t}=\\left[\\underbrace{\\mathbf{f}(\\mathbf{x}_{t},t)-g^{2}(t)\\mathbf{\\nabla}\\log q(\\mathbf{x}_{t},t^{\\prime})}_{-\\mathbf{\\mu}(\\mathbf{x}_{t},t^{\\prime})}+\\underbrace{g(t)}_{\\sigma(t^{\\prime})}\\mathbf{a}(\\mathbf{x}_{t},t)\\right]\\mathrm{d}t+g(t)\\mathrm{d}\\mathbf{w}(t),\\tag{10}$$ \\[\\mathbf{\\mu}\\\n$$(8)$$\n\nwhere Eq. (22) is recovered considering a(x, t) = σ(t\n′)∇ log q(x, t′) = g(t)∇ log q(x, t′). Eq. (23) is associated to an elbo (Huang et al. (2021), Thm 3) that is attained with equality if and only if a(x, t) = σ(t\n′)∇ log q(x, t′). Consequently, we can write the following equality associated to the backward process of Eq. (22)\n\n$$\\log q(\\mathbf{x},T)=\\mathbb{E}\\left[-\\frac{1}{2}\\int\\limits_{0}^{T}||\\mathbf{a}(\\mathbf{x}_{t},t)||^{2}+2\\nabla^{\\top}\\mathbf{\\mu}(\\mathbf{x}_{t},t^{\\prime})ds+\\log q(\\mathbf{x}_{T},0)\\quad\\left|\\mathbf{x}_{0}=\\mathbf{x}\\right|\\right],\\tag{1}$$\n$$(23)$$\n$$(24)$$\n\nwhere expected value is taken w.r.t. dynamics of the associated forward process.\n\nBy careful inspection of the couple of equations we notice that in the process xt the drift includes the\n∇ log q(xt, t) term, while in our main (1) we have ∇ log p(xt, t′). In general the two vector fields do not agree. However, if we select as starting distribution of the generating process p(x, T), i.e. q(x, 0) = p(x, T),\nthen ∀t, q(x, t) = p(x, t′).\n\nGiven initial conditions, the time evolution of the density p is fully described by the Fokker-Planck equation\n\n$$\\frac{d}{dt}p(\\mathbf{x},t)=-\\nabla^{\\top}\\left(\\mathbf{f}(\\mathbf{x},t)p(\\mathbf{x},t)\\right)+\\frac{g^{2}(t)}{2}\\Delta(p(\\mathbf{x},t)),\\quad p(\\mathbf{x},0)=p_{\\mathsf{data}}\\left(\\mathbf{x}\\right).\\tag{25}$$ In fact,\nSimilarly, for the density q,\n$$\\frac{d}{dt}q(\\mathbf{x},t)=-\\mathbf{\\nabla}^{\\top}\\left(-\\mathbf{f}(\\mathbf{x},t^{\\prime})q(\\mathbf{x},t)+g^{2}(t^{\\prime})\\mathbf{\\nabla}\\log q(\\mathbf{x},t)q(\\mathbf{x},t)\\right)+\\frac{g^{2}(t^{\\prime})}{2}\\Delta(q(\\mathbf{x},t)),\\quad q(\\mathbf{x},0)=p(\\mathbf{x},T).\\tag{26}$$\n\nBy Taylor expansion we have\n\n$$q(\\mathbf{x},\\delta t)=q(\\mathbf{x},0)+\\delta t\\left(\\frac{d}{\\mathrm{d}t}q(\\mathbf{x},t)\\right)_{t=0}+\\mathcal{O}(\\delta t^{2})=$$ $$q(\\mathbf{x},0)+\\delta t\\left(-\\nabla^{\\top}\\left(-\\mathbf{f}(\\mathbf{x},T)q(\\mathbf{x},0)+g^{2}(T)\\nabla\\log q(\\mathbf{x},0)q(\\mathbf{x},0)\\right)+\\frac{g^{2}(T)}{2}\\Delta(q(\\mathbf{x},0))\\right)+\\mathcal{O}(\\delta t^{2})=$$ $$q(\\mathbf{x},0)+\\delta t\\left(\\nabla^{\\top}\\left(\\mathbf{f}(\\mathbf{x},T)q(\\mathbf{x},0)\\right)-\\frac{g^{2}(T)}{2}\\Delta(q(\\mathbf{x},0))\\right)+\\mathcal{O}(\\delta t^{2}),$$\n\nand\n\n$$\\begin{array}{l}{{p(\\mathbf{x},T-\\delta t)=p(\\mathbf{x},T)-\\delta t\\left(\\frac{d}{d t}p(\\mathbf{x},t)\\right)_{t=T}+\\mathcal{O}(\\delta t^{2})=}}\\\\ {{p(\\mathbf{x},T)-\\delta t\\left(-\\nabla^{\\top}\\left(\\mathbf{f}(\\mathbf{x},T)p(\\mathbf{x},T)\\right)+\\frac{g^{2}(T)}{2}\\Delta(p(\\mathbf{x},T))\\right)+\\mathcal{O}(\\delta t^{2})=}}\\\\ {{p(\\mathbf{x},T)+\\delta t\\left(\\nabla^{\\top}\\left(\\mathbf{f}(\\mathbf{x},T)p(\\mathbf{x},T)\\right)-\\frac{g^{2}(T)}{2}\\Delta(p(\\mathbf{x},T))\\right)+\\mathcal{O}(\\delta t^{2})}}\\end{array}$$\n\nSince q(x, 0) = p(x, T), we finally have q(x, δt) − p(x, T − δt) = O(δt2). This holds for arbitrarily small δt.\n\nBy induction, with similar reasoning, we claim that q(x, t) = p(x, t′).\n\nThis last result allows us to rewrite Eq. (22) as the pair of sdes\n\n$$\\begin{array}{l}{\\rm d}\\mathbf{x}_{t}=\\left[\\,-\\mathbf{f}(\\mathbf{x}_{t},t^{\\prime})+g^{2}(t^{\\prime})\\mathbf{\\nabla}\\log p(\\mathbf{x}_{t},t^{\\prime})\\right]{\\rm d}t+g(t^{\\prime}){\\rm d}\\mathbf{w}(t),\\\\ {\\rm d}\\mathbf{x}_{t}=\\mathbf{f}(\\mathbf{x}_{t},t){\\rm d}t+g(t){\\rm d}\\mathbf{w}(t).\\end{array}\\tag{27}$$\n\nMoreover, since q(x, T) = p(x, 0) = p*data* (x), together with the result Eq. (24), we have the following equality\n\n$$\\log p_{\\mathrm{data}}(\\mathbf{x})=\\mathbb{E}\\left[-1\\right]$$\n$$\\frac{1}{2}\\int\\limits_{0}^{T}||\\mathbf{a}(\\mathbf{x}_{t},t)||^{2}+2\\nabla^{\\top}\\mathbf{\\mu}(\\mathbf{x}_{t},t^{\\prime})\\mbox{d}t+\\log p(\\mathbf{x}_{T},T)\\quad|\\mathbf{x}_{0}=\\mathbf{x}\\Bigg{]}\\,.\\tag{28}$$\n\nConsequently\n\n− 1 2 Z T Ex∼pdata [log pdata (x)] = E [log p(xT , T)] + E 0 ||a(xt, t)||2 + 2∇⊤µ(xt, t′)dt = E [log p(xT , T)] + E − 1 2 Z T 0 g(t) 2||∇ log p(xt, t)||2 + 2∇⊤−f(xt, t) + g 2(t)∇ log p(xt, t)dt = E [log p(xT , T)] + E − 1 2 Z T 0 g(t) 2||∇ log p(xt, t)||2 − 2g 2(t)∇⊤ x log p(xt, t)∇ log p(xt, t|x0)dt + E − 1 2 Z T 0 2f ⊤(xt, t)∇ log p(xt, t|x0)dt = E [log p(xT , T)] + E − 1 2 Z T 0 g(t) 2||∇ log p(xt, t) − ∇ log p(xt, t|x0)||2dt + − 1 2 Z T E 0 −g(t) 2||∇ log p(xt, t|x0)||2 + 2f ⊤(xt, t)∇ log p(xt, t|x0)dt .\n\nRemembering the definitions\n\n$$\\begin{array}{l}{{K(T)=\\frac{1}{2}\\int\\limits_{t=0}^{T}g^{2}(t)\\mathbb{E}_{\\sim(1)}\\left[||\\nabla\\log p(\\mathbf{x}_{t},t)-\\nabla\\log p(\\mathbf{x}_{t},t|\\mathbf{x}_{0})||\\right]^{2}\\mathrm{d}t}}\\\\ {{R(T)=\\frac{1}{2}\\int\\limits_{t=0}^{T}\\mathbb{E}_{\\sim(1)}\\left[g^{2}(t)||\\mathbf{\\nabla}\\log p(\\mathbf{x},t\\,|\\,\\mathbf{x}_{0})||\\right]^{2}-2\\mathbf{f}^{\\top}(\\mathbf{x},t)\\mathbf{\\nabla}\\log p(\\mathbf{x},t\\,|\\,\\mathbf{x}_{0})\\mathrm{d}t,}}\\end{array}$$\n\nwe finally conclude the proof that\n\n$$\\mathbb{E}_{\\sim(1)}[\\log p(\\mathbf{x}_{T},T)]-K(T)+R(T)=\\mathbb{E}_{\\mathbf{x}\\sim p_{\\texttt{data}}}\\left[\\log p_{\\texttt{data}}(\\mathbf{x})\\right].\\tag{1}$$\n$$(29)$$\n\n## E Proof Of **Lemma 1**\n\nIn this Section we prove the validity of Lemma 1 for the case of Variance Preserving (VP) and Variance Exploding (VE) sdes. Remember, as reported also in main Table 1, that the above mentioned classes correspond to α(t) = −\n1 2 β(t), g(t) = pβ(t), β(t) = β0 + (β1 − β0)t and α(t) = 0, g(t) = qdσ2(t)\ndt, σ2(t) =\nσmax σmin trespectively.\n\nLemma 1. For the classes of sdes considered (Table 1), the discrepancy between p(x, T) and the p*noise*(x)\ncan be bounded as follows.\n\nFor Variance Preserving sdes, it holds that: kl [p(x, T) ∥ p*noise*(x)] ≤ C1 exp−R T\n0 β(t)dt.\n\nFor Variance Exploding sdes, it holds that: kl [p(x, T) ∥ p*noise*(x)] ≤ C21 σ2(T)−σ2(0) .\n\n## E.1 The Variance Preserving (Vp) Convergence\n\nWe associate this class of sdes to the Fokker Planck operator\n\n$${\\mathcal{L}}^{\\dagger}(t)={\\frac{1}{2}}\\beta(t)\\nabla^{\\top}\\left(\\mathbf{x}\\cdot+\\nabla(\\cdot)\\right),$$\n\n$$(30)$$\nβ(t)∇⊤ (x · +∇(·)), (30)\nand consequently dp(x,t)\ndt = L\n†(t)p(x, t). Simple calculations show that lim T→∞\np(x, T) = N1(x).\n\nWe compute bound the time derivative of the kl term as\n\nd dt kl [p(x, T) ∥ N1(x)] = Zdp(x, t) dtlogp(x, t) N1(x) dx + Zp(x, t) p(x, t) dp(x, t) dtdx = 1 2 β(t) Z∇⊤ (−∇ log(N1(x))p(x, t)) + ∇p(x, t))) logp(x, t) N1(x) dx = − 1 2 β(t) Zp(x, t) (−∇ log(N1(x)) + ∇ log p(x, t)))⊤ ∇(logp(x, t) N1(x) )dx = − 1 2 β(t) Zp(x, t)∇(logp(x, t) N1(x) ) ⊤∇(logp(x, t) N1(x) )dx = − 1 2 β(t) Zp(x, t)||∇(logp(x, t) N1(x) )||2dx ≤ −β(t)kl [p(x, T) ∥ N1(x)] . (31)\nWe then apply Gronwall's inequality (Villani, 2009) to d dtkl [p(x, T) ∥ N1(x)] ≤ −β(t)kl [p(x, T) ∥ N1(x)]\nto claim\n\n$$\\operatorname{KL}\\left[p(\\mathbf{x},T)\\parallel{\\mathcal{N}}_{1}(\\mathbf{x})\\right]\\leq\\operatorname{KL}\\left[p(\\mathbf{x},0)\\parallel{\\mathcal{N}}_{1}(\\mathbf{x})\\right]\\exp\\Biggl(-\\int_{0}^{T}\\beta(s)d s\\Biggr).$$\n\nTo claim validity of the result, we need to assume that p(x, t) has finite first and second order derivatives, and that kl [p(x, 0) ∥ N1(x)] < ∞.\n\n$$(32)$$\n\n## E.2 The Variance Exploding (Ve) Convergence\n\nThe first step is to bound the derivative w.r.t to ω of the divergence kl [pω(x) ∥ Nω(x)], i.e.\n\nd\ndωkl [pω(x) ∥ Nω(x)] = Zdpω(x)\ndω logpω(x)\nNω(x)\ndx +\nZpω(x)\npω(x)\ndpω(x)\ndω dx −\nZpω(x)\nNω(x)\ndNω(x)\ndω dx =\n1\n2\nZ(∆pω(x)) logpω(x)\nNω(x)\n− (∆Nω(x)) pω(x)\nNω(x)\ndx =\n1\n2\nZ∇⊤ (pω(x)∇ log pω(x)) logpω(x)\nNω(x)\n− ∇⊤ (Nω(x)∇ log Nω(x)) pω(x)\nNω(x)\ndx =\n−\n1\n2\nZ(pω(x)∇ log pω(x))⊤ ∇(logpω(x)\nNω(x)\n) − (Nω(x)∇ log Nω(x))⊤ ∇(\npω(x)\nNω(x)\n)dx =\n−\n1\n2\nZ(pω(x)∇ log pω(x))⊤ ∇(logpω(x)\nNω(x)\n) − (pω(x)∇ log Nω(x))⊤ ∇(logpω(x)\nNω(x)\n)dx =\n−\n1\n2\nZpω(x)||∇(logpω(x)\nNω(x)\n)||2dx ≤ −\n1\nω\nkl [pω(x) ∥ Nω(x)] . (33)\nConsequently, using again Gronwall inequality, for all ω1 > ω0 > 0 we have\n$\\mathrm{KL}\\left[p_{\\omega_{1}}(\\mathbf{x})\\parallel\\mathcal{N}_{\\omega_{1}}(\\mathbf{x})\\right]\\leq\\mathrm{KL}\\left[p_{\\omega_{0}}(\\mathbf{x})\\parallel\\mathcal{N}_{\\omega_{0}}(\\mathbf{x})\\right]\\exp(-(\\log(\\omega_{1})-\\log(\\omega_{0})))=\\mathrm{KL}\\left[p_{\\omega_{0}}(\\mathbf{x})\\parallel\\mathcal{N}_{\\omega_{0}}(\\mathbf{x})\\right]\\omega_{0}\\frac{1}{\\omega_{1}}.$\nThis can be directly applied to obtain the bound for VE sde. Consider ω1 = σ 2(T) − σ 2(0) and ω0 =\nσ 2(τ ) − σ 2(0) for an arbitrarily small *τ < T*. Then, since for the considered class of variance exploding sde we have p(x, T) = pσ2(T)−σ2(0)(x)\n\n$${\\rm KL}\\left[p(\\mathbf{x},T)\\parallel{\\cal N}_{\\sigma^{2}(T)-\\sigma^{2}(0)}(\\mathbf{x})\\right]\\leq C\\frac{1}{\\sigma^{2}(T)-\\sigma^{2}(0)}\\tag{34}$$\n\nwhere C = kl -p(x, τ ) ∥ Nσ2(τ)−σ2(0)(x)(σ 2(τ ) − σ 2(0)).\n\nSimilarly to the previous case, we assume that p(x, t) has finite first and second order derivatives, and that C < ∞.\n\n## F Proof Of **Lemma 2**\n\nLemma 2. The optimal score gap term G(bsθ, T) is a non-decreasing function in T. That is, given T2 > T1, and θ1 = arg minθ I(sθ, T1), θ2 = arg minθ I(sθ, T2)*, then* G(sθ2\n, T2) ≥ G(sθ1\n, T1).\n\nProof. For θ1 defined as in the lemma, I(sθ1\n, T1) = K(T1) + G(sθ1\n, T1). Next, select T2 > T1. Then, for a generic θ, including θ2,\n\nI(sθ, T2) = Z T1 t=0 g 2(t)E∼(1) -||sθ(xt, t) − ∇ log p(xt, t|x0)||2dt + | {z } =I(sθ,T1)≥K(T1)+G(sθ1 ,T1)=I(sθ1 ,T1) Z T2 t=T1 g 2(t)E∼(1) -||sθ(xt, t) − ∇ log p(xt, t|x0)||2dt t=T1 g 2(t)E∼(1)[||∇ log p(xt,t)−∇ log p(xt,t|x0)||2]dt=K(T2)−K(T1) ≥ G(sθ1 , T1) + K(T2), | {z } ≥R T2\nfrom which $\\mathcal{G}(\\mathbf{s_{\\theta}},T_{2})=I(\\mathbf{s_{\\theta}},T_{2})-K(T_{2})\\geq\\mathcal{G}(\\mathbf{s_{\\theta_{1}}},T_{1})$. \n\n## G Proof Of **Proposition 2**\n\nProposition 2. Consider the elbo decomposition in Eq. (9). We study it as a function of time T, and seek its optimal argument T\n⋆ = arg maxT Lelbo(bsθ, T)*. Then, the optimal diffusion time* T\n⋆ ∈ R\n+, and thus not necessarily T\n⋆ = ∞. Additional assumptions on the gap term G(·) *can be used to guarantee strict finiteness* of T\n⋆. There exists at least one optimal diffusion *time* T\n⋆in the interval [0, ∞], which maximizes the elbo, that is T\n⋆ = arg maxT Lelbo(bsθ, T).\n\nProof. It is trivial to verify that since the optimal gap term G(bsθ, T) is an increasing function in T Lemma 2, then ∂G\n∂T ≥ 0.Then, we study the sign of the kl derivative, which is always negative as shown by Eq. (31) and Eq. (33) (where we also notice d dt =\ndω dt d dω keep the sign). Moreover, we know that that lim T→∞\n∂kl\n∂T = 0. Then, the function ∂Lelbo\n∂T =\n∂G\n∂T +\n∂kl\n∂T has at least one zero in [0, ∞].\n\n## H Optimization Of T ⋆\n\nIt is possible to treat the diffusion time T as an hyper-parameter and perform gradient based optimization jointly with the score model parameters θ. Indeed, simple calculations show that\n\n$$\\frac{\\partial\\mathcal{L}_{\\texttt{EED}}(\\mathbf{s_{\\theta}},T)}{\\partial T}=\\mathbb{E}\\left[\\left(\\mathbf{f}^{\\top}(\\mathbf{x}_{T},T)\\mathbf{\\nabla}+g^{2}(T)\\Delta\\right)\\log p_{\\texttt{Bres}}(\\mathbf{x}_{T})\\right]+$$ $$-\\frac{1}{2}\\mathbb{E}\\left[\\left\\|\\mathbf{s_{\\theta}}(\\mathbf{x}_{T},T)-\\mathbf{\\nabla}\\log p(\\mathbf{x}_{T},T\\,|\\,\\mathbf{x}_{0})\\right\\|^{2}\\right]+$$ $$\\frac{1}{2}\\mathbb{E}\\left[g^{2}(T)\\|\\mathbf{\\nabla}\\log p(\\mathbf{x}_{T},T\\,|\\,\\mathbf{x}_{0})\\right\\|^{2}-2\\mathbf{f}^{\\top}(\\mathbf{x}_{T},T)\\mathbf{\\nabla}\\log p(\\mathbf{x}_{T},T\\,|\\,\\mathbf{x}_{0})\\right]$$\n$$\\begin{array}{l}{(35)}\\\\ {\\qquad(36)}\\end{array}$$\n$$(37)$$\n\n## I Proof Of **Proposition 4**\n\nProposition 4. *Given the existence of* T\n⋆, defined as the diffusion time such that the elbo *is maximized*\n(Proposition 2), there exists at least one diffusion time τ ≤ T\n⋆*, such that* L\nϕ\n⋆\nelbo(bsθ, τ ) ≥ Lelbo(bsθ, T ∗).\n\nProof. Since ∀T we have L\nϕ elbo(sθ, T) ≥ Lelbo(sθ, T), there exists a countable set of intervals I contained in [0, T ⋆] of variable supports, where L\nϕ elbo is greater than Lelbo(sθ, T). Assuming continuity of L\nϕ elbo, in these intervals is possible to find at least one τ ≤ T\n⋆ where L\nϕ\n⋆\nelbo(bsθ, τ ) ≥ Lelbo(bsθ, T ∗).\n\nWe notice that the degenerate case I = T\n⋆is obtained only when ∀T ≤ T\n⋆,kl [p(x, T) ∥ νϕ∗ (x)] =\nkl [p(x, T) ∥ p*noise* (x)]. We expect this condition to never occur in practice.\n\n## J Invariance To Noise Schedule\n\nWe here discuss about the claims made in § 2.4 about the invariance of the elbo to the particular choice of noise schedule. First in Appendix J.1 we explain how different sdes corresponding to different noise schedules can be translated one into the other. We introduce the concept of signal-to-noise ratio (snr). We clarify the unified score parametrization used in practice in the literature Karras et al. (2022); Kingma et al. (2021).\n\nThen, in Appendix J.2, we prove how the single elements of the elbo depend only on the value of the snr at the final diffusion time T, as claimed in the main paper.\n\n## J.1 Preliminaries\n\nWe consider as reference sde a pure Wiener process diffusion,\n\n$$\\mathrm{d}\\mathbf{x}_{t}=\\mathrm{d}\\mathbf{w}_{t}\\quad\\mathrm{with}\\quad\\mathbf{x}_{0}\\sim p_{\\mathrm{data}}\\ ,$$\ndxt = dwt with x0 ∼ p*data* , (38)\nIt is easily seen that the solution of the random process admits representation\n\n$${\\mathbf{x}}_{t}={\\mathbf{x}}_{0}+{\\sqrt{t}}{\\boldsymbol{\\epsilon}},\\quad{\\boldsymbol{\\epsilon}}\\sim{\\mathcal{N}}({\\mathbf{0}},{\\boldsymbol{I}})$$\n$$(39)$$\n$$(40)$$\n$$(41)$$\n√tϵ, ϵ ∼ N (0, I) (39)\nIn this case the time varying probability density, that we indicate with ψ, satisfies\n\n$$\\psi(\\mathbf{x},t)=\\exp\\biggl({\\frac{t}{2}}\\Delta\\biggr)p_{\\texttt{data}}(\\mathbf{x}),\\quad\\psi(\\mathbf{x},t\\,|\\,\\mathbf{x}_{0})=\\exp\\biggl({\\frac{t}{2}}\\Delta\\biggr)\\delta(\\mathbf{x}-\\mathbf{x}_{0})$$\n\nSimple calculations show that\n\n$$\\nabla\\log\\psi(\\mathbf{x},\\sigma^{2})={\\frac{\\mathbb{E}[\\mathbf{x}_{0}\\,|\\,\\mathbf{x}_{0}+\\sigma\\epsilon=\\mathbf{x}]-\\mathbf{x}}{\\sigma^{2}}}\\doteq{\\frac{d(\\mathbf{x};\\sigma^{2})-\\mathbf{x}}{\\sigma^{2}}},$$\n\nwhere again x0 ∼ p*data* and the function d can be interpreted as a *denoiser*.\n\nOur goal is to show the relationship between equations like Equation (1), and Equation (38). In particular,\nwe focus on *affine* sdes, as classically done with Diffusion models. The class of considered affine sdes is the\nfollowing:\n$$\\mathrm{d}\\mathbf{x}_{t}=\\alpha(t)\\mathbf{x}_{t}\\mathrm{d}t+g(t)\\mathrm{d}\\mathbf{w}_{t}\\quad\\mathrm{with}\\quad\\mathbf{x}_{0}\\sim p_{\\mathrm{data}}\\ ,$$\nIn this simple linear case the process admits representation\n$$(42)$$\n$$(43)$$\n$$\\mathbf{x}_{t}=k(t)\\mathbf{x}_{0}+\\sigma(t)\\mathbf{\\epsilon},\\quad\\mathbf{\\epsilon}\\sim\\mathcal{N}(\\mathbf{0},\\mathbf{I})$$\n$$(44)$$\nxt = k(t)x0 + σ(t)ϵ, ϵ ∼ N (0, I) (43)\nwhere $k(t)=\\exp\\left(\\int\\limits_{0}^{t}\\sigma(s)ds\\right),\\sigma^{2}(t)=k^{2}(t)\\int\\limits_{0}^{t}\\frac{\\frac{1}{2}\\,t^{2}(s)}{t^{2}(s)}ds$. We can rewrite Equation (43) as $\\mathbf{x}_{t}=k(t)(\\mathbf{x}_{0}+\\delta(t)\\mathbf{\\epsilon})$, and define the SNR as $\\delta(t)=\\frac{\\sigma(t)}{k(t)}$. The density associated to Equation (42) can be expressed as a function of time $t$. \nψ as follows\n\n$$p(\\mathbf{x},t)=k(t)^{-D}\\left[\\exp\\biggl{(}\\frac{\\partial^{2}(t)}{2}\\Delta\\biggr{)}p_{\\mathsf{Bias}}(\\mathbf{x})\\right]_{\\frac{\\mathbf{x}}{k(t)}}=k(t)^{-D}\\psi(\\frac{\\mathbf{x}}{k(t)},\\sigma^{2}(t)).$$\n\nThe score function associated to Equation (43) has consequently expression\n\n$$\\nabla_{\\mathbf{x}}\\log p(\\mathbf{x},t)=\\nabla_{\\mathbf{x}}\\log\\psi({\\frac{\\mathbf{x}}{k(t)}},\\sigma^{2}(t))={\\frac{1}{k(t)}}\\nabla_{{\\frac{\\mathbf{x}}{k(t)}}}\\log\\psi({\\frac{\\mathbf{x}}{k(t)}},\\sigma^{2}(t))={\\frac{k(t)d({\\frac{\\mathbf{x}}{k(t)}};\\sigma^{2}(t))-\\mathbf{x}}{\\sigma^{2}(t)}}.$$\n\n## J.2 Different Noise Schedules\n\nConsider a diffusion of the form Equation (38) and a score network s¯θ that approximate the true score.\n\nInspecting Equation (45), we parametrize the score network associated to a generic diffusion Equation (42)\nas a function of the score of the reference diffusion. The score parametrization considered in Kingma et al.\n\n(2021), can be generalized to arbitrary sdes Karras et al. (2022). In particular, as suggested by Equation (41),\nwe select\n\n$$(45)$$\n$$\\bar{\\mathbf{s}}_{\\mathbf{\\theta}}(\\mathbf{x},t)=\\frac{k(t)\\mathbf{d}_{\\mathbf{\\theta}}(\\frac{\\mathbf{x}}{k(t)};\\,\\hat{\\sigma}^{2}(t))-\\mathbf{x}}{\\sigma^{2}(t)}\\tag{46}$$\n\nWe proceed by showing that the different components of the elbo depends on the diffusion time T only through σ˜(T), but not on k(t), σ(t) singularly for any time *t < T*.\n\nTheorem 1. Consider a generic diffusion *Equation* (42) *and parametrize the score network as* s¯θ( x k(t)\n, σ˜(t)).\n\nThen, the gap term G(s¯θ, T) associated to *Equation* (42) for a diffusion time T depends only on σ˜(T) *but not* on k(t), σ(t) singularly for any time *t < T*.\n\nProof. We first rearrange the gap term\n\n$$2{\\mathcal{G}}({\\bar{\\mathbf{s}}}_{\\theta},T)=\\int\\limits_{t=0}^{T}g^{2}(t){\\mathbb{E}}_{\\sim({\\pm}{\\mathbb{Z}})}\\left[\\left\\|{\\bar{\\mathbf{s}}}_{\\theta}(\\mathbf{x}_{t},t)-\\mathbf{\\nabla}\\log p(\\mathbf{x}_{t},t\\,|\\,\\mathbf{x}_{0})\\right\\|^{2}\\right]{\\mathrm{d}}t-$$\n$$\\int\\limits_{t=0}^{T}g^{2}(t)\\mathbb{E}_{\\sim(\\frac{t}{2})}\\left[\\left\\|\\,\\boldsymbol{\\nabla}\\log p(\\boldsymbol{x}_{t},t)-\\boldsymbol{\\nabla}\\log p(\\boldsymbol{x}_{t},t\\,|\\,\\boldsymbol{x}_{0})\\right\\|^{2}\\right]\\mathrm{d}t=$$ $$\\int\\limits_{t=0}^{T}g^{2}(t)\\mathbb{E}_{\\sim(\\frac{t}{2})}\\left[\\left\\|\\,\\bar{\\boldsymbol{s}}_{\\boldsymbol{\\theta}}(\\boldsymbol{x}_{t},t)-\\boldsymbol{\\nabla}\\log p(\\boldsymbol{x}_{t},t)\\right\\|^{2}\\right]\\mathrm{d}t$$\n\nas shown in []2. Then\n\nZ\nT\nt=0\ng\n2(t)∥s¯θ(x, t) − ∇ log p(x, t)∥\n2p(x, t| x0)p*data* (x0)dxdx0dt =\nZ\nT\nt=0\ng\n2(t)\n\nk(t)dθ(\nx\nk(t)\n; ˜σ\n2(t)) − x\nσ\n2(t)−\nk(t)dθ(\nx\nk(t)\n; ˜σ\n2(t)) − x\nσ\n2(t)\n\n2\np(x, t| x0)p*data* (x0)dxdx0dt =\nZ\nT\nt=0\ng\n2(t)\n\nk(t)dθ(\nx\nk(t)\n; ˜σ\n2(t)) − k(t)d(\nx\nk(t)\n; ˜σ\n2(t))\nσ\n2(t)\n\n2\np(x, t| x0)p*data* (x0)dxdx0dt =\nt=0\ng\n2(t)\nk\n2(t)\n\ndθ(\nx\nk(t)\n; ˜σ\n2(t)) − d(\nx\nk(t)\n; ˜σ\n2(t))\nσ˜\n2(t)\n\n2\np(x, t| x0)p*data* (x0)dxdx0dt =\nZ\nT\nt=0\ng\n2(t)\nk\n2(t)\n\ndθ(\nx\nk(t)\n; ˜σ\n2(t)) − d(\nx\nk(t)\n; ˜σ\n2(t))\nσ˜\n2(t)\n\n2\nψ(\nx\nk(t)\n, σ˜\n2(t)| x0)p*data* (x0)k(t)\n−Ddxdx0dt =\nZ\nT\nsubst. x˜ =x\nk(t)\n, dx˜ = dxk\n−D(t)\nZ\nT\nt=0\ng\n2(t)\nk\n2(t)\n\ndθ(x˜; ˜σ\n2(t)) − d(x˜; ˜σ\n2(t))\nσ˜\n2(t)\n\n2\nψ(x˜, σ˜\n2(t)| x0)p*data* (x0)dxdx0dt =\nsubst. r = ˜σ\n2(t), dr =\ng\n2(t)\nk\n2(t)\ndt\nσ˜\n2(T)\nZ\nt=0\n∥s¯θ(x˜, r) − ∇ log ψ(x˜, r | x0)∥\n2ψ(x˜, r)p*data* (x0)dx˜dx0dr\nFor any k(t), σ(t) such that σ˜(T) is the same, the score matching loss is the same Theorem 2. Suppose that for any ϕ of the auxiliary model νϕ(x) *it exists one* ϕ\n′*such that* νϕ\n′ (x) =\nk\n−Dνϕ( x k\n), for any k > 0*. Notice that this condition is trivially satisfied if the considered parametric model* has the expressiveness to multiply its output by the scalar k*. Then the minimum of Kullback-Leibler divergence* betweeen p(x, T) associated to a generic diffusion *Equation* (42) *and the density of an auxiliary model* νϕ(x)\ndepends only on σ˜(T) and not on σ(T) *alone.*\n\n2Citation not included to avoid breaking anonymity\n\n\n\nFigure 8: Visualization of few samples at different diffusion times T.\nProof. We start with the equality\n\nkl [p(x, T) ∥ νϕ(x)] = kl k(T) −Dψ(x k(T) , σ˜(T)) ∥ νϕ(x) = kl k(T) −Dψ(x k(T) , σ˜(T)) ∥ k(T) −Dνϕ ′ (x k(T) ) = Zk(T) −Dψ(x k(T) , σ˜(T)) log ψ(x k(T) , σ˜(T)) νϕ ′ (x k(T) ) ! dx = Zψ(x˜, σ˜(T)) log ψ(x˜, σ˜(T)) νϕ ′ (x˜) ! dx˜ = kl hψ(x, σ˜(T)) ∥ νϕ ′ (x) i\nThen the minimimum only depends on σ˜(T), as it is always possible to achieve the same value independently on the sde by rescaling the auxiliary model output.\n\n## K Experimental Details\n\nWe here give some additional details concerning the experimental (§ 4) settings.\n\n## K.1 Toy Example Details\n\nIn the toy example, we use 8192 samples from a simple Gaussian mixture with two components as target p*data* (x). In detail, we have p*data* (x) = πN (1, 0.1 2) + (1 − π)N (3, 0.5 2), with π = 0.3. The choice of Gaussian mixture allows to write down explicitly the time-varying density\n\n$$p(\\mathbf{x}_{t},t)=\\pi{\\mathcal{N}}(1,s^{2}(t)+0.1^{2})+(1-\\pi){\\mathcal{N}}(3,s^{2}(t)+0.5^{2}),$$\n\nwhere s 2(t) is the marginal variance of the process at time t. We consider a variance exploding sde of the type dxt = σ tdwt, which corresponds to s 2(t) = σ 2t−1 2 log σ\n.\n\n## K.2 § 4 **Details**\n\nWe considered Variance Preserving sde with default β0, β1 parameter settings. When experimenting on cifar10 we considered the NCSN++ architecture as implemented in Song et al. (2021c). Training of the score matching network has been carried out with the default set of optimizers and schedulers of Song et al.\n\n(2021c), independently of the selected T.\n\nFor the mnist dataset we reduced the architecture by considering 64 features, ch_mult = (1, 2) and attention resolutions equal to 8. The optimizer has been selected as the one in the cifar10 experiment but the warmup has been reduced to 1000 and the total number of iterations to 65000.\n\n## K.3 Varying T\n\nWe clarify about the T truncation procedure during both training and testing. The sde parameters are kept unchanged irrespective of T. During training, as evident from Eq. (3), it is sufficient to sample randomly\n\n$$(47)$$\n\nthe diffusion time from distribution U(0, T) where T can take any positive value. For testing (sampling) we simply modified the algorithmic routines to begin the reverse diffusion processes from a generic T instead of the default 1.0.\n\n## L Non Curated Samples\n\nWe provide for completeness collection of non curated samples for the cifar10 Figs. 9 to 12, mnist dataset Figs. 13 to 16 and celebadataset Fig. 17 and Table 6\n\nTable 6: celeba: fid scores for our method and baseline (T =\n1.0)\n.\n\n| fid (↓) | nfe (↓) | |\n|---------------------------------------|-----------|------|\n| Model | (sde) | |\n| ScoreSDE Song et al. (2021c) | 3.90 | 1000 |\n| Our (T = 0.5) | 8.06 | 500 |\n| Our (T = 0.2) | 86.9 | 200 |\n| Our with pretrain diffusion (T = 0.5) | 8.58 | 500 |\n| Our with pretrain diffusion (T = 0.2) | 86.7 | 200 |\n\n\n\nFigure 9: cifar10:Our(left) and Vanilla(right) method at T = 0.2\n\n\n\nFigure 10: cifar10:Our(left) and Vanilla(right) method at T = 0.4\n\n\n\nFigure 11: cifar10:Our(left) and Vanilla(right) method at T = 0.6\n\n\n\nFigure 12: Vanilla method at T = 1.0\n\n\n\nFigure 13: MNIST:Our(left) and Vanilla(right) method at T = 0.2\n\n\n\nFigure 14: MNIST:Our(left) and Vanilla(right) method at T = 0.4\n\n\n\nFigure 15: MNIST: Our(left) and Vanilla(right) method at T = 0.6\n\n\n\nFigure 16: MNIST: Vanilla method at T = 1.0\n\n\n\nwith pretrained score model and Glow (T = 0.5) and Bottom: baseline diffusion (T = 1.0)"},"reviews":{"kind":"string","value":"Review 1:\nSummary: In this paper, the authors provide a theoretical analysis of the role of the forward time T in denoising diffusion models. The authors advocate that there is a trade-off between the time needed for the forward diffusion to converge for which a large time T is needed and the score approximation which requires a small time T. The authors quantitfy this trade-off and propose a new method to modify the forward process. Experiments on MNIST and CIFAR10 are provided. The new proposed method suggests not to start from a Gaussian distribution but from a learned prior that approximate the end distribution of the forward process.\n\nStrengths and Weaknesses: STRENGTHS:\n\n* Diffusion models are a very active area of research and obtaining a better understanding of their property is a very appealing challenge. In particular the time T is of practical and theoretical importance.\n\n* The paper is mostly well-written and the proofs seem correct to me.\n\n* I like the idea of the auxilary model to replace the pure noise model (even though the novelty of this contribution could be discussed).\n\nWEAKNESSES:\n\n* (C) I strongly disagree with statements such as \"While recent works have started to lay down a theoretical foundations for these models, an analytical understanding of the role of the diffusion time T is still lacking\". This is not true. For example in [1, Theorem 1] there is a clear trade-off between the two terms in the upper-bound. One term corresponds to the distance between $p_T$ and the noise distribution and the other to the discretization and approximation of the backward process which includes the score approximation (we note that in the current work the discretization is not taken into account). Similarly in [2], see Theorem 4.2 for instance, the authors split the error term into a convergence term and an approximation term $\\varepsilon_1$. In [3, Theorem 2] there is a clear balance between the convergence term and an approximation term (I am aware that [3] was submitted approximately at the same time as the current paper but this is not the case for [1] and [2]).\n \n* (C) I disagree that the ELBO decomposition in 2.1 is new. In fact these results are already obtained in [4] as shown in [5, Appendix H]. I think the current way of writing the decomposition is unecessarily complicated. This can be seen as there is an explicit expression for the gap $\\mathcal{G}$, see [5, Appendix H] or the proof of Theorem 1 (with the missing citation). Once we rewrite $\\mathcal{G}$ like this it turns out that Equation (9) which is the key result of Section 2.1 is exactly Equation (8) in [4, Theorem 1]. I think the claims of novelty in this case are inflated. \n\n* I am quite surprised by the FID score provided in Table 6 as they don't seem good at all compared to the one of [6] (even though the number of NFE is twice as low).\n\n* It is well-known that log-likelihood score do not necessarily correlate with image quality (log-likelihood is not necessarily correlated with FID score), see [7] for example in the context of diffusion models. I think the authors should discuss this discrepancy as (at least in the case of image generative modeling) the most important metric is the image quality.\n\n* There is no confidence interval in the experimental results. Given that some of the improvement is small it is hard to know if it is negligeable or not. Some of the numbers are also quite misleading (see for example Table 2 where the authors chose to bold the second 1.16 but not the first one).\n \n* Why do the authors use VE-SDE when they turn to the CelebA dataset. It seems to me that VP-SDE performs better.\n\n* (C) The idea of changing the prior of the model is quite interesting I found. However I think the authors could do a better job at citing the related literature. For example [8] changes the forward kernel and does not target a Gaussian distribution. Instead a kernel with fatter tails is chosen (this is in an ODE setting). More closely related to this work, Schrodinger bridges are models based on diffusions which interpolate between arbitrary distributions [3,9,10], solving effectively the need of setting the prior distribution to be Gaussian. I think there exist deep links between the method proposed by the authors and this method. Also related is the concept of latent diffusion [11]. In the case where the encoder/decoder are invertible (normalizing flows) these can be seen as changing the forward process and the target distribution.\n\nMINOR POINTS:\n\n* I could not find the assumption on $p_{data}$ in Lemma 1. I may have missed something here.\n\n* Why citation p.23 cannot be included?\n\n* Can the influence of $g(t)$ be discussed by the authors in their setting? I have read the discussion 2.4 but I don't really see how the theory introduced by the authors fit in this discussion.\n\n[1] Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling -- De Bortoli, Thornton, Heng, Doucet.\n\n[2] Convergence for score-based generative modeling with polynomial complexity -- Lee, Lu, Tan\n\n[3] Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions -- Chen, Chewi, Li, Li, Salim, Zhang\n\n[4] Maximum Likelihood Training of Score-Based Diffusion Models -- Song, Durkan, Ermon\n\n[5] Convergence of denoising diffusion models under the manifold hypothesis -- De Bortoli\n\n[6] Score-Based Generative Modeling through Stochastic Differential Equation -- Song, Sohl-Dickstein, Kingma, Kumar, Ermon, Poole\n\n[7] Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Quality -- Watson, Chan, Ho, Norouzi\n\n[8] Poisson Flow Generative Models - Xu, Liu, Tegmark, Jaakkola\n\n[9] Deep Generalized Schrödinger Bridge - Liu, Chen, Theodorou\n\n[10] Solving Schrödinger Bridges via Maximum Likelihood - Vargas, Thodoroff, Lawrence, Lamacraft\n\n[11] Score-based Generative Modeling in Latent Space - Vahdat, Kreis, Kautz\n\nRequested Changes: My main criticism of the paper is:\n\n* The claim that no theory exists for the analysis of the time.\n\n* The novelty of the ELBO which could be derived in a simpler manner.\n\n* The lack of related work for the proposed methodology\n\nThese main concerns are detailed in the \"Strength and Weaknesses\" paragraph and highlighted with a (C). \nAddressing these issues is critical to secure my recommendation.\n\nBroader Impact Concerns: None.\n\n==================================================\n\nReview 2:\nSummary: In the paper \"Taming Diffusion Times in Score-based Generative Models: Trade-offs and Solutions\", the authors quantify the role of the diffusion time T in score based generative models. Whereas existing practice works with the case of large T, the authors in this paper show that increasing T causes a tradeoff between efficiency and model quality / approximation errors. Precisely, the authors study an ELBO decomposition, which they show has two terms that quantify this tradeoff. The authors establish existence of optimal diffusion times and use that to motivate the development of novel diffusion models with small diffusion times. The authors show that their novel diffusion models has competitive performance with sota methods. \n\n\nStrengths and Weaknesses: Strengths\n- The result about optimal diffusion times is a surprising, non-intuitive theoretical development on a system that is of significant current importance and impact. \n- The result could inspire additional work by related to diffusion models in a variety of contexts\n- The writing in the paper is exceptional, and the authors do a fantastic job of conveying the mathematical system they are studying, the fundamental tradeoff they are addressing, and the developments of the paper\n- The experiments provide reasonable validation for the proposed approach\n\n\nWeaknesses\n\n- I am slightly confused by a detail in Proposition 2. The theorem states that there is an optimal diffusion time T* in [0, infty] . Is including the possibility of T*=infty a typo? Do you mean to say [0, infty)? If T*=infty is included in the possible T*'s, then the proposition doesn't seem to be that meaningful, as any function over t>=0 would have an optimal value that is in [0,infty]\n\n\nRequested Changes: See weakness above regarding Proposition 2. Some modification or commentary in the paper is needed.\n\nOn page 4 there is a type logaritmic -> logarithmic\n\n\nBroader Impact Concerns: None.\n\n==================================================\n\nReview 3:\nSummary: This paper studies the effect of diffusion time has on score-based models. The paper proposes to learn a parametric surrogate for the empirically destroyed density, which is a sensible and surprisingly has not been considered before. This is a good although incremental contribution. Second, the paper decomposes the diffusion loss and shows theoretically that the optimal total diffusion time is actually not infinite, but strikes a balance over the regularising and fit terms of ELBO. The results demonstrate the claims sufficiently. It seems that optimal time is not that small for the vanilla model, while the auxiliary bridge is much more helpful. Both are useful and interesting findings.\n\nStrengths and Weaknesses: Paper is generally clearly presented (except section 2). The problem definition is clear, methods are principled and results are appropriate.\n\nThe main weakness is the poor presentation of section 2 that is almost illegible. This needs to be revised.\n\nRequested Changes: The paper is generally well written (especially sections 1 and 3), however section 2 is frankly in a poor state, and needs a major rewrite.\n\n* In q(x,t) the time variable “t” is reverse time, but this wasn’t defined. This is confusing as q(x,T) is claimed to be close to data. It would be clearer if one would use eg q(x,t’) instead, or perhaps go back to using q(x,0) to denote a fully reversed density. \n* It is not explained where eq 3 is coming from. This is not the score loss, but instead one of the later variants. The p(x,t|x0) looks “magic” from the presentation alone. Given that there are many losses that have been introduced, it would be good to position this loss a bit more to literature. It would also be good to discuss if the results of this paper are dependent on this particular model variant, or if the results also generalise to other variants. \n* It’s unclear where x0 is coming from in eq 3. Is there an outer E_{x0} expectation missing, since one should iterate the loss over all observations. I assume this is merged into the E_{1} thingy [please write this explicitly, it might also be useful to move to \\int notation if \\E gets too cumbersome], but that should then expand into two expectations: one over the starting point, another over time. I would then expect that one should replace the E_1 with E_x0 alone (since we already integrate over time).\n* Fig1 y-axis should perhaps be log likelihood\n* It would be good write the p_noise explicitly, I believe it has a fixed form given eq 1.\n* Can you explain why we start from cross-entropy instead of evidence in eq 4? Usually ELBOs are evidence lower bounds, but this seems a CE bound.\n* It’s unclear how does the overall loss eq 3 connect to the elbo of eq 4. These seem very different. Please make the connection explicit.\n* The sec 2.1. is the “meat” of the paper, and yet I couldn’t follow because of insufficient explanations. The problem starts immediately at eq 4: None of this is explained. We have a cross-entropy, an ELBO, and the exotic ~1 expectation and R and I terms. The cross-entropy is not explained, so I’m not sure where it’s coming from. The ELBO is undefined, and does not match eq 3, so I’m unsure where it comes from. The log noise term seems to be another cross-entropy. The I is not explained, but seems to be score loss, and R is also not explained, but seems to be some kind of reverse stepping error. Later the K and gap are undefined, and we also get more entropies emerging in eqs 8,9 (why?). Please rewrite this section such that the equations can be understood without first reading Huang’21.\n* After all the eq 9 seems sensible. I wonder where it comes from: is this novel or from Huang’21?\n* In eq 9 the entropy seems constant: do we need this term?\n* Is lemma 2 novel? \n* There is likely a typo in table 1 at sigmaT-sigma0I. \n* I can’t follow the gap discussion in 2.2 and 2.3. because gap was never defined: not sure what it is. \n* I’m a bit surprised that the equations contain (cross)entropies several times, yet they are not acknowledged in text almost at all. It would be good to discuss the role of (cross)entropies in the text. \n* I don’t understand fig 4. So blue is entropy that goes up and down over time. Red is number of mixture components which goes down as diffusion time increases (?). Do the red and blue curves have something to do with each other?\n\n\nBroader Impact Concerns: None\n\n==================================================\n\nMetareview:\nRecommendation: Reject\n\nComment: In the current version (revision), the authors claim that they found an interesting trade-off in setting the (absolute) diffusion time T for generative diffusion models. They also claim that a bound of the likelihood is theoretically analyzed. Based on the theoretical analysis, the authors propose to set T to a smaller value than typical values in the literature, and use a trainable distribution for the initial distribution of the reverse process. Some empirical results are shown.\n\nReviewers raised concerns on clarity, novelty, and technical soundness, most of which have been addressed by the authors. Two reviewers recommended accept, while the other leaning to reject.\nI raised major concerns (see below), and asked questions to the authors and the reviewers. The authors rebuttal didn't fully address the concerns, the two accepting reviewers didn't support the paper by addressing my concerns in one week, and the reviewer leaning to reject sympathized me.\n\nI have to say the current paper is highly misleading and publishing it wouldn't give positive impact to the community. Also I wan't convinced by the authors' last comments that they revise the paper so that the paper wouldn't mislead readers. Therefore, I reject the paper. Details follow.\n\nMy major concerns in the last communications were these three.\n\n\t1\tRemove ambiguity of the setting. I didn't suggest to delve into the discretization, but suggest to make it clear that the analysis makes sense. You say you use a fixed step-size eta = T/N. But what do you mean by fixed? Fixed for different data? Fixed for different T? Why don't you say the value of N if it's fixed for all cases? Even in this communication, your answer is ambiguous. What I meant is that because of the invariance on the scale (fast diffusion for short time vs. slow diffusion for long time), if you increase T and decrease N and noise schedule is optimized for each case, you could show opposite results. My suggestion is simply to make things explicit so that readers see that the invariance can't absorb the effect of decreasing T both in SDE and ODE.\n\t2\tNo theoretical contribution can be found. You say \"the first to formally and extensively discuss the tradeoff that emerges from the ELBO formulation, and corroborate observations with a thorough experimental validation.\" But discussion without conclusion is not theoretical contribution. Without knowledge, one expects that optimal T can be any non-negative value. The known decomposition into non-increasing and non-descreasing functions implies the trade-off, which however only applies to a bound, and also doesn't narrow the range of possible value for T.
Also, this decomposition is not novel, so not the contribution of the current work. You make some statements as lemmas, which also don't narrow the range. What is the conclusion of your discussion? If you'd say you \"corroborate\" observations, specify which statements theoretically corroborate the observation. What unknown statements you make in the discussion, and how they corroborate observations?\n\t3\tNot well-designed experiments I see your main claims are\n\t•\tdecreasing T from the typical setting in the literature improves the performance.\n\t•\tReplacing the top diffusion layers with NF is beneficial. So these should be empirically supported at least. For the first point, it is unclear what is the typical setting in the literature. Probably the easiest is to take a few SOTA diffusion models, and show that shrinking T improves the performance. If the story of the paper is true, readers will expect this. Why don't you show it? For the second point, the baseline should be a carefully optimized (in terms of hyperparameters including noise schedule, step size, etc.) diffusion model. After you have an optimal diffusion model for given data, you should replace the top part with NF and show gains in terms of accuracy and/or computation time. Current experiments do not well disentangle different factors.\n\nThe first point was addressed. The authors should clearly describe how all the parameters are set so that readers are convinced that the plots show what should be shown.\nOn the second point, the authors insisted nothing about the theoretical contribution in the last comments. I suggest to change the abstract and introduction so that readers won't expect any theoretical contribution. The current abstract says \"lack of theory\", which usually implies that this paper would have some contribution. Also, it does not make sense to spend several pages with \"discussion\" without any conclusion or theoretical contribution. Most of the theory parts should be removed. For the claim the authors can make from their results, it is sufficient to show an equation that the ELBO can be decomposed into non-increasing part and non-decreasing part, and give some intuitive argument (perhaps in a few paragraphs). In any case, the authors can't make any reasonably strong theoretical claim because they analyzed only a lower-bound and even failed to prove the existence of trade-off (optimal T is finite). Note that any function can be decomposed into non-increasing and non-decreasing functions, so this decomposition itself says nothing about the behavior of the function at large T.\nOn the third point, the rebuttal was not convincing. I suggested to show shrinking T improves \"a few\" sota models, because it seems that the authors assume that experts would believe that T should be set as large as possible as long as computation resource allows. I don't really believe this assumption (the authors didn't explicitly claim but it's not appropriate to imply it). Even the authors' chosen model, Song21, this didn't really happen. The authors only show that, in some cases, bpd can be improved by reducing T from the original setting. First of all, the gain in bpd is very tiny in Tables 2 and 4, and the authors even refused to do statistical test in the communication with a reviewer. This is not acceptable, and at least the authors should show statistical significance. A more serious problem is that shrinking T degrades FID significantly in Table 4 and 5. Do you still say reducing T improves the performance, and recommend readers to use smaller T? In what applications bpd is more important than FID? This must be justified. Don't mix up this point with computational gain, which only confuses readers on what the authors want to claim. Lastly, in the auxiliary models, the experiments simply show that DPGMM and flow are fast and bad method, and therefore replacing the last part of diffusion model with them degrades the performance while speed up, which is not surprising.\n\nIn my understanding, what the authors can claim with the current results are\n\n1. It was empirically shown that too large T can harm the density estimation performance. But it shouldn't be implied that experts believe T-> infinity is the best if computation cost is ignored. The result is not surprising, but showing a clear peak in Figure 1 can be counted as a contribution. \n2. Replacing the top part of the diffusion model with simple models can speed up with marginal performance degradation. This is again not surprising, but if the authors conduct more detailed experiments in more examples, this could be counted as a contribution.\n\nIn addition, if the authors really prove that optimal T is finite under mild conditions, this will be a theoretical contribution. I'd suggest the authors to make 2. as the main contribution, conduct extensive experiments, and resubmit a major revision to TMLR.\n\n\n\n==================================================\n\n"}}},{"rowIdx":94,"cells":{"paper":{"kind":"string","value":"# Black Box Uncertainty Analysis For Semantic Segmentation\n\nAnonymous authors\n\n\n\nPaper under double-blind review Figure 1: A couple of examples show how well the entropy of predicted outputs correlates with misclassified pixels. Green indicates misclassified pixels with high entropy, red shows misclassified pixels with low entropy, and blue shows correctly classified pixels with high entropy. Images in each row show the input and outputs from DRN D-22, OneFormer ConvNeXt-L and SegFormer B5, respectively. We note that entropy captures misclassified pixels with high recall.\n\n## Abstract\n\nSemantic segmentation has become an important task in computer vision with the growth of self-driving cars, medical image segmentation, etc. Although current models provide excellent results, they are still far from perfect. While there has been significant work in trying to improve the performance, both with respect to accuracy and speed of segmentation, there has been little work which analyses the failure cases of such systems. In this work, we aim to provide an analysis of how segmentation fails across different models and consider the question of whether these can be predicted reasonably at test time. To do so, we explore existing uncertainty-based metrics and see how well they correlate with misclassifications, allowing us to define the degree of trust we put in the output of our prediction models.\n\nThrough several experiments on three different models across three datasets, we show that simple measures such as entropy can be used to capture misclassification with high recall.\n\n## 1 Introduction\n\nSemantic segmentation is defined as a task which involves taking an image and labelling each pixel of the image as belonging to one of a set of predefined classes. With the advent of self-driving cars, path-finding robots, and various other such problems, semantic segmentation has become essential in the field of computer vision, as all of these applications rely on being able to segment, parse and understand the scene they see in order to take appropriate action. Semantic segmentation is even used in the medical field to segment areas of medical images for analysis. When used for such critical tasks, it is of utmost importance that the models we propose provide us with accurate results or some confidence metrics which can be used to rely on their outputs.\n\nMost of the earlier approaches to semantic segmentation involve using Convolutional Neural Networks (CNN)\nto extract features and predict the segmentation class for each pixel. Examples of such approaches include Chen et al. (2014); Yu et al. (2017); Chen et al. (2017). Some of these approaches also utilise Conditional Random Fields (CRF) to improve their results further. More recent approaches use Transformer-based architectures such as Cheng et al. (2021); Xie et al. (2021); Jain et al. (2023b;a). Cheng et al. (2021) also utilises masked predictions for each class rather than simple pixel-wise multi-class predictions.\n\nAlthough we now have a wealth of approaches that target semantic segmentation, the uncertainty quantification for their predictions is relatively less well-studied. This is especially true during test time since we cannot judge the model against ground truth predictions. For example, consider the case when there is a domain shift in the inputs during test time; how do we trust the output of our network without any form of confidence or trust metric? There have been works which look at uncertainty in deep neural networks in general (Gawlikowski et al.,\n2023; Grathwohl et al., 2019; Hendrycks & Gimpel, 2016), including relatively new ones like Vazhentsev et al.\n\n(2022), which specifically considers uncertainty in Transformer networks. For semantic segmentation, we have works such as Xia et al. (2020); Rahman et al. (2022), which aim to detect failures and out-of-distribution samples. However, they use separate networks that require additional training. Our work explicitly focuses on obtaining such information at test time without any extra networks or training. By this, we mean that without any knowledge of the architecture, at test time, we wish to know whether our model has succeeded in semantic segmentation or not. Considering the gravity of this task, we believe that there should be a study of the same for the sake of understanding these networks better.\n\nIn this paper, we first look at where these segmentation networks fail to predict the correct classes (Section 3). We then ask questions such as, *\"Knowing where these networks fail, can we predict these failures?\"*, and \"How well do current approaches of gauging uncertainty correlate with misclassification?\". To this end, we perform a series of experiments, analyse the results and show that certain uncertainty methods can be used to decide how much we should trust the output of our networks (Section 5.3) and that we can improve upon our results by using calibrated models (Section 5.4). We show that these techniques work even when we consider domain shifts (Section 5.5) or noise (Section 5.6) in the input. For instance, when performing transfer learning on a network trained on Cityscapes to the Dark Zurich dataset, we validate that we can predict the specific cases of failure in segmentation well. We hope our work provides insight into existing segmentation networks and helps drive further research to ensure their trustworthiness. Figure 1 shows some examples of our approach where we use entropy to judge whether a pixel is likely to be misclassified. We note that simple entropy can identify the likely misclassified regions with high recall. Further detailed analysis is presented in Section 5.\n\n## 2 Related Works\n\nSemantic segmentation has been researched by the vision community for over a decade, and has been well documented in survey papers such as Guo et al. (2018); Hao et al. (2020). However, we will primarily concern ourselves with relatively recent works that use deep neural networks for performing this task. Among these, most earlier works, such as Chen et al. (2014); Long et al. (2015); Yu et al. (2017); Zhao et al. (2017);\nChen et al. (2017); Yang et al. (2018), are convolutional in structure and generally make use of existing Convolutional Neural Networks (CNN) such as VGG (Simonyan & Zisserman, 2014), ResNet (He et al.,\n2016), ResNeXt (Xie et al., 2017), for feature extraction purposes followed by combining extracted features, possibly at multiple scales, in novel ways. Generally, the classification is done at a lower resolution than the input image and is upsampled back to the original dimensions near the end of the pipeline. The improvement in performance across these approaches mainly derives from coming up with new novel ways of pooling and combining the extracted features, such as the usage of Dilated Convolutions, Pyramid Pooling and Atrous Spatial Pyramids, among others. Newer approaches such as Cheng et al. (2021); Xie et al. (2021); Strudel et al. (2021); Gu et al. (2022); Jain et al. (2023b;a), are based on Transformer networks introduced in Vaswani et al. (2017). Initially proposed for natural language processing tasks, Transformers also proved effective in computer vision. These approaches follow two major pipelines for extracting features from the input. Either they use CNN feature extractors, much like earlier works, followed by Transformer decoders for refinement. Alternatively, they use ViTlike (Dosovitskiy et al., 2020) architectures where the input image is divided into patches, which are directly fed to a Transformer encoder for feature generation, followed by a Transformer decoder and segmentation head.\n\n\n\n\n\nFigure 2: Comparison of misclassified pixels and entropy of DRN D-22 (DRN), OneFormer ConvNeXt-L\n(OF) and SegFormer B5 (SF) networks on a couple of Cityscapes validation images, along with the images themselves and edges detected using the Scharr operator. It is to be noted that OneFormer generally produces high entropy outputs, which cause most of the image to be grey. However, we can see that the highest entropy regions, in bright white, still correspond well to misclassified pixels.\nWe also consider how to extract uncertainty information from deep neural networks and refer to some existing works. Works like Deep Ensembles (Lakshminarayanan et al., 2017) and Bayesian Networks (Blundell et al.,\n2015) allow us to measure the uncertainty of predictions relatively easily, but they require special training procedures and are not applicable to already existing networks. However, Gal & Ghahramani (2016) shows that Dropout layers can be used to generate multiple predictions for a given input and thus allow us to compute a certain measure of model uncertainty. Similarly, Houlsby et al. (2011) proposed a metric originally meant for active learning but can also be used as an uncertainty measure for existing networks. To the best of our knowledge, this type of work focusing on semantic segmentation has not been done earlier.\n\nWorks such as Jammalamadaka et al. (2012); Parikh & Zitnick (2011); Vazhentsev et al. (2022); Xia et al.\n\n(2020); Rahman et al. (2022) are closest to our work. Vazhentsev et al. (2022) proposes and evaluates several uncertainty metrics for gauging the uncertainty of Transformer-based architectures and has similar goals of misclassification detection using uncertainty; however, the paper only explores these within the setting of Transformer architectures, and while most newer segmentation architectures are Transformer-based, older CNN-based networks are still used. Thus, we focus on making our evaluation and analysis as general as possible.\n\n## 3 Failure Analysis\n\nBefore we develop a trust score or metric, we must first look at and try to understand the failure cases of our models. We consider a few standard segmentation networks - Dilated Residual Networks (Yu et al., 2017)\n\n\n\nFigure 3: Cumulative histogram of entropy values for correctly classified (blue) and misclassified pixels\n(orange) for a couple of experimental settings. The graph is plotted such that if we choose any threshold value along the x-axis we can directly read what fraction of correctly classified and misclassified pixels are captured.\n(DRN), OneFormer (Jain et al., 2023a) and SegFormer (Xie et al., 2021) and look at their outputs. We specifically choose a mix of older and newer networks for this since they are quite different architecturally and should allow us to focus on generalities rather than the peculiarities of any single architectural family.\n\nWe use pre-trained weights on the Cityscapes dataset (Cordts et al., 2016) for all networks and observe their outputs on the validation set. Figures 2b to 2d and 2j to 2l highlight the misclassified pixels for the networks on a couple of such images. Black indicates correctly classified or ignored pixels, whereas white indicates misclassified pixels.\n\nFrom the images, it is clear that OneFormer and SegFormer are better than DRN at segmenting images.\n\nIndeed, mIoU for DRN D-22, OneFormer ConvNeXT-L, and SegFormer B5 are 0.6790, 0.8287, and 0.8239, respectively. However, we note that the failure modes for the networks are very similar. All of these networks misclassify pixels in similar regions of the images. It is also evident that several of these misclassifications lie along the edges of various objects in the images. This makes intuitive sense since it can be challenging, even for humans, to assign the boundary pixels to a particular object exactly. This naturally brings us to the question, *\"Can we reasonably predict which pixels are most likely to be* misclassified?\". Even if we cannot accurately predict which pixels will be misclassified, we would like to have some measure of confidence in these classifications. Since we would like to apply whatever approach we come up with on existing networks without any re-training or finetuning, we need to consider what kind of information we can extract and use from just the images and their segmentation predictions since that is all we have during test time. Edge detection may seem an obvious first choice, as we have already established that several misclassified pixels lie along object boundaries. A better choice is some measure of uncertainty, such as entropy.\n\nFigures 2e and 2m show the edges obtained when the Scharr filter is applied to the images. The gradient magnitudes thus obtained are scaled between [0, 1], with 0 being black and 1 being white. All ignored pixels are also in black. Since the edges are obtained from just the image rather than the predictions, the network used does not matter. This already tells us that edge detection may not be particularly effective in predicting misclassifications, which depend on the segmentation network used. While some edges align well with the misclassifications, most are correctly classified, especially the edges within each object. The fact remains that only the edges that align with ground truth label boundaries are misclassified. Moreover, by its very nature, edge detection completely fails to capture any misclassified pixels within objects; this is especially evident in the second image (Figure 2m).\n\nOn the other hand, the simple entropy of predicted labels, as shown in Figures 2f to 2h and 2n to 2p, is far more aligned with the misclassifications. Similar to edges, the entropy is scaled between [0, 1] for each image and ignored pixels are set to 0. Here, we see a marked difference between OneFormer and the other two networks; DRN and SegFormer provide very confident predictions and result in most pixels having very low entropy, whereas OneFormer's classifications are under-confident, resulting in most pixels having higher entropy. However, even then, at a glance, we notice that the highest entropy pixels align well with misclassifications. For a more quantitative look, we plot the cumulative histogram of entropy values for correctly classified and misclassified pixels in Figure 3. The plot accumulates histogram counts for correctly classified pixels as the threshold increases and accumulates counts for misclassified pixels as it decreases. This allows us to directly read the percentage of correct and incorrect pixels predicted if we threshold entropy at any given value and consider anything below the threshold to be correctly classified and anything above it to be misclassified. With these observations in mind, we focus on quantitative analysis of various uncertainty measures, which can be derived at test time, on pre-trained networks and consider how well they manage to predict misclassified pixels.\n\n## 4 Uncertainty Metrics\n\nWe now describe the uncertainty metrics used for our experiments and how they are calculated. We only consider metrics that can be obtained purely at test time and those that work with nearly all neural networks.\n\nThis rules out any approach for gauging uncertainty that requires us to change the network structure or fine-tune it. Specifically, we consider the following techniques.\n\n- **Probability Scores** - Some of the simplest possible uncertainty metrics we can obtain involve directly using the highest probability scores output by the network (Hendrycks & Gimpel, 2016).\n\nWe consider two different but related metrics. The first one is often called the Variation Ratio and is defined as follows:\nVR = 1 − pm (1)\nwhere, pm = maxK\nk=1(pk) represents the highest probability across all classes, and pk represents the probability of a pixel belonging to class k. The second metric, Probability Margin, takes into account the difference between the highest and the second highest probability and is defined as:\n\n$$\\mathrm{VR}=1-p_{m}$$\n$$\\mathrm{PM}=1-\\left\\{p_{m}-\\operatorname*{max}_{k=1,k\\neq m}^{K}(p_{k})\\right\\}$$\n(2)\n- **Entropy** - Another simple and possibly the most commonly used uncertainty metric we can obtain from a model. The only requirement is that the model outputs a probability distribution per pixel over all classes, which is true for nearly all (if not all) semantic segmentation models. Formally, the entropy for a single pixel is defined as:\n\n$$\\mathrm{H}=-\\sum_{k=1}^{K}p_{k}\\log p_{k}$$\n\n$\\downarrow$ . \n\n$\\downarrow$ . \n\npk log pk (3)\nwhere, K is the number of classes and pk represents the probability of the pixel belonging to class k.\n\nThe above-mentioned techniques can be used with just a single output mask per image, but for metrics like BALD we need multiple outputs. To generate such outputs for each input image, we consider the following techniques.\n\n- **Monte Carlo Dropout (MC Dropout) (Gal & Ghahramani, 2016)** - Typically, Dropout layers (Hinton et al., 2012) in neural networks are disabled during test time and replaced with a simple scaling of the features. MC Dropout utilises these layers to generate multiple predictions for the same image by keeping the Dropout layers active during test time. This introduces a source of randomness in the network, which allows it to produce different outputs for the same input on each forward pass. However, one drawback of the technique is that it can only be used with networks trained with Dropout layers.\n\n- **Noise** - Yet another approach to obtaining multiple outputs is to introduce a tiny amount of random noise in the input, which perturbs the output. Like MC Dropout, this allows us to generate multiple predictions for the same input by using multiple forward passes (each with a different noise). For our purposes, we use zero mean Gaussian noise, with a small standard deviation.\n\n- **Scaling** - Segmentation networks already use multi-scale inputs to improve their performance (Jain et al., 2023a; Yu et al., 2017). Essentially, it involves running forward passes on multiple scales of the input, which generate multiple segmentations for the same image. The networks can then scale back the outputs to the original resolution and average them to get better results than a single pass.\n\nWe can, however, keep these generated masks for computing uncertainty.\n\nOnce we obtain multiple predictions for each input image, we can extract uncertainty information from these in the following manner.\n\n- **Averaged Probability Scores** - Given multiple probability distributions over a pixel, we can take their mean to obtain a single representative probability distribution for said pixel. Mathematically we have:\n\n$$\\overline{p_{k}}=\\frac{1}{N}\\sum_{i=1}^{N}p_{k}^{i}\\tag{1}$$\n$$\\left(4\\right)$$\n$$\\left(5\\right)$$\n\nwhere pk represents the average probability of a pixel belonging to class k, and p i k represents the probability of the pixel belonging to class k for the i th prediction. N is the total number of predictions. Once we have the average probability distribution per pixel, we can compute Variation Ratio and Probability Margin values over them as follows:\n\n$\\mathrm{VR}=1-\\overline{p_{m}},\\quad\\mathrm{where}\\quad\\overline{p_{m}}=\\max_{k=1}^{K}(\\overline{p_{k}})$ $$\\mathrm{PM}=1-\\left\\{\\overline{p_{m}}-\\max_{k=1,i\\neq m}^{K}(\\overline{p_{k}})\\right\\}$$\n- **Averaged Entropy** - Similar to averaged probability scores, we can use Equation (4) to obtain the average probability distribution per pixel and compute entropy over it as:\n\n$$\\mathrm{H}=-\\sum_{k=1}^{K}{\\overline{{p_{k}}}}\\log{\\overline{{p_{k}}}}$$\n$$({\\mathfrak{f}}{\\mathfrak{o}})$$\n\n$$\\left(7\\right)$$\npk log pk (7)\n- **Variance** - Variance of the probability distribution has also been used as a measure of uncertainty in Bayesian settings (Gal et al., 2017; Smith & Gal, 2018). Since we have multiple predictions, we can compute the empirical variance of class k for any given pixel as:\n\n$$\\sigma_{k}^{2}=\\frac{1}{N}\\sum_{i=1}^{N}(p_{k}^{i}-\\overline{{{p_{k}}}})^{2}\\tag{1}$$\n\nwhere, pk is calculated as in eq. (4). To obtain a single value per pixel, we consider taking both the average as well as the maximum across across all classes.\n\n- **Bayesian Active Learning by Disagreement (BALD)** (Houlsby et al., 2011) - Originally proposed as an acquisition function for selecting samples in a Bayesian active learning setting, BALD\ncan also be used as an uncertainty metric. The BALD score is defined as:\n\n$$\\mathrm{BALD}=-\\sum_{k=1}^{K}{\\overline{{p_{k}}}}\\log{\\overline{{p_{k}}}}+{\\frac{1}{N}}\\sum_{k=1,i=1}^{K,N}p_{k}^{i}\\log p_{k}^{i}$$\n$$({\\boldsymbol{8}})$$\n\n$$({\\mathfrak{g}})$$\nk(9)\nwhere, pk and p i k are same as defined earlier. Essentially, it checks the disagreement between the entropy of the expected prediction and the expected entropy across all predictions. Higher disagreement represents more uncertainty.\n\n## 5 Experiments And Observation\n\nIn this section, we discuss the various experiments performed along with their results and observe the degree of correlation between existing uncertainty metrics and misclassifications in semantic image segmentation.\n\nWe also look at how well we are able to capture misclassifications when there are domain shifts in the inputs or when the input is noisy. We also consider calibrating the models to improve our results further.\n\n## 5.1 Datasets And Models\n\nWe consider the following datasets for our experiments, as they are quite standard for image segmentation tasks and the networks we have chosen provide pre-trained weights for these datasets.\n\n- **Cityscapes (Cordts et al., 2016)** - The primary dataset used for our experiments consists of dashcam videos from cars in several German cities targeted towards understanding urban street scenes. The validation set is comprised of 500 finely annotated images of 2048 × 1024 resolution.\n\nPixels in each image are annotated as belonging to one of 30 defined classes, of which 19 are considered for classification and evaluation. Any pixels not belonging to one of these classes are effectively ignored for evaluation purposes.\n\n- **Dark Zurich (Sakaridis et al., 2019)** - A dataset analogous to Cityscapes with a small validation set of only 50 images with resolution of 1920 × 1080; it consists of urban scenes from the city of Zurich at night time as opposed to Cityscapes where the images are taken at day time. We use this dataset to check how well our metrics perform when there is a domain shift of the input.\n\n- **ADE20K (Zhou et al., 2019)** - A rather large dataset with a validation set of 2000 images of varying resolutions. Unlike Cityscapes and Dark Zurich, ADE20K has a much larger number of segmentation classes, 150 and the images are not only from dashcam videos but cover a wide range of subjects.\n\nWe now describe the segmentation networks that were used in our experiments. We choose our models such that they range from small (DRN D-22) to large (OneFormer Swin-L) and cover a wide variety of architectures.\n\n- **Dilated Residual Networks (Yu et al., 2017)** - Convolutional Neural Network (CNN) based on ResNet (He et al., 2016) but using Dilated Convolutions instead, which helps it achieve better performance in semantic segmentation tasks. We use the pre-trained weights for DRN D-22 and DRN D-105 variants provided by the authors for the Cityscapes dataset.\n\n- **OneFormer (Jain et al., 2023a)** - Transformer-based architecture that utilises features from a backbone network and passes them through a Transformer decoder to get pixel-level classifications.\n\nWe use pre-trained weights with ConvNeXt-L and Swin-L backbones on Cityscapes and ADE20K\ndatasets, as provided by the authors.\n\n- **SegFormer (Xie et al., 2021)** - Another Transformer-based architecture that uses a Hierarchical Transformer encoder and a Multi-Layer Perceptron (MLP) decoder. We use pre-trained weights of the B5 variant for Cityscapes and ADE20K datasets as provided by the authors.\n\n## 5.2 Evaluation Metrics And Scenarios\n\nIn order to measure how well a particular uncertainty metric corresponds to misclassifications, we use Precision, Recall, Area Under the Receiver Operating Characteristic (AUROC), Area Under the Precision Recall Curve (AUPRC) and False Positive Rate at 95% True Positive Rate metrics. AUROC involves plotting the False Positive Rate against the True Positive Rate at different thresholds and then computing the area covered by the curve. Similarly AUPRC computes the area under the Precision Recall Curve at different thresholds. While AUROC and AUPRC consider multiple threshold values, we need to choose a single threshold value to binarise the uncertainty scores for computing Precision and Recall. Unfortunately, as seen in Figure 2, the scales of these uncertainty values can vary quite a lot across different models and even across different images for a single model. As such, we cannot use a single constant threshold value across all models or images. Therefore, we define a couple of processes to dynamically choose a threshold for each image. Both methods rely on the fact that the fraction of misclassified pixels is not very high.\n\n1. For the first method, we check the ratio of predicted misclassified pixels to total pixels at multiple uniformly spaced threshold levels and select the threshold that causes the largest change in this value.\n\n2. We note that the ratio of predicted misclassified pixels to total pixels increases as the threshold increases. For the second method, we specify a maximum value allowed for this ratio and choose the highest threshold, which results in a ratio lower than the specified value.\n\nBefore we discuss the results, we clarify the meaning of each scenario:\n- **Base** - The simplest scenario where we only consider a single forward pass. As such, only Variation Ratio, Probability Margin and Entropy can be used as uncertainty metrics for this setting.\n\n- **Noise** - The images are injected with a Gaussian noise of zero mean and 0.01 standard deviation.\n\nThe noise is injected after normalizing the images. We use ten forward passes per image in this setting.\n\n- **Scale** - This represents using multi-scaled inputs for generating multiple outputs. We use scales of\n{0.5, 0.75, 1.0, 1.25, 1.5} for our purposes.\n\n- **Drop** - MC Dropout is applied in this scenario. Since it requires the network to have Dropout layers, we did not use this technique with DRN. Similar to the Noise scenario, we use ten passes per image.\n\nBefore moving ahead, we would like to state that although a large number of experiments under various scenarios were performed, we only highlight some of those results for brevity and compactness in the following text. The complete set of results for all the experiments is provided in the appendix for this paper. Similarly, since we obtain the metrics for each pixel rather than each image, we consider both micro-averaging and macro-averaging over the images. However, we only show the micro-averaged results in this text and provide the rest in the appendix. The appendix also includes classwise results for several experiments.\n\n## 5.3 Primary Analysis\n\nAs AUROC allows us to get a good overview of how our uncertainty metrics correlate with misclassification, we first look at these in Table 1 for the Cityscapes dataset. We observe that simple metrics like Entropy and Probability Scores perform better than more involved metrics such as Variance and BALD. We also look at the values of AUROC on ADE20K dataset in Table 2. The differences are smaller between various metrics on this dataset. Across both datasets the Scale scenario performs the best.\n\nWe show the AUPRC-Error, AUPRC-Success and False Positive Rate at 95% True Positive Rate results on the Cityscapes dataset using Entropy as an uncertainty metric in Table 3. AUPRC-Error denotes the AUPRC obtained when the misclassifications are treated as the positive class, whereas AUPRC-Success is\n\n| Scenario | Var. Ratio | Prob. Margin | Entropy | Avg. Var. | Max. Var. | BALD |\n|------------|----------------------|----------------|-----------|-------------|-------------|--------|\n| | DRN D-22 | | | | | |\n| Base | 0.9118 | 0.9152 | 0.9205 | | | |\n| Noise | 0.9119 | 0.9152 | 0.9205 | 0.7609 | 0.7614 | 0.8196 |\n| Scale | 0.9410 | 0.9414 | 0.9398 | 0.8973 | 0.8976 | 0.9066 |\n| | OneFormer ConvNeXt-L | | | | | |\n| Base | 0.7265 | 0.7893 | 0.7089 | | | |\n| Noise | 0.7334 | 0.7927 | 0.7153 | 0.6458 | 0.6323 | 0.6727 |\n| Scale | 0.7647 | 0.8305 | 0.7350 | 0.7835 | 0.7637 | 0.8173 |\n| Drop | 0.7471 | 0.8059 | 0.7278 | 0.7219 | 0.6994 | 0.7624 |\n| | SegFormer B5 | | | | | |\n| Base | 0.8830 | 0.8892 | 0.8997 | | | |\n| Noise | 0.8833 | 0.8896 | 0.9002 | 0.6972 | 0.6965 | 0.7520 |\n| Scale | 0.9212 | 0.9245 | 0.9290 | 0.8678 | 0.8687 | 0.8947 |\n| Drop | 0.8842 | 0.8904 | 0.9008 | 0.7677 | 0.7660 | 0.8255 |\n\nTable 1: Area under the Receiver Operating Characteristics (AUROC) on the Cityscapes dataset. We observe that we can obtain high AUROC scores even with simple uncertainty metrics such as Entropy.\n\n| Scenario | Var. Ratio | Prob. Margin | Entropy | Avg. Var. | Max. Var. | BALD |\n|------------|----------------------|----------------|-----------|-------------|-------------|--------|\n| | OneFormer ConvNeXt-L | | | | | |\n| Base | 0.6067 | 0.7048 | 0.5588 | | | |\n| Noise | 0.6052 | 0.7039 | 0.5576 | 0.6328 | 0.6184 | 0.6218 |\n| Scale | 0.6243 | 0.7285 | 0.5693 | 0.6880 | 0.6635 | 0.7473 |\n| Drop | 0.6136 | 0.7151 | 0.5618 | 0.6732 | 0.6517 | 0.7359 |\n| | | SegFormer B5 | | | | |\n| Base | 0.8000 | 0.8024 | 0.8113 | | | |\n| Noise | 0.8003 | 0.8028 | 0.8115 | 0.7123 | 0.7116 | 0.7627 |\n| Scale | 0.8356 | 0.8353 | 0.8384 | 0.7929 | 0.7929 | 0.8141 |\n| Drop | 0.8016 | 0.8039 | 0.8125 | 0.7521 | 0.7520 | 0.7914 |\n\nTable 2: Area under the Receiver Operating Characteristics (AUROC) on the ADE20K dataset. While we do not perform as well as on Cityscapes our AUROC values are still reasonably good. obtained when correctly classified pixels are treated as the positive class. Since all of our uncertainty metrics are normalized between [0, 1] and higher values represent more uncertainty, we flip these values as 1 − x for calculating AUPRC-Success. Similar to AUROC, the Scale setting performs the best. We now take a look at how well our first method for dynamic thresholding works. Table 4 shows the results of using the largest difference as a thresholding technique on the Cityscapes dataset. We use Entropy as the uncertainty metric. We observe that while we get very high recall rates, we also select quite a large fraction of pixels with values reaching as high as 40% for the Scale setting with DRN. As a result, precision suffers quite a bit in these cases. However, we will see later that this technique is useful when the network produces highly inaccurate outputs, such as when there is a domain shift in the input.\n\nWe now consider the performance of the second thresholding method. Table 5 shows the results for the same on the Cityscapes dataset. As expected, recall improves at the cost of precision across all scenarios when we increase the maximum fraction of pixels allowed. However, even with as few as 10% pixels, we can get 60-70% recall using Entropy as an uncertainty metric. Among the different scenarios, Scale performs better than the others, especially for Transformer-based networks.\n\n| Scenario | AUPRC-Error ↑ | AUPRC-Success ↑ | FPR@95%TPR ↓ |\n|------------|----------------------|-------------------|----------------|\n| | DRN D-22 | | |\n| Base | 0.4249 | 0.9953 | 25.15 |\n| Noise | 0.4246 | 0.9953 | 25.15 |\n| Scale | 0.4469 | 0.9963 | 19.88 |\n| | OneFormer ConvNeXt-L | | |\n| Base | 0.1193 | 0.9847 | 80.34 |\n| Noise | 0.1244 | 0.9844 | 79.26 |\n| Scale | 0.1424 | 0.9860 | 79.61 |\n| Drop | 0.1332 | 0.9853 | 79.91 |\n| | SegFormer B5 | | |\n| Base | 0.3840 | 0.9966 | 100.00 |\n| Noise | 0.3826 | 0.9967 | 100.00 |\n| Scale | 0.4099 | 0.9975 | 100.00 |\n| Drop | 0.3843 | 0.9966 | 100.00 |\n\nTable 3: AUPRC-Error, AUPRC-Success, and FPR@95%TPR on the Cityscapes dataset using Entropy as the uncertainty metric.\n\n| Scenario | Precision | Recall | F1 Score | Pixel % |\n|----------------------|--------------|----------|------------|-----------|\n| | DRN D-22 | | | |\n| Base | 17.52 | 95.04 | 29.58 | 28.94 |\n| Noise | 17.52 | 95.05 | 29.58 | 28.95 |\n| Scale | 13.04 | 98.74 | 23.04 | 40.38 |\n| OneFormer ConvNeXt-L | | | | |\n| Base | 14.13 | 81.30 | 24.07 | 18.40 |\n| Noise | 14.25 | 80.60 | 24.22 | 18.08 |\n| Scale | 12.11 | 82.76 | 21.13 | 21.84 |\n| Drop | 13.76 | 82.96 | 23.60 | 19.28 |\n| | SegFormer B5 | | | |\n| Base | 21.43 | 86.20 | 34.33 | 13.14 |\n| Noise | 21.37 | 86.34 | 34.26 | 13.20 |\n| Scale | 17.37 | 92.46 | 29.24 | 17.39 |\n| Drop | 21.28 | 86.46 | 34.16 | 13.27 |\n\nTable 4: Precision, Recall and F1 Score on Cityscapes dataset using largest difference as thresholding strategy. All of the results shown use Entropy as the uncertainty metric.\n\n## 5.4 Calibration Experiments\n\nIn order to improve our scores, we also consider calibrating the networks we are using. We compute the Expected Calibration Error and Brier Score of all the networks tested and find that OneFormer, especially, is not well-calibrated. We use temperature scaling on the logits while computing softmax to counter this and generally find that our AUROC scores using Entropy as the uncertainty metric do improve on the calibrated models. This experiment indicates that considering the underlying causes for uncertainty, such as calibration, and fixing these can help further identify the misclassified pixels. We show the results of this experiment in Table 6. We use temperatures of 0.2 and 2 for OneFormer and SegFormer models respectively.\n\nWe found that DRN models were already well calibrated and further temperature tuning was unnecessary.\n\n| Scenario | Max 5% pixels | Max 10% pixels | Max 15% pixels | | | | | | |\n|---------------------|-----------------|------------------|------------------|--------|---------|-----------|--------|---------|-------|\n| Precision | Recall | Pixel % | Precision | Recall | Pixel % | Precision | Recall | Pixel % | |\n| DRN D-22 | | | | | | | | | |\n| Base | 44.04 | 39.76 | 4.82 | 34.80 | 63.64 | 9.75 | 28.11 | 76.42 | 14.50 |\n| Noise | 44.02 | 39.75 | 4.82 | 34.79 | 63.64 | 9.76 | 28.12 | 76.45 | 14.50 |\n| Scale | 43.68 | 39.53 | 4.83 | 35.26 | 63.98 | 9.68 | 28.67 | 78.57 | 14.62 |\n| OneFormer ConNeXt-L | | | | | | | | | |\n| Base | 30.89 | 44.48 | 4.60 | 23.82 | 64.18 | 8.61 | 20.07 | 70.79 | 11.28 |\n| Noise | 31.29 | 45.12 | 4.61 | 24.19 | 65.25 | 8.62 | 20.46 | 71.70 | 11.20 |\n| Scale | 31.49 | 46.19 | 4.69 | 23.63 | 66.65 | 9.02 | 19.21 | 73.98 | 12.31 |\n| Drop | 31.03 | 45.20 | 4.66 | 23.63 | 65.94 | 8.92 | 19.41 | 72.95 | 12.01 |\n| SegFormer B5 | | | | | | | | | |\n| Base | 36.61 | 54.78 | 4.89 | 26.09 | 74.94 | 9.38 | 22.42 | 82.52 | 12.03 |\n| Noise | 36.49 | 54.55 | 4.88 | 26.04 | 74.84 | 9.39 | 22.36 | 82.56 | 12.06 |\n| Scale | 37.88 | 56.12 | 4.84 | 26.49 | 78.44 | 9.67 | 20.75 | 86.91 | 13.68 |\n| Drop | 36.61 | 54.74 | 4.88 | 26.05 | 74.96 | 9.40 | 22.31 | 82.68 | 12.10 |\n\nTable 5: Precision and Recall on Cityscapes dataset using maximum percentage of pixels as thresholding strategy. All of the results shown use Entropy as the uncertainty metric.\n\n| Model | ECE ↓ | BS ↓ | AUROC ↑ | AUPRC-Error ↑ | AUPRC-Success ↑ |\n|----------------------|---------|--------|-----------|-----------------|-------------------|\n| OneFormer ConvNeXT-L | | | | | |\n| Uncalibrated | 0.7887 | 0.8012 | 0.7089 | 0.1193 | 0.9847 |\n| Calibrated | 0.1173 | 0.0668 | 0.8387 | 0.2685 | 0.9885 |\n| SegFormer B5 | | | | | |\n| Uncalibrated | 0.0631 | 0.0516 | 0.8997 | 0.3840 | 0.9966 |\n| Calibrated | 0.0282 | 0.0506 | 0.9357 | 0.3382 | 0.9975 |\n\nTable 6: ECE, Brier Score (BS), AUROC, AUPRC-Error and AUPRC-Success for uncalibrated and calibrated models in Base setting with Entropy as uncertainty metric for the Cityscapes dataset.\n\n## 5.5 Domain Shift\n\nIn order to check whether these uncertainty metrics can capture misclassification in the case of input domain shift, we consider testing the models trained with the Cityscapes dataset on the Dark Zurich dataset. Dark Zurich is similar to Cityscapes in the sense that both include dashcam images from German cities. The major difference is that in Dark Zurich the images are taken during nighttime while in Cityscapes the images are taken during the day. This represents a natural domain shift one may encounter in the real world. Table 7 shows the results of using Cityscape- trained DRN and OneFormer models on the Dark Zurich dataset using Entropy as the uncertainty metric. The models perform poorly on this dataset, with mIoU of 0.0710 for DRN and 0.3994 for OneFormer. Even though the models perform poorly, we note that we can still use Entropy to get good Precision and Recall rates using the largest difference method for dynamic thresholding.\n\nWe also note that using multiple scales generally performs better than other scenarios.\n\n## 5.6 Noisy Inputs\n\nWe also consider introducing noise in the input images and observing how it affects our results. Effectively, this represents another form of domain shift. We introduce varying levels of noise in the input and check the performance of the networks on them. The noise introduced is Gaussian in nature with standard deviations of {5, 10, 25, 50}. The noise is introduced into the image before any preprocessing to ensure we do not subvert any mechanisms used by the networks to protect against such cases. Table 8 shows the AUROC values in this scenario on DRN D-22 and OneFormer ConvNeXt-L models. We also provide a visual example of the\n\n\n\nFigure 4: An example of how DRN misclassifies inputs when noise is added and how well entropy can capture it. From left to right, we have the original image followed by increasing amounts of noise added to it. The bottom row shows misclassified pixels with high entropy in green, misclassified pixels with low entropy in red and correctly classified pixels with high entropy in blue.\n\n| Scenario | Largest Difference | | | AUROC | |\n|------------|----------------------|----------|---------|---------|--------|\n| Precision | Recall | F1 Score | Pixel % | | |\n| | DRN D 22 | | | | |\n| Base | 68.92 | 80.93 | 74.44 | 77.21 | 0.6054 |\n| Noise | 68.77 | 80.95 | 74.37 | 77.39 | 0.6024 |\n| Scale | 71.92 | 79.24 | 75.40 | 72.45 | 0.6680 |\n| | OneFormer ConNeXt-L | | | | |\n| Base | 38.41 | 56.71 | 45.80 | 42.67 | 0.5667 |\n| Noise | 41.33 | 56.58 | 47.77 | 39.57 | 0.5641 |\n| Scale | 42.47 | 73.21 | 53.76 | 49.82 | 0.6377 |\n| Drop | 39.10 | 57.44 | 46.53 | 42.46 | 0.5755 |\n\nTable 7: Precision, Recall and AUROC on the Dark Zurich dataset, with Entropy as the uncertainty metric.\n\n| Scenario | DRN D-22 | OneFormer ConvNeXt-L |\n|------------|------------|------------------------|\n| Base | 0.9205 | 0.7089 |\n| Noise 5 | 0.8554 | 0.6975 |\n| Noise 10 | 0.7866 | 0.7008 |\n| Noise 25 | 0.7156 | 0.6810 |\n| Noise 50 | 0.6427 | 0.6375 |\n\nTable 8: AUROC for increasing noise levels on the Cityscapes dataset for DRN and OneFormer models.\n\nsame in Figure 4. As can be seen with increasing noise levels, the outputs become highly inaccurate, but we can still capture most of these using Entropy as the uncertainty metric.\n\n## 6 Conclusion\n\nIn this paper, we present a new form of analysis for semantic segmentation tasks focused on trusting the output of such networks during test time. Our work is primarily concerned with evaluating existing uncertainty metrics and seeing how well they correlate with and are able to capture misclassifications. Our analysis suggests that it is indeed possible to obtain valid evaluation measures that can be used to predict the test time performance of existing segmentation networks. We have shown that even simple measures such as Probability Margins and Entropy can be used as a proxy for capturing misclassified pixels with high recall rates. 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Denseaspp for semantic segmentation in street scenes. In *Proceedings of the IEEE conference on computer vision and pattern recognition*, pp. 3684–3692, 2018.\n\nFisher Yu, Vladlen Koltun, and Thomas Funkhouser. Dilated residual networks. In *Proceedings of the IEEE*\nconference on computer vision and pattern recognition, pp. 472–480, 2017.\n\nHengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia. Pyramid scene parsing network.\n\nIn *Proceedings of the IEEE conference on computer vision and pattern recognition*, pp. 2881–2890, 2017.\n\nBolei Zhou, Hang Zhao, Xavier Puig, Tete Xiao, Sanja Fidler, Adela Barriuso, and Antonio Torralba.\n\nSemantic understanding of scenes through the ade20k dataset. *International Journal of Computer Vision*,\n127:302–321, 2019.\n\n## A Cityscapes And Ade20K\n\nFirst, we provide all results related to Cityscapes and ADE20K datasets. We provide results for Dark Zurich in a separate section, as experiments on it involve input domain shift. Table 9 collates all the performance metrics for the Base setting of all networks. We compute Mean Intersection Over Union (mIoU), Expected Calibration Error (ECE) and Brier Score (BS). For all of the metrics, we show both micro-averaged (µ Avg.)\nas well as macro-averaged (M Avg.) results.\n\nAs expected, DRN performs the worst among all the models, while OneFormer and SegFormer perform much better. We also note that DRN and SegFormer are relatively well-calibrated, but OneFormer is not. Results on Cityscapes are much better than on ADE20K since the latter is a much more challenging dataset with more variety and a much larger number of classes.\n\n| Model | mIoU ↑ | ECE ↓ | BS ↓ | | | |\n|----------------------|----------|---------|--------|--------|--------|--------|\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | |\n| Cityscapes | | | | | | |\n| DRN D-22 | 0.6790 | 0.5479 | 0.0380 | 0.0177 | 0.0808 | 0.0824 |\n| DRN D-105 | 0.7551 | 0.6330 | 0.0337 | 0.0171 | 0.0617 | 0.0631 |\n| OneFormer ConvNeXt-L | 0.8287 | 0.6733 | 0.7887 | 0.8383 | 0.8012 | 0.8011 |\n| OneFormer Swin-L | 0.8288 | 0.6682 | 0.7973 | 0.8400 | 0.8034 | 0.8034 |\n| SegFormer B5 | 0.8239 | 0.6875 | 0.0631 | 0.0160 | 0.0516 | 0.0524 |\n| ADE20K | | | | | | |\n| OneFormer ConvNeXt-L | 0.5684 | 0.4854 | 0.8753 | 0.8195 | 0.9690 | 0.9686 |\n| OneFormer Swin-L | 0.5698 | 0.4921 | 0.8948 | 0.8261 | 0.9689 | 0.9687 |\n| SegFormer B5 | 0.5100 | 0.4434 | 0.1051 | 0.1068 | 0.2646 | 0.2845 |\n\nTable 9: Micro and macro-averaged (µ and M, respectively) Mean Intersection Over Union (mIoU), Expected Calibration Error (ECE) and Brier Score (BS) on Cityscapes and ADE20K datasets.\n\nTable 10 and Table 11 show the Area Under Receiver Operating Characteristic (AUROC) as well as the Precision, Recall and fraction of selected pixels on Cityscapes and ADE20K datasets, respectively.\n\nWe observe that in most cases, Scale seems to provide the best AUROC and Recall, while Noise generally provides good Precision. However, in most cases, Base comes quite close to these values and has the advantage of being computationally cheaper. We also note that simpler metrics such as Variation Ratio, Probability Margin and Entropy are often better than or very close to more involved metrics such as Variance and BALD. The Recall values often reach higher than 90% for Cityscapes and 70% for ADE20K, although often at the cost of selecting too many pixels in the image.\n\nTable 10: Micro and macro-averaged (µ and M, respectively) Area Under Receiver Operating Characteristic (AUROC) and Precision, Recall and the fraction of pixels selected for largest difference thresholding on the Cityscapes dataset.\n\n| Scenario | Uncertainty | AUROC ↑ | Largest Difference Thresholding | | | | | | |\n|----------------------|---------------|-----------|-----------------------------------|--------|--------|--------|--------|-------|-------|\n| Metric | Precision ↑ | Recall ↑ | Pixel % | | | | | | |\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | | |\n| DRN D-22 | | | | | | | | | |\n| Base | Var. Ratio | 0.9118 | 0.9277 | 21.10 | 20.18 | 92.13 | 94.71 | 23.29 | 23.51 |\n| Prob. Margin | 0.9152 | 0.9302 | 19.71 | 18.84 | 93.31 | 95.62 | 25.26 | 25.48 | |\n| Entropy | 0.9205 | 0.9337 | 17.52 | 16.74 | 95.04 | 96.89 | 28.94 | 29.21 | |\n| Noise | Var. Ratio | 0.9119 | 0.9278 | 21.10 | 20.18 | 92.15 | 94.73 | 23.30 | 23.52 |\n| Prob. Margin | 0.9152 | 0.9303 | 19.71 | 18.85 | 93.32 | 95.63 | 25.26 | 25.49 | |\n| Entropy | 0.9205 | 0.9337 | 17.52 | 16.74 | 95.05 | 96.90 | 28.95 | 29.22 | |\n| Avg. Var. | 0.7609 | 0.7722 | 37.97 | 36.82 | 57.26 | 59.58 | 08.04 | 08.13 | |\n| Max. Var. | 0.7614 | 0.7724 | 37.71 | 36.55 | 57.42 | 59.71 | 08.12 | 08.21 | |\n| BALD | 0.8196 | 0.8276 | 29.28 | 28.36 | 72.32 | 73.90 | 13.18 | 13.30 | |\n| Scale | Var. Ratio | 0.9410 | 0.9466 | 15.90 | 15.22 | 97.74 | 98.46 | 32.80 | 33.08 |\n| Prob. Margin | 0.9414 | 0.9465 | 14.80 | 14.19 | 98.14 | 98.76 | 35.36 | 35.66 | |\n| Entropy | 0.9398 | 0.9444 | 13.04 | 12.52 | 98.74 | 99.19 | 40.38 | 40.72 | |\n| Avg. Var. | 0.8973 | 0.8998 | 23.50 | 22.51 | 91.38 | 92.38 | 20.74 | 20.95 | |\n| Max. Var. | 0.8976 | 0.8995 | 23.18 | 22.20 | 91.69 | 92.65 | 21.11 | 21.31 | |\n| BALD | 0.9066 | 0.9075 | 16.49 | 15.82 | 96.37 | 97.10 | 31.17 | 31.44 | |\n| DRN D-105 | | | | | | | | | |\n| Base | Var. Ratio | 0.8939 | 0.9206 | 23.04 | 22.19 | 85.84 | 90.61 | 14.76 | 14.87 |\n| Prob. Margin | 0.8995 | 0.9247 | 21.77 | 20.94 | 87.36 | 91.77 | 15.90 | 16.02 | |\n| Entropy | 0.9094 | 0.9322 | 19.75 | 18.95 | 89.88 | 93.64 | 18.03 | 18.17 | |\n| Noise | Var. Ratio | 0.8949 | 0.9209 | 23.09 | 22.22 | 86.03 | 90.70 | 14.76 | 14.87 |\n| Prob. Margin | 0.9004 | 0.9250 | 21.82 | 20.98 | 87.54 | 91.84 | 15.90 | 16.01 | |\n| Entropy | 0.9103 | 0.9325 | 19.79 | 18.98 | 90.05 | 93.70 | 18.03 | 18.16 | |\n| Avg. Var. | 0.7471 | 0.7666 | 39.53 | 38.65 | 52.68 | 56.64 | 05.28 | 05.34 | |\n| Max. Var. | 0.7472 | 0.7664 | 39.46 | 38.56 | 52.71 | 56.61 | 05.29 | 05.36 | |\n| BALD | 0.8077 | 0.8249 | 32.48 | 31.63 | 66.59 | 70.05 | 08.12 | 08.21 | |\n| Scale | Var. Ratio | 0.9341 | 0.9474 | 19.48 | 18.60 | 93.73 | 95.92 | 19.07 | 19.21 |\n| Prob. Margin | 0.9366 | 0.9490 | 18.33 | 17.51 | 94.52 | 96.48 | 20.43 | 20.57 | |\n| Entropy | 0.9400 | 0.9509 | 16.45 | 15.71 | 95.79 | 97.33 | 23.07 | 23.23 | |\n| Avg. Var. | 0.8893 | 0.9007 | 27.81 | 26.58 | 84.51 | 87.00 | 12.04 | 12.15 | |\n| Max. Var. | 0.8900 | 0.9009 | 27.59 | 26.37 | 84.74 | 87.17 | 12.17 | 12.28 | |\n| BALD | 0.9111 | 0.9197 | 20.60 | 19.70 | 91.20 | 93.16 | 17.54 | 17.67 | |\n| OneFormer ConvNeXt-L | | | | | | | | | |\n| Base | Var. Ratio | 0.7265 | 0.8822 | 15.02 | 16.52 | 84.68 | 87.75 | 18.03 | 18.17 |\n| Prob. Margin | 0.7893 | 0.9073 | 14.37 | 15.63 | 89.19 | 91.32 | 19.84 | 19.98 | |\n| Entropy | 0.7089 | 0.8569 | 14.13 | 16.23 | 81.30 | 84.88 | 18.40 | 18.65 | |\n| Noise | Var.Ratio | 0.7334 | 0.8851 | 15.04 | 16.66 | 84.72 | 87.85 | 18.00 | 18.13 |\n| Prob. Margin | 0.7927 | 0.9087 | 14.15 | 15.72 | 89.07 | 91.37 | 20.12 | 20.25 | |\n| Entropy | 0.7153 | 0.8612 | 14.25 | 16.54 | 80.60 | 85.05 | 18.08 | 18.21 | |\n| Avg. Var. | 0.6458 | 0.6474 | 18.71 | 28.78 | 34.52 | 35.06 | 05.90 | 05.99 | |\n| Max. Var. | 0.6323 | 0.6324 | 17.11 | 27.49 | 31.93 | 32.20 | 05.97 | 06.05 | |\n| BALD | 0.6727 | 0.6742 | 20.21 | 29.16 | 40.01 | 40.55 | 06.33 | 06.42 | |\n| Scale | Var. Ratio | 0.7647 | 0.8972 | 12.96 | 14.53 | 88.64 | 90.63 | 21.87 | 22.06 |\n| Prob. Margin | 0.8305 | 0.9215 | 12.58 | 13.53 | 92.36 | 93.71 | 23.46 | 23.68 | |\n| Entropy | 0.7350 | 0.8736 | 12.11 | 14.36 | 82.76 | 87.42 | 21.84 | 21.90 | |\n\n| Avg. Var. | 0.7835 | 0.7880 | 16.78 | 21.70 | 65.40 | 67.80 | 12.46 | 12.54 | |\n|------------------|------------|----------|---------|---------|---------|---------|---------|---------|-------|\n| Max. Var. | 0.7637 | 0.7677 | 15.75 | 20.82 | 62.04 | 64.34 | 12.59 | 12.67 | |\n| BALD | 0.8173 | 0.8213 | 17.66 | 21.76 | 71.71 | 74.12 | 12.98 | 13.05 | |\n| Drop | Var. Ratio | 0.7471 | 0.8936 | 14.89 | 15.82 | 85.54 | 89.39 | 18.36 | 18.44 |\n| Prob. Margin | 0.8059 | 0.9166 | 13.51 | 14.47 | 91.12 | 93.09 | 21.56 | 21.71 | |\n| Entropy | 0.7278 | 0.8712 | 13.76 | 15.12 | 82.96 | 87.21 | 19.28 | 19.40 | |\n| Avg. Var. | 0.7219 | 0.7264 | 21.08 | 27.12 | 49.93 | 52.42 | 07.57 | 07.68 | |\n| Max. Var. | 0.6994 | 0.7023 | 19.55 | 25.98 | 45.49 | 47.68 | 07.44 | 07.54 | |\n| BALD | 0.7624 | 0.7667 | 21.71 | 27.02 | 58.30 | 60.90 | 08.58 | 08.69 | |\n| OneFormer Swin-L | | | | | | | | | |\n| Base | Var. Ratio | 0.7379 | 0.8909 | 13.68 | 15.60 | 86.62 | 89.80 | 20.24 | 20.37 |\n| Prob. Margin | 0.7937 | 0.9127 | 13.30 | 14.59 | 89.82 | 92.61 | 21.57 | 21.72 | |\n| Entropy | 0.7238 | 0.8677 | 13.57 | 15.52 | 82.85 | 86.61 | 19.51 | 19.73 | |\n| Noise | Var.Ratio | 0.7432 | 0.8927 | 14.34 | 15.84 | 86.49 | 89.67 | 19.28 | 19.48 |\n| Prob. Margin | 0.7976 | 0.9139 | 13.76 | 14.87 | 90.09 | 92.60 | 20.93 | 21.10 | |\n| Entropy | 0.7288 | 0.8709 | 13.95 | 15.60 | 83.60 | 87.08 | 19.15 | 19.39 | |\n| Avg. Var. | 0.6603 | 0.6611 | 23.56 | 29.87 | 36.01 | 36.33 | 04.88 | 05.00 | |\n| Max. Var. | 0.6467 | 0.6463 | 22.08 | 28.63 | 33.36 | 33.46 | 04.83 | 04.95 | |\n| BALD | 0.6866 | 0.6874 | 25.56 | 30.50 | 41.36 | 41.61 | 05.17 | 05.24 | |\n| Scale | Var. Ratio | 0.7816 | 0.9109 | 12.76 | 13.84 | 89.92 | 92.58 | 22.52 | 22.69 |\n| Prob. Margin | 0.8415 | 0.9304 | 11.87 | 12.90 | 93.29 | 95.15 | 25.12 | 25.40 | |\n| Entropy | 0.7557 | 0.8951 | 12.30 | 13.65 | 87.69 | 90.83 | 22.78 | 22.95 | |\n| Avg. Var. | 0.8093 | 0.8159 | 19.67 | 22.86 | 70.28 | 71.57 | 11.42 | 11.59 | |\n| Max. Var. | 0.7929 | 0.7985 | 18.86 | 22.18 | 67.53 | 68.59 | 11.44 | 11.61 | |\n| BALD | 0.8352 | 0.8421 | 20.03 | 22.73 | 75.31 | 76.74 | 12.02 | 12.20 | |\n| Drop | Var. Ratio | 0.7543 | 0.8972 | 13.39 | 14.67 | 87.99 | 90.76 | 20.99 | 21.20 |\n| Prob. Margin | 0.8078 | 0.9179 | 12.95 | 14.15 | 90.91 | 93.25 | 22.43 | 22.68 | |\n| Entropy | 0.7396 | 0.8779 | 12.60 | 14.31 | 85.42 | 88.66 | 21.65 | 21.85 | |\n| Avg. Var. | 0.7211 | 0.7397 | 23.02 | 28.08 | 49.79 | 53.67 | 06.91 | 07.04 | |\n| Max. Var. | 0.6994 | 0.7164 | 21.31 | 27.09 | 45.57 | 49.20 | 06.83 | 07.01 | |\n| BALD | 0.7593 | 0.7765 | 24.27 | 28.27 | 57.69 | 61.29 | 07.60 | 07.69 | |\n| SegFormer B5 | | | | | | | | | |\n| Base | Var. Ratio | 0.8830 | 0.9087 | 24.60 | 24.07 | 82.10 | 86.94 | 10.90 | 10.95 |\n| Prob. Margin | 0.8892 | 0.9142 | 23.40 | 22.87 | 83.65 | 88.29 | 11.68 | 11.73 | |\n| Entropy | 0.8997 | 0.9234 | 21.43 | 20.89 | 86.20 | 90.47 | 13.14 | 13.20 | |\n| Noise | Var.Ratio | 0.8833 | 0.9089 | 24.53 | 24.00 | 82.22 | 87.02 | 10.95 | 11.00 |\n| Prob. Margin | 0.8896 | 0.9143 | 23.33 | 22.81 | 83.77 | 88.37 | 11.73 | 11.78 | |\n| Entropy | 0.9002 | 0.9236 | 21.37 | 20.84 | 86.34 | 90.55 | 13.20 | 13.26 | |\n| Avg. Var. | 0.6972 | 0.7147 | 40.19 | 40.11 | 41.61 | 45.11 | 03.38 | 03.40 | |\n| Max. Var. | 0.6965 | 0.7136 | 40.18 | 40.07 | 41.46 | 44.90 | 03.37 | 03.39 | |\n| BALD | 0.7520 | 0.7697 | 34.08 | 34.03 | 53.80 | 57.30 | 05.16 | 05.19 | |\n| Scale | Var. Ratio | 0.9212 | 0.9391 | 20.33 | 19.77 | 90.13 | 93.35 | 14.48 | 14.56 |\n| Prob. Margin | 0.9245 | 0.9418 | 19.21 | 18.67 | 91.02 | 94.10 | 15.48 | 15.56 | |\n| Entropy | 0.9290 | 0.9459 | 17.37 | 16.85 | 92.46 | 95.29 | 17.39 | 17.48 | |\n| Avg. Var. | 0.8678 | 0.8826 | 28.23 | 27.55 | 78.95 | 81.98 | 09.13 | 09.19 | |\n| Max. Var. | 0.8687 | 0.8832 | 28.04 | 27.36 | 79.20 | 82.18 | 09.23 | 09.28 | |\n| BALD | 0.8947 | 0.9091 | 21.36 | 20.81 | 86.42 | 89.33 | 13.22 | 13.28 | |\n| Drop | Var. Ratio | 0.8842 | 0.9097 | 24.44 | 23.91 | 82.38 | 87.17 | 11.01 | 11.06 |\n| Prob. Margin | 0.8904 | 0.9151 | 23.24 | 22.72 | 83.92 | 88.50 | 11.80 | 11.85 | |\n| Entropy | 0.9008 | 0.9242 | 21.28 | 20.74 | 86.46 | 90.66 | 13.27 | 13.33 | |\n| Avg. Var. | 0.7677 | 0.7931 | 37.02 | 36.66 | 56.64 | 61.71 | 05.00 | 05.02 | |\n| Max. Var. | 0.7660 | 0.7915 | 37.06 | 36.69 | 56.31 | 61.38 | 04.96 | 04.98 | |\n| BALD | 0.8255 | 0.8519 | 28.97 | 28.73 | 70.12 | 75.25 | 07.91 | 07.93 | |\n\nTable 11: Micro and macro-averaged (µ and M, respectively) Area Under Receiver Operating Characteristic (AUROC) and Precision, Recall and the fraction of pixels selected for largest difference thresholding on the ADE20K dataset.\n\n| Scenario | Uncertainty | AUROC ↑ | Largest Difference Thresholding | | | | | | |\n|----------------------|---------------|-----------|-----------------------------------|--------|--------|--------|--------|-------|-------|\n| Metric | Precision ↑ | Recall ↑ | Pixel % | | | | | | |\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | | |\n| OneFormer ConvNeXt-L | | | | | | | | | |\n| Base | Var. Ratio | 0.6067 | 0.6941 | 25.45 | 28.92 | 50.31 | 63.73 | 29.11 | 30.34 |\n| Prob. Margin | 0.7048 | 0.7750 | 25.15 | 29.12 | 63.92 | 75.96 | 37.43 | 38.25 | |\n| Entropy | 0.5588 | 0.6164 | 23.07 | 27.39 | 41.38 | 54.30 | 26.42 | 27.64 | |\n| Noise | Var. Ratio | 0.6052 | 0.6927 | 25.73 | 28.89 | 50.62 | 63.82 | 28.97 | 30.43 |\n| Prob. Margin | 0.7039 | 0.7727 | 25.48 | 29.24 | 63.66 | 75.54 | 36.79 | 37.66 | |\n| Entropy | 0.5576 | 0.6151 | 23.34 | 27.23 | 41.43 | 54.34 | 26.13 | 27.57 | |\n| Avg. Var. | 0.6328 | 0.6235 | 30.67 | 35.27 | 41.61 | 45.27 | 19.98 | 22.40 | |\n| Max. Var. | 0.6184 | 0.6084 | 29.52 | 34.66 | 38.94 | 42.18 | 19.42 | 21.76 | |\n| BALD | 0.6218 | 0.6672 | 26.34 | 29.24 | 54.24 | 58.38 | 30.32 | 33.04 | |\n| Scale | Var. Ratio | 0.6243 | 0.7078 | 24.23 | 28.00 | 52.29 | 65.56 | 31.78 | 32.76 |\n| Prob. Margin | 0.7285 | 0.7883 | 24.87 | 28.95 | 65.42 | 76.88 | 38.74 | 39.19 | |\n| Entropy | 0.5693 | 0.6304 | 22.37 | 26.39 | 42.37 | 55.78 | 27.89 | 29.02 | |\n| Avg. Var. | 0.6880 | 0.6728 | 27.69 | 30.54 | 61.78 | 66.51 | 32.85 | 35.02 | |\n| Max. Var. | 0.6635 | 0.6489 | 26.92 | 29.84 | 58.47 | 62.64 | 31.98 | 34.25 | |\n| BALD | 0.7473 | 0.7279 | 28.66 | 30.16 | 73.48 | 77.93 | 37.76 | 39.69 | |\n| Drop | Var. Ratio | 0.6136 | 0.6956 | 25.19 | 28.55 | 50.22 | 64.06 | 29.36 | 30.82 |\n| Prob. Margin | 0.7151 | 0.7778 | 25.10 | 29.06 | 63.51 | 76.03 | 37.26 | 38.10 | |\n| Entropy | 0.5618 | 0.6169 | 22.72 | 26.59 | 40.91 | 54.33 | 26.51 | 27.99 | |\n| Avg. Var. | 0.6732 | 0.6636 | 29.53 | 33.02 | 54.03 | 59.30 | 26.95 | 29.28 | |\n| Max. Var. | 0.6517 | 0.6415 | 28.65 | 32.46 | 50.24 | 54.95 | 25.82 | 28.23 | |\n| BALD | 0.7359 | 0.7258 | 29.66 | 31.50 | 68.32 | 74.11 | 33.92 | 36.23 | |\n| OneFormer Swin-L | | | | | | | | | |\n| Base | Var. Ratio | 0.6115 | 0.6977 | 24.79 | 28.04 | 52.66 | 64.95 | 30.16 | 31.21 |\n| Prob. Margin | 0.7066 | 0.7770 | 25.18 | 28.13 | 66.49 | 76.84 | 37.49 | 38.22 | |\n| Entropy | 0.5661 | 0.6222 | 22.96 | 26.26 | 43.63 | 55.65 | 26.97 | 28.53 | |\n| Noise | Var. Ratio | 0.6110 | 0.6958 | 24.86 | 27.82 | 53.28 | 65.05 | 30.42 | 31.46 |\n| Prob. Margin | 0.7066 | 0.7748 | 24.83 | 27.98 | 65.73 | 76.59 | 37.58 | 38.38 | |\n| Entropy | 0.5652 | 0.6208 | 22.82 | 26.20 | 43.62 | 55.28 | 27.13 | 28.37 | |\n| Avg. Var. | 0.6331 | 0.6281 | 28.40 | 34.11 | 43.92 | 47.86 | 21.95 | 24.05 | |\n| Max. Var. | 0.6180 | 0.6114 | 27.42 | 33.52 | 40.58 | 44.08 | 21.01 | 23.09 | |\n| BALD | 0.6510 | 0.6806 | 26.38 | 29.88 | 57.68 | 62.16 | 31.04 | 33.33 | |\n| Scale | Var. Ratio | 0.6296 | 0.7142 | 23.84 | 27.27 | 55.82 | 67.39 | 33.23 | 33.60 |\n| Prob. Margin | 0.7322 | 0.7933 | 24.05 | 27.69 | 68.38 | 78.83 | 40.37 | 40.47 | |\n| Entropy | 0.5742 | 0.6358 | 21.99 | 25.40 | 45.23 | 57.34 | 29.20 | 30.27 | |\n| Avg. Var. | 0.6994 | 0.6891 | 27.97 | 30.37 | 64.48 | 68.74 | 32.72 | 34.31 | |\n| Max. Var. | 0.6735 | 0.6643 | 26.78 | 29.50 | 60.42 | 64.77 | 32.02 | 33.75 | |\n| BALD | 0.7563 | 0.7427 | 28.37 | 29.68 | 74.23 | 79.18 | 37.15 | 38.73 | |\n| Drop | Var. Ratio | 0.6173 | 0.7005 | 24.31 | 27.57 | 53.69 | 65.76 | 31.35 | 32.33 |\n| Prob. Margin | 0.7156 | 0.7804 | 24.41 | 28.04 | 65.93 | 76.85 | 38.34 | 38.83 | |\n| Entropy | 0.5693 | 0.6241 | 22.34 | 25.76 | 43.40 | 55.83 | 27.57 | 28.91 | |\n| Avg. Var. | 0.6665 | 0.6565 | 27.86 | 31.72 | 55.14 | 59.57 | 28.10 | 30.11 | |\n| Max. Var. | 0.6459 | 0.6333 | 26.98 | 31.18 | 51.19 | 54.98 | 26.93 | 28.93 | |\n\n| BALD | 0.7302 | 0.7182 | 28.76 | 30.76 | 69.12 | 73.75 | 34.11 | 35.92 | |\n|--------------|------------|----------|---------|---------|---------|---------|---------|---------|-------|\n| SegFormer B5 | | | | | | | | | |\n| Base | Var. Ratio | 0.8000 | 0.8388 | 35.54 | 33.27 | 80.01 | 87.38 | 36.81 | 38.62 |\n| Prob. Margin | 0.8024 | 0.8397 | 34.56 | 32.45 | 81.51 | 88.36 | 38.56 | 40.28 | |\n| Entropy | 0.8113 | 0.8484 | 32.20 | 30.34 | 84.63 | 90.97 | 42.96 | 44.99 | |\n| Noise | Var. Ratio | 0.8356 | 0.8628 | 31.86 | 30.41 | 88.13 | 93.03 | 45.22 | 46.72 |\n| Prob. Margin | 0.8353 | 0.8615 | 30.93 | 29.67 | 88.88 | 93.51 | 46.97 | 48.33 | |\n| Entropy | 0.8384 | 0.8661 | 28.80 | 27.87 | 90.24 | 94.82 | 51.21 | 52.85 | |\n| Avg. Var. | 0.7929 | 0.8096 | 39.47 | 37.55 | 77.79 | 83.11 | 32.22 | 33.33 | |\n| Max. Var. | 0.7929 | 0.8089 | 39.28 | 37.40 | 78.06 | 83.31 | 32.48 | 33.60 | |\n| BALD | 0.8141 | 0.8266 | 32.97 | 31.46 | 86.96 | 91.18 | 43.11 | 44.50 | |\n| Scale | Var. Ratio | 0.8003 | 0.8388 | 35.55 | 33.26 | 80.02 | 87.38 | 36.79 | 38.59 |\n| Prob. Margin | 0.8028 | 0.8397 | 34.57 | 32.44 | 81.51 | 88.35 | 38.54 | 40.24 | |\n| Entropy | 0.8115 | 0.8483 | 32.12 | 30.34 | 84.27 | 90.91 | 42.89 | 44.93 | |\n| Avg. Var. | 0.7123 | 0.7490 | 48.46 | 45.09 | 52.74 | 61.89 | 17.79 | 19.46 | |\n| Max. Var. | 0.7116 | 0.7483 | 48.38 | 45.03 | 52.65 | 61.80 | 17.79 | 19.45 | |\n| BALD | 0.7627 | 0.7952 | 41.45 | 38.38 | 68.51 | 76.96 | 27.02 | 29.59 | |\n| Drop | Var. Ratio | 0.8016 | 0.8398 | 35.32 | 33.08 | 80.47 | 87.70 | 37.24 | 39.05 |\n| Prob. Margin | 0.8039 | 0.8407 | 34.34 | 32.26 | 81.92 | 88.65 | 38.99 | 40.71 | |\n| Entropy | 0.8125 | 0.8491 | 31.92 | 30.16 | 84.70 | 91.18 | 43.37 | 45.43 | |\n| Avg. Var. | 0.7521 | 0.7955 | 44.41 | 41.56 | 63.76 | 73.73 | 23.47 | 24.96 | |\n| Max. Var. | 0.7520 | 0.7955 | 44.35 | 41.52 | 63.85 | 73.85 | 23.54 | 25.04 | |\n| BALD | 0.7914 | 0.8299 | 36.50 | 34.17 | 78.07 | 86.38 | 34.96 | 37.22 | |\n\nWe also show the Area Under Precision Recall Curve (AUPRC) and False Positive Rate at 95% True Positive Rate (FPR@95%TPR) for selected models on the Cityscapes dataset in Table 12. For AUPRC we show both AUPRC Error and Success. In AUPRC-Error the misclassified pixels are treated as the positive class whereas in AUPRC-Success correctly classified pixels are treated as the positive class. We observe that much like in the case of AUROC, Scale setting peforms the best.\n\n| Scenario | Uncertainty | AUPRC-Error ↑ | AUPRC-Success ↑ | FPR@95%TPR ↓ | | | |\n|--------------|---------------|-----------------|-------------------|----------------|--------|--------|--------|\n| Metric | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | |\n| DRN D-22 | | | | | | | |\n| Base | Var. Ratio | 0.4383 | 0.4483 | 0.9950 | 0.9933 | 1.0000 | 0.4042 |\n| Prob. Margin | 0.4208 | 0.4308 | 0.9951 | 0.9934 | 1.0000 | 0.3716 | |\n| Entropy | 0.4249 | 0.4316 | 0.9953 | 0.9936 | 0.2515 | 0.3104 | |\n| Noise | Var. Ratio | 0.4375 | 0.4483 | 0.9951 | 0.9933 | 1.0000 | 0.4037 |\n| Prob. Margin | 0.4198 | 0.4307 | 0.9951 | 0.9934 | 1.0000 | 0.3702 | |\n| Entropy | 0.4246 | 0.4317 | 0.9953 | 0.9936 | 0.2515 | 0.3089 | |\n| Avg. Var. | 0.3364 | 0.3320 | 0.9926 | 0.9912 | 1.0000 | 1.0000 | |\n| Max. Var. | 0.3347 | 0.3300 | 0.9925 | 0.9912 | 1.0000 | 1.0000 | |\n| BALD | 0.3231 | 0.3162 | 0.9892 | 0.9883 | 1.0000 | 0.9982 | |\n| Scale | Var. Ratio | 0.4901 | 0.4850 | 0.9964 | 0.9954 | 0.2134 | 0.2177 |\n| Prob. Margin | 0.4665 | 0.4624 | 0.9964 | 0.9954 | 0.2040 | 0.2070 | |\n| Entropy | 0.4469 | 0.4402 | 0.9963 | 0.9952 | 0.1988 | 0.1938 | |\n| Avg. Var. | 0.3167 | 0.3048 | 0.9946 | 0.9933 | 1.0000 | 0.6696 | |\n| Max. Var. | 0.3153 | 0.2983 | 0.9946 | 0.9932 | 1.0000 | 0.6395 | |\n| BALD | 0.3052 | 0.2940 | 0.9942 | 0.9931 | 0.2742 | 0.3224 | |\n\nTable 12: Micro and macro-averaged (µ and M, respectively) Area Under Precision Recall Curve (AUPRC) and False Positive Rate at 95% True Positive Rate (FPR@95%TPR) on the Cityscapes dataset.\n\n| OneFormer ConvNeXt-L | | | | | | | |\n|------------------------|------------|--------|--------|--------|--------|--------|--------|\n| Base | Var. Ratio | 0.1310 | 0.3090 | 0.9853 | 0.9633 | 0.8001 | 0.6999 |\n| Prob. Margin | 0.2300 | 0.3562 | 0.9885 | 0.9722 | 0.7587 | 0.5619 | |\n| Entropy | 0.1193 | 0.2832 | 0.9847 | 0.9548 | 0.8034 | 0.7569 | |\n| Noise | Var.Ratio | 0.1356 | 0.3134 | 0.9849 | 0.9628 | 0.7973 | 0.6880 |\n| Prob. Margin | 0.2319 | 0.3590 | 0.9877 | 0.9691 | 0.7601 | 0.5556 | |\n| Entropy | 0.1244 | 0.2886 | 0.9844 | 0.9540 | 0.7926 | 0.7674 | |\n| Avg. Var. | 0.1255 | 0.2178 | 0.9913 | 0.9874 | 1.0000 | 0.9981 | |\n| Max. Var. | 0.1141 | 0.2038 | 0.9909 | 0.9865 | 1.0000 | 0.9981 | |\n| BALD | 0.1426 | 0.2335 | 0.9877 | 0.9858 | 1.0000 | 0.9981 | |\n| Scale | Var. Ratio | 0.1704 | 0.3258 | 0.9871 | 0.9741 | 0.7766 | 0.5904 |\n| Prob. Margin | 0.2921 | 0.3808 | 0.9905 | 0.9794 | 0.7167 | 0.4413 | |\n| Entropy | 0.1424 | 0.2903 | 0.9860 | 0.9702 | 0.7961 | 0.6500 | |\n| Avg. Var. | 0.1567 | 0.2303 | 0.9931 | 0.9886 | 1.0000 | 0.9970 | |\n| Max. Var. | 0.1391 | 0.2101 | 0.9924 | 0.9883 | 1.0000 | 0.9971 | |\n| BALD | 0.1916 | 0.2515 | 0.9928 | 0.9897 | 1.0000 | 0.9957 | |\n| Drop | Var. Ratio | 0.1454 | 0.3154 | 0.9855 | 0.9634 | 0.7865 | 0.6321 |\n| Prob. Margin | 0.2453 | 0.3614 | 0.9886 | 0.9703 | 0.7427 | 0.4868 | |\n| Entropy | 0.1332 | 0.2888 | 0.9853 | 0.9537 | 0.7991 | 0.6994 | |\n| Avg. Var. | 0.1480 | 0.2360 | 0.9921 | 0.9884 | 1.0000 | 0.9987 | |\n| Max. Var. | 0.1353 | 0.2164 | 0.9916 | 0.9878 | 1.0000 | 0.9987 | |\n| BALD | 0.1673 | 0.2557 | 0.9910 | 0.9890 | 1.0000 | 0.9987 | |\n| SegFormer B5 | | | | | | | |\n| Base | Var. Ratio | 0.3873 | 0.4049 | 0.9969 | 0.9963 | 1.0000 | 0.7554 |\n| Prob. Margin | 0.3771 | 0.3935 | 0.9969 | 0.9963 | 1.0000 | 0.6954 | |\n| Entropy | 0.3840 | 0.4038 | 0.9966 | 0.9961 | 1.0000 | 0.5757 | |\n| Noise | Var. Ratio | 0.3838 | 0.4021 | 0.9969 | 0.9964 | 1.0000 | 0.7575 |\n| Prob. Margin | 0.3724 | 0.3901 | 0.9969 | 0.9963 | 1.0000 | 0.6901 | |\n| Entropy | 0.3826 | 0.4023 | 0.9967 | 0.9961 | 1.0000 | 0.5704 | |\n| Avg. Var. | 0.3125 | 0.3210 | 0.9966 | 0.9962 | 1.0000 | 1.0000 | |\n| Max. Var. | 0.3123 | 0.3204 | 0.9966 | 0.9962 | 1.0000 | 1.0000 | |\n| BALD | 0.3073 | 0.3162 | 0.9915 | 0.9912 | 1.0000 | 1.0000 | |\n| Scale | Var. Ratio | 0.4335 | 0.4422 | 0.9976 | 0.9971 | 1.0000 | 0.3976 |\n| Prob. Margin | 0.4185 | 0.4256 | 0.9976 | 0.9971 | 1.0000 | 0.3642 | |\n| Entropy | 0.4099 | 0.4233 | 0.9975 | 0.9965 | 1.0000 | 0.2990 | |\n| Avg. Var. | 0.3371 | 0.3334 | 0.9975 | 0.9971 | 1.0000 | 0.9981 | |\n| Max. Var. | 0.3352 | 0.3303 | 0.9975 | 0.9971 | 1.0000 | 0.9981 | |\n| BALD | 0.3186 | 0.3155 | 0.9960 | 0.9955 | 1.0000 | 0.7832 | |\n| Drop | Var. Ratio | 0.3876 | 0.4051 | 0.9969 | 0.9963 | 1.0000 | 0.7481 |\n| Prob. Margin | 0.3770 | 0.3934 | 0.9968 | 0.9962 | 1.0000 | 0.6829 | |\n| Entropy | 0.3843 | 0.4036 | 0.9966 | 0.9961 | 1.0000 | 0.5651 | |\n| Avg. Var. | 0.3334 | 0.3425 | 0.9969 | 0.9965 | 1.0000 | 1.0000 | |\n| Max. Var. | 0.3320 | 0.3410 | 0.9969 | 0.9965 | 1.0000 | 1.0000 | |\n| BALD | 0.3211 | 0.3327 | 0.9917 | 0.9914 | 1.0000 | 0.9925 | |\n\nFinally, Table 13 and Table 14 show Precision and Recall for thresholding based on a maximum fraction of pixels allowed on Cityscapes and ADE20K datasets, respectively. We show results across three threshold values.\n\nAs expected, we see that as we increase the maximum number of pixels allowed, we get higher Recall at the cost of Precision. This observation holds true across all of the experiments. Again, we note that Scale often has the best performance with simple uncertainty metrics across both datasets.\n\n| a maximum fraction of pixels allowed on the Cityscapes dataset. | | | | | | | | | | | | | |\n|-------------------------------------------------------------------|-------------|---------------|----------------|----------------|-------------|----------|-------|-------|-------|-------|-------|-------|-------|\n| Scene | Unc. | Max 5% pixels | Max 10% pixels | Max 15% pixels | | | | | | | | | |\n| Metric | Precision ↑ | Recall ↑ | Precision ↑ | Recall ↑ | Precision ↑ | Recall ↑ | | | | | | | |\n| µ | M | µ | M | µ | M | µ | M | µ | M | µ | M | | |\n| DRN D-22 | | | | | | | | | | | | | |\n| Base | VR | 45.19 | 45.40 | 41.52 | 50.74 | 35.04 | 35.18 | 64.13 | 73.92 | 28.48 | 28.45 | 76.44 | 84.64 |\n| PM | 44.66 | 44.83 | 41.35 | 50.58 | 34.77 | 34.92 | 64.13 | 73.95 | 28.14 | 28.19 | 76.51 | 84.74 | |\n| E | 44.04 | 44.23 | 39.76 | 48.74 | 34.80 | 35.03 | 63.64 | 73.61 | 28.11 | 28.23 | 76.42 | 84.73 | |\n| Noise | VR | 45.15 | 45.35 | 41.51 | 50.75 | 35.01 | 35.16 | 64.10 | 73.94 | 28.46 | 28.44 | 76.39 | 84.64 |\n| PM | 44.60 | 44.78 | 41.27 | 50.56 | 34.76 | 34.91 | 64.07 | 73.93 | 28.15 | 28.18 | 76.50 | 84.77 | |\n| E | 44.02 | 44.20 | 39.75 | 48.77 | 34.79 | 35.02 | 63.64 | 73.64 | 28.12 | 28.23 | 76.45 | 84.78 | |\n| AV | 40.06 | 39.93 | 32.11 | 38.45 | 38.03 | 37.47 | 50.03 | 55.62 | 37.92 | 36.97 | 55.23 | 58.76 | |\n| MV | 39.79 | 39.66 | 31.88 | 38.17 | 37.81 | 37.22 | 49.85 | 55.45 | 37.65 | 36.70 | 55.34 | 58.86 | |\n| B | 36.39 | 36.38 | 30.65 | 36.02 | 32.57 | 32.42 | 50.90 | 57.46 | 30.13 | 29.82 | 63.10 | 68.56 | |\n| Scale | VR | 47.90 | 48.21 | 43.77 | 52.63 | 36.67 | 36.94 | 67.41 | 76.23 | 29.11 | 29.28 | 79.80 | 86.84 |\n| PM | 47.17 | 47.44 | 43.46 | 52.41 | 36.33 | 36.59 | 67.23 | 76.08 | 28.92 | 29.10 | 79.84 | 86.82 | |\n| E | 43.68 | 43.86 | 39.53 | 47.72 | 35.26 | 35.54 | 63.98 | 73.68 | 28.67 | 28.92 | 78.57 | 86.18 | |\n| AV | 32.12 | 32.29 | 29.36 | 34.35 | 30.02 | 30.11 | 54.35 | 62.02 | 27.29 | 27.15 | 71.54 | 78.61 | |\n| MV | 31.42 | 31.55 | 28.76 | 33.67 | 29.60 | 29.67 | 53.63 | 61.29 | 27.02 | 26.87 | 71.09 | 78.23 | |\n| B | 31.13 | 31.32 | 28.39 | 32.45 | 28.38 | 28.55 | 51.55 | 57.83 | 25.34 | 25.41 | 68.24 | 74.57 | |\n| DRN D-105 | | | | | | | | | | | | | |\n| Base | VR | 39.63 | 39.78 | 49.13 | 59.67 | 29.24 | 29.21 | 70.10 | 80.33 | 24.61 | 24.23 | 79.31 | 87.50 |\n| PM | 39.32 | 39.46 | 48.98 | 59.54 | 28.83 | 28.86 | 70.23 | 80.53 | 23.91 | 23.60 | 79.69 | 87.97 | |\n| E | 39.72 | 39.95 | 48.70 | 59.33 | 28.85 | 28.94 | 70.48 | 80.80 | 23.15 | 22.99 | 80.26 | 88.63 | |\n| Noise | VR | 39.57 | 39.70 | 49.01 | 59.57 | 29.24 | 29.19 | 70.09 | 80.33 | 24.62 | 24.24 | 79.34 | 87.55 |\n| PM | 39.23 | 39.35 | 48.89 | 59.48 | 28.82 | 28.83 | 70.23 | 80.55 | 23.94 | 23.62 | 79.72 | 88.01 | |\n| E | 39.66 | 39.90 | 48.58 | 59.24 | 28.83 | 28.92 | 70.51 | 80.82 | 23.17 | 23.00 | 80.32 | 88.67 | |\n| AV | 40.51 | 40.32 | 40.62 | 48.14 | 39.37 | 38.74 | 50.96 | 56.09 | 39.41 | 38.64 | 52.14 | 56.53 | |\n| MV | 40.36 | 40.17 | 40.64 | 48.16 | 39.28 | 38.64 | 51.00 | 56.08 | 39.34 | 38.55 | 52.17 | 56.51 | |\n| B | 37.20 | 37.19 | 40.30 | 47.75 | 32.99 | 32.68 | 59.12 | 66.07 | 32.29 | 31.73 | 64.62 | 69.48 | |\n| Scale | VR | 42.41 | 42.67 | 52.51 | 62.94 | 30.32 | 30.47 | 74.36 | 83.62 | 23.97 | 23.82 | 84.09 | 91.08 |\n| PM | 42.07 | 42.31 | 52.42 | 62.88 | 30.07 | 30.23 | 74.33 | 83.65 | 23.48 | 23.45 | 84.15 | 91.18 | |\n| E | 41.11 | 41.40 | 49.76 | 60.38 | 30.00 | 30.29 | 73.84 | 83.50 | 23.16 | 23.24 | 84.14 | 91.32 | |\n| AV. | 35.74 | 35.87 | 43.55 | 51.63 | 30.48 | 30.24 | 68.84 | 77.10 | 28.21 | 27.52 | 79.36 | 84.93 | |\n| MV. | 35.35 | 35.49 | 43.03 | 51.07 | 30.32 | 30.07 | 68.69 | 76.96 | 28.03 | 27.36 | 79.33 | 84.95 | |\n| B | 33.45 | 33.64 | 40.73 | 47.25 | 27.51 | 27.57 | 65.47 | 73.22 | 23.51 | 23.33 | 79.31 | 85.81 | |\n| OneFormer ConvNeXt-L | | | | | | | | | | | | | |\n| Base | VR | 32.16 | 32.31 | 47.73 | 57.40 | 24.77 | 24.84 | 67.92 | 77.28 | 20.80 | 21.03 | 74.47 | 82.96 |\n| PM | 33.41 | 33.78 | 50.18 | 60.30 | 24.95 | 24.99 | 71.11 | 80.37 | 20.58 | 20.68 | 79.04 | 86.67 | |\n| E | 30.89 | 31.47 | 44.48 | 54.09 | 23.82 | 24.11 | 64.18 | 74.15 | 20.07 | 20.51 | 70.79 | 79.86 | |\n| Noise | VR | 32.43 | 32.39 | 48.17 | 57.91 | 24.70 | 24.74 | 67.97 | 77.59 | 20.72 | 20.89 | 74.96 | 83.38 |\n| PM | 33.53 | 33.86 | 50.60 | 60.76 | 24.90 | 25.00 | 70.69 | 80.41 | 20.82 | 20.85 | 79.28 | 86.78 | |\n| E | 31.29 | 31.41 | 45.12 | 54.88 | 24.19 | 24.24 | 65.25 | 74.79 | 20.46 | 20.73 | 71.70 | 80.37 | |\n| AV | 28.32 | 30.65 | 23.29 | 27.49 | 25.39 | 29.84 | 27.17 | 30.78 | 24.23 | 29.78 | 28.87 | 32.20 | |\n| MV | 26.90 | 29.38 | 21.05 | 24.74 | 23.40 | 28.55 | 24.70 | 27.96 | 22.19 | 28.49 | 26.33 | 29.30 | |\n| B | 29.88 | 31.11 | 27.14 | 31.75 | 26.94 | 30.33 | 31.97 | 35.82 | 25.64 | 30.16 | 34.28 | 37.59 | |\n| Scale | VR | 33.06 | 33.12 | 49.59 | 59.05 | 24.31 | 24.37 | 70.19 | 79.48 | 20.03 | 20.18 | 77.58 | 85.63 |\n| PM | 34.95 | 35.08 | 53.07 | 62.79 | 24.83 | 24.94 | 73.38 | 82.40 | 20.01 | 20.04 | 81.97 | 88.92 | |\n| E | 31.49 | 31.81 | 46.19 | 55.45 | 23.63 | 23.77 | 66.65 | 76.48 | 19.21 | 19.49 | 73.98 | 82.95 | |\n| AV | 26.24 | 26.37 | 34.74 | 41.71 | 23.57 | 24.22 | 50.06 | 57.43 | 22.03 | 23.46 | 54.38 | 61.20 | |\n\nTable 13: Micro and macro-averaged (µ and M, respectively) Precision and Recall for thresholding based on a maximum fraction of pixels allowed on the Cityscapes dataset.\n\n| MV | 24.47 | 24.64 | 31.43 | 38.02 | 22.32 | 23.08 | 46.09 | 53.41 | 20.73 | 22.42 | 50.57 | 57.34 | |\n|------------------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|-------|\n| B | 27.69 | 27.71 | 38.17 | 45.61 | 24.49 | 24.68 | 55.98 | 63.55 | 22.87 | 23.67 | 60.90 | 67.73 | |\n| Drop | VR | 32.26 | 32.34 | 48.26 | 57.81 | 24.32 | 24.33 | 68.79 | 78.27 | 19.99 | 20.11 | 76.25 | 84.80 |\n| PM | 33.41 | 33.65 | 50.55 | 60.60 | 24.66 | 24.76 | 71.27 | 80.90 | 20.26 | 20.21 | 80.70 | 87.94 | |\n| E | 31.03 | 31.47 | 45.20 | 54.88 | 23.63 | 23.72 | 65.94 | 75.90 | 19.41 | 19.59 | 72.95 | 82.24 | |\n| AV | 27.22 | 28.30 | 29.97 | 37.42 | 25.57 | 27.51 | 39.85 | 47.02 | 24.53 | 27.53 | 41.76 | 48.59 | |\n| MV | 25.91 | 26.88 | 27.07 | 33.79 | 24.04 | 26.38 | 35.73 | 42.43 | 23.02 | 26.34 | 37.48 | 43.91 | |\n| B | 28.84 | 29.05 | 34.44 | 42.68 | 26.69 | 27.84 | 46.91 | 54.52 | 25.52 | 27.71 | 49.72 | 56.72 | |\n| OneFormer Swin-L | | | | | | | | | | | | | |\n| Base | VR | 32.94 | 33.23 | 49.05 | 57.91 | 24.85 | 24.96 | 69.57 | 77.97 | 20.84 | 20.77 | 78.74 | 84.70 |\n| PM | 34.55 | 34.74 | 51.98 | 61.02 | 25.30 | 25.40 | 72.51 | 80.80 | 20.86 | 20.73 | 81.77 | 87.42 | |\n| E | 31.81 | 32.27 | 45.96 | 54.81 | 24.24 | 24.56 | 66.79 | 75.25 | 20.25 | 20.52 | 75.56 | 81.89 | |\n| Noise | VR | 33.11 | 33.47 | 49.38 | 58.24 | 24.97 | 25.05 | 70.12 | 78.27 | 20.99 | 20.94 | 79.10 | 84.90 |\n| PM | 34.73 | 34.88 | 52.24 | 61.27 | 25.36 | 25.39 | 72.93 | 80.98 | 20.70 | 20.58 | 81.90 | 87.55 | |\n| E | 32.04 | 32.58 | 46.64 | 55.36 | 24.33 | 24.57 | 67.07 | 75.50 | 20.33 | 20.55 | 76.08 | 82.28 | |\n| AV | 29.87 | 31.00 | 26.77 | 30.33 | 27.68 | 30.23 | 31.40 | 34.14 | 26.58 | 30.07 | 32.79 | 35.08 | |\n| MV | 28.28 | 29.58 | 24.38 | 27.57 | 26.22 | 28.98 | 28.79 | 31.26 | 25.39 | 28.89 | 30.04 | 32.11 | |\n| B | 31.24 | 31.81 | 30.48 | 34.60 | 28.89 | 30.78 | 36.27 | 39.23 | 27.91 | 30.66 | 38.01 | 40.31 | |\n| Scale | VR | 33.94 | 34.32 | 50.84 | 59.78 | 25.02 | 25.08 | 73.38 | 80.83 | 20.08 | 19.99 | 82.28 | 87.82 |\n| PM | 35.94 | 36.36 | 54.53 | 63.51 | 25.64 | 25.65 | 76.29 | 83.36 | 19.98 | 19.84 | 84.67 | 90.01 | |\n| E | 32.44 | 32.95 | 47.77 | 56.50 | 24.29 | 24.38 | 70.42 | 78.41 | 19.44 | 19.53 | 79.54 | 85.80 | |\n| AV | 27.75 | 27.90 | 37.81 | 44.46 | 24.89 | 25.01 | 57.01 | 62.72 | 23.77 | 24.16 | 63.79 | 67.53 | |\n| MV | 25.97 | 26.06 | 34.85 | 40.95 | 23.92 | 24.09 | 53.77 | 59.28 | 22.74 | 23.24 | 60.39 | 64.22 | |\n| B | 29.08 | 29.14 | 41.04 | 47.95 | 25.57 | 25.34 | 61.88 | 67.78 | 24.06 | 24.11 | 69.03 | 72.82 | |\n| Drop | VR | 32.99 | 33.47 | 49.35 | 58.22 | 24.84 | 24.91 | 70.67 | 78.76 | 20.58 | 20.55 | 79.87 | 85.54 |\n| PM | 34.51 | 34.86 | 52.25 | 61.35 | 25.26 | 25.31 | 73.34 | 81.33 | 20.42 | 20.32 | 82.53 | 88.16 | |\n| E | 31.84 | 32.43 | 46.51 | 55.32 | 23.95 | 24.13 | 67.69 | 76.26 | 19.89 | 19.97 | 77.17 | 83.17 | |\n| AV | 29.79 | 30.33 | 33.67 | 40.77 | 27.84 | 29.11 | 43.91 | 50.12 | 26.93 | 28.82 | 46.11 | 51.68 | |\n| MV | 28.13 | 28.72 | 30.60 | 37.09 | 26.50 | 27.96 | 40.00 | 45.77 | 25.88 | 27.83 | 41.88 | 47.13 | |\n| B | 31.21 | 31.36 | 38.42 | 46.14 | 28.73 | 29.50 | 50.96 | 57.41 | 27.71 | 29.06 | 53.38 | 59.08 | |\n| SegFormer B5 | | | | | | | | | | | | | |\n| Base | VR | 36.62 | 36.67 | 54.75 | 63.87 | 27.19 | 26.95 | 74.08 | 81.94 | 24.94 | 24.49 | 80.02 | 86.05 |\n| PM | 36.39 | 36.44 | 54.79 | 63.96 | 26.68 | 26.48 | 74.42 | 82.37 | 23.88 | 23.45 | 81.16 | 87.19 | |\n| E | 36.61 | 36.71 | 54.78 | 63.98 | 26.09 | 25.99 | 74.94 | 83.01 | 22.42 | 22.02 | 82.52 | 88.66 | |\n| Noise | VR | 36.50 | 36.54 | 54.61 | 63.79 | 27.16 | 26.91 | 74.01 | 81.90 | 24.87 | 24.42 | 80.04 | 86.09 |\n| PM | 36.29 | 36.34 | 54.60 | 63.82 | 26.65 | 26.45 | 74.41 | 82.36 | 23.82 | 23.39 | 81.15 | 87.21 | |\n| E | 36.49 | 36.60 | 54.55 | 63.83 | 26.04 | 25.94 | 74.84 | 82.98 | 22.36 | 21.96 | 82.56 | 88.71 | |\n| AV | 40.42 | 40.39 | 38.53 | 43.44 | 40.25 | 40.13 | 41.31 | 45.00 | 40.19 | 40.11 | 41.61 | 45.11 | |\n| MV | 40.42 | 40.36 | 38.38 | 43.21 | 40.19 | 40.07 | 41.25 | 44.84 | 40.18 | 40.07 | 41.46 | 44.90 | |\n| B | 35.96 | 36.06 | 42.17 | 48.35 | 34.23 | 34.20 | 52.27 | 56.64 | 34.07 | 34.05 | 53.50 | 57.22 | |\n| Scale | VR | 38.52 | 38.66 | 57.84 | 66.40 | 27.03 | 26.98 | 78.45 | 85.47 | 22.33 | 21.92 | 86.20 | 91.23 |\n| PM | 38.36 | 38.49 | 57.89 | 66.44 | 26.68 | 26.68 | 78.68 | 85.68 | 21.55 | 21.23 | 86.60 | 91.67 | |\n| E | 37.88 | 38.15 | 56.12 | 65.23 | 26.49 | 26.57 | 78.44 | 85.68 | 20.75 | 20.54 | 86.91 | 92.12 | |\n| AV | 34.28 | 34.28 | 49.87 | 57.04 | 29.54 | 29.07 | 72.25 | 78.02 | 28.45 | 27.81 | 77.68 | 81.50 | |\n| MV | 34.09 | 34.09 | 49.70 | 56.83 | 29.38 | 28.91 | 72.37 | 78.12 | 28.26 | 27.62 | 77.88 | 81.69 | |\n| B | 31.46 | 31.51 | 46.09 | 52.15 | 25.41 | 25.31 | 71.04 | 77.64 | 22.48 | 22.11 | 81.72 | 86.61 | |\n| Drop | VR | 36.62 | 36.67 | 54.79 | 63.92 | 27.17 | 26.93 | 74.15 | 82.01 | 24.81 | 24.35 | 80.25 | 86.25 |\n| PM | 36.42 | 36.47 | 54.80 | 63.96 | 26.64 | 26.45 | 74.51 | 82.45 | 23.76 | 23.33 | 81.35 | 87.36 | |\n| E | 36.61 | 36.73 | 54.74 | 63.96 | 26.05 | 25.96 | 74.96 | 83.06 | 22.31 | 21.92 | 82.68 | 88.80 | |\n| AV | 38.44 | 38.37 | 47.21 | 54.70 | 37.02 | 36.69 | 56.13 | 61.53 | 37.02 | 36.66 | 56.64 | 61.71 | |\n| MV | 38.39 | 38.33 | 46.98 | 54.43 | 37.06 | 36.72 | 55.80 | 61.20 | 37.06 | 36.69 | 56.31 | 61.38 | |\n| B | 34.73 | 34.82 | 45.35 | 52.84 | 29.60 | 29.53 | 65.38 | 72.44 | 28.96 | 28.77 | 69.43 | 75.02 | |\n\nTable 14: Micro and macro-averaged (µ and M, respectively) Precision and Recall for thresholding based on a maximum fraction of pixels allowed on the ADE20K dataset.\n\n| a maximum fraction of pixels allowed on the ADE20K dataset. | | | | | | | | | | | | | |\n|---------------------------------------------------------------|-------------|---------------|----------------|----------------|-------------|----------|-------|-------|-------|-------|-------|-------|-------|\n| Scene | Unc. | Max 5% pixels | Max 10% pixels | Max 15% pixels | | | | | | | | | |\n| Metric | Precision ↑ | Recall ↑ | Precision ↑ | Recall ↑ | Precision ↑ | Recall ↑ | | | | | | | |\n| µ | M | µ | M | µ | M | µ | M | µ | M | µ | M | | |\n| OneFormer ConvNeXt-L | | | | | | | | | | | | | |\n| Base | VR | 47.06 | 48.64 | 13.71 | 26.63 | 41.24 | 42.80 | 22.95 | 39.37 | 37.98 | 39.10 | 30.06 | 47.02 |\n| PM | 50.10 | 52.72 | 13.67 | 27.27 | 44.99 | 47.46 | 24.82 | 42.06 | 41.45 | 43.45 | 34.19 | 51.95 | |\n| E | 43.42 | 45.63 | 11.38 | 23.14 | 38.09 | 40.24 | 19.18 | 34.30 | 34.36 | 36.72 | 24.40 | 40.23 | |\n| Noise | VR | 46.82 | 48.40 | 13.60 | 26.50 | 40.80 | 42.62 | 22.69 | 39.25 | 37.48 | 38.92 | 29.51 | 46.74 |\n| PM | 49.72 | 51.84 | 13.56 | 26.82 | 44.48 | 46.83 | 24.65 | 41.97 | 41.35 | 43.13 | 34.27 | 51.96 | |\n| E | 43.29 | 45.73 | 11.36 | 23.11 | 37.44 | 39.87 | 18.91 | 34.23 | 34.22 | 36.50 | 24.11 | 40.07 | |\n| AV | 35.66 | 37.79 | 08.32 | 14.39 | 34.12 | 36.78 | 14.30 | 21.51 | 32.70 | 36.13 | 18.94 | 26.12 | |\n| MV | 34.75 | 37.29 | 07.96 | 13.66 | 33.20 | 36.16 | 13.66 | 20.13 | 31.75 | 35.70 | 17.86 | 24.32 | |\n| B | 37.41 | 37.75 | 09.20 | 14.77 | 34.55 | 35.70 | 16.07 | 23.16 | 33.63 | 34.57 | 22.48 | 29.42 | |\n| Scale | VR | 48.64 | 50.01 | 14.37 | 27.55 | 41.80 | 43.48 | 23.98 | 40.57 | 38.22 | 39.40 | 31.20 | 48.36 |\n| PM | 51.85 | 54.22 | 14.63 | 28.57 | 45.91 | 48.29 | 26.21 | 43.75 | 41.99 | 44.14 | 35.70 | 53.54 | |\n| E | 44.02 | 46.12 | 11.83 | 23.81 | 37.77 | 39.70 | 19.77 | 35.24 | 34.16 | 35.94 | 25.65 | 41.55 | |\n| AV | 33.02 | 33.00 | 08.90 | 15.87 | 31.13 | 32.04 | 16.00 | 25.62 | 30.02 | 31.46 | 22.35 | 32.39 | |\n| MV | 31.15 | 31.46 | 08.22 | 14.56 | 29.56 | 30.60 | 14.87 | 23.66 | 28.26 | 30.01 | 20.47 | 29.77 | |\n| B | 37.05 | 37.00 | 10.57 | 18.79 | 34.67 | 34.90 | 19.06 | 30.30 | 32.94 | 33.54 | 26.15 | 38.40 | |\n| Drop | VR | 47.23 | 48.70 | 13.88 | 26.96 | 41.36 | 42.79 | 23.23 | 39.93 | 37.82 | 38.97 | 30.23 | 47.41 |\n| PM | 50.62 | 52.55 | 14.06 | 27.53 | 45.01 | 47.40 | 25.44 | 42.56 | 41.51 | 43.43 | 34.83 | 52.39 | |\n| E | 43.48 | 45.72 | 11.61 | 23.58 | 37.55 | 39.90 | 19.32 | 34.64 | 34.41 | 36.36 | 25.01 | 40.94 | |\n| AV | 32.91 | 34.23 | 08.39 | 15.68 | 31.96 | 33.95 | 15.30 | 24.70 | 30.97 | 33.53 | 21.32 | 31.35 | |\n| MV | 31.38 | 33.16 | 07.89 | 14.49 | 30.31 | 32.86 | 14.19 | 22.54 | 29.59 | 32.70 | 19.63 | 28.57 | |\n| B | 35.57 | 36.53 | 09.66 | 18.33 | 34.25 | 35.34 | 18.07 | 29.48 | 33.37 | 34.64 | 25.62 | 38.02 | |\n| OneFormer Swin-L | | | | | | | | | | | | | |\n| Base | VR | 46.97 | 48.39 | 14.18 | 27.16 | 41.28 | 42.31 | 23.92 | 40.21 | 38.27 | 38.78 | 31.45 | 47.94 |\n| PM | 49.35 | 51.85 | 14.19 | 27.81 | 44.07 | 46.13 | 25.34 | 42.57 | 41.18 | 42.29 | 35.43 | 52.68 | |\n| E | 43.36 | 45.43 | 11.87 | 23.72 | 37.71 | 39.28 | 20.01 | 34.95 | 34.84 | 35.71 | 26.00 | 41.07 | |\n| Noise | VR | 46.76 | 48.02 | 14.18 | 26.98 | 41.26 | 42.21 | 24.00 | 40.25 | 38.13 | 38.55 | 31.40 | 47.83 |\n| PM | 49.39 | 51.30 | 14.30 | 27.85 | 44.03 | 45.67 | 25.47 | 42.60 | 40.67 | 41.89 | 35.15 | 52.55 | |\n| E | 42.99 | 45.13 | 11.89 | 23.66 | 37.56 | 39.34 | 19.86 | 34.81 | 34.67 | 35.65 | 26.07 | 41.10 | |\n| AV | 33.66 | 35.65 | 08.40 | 14.71 | 32.44 | 35.30 | 14.46 | 22.00 | 30.69 | 34.96 | 19.15 | 27.13 | |\n| MV | 32.74 | 34.93 | 07.97 | 13.72 | 31.41 | 34.61 | 13.67 | 20.48 | 29.65 | 34.08 | 18.00 | 25.09 | |\n| B | 36.48 | 37.19 | 09.60 | 16.23 | 33.93 | 35.36 | 17.18 | 25.26 | 32.10 | 34.11 | 23.16 | 31.76 | |\n| Scale | VR | 48.38 | 49.56 | 14.85 | 28.10 | 41.74 | 42.90 | 25.00 | 41.66 | 38.39 | 39.05 | 32.99 | 49.81 |\n| PM | 51.02 | 52.71 | 15.09 | 28.78 | 45.13 | 46.90 | 27.08 | 44.40 | 41.36 | 42.87 | 36.75 | 54.48 | |\n| E | 43.48 | 45.43 | 12.25 | 24.32 | 37.57 | 39.17 | 20.63 | 35.87 | 34.33 | 35.33 | 26.97 | 42.45 | |\n| AV | 34.14 | 34.12 | 09.65 | 17.18 | 32.33 | 32.78 | 17.57 | 27.72 | 30.42 | 31.87 | 23.84 | 34.88 | |\n| MV | 32.14 | 32.40 | 08.90 | 15.81 | 30.23 | 31.10 | 16.14 | 25.39 | 28.56 | 30.32 | 21.88 | 31.85 | |\n| B | 37.85 | 37.71 | 11.38 | 20.03 | 35.90 | 35.87 | 20.74 | 32.48 | 33.65 | 34.27 | 28.15 | 41.15 | |\n| Drop | VR | 47.08 | 48.53 | 14.39 | 27.37 | 41.44 | 42.34 | 24.48 | 40.77 | 38.36 | 38.68 | 31.94 | 48.42 |\n| PM | 50.04 | 52.30 | 14.62 | 28.21 | 44.52 | 46.35 | 26.12 | 43.17 | 41.48 | 42.47 | 36.28 | 53.25 | |\n| E | 43.07 | 45.14 | 12.06 | 23.97 | 37.85 | 39.12 | 20.53 | 35.48 | 34.68 | 35.35 | 26.65 | 41.70 | |\n| AV | 32.02 | 33.24 | 08.56 | 15.45 | 30.49 | 32.83 | 15.41 | 24.45 | 29.64 | 32.45 | 21.13 | 30.64 | |\n| MV | 30.58 | 31.90 | 07.99 | 14.26 | 29.13 | 31.69 | 14.23 | 22.27 | 28.17 | 31.49 | 19.49 | 27.95 | |\n| B | 34.35 | 35.55 | 09.68 | 17.86 | 33.17 | 34.43 | 18.12 | 29.11 | 32.11 | 33.38 | 25.36 | 37.35 | |\n| SegFormer B5 | | | | | | | | | | | | | |\n| Base | VR | 55.19 | 55.90 | 16.38 | 29.25 | 48.88 | 49.68 | 29.08 | 46.12 | 44.74 | 45.38 | 39.35 | 57.38 |\n\n| PM | 53.48 | 54.15 | 15.97 | 28.81 | 48.18 | 48.93 | 28.83 | 45.86 | 44.27 | 44.92 | 39.22 | 57.25 | |\n|-------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|-------|\n| E | 54.69 | 55.50 | 16.07 | 28.51 | 48.63 | 49.74 | 28.67 | 45.76 | 44.33 | 45.40 | 39.16 | 57.46 | |\n| Noise | VR | 55.21 | 55.90 | 16.39 | 29.24 | 48.90 | 49.68 | 29.10 | 46.14 | 44.73 | 45.37 | 39.34 | 57.38 |\n| PM | 53.50 | 54.13 | 15.97 | 28.81 | 48.19 | 48.93 | 28.82 | 45.87 | 44.26 | 44.89 | 39.19 | 57.25 | |\n| E | 54.61 | 55.45 | 16.02 | 28.48 | 48.60 | 49.71 | 28.66 | 45.74 | 44.29 | 45.35 | 39.13 | 57.45 | |\n| AV | 49.94 | 50.55 | 13.87 | 24.11 | 48.42 | 48.36 | 24.92 | 38.74 | 47.80 | 47.03 | 33.24 | 47.63 | |\n| MV | 49.83 | 50.43 | 13.79 | 23.94 | 48.28 | 48.24 | 24.78 | 38.57 | 47.61 | 46.92 | 33.19 | 47.52 | |\n| B | 48.63 | 49.49 | 13.84 | 23.18 | 45.82 | 46.51 | 25.20 | 38.80 | 44.00 | 44.23 | 34.62 | 49.73 | |\n| Scale | VR | 57.22 | 58.41 | 16.86 | 29.97 | 50.55 | 51.87 | 30.04 | 47.26 | 45.83 | 47.10 | 40.67 | 58.91 |\n| PM | 54.98 | 56.10 | 16.23 | 29.29 | 49.43 | 50.66 | 29.52 | 46.70 | 45.15 | 46.38 | 40.34 | 58.50 | |\n| E | 55.19 | 56.35 | 16.16 | 28.46 | 49.43 | 50.91 | 29.07 | 46.08 | 45.05 | 46.61 | 39.71 | 58.12 | |\n| AV | 44.61 | 46.20 | 13.17 | 22.04 | 43.34 | 44.56 | 25.23 | 38.40 | 42.16 | 42.98 | 35.75 | 50.37 | |\n| MV | 44.17 | 45.76 | 13.04 | 21.84 | 42.86 | 44.17 | 24.99 | 38.12 | 41.82 | 42.68 | 35.53 | 50.13 | |\n| B | 45.28 | 47.00 | 13.38 | 21.49 | 42.71 | 44.29 | 25.22 | 37.42 | 40.59 | 41.95 | 35.64 | 49.61 | |\n| Drop | VR | 55.26 | 55.97 | 16.40 | 29.26 | 48.92 | 49.72 | 29.12 | 46.18 | 44.71 | 45.40 | 39.38 | 57.45 |\n| PM | 53.53 | 54.18 | 15.98 | 28.81 | 48.20 | 48.95 | 28.85 | 45.89 | 44.27 | 44.93 | 39.23 | 57.30 | |\n| E | 54.62 | 55.46 | 16.06 | 28.46 | 48.65 | 49.78 | 28.68 | 45.75 | 44.32 | 45.42 | 39.14 | 57.47 | |\n| AV | 51.84 | 52.71 | 15.04 | 26.50 | 48.66 | 49.12 | 27.29 | 43.02 | 46.72 | 46.54 | 37.01 | 53.72 | |\n| MV | 51.47 | 52.41 | 14.92 | 26.34 | 48.43 | 48.90 | 27.09 | 42.82 | 46.49 | 46.39 | 36.88 | 53.62 | |\n| B | 51.86 | 52.96 | 15.10 | 25.75 | 47.11 | 48.29 | 27.14 | 42.33 | 43.76 | 44.77 | 37.09 | 53.92 | |\n\nConsidering all these results, we would suggest using either Base or Scale to gauge the trustworthiness of a network. Base is computationally much cheaper, but Scale generally provides better results. As for the uncertainty metric, it is best to use the simpler metrics like Variation Ratio, Probability Margin and Entropy since they seem to provide the best results in most cases. It also seems that for in-domain data, thresholding based on a maximum fraction of pixels is better than using the largest difference.\n\n## B Dark Zurich\n\nWe now provide all results on the Dark Zurich dataset. Since Dark Zurich was used to test how well the metrics work when there is a shift in the input domain, all networks perform poorly on the dataset. Table 15 shows the base performance of the models on the dataset.\n\nDRN performs very poorly, while OneFormer and SegFormer provide much more respectable results. Overall, the results are quite poor and show that, in general, these networks do not handle large changes in input domain well.\n\n| Model | mIoU ↑ | ECE ↓ | BS ↓ | | | |\n|----------------------|----------|---------|--------|--------|--------|--------|\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | |\n| DRN D-22 | 0.0710 | 0.0737 | 0.3067 | 0.5142 | 1.1303 | 1.1361 |\n| DRN D-105 | 0.0951 | 0.0990 | 0.2877 | 0.4173 | 0.9259 | 0.9315 |\n| OneFormer ConvNeXt-L | 0.3994 | 0.3752 | 0.5964 | 0.5755 | 0.8340 | 0.8345 |\n| OneFormer Swin-L | 0.3994 | 0.3949 | 0.5988 | 0.6678 | 0.8259 | 0.8263 |\n| SegFormer B5 | 0.3128 | 0.2686 | 0.2212 | 0.2715 | 0.6053 | 0.6063 |\n\nTable 15: Micro and macro-averaged (µ and M, respectively) Mean Intersection Over Union (mIoU), Expected Calibration Error (ECE) and Brier Score (BS) on the Dark Zurich dataset.\nTable 16 shows the AUROC and the performance of thresholding based on the largest difference, and Table 17 shows the results when we use a maximum fraction of pixels as the thresholding metric.\n\nEven with such poor performance, we observe that the largest difference thresholding is able to achieve AUROC of 0.7 or greater. Precision suffers a bit in this scenario, but Recall is excellent, and while a large fraction of pixels are selected, it is often correct since most pixels are actually misclassified. Scale with Probability Margin as the uncertainty metric often performs the best, similar to earlier experiments.\n\nOn the other hand, thresholding based on a maximum fraction of pixels is less useful in this case, with Recall being much lower than the largest difference method. This is because, as explained earlier, most pixels are actually misclassified and restricting it to a maximum of 15% actually hurts performance.\n\nTable 16: Micro and macro-averaged (µ and M, respectively) Area Under Receiver Operating Characteristic (AUROC) and Precision, Recall and the fraction of pixels selected for largest difference thresholding on the Dark Zurich dataset.\n\n| Scenario | Uncertainty | AUROC ↑ | Largest Difference Thresholding | | | | | | |\n|----------------------|---------------|-----------|-----------------------------------|--------|--------|--------|--------|-------|-------|\n| Metric | Precision ↑ | Recall ↑ | Pixel % | | | | | | |\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | | |\n| DRN D-22 | | | | | | | | | |\n| Base | Var. Ratio | 0.6060 | 0.6061 | 70.14 | 70.63 | 74.73 | 75.07 | 70.05 | 70.01 |\n| Prob. Margin | 0.6056 | 0.6067 | 69.67 | 70.18 | 77.53 | 77.85 | 73.16 | 73.11 | |\n| Entropy | 0.6054 | 0.6062 | 68.92 | 69.46 | 80.93 | 81.24 | 77.21 | 77.15 | |\n| Noise | Var. Ratio | 0.6027 | 0.6038 | 69.96 | 70.49 | 74.81 | 75.17 | 70.31 | 70.26 |\n| Prob. Margin | 0.6021 | 0.6041 | 69.50 | 70.05 | 77.57 | 77.91 | 73.39 | 73.33 | |\n| Entropy | 0.6024 | 0.6038 | 68.77 | 69.35 | 80.95 | 81.28 | 77.39 | 77.32 | |\n| Avg. Var. | 0.5892 | 0.5918 | 75.12 | 75.56 | 46.70 | 46.99 | 40.87 | 40.89 | |\n| Max. Var. | 0.5898 | 0.5922 | 75.04 | 75.47 | 47.31 | 47.60 | 41.46 | 41.47 | |\n| BALD | 0.6113 | 0.6185 | 73.39 | 74.13 | 63.25 | 63.62 | 56.66 | 56.62 | |\n| Scale | Var. Ratio | 0.6835 | 0.6853 | 69.46 | 70.14 | 94.25 | 94.42 | 89.22 | 89.21 |\n| Prob. Margin | 0.6849 | 0.6864 | 68.47 | 69.26 | 95.12 | 95.40 | 91.34 | 91.31 | |\n| Entropy | 0.6680 | 0.6714 | 71.92 | 72.34 | 79.24 | 79.24 | 72.45 | 72.67 | |\n| Avg. Var. | 0.6769 | 0.6771 | 71.45 | 71.78 | 91.80 | 91.93 | 84.48 | 84.48 | |\n| Max. Var. | 0.6763 | 0.6766 | 71.16 | 71.51 | 92.49 | 92.61 | 85.46 | 85.46 | |\n| BALD | 0.6808 | 0.6809 | 69.02 | 69.77 | 93.96 | 94.32 | 89.51 | 89.41 | |\n| DRN D-105 | | | | | | | | | |\n| Base | Var. Ratio | 0.6909 | 0.6722 | 65.35 | 64.55 | 77.75 | 77.90 | 64.00 | 64.30 |\n| Prob. Margin | 0.6944 | 0.6749 | 64.45 | 63.68 | 80.71 | 80.82 | 67.35 | 67.66 | |\n| Entropy | 0.6954 | 0.6756 | 63.41 | 62.68 | 83.74 | 83.88 | 71.03 | 71.35 | |\n| Noise | Var. Ratio | 0.6951 | 0.6765 | 65.42 | 64.54 | 78.16 | 78.25 | 64.26 | 64.55 |\n| Prob. Margin | 0.6986 | 0.6793 | 64.50 | 63.65 | 81.08 | 81.13 | 67.61 | 67.91 | |\n| Entropy | 0.6994 | 0.6795 | 63.44 | 62.66 | 84.12 | 84.18 | 71.31 | 71.62 | |\n| Avg. Var. | 0.6597 | 0.6499 | 72.46 | 71.56 | 53.21 | 53.57 | 39.50 | 39.70 | |\n| Max. Var. | 0.6594 | 0.6496 | 72.30 | 71.41 | 53.50 | 53.90 | 39.80 | 40.00 | |\n| BALD | 0.6951 | 0.6820 | 68.93 | 68.17 | 69.57 | 69.91 | 54.28 | 54.56 | |\n| Scale | Var. Ratio | 0.7514 | 0.7291 | 61.32 | 60.82 | 91.13 | 91.52 | 79.94 | 80.29 |\n| Prob. Margin | 0.7520 | 0.7295 | 60.48 | 60.08 | 91.99 | 92.49 | 81.80 | 82.15 | |\n| Entropy | 0.7410 | 0.7184 | 59.96 | 60.21 | 88.06 | 89.04 | 78.99 | 79.12 | |\n| Avg. Var. | 0.7369 | 0.7168 | 65.89 | 64.95 | 83.86 | 83.77 | 68.46 | 68.80 | |\n| Max. Var. | 0.7368 | 0.7149 | 65.51 | 64.57 | 84.64 | 84.53 | 69.49 | 69.83 | |\n| BALD | 0.7433 | 0.7225 | 62.10 | 61.33 | 90.92 | 90.90 | 78.75 | 79.09 | |\n| OneFormer ConvNeXt-L | | | | | | | | | |\n| Base | Var. Ratio | 0.5855 | 0.6168 | 38.21 | 39.15 | 60.89 | 64.90 | 46.06 | 46.13 |\n| Prob. Margin | 0.6221 | 0.6604 | 40.17 | 41.14 | 66.90 | 70.26 | 48.14 | 48.38 | |\n| Entropy | 0.5667 | 0.5949 | 38.41 | 39.55 | 56.71 | 60.16 | 42.67 | 42.79 | |\n| Noise | Var. Ratio | 0.5810 | 0.6297 | 41.12 | 41.79 | 62.40 | 64.63 | 43.86 | 44.11 |\n| Prob. Margin | 0.6235 | 0.6731 | 44.26 | 44.16 | 70.58 | 71.58 | 46.09 | 46.43 | |\n| Entropy | 0.5641 | 0.6036 | 41.33 | 42.15 | 56.58 | 58.65 | 39.57 | 39.84 | |\n\n| Avg. Var. | 0.6185 | 0.5884 | 45.33 | 43.47 | 49.41 | 46.04 | 31.51 | 31.94 | |\n|------------------|------------|----------|---------|---------|---------|---------|---------|---------|-------|\n| Max. Var. | 0.6061 | 0.5770 | 44.69 | 42.99 | 46.14 | 42.97 | 29.84 | 30.22 | |\n| BALD | 0.6408 | 0.6098 | 46.09 | 44.01 | 54.68 | 51.24 | 34.29 | 34.82 | |\n| Scale | Var. Ratio | 0.6645 | 0.7152 | 42.57 | 43.90 | 77.82 | 80.84 | 52.83 | 52.83 |\n| Prob. Margin | 0.7136 | 0.7483 | 42.30 | 42.50 | 86.97 | 88.44 | 59.42 | 59.58 | |\n| Entropy | 0.6377 | 0.7012 | 42.47 | 43.69 | 73.21 | 77.11 | 49.82 | 49.79 | |\n| Avg. Var. | 0.7535 | 0.7529 | 47.01 | 47.51 | 79.58 | 79.77 | 48.93 | 49.27 | |\n| Max. Var. | 0.7375 | 0.7389 | 46.93 | 47.73 | 75.94 | 76.41 | 46.78 | 47.11 | |\n| BALD | 0.7782 | 0.7702 | 46.55 | 48.37 | 84.16 | 83.93 | 52.26 | 52.63 | |\n| Drop | Var. Ratio | 0.5911 | 0.6229 | 38.94 | 39.56 | 62.37 | 65.38 | 46.30 | 46.49 |\n| Prob. Margin | 0.6249 | 0.6608 | 40.57 | 41.36 | 70.08 | 73.29 | 49.93 | 50.09 | |\n| Entropy | 0.5755 | 0.6089 | 39.10 | 40.23 | 57.44 | 61.24 | 42.46 | 42.61 | |\n| Avg. Var. | 0.6007 | 0.5568 | 41.51 | 42.21 | 50.29 | 46.64 | 35.02 | 35.36 | |\n| Max. Var. | 0.5884 | 0.5424 | 39.94 | 40.71 | 47.21 | 42.98 | 34.16 | 34.50 | |\n| BALD | 0.6246 | 0.5831 | 43.23 | 43.76 | 61.41 | 57.44 | 41.06 | 41.61 | |\n| OneFormer Swin-L | | | | | | | | | |\n| Base | Var. Ratio | 0.6779 | 0.7245 | 30.32 | 31.88 | 76.77 | 79.86 | 51.50 | 51.69 |\n| Prob. Margin | 0.7280 | 0.7592 | 30.33 | 31.68 | 82.61 | 85.43 | 55.40 | 55.62 | |\n| Entropy | 0.6555 | 0.7020 | 30.87 | 31.99 | 72.48 | 75.92 | 47.75 | 48.00 | |\n| Noise | Var. Ratio | 0.6483 | 0.6946 | 29.73 | 31.70 | 75.11 | 78.96 | 51.38 | 51.44 |\n| Prob. Margin | 0.6926 | 0.7253 | 30.44 | 31.83 | 82.02 | 84.50 | 54.80 | 54.92 | |\n| Entropy | 0.6293 | 0.6770 | 29.85 | 31.36 | 70.65 | 74.42 | 48.13 | 48.15 | |\n| Avg. Var. | 0.6410 | 0.6416 | 33.52 | 35.01 | 56.22 | 56.33 | 34.11 | 34.37 | |\n| Max. Var. | 0.6307 | 0.6288 | 33.44 | 35.05 | 53.37 | 53.36 | 32.46 | 32.70 | |\n| BALD | 0.6670 | 0.6635 | 34.19 | 35.52 | 63.24 | 62.72 | 37.62 | 37.98 | |\n| Scale | Var. Ratio | 0.7211 | 0.7414 | 29.35 | 30.90 | 87.33 | 89.37 | 60.52 | 60.78 |\n| Prob. Margin | 0.7577 | 0.7662 | 30.10 | 31.51 | 93.48 | 94.12 | 63.17 | 63.49 | |\n| Entropy | 0.7008 | 0.7259 | 29.76 | 31.32 | 80.13 | 82.50 | 54.77 | 54.87 | |\n| Avg. Var. | 0.6925 | 0.6958 | 29.62 | 31.29 | 82.56 | 83.32 | 56.69 | 57.03 | |\n| Max. Var. | 0.6752 | 0.6806 | 29.78 | 31.03 | 78.57 | 79.66 | 53.65 | 53.92 | |\n| BALD | 0.7250 | 0.7221 | 30.32 | 31.52 | 85.74 | 86.40 | 57.50 | 57.76 | |\n| Drop | Var. Ratio | 0.6956 | 0.7290 | 29.63 | 31.56 | 79.85 | 83.26 | 54.81 | 54.88 |\n| Prob. Margin | 0.7379 | 0.7591 | 29.56 | 31.19 | 86.79 | 88.97 | 59.71 | 59.94 | |\n| Entropy | 0.6736 | 0.7098 | 30.07 | 31.84 | 74.97 | 78.39 | 50.70 | 50.75 | |\n| Avg. Var. | 0.6046 | 0.6040 | 31.75 | 35.61 | 49.89 | 50.91 | 31.95 | 31.96 | |\n| Max. Var. | 0.5834 | 0.5822 | 30.39 | 34.46 | 44.06 | 45.19 | 29.49 | 29.42 | |\n| BALD | 0.6471 | 0.6387 | 33.59 | 36.63 | 59.41 | 59.76 | 35.97 | 36.06 | |\n| SegFormer B5 | | | | | | | | | |\n| Base | Var. Ratio | 0.7999 | 0.7978 | 61.71 | 60.47 | 83.55 | 83.24 | 48.95 | 49.18 |\n| Prob. Margin | 0.8053 | 0.8033 | 60.56 | 59.34 | 85.81 | 85.53 | 51.23 | 51.46 | |\n| Entropy | 0.8155 | 0.8156 | 58.55 | 57.43 | 89.42 | 89.24 | 55.22 | 55.47 | |\n| Noise | Var. Ratio | 0.7949 | 0.7903 | 62.18 | 60.54 | 82.73 | 82.46 | 48.10 | 48.26 |\n| Prob. Margin | 0.8012 | 0.7966 | 61.02 | 59.40 | 85.26 | 84.98 | 50.52 | 50.69 | |\n| Entropy | 0.8130 | 0.8092 | 58.85 | 57.44 | 89.09 | 88.89 | 54.74 | 54.93 | |\n| Avg. Var. | 0.6740 | 0.6721 | 67.72 | 67.33 | 47.74 | 47.73 | 25.48 | 25.56 | |\n| Max. Var. | 0.6743 | 0.6724 | 67.80 | 67.37 | 47.72 | 47.71 | 25.45 | 25.51 | |\n| BALD | 0.7243 | 0.7217 | 65.13 | 64.02 | 63.62 | 63.44 | 35.31 | 35.39 | |\n| Scale | Var. Ratio | 0.8474 | 0.8490 | 56.87 | 55.85 | 94.71 | 94.56 | 60.21 | 60.54 |\n| Prob. Margin | 0.8483 | 0.8491 | 55.76 | 54.76 | 95.73 | 95.62 | 62.06 | 62.38 | |\n| Entropy | 0.8519 | 0.8566 | 53.81 | 52.90 | 97.22 | 97.22 | 65.32 | 65.63 | |\n| Avg. Var. | 0.8064 | 0.8045 | 62.63 | 61.51 | 84.98 | 84.79 | 49.06 | 49.35 | |\n| Max. Var. | 0.8088 | 0.8060 | 62.39 | 61.25 | 85.59 | 85.37 | 49.60 | 49.90 | |\n| BALD | 0.8247 | 0.8249 | 57.76 | 56.68 | 93.09 | 92.86 | 58.27 | 58.59 | |\n\n| Drop | Var. Ratio | 0.8013 | 0.7990 | 61.58 | 60.33 | 83.92 | 83.58 | 49.27 | 49.50 |\n|--------------|--------------|----------|----------|---------|---------|---------|---------|---------|---------|\n| Prob. Margin | 0.8065 | 0.8043 | 60.42 | 59.19 | 86.15 | 85.83 | 51.55 | 51.78 | |\n| Entropy | 0.8165 | 0.8163 | 58.37 | 57.24 | 89.68 | 89.48 | 55.55 | 55.80 | |\n| Avg. Var. | 0.6980 | 0.6990 | 69.68 | 68.43 | 51.47 | 51.90 | 26.70 | 26.78 | |\n| Max. Var. | 0.6979 | 0.6976 | 70.03 | 68.67 | 51.20 | 51.42 | 26.43 | 26.50 | |\n| BALD | 0.7539 | 0.7532 | 65.18 | 64.02 | 69.57 | 69.40 | 38.59 | 38.75 | |\n\n| Scene | Unc. | Max 5% pixels | Max 10% pixels | Max 15% pixels | | | | | | | | | |\n|----------------------|-------------|-----------------|------------------|------------------|-------------|----------|-------|-------|-------|-------|-------|-------|-------|\n| Metric | Precision ↑ | Recall ↑ | Precision ↑ | Recall ↑ | Precision ↑ | Recall ↑ | | | | | | | |\n| µ | M | µ | M | µ | M | µ | M | µ | M | µ | M | | |\n| DRN D-22 | | | | | | | | | | | | | |\n| Base | VR | 80.81 | 80.86 | 05.86 | 05.94 | 79.04 | 79.14 | 11.65 | 11.79 | 77.96 | 78.11 | 17.41 | 17.61 |\n| PM | 78.91 | 79.08 | 05.74 | 05.81 | 78.24 | 78.41 | 11.63 | 11.76 | 77.62 | 77.77 | 17.41 | 17.59 | |\n| E | 81.07 | 81.12 | 05.92 | 06.01 | 78.88 | 78.96 | 11.57 | 11.74 | 77.51 | 77.58 | 17.03 | 17.23 | |\n| Noise | VR | 79.99 | 80.04 | 05.75 | 05.83 | 78.31 | 78.44 | 11.49 | 11.63 | 77.40 | 77.56 | 17.24 | 17.43 |\n| PM | 77.93 | 78.09 | 05.63 | 05.69 | 77.36 | 77.56 | 11.45 | 11.58 | 76.98 | 77.16 | 17.27 | 17.45 | |\n| E | 80.71 | 80.78 | 05.92 | 06.01 | 78.48 | 78.56 | 11.59 | 11.74 | 77.12 | 77.22 | 16.98 | 17.19 | |\n| AV | 77.78 | 77.89 | 05.65 | 05.69 | 77.46 | 77.62 | 11.13 | 11.20 | 77.28 | 77.46 | 16.58 | 16.68 | |\n| MV | 77.75 | 77.81 | 05.64 | 05.68 | 77.27 | 77.53 | 11.18 | 11.25 | 77.17 | 77.43 | 16.57 | 16.69 | |\n| B | 77.99 | 78.09 | 05.65 | 05.70 | 77.99 | 78.30 | 11.22 | 11.32 | 77.93 | 78.23 | 16.78 | 16.92 | |\n| Scale | VR | 83.99 | 84.07 | 06.03 | 06.13 | 82.50 | 82.54 | 11.97 | 12.16 | 81.54 | 81.62 | 17.83 | 18.10 |\n| PM | 82.04 | 82.13 | 05.49 | 05.56 | 81.63 | 81.73 | 11.75 | 11.92 | 81.09 | 81.22 | 17.87 | 18.13 | |\n| E | 84.60 | 84.61 | 06.08 | 06.18 | 81.69 | 81.80 | 11.88 | 12.08 | 80.24 | 80.34 | 17.67 | 17.94 | |\n| AV | 76.09 | 76.20 | 05.45 | 05.50 | 76.80 | 76.91 | 11.22 | 11.33 | 77.37 | 77.53 | 16.96 | 17.13 | |\n| MV | 75.39 | 75.64 | 05.48 | 05.52 | 76.65 | 76.87 | 11.27 | 11.37 | 77.42 | 77.68 | 17.04 | 17.21 | |\n| B | 81.40 | 81.51 | 05.87 | 05.94 | 79.81 | 79.95 | 11.60 | 11.73 | 79.19 | 79.32 | 17.43 | 17.61 | |\n| DRN D-105 | | | | | | | | | | | | | |\n| Base | VR | 79.49 | 79.57 | 06.99 | 07.54 | 77.15 | 77.21 | 13.95 | 14.92 | 75.53 | 75.57 | 20.69 | 21.99 |\n| PM | 78.21 | 78.21 | 07.00 | 07.49 | 76.79 | 76.80 | 14.02 | 14.95 | 75.38 | 75.42 | 20.73 | 22.02 | |\n| E | 78.96 | 79.07 | 07.03 | 07.59 | 76.02 | 76.04 | 13.50 | 14.44 | 74.43 | 74.49 | 19.73 | 21.04 | |\n| Noise | VR | 79.45 | 79.67 | 07.01 | 07.55 | 77.69 | 77.87 | 14.08 | 15.07 | 76.24 | 76.40 | 20.83 | 22.21 |\n| PM | 78.74 | 78.86 | 07.01 | 07.54 | 77.46 | 77.60 | 14.10 | 15.06 | 76.13 | 76.30 | 20.91 | 22.28 | |\n| E | 78.57 | 78.76 | 06.98 | 07.54 | 76.16 | 76.30 | 13.49 | 14.45 | 74.78 | 74.96 | 19.91 | 21.27 | |\n| AV | 82.54 | 82.55 | 07.32 | 07.80 | 80.39 | 80.35 | 14.29 | 15.18 | 78.53 | 78.51 | 20.94 | 22.18 | |\n| MV | 82.35 | 82.37 | 07.34 | 07.82 | 80.21 | 80.12 | 14.26 | 15.14 | 78.36 | 78.36 | 20.82 | 22.01 | |\n| B | 83.58 | 83.48 | 07.41 | 07.91 | 81.03 | 81.08 | 14.36 | 15.31 | 79.27 | 79.24 | 21.10 | 22.38 | |\n| Scale | VR | 85.07 | 85.04 | 07.44 | 08.09 | 83.39 | 83.39 | 14.81 | 15.95 | 81.64 | 81.69 | 21.89 | 23.48 |\n| PM | 83.96 | 83.95 | 07.22 | 07.84 | 82.98 | 82.98 | 14.79 | 15.93 | 81.75 | 81.75 | 22.28 | 23.85 | |\n| E | 81.47 | 81.53 | 07.09 | 07.70 | 78.75 | 78.91 | 13.96 | 15.06 | 76.88 | 77.00 | 20.53 | 21.96 | |\n| AV | 80.17 | 80.07 | 07.04 | 07.52 | 79.30 | 79.18 | 14.21 | 15.06 | 78.35 | 78.17 | 21.11 | 22.31 | |\n| MV | 79.46 | 79.40 | 07.02 | 07.49 | 78.23 | 78.17 | 14.07 | 14.93 | 77.44 | 77.35 | 20.89 | 22.10 | |\n| B | 83.78 | 83.56 | 07.35 | 07.95 | 81.43 | 81.31 | 14.51 | 15.53 | 79.77 | 79.67 | 21.41 | 22.75 | |\n| OneFormer ConvNeXt-L | | | | | | | | | | | | | |\n| Base | VR | 55.89 | 55.76 | 08.77 | 09.96 | 50.95 | 51.16 | 16.68 | 18.75 | 49.33 | 49.48 | 24.12 | 27.14 |\n| PM | 56.90 | 57.26 | 08.92 | 10.33 | 52.74 | 52.69 | 17.00 | 19.15 | 50.50 | 50.72 | 25.34 | 28.66 | |\n| E | 53.66 | 57.98 | 07.10 | 07.95 | 50.51 | 52.77 | 15.43 | 17.23 | 48.45 | 48.76 | 23.19 | 26.27 | |\n| Noise | VR | 50.88 | 50.21 | 08.01 | 09.27 | 48.60 | 48.81 | 15.88 | 18.17 | 48.02 | 48.27 | 23.52 | 26.45 |\n\nTable 17: Micro and macro-averaged (µ and M, respectively) Precision and Recall for thresholding based on a maximum fraction of pixels allowed on the Dark Zurich dataset.\n\nPM 53.23 53.87 08.42 09.64 51.40 51.91 16.85 19.14 51.08 51.35 25.31 28.26\n\nE 49.02 51.82 06.52 07.39 49.29 48.66 15.02 17.19 48.25 48.43 22.81 25.61 AV 46.85 45.04 06.36 07.00 44.08 43.39 12.61 13.80 43.62 43.65 17.48 19.20 MV 45.42 43.87 06.00 06.62 43.74 42.74 11.67 12.69 42.86 42.59 16.37 17.96 B 48.90 47.01 07.34 08.14 46.53 45.87 13.75 15.10 46.43 46.20 20.19 22.12\n\nScale VR 52.04 51.60 07.88 08.93 48.94 49.20 15.42 17.39 50.30 50.41 24.27 27.58\n\nPM 46.41 47.38 06.59 07.49 48.77 48.99 15.34 17.39 50.86 50.86 24.99 28.57\n\nE 51.66 50.42 06.92 08.08 51.24 51.17 14.99 17.19 51.33 51.43 23.35 26.61\n\nAV 52.01 51.73 07.79 08.84 56.87 55.85 16.98 19.21 57.48 56.90 25.91 29.41\n\nMV 51.34 49.82 06.96 07.83 55.74 53.54 14.94 17.66 57.64 55.91 24.48 28.40\n\nB **57.51 57.99** 09.13 10.25 58.43 58.94 18.77 21.02 58.65 59.66 **28.41 31.66**\n\nDrop VR 51.98 51.69 08.09 09.25 49.13 49.10 16.15 18.31 48.15 48.37 23.93 26.97\n\nPM 51.11 51.05 07.99 09.14 49.69 49.70 16.39 18.56 49.00 49.12 24.54 27.56 E 52.52 50.47 06.85 07.76 50.17 48.08 15.28 17.34 47.78 47.82 22.84 25.84\n\nAV 38.57 37.21 04.57 05.25 39.58 38.91 09.37 10.53 37.29 36.68 13.84 15.12\n\nMV 38.98 36.21 04.45 05.15 37.04 34.95 08.49 09.52 36.13 35.49 13.12 14.32\n\nB 39.47 38.96 04.86 05.60 39.04 40.32 10.22 11.43 39.46 39.81 15.44 17.16\n\nOneFormer Swin-L\n\nBase VR 56.31 57.18 12.60 13.72 52.36 52.90 24.12 26.45 48.69 49.42 33.07 36.21\n\nPM **56.72 57.57** 11.37 13.35 55.15 55.24 25.23 28.12 51.94 52.22 **36.61 40.07**\n\nE 52.69 55.74 09.64 11.01 50.94 53.44 21.54 23.85 47.18 49.77 29.67 32.64\n\nNoise VR 47.88 49.03 10.66 11.51 45.92 46.34 21.15 22.78 42.87 43.36 29.80 31.88\n\nPM 47.28 49.55 10.06 11.08 46.03 46.35 21.56 23.40 44.13 44.37 31.53 33.76 E 47.89 50.21 09.52 10.16 45.17 47.59 19.38 21.13 42.15 42.67 28.22 30.05 AV 40.68 39.68 08.21 08.92 37.55 38.22 15.33 16.84 37.95 38.76 22.46 24.85\n\nMV 37.52 38.19 07.38 08.13 37.55 38.72 14.63 16.60 35.51 36.92 21.04 23.13\n\nB 42.65 42.53 09.14 09.80 41.62 41.63 17.70 19.21 39.95 40.22 25.51 27.80\n\nScale VR 48.65 49.65 10.46 11.12 46.42 47.06 20.92 22.38 45.17 45.57 31.19 33.47\n\nPM 43.23 46.15 08.91 09.53 44.32 44.66 20.25 21.83 45.21 45.28 31.92 34.26\n\nE 46.59 48.57 09.34 10.13 46.59 48.38 19.30 21.02 44.81 45.35 29.41 31.71\n\nAV 36.17 35.70 08.00 08.83 36.06 35.52 16.22 17.54 36.04 36.23 24.61 26.69 MV 33.22 32.79 07.13 08.06 34.45 33.58 14.99 16.49 35.30 34.60 23.07 25.30 B 42.63 42.15 09.55 09.89 41.67 41.58 19.41 20.26 40.87 41.04 28.37 29.89\n\nDrop VR 53.68 54.48 11.89 12.78 50.24 50.64 23.10 25.08 47.42 47.96 33.15 35.84\n\nPM 50.47 51.85 10.43 11.89 51.13 51.63 23.81 26.40 49.95 50.33 35.11 38.40\n\nE 52.31 56.24 09.76 10.67 49.54 52.10 21.05 23.13 46.24 48.85 30.17 33.00\n\nAV. 32.22 36.01 05.65 06.56 35.33 37.06 12.58 14.02 33.60 35.12 16.58 18.30\n\nMV. 31.64 35.13 05.12 05.95 32.18 33.85 10.85 12.03 30.54 32.84 15.57 17.30\n\nB 36.19 37.58 07.08 08.17 38.70 40.07 13.87 15.61 37.65 39.60 19.59 21.82\n\nSegFormer B5\n\nBase VR 72.70 72.67 09.58 11.22 71.27 71.21 19.25 22.15 69.95 69.87 28.38 32.22\n\nPM 72.24 72.19 09.73 11.32 71.05 71.00 19.32 22.18 69.75 69.64 28.57 32.32 E 72.93 72.80 09.47 11.01 71.84 71.79 18.83 21.77 70.47 70.46 28.09 32.03\n\nNoise VR 70.16 70.08 09.29 10.96 69.24 69.22 18.66 21.70 68.26 68.20 27.78 31.80\n\nPM 69.36 69.36 09.30 10.92 69.00 68.94 18.82 21.81 68.08 68.02 27.87 31.87\n\nE 71.56 71.48 09.34 10.99 70.72 70.54 18.61 21.78 68.74 68.77 27.30 31.54\n\nAV 69.60 69.58 08.94 10.26 68.89 68.89 17.37 19.75 68.13 68.16 25.14 28.08\n\nMV 69.83 69.78 09.03 10.39 68.77 68.92 17.49 20.04 68.30 68.16 25.17 28.07\n\nB 68.81 68.80 08.88 10.21 67.70 67.64 16.99 19.23 66.81 66.66 24.81 27.69\n\nScale VR **77.93 77.94** 10.22 11.91 76.02 76.08 20.21 23.45 74.41 74.45 **30.01 34.19**\n\nPM 75.47 75.56 09.87 11.43 74.68 74.77 20.04 23.06 73.60 73.63 29.95 34.02 E 75.50 75.60 09.76 11.42 75.39 75.33 19.60 22.63 74.43 74.41 29.45 33.66 AV. 67.95 68.02 08.96 10.33 68.42 68.45 18.39 21.15 69.37 69.32 28.13 31.94\n\n| MV. | 66.71 | 66.83 | 08.83 | 10.13 | 68.17 | 68.16 | 18.40 | 21.12 | 69.45 | 69.33 | 28.25 | 31.95 | |\n|-------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|---------|-------|\n| B | 69.76 | 69.68 | 09.21 | 10.68 | 68.56 | 68.62 | 18.32 | 21.08 | 68.93 | 68.88 | 27.78 | 31.49 | |\n| Drop | VR | 72.80 | 72.78 | 09.66 | 11.28 | 71.46 | 71.37 | 19.38 | 22.24 | 70.06 | 69.97 | 28.56 | 32.37 |\n| PM | 72.48 | 72.43 | 09.75 | 11.30 | 71.25 | 71.16 | 19.40 | 22.19 | 69.84 | 69.74 | 28.59 | 32.34 | |\n| E | 72.92 | 72.78 | 09.47 | 10.98 | 71.92 | 71.81 | 18.99 | 21.88 | 70.55 | 70.56 | 27.98 | 31.86 | |\n| AV | 75.17 | 74.96 | 09.59 | 10.93 | 73.69 | 73.33 | 18.39 | 20.67 | 72.07 | 71.87 | 26.53 | 29.74 | |\n| MV | 75.22 | 74.96 | 09.58 | 10.84 | 73.62 | 73.33 | 18.30 | 20.62 | 72.26 | 72.06 | 26.56 | 29.70 | |\n| B | 74.30 | 73.99 | 09.48 | 10.74 | 72.36 | 72.05 | 18.21 | 20.52 | 70.46 | 70.54 | 26.30 | 29.47 | |\n\nThe results on Dark Zurich suggest much the same things as those on Cityscapes and ADE20K that Scale with simpler metrics is probably the best choice. One major difference is that if domain shift is expected, it is better to use the largest difference thresholding method than a maximum fraction of allowed pixels.\n\n## C Calibration Experiments\n\nIn order to improve our results we run experiments on calibrated OneFormer ConvNeXt-L and SegFormer B5 models. Specifically, we use temperature scaling on the logits during softmax computation to calibrate the models. We use temperatures of 0.2 and 2 for OneFormer and SegFormer models respectively. We first show the calibration results of such scaling in Table 18. Model calibration is evaluated using Expected Calibration Error (ECE) and Brier Score (BS). As can be seen after temperature scaling both models are better calibrated.\n\n| Calibration | ECE ↓ | BS ↓ | | |\n|----------------------|---------|--------|--------|--------|\n| Status | µ Avg. | M Avg. | µ Avg. | M Avg. |\n| OneFormer ConvNeXt-L | | | | |\n| Uncalibrated | 0.7887 | 0.8383 | 0.8012 | 0.8011 |\n| Calibrated | 0.1173 | 0.1084 | 0.0668 | 0.0681 |\n| SegFormer B5 | | | | |\n| Uncalibrated | 0.0631 | 0.0160 | 0.0516 | 0.0524 |\n| Calibrated | 0.0282 | 0.0125 | 0.0506 | 0.0514 |\n\nTable 18: Micro and macro-averaged (µ and M, respectively) Expected Calibration Error (ECE) and Brier Score (BS) for OneFormer ConvNeXt-L and SegFormer B5 models on the Cityscapes dataset\n\n| Scenario | Uncertainty | Uncalibrated | Calibrated | | |\n|----------------------|---------------|----------------|--------------|--------|--------|\n| Metric | AUROC ↑ | AUROC ↑ | | | |\n| µ Avg. | M Avg. | µ Avg. | M Avg. | | |\n| OneFormer ConvNeXt-L | | | | | |\n| Base | Var. Ratio | 0.7265 | 0.8822 | 0.8458 | 0.8848 |\n| Prob. Margin | 0.7893 | 0.9073 | 0.8543 | 0.8913 | |\n| Entropy | 0.7089 | 0.8569 | 0.8387 | 0.8896 | |\n\nWe now show the Area Under the Receiver Operating Characteristics (AUROC) as well as the Area Under the Precision Recall Curve (AUPRC) in Table 19 and Table 20, respectively. We observe that in almost all cases the calibrated models perform better than uncalibrated ones. The only departure from this pattern is AUPRC for SegFormer B5 with the Scale setting. Table 19: Micro and macro-averaged (µ and M, respectively) Area Under Receiver Operating Characteristic (AUROC) for uncalibrated and calibrated models on the Cityscapes dataset.\n\n| Scale | Var. Ratio | 0.7647 | 0.8972 | 0.8896 | 0.9163 |\n|--------------|--------------|----------|----------|----------|----------|\n| Prob. Margin | 0.8305 | 0.9215 | 0.8959 | 0.9221 | |\n| Entropy | 0.7350 | 0.8736 | 0.8805 | 0.9166 | |\n| Avg. Var. | 0.7835 | 0.7880 | 0.8268 | 0.8324 | |\n| Max. Var. | 0.7637 | 0.7677 | 0.8383 | 0.8453 | |\n| BALD | 0.8173 | 0.8213 | 0.8496 | 0.8563 | |\n| SegFormer B5 | | | | | |\n| Base | Var. Ratio | 0.8830 | 0.9087 | 0.9328 | 0.9468 |\n| Prob. Margin | 0.8892 | 0.9142 | 0.9351 | 0.9485 | |\n| Entropy | 0.8997 | 0.9234 | 0.9357 | 0.9467 | |\n| Scale | Var. Ratio | 0.9212 | 0.9391 | 0.9459 | 0.9569 |\n| Prob. Margin | 0.9245 | 0.9418 | 0.9477 | 0.9582 | |\n| Entropy | 0.9290 | 0.9459 | 0.9437 | 0.9522 | |\n| Avg. Var. | 0.8678 | 0.8826 | 0.8887 | 0.9010 | |\n| Max. Var. | 0.8687 | 0.8832 | 0.8864 | 0.8978 | |\n| BALD | 0.8947 | 0.9091 | 0.8990 | 0.9067 | |\n\n| Scenario | Uncertainty | Uncalibrated | Calibrated | | | | | | |\n|----------------------|---------------|-----------------|---------------|-----------------|--------|--------|--------|--------|--------|\n| Metric | AUPRC-Error ↑ | AUPRC-Success ↑ | AUPRC-Error ↑ | AUPRC-Success ↑ | | | | | |\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | | |\n| OneFormer ConvNeXt-L | | | | | | | | | |\n| Base | Var. Ratio | 0.1310 | 0.3090 | 0.9853 | 0.9633 | 0.2834 | 0.3173 | 0.9888 | 0.9611 |\n| Prob. Margin | 0.2300 | 0.3562 | 0.9885 | 0.9722 | 0.3107 | 0.3397 | 0.9892 | 0.9645 | |\n| Entropy | 0.1193 | 0.2832 | 0.9847 | 0.9548 | 0.2685 | 0.3078 | 0.9885 | 0.9669 | |\n| Scale | Var. Ratio | 0.1704 | 0.3258 | 0.9871 | 0.9741 | 0.3169 | 0.3417 | 0.9918 | 0.9786 |\n| Prob. Margin | 0.2921 | 0.3808 | 0.9905 | 0.9794 | 0.3565 | 0.3717 | 0.9922 | 0.9797 | |\n| Entropy | 0.1424 | 0.2903 | 0.9860 | 0.9702 | 0.2858 | 0.3198 | 0.9922 | 0.9822 | |\n| Avg. Var. | 0.1567 | 0.2303 | 0.9931 | 0.9886 | 0.3136 | 0.2989 | 0.9949 | 0.9917 | |\n| Max. Var. | 0.1391 | 0.2101 | 0.9924 | 0.9883 | 0.3029 | 0.2885 | 0.9951 | 0.9915 | |\n| BALD | 0.1916 | 0.2515 | 0.9928 | 0.9897 | 0.2984 | 0.2959 | 0.9941 | 0.9918 | |\n| SegFormer B5 | | | | | | | | | |\n| Base | Var. Ratio | 0.3873 | 0.4049 | 0.9969 | 0.9963 | 0.3901 | 0.4062 | 0.9975 | 0.9962 |\n| Prob. Margin | 0.3771 | 0.3935 | 0.9969 | 0.9963 | 0.3831 | 0.3998 | 0.9976 | 0.9963 | |\n| Entropy | 0.3840 | 0.4038 | 0.9966 | 0.9961 | 0.3382 | 0.3486 | 0.9975 | 0.9965 | |\n| Scale | Var. Ratio | 0.4335 | 0.4422 | 0.9976 | 0.9971 | 0.4204 | 0.4327 | 0.9980 | 0.9968 |\n| Prob. Margin | 0.4185 | 0.4256 | 0.9976 | 0.9971 | 0.4146 | 0.4258 | 0.9980 | 0.9969 | |\n| Entropy | 0.4099 | 0.4233 | 0.9975 | 0.9965 | 0.3439 | 0.3524 | 0.9977 | 0.9970 | |\n| Avg. Var. | 0.3371 | 0.3334 | 0.9975 | 0.9971 | 0.2716 | 0.2673 | 0.9961 | 0.9954 | |\n| Max. Var. | 0.3352 | 0.3303 | 0.9975 | 0.9971 | 0.2596 | 0.2546 | 0.9960 | 0.9952 | |\n| BALD | 0.3186 | 0.3155 | 0.9960 | 0.9955 | 0.2409 | 0.2391 | 0.9958 | 0.9949 | |\n\nTable 20: Micro and macro-averaged (µ and M, respectively) Area Under Precision Recall Curve (AUPRC) for uncalibrated and calibrated models on the Cityscapes dataset.\n\n## D Experiments With Noisy Inputs\n\nHere, we provide the results of our experiments with noisy inputs. This experiment was only performed on the Cityscapes dataset with DRN D-22 and OneFormer ConvNeXt-L models. Table 21 shows the base performances across varying noise levels.\n\nAs noise levels increase, we see a decrease in performance for both models, although, as seen earlier DRN\ngives much worse outputs than OneFormer. DRN's calibration also worsens with each noise level, whereas OneFormer's calibration stays mostly the same.\n\nTable 21: Micro and macro-averaged (µ and M, respectively) Mean Intersection Over Union (mIoU), Expected Calibration Error (ECE) and Brier Score (BS) on the Cityscapes dataset for noisy inputs.\n\n| Noise Level | mIoU ↑ | ECE ↓ | BS ↓ | | | |\n|----------------------|----------|---------|--------|--------|--------|--------|\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | |\n| DRN D-22 | | | | | | |\n| None | 0.6790 | 0.5479 | 0.0380 | 0.0177 | 0.0808 | 0.0824 |\n| Std. 5 | 0.4961 | 0.4497 | 0.0897 | 0.0619 | 0.1810 | 0.1828 |\n| Std. 10 | 0.3066 | 0.2928 | 0.0288 | 0.1733 | 0.4327 | 0.4325 |\n| Std. 25 | 0.1071 | 0.1034 | 0.0700 | 0.3995 | 0.9390 | 0.9349 |\n| Std. 50 | 0.0311 | 0.0312 | 0.0917 | 0.5574 | 1.2908 | 1.2878 |\n| OneFormer ConvNeXt-L | | | | | | |\n| None | 0.8287 | 0.6733 | 0.7887 | 0.8383 | 0.8012 | 0.8011 |\n| Std. 5 | 0.8149 | 0.6639 | 0.7912 | 0.8365 | 0.8014 | 0.8014 |\n| Std. 10 | 0.7979 | 0.6525 | 0.7874 | 0.8325 | 0.8013 | 0.8013 |\n| Std. 25 | 0.7413 | 0.6053 | 0.7693 | 0.8176 | 0.8012 | 0.8009 |\n| Std. 50 | 0.6202 | 0.5179 | 0.7505 | 0.7770 | 0.7996 | 0.7992 |\n\nTable 22 shows the results of using the largest difference as a thresholding technique on noisy inputs and AUROC values. Table 23 shows the results of using a maximum fraction of pixels allowed as thresholding.\n\nAs seen with Dark Zurich, the largest thresholding technique becomes better as noise levels increase, with better Precision and Recall. In contrast, a maximum fraction of pixels thresholding works better at lower noise.\n\n| Noise Level | Uncertainty | AUROC ↑ | Largest Difference Thresholding | | | | | | |\n|---------------|---------------|-----------|-----------------------------------|--------|--------|--------|--------|-------|-------|\n| Metric | Precision ↑ | Recall ↑ | Pixel % | | | | | | |\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | | |\n| DRN D-22 | | | | | | | | | |\n| None | Var. Ratio | 0.9118 | 0.9277 | 21.10 | 20.18 | 92.13 | 94.71 | 23.29 | 23.51 |\n| Prob. Margin | 0.9152 | 0.9302 | 19.71 | 18.84 | 93.31 | 95.62 | 25.26 | 25.48 | |\n| Entropy | 0.9205 | 0.9337 | 17.52 | 16.74 | 95.04 | 96.89 | 28.94 | 29.21 | |\n| Std. 5 | Var. Ratio | 0.8413 | 0.8618 | 30.23 | 29.08 | 84.43 | 87.70 | 31.78 | 31.86 |\n| Prob. Margin | 0.8450 | 0.8650 | 28.71 | 27.64 | 86.27 | 89.29 | 34.19 | 34.27 | |\n| Entropy | 0.8554 | 0.8732 | 26.34 | 25.36 | 89.41 | 91.86 | 38.63 | 38.74 | |\n| Std. 10 | Var. Ratio | 0.7718 | 0.7805 | 43.73 | 42.55 | 82.88 | 84.59 | 50.24 | 50.14 |\n| Prob. Margin | 0.7735 | 0.7816 | 42.39 | 41.23 | 85.17 | 86.68 | 53.25 | 53.15 | |\n| Entropy | 0.7866 | 0.7934 | 40.22 | 39.09 | 89.29 | 90.40 | 58.84 | 58.75 | |\n| Std. 25 | Var. Ratio | 0.6966 | 0.6857 | 67.14 | 66.38 | 92.13 | 92.24 | 80.87 | 80.71 |\n| Prob. Margin | 0.6887 | 0.6776 | 66.38 | 65.63 | 93.47 | 93.55 | 82.98 | 82.83 | |\n| Entropy | 0.7156 | 0.7041 | 65.11 | 64.43 | 96.37 | 96.45 | 87.23 | 87.10 | |\n| Std. 50 | Var. Ratio | 0.6294 | 0.6082 | 84.63 | 84.46 | 95.09 | 95.14 | 92.66 | 92.61 |\n| Prob. Margin | 0.6240 | 0.6027 | 84.40 | 84.22 | 95.88 | 95.92 | 93.68 | 93.64 | |\n\nTable 22: Micro and macro-averaged (µ and M, respectively) Area Under Receiver Operating Characteristic (AUROC) and Precision, Recall and the fraction of pixels selected for largest difference thresholding on the Cityscapes dataset for noisy inputs.\n\n| Entropy | 0.6427 | 0.6213 | 83.67 | 84.06 | 88.34 | 89.15 | 87.06 | 87.31 |\n|----------------------|----------|----------|---------|---------|---------|---------|---------|---------|\n| OneFormer ConvNeXt-L | | | | | | | | |\n| Prob. Margin | 0.7893 | 0.9073 | 14.37 | 15.63 | 89.19 | 91.32 | 19.84 | 19.98 |\n| Entropy | 0.7089 | 0.8569 | 14.13 | 16.23 | 81.30 | 84.88 | 18.40 | 18.65 |\n| Prob. Margin | 0.7846 | 0.9082 | 14.70 | 15.91 | 90.27 | 91.91 | 20.82 | 20.98 |\n| Entropy | 0.6975 | 0.8504 | 14.96 | 16.88 | 80.37 | 84.16 | 18.23 | 18.31 |\n| Prob. Margin | 0.7902 | 0.9039 | 15.00 | 16.13 | 90.42 | 91.79 | 22.64 | 22.83 |\n| Entropy | 0.7008 | 0.8431 | 15.25 | 17.26 | 80.37 | 83.66 | 19.79 | 20.01 |\n| Prob. Margin | 0.7771 | 0.8858 | 16.01 | 17.45 | 88.52 | 90.84 | 28.08 | 28.29 |\n| Entropy | 0.6810 | 0.8173 | 16.21 | 18.35 | 78.04 | 81.50 | 24.46 | 24.61 |\n| Prob. Margin | 0.7432 | 0.8416 | 20.11 | 21.58 | 84.53 | 86.92 | 36.74 | 36.94 |\n| Entropy | 0.6375 | 0.7544 | 19.36 | 22.07 | 68.96 | 73.90 | 31.14 | 31.33 |\n\nOneFormer ConvNeXt-L\n\nNone Var. Ratio 0.7265 0.8822 15.02 16.52 84.68 87.75 18.03 18.17\n\nProb. Margin 0.7893 0.9073 14.37 15.63 89.19 91.32 19.84 19.98 Entropy 0.7089 0.8569 14.13 16.23 81.30 84.88 18.40 18.65\n\nStd. 5 Var. Ratio 0.7189 0.8828 15.66 17.20 85.86 88.09 18.59 18.74\n\nProb. Margin 0.7846 **0.9082** 14.70 15.91 90.27 **91.91** 20.82 20.98\n\nEntropy 0.6975 0.8504 14.96 16.88 80.37 84.16 18.23 18.31\n\nStd. 10 Var. Ratio 0.7225 0.8737 15.70 17.35 85.73 87.67 20.51 20.69\n\nProb. Margin **0.7902** 0.9039 15.00 16.13 **90.42** 91.79 22.64 22.83 Entropy 0.7008 0.8431 15.25 17.26 80.37 83.66 19.79 20.01\n\nStd. 25 Var. Ratio 0.7071 0.8506 16.76 18.45 82.89 85.96 25.13 25.36\n\nProb. Margin 0.7771 0.8858 16.01 17.45 88.52 90.84 28.08 28.29 Entropy 0.6810 0.8173 16.21 18.35 78.04 81.50 24.46 24.61\n\nStd. 50 Var. Ratio 0.6706 0.7929 19.69 21.88 75.11 79.44 33.35 33.54\n\nProb. Margin 0.7432 0.8416 **20.11** 21.58 84.53 86.92 36.74 36.94 Entropy 0.6375 0.7544 19.36 **22.07** 68.96 73.90 31.14 31.33\n\n| Noise | Unc. | Max 5% pixels | Max 10% pixels | Max 15% pixels | | | | | | | | | |\n|----------------------|--------|-----------------|------------------|------------------|----------|-------------|----------|-------|-------|-------|-------|-------|-------|\n| Level | Metric | Precision ↑ | Recall ↑ | Precision ↑ | Recall ↑ | Precision ↑ | Recall ↑ | | | | | | |\n| µ | M | µ | M | µ | M | µ | M | µ | M | µ | M | | |\n| DRN D-22 | | | | | | | | | | | | | |\n| None | VR | 45.19 | 45.40 | 41.52 | 50.74 | 35.04 | 35.18 | 64.13 | 73.92 | 28.48 | 28.45 | 76.44 | 84.64 |\n| PM | 44.66 | 44.83 | 41.35 | 50.58 | 34.77 | 34.92 | 64.13 | 73.95 | 28.14 | 28.19 | 76.51 | 84.74 | |\n| E | 44.04 | 44.23 | 39.76 | 48.74 | 34.80 | 35.03 | 63.64 | 73.61 | 28.11 | 28.23 | 76.42 | 84.73 | |\n| Std. 5 | VR | 54.97 | 55.12 | 23.54 | 29.69 | 47.63 | 47.73 | 41.16 | 49.70 | 42.10 | 42.13 | 54.23 | 63.15 |\n| PM | 53.45 | 53.56 | 23.09 | 29.26 | 47.06 | 47.17 | 40.89 | 49.50 | 41.71 | 41.77 | 54.16 | 63.14 | |\n| E | 55.09 | 55.13 | 23.23 | 28.93 | 48.08 | 48.25 | 40.97 | 49.43 | 42.23 | 42.35 | 54.47 | 63.39 | |\n| Std. 10 | VR | 67.13 | 67.11 | 12.20 | 14.46 | 62.14 | 62.10 | 22.95 | 26.78 | 58.77 | 58.69 | 32.73 | 37.61 |\n| PM | 63.41 | 63.41 | 11.63 | 13.83 | 60.68 | 60.64 | 22.57 | 26.38 | 57.95 | 57.88 | 32.44 | 37.32 | |\n| E | 68.83 | 68.73 | 12.52 | 14.77 | 64.07 | 63.95 | 23.34 | 27.10 | 60.27 | 60.19 | 32.97 | 37.82 | |\n| Std. 25 | VR | 81.70 | 81.65 | 06.61 | 06.99 | 79.06 | 79.02 | 12.92 | 13.61 | 77.02 | 76.97 | 19.09 | 20.07 |\n| PM | 75.67 | 75.68 | 06.03 | 06.37 | 75.10 | 75.09 | 12.37 | 13.03 | 74.55 | 74.52 | 18.63 | 19.58 | |\n| E | 83.90 | 83.86 | 06.77 | 07.16 | 82.12 | 82.05 | 13.40 | 14.13 | 80.52 | 80.45 | 19.86 | 20.87 | |\n| Std. 50 | VR | 88.49 | 88.46 | 05.06 | 05.11 | 88.09 | 88.06 | 10.24 | 10.33 | 87.80 | 87.76 | 15.46 | 15.58 |\n| PM | 86.81 | 86.76 | 04.82 | 04.85 | 86.71 | 86.65 | 10.07 | 10.14 | 86.63 | 86.56 | 15.35 | 15.46 | |\n| E | 89.38 | 89.35 | 05.12 | 05.17 | 89.02 | 88.97 | 10.32 | 10.40 | 88.78 | 88.72 | 15.56 | 15.69 | |\n| OneFormer ConvNeXt-L | | | | | | | | | | | | | |\n| None | VR | 32.16 | 32.31 | 47.73 | 57.40 | 24.77 | 24.84 | 67.92 | 77.28 | 20.80 | 21.03 | 74.47 | 82.96 |\n| PM | 33.41 | 33.78 | 50.18 | 60.30 | 24.95 | 24.99 | 71.11 | 80.37 | 20.58 | 20.68 | 79.04 | 86.67 | |\n| E | 30.89 | 31.47 | 44.48 | 54.09 | 23.82 | 24.11 | 64.18 | 74.15 | 20.07 | 20.51 | 70.79 | 79.86 | |\n| Std. 5 | VR | 33.39 | 33.47 | 46.82 | 55.89 | 25.65 | 25.84 | 67.10 | 76.08 | 21.72 | 21.79 | 76.17 | 83.03 |\n| PM | 34.87 | 35.01 | 49.54 | 58.84 | 26.23 | 26.34 | 70.75 | 79.39 | 21.61 | 21.63 | 79.90 | 86.48 | |\n| E | 32.06 | 32.15 | 43.67 | 52.72 | 24.64 | 24.94 | 63.04 | 72.49 | 20.95 | 21.27 | 71.34 | 79.12 | |\n| Std. 10 | VR | 34.62 | 34.73 | 44.15 | 52.27 | 26.98 | 27.09 | 65.58 | 73.98 | 22.53 | 22.66 | 74.42 | 81.57 |\n| PM | 36.58 | 36.73 | 47.19 | 55.44 | 27.76 | 27.82 | 69.51 | 77.30 | 22.83 | 22.83 | 79.17 | 85.34 | |\n| E | 33.13 | 33.47 | 41.06 | 49.09 | 25.98 | 26.28 | 61.66 | 70.25 | 21.70 | 21.95 | 70.92 | 78.15 | |\n\nTable 23: Micro and macro-averaged (µ and M, respectively) Precision and Recall for thresholding based on a maximum fraction of pixels allowed on the Cityscapes dataset for noisy inputs.\n\nStd. 25 VR 38.64 38.71 36.11 42.76 31.73 31.80 58.76 66.09 26.75 26.78 69.95 76.44\n\nPM 41.41 41.45 39.21 46.05 33.05 33.15 62.47 69.61 27.47 27.47 74.83 80.59 E 36.77 36.65 32.98 39.34 30.38 30.56 54.93 62.39 25.65 25.85 65.64 72.39\n\nStd. 50 VR 47.30 47.37 25.49 31.07 41.07 41.14 44.32 51.89 35.91 35.85 56.27 63.68\n\nPM **49.59 49.81** 27.03 32.95 43.02 42.94 47.25 54.86 37.62 37.51 61.20 68.13\n\nE 44.83 44.97 22.80 28.00 39.28 39.36 40.75 48.10 34.38 34.49 51.40 59.02\n\n## E Classwise Results\n\nFinally, we provide classwise results for some of the experiments. Table 24 shows classwise Mean Intersection Over Union (mIoU) for all tested networks on the Cityscapes dataset, and Table 25 shows the same for ADE20K dataset. We observe that the best and worst classes in each dataset are consistent across models.\n\nTable 24: Classwise micro and macro averaged (µ and M respectively) Mean Intersection Over Union (mIou)\non the Cityscapes dataset across different models. Higher is better. The top five classes for every model are highlighted in bold green, and the worst five classes are highlighted in red italics.\n\n| Class | DRN D-22 | DRN D-105 | OF ConvNeXt-L | OF Swin-L | Segformer B5 | | | | | |\n|---------------|------------|-------------|-----------------|-------------|----------------|--------|--------|--------|--------|--------|\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | |\n| road | 0.9721 | 0.9390 | 0.9815 | 0.9492 | 0.9858 | 0.9553 | 0.9847 | 0.9561 | 0.9847 | 0.9524 |\n| sidewalk | 0.7966 | 0.6962 | 0.8498 | 0.7424 | 0.8775 | 0.7710 | 0.8673 | 0.7646 | 0.8735 | 0.7671 |\n| building | 0.9019 | 0.8461 | 0.9249 | 0.8744 | 0.9371 | 0.8926 | 0.9401 | 0.8965 | 0.9375 | 0.8947 |\n| wall | 0.3805 | 0.1526 | 0.4826 | 0.2291 | 0.5081 | 0.3053 | 0.6663 | 0.2973 | 0.6960 | 0.3196 |\n| fence | 0.4654 | 0.1716 | 0.5877 | 0.2515 | 0.6937 | 0.2809 | 0.6951 | 0.2886 | 0.6754 | 0.3224 |\n| pole | 0.5877 | 0.5258 | 0.6632 | 0.6062 | 0.7235 | 0.6665 | 0.7217 | 0.6683 | 0.6964 | 0.6454 |\n| traffic light | 0.6301 | 0.3537 | 0.7289 | 0.4734 | 0.7678 | 0.5816 | 0.7644 | 0.5589 | 0.7547 | 0.5632 |\n| traffic sign | 0.7300 | 0.6192 | 0.8053 | 0.7024 | 0.8445 | 0.7566 | 0.8517 | 0.7639 | 0.8231 | 0.7385 |\n| vegetation | 0.9114 | 0.8696 | 0.9261 | 0.8914 | 0.9352 | 0.9058 | 0.9322 | 0.9006 | 0.9316 | 0.8999 |\n| terrain | 0.5697 | 0.2557 | 0.6229 | 0.2790 | 0.6968 | 0.3332 | 0.6589 | 0.3072 | 0.6684 | 0.3351 |\n| sky | 0.9372 | 0.8121 | 0.9511 | 0.8418 | 0.9573 | 0.8787 | 0.9588 | 0.8719 | 0.9567 | 0.8711 |\n| person | 0.7741 | 0.5214 | 0.8316 | 0.6098 | 0.8743 | 0.6760 | 0.8692 | 0.6546 | 0.8494 | 0.6344 |\n| rider | 0.5208 | 0.3254 | 0.6300 | 0.4560 | 0.7465 | 0.5362 | 0.7267 | 0.5055 | 0.6856 | 0.4777 |\n| car | 0.9234 | 0.8230 | 0.9484 | 0.8794 | 0.9649 | 0.9029 | 0.9642 | 0.8985 | 0.9571 | 0.8971 |\n| truck | 0.4067 | 0.1188 | 0.5999 | 0.2143 | 0.9029 | 0.3239 | 0.9004 | 0.3501 | 0.8632 | 0.3581 |\n| bus | 0.6826 | 0.2365 | 0.8080 | 0.4289 | 0.9410 | 0.5502 | 0.9299 | 0.5614 | 0.9214 | 0.5964 |\n| train | 0.5291 | 0.0765 | 0.5681 | 0.2004 | 0.8790 | 0.4827 | 0.8473 | 0.4778 | 0.8354 | 0.3840 |\n| motorcycle | 0.4534 | 0.1361 | 0.6456 | 0.2572 | 0.7136 | 0.2762 | 0.6961 | 0.3006 | 0.7412 | 0.3570 |\n| bicycle | 0.7287 | 0.4793 | 0.7918 | 0.5730 | 0.7953 | 0.6099 | 0.7717 | 0.5694 | 0.8022 | 0.5964 |\n\n| Class | OF ConvNeXt-L | OF Swin-L | Segformer B5 | | | |\n|----------|-----------------|-------------|----------------|--------|--------|--------|\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | |\n| wall | 0.8067 | 0.6516 | 0.8145 | 0.6618 | 0.7856 | 0.6381 |\n| building | 0.8439 | 0.6402 | 0.8566 | 0.6479 | 0.8214 | 0.6160 |\n| sky | 0.9483 | 0.8759 | 0.9506 | 0.8790 | 0.9453 | 0.8648 |\n| floor | 0.8459 | 0.7415 | 0.8456 | 0.7434 | 0.8220 | 0.7262 |\n\nTable 25: Classwise micro and macro averaged (µ and M respectively) Mean Intersection Over Union (mIou) on the ADE20K dataset across different models. Higher is better. The top five classes for every model are highlighted in bold green, and the worst five classes are highlighted in red italics.\n\n| tree | 0.7759 | 0.6058 | 0.7753 | 0.6108 | 0.7477 | 0.5786 |\n|--------------|----------|----------|----------|----------|----------|----------|\n| ceiling | 0.8583 | 0.7567 | 0.8542 | 0.7483 | 0.8490 | 0.7379 |\n| road | 0.8463 | 0.6934 | 0.8603 | 0.7142 | 0.8580 | 0.6668 |\n| bed | 0.9170 | 0.8345 | 0.9106 | 0.8051 | 0.8991 | 0.7446 |\n| window | 0.6355 | 0.5212 | 0.6638 | 0.5158 | 0.6120 | 0.4749 |\n| grass | 0.7153 | 0.4215 | 0.7346 | 0.4164 | 0.7024 | 0.3966 |\n| cabinet | 0.6241 | 0.3613 | 0.6379 | 0.4008 | 0.6327 | 0.3527 |\n| sidewalk | 0.7021 | 0.5364 | 0.7059 | 0.5269 | 0.6682 | 0.4679 |\n| person | 0.8623 | 0.6387 | 0.8622 | 0.6348 | 0.8258 | 0.5858 |\n| ground | 0.3729 | 0.1986 | 0.4212 | 0.2239 | 0.3781 | 0.2129 |\n| door | 0.5428 | 0.3631 | 0.5742 | 0.3691 | 0.4766 | 0.3215 |\n| table | 0.6608 | 0.4943 | 0.6908 | 0.5068 | 0.6112 | 0.4095 |\n| mountain | 0.6218 | 0.4125 | 0.6210 | 0.4709 | 0.6123 | 0.4474 |\n| plant | 0.5788 | 0.3617 | 0.5737 | 0.3593 | 0.4870 | 0.3139 |\n| curtain | 0.7856 | 0.6775 | 0.8282 | 0.7015 | 0.7346 | 0.5954 |\n| chair | 0.6585 | 0.4655 | 0.6731 | 0.4784 | 0.5906 | 0.3688 |\n| car | 0.8771 | 0.6539 | 0.8848 | 0.6554 | 0.8599 | 0.5979 |\n| water | 0.5550 | 0.3294 | 0.5965 | 0.3596 | 0.5875 | 0.3113 |\n| painting | 0.7703 | 0.6583 | 0.7710 | 0.6517 | 0.7460 | 0.5801 |\n| sofa | 0.7240 | 0.5361 | 0.7717 | 0.6058 | 0.6761 | 0.4785 |\n| shelf | 0.4346 | 0.2814 | 0.5085 | 0.2834 | 0.4334 | 0.1944 |\n| house | 0.4782 | 0.2273 | 0.4705 | 0.2188 | 0.3589 | 0.2050 |\n| sea | 0.6542 | 0.6539 | 0.6563 | 0.6657 | 0.6163 | 0.6404 |\n| mirror | 0.7409 | 0.4859 | 0.7459 | 0.5405 | 0.6822 | 0.4619 |\n| rug | 0.6931 | 0.5074 | 0.6697 | 0.5201 | 0.5712 | 0.4658 |\n| field | 0.3654 | 0.2385 | 0.3744 | 0.2600 | 0.3234 | 0.2377 |\n| armchair | 0.4767 | 0.3774 | 0.5408 | 0.4324 | 0.4380 | 0.3323 |\n| seat | 0.6794 | 0.4377 | 0.6271 | 0.3935 | 0.6062 | 0.2640 |\n| fence | 0.4816 | 0.2270 | 0.4778 | 0.2090 | 0.4450 | 0.1653 |\n| desk | 0.5720 | 0.3196 | 0.5498 | 0.3065 | 0.5174 | 0.2196 |\n| rock | 0.6209 | 0.3134 | 0.6251 | 0.3511 | 0.4836 | 0.2421 |\n| wardrobe | 0.4637 | 0.3900 | 0.5461 | 0.3966 | 0.4957 | 0.2762 |\n| lamp | 0.7549 | 0.5837 | 0.7439 | 0.5656 | 0.6543 | 0.4512 |\n| bathtub | 0.7890 | 0.6519 | 0.7881 | 0.7054 | 0.7722 | 0.5617 |\n| railing | 0.4077 | 0.1737 | 0.4024 | 0.1923 | 0.3233 | 0.1239 |\n| cushion | 0.7104 | 0.6211 | 0.7144 | 0.6059 | 0.5829 | 0.4591 |\n| pedestal | 0.3440 | 0.1316 | 0.3931 | 0.1337 | 0.3005 | 0.1151 |\n| box | 0.4007 | 0.2085 | 0.3680 | 0.1998 | 0.3042 | 0.1335 |\n| column | 0.5654 | 0.2827 | 0.5452 | 0.2992 | 0.3971 | 0.1823 |\n| sign | 0.4285 | 0.3156 | 0.4494 | 0.3097 | 0.3753 | 0.2077 |\n| chest | 0.4499 | 0.3279 | 0.3611 | 0.3373 | 0.4488 | 0.2941 |\n| counter | 0.5034 | 0.2014 | 0.4112 | 0.2114 | 0.2756 | 0.1485 |\n| sand | 0.5074 | 0.4246 | 0.4304 | 0.4166 | 0.3661 | 0.2842 |\n| sink | 0.7603 | 0.6281 | 0.8013 | 0.6238 | 0.7293 | 0.4948 |\n| skyscraper | 0.3880 | 0.4413 | 0.4748 | 0.4636 | 0.6001 | 0.4912 |\n| fireplace | 0.7373 | 0.5610 | 0.7466 | 0.5621 | 0.7959 | 0.5376 |\n| refrigerator | 0.8125 | 0.6262 | 0.7967 | 0.6240 | 0.7766 | 0.5658 |\n| grandstand | 0.5877 | 0.4089 | 0.4502 | 0.3187 | 0.4576 | 0.2347 |\n| path | 0.2789 | 0.1794 | 0.2696 | 0.1930 | 0.2415 | 0.1313 |\n| stairs | 0.3472 | 0.1767 | 0.3306 | 0.1939 | 0.3179 | 0.1396 |\n| runway | 0.6795 | 0.5788 | 0.7157 | 0.6200 | 0.7075 | 0.4809 |\n| case | 0.6180 | 0.2870 | 0.5921 | 0.3194 | 0.4926 | 0.1707 |\n| pool table | 0.9529 | 0.7290 | 0.9521 | 0.8312 | 0.9378 | 0.7310 |\n| pillow | 0.6874 | 0.5569 | 0.6626 | 0.5333 | 0.5727 | 0.4080 |\n\n| screen door | 0.6210 | 0.6015 | 0.7174 | 0.7501 | 0.6945 | 0.4878 |\n|-----------------|----------|----------|----------|----------|----------|----------|\n| stairway | 0.3473 | 0.2017 | 0.4628 | 0.1861 | 0.2846 | 0.1354 |\n| river | 0.2326 | 0.2446 | 0.1674 | 0.1943 | 0.1530 | 0.1384 |\n| bridge | 0.7893 | 0.3932 | 0.8101 | 0.4119 | 0.6948 | 0.2788 |\n| bookcase | 0.3632 | 0.3125 | 0.3668 | 0.3179 | 0.3821 | 0.2812 |\n| blind | 0.4365 | 0.2617 | 0.5201 | 0.2866 | 0.4640 | 0.2325 |\n| coffee table | 0.6210 | 0.4195 | 0.6698 | 0.4621 | 0.5341 | 0.3168 |\n| toilet | 0.9087 | 0.8842 | 0.8984 | 0.8076 | 0.8761 | 0.7845 |\n| flower | 0.5273 | 0.3844 | 0.5180 | 0.3990 | 0.4448 | 0.3256 |\n| book | 0.5470 | 0.2751 | 0.5712 | 0.2600 | 0.4917 | 0.1820 |\n| hill | 0.1413 | 0.0699 | 0.0882 | 0.0782 | 0.0699 | 0.0596 |\n| bench | 0.7182 | 0.2913 | 0.4896 | 0.2537 | 0.4612 | 0.1472 |\n| countertop | 0.6699 | 0.3158 | 0.6848 | 0.3225 | 0.5890 | 0.3261 |\n| stove | 0.8335 | 0.6621 | 0.8505 | 0.7050 | 0.7762 | 0.5391 |\n| palm | 0.5766 | 0.4356 | 0.5272 | 0.4224 | 0.4802 | 0.3737 |\n| kitchen island | 0.4398 | 0.1622 | 0.3594 | 0.1433 | 0.4026 | 0.2282 |\n| computer | 0.7565 | 0.4210 | 0.7788 | 0.4522 | 0.7503 | 0.3573 |\n| swivel chair | 0.6038 | 0.4412 | 0.5583 | 0.3940 | 0.3974 | 0.3466 |\n| boat | 0.4278 | 0.3011 | 0.6905 | 0.3579 | 0.5051 | 0.2265 |\n| bar | 0.6650 | 0.3924 | 0.5365 | 0.2896 | 0.3440 | 0.2020 |\n| arcade machine | 0.8454 | 0.2722 | 0.6547 | 0.2077 | 0.8334 | 0.2863 |\n| hut | 0.4318 | 0.2134 | 0.3258 | 0.2070 | 0.3131 | 0.1490 |\n| bus | 0.9460 | 0.3583 | 0.9115 | 0.2936 | 0.9068 | 0.2478 |\n| towel | 0.7651 | 0.5525 | 0.7642 | 0.5077 | 0.6295 | 0.3371 |\n| light | 0.6364 | 0.4461 | 0.6482 | 0.4464 | 0.5536 | 0.3497 |\n| truck | 0.4274 | 0.1781 | 0.4698 | 0.2005 | 0.4223 | 0.1556 |\n| tower | 0.2338 | 0.1999 | 0.3230 | 0.2109 | 0.1143 | 0.1274 |\n| chandelier | 0.7500 | 0.5696 | 0.7364 | 0.5751 | 0.6495 | 0.4677 |\n| awning | 0.4043 | 0.2681 | 0.3715 | 0.2658 | 0.2873 | 0.1627 |\n| streetlight | 0.4746 | 0.3086 | 0.4414 | 0.3132 | 0.2739 | 0.1378 |\n| booth | 0.5275 | 0.2094 | 0.6864 | 0.2372 | 0.3458 | 0.2258 |\n| television | 0.7962 | 0.6454 | 0.7413 | 0.6022 | 0.7161 | 0.4806 |\n| airplane | 0.6213 | 0.4914 | 0.6869 | 0.4720 | 0.6163 | 0.3258 |\n| dirt track | 0.0322 | 0.0492 | 0.0306 | 0.0237 | 0.0490 | 0.0274 |\n| clothes | 0.3764 | 0.1749 | 0.4861 | 0.2240 | 0.3336 | 0.1717 |\n| pole | 0.3596 | 0.1927 | 0.3444 | 0.1895 | 0.2177 | 0.1025 |\n| land | 0.0627 | 0.0412 | 0.0747 | 0.0999 | 0.0058 | 0.0056 |\n| bannister | 0.2125 | 0.1589 | 0.2319 | 0.1424 | 0.1263 | 0.0660 |\n| escalator | 0.2659 | 0.3568 | 0.5813 | 0.3277 | 0.5315 | 0.2616 |\n| ottoman | 0.4856 | 0.2862 | 0.5834 | 0.3402 | 0.5115 | 0.1919 |\n| bottle | 0.4579 | 0.2554 | 0.4654 | 0.2594 | 0.3638 | 0.1409 |\n| buffet | 0.4452 | 0.2182 | 0.4719 | 0.2150 | 0.3274 | 0.1564 |\n| poster | 0.3500 | 0.1344 | 0.2816 | 0.1522 | 0.2479 | 0.0986 |\n| stage | 0.1463 | 0.1667 | 0.1337 | 0.2894 | 0.1580 | 0.1140 |\n| van | 0.4994 | 0.1954 | 0.5293 | 0.2069 | 0.4345 | 0.1864 |\n| ship | 0.6307 | 0.3798 | 0.8245 | 0.4695 | 0.7014 | 0.2476 |\n| fountain | 0.3263 | 0.2975 | 0.4673 | 0.4499 | 0.2149 | 0.1999 |\n| conveyer belt | 0.7296 | 0.3313 | 0.6991 | 0.2257 | 0.7720 | 0.2220 |\n| canopy | 0.4703 | 0.2735 | 0.3512 | 0.2969 | 0.4163 | 0.2003 |\n| washing machine | 0.7302 | 0.7117 | 0.8698 | 0.7630 | 0.7532 | 0.6962 |\n| toy | 0.4918 | 0.2208 | 0.3575 | 0.2384 | 0.2614 | 0.1435 |\n| swimming pool | 0.8168 | 0.6303 | 0.7866 | 0.4143 | 0.5739 | 0.3583 |\n| stool | 0.5461 | 0.2621 | 0.5905 | 0.2739 | 0.3919 | 0.1236 |\n| barrel | 0.8925 | 0.8798 | 0.6715 | 0.5162 | 0.4894 | 0.3102 |\n\n| basket | 0.3604 | 0.2936 | 0.4157 | 0.3148 | 0.3615 | 0.2327 |\n|----------------|----------|----------|----------|----------|----------|----------|\n| waterfall | 0.7514 | 0.5450 | 0.5517 | 0.5178 | 0.6443 | 0.3568 |\n| tent | 0.9594 | 0.4856 | 0.9292 | 0.5659 | 0.9487 | 0.3551 |\n| bag | 0.2520 | 0.1483 | 0.2170 | 0.1410 | 0.1267 | 0.0616 |\n| motorbike | 0.7769 | 0.3167 | 0.7472 | 0.3723 | 0.7456 | 0.3757 |\n| cradle | 0.8654 | 0.7638 | 0.8667 | 0.8741 | 0.7678 | 0.7065 |\n| oven | 0.5860 | 0.2807 | 0.5741 | 0.3720 | 0.4922 | 0.2726 |\n| ball | 0.4123 | 0.2837 | 0.5238 | 0.2504 | 0.5260 | 0.1281 |\n| food | 0.6462 | 0.2817 | 0.6334 | 0.3565 | 0.3681 | 0.2434 |\n| step | 0.1872 | 0.0850 | 0.0899 | 0.0758 | 0.1861 | 0.0869 |\n| tank | 0.5648 | 0.1351 | 0.5138 | 0.1449 | 0.5763 | 0.1914 |\n| trade name | 0.3477 | 0.2577 | 0.3493 | 0.2611 | 0.2678 | 0.1511 |\n| microwave | 0.8695 | 0.6115 | 0.8657 | 0.6096 | 0.7648 | 0.5329 |\n| pot | 0.5798 | 0.3344 | 0.5933 | 0.3199 | 0.4460 | 0.2539 |\n| animal | 0.6246 | 0.4527 | 0.6330 | 0.5182 | 0.5985 | 0.4072 |\n| bicycle | 0.6379 | 0.3518 | 0.6383 | 0.3425 | 0.5816 | 0.2529 |\n| lake | 0.0000 | 0.0000 | 0.4752 | 0.0994 | 0.6027 | 0.1241 |\n| dishwasher | 0.7900 | 0.5149 | 0.7234 | 0.4633 | 0.6610 | 0.4105 |\n| screen | 0.6464 | 0.4130 | 0.6395 | 0.4824 | 0.7059 | 0.3711 |\n| blanket | 0.3817 | 0.3033 | 0.3442 | 0.3109 | 0.1784 | 0.1382 |\n| sculpture | 0.6732 | 0.1532 | 0.6341 | 0.1354 | 0.6072 | 0.0820 |\n| hood | 0.6624 | 0.6031 | 0.6562 | 0.6131 | 0.5749 | 0.5388 |\n| sconce | 0.6015 | 0.3542 | 0.6167 | 0.3599 | 0.4927 | 0.2539 |\n| vase | 0.5359 | 0.3630 | 0.5246 | 0.3155 | 0.4137 | 0.2371 |\n| traffic light | 0.4124 | 0.3142 | 0.4778 | 0.2960 | 0.3619 | 0.1691 |\n| tray | 0.1839 | 0.1243 | 0.2635 | 0.1621 | 0.1173 | 0.0869 |\n| ashcan | 0.4290 | 0.2278 | 0.4943 | 0.2422 | 0.4235 | 0.1743 |\n| fan | 0.7102 | 0.4734 | 0.7327 | 0.5408 | 0.6441 | 0.3386 |\n| pier | 0.3749 | 0.1524 | 0.3767 | 0.2290 | 0.6224 | 0.1378 |\n| crt screen | 0.2180 | 0.0496 | 0.0694 | 0.0374 | 0.1122 | 0.0338 |\n| plate | 0.6085 | 0.2960 | 0.6328 | 0.2942 | 0.5125 | 0.1629 |\n| monitor | 0.6631 | 0.2648 | 0.1195 | 0.1170 | 0.0759 | 0.0717 |\n| bulletin board | 0.6488 | 0.2437 | 0.5838 | 0.1777 | 0.4835 | 0.1756 |\n| shower | 0.0315 | 0.1646 | 0.0358 | 0.2362 | 0.0665 | 0.1209 |\n| radiator | 0.5876 | 0.3435 | 0.6938 | 0.3621 | 0.6611 | 0.4215 |\n| glass | 0.2507 | 0.1983 | 0.2497 | 0.1725 | 0.1732 | 0.1312 |\n| clock | 0.3627 | 0.3053 | 0.5175 | 0.3060 | 0.4064 | 0.2080 |\n| flag | 0.7495 | 0.4274 | 0.5406 | 0.3913 | 0.5256 | 0.3048 |\n\nTable 26 and Table 27 show the Area Under Receiver Operating Characteristic (AUROC) on Cityscapes and ADE20K datasets across a few scenarios. All of the results use entropy as the uncertainty metric. Here also we observe that the best and worst performing classes are generally consistent across different scenarios, especially for Cityscapes.\n\n| Class | DRN D-105 Base | DRN D-105 Scale | Segformer B5 Base | Segformer B5 Scale | | | |\n|---------|------------------|-------------------|---------------------|----------------------|--------|--------|--------|\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. |\n\nTable 26: Classwise micro and macro averaged (µ and M respectively) Area Under Receiver Operating Characteristic (AUROC) on the Cityscapes dataset across a couple different models and settings. Higher is better. The top five classes for every model are highlighted in bold green, and the worst five classes are highlighted in red italics.\n\n| road | 0.9241 | 0.9783 | 0.9615 | 0.9861 | 0.9118 | 0.9693 | 0.9335 | 0.9801 |\n|---------------|----------|----------|----------|----------|----------|----------|----------|----------|\n| sidewalk | 0.8606 | 0.8756 | 0.8996 | 0.8742 | 0.8642 | 0.8826 | 0.8962 | 0.8795 |\n| building | 0.9235 | 0.9332 | 0.9349 | 0.9346 | 0.9148 | 0.9293 | 0.9280 | 0.9353 |\n| wall | 0.6513 | 0.4907 | 0.6980 | 0.4717 | 0.7152 | 0.5491 | 0.7167 | 0.5132 |\n| fence | 0.6946 | 0.5896 | 0.7400 | 0.5685 | 0.7387 | 0.6148 | 0.8026 | 0.5933 |\n| pole | 0.7529 | 0.7270 | 0.7550 | 0.7205 | 0.7565 | 0.7447 | 0.7680 | 0.7438 |\n| traffic light | 0.8339 | 0.7137 | 0.8336 | 0.6879 | 0.8376 | 0.7542 | 0.8491 | 0.7127 |\n| traffic sign | 0.8718 | 0.8388 | 0.8773 | 0.8174 | 0.8523 | 0.8506 | 0.8775 | 0.8292 |\n| vegetation | 0.9362 | 0.9366 | 0.9534 | 0.9453 | 0.9286 | 0.9341 | 0.9511 | 0.9484 |\n| terrain | 0.7133 | 0.5592 | 0.7487 | 0.5510 | 0.7117 | 0.5568 | 0.7751 | 0.5526 |\n| sky | 0.9681 | 0.9226 | 0.9708 | 0.9169 | 0.9660 | 0.9223 | 0.9628 | 0.9064 |\n| person | 0.9078 | 0.7937 | 0.9152 | 0.7751 | 0.8952 | 0.7961 | 0.9134 | 0.7807 |\n| rider | 0.7214 | 0.6861 | 0.7483 | 0.6808 | 0.8223 | 0.7011 | 0.8396 | 0.6993 |\n| car | 0.9672 | 0.9422 | 0.9727 | 0.9428 | 0.9464 | 0.9451 | 0.9654 | 0.9495 |\n| truck | 0.7554 | 0.4891 | 0.8498 | 0.4899 | 0.8737 | 0.5390 | 0.9379 | 0.5555 |\n| bus | 0.9031 | 0.7551 | 0.9349 | 0.7501 | 0.9203 | 0.7927 | 0.9631 | 0.7892 |\n| train | 0.7721 | 0.5658 | 0.7414 | 0.5000 | 0.8993 | 0.6744 | 0.9125 | 0.6772 |\n| motorcycle | 0.8312 | 0.5072 | 0.8179 | 0.5009 | 0.8516 | 0.5942 | 0.8607 | 0.5732 |\n| bicycle | 0.8795 | 0.7646 | 0.8984 | 0.7691 | 0.8828 | 0.7628 | 0.9021 | 0.7596 |\n\n| Class | OF ConvNeXt-L Base | OF ConvNeXt-L Scale | Segformer B5 Base | Segformer B5 Scale | | | | |\n|----------|----------------------|-----------------------|---------------------|----------------------|--------|--------|--------|--------|\n| µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | µ Avg. | M Avg. | |\n| wall | 0.5739 | 0.6680 | 0.5838 | 0.6911 | 0.8406 | 0.8069 | 0.8743 | 0.8182 |\n| building | 0.5608 | 0.6452 | 0.5531 | 0.6701 | 0.8300 | 0.8045 | 0.8605 | 0.8039 |\n| sky | 0.5670 | 0.7676 | 0.5786 | 0.7708 | 0.9233 | 0.9184 | 0.9339 | 0.9159 |\n| floor | 0.5547 | 0.7259 | 0.5663 | 0.7346 | 0.8556 | 0.8752 | 0.8674 | 0.8720 |\n| tree | 0.6203 | 0.7113 | 0.6428 | 0.7331 | 0.8665 | 0.7693 | 0.8745 | 0.7638 |\n| ceiling | 0.5189 | 0.7149 | 0.5099 | 0.7141 | 0.8375 | 0.8589 | 0.8665 | 0.8536 |\n| road | 0.4516 | 0.7259 | 0.4608 | 0.7366 | 0.8314 | 0.8193 | 0.8476 | 0.8192 |\n| bed | 0.7180 | 0.7278 | 0.6719 | 0.7245 | 0.9456 | 0.9118 | 0.9646 | 0.9123 |\n| window | 0.5302 | 0.6360 | 0.5263 | 0.6552 | 0.7547 | 0.6916 | 0.7667 | 0.6798 |\n| grass | 0.5200 | 0.6441 | 0.5906 | 0.6610 | 0.8606 | 0.7107 | 0.8761 | 0.7139 |\n| cabinet | 0.6989 | 0.5732 | 0.7458 | 0.5806 | 0.7957 | 0.5975 | 0.8219 | 0.5716 |\n| sidewalk | 0.5447 | 0.7314 | 0.5405 | 0.7414 | 0.7642 | 0.7200 | 0.8085 | 0.6995 |\n| person | 0.6048 | 0.7202 | 0.6161 | 0.7258 | 0.9108 | 0.7334 | 0.9062 | 0.7070 |\n| ground | 0.5967 | 0.4664 | 0.6169 | 0.4740 | 0.6065 | 0.4299 | 0.6027 | 0.4227 |\n| door | 0.6446 | 0.6255 | 0.7004 | 0.6337 | 0.6358 | 0.5255 | 0.6586 | 0.4951 |\n| table | 0.5518 | 0.6847 | 0.5349 | 0.6946 | 0.7932 | 0.6604 | 0.8172 | 0.6386 |\n| mountain | 0.7045 | 0.6829 | 0.6920 | 0.7049 | 0.8066 | 0.7067 | 0.8241 | 0.6949 |\n| plant | 0.5941 | 0.5722 | 0.5973 | 0.5666 | 0.7097 | 0.4936 | 0.7293 | 0.4755 |\n| curtain | 0.5007 | 0.7722 | 0.4830 | 0.7625 | 0.8088 | 0.8119 | 0.8318 | 0.8035 |\n| chair | 0.5607 | 0.6705 | 0.5258 | 0.6773 | 0.6841 | 0.6138 | 0.7018 | 0.5877 |\n| car | 0.6087 | 0.8017 | 0.6218 | 0.8141 | 0.9002 | 0.7805 | 0.9236 | 0.7674 |\n| water | 0.5340 | 0.4311 | 0.5104 | 0.4271 | 0.7816 | 0.5628 | 0.8001 | 0.5702 |\n| painting | 0.5304 | 0.7683 | 0.5556 | 0.7635 | 0.8534 | 0.7571 | 0.8708 | 0.7384 |\n| sofa | 0.6279 | 0.6968 | 0.6681 | 0.7309 | 0.8479 | 0.7624 | 0.8404 | 0.7457 |\n\nTable 27: Classwise micro and macro averaged (µ and M respectively) Area Under Receiver Operating Characteristic (AUROC) on the ADE20K dataset across a couple different models and settings. Higher is better. The top five classes for every model are highlighted in bold green, and the worst five classes are highlighted in red italics.\n\n| shelf | 0.6358 | 0.4825 | 0.6114 | 0.4893 | 0.7001 | 0.4210 | 0.7571 | 0.4163 |\n|----------------|----------|----------|----------|----------|----------|----------|----------|----------|\n| house | 0.4963 | 0.4237 | 0.5240 | 0.4215 | 0.5942 | 0.3930 | 0.5882 | 0.3653 |\n| sea | 0.8416 | 0.7356 | 0.8649 | 0.7661 | 0.5105 | 0.7815 | 0.6722 | 0.7845 |\n| mirror | 0.5035 | 0.6449 | 0.4964 | 0.6652 | 0.7857 | 0.6698 | 0.7806 | 0.6288 |\n| rug | 0.4027 | 0.5136 | 0.4490 | 0.5264 | 0.6194 | 0.6740 | 0.6000 | 0.6428 |\n| field | 0.6289 | 0.4627 | 0.6797 | 0.4573 | 0.7246 | 0.4810 | 0.7626 | 0.4843 |\n| armchair | 0.6309 | 0.6297 | 0.6719 | 0.6332 | 0.5639 | 0.5896 | 0.6109 | 0.5765 |\n| seat | 0.5390 | 0.5249 | 0.6303 | 0.5425 | 0.8414 | 0.5146 | 0.8518 | 0.5117 |\n| fence | 0.6256 | 0.4773 | 0.6257 | 0.4841 | 0.7178 | 0.3680 | 0.7237 | 0.3705 |\n| desk | 0.5602 | 0.6556 | 0.6061 | 0.6902 | 0.7862 | 0.6218 | 0.8023 | 0.6075 |\n| rock | 0.6126 | 0.4923 | 0.6155 | 0.5125 | 0.7262 | 0.5525 | 0.7262 | 0.5262 |\n| wardrobe | 0.6437 | 0.5709 | 0.7209 | 0.5765 | 0.7010 | 0.5167 | 0.7658 | 0.5172 |\n| lamp | 0.7204 | 0.7042 | 0.7100 | 0.7101 | 0.8199 | 0.6966 | 0.8165 | 0.6669 |\n| bathtub | 0.6990 | 0.7776 | 0.6818 | 0.8174 | 0.8999 | 0.8481 | 0.8828 | 0.8086 |\n| railing | 0.6131 | 0.4368 | 0.6146 | 0.4408 | 0.6652 | 0.3275 | 0.6494 | 0.3161 |\n| cushion | 0.4820 | 0.6654 | 0.4748 | 0.6584 | 0.6816 | 0.7281 | 0.7192 | 0.7051 |\n| pedestal | 0.6809 | 0.4428 | 0.6873 | 0.4416 | 0.5481 | 0.3212 | 0.5147 | 0.3059 |\n| box | 0.6084 | 0.3712 | 0.5978 | 0.3630 | 0.6921 | 0.2932 | 0.6581 | 0.2552 |\n| column | 0.3089 | 0.4293 | 0.4210 | 0.4016 | 0.6474 | 0.4266 | 0.5982 | 0.3984 |\n| sign | 0.5225 | 0.6147 | 0.5353 | 0.5977 | 0.6341 | 0.3961 | 0.5934 | 0.3425 |\n| chest | 0.4561 | 0.5929 | 0.4538 | 0.6150 | 0.5828 | 0.5883 | 0.6105 | 0.5485 |\n| counter | 0.5578 | 0.3881 | 0.6440 | 0.3824 | 0.6070 | 0.3743 | 0.6560 | 0.3476 |\n| sand | 0.7243 | 0.5635 | 0.7637 | 0.5650 | 0.7968 | 0.5653 | 0.8045 | 0.5613 |\n| sink | 0.5907 | 0.7345 | 0.5826 | 0.7294 | 0.8554 | 0.6905 | 0.8545 | 0.6610 |\n| skyscraper | 0.4182 | 0.6289 | 0.3830 | 0.5981 | 0.6903 | 0.7200 | 0.7077 | 0.7068 |\n| fireplace | 0.5034 | 0.7696 | 0.6717 | 0.8441 | 0.9120 | 0.8033 | 0.9060 | 0.7539 |\n| refrigerator | 0.8346 | 0.7137 | 0.8122 | 0.7293 | 0.8574 | 0.7904 | 0.8890 | 0.7767 |\n| grandstand | 0.8782 | 0.5326 | 0.7779 | 0.4291 | 0.8273 | 0.5412 | 0.8325 | 0.4975 |\n| path | 0.4835 | 0.3132 | 0.4781 | 0.3176 | 0.5924 | 0.3287 | 0.5574 | 0.3038 |\n| stairs | 0.5280 | 0.3520 | 0.5338 | 0.3616 | 0.7095 | 0.3098 | 0.5954 | 0.2824 |\n| runway | 0.2676 | 0.4812 | 0.3359 | 0.4756 | 0.9490 | 0.6891 | 0.9581 | 0.6515 |\n| case | 0.6746 | 0.5060 | 0.7389 | 0.5134 | 0.8008 | 0.6118 | 0.8432 | 0.6066 |\n| pool table | 0.7485 | 0.8723 | 0.7391 | 0.8845 | 0.9305 | 0.8488 | 0.9528 | 0.8458 |\n| pillow | 0.5753 | 0.7081 | 0.5782 | 0.7042 | 0.7406 | 0.6275 | 0.7278 | 0.5904 |\n| screen door | 0.3675 | 0.6923 | 0.3557 | 0.6841 | 0.6830 | 0.6751 | 0.6335 | 0.6286 |\n| stairway | 0.3363 | 0.3674 | 0.3572 | 0.3712 | 0.7469 | 0.2621 | 0.7850 | 0.2590 |\n| river | 0.4717 | 0.3615 | 0.4526 | 0.3756 | 0.3881 | 0.3104 | 0.3797 | 0.2734 |\n| bridge | 0.6281 | 0.6242 | 0.7111 | 0.6332 | 0.7987 | 0.6093 | 0.8159 | 0.6052 |\n| bookcase | 0.5989 | 0.6119 | 0.5601 | 0.5919 | 0.6138 | 0.5141 | 0.6564 | 0.5434 |\n| blind | 0.7091 | 0.3400 | 0.3707 | 0.3050 | 0.5208 | 0.3672 | 0.4756 | 0.3452 |\n| coffee table | 0.5752 | 0.6190 | 0.5177 | 0.6519 | 0.8910 | 0.6605 | 0.8744 | 0.6495 |\n| toilet | 0.7409 | 0.7570 | 0.7696 | 0.7848 | 0.9246 | 0.9070 | 0.9500 | 0.8863 |\n| flower | 0.6332 | 0.5608 | 0.6670 | 0.5665 | 0.6797 | 0.5289 | 0.6650 | 0.4774 |\n| book | 0.4242 | 0.5053 | 0.5302 | 0.5146 | 0.7241 | 0.4159 | 0.7423 | 0.3906 |\n| hill | 0.6818 | 0.2886 | 0.8445 | 0.3153 | 0.3824 | 0.1686 | 0.4030 | 0.1693 |\n| bench | 0.6265 | 0.5043 | 0.6425 | 0.4843 | 0.7980 | 0.3212 | 0.7693 | 0.2662 |\n| countertop | 0.3637 | 0.5977 | 0.4136 | 0.6233 | 0.7284 | 0.7519 | 0.7343 | 0.7309 |\n| stove | 0.4978 | 0.7285 | 0.6257 | 0.7418 | 0.9168 | 0.7212 | 0.9310 | 0.6986 |\n| palm | 0.6845 | 0.6145 | 0.6536 | 0.6336 | 0.6720 | 0.5673 | 0.7011 | 0.5387 |\n| kitchen island | 0.5199 | 0.4266 | 0.6521 | 0.4332 | 0.7225 | 0.6503 | 0.7809 | 0.6426 |\n| computer | 0.8235 | 0.5815 | 0.8232 | 0.5919 | 0.9081 | 0.7305 | 0.8766 | 0.7051 |\n| swivel chair | 0.5784 | 0.6860 | 0.6574 | 0.7054 | 0.6079 | 0.5702 | 0.6943 | 0.5242 |\n| boat | 0.8885 | 0.5479 | 0.9060 | 0.5646 | 0.7252 | 0.4369 | 0.7538 | 0.4301 |\n| bar | 0.6554 | 0.4605 | 0.6270 | 0.4680 | 0.7454 | 0.3915 | 0.7891 | 0.3932 | | arcade machine | 0.4072 | 0.5503 | 0.5758 | 0.6500 | 0.8680 | 0.7148 | 0.6931 | 0.6238 |\n|------------------|----------|----------|----------|----------|----------|----------|----------|----------|\n| hut | 0.5054 | 0.4863 | 0.4311 | 0.4959 | 0.6138 | 0.3689 | 0.6393 | 0.3997 |\n| bus | 0.3467 | 0.4428 | 0.3641 | 0.4371 | 0.9765 | 0.3753 | 0.9795 | 0.3688 |\n| towel | 0.7283 | 0.6239 | 0.7306 | 0.6176 | 0.8701 | 0.6178 | 0.8842 | 0.5967 |\n| light | 0.6130 | 0.7420 | 0.6070 | 0.7311 | 0.6760 | 0.4859 | 0.6491 | 0.4352 |\n| truck | 0.5780 | 0.4361 | 0.7299 | 0.4487 | 0.7291 | 0.2699 | 0.5865 | 0.2357 |\n| tower | 0.6639 | 0.2167 | 0.7170 | 0.2161 | 0.3517 | 0.3008 | 0.3304 | 0.2651 |\n| chandelier | 0.6425 | 0.7446 | 0.6891 | 0.7516 | 0.7727 | 0.7477 | 0.7470 | 0.7108 |\n| awning | 0.7975 | 0.4976 | 0.7861 | 0.4995 | 0.4719 | 0.3072 | 0.4494 | 0.2566 |\n| streetlight | 0.6850 | 0.6345 | 0.6393 | 0.6176 | 0.6430 | 0.2739 | 0.6165 | 0.2448 |\n| booth | 0.3981 | 0.2835 | 0.5152 | 0.2781 | 0.6733 | 0.4017 | 0.8289 | 0.4596 |\n| television | 0.4803 | 0.7019 | 0.5252 | 0.6495 | 0.9151 | 0.7664 | 0.9109 | 0.7065 |\n| airplane | 0.8537 | 0.6929 | 0.8063 | 0.6951 | 0.5502 | 0.6165 | 0.5442 | 0.5876 |\n| dirt track | 0.4029 | 0.0504 | 0.4449 | 0.0566 | 0.2451 | 0.1159 | 0.2267 | 0.1355 |\n| clothes | 0.8325 | 0.4714 | 0.7844 | 0.3999 | 0.8503 | 0.4640 | 0.8761 | 0.4604 |\n| pole | 0.5662 | 0.5002 | 0.6806 | 0.5169 | 0.5469 | 0.2786 | 0.5646 | 0.2776 |\n| land | 0.4633 | 0.1282 | 0.5031 | 0.1432 | 0.1227 | 0.0465 | 0.1449 | 0.0565 |\n| bannister | 0.3417 | 0.3438 | 0.3737 | 0.3525 | 0.4584 | 0.1728 | 0.3413 | 0.1535 |\n| escalator | 0.6141 | 0.4464 | 0.6299 | 0.3113 | 0.7641 | 0.5747 | 0.6754 | 0.5215 |\n| ottoman | 0.6320 | 0.4822 | 0.6135 | 0.4618 | 0.8452 | 0.4835 | 0.8102 | 0.4721 |\n| bottle | 0.6183 | 0.4295 | 0.5480 | 0.4040 | 0.6643 | 0.2820 | 0.5900 | 0.2440 |\n| buffet | 0.5471 | 0.3650 | 0.3454 | 0.3767 | 0.6931 | 0.2672 | 0.7233 | 0.2774 |\n| poster | 0.7482 | 0.3111 | 0.8322 | 0.2891 | 0.6782 | 0.2700 | 0.6315 | 0.2369 |\n| stage | 0.7343 | 0.4193 | 0.7956 | 0.4082 | 0.4942 | 0.4548 | 0.4788 | 0.4133 |\n| van | 0.4960 | 0.5453 | 0.5783 | 0.5172 | 0.7748 | 0.3088 | 0.7363 | 0.2792 |\n| ship | 0.7172 | 0.6277 | 0.7486 | 0.6264 | 0.9629 | 0.5954 | 0.9360 | 0.5856 |\n| fountain | 0.2365 | 0.6323 | 0.3632 | 0.6578 | 0.7300 | 0.4035 | 0.6050 | 0.3776 |\n| conveyer belt | 0.7016 | 0.7340 | 0.7865 | 0.7000 | 0.8565 | 0.6961 | 0.7608 | 0.6925 |\n| canopy | 0.6832 | 0.5018 | 0.7821 | 0.6180 | 0.7657 | 0.5034 | 0.7937 | 0.4958 |\n| washing machine | 0.9583 | 0.9192 | 0.9575 | 0.9109 | 0.9189 | 0.7947 | 0.9414 | 0.8013 |\n| toy | 0.4970 | 0.4003 | 0.5326 | 0.4043 | 0.5191 | 0.4129 | 0.4756 | 0.3660 |\n| swimming pool | 0.8311 | 0.8127 | 0.8263 | 0.7989 | 0.8580 | 0.8935 | 0.8550 | 0.8604 |\n| stool | 0.6112 | 0.4539 | 0.6849 | 0.4576 | 0.7468 | 0.3398 | 0.7656 | 0.3160 |\n| barrel | 0.8543 | 0.8331 | 0.8104 | 0.7544 | 0.9399 | 0.7760 | 0.9281 | 0.7556 |\n| basket | 0.6128 | 0.5168 | 0.5646 | 0.5100 | 0.7601 | 0.4798 | 0.6847 | 0.3967 |\n| waterfall | 0.6714 | 0.6821 | 0.7421 | 0.7442 | 0.8017 | 0.7390 | 0.8338 | 0.7768 |\n| tent | 0.6744 | 0.8092 | 0.8069 | 0.8461 | 0.9881 | 0.9836 | 0.9890 | 0.9885 |\n| bag | 0.5270 | 0.3574 | 0.5706 | 0.3445 | 0.5483 | 0.1569 | 0.5100 | 0.1343 |\n| motorbike | 0.7851 | 0.6201 | 0.7966 | 0.6435 | 0.9041 | 0.5442 | 0.8624 | 0.4961 |\n| cradle | 0.4388 | 0.7722 | 0.5862 | 0.8150 | 0.9401 | 0.9206 | 0.9374 | 0.9253 |\n| oven | 0.6352 | 0.4992 | 0.7394 | 0.4804 | 0.6788 | 0.4870 | 0.6820 | 0.4598 |\n| ball | 0.7694 | 0.4742 | 0.7446 | 0.4316 | 0.7972 | 0.2225 | 0.8621 | 0.2144 |\n| food | 0.6721 | 0.4196 | 0.6289 | 0.4441 | 0.6922 | 0.4929 | 0.6790 | 0.4987 |\n| step | 0.3730 | 0.1241 | 0.4539 | 0.1345 | 0.7090 | 0.1597 | 0.6085 | 0.1420 |\n| tank | 0.7262 | 0.4008 | 0.3890 | 0.3767 | 0.4738 | 0.3926 | 0.4686 | 0.3903 |\n| trade name | 0.6069 | 0.4242 | 0.6257 | 0.4519 | 0.3866 | 0.2330 | 0.3488 | 0.2042 |\n| microwave | 0.6897 | 0.6949 | 0.6350 | 0.6884 | 0.8810 | 0.6676 | 0.9135 | 0.6587 |\n| pot | 0.5060 | 0.5151 | 0.4797 | 0.5167 | 0.7731 | 0.4351 | 0.7853 | 0.3942 |\n| animal | 0.5093 | 0.2945 | 0.4863 | 0.2946 | 0.7762 | 0.5719 | 0.7497 | 0.5512 |\n| bicycle | 0.6692 | 0.5936 | 0.7153 | 0.5852 | 0.8083 | 0.3472 | 0.8002 | 0.3154 |\n| lake | 0.0234 | 0.0002 | 0.0860 | 0.0001 | 0.7880 | 0.1997 | 0.8605 | 0.1997 |\n| dishwasher | 0.6327 | 0.6098 | 0.8802 | 0.6309 | 0.9222 | 0.6692 | 0.8628 | 0.6224 |\n| screen | 0.8307 | 0.4992 | 0.7236 | 0.5010 | 0.8256 | 0.6681 | 0.7968 | 0.6055 |\n| blanket | 0.3294 | 0.3658 | 0.3940 | 0.3488 | 0.4154 | 0.2845 | 0.3097 | 0.1913 |\n\n| sculpture | 0.5847 | 0.3240 | 0.5768 | 0.3206 | 0.8343 | 0.2758 | 0.8373 | 0.2807 |\n|----------------|----------|----------|----------|----------|----------|----------|----------|----------|\n| hood | 0.5973 | 0.6423 | 0.7049 | 0.6036 | 0.5808 | 0.6882 | 0.5094 | 0.5935 |\n| sconce | 0.7347 | 0.5145 | 0.7293 | 0.5030 | 0.6491 | 0.4032 | 0.6226 | 0.3479 |\n| vase | 0.6813 | 0.6166 | 0.6540 | 0.5930 | 0.7775 | 0.4159 | 0.7499 | 0.3745 |\n| traffic light | 0.5853 | 0.6974 | 0.6836 | 0.6796 | 0.6773 | 0.3443 | 0.6240 | 0.3111 |\n| tray | 0.6858 | 0.3117 | 0.6344 | 0.3108 | 0.5955 | 0.2606 | 0.5262 | 0.2263 |\n| ashcan | 0.7771 | 0.4834 | 0.7593 | 0.4636 | 0.8191 | 0.3523 | 0.8039 | 0.3130 |\n| fan | 0.6371 | 0.7163 | 0.6260 | 0.6670 | 0.7506 | 0.5828 | 0.7302 | 0.5493 |\n| pier | 0.8935 | 0.3398 | 0.8430 | 0.3722 | 0.9237 | 0.4679 | 0.8972 | 0.4443 |\n| crt screen | 0.4179 | 0.1100 | 0.5170 | 0.1263 | 0.3179 | 0.1178 | 0.2905 | 0.1186 |\n| plate | 0.6950 | 0.5008 | 0.7655 | 0.4871 | 0.8537 | 0.3461 | 0.8517 | 0.3406 |\n| monitor | 0.7184 | 0.3013 | 0.7747 | 0.2437 | 0.1745 | 0.2068 | 0.1027 | 0.1705 |\n| bulletin board | 0.8242 | 0.4924 | 0.7822 | 0.4840 | 0.8263 | 0.4846 | 0.6846 | 0.4034 |\n| shower | 0.3092 | 0.4837 | 0.5107 | 0.4570 | 0.1334 | 0.2059 | 0.1273 | 0.1660 |\n| radiator | 0.7408 | 0.5331 | 0.7187 | 0.5401 | 0.8359 | 0.6907 | 0.8150 | 0.6724 |\n| glass | 0.5036 | 0.5833 | 0.5382 | 0.5610 | 0.4016 | 0.2441 | 0.4431 | 0.2217 |\n| clock | 0.4518 | 0.4883 | 0.3102 | 0.4624 | 0.6924 | 0.3115 | 0.6216 | 0.2572 |\n| flag | 0.5878 | 0.6707 | 0.5870 | 0.6537 | 0.6334 | 0.3984 | 0.5834 | 0.3099 |\n\n## F Qualitative Results On Ade20K\n\nWe show a few qualitative results on the ADE20K dataset for a couple of best and worst performing classes\n\n\n\nin Figure 5 and Figure 6, respectively. The figures show the original image, the class label being looked at, misclassified pixels, entropy, thresholded entropy values (using the largest difference technique), and the detection performance (green areas show correctly identified misclassified pixels, blue areas are false positives and red areas are false negatives). All of these results are from the Base setting of OneFormer ConvNeXt-L\nmodel.\n\nFigure 5: A few images showing a couple of best performing classes in the ADE20K dataset. The first two images include the washing machine class while the last two include the tent class. For each image, we show from left to right, the image, the highlighted class, misclassified pixels, entropy, thresholded entropy, detection mask.\n\n\n\nFigure 6: A few images showing a couple of worst performing classes in the ADE20K dataset. The first two images include dirt track class while the last two include the lake class. For each image, we show from left to right, the image, the highlighted class, misclassified pixels, entropy, thresholded entropy, detection mask."},"reviews":{"kind":"string","value":""}}},{"rowIdx":95,"cells":{"paper":{"kind":"string","value":"# Partial Optimal Transport For Support Subset Selection\n\nBilal Riaz bilalria@udel.edu Department of Electrical & Computer Engineering University of Delaware Yüksel Karahan ykarahan@udel.edu Department of Electrical & Computer Engineering University of Delaware Austin J. Brockmeier ajbrock@udel.edu Department of Electrical & Computer Engineering Department of Computer & Information Sciences University of Delaware Reviewed on OpenReview: *https: // openreview. net/ forum? id= 75CcopPxIr*\n\n## Abstract\n\nIn probabilistic terms, optimal transport aims to find a joint distribution that couples two distributions and minimizes the cost of transforming one distribution to another. Any feasible coupling necessarily maintains the support of both distributions. However, maintaining the entire support is not ideal when only a subset of one of the distributions, namely the source, is assumed to align with the other target distribution. For these cases, which are common in machine learning applications, we study the semi-relaxed partial optimal transport problem that relaxes the constraints on the joint distribution allowing it to under-represent a subset of the source by over-representing other subsets of the source by a constant factor. In the discrete distribution case, such as in the case of two samples from continuous random variables, optimal transport with the relaxed constraints is a linear program. When sufficiently relaxed, the solution has a source marginal with only a subset of its original support.\n\nWe investigate the scaling path of solutions, specifically the relaxed marginal distribution for the source, across different relaxations and show that it is distinct from the solutions from penalty-based semi-relaxed unbalanced optimal transport problems and fully-relaxed partial optimal transport, which have previously been explored. We demonstrate the usefulness of this support subset selection in applications such as color transfer, partial point cloud alignment, and semisupervised machine learning, where a part of data is curated to have reliable labels and another part is unlabeled or has unreliable labels. Our experiments show that optimal transport under the relaxed constraint can improve the performance of these applications by allowing for more flexible alignment between distributions.\n\n## 1 Introduction\n\nMeasuring, and subsequently minimizing, the dissimilarity between two distributions or data samples are ubiquitous tasks in machine learning. Recently, the theory of optimal transport and the family of Wasserstein distances have seen applications across the spectrum of machine learning problems including computer vision\n(Solomon et al., 2014; Kolouri et al., 2017; Rabin et al., 2011; Garg et al., 2020), generative modeling (Arjovsky et al., 2017; Gulrajani et al., 2017; Salimans et al., 2018; Genevay et al., 2018; Tolstikhin et al.,\n2018; Kolouri et al., 2018; Deshpande et al., 2018; Rout et al., 2022; Korotin et al., 2023; Mokrov et al.,\n2021), natural language processing (Xu et al., 2018), and domain adaptation (Kirchmeyer et al., 2022).\n\nOptimal transport is widely applicable since it combines the statistical and geometric aspects of data and provides a correspondence to couple two samples or distributions.\n\nHowever, a limitation of standard optimal transport is the strict constraint on the complete transfer of mass between the two distributions being compared (Frogner et al., 2015). This can be problematic in cases where such a transfer is not necessary or desirable, such as when dealing with distributions with different support, over- or under-representation, or the presence of outliers in a portion of the data. In order to deal with these scenarios partial optimal transport problems (Rubner et al., 1997; Figalli, 2010; Caffarelli & McCann, 2010; Bonneel & Coeurjolly, 2019; Chapel et al., 2020) have been proposed. Partial optimal transport relaxes the marginal constraints on the transport plan to inequalities, allowing transportation plans that cover only a fraction of the total mass. Similarly, unbalanced optimal transport corresponds to the case of unbalanced masses, where only a portion of the mass is transported. One or both of the marginal constraints can also be relaxed by divergence-based regularizations, such as Kullback-Leibler divergence, total-variation distance (equivalent to the ℓ1-norm of the difference in mass vectors in the discrete case), or squared ℓ2norm (Benamou, 2003; Chizat et al., 2018; Blondel et al., 2018; Peyré & Cuturi, 2019; Séjourné et al., 2019; Chapel et al., 2021). The solution of partial or unbalanced optimal transport often selects a subset (also known as an active region) of the support. This property is exploited in machine learning (Chapel et al.,\n2020; 2021; Phatak et al., 2023), including the partial Wasserstein covering problem, which has applications in active learning (Kawano et al., 2022). In practice, a key limiting factor is the scalability of solving the linear programs corresponding to partial, unbalanced, or standard optimal transport for large data sets. While efficient gradient based methods cannot be applied to linear programs directly, a number of regularization based remedies, including entropic (Cuturi, 2013; Cuturi & Peyré, 2018), quadratic, and group-LASSO regularization (Flamary et al., 2016; Blondel et al., 2018) have been shown to give approximate solutions. In particular, the Sinkhorn algorithm provides a solution to the entropically regularized standard optimal transport problem. The Sinkhorn algorithm has been widely applied due to its simple implementation consisting of alternating projections to the feasible sets of marginal constraints. In the work by Chizat et al. (2018), Dykstra's algorithm1is used to solve the entropically regularized partial optimal transport problem. Similarly, Sinkhorn-like iterations are used for entropically regularized unbalanced optimal transport in the work by Frogner et al. (2015), and the work by Séjourné et al. (2019) adapts the Sinkhorn algorithm via asymmetric proximal operators to handle a variety of divergence-based relaxations. Scalability is even more important in cases where the solution at different levels of relaxation are sought. Recent works have proposed algorithms for computing the entire scaling or regularization path of solution for fully-relaxed partial optimal transport (Phatak et al., 2023)\nand unbalanced optimal transport problems with fully or semi-relaxed marginal constraints (Chapel et al.,\n2021).\n\nIn this paper, we study a discrete case of partial optimal transport (Figalli, 2010; Caffarelli & McCann, 2010),\nwhich is posed as a linear program, where the solution is a joint distribution with one fixed marginal and one marginal that is constrained to be pointwise less than or equal to a constant factor c ≥ 1 of the original, typically uniform, marginal. The generalized form of this constraint where each point has its own capacity factor was proposed in the work by Rabin et al. (2014). Initially, we propose an entropically regularized version to efficiently find an approximate solution, which is equivalent to a specific case within the framework covered in the work of Séjourné et al. (2019), that we solve with an algorithm that combines Sinkhorn-like projections with an accelerated proximal gradient method.\n\nAs the regularized solution is not sparse, and sparsity of transport map is essential for support subset selection, we adopt an inexact Bregman proximal point method (Xie et al., 2020) to yield solutions closer to the original, unregularized, linear program. The work by Xie et al. (2020) is based on the observation that solving the entropically regularized optimal transport problem is equivalent to a Bregman proximal point evaluation with Kullback-Leilber divergence as proximal function. As a proximal point method that uses exact proximal point evaluations is computationally expensive, an inexact proximal point evaluation, where only a few inner Sinkhorn iterations are used, is used to make the algorithm efficient. In our case, the computational complexity is on the same order as the accelerated proximal gradient method, but it returns solution much closer to the linear program for the semi-relaxed partial optimal transport problem. While other regularization-based methods may induce sparse support, the regularized solution will generally be distinct from the linear program's solutions.\n\n1Dykstra's algorithm can find solutions at the intersection of convex, not necessarily affine, sets.\nThe contributions of this paper are the following: - We motivate and study support susbset selection (SS) a specific formulation of a semi-relaxed partial optimal transport problem for selecting a subset of a source distribution, parameterized in terms of a single scalar c ≥ 1 for a fixed target distribution.\n\n- We study the solutions obtained along the scaling path for various values of c, and compare the solution path to fully-relaxed partial and divergence-based semi-relaxed unbalanced optimal transport problems.\n\n- We develop an accelerated proximal gradient method-based algorithm to solve the entropically regularized version and adapt the inexact Bregman proximal method-based approach for optimal transport (POT) (Xie et al., 2020) to mitigate the effects of entropic regularization detailing an algorithm SS-Bregman that yields solutions close to the linear program formulation for subset selection and is as scalable as the Sinkhorn algorithm.\n\n- We apply SS-Bregman to applications including color adaptation, partial distribution alignment, partial point cloud registration problems, and positive-unlabeled learning (Bekker & Davis, 2020).\n\n- We incorporate the subset selection-based approach into a semi-supervised loss function for training a neural network-based classifier, which computes the optimal transport based on the learning representation.\n\n## 2 Methodology\n\nIn Section 2.1, relevant preliminaries related to discrete optimal transport along with formulation of subset selection problem are discussed. In Section 2.3, the entropically regularized support subset selection problem and an algorithm to solve it are discussed. In Section 2.4, an inexact Bregman proximal point method to better approximate the solution of the unregularized support selection problem is detailed.\n\nNotation: The set of the first n natural numbers {1, 2*, . . . , n*}, is denoted by [n]. The set of integers is denoted by Z. The ceiling function defined on real numbers x ∈ R is ⌈x⌉ = min{n ∈ Z : n ≥ x}. The floor function defined on real numbers x ∈ R is ⌊x⌋ = max{n ∈ Z : n ≤ x}. The n-dimensional real vector space is denoted by R\nn. Vectors are typeset in lowercase bold (x); matrices are in uppercase bold (X); and bold is dropped when an element are referenced by subscripts (xi, Xij ). When needed for clarity, elements will be referenced by subscripts on square brackets ([x1]i, [X2]ij ). The set of non-negative vectors in R\nn, known as the non-negative orthant, is denoted by R\nn\n+. The n-dimensional vector with all elements equal to unity is denoted by 1n and the m-by-n matrix with all unity elements is denoted by 1m×n. For vectors and matrices, the symbol ≼ denotes element-wise less than or equal to, and ≽ denotes element-wise greater than or equal to. The set denoted by ∆n = {x ∈ R\nn\n+ :Pn i=1 xi = 1} is the probability simplex. The element-wise product for vectors and matrices is denoted by the ⊙ symbol. The element-wise division for vectors and matrices is denoted by the ⊘ symbol. The diagonal operator is a matrix valued map D : R\nn → R\nn×n, such that [D(x)]ii = xi ∀ i ∈ [n] and [D(x)]ij = 0 ∀ i ̸= j ∈ [n]. For x ∈ R\nn, the ℓ1, ℓ2 and ℓ∞\nnorms are given by ∥x∥1 =Pi |xi|, ∥x∥2 = (Pi |xi| 2)\n1 2 and ∥x∥∞ = max i|xi|, respectively. Both, the Euclidean inner-product for x, y ∈ R\nn given by Pi xiyi, and the Frobenius inner-product for X,Y ∈ R\nm×n given by Pm i=1 Pn j=1 XijYij , are denoted by ⟨·, ·⟩. The element-wise exponent of a vector or a matrix is denoted by exp(·) and the element-wise logarithm of a vector or a matrix is denoted by log(·). For a matrix X ∈ R\nm×n, its non-negative part is represented by X+ or [X]+, with matrix components given by -X+\nij = max{Xij , 0} ∀ i ∈ [m], j ∈ [n]. Similarly, the non-positive part of matrix X ∈ R\nm×n is denoted by X− or [X]− and contains matrix elements -X−\nij = min{Xij , 0} ∀ i ∈ [m] and j ∈ [n]. For a vector x ∈ R\nn, both x+ and x− denote the non-negative and non-positive parts respectively. We denote the indicator function of the singleton set {z} as δz(x) = (1, x = z 0, x ̸= z\n. The set of vectors {ei}\nn i=1 form the standard basis for R\nn, where [ei]i = 1 and [ei]j = 0 for i ̸= j.\n\n## 2.1 Problem Formulation\n\nWe consider the discrete optimal transport between two weighted samples of size m and n corresponding to random variables X ∼ µ defined on {x\n(i)}\nm i=1 ⊂ R\nd and Y ∼ ν defined on {y\n(j)}\nn j=1 ⊂ R\nd, with probability measures µ =Pm i=1 µiδx(i) and ν =Pn j=1 νj δy(j) for probability masses µ ∈ ∆m (with {µi = µ(x\n(i))}\nm i=1)\nand ν ∈ ∆n (with {νj = ν(y\n(j))}\nn j=1), respectively. Let d : R\nd × R\nd → R+ denote a distance function. In practice, this is often the Euclidean distance metric between the points d(x, y) = ∥x−y∥2. Given 1 ≤ p ≤ ∞,\nthe p-Wasserstein distance (to the p-power) between µ and ν is expressed in terms of the cost matrix M,\nwhere Mij = dp(x\n(i), y\n(j)) ∀i ∈ [m], j ∈ [n] is the cost associated with transporting x\n(i)to y\n(j), as\n\n$${\\mathcal W}_{p}^{p}(\\mu,\\nu):=\\operatorname*{min}_{P\\succcurlyeq0}\\;\\;\\langle P,M\\rangle\\quad\\mathrm{s.t.}\\quad P{\\mathbf1}_{n}=\\mu,\\;P^{\\top}{\\mathbf1}_{m}=\\nu,$$\n$$(1)$$\n$$\\left(2\\right)$$\n⊤1m = ν, (1)\nwhere P is the transport map and 1\n⊤\nmP 1n = 1\n⊤\nmµ = ν\n⊤1n = 1. The constraints ensure that any solution P\n∗\nis a joint distribution that couples the target marginal µ and the source marginal ν. (In the computational optimal transport literature, µ is referred to as the target and ν as the source.) Therefore, any Wasserstein distance requires the complete mass transfer between a fixed source and target.\n\nIn partial optimal transport (Figalli, 2010), the marginal equality constraints are replaced by the inequalities\nP 1n ≼ µ, P\n⊤1m ≼ ν, and the equality 1\n⊤\nmP 1n = s; µ ∈ R\nm\n+ and ν ∈ R\nn\n+ may not have equal mass; and\nthe transport map need only transport a fraction of the total mass s ∈ [0, min{∥µ∥1, ∥ν∥1}].\nMotivated by machine learning scenarios with a trusted target sample of data and an additional source\nof data which cannot be assumed to be of uniform quality, we focus on the semi-relaxed case, where the\nconstraint on the target is fixed P 1n = µ with ∥µ∥1 = 1, ensuring the total mass constraint, but relax the\nconstraint on the source, allowing mass to redistribute among the source points ν\n∗ = P\n⊤1m ≤ cν, where\nc ≥ 1 is a scaling factor and ∥ν∥1 = 1. The resulting partial optimal transport problem,2 which we refer to\nas subset selection (SS), is\n$$\\operatorname*{min}_{\\mathbf{P}\\succcurlyeq0}\\;\\;\\langle\\mathbf{P},\\mathbf{M}\\rangle\\quad\\mathrm{s.t.}\\quad\\mathbf{P1}_{n}=\\mathbf{\\mu},\\ \\mathbf{P}^{\\top}\\mathbf{1}_{m}\\preccurlyeq\\mathbf{\\nu}.$$\n⊤1m ≼ cν. (2)\nLet P\n∗\nc denote an optimal solution, then the source's new mass is ν\n∗\nc = P\n∗⊤\nc 1m. Since 1\n⊤\nmµ = 1\n⊤\nmP\n∗\nc 1n = 1,\n∥ν\n∗\nc ∥1 = 1. Intuitively, this problem allows the new mass of some source points that have relatively lower cost to increase by a factor of c of the original mass, which enables higher cost source points to have less or even zero mass. In other words, due to total unit mass constraint, the mass increment at one source point results in its decrement at other source points. The subset of the source points selected is supp(ν\n∗\nc\n), where supp(·) indicates the support of a vector, i.e., the indices of the points with non-zero mass. To explore the relaxed constraint set, we consider the case of a uniformly distributed mass ν =\n1 n 1n and express the constraint as P\n⊤1m ≼\n1 L\n1n, where 0 < L ≤ n and c =\nn L\n. For a fixed value of L, the set of feasible source marginal distributions form a polyhedral set Ξ\n(L)\nn ⊆ ∆n bounded by linear inequalities parameterized by L. The set of feasible source marginals Ξ\n(L)\nn is defined as Ξ\n(L)\nn = {x ∈ ∆n : x ≼\n1 L\n1n}.\n\nThe set Ξ\n(L)\n3for different values of L are given in Figure 1. Using combinatoric reasoning we deduce the number of vertices of Ξ\n(L)\nn defining the feasible set for the source marginals.\n\nRemark 1. For the support selection partial optimal transport problem 2 with n > 2 and ν =\n1 n 1n, by defining L =\nn c\n, the inequality constraint can be written as P\n⊤1m ≼\n1 L\n1n*. Extreme points of* Ξ\n(L)\nn *can be* characterized as follows:\n- For 0 < L ≤ 1, the entire probability simplex ∆n is feasible due to the fact that in this case the vertices of the probability simplex correspond to extreme points of feasible set.\n\n- For 1 < L < 2 and n > 2*, the feasible set* Ξ\n(L)\nn has n(n − 1) *number of vertices, which can can be written*\n\nas as convex combination 1L\nei + (1 −\n1 L\n)ej , where i, j ∈ [n] and i ̸= j.\n\n- For L = 2*, the feasible set* Ξ\n(2)\nn has n(n−1)\n2*vertices given as* 12\n(ei + ej ) for i ̸= j.\n2An equivalent set of constraints are P 1n ≼ µ, P ⊤1m ≼ cν, 1⊤mP 1n = 1.\n- *More generally, the number of extreme points of the feasible set* Ξ\n(L)\nn is\n\n$${\\frac{n!}{\\lfloor L\\rfloor!\\lceil1-{\\frac{\\lfloor L\\rfloor}{L}}\\rceil!(n-\\lfloor L\\rfloor-\\lceil1-{\\frac{\\lfloor L\\rfloor}{L}}\\rceil)!}},$$\n\nwith the vertices given as the set of possible multi-set permutations of the vector:\n\nz }| { 0 0 · · · 0 i⊤=- ⌊ n c ⌋ terms z }| { c ncn· · ·cn1 − c n ⌊ n c ⌋ n − 1 − ⌊ n c ⌋ terms z }| { 0 0 · · · 0⊤. (3) h ⌊L⌋ terms z }| { 1 L1L· · ·1L1− ⌊L⌋ L n − 1 − ⌊L⌋ terms\n$$\\mathbf{\\Sigma}$$\n$$\\mathbf{\\Sigma}$$\nThus, when L ≥ 2 *is an integer, which corresponds to* c ≤\nn 2 and 1 c mod 1n = 0 the vertices correspond to uniform distribution of mass across a support of cardinality L =\nn c\n. However, due to the equality constraints on the other marginal the optimal solution may be a convex combination of these vertices. Since the amount of mass to be transported is constant, when the scaling parameter c is increased (or equivalently, when the value of L is decreased) starting from c = 1, the behavior of the coupling P\n∗\nc\n,\nand the marginal ν\n∗\nc = (P\n∗\nc\n)\n⊤1n in the transportation problem is affected. This behavior depends on the structure of the cost matrix M and leads to a redistribution of mass, assigning more mass to certain points and less to others. At c = 1, corresponding to standard optimal transport, all source constraints are active, meaning that all constraints in the problem are considered. As the value of c is increased, constraints become inactive, which constraints depends on the structure of the cost matrix M and the distributions. Once a point's mass goes to zero it never re-enters the support for larger values of c. These observations follow from the linear nature of the problem. Analogous solution paths have been studied for fully-relaxed optimal transport (Phatak et al., 2023) and fully or semi-relaxed regularized unbalanced optimal transport (Chapel et al., 2021). Once c reaches c\n∗, all inequality constraints are inactive and can be discarded. After this breakpoint c\n∗, any further increments in c do not affect the resulting transport plans. In other words, for c ≥ c\n∗, the transport plan P\n∗\ncremains the same as P\n∗\nc\n∗ , which is a solution that can found by a greedy algorithm. The exact value of c\n∗can be determined analytically, based on the possible greedy solutions. The analytical expression for c\n∗ depends on the properties of the cost matrix M and the constraints involved,\n\n$$c^{*}=\\max_{j\\in[n]}\\frac{\\sum_{i}Q_{ij}}{\\nu_{j}},\\tag{1}$$\nwhere the matrix Q is found by nearest neighbor search,\n$$Q_{ij}=\\begin{cases}\\mu_{i},&j\\in\\arg\\min_{k\\in[n]}M_{ik}\\\\ 0,&\\text{otherwise}\\end{cases},\\quad i\\in[m],j\\in[n].\\tag{1}$$\n\nP\n∗\nc\n∗ is a greedy solution with non-zero entries taken from Q by taking one non-zero element in each row, breaking ties arbitrarily. Thus, as c varies from 1 to c\n∗, SS varies from standard optimal transport to a nearest-neighbor transport.\n\nFor more flexibility, a designer can provide an upper-bound on the mass assignments to source points by ζ ∈ R\nn\n+ with ∥ζ∥1 ≥ 1 (equality corresponds to standard optimal transport), which adds the flexibility in designing partial optimal transport problems that allow variable ranges of masses for the source points. The upper-bound can be expressed in terms of a capacity vector κ ∈ R\nn\n+ such that ζ = κ⊙ν as introduced in the work by Rabin et al. (2014). For target distribution µ and source upper-bounding measure ζ =Pm j=1 ζj δy(j) ,\nwhich is not necessarily a probability measure, the support subset selection problem can be stated as\n\n$$\\mathcal{S}_{p}(\\mu,\\zeta):=\\min_{\\mathbf{P}\\geq0}\\quad\\langle\\mathbf{P},\\mathbf{M}\\rangle\\quad\\text{s.t.}\\quad\\mathbf{P1}_{n}=\\mu,\\ \\mathbf{P}^{\\top}\\mathbf{1}_{m}\\prec\\zeta,\\tag{1}$$\n\nand related to the p-Wasserstein distance (to the p-power) by Sp(*µ, ζ*)\f\fζ=cν\n$$\\left.\\zeta\\right)\\right|_{\\zeta=c\\nu}=\\mathcal{S}_{p}(\\mu,c\\nu)\\leq\\mathcal{W}_{p}^{p}(\\mu,\\nu)\\;\\mathrm{for}\\;$$\n\n## (*Μ, Ν*) For C ≥ 1. 2.2 Relation To Prior Work\n\nWhile we consider a purely linear program consisting of a semi-relaxed partial optimal transport, prior work includes fully-relaxed partial optimal transport and a variety of non-linear approaches for fully or semi-relaxed partial and unbalanced optimal transport.\n\n$$(6)$$\n\n\n\nFigure 1: The feasible set Ξ\n(L)\n3of the source's new marginal distribution given a uniform distribution ν for different values of L. For 0 < L ≤ 1 the whole probability simplex is feasible. (a) L = 1.3, six vertices will exist for 1 *< L <* 2 . (b) L = 2 yields 3 vertices. (c) L = 2.2, 2 *< L <* 3 also give 3 vertices. (d) L = 3 yields a singleton set corresponding to the original uniform distribution.\n\n## 2.2.1 Divergence-Based Semi-Relaxed Partial And Unbalanced Optimal Transport\n\nMany relaxed approaches for unbalanced and partial optimal transport penalize the divergence of the marginal from the source and/or target distribution (Rabin et al., 2014; Frogner et al., 2015; Chizat et al.,\n2018; Séjourné et al., 2019). Many of these also use entropic regularization for scalable algorithms since entropic regularization of the joint distribution defining the transport plan can be cast as the Kullback-Leibler divergence between the joint and the product of the marginals. The works by Chizat et al. (2018) and Séjourné et al. (2019) provides an extensive framework for entropic regularization and divergence-based relaxations. The latter work (Séjourné et al., 2019) mentions the adjustments necessary to perform asymmetric marginal penalty case for the semi-relaxed case. The Kullback-Leibler divergence is a member of the wider family of f-divergences that have been adopted in relaxed and semi-relaxed optimal transport (Chizat et al., 2018; Séjourné et al., 2019). In the discrete distribution case p, q ∈ ∆n, these divergences can be expressed as Dφ(p∥q) := Pn i=1 qiφ( pi qi\n), where φ is the generating function of the divergence. Notably, φ(r) = KL(r) := r log r − r + 1 is the generating function for the KL divergence, and φ(r) = TV(r) := 12 |r − 1| is the generating function for total variation, which in the discrete case is equivalent to an ℓ1-norm based distance: DTV(p∥q) = 12 Pn i=1 qi| pi qi\n− 1| =\n1 2\n∥p − q∥1.\n\nThe family of semi-relaxed optimal transport problems with divergence penalty 1ρ > 0 on the marginal is\n\n$${\\mathcal{S R W}}_{\\varphi}^{(\\rho)}(\\mu,\\nu):=\\operatorname*{min}_{\\mathbf{P}\\nearrow0}\\quad\\langle\\mathbf{P},\\mathbf{M}\\rangle+\\frac{1}{\\rho}{\\mathcal{D}}_{\\varphi}(\\mathbf{P}^{\\top}\\mathbf{1}_{m}\\|\\ \\nu)\\quad\\mathrm{s.t.}\\quad\\mathbf{P1}_{n}=\\mu.$$\n⊤1m∥ νs.t. P 1n = µ. (7)\nThe choice φ(r) = ı[0,c](r) := (0, r ∈ [0, c],\n∞, otherwise, yields a range constraint to the interval of [0, c] for the ratio of the source marginal (Chizat et al., 2018; Séjourné et al., 2019), which is equivalent to Sp(*µ, cν*) for any value of ρ and c ≥ 1: SRWı[0,c]\n(*µ, ν*) = S\np p\n(*µ, cν*).\n\nMore generally, one can consider penalties which are not f-divergences such as those based on ℓ∞-norm Dℓ∞(P\n⊤1m∥ ν:= ∥(P\n⊤1m)⊘ν−1∥∞ and the squared ℓ2-norm Dℓ2\n(P\n⊤1m∥ν:= ∥P\n⊤1m−ν∥\n2 2\n(Benamou, 2003; Blondel et al., 2018; Chapel et al., 2021). The work by Chapel et al. (2021) details a regularization path algorithm for finding the breakpoints in terms of ρ of the piecewise linear related solutions along the path, and efficiently calculating the solutions via rank-1 updates of matrix inverse required to solve the sequence of non-negative, penalized linear regression problems.\n\nSolutions to all of the penalty forms can also be found via equivalent constraint-based optimizations with the constraint Dφ P\n⊤1m∥ ν≤ a, where a ≥ 0. In particular, for c ≥ 2, Sp(*µ, cν*), the SS problem 2, is\n\n(7) $\\frac{1}{2}$\nequivalent to the constraint-based optimization with the ℓ∞-norm based penalty by letting a = c − 1, since\n\nment to the constraint-based optimization with the $\\varepsilon_{\\infty}$-norm based penalty by setting $u=c-1$, $$\\|(\\boldsymbol{P}^{\\top}\\boldsymbol{1}_{m})\\otimes\\nu-\\boldsymbol{1}_{n}\\|_{\\infty}\\leq a\\implies\\forall j\\in[n],\\quad-av_{j}\\leq[\\boldsymbol{P}^{\\top}\\boldsymbol{1}_{m}]_{j}-\\nu_{j}\\leq av_{j}$$ $$\\implies\\forall j\\in[n],\\quad(1-a)\\nu_{j}\\leq[\\boldsymbol{P}^{\\top}\\boldsymbol{1}_{m}]_{j}\\leq(a+1)\\nu_{j}\\implies(2-c)\\nu\\leq\\boldsymbol{P}^{\\top}\\boldsymbol{1}_{m}\\ll\\alpha,$$ the above decision inequality for $\\varepsilon>0$. However, for $1$, the set $\\mathcal{D}$, for the choice of $\\mathcal{D}$\nwhere the lower bound is non-positive for c ≥ 2. However, for 1 *< c <* 2 or for other choices of Dφ the feasible set of the marginal differs from the linear program.\n\nThe work by Rabin et al. (2014) presents a semi-relaxed optimal transport that combines an additional ℓ1-norm penalty on the capacity's deviation with an inequality constraint on the marginal, which ensures the problem in terms of the transport plan P stays linear,\n\n⟨P ,M⟩ + 1 ρ ∥κ − 1n∥1 ⟨P ,M⟩ + 1 ρ ∥ζ ⊘ ν − 1n∥1 min P ≽0 κ∈R n + ζ=κ⊙ν = min P ≽0 ζ∈R n + s.t. P 1n = µ, P ⊤1m ≼ κ ⊙ ν, ⟨κ, ν⟩ ≥ 1 s.t. P 1n = µ, P ⊤1m ≼ ζ, ζ ⊤1n ≥ 1.\n$$({\\boldsymbol{\\delta}})$$\nFor a given ρ, there is a value of a ≥ 0 such that the constraint-based optimization problem\n\nthere is a value of $a\\geq0$ such that the constraint based optimization problem is $$\\min_{\\begin{subarray}{c}\\mathbf{P}^{n}\\geq0\\\\ \\mathbf{\\zeta}\\in\\mathbb{R}_{+}^{n}\\end{subarray}}\\quad(\\mathbf{P},\\mathbf{M})\\quad\\text{s.t.}\\quad\\mathbf{P1}_{n}=\\mathbf{\\mu},\\ \\mathbf{P}^{\\top}\\mathbf{1}_{m}\\preccurlyeq\\mathbf{\\zeta},\\ \\zeta^{\\top}\\mathbf{1}_{n}\\geq1,\\ \\|\\mathbf{\\zeta}\\oslash\\nu-\\mathbf{1}_{n}\\|_{1}\\leq a,$$\nhas an equivalent solution. The ℓ1-norm based divergence constraint will induce sparsity in the deviations between ζ ⊘ ν and 1n, such that many of the points maintain the corresponding value of ν. In the uniform distribution case ν =\n1 n 1n, the solution for ζ has a maximum value of an such that ζ ⊘ ν = a , the vertices of the feasible set for the constraint-based formulations include permutations of the vector\n\n$${\\overset{\\begin{array}{r l}{\\overbrace{\\left[{\\frac{a}{n}}\\quad\\quad{\\frac{1}{n}}\\quad\\quad{\\frac{1}{n}}\\quad\\cdots\\quad{\\frac{1}{n}}\\quad\\quad{\\frac{[a]-a}{n}}\\quad\\quad{\\frac{[a]-a}{0}}\\quad\\quad\\cdots\\quad0\\right]}^{\\top}}}.$$\n\nThis can be compared to the vertices of the feasible set for the marginals of the SS problem 2 explored in Remark 1, which have more uniform distribution of mass. More uniform mass distribution is motivated by the maximum entropy principle and desirable such as machine learning tasks such as semi-supervised learning where the goal is to augment the learning based on additional diversity.\n\nBy replacing the ℓ1-norm with the ℓ∞-norm in the capacity-based optimization problem 8 can be related to Sp(*µ, cν*) for all values of c ≥ 1 by setting a = c − 1 since ζj is only involved as an upper bound, no longer considering the lower bound of (2 − c)νj , for [P\n⊤1m]j =Pm i=1 Pij ≤ ζj ≤ cνj ∀j ∈ [n], yielding the SS\nproblem 2.\n\n## 2.2.2 Strictly Uniform, Semi-Relaxed Partial Optimal Transport\n\nThe work by Chapel et al. (2020), introduced in the context of positive-unlabeled (PU) learning, further constrains the semi-relaxed partial optimal transport problem such that the non-zero source masses must be 1 n\n, the PU Wasserstein optimal transport problem is\n\n$$\\mathcal{P}\\mathcal{U}\\mathcal{W}_{p}^{\\mathrm{w}}(\\mu,\\nu;s):=\\min_{\\mathbf{T}\\ \\neq0}\\quad\\langle\\mathbf{T},\\mathbf{M}\\rangle\\quad\\text{s.t.}\\quad\\mathbf{1}_{m}^{\\top}\\mathbf{T1}_{n}=s,\\ \\mathbf{T1}_{n}\\ \\ll\\frac{s}{m}\\mathbf{1}_{m},\\ \\mathbf{T}^{\\top}\\mathbf{1}_{m}\\in\\{0,\\frac{1}{n}\\}^{n}.\\tag{9}$$\n\nDue to the constraint that the marginal source masses are in the set {0, 1 n\n} this problem is not a linear program and only has a non-empty feasible set when s mod 1/n = 0, since s must be an integer multiple of 1n\n, which is an analogous condition to having a uniform distribution among the support as discussed in Remark 1. If this constraint is relaxed to T\n⊤1m ≼\n1 n 1n, then the problem is equivalent to semi-relaxed SS\nproblem 2, which by the linearity of the problem may result in a solution satisfying the original constraint T\n⊤1m ∈ {0, 1 n\n}\nn. To solve this combinatoric problem, the work by Chapel et al. (2020) obtains the solution to a convex minimization problem involving group LASSO regularization and additional dummy points, as in problems for unbalanced optimal transport (Guittet, 2002) to account for dropped mass. The solution to this problem will create a strictly uniform distribution (after renormalization) amongst the selected subset of the source marginal. That is, the group LASSO regularization induces a solution whose marginal is a vertex of the feasible set discussed in Remark 1.\n\n## 2.2.3 Fully-Relaxed Partial Optimal Transport\n\nWhile the formulation we adopt for subset selection is only for the source marginal (a semi-relaxed formulation of optimal transport), partial optimal transport formulations can achieve support subset selection on both marginals using a fully-relaxed optimization (Figalli, 2010; Phatak et al., 2023). Adapting the notation in the work by Chapel et al. (2020), the partial optimal transport problem is\n\n$$\\mathcal{P}\\mathcal{W}_{p}^{\\mathsf{s}}(\\mu,\\nu;s):=\\min_{T\\ \\geqslant0}\\ (T,\\mathbf{M})\\quad\\text{s.t.}\\quad T\\mathbf{1}_{n}\\preccurlyeq\\mu,\\ T^{\\top}\\mathbf{1}_{m}\\preccurlyeq\\nu,\\ \\mathbf{1}_{m}^{\\top}\\mathbf{T}\\mathbf{1}_{n}=s,\\tag{10}$$\n\nwhere s ∈ (0, 1] is the fraction of the mass transported. When the source distribution is uniform µ =\n1 n 1n, after renormalization P =\n1 s T , the feasible set for the source marginal P\n⊤1m consists of permutations of the vector\n\n$$\\overbrace{\\left[{\\frac{1}{n s}}\\quad{\\frac{1}{n s}}\\quad\\cdots\\quad{\\frac{1}{n s}}\\right]}^{\\mathrm{ns\\terms}}\\quad\\overbrace{0\\quad0\\quad\\cdots\\quad0}^{\\mathrm{n(1-s)\\terms}}\\ ,$$\n\nwhich are also the vertices of the feasible set of the marginals for the semi-relaxed SS problem 2 explored in Remark 1 with L = ns. We note that solutions for the fully-relaxed problem are more likely to have marginals with this form due to the lack of the equality constraints for the target marginal, as compared to the semi-relaxed SS problem. However, the main difference of the fully-relaxed approaches compared to our semi-relaxed SS approach (or other semi-relaxed approaches) is that points in the target may lose mass or be completely dropped, which is not ideal when the goal is to filter the source distribution for any points similar to the target distribution.\n\nThe fully-relaxed optimal transport can be directly related to the divergence-based unbalanced optimal transport problem,\n\n$${\\mathcal{F R W}}_{\\varphi}^{(\\rho)}(\\mu,\\nu):=\\operatorname*{min}_{\\mathbf{P}\\ni0}\\quad\\langle\\mathbf{P},\\mathbf{M}\\rangle+{\\frac{1}{\\rho}}\\left({\\mathcal{D}}_{\\varphi}(\\mathbf{P1}_{n}\\|\\ \\mu)+{\\mathcal{D}}_{\\varphi}(\\mathbf{P}^{\\top}\\mathbf{1}_{m}\\|\\ \\nu))\\right).$$\n\nThere exists a value of ρ ≥ 0 such that the fully-relaxed optimal transport with modified cost matrix M′ = M−\n1 ρ 1m1\n⊤\nn and total variation divergence penalties on both marginals will yield the same solution as PWpp\n(*µ, ν*; s) Caffarelli & McCann (2010); Chizat et al. (2018); Séjourné et al. (2019). However, as discussed above, a total variation or ℓ1-based penalty on only the one marginal with an equality constraint on the other induces a different solution. For c =\n1 s ≥ 1, the fully-relaxed partial optimal transport problem is\n\n$$\\begin{array}{l l l}{{\\operatorname*{min}_{P\\ni0}}}&{{\\langle P,M\\rangle}}&{{}}&{{=}}&{{\\operatorname*{min}_{T\\ni0}\\;\\;c\\langle T,M\\rangle}}\\\\ {{\\mathrm{s.t.}}}&{{P{\\bf1}_{n}\\leqslant c\\mu,\\;P^{\\top}{\\bf1}_{m}\\leqslant c\\nu,\\;{\\bf1}_{m}^{\\top}P{\\bf1}_{n}=1,}}&{{}}&{{\\mathrm{s.t.}\\;\\;\\;T{\\bf1}_{n}\\preccurlyeq\\mu,\\;T^{\\top}{\\bf1}_{m}\\preccurlyeq\\nu,{\\bf1}_{m}^{\\top}T{\\bf1}_{n}={\\frac{1}{c}},}}\\end{array}$$\n$$(11)$$\n$$\\left(12\\right)$$\n\nwhere T =\n1 cP . Based on equation 12, c *· PW*pp\n(*µ, ν*;\n1 c\n) is the cost for an optimal transport plan with fully-relaxed constraints P 1n ≼ cµ and P\n⊤1m ≼ cν.\n\nThe recent work by Phatak et al. (2023), studies the value of ω(s) := PWpp\n(*µ, ν*; s) across all values of s ∈ (0, 1], which is known as the OT-profile as introduced in the work of Figalli (2010). In the work by Phatak et al. (2023), ω(s) is shown to be a piece-wise linear convex function of s, and the entire profile can be computed exactly and approximated efficiently. Furthermore, derivative of ω with respect to s can be used find a mass fraction where the partial transport separates inliers and outliers Phatak et al. (2023). To relate the fully-relaxed to the semi-relaxed partial optimal transport, we consider the unbalanced partial OT-profile function to denote the case where the target marginal is also scaled down by s such that it is wholly transported\n\n$\\mathcal{P}\\mathcal{W}_{p}^{p}(s\\mu,\\nu;s):=\\min_{\\mathbf{T}\\ \\succcurlyeq0}\\quad\\langle\\mathbf{T},\\mathbf{M}\\rangle\\quad\\text{s.t.}\\quad\\mathbf{T1}_{n}\\prec s\\mu,\\ \\mathbf{T}^{\\top}\\mathbf{1}_{m}\\prec\\nu,\\ \\mathbf{1}_{m}^{\\top}\\mathbf{T1}_{n}=s.$\nLet c =\n1 s\n, then our proposed subset support cost is\n\n$$\\mathcal{S}_{p}(\\mu,\\frac{1}{s}\\nu)=\\min_{\\mathbf{P}\\succcurlyeq\\mathbf{0}}\\langle\\mathbf{P},\\mathbf{M}\\rangle$$ $$\\text{s.t.}\\mathbf{1}_{m}^{\\top}\\mathbf{P1}_{n}=1,$$ $$\\mathbf{P}\\preccurlyeq\\mathbf{\\mu},\\ \\mathbf{P}^{\\top}\\mathbf{1}_{m}\\preccurlyeq\\frac{1}{s}\\nu$$ $$\\text{s.t.}\\mathbf{1}_{m}^{\\top}\\mathbf{T1}_{n}=s,$$ $$\\mathbf{T1}_{n}\\preccurlyeq s\\mu,\\ \\mathbf{T}^{\\top}\\mathbf{1}_{m}\\preccurlyeq\\nu.$$\n$$(13)$$\n\nThus, Sp(µ, 1 s ν) = 1 sPWpp\n(*sµ, ν*; s) and 1 c Sp(*µ, cν*) = PWpp\n(\n1 c µ, ν; 1 c\n), with optimal solutions related by P\n∗ =\n1 s T\n∗. Based on this relation we explore the adoption of the knee finding algorithms applied to selection of s (Phatak et al., 2023) to optimize the selection of c in cases where the source differs from the target by the presence of outliers.\n\n## 2.3 Support Subset Selection With Entropic Regularization\n\nThe support subset selection problem 6 is a linear program, which can be exactly solved by the simplex method or interior point methods, both of which do not scale well with the dimension of transport map\n(Cuturi, 2013). In order to apply efficient gradient-based optimization to linear programs, entropic regularization has been added to linear objective functions (Li & Fang, 1997). In the work by Cuturi (2013),\nentropic regularization is added to the optimal transport problem to efficiently approximate the Wasserstein distance using Sinhkorn's matrix scaling algorithm (Cuturi, 2013; Sinkhorn, 1964). For fixed target distribution µ and upper-bounding source measure ζ with mass ζ ∈ R\nn\n+, the proposed entropically regularized support subset selection problem is\n\n$${\\mathcal{S}}_{p}^{(\\gamma)}(\\mu,\\zeta):=\\operatorname*{min}_{\\mathbf{P}\\neq0}\\quad\\langle\\mathbf{P},\\mathbf{M}\\rangle+\\gamma\\langle\\mathbf{P},\\mathbf{\\log}(\\mathbf{P})-\\mathbf{1}_{m\\times n}\\rangle\\quad{\\mathrm{s.t.}}\\quad\\mathbf{P1}_{n}=\\mu,\\ \\mathbf{P}^{\\top}\\mathbf{1}_{m}\\ll\\zeta,$$\n\nwhere γ is the regularization parameter. In the case of a uniform distribution ν =\n1 n 1n, the entropically regularization problem will be equivalent to using a Kullback-Leibler divergence between the joint and product of the given marginals and additional identity and range constraints on the solution's marginals Séjourné et al.\n\n(2019), as shown in Appendix D. It is important to mention that the regularization term ⟨P , log(P )−1m×n⟩\nis negative entropy, which is 1-strongly convex with respect to the ℓ1 and ℓ2 norms in the feasible set:\n{P : P ≽ 0, P 1n = µ, P\n⊤1m ≼ ζ} (Beck, 2017). The Lagrangian of problem 13 is\n\n$${\\mathcal{L}}(P,\\alpha,\\beta)=\\langle P,\\ M+\\gamma(\\log P-{\\mathbf{1}_{m}}{\\mathbf{1}_{n}^{\\top}})+\\alpha{\\mathbf{1}_{n}^{\\top}}+{\\mathbf{1}_{m}}\\beta^{\\top}\\rangle-\\langle\\alpha,\\mu\\rangle-\\langle\\beta,\\zeta\\rangle,$$\n\nwhere α, β are the Lagrange multipliers. Note that we have adopted the approach of Cuturi (2013) and do not explicitly enforce the simplex constraint on P , which would lead to the log-sum-exp formulation as in the works by Cuturi & Peyré (2018); Lin et al. (2022); Guminov et al. (2021). Taking the element-wise derivative of L with respect to P and setting it to zero yields\n\n$$\\tilde{P}(\\alpha,\\beta)=D\\bigg(\\exp{(-\\alpha/\\gamma)}\\bigg)\\exp{(-M/\\gamma)}D\\bigg(\\exp{(-\\beta/\\gamma)}\\bigg).$$\n\nSubstituting P˜ back into Lagrangian results in the dual problem\n\n Lagrangian results in the dual problem $\\min\\limits_{\\pmb{\\alpha},\\pmb{\\beta}}\\quad\\left\\{f(\\pmb{\\alpha},\\pmb{\\beta}):=\\gamma\\mathbf{1}_m^\\top\\pmb{P}(\\pmb{\\alpha},\\pmb{\\beta})\\mathbf{1}_n+\\langle\\pmb{\\alpha},\\pmb{\\mu}\\rangle+\\langle\\pmb{\\beta},\\pmb{\\zeta}\\rangle\\right\\}$ s.t. $\\quad\\beta\\geqslant0$. \n$$(16)$$\nThe constraint set β ≽ 0 is closed. The indicator function of the constraint set β ≽ 0 is defined as\n\n$$(14)$$\n$$(15)$$\n$$\\iota_{+}(\\beta):={\\begin{cases}0,&{\\mathrm{for~}}\\beta\\succ0\\\\ \\infty,&{\\mathrm{otherwise.}}\\end{cases}}$$\n\nTherefore, we can convert problem 16 into an unconstrained composite optimization problem,\n\n$$\\operatorname*{min}_{\\alpha,\\beta}\\quad f(\\alpha,\\beta)+\\iota_{+}(\\beta).$$\nα,βf(α, β) + ı+(β). (17)\nSince f(α, β) is convex and ı+(β) is proper, closed, and convex, we can apply the accelerated proximal gradient algorithm to solve the composite optimization problem. Defining the Gibbs kernel K = exp (−M\nγ\n),\nthe partial gradient ∇βf(α, β) is\n\n$$\\nabla_{\\beta}f(\\alpha,\\beta)=\\zeta-\\exp\\left(-\\beta/\\gamma\\right)\\odot\\left(K^{\\top}\\exp\\left(-\\alpha/\\gamma\\right)\\right).\\tag{1}$$\n\nThe proximal projection for the non-negative orthant's indicator function ı+ is computed by setting any negative entries to zero.\n\nAlgorithm 1 (SS-Entropic) outlines our accelerated proximal gradient algorithm to solve the dual form of subset selection problem with entropic regularization. Similar to the standard entropically regularized optimal transport problem, the dual variable α is updated with a Sinkhorn-like update at iteration k as\n\n$$(18)$$\n$$\\alpha^{(k+1)}=\\gamma\\log\\bigg(\\big(K\\exp(-\\beta^{(k)}/\\gamma)\\big)\\oslash\\mu\\bigg).$$\n$$(19)$$\n\nWhereas, β is updated at iteration k using accelerated proximal gradient based update rule (Beck, 2017; Beck & Teboulle, 2009) using the extrapolated point ξ with step size 1/η(k)\ns\n\n$$\\boldsymbol{\\beta}^{(k+1)}=\\big{[}\\boldsymbol{\\xi}^{(k)}-\\frac{1}{\\eta_{1}^{(k)}}\\nabla_{\\boldsymbol{\\xi}}f(\\boldsymbol{\\alpha}^{(k+1)},\\boldsymbol{\\xi}^{(k)})\\big{]}_{+}=\\bigg{[}\\boldsymbol{\\xi}^{(k)}-\\frac{1}{\\eta_{1}^{(k)}}\\left(\\boldsymbol{\\zeta}-\\exp(-\\boldsymbol{\\xi}^{(k)}/\\gamma)\\ominus\\boldsymbol{K}^{\\top}\\exp\\left(-\\boldsymbol{\\alpha}^{(k+1)}/\\gamma\\right)\\right),$$ $$\\boldsymbol{\\xi}^{(k+1)}=\\boldsymbol{\\beta}^{(k+1)}+\\frac{t_{k}-1}{t_{k+1}}(\\boldsymbol{\\beta}^{(k+1)}-\\boldsymbol{\\beta}^{(k)}),\\text{with}t_{k+1}=\\frac{1+\\sqrt{1+4t_{k}^{2}}}{2},$$\n$$\\left.\\begin{array}{l}{{+\\,}}\\\\ {{}}\\\\ {{}}\\end{array}\\right]_{+}$$\n\nwhich uses equation 18 to compute the gradient with respect to the variable ξ\n(k) before applying the proximal\n\n```\noperator [·]+. In SS-Entropic, we use a constant step size 1\n η\n (k)\n s\n = γ (since the primal problem 16 is γ-strongly\n\n```\n\nconvex and its semi-dual is 1 γ\n-Lipschitz smooth Cuturi & Peyré (2016), see Appendix A for details), but another option is a backtracking line search (Beck, 2017).\n\nBy incorporating the update of α(k+1) as in equation 19 directly into the gradient ∇βf(α(k+1), β\n(k)), the algorithm can be written entirely in terms of β\n(k). This shows that SS-Entropic consists of standard accelerated proximal gradient updates and has a O(1/k2) convergence rate. As shown in the work by Beck\n(2017), the required number of iterations kε to achieve an ε sub-optimal solution of the optimization problem 17 using SS-Entropic is upper-bounded as\n\n$$(21)$$\n$$k_{\\varepsilon}+1\\leq\\sqrt{\\frac{2}{\\gamma_{\\varepsilon}}}\\cdot\\|\\beta^{(i)}-\\beta^{*}\\|,\\tag{22}$$\n\nwhere β\n(i)is the initialization and β\n∗is the optimal solution.\n\nIf SS-Entropic is allowed to run until its convergence, it returns the optimal coupling P\n∗, but in practice, if SS-Entropic does not reach convergence, Pˆ∗ ∈ R\nm×n\n+ may violate the primal constraints on its marginals as these are not ensured by an approximate dual solution. For some applications a projection of Pˆ∗ to satisfy one or both of the marginal constraints may be required. While not explored in this paper due to the additional computational cost, projection to the feasible set can be done by the fast dual proximal gradient (FDPG) algorithm from the works by Beck & Teboulle (2014) and Beck (2017) in conjunction with Algorithm-2 in the work of Altschuler et al. (2017).\n\n## 2.4 Support Subset Selection With The Inexact-Bregman Proximal-Point Method\n\nAlthough the entropic regularization of the coupling distribution enables an efficient approximation of the support subset selection problem 6, the entropic regularization yields denser coupling distributions as compared to the unregularized problem. The denser coupling distributions result in a new marginal mass ν\n∗that is also not sparse, yielding complete support rather than a subset of the source points. Different approaches Algorithm 1: (SS-Entropic) Fast proximal gradient algorithm to solve the dual problem 17 of the entropically regularized support subset selection problem 13 Inputs : Target distribution µ, mass assignment bounding vector ζ, cost matrix M,\nentropic regularization parameter γ, initial dual variable, β\n(i) ∈ R\nn\n+, and iteration\n\n$$\\leftarrow\\mathbf{exp}(-{\\frac{1}{\\gamma}}M)$$\nlimit max-iter. Outputs : Pˆ∗, which approaches the optimal coupling P ∗ 1 Function EntropicSS(µ, ζ, β (i), γ, M, max-iter): Initialization: t0 ← 1, β (0) ← β (i), ξ (0) ← β (i), K ← exp(− 2 for k ← 0 to max-iter − 1 do 3 α(k+1) ← γ log K exp(− 1 γ β (k))⊘ µ 4 β (k+1) ← ξ (k) − γ∇ξf(α(k+1), ξ (k)) + 5 tk+1 ← 1+√1+4t 2 k 2 6 ξ (k+1) ← β (k+1) + ( tk−1 tk+1 )(β (k+1) − β (k)) 7 end 8 α∗ ← α(k+1)\n \\begin{tabular}{|l|l|} \\hline **7** & **end** & **end** \\\\ **8** & $\\alpha^{*}\\leftarrow\\alpha^{(k+1)}$ \\\\ **9** & $\\beta^{*}\\leftarrow\\beta^{(k+1)}$ \\\\ **10** & $\\hat{\\mathbf{P}}^{*}\\leftarrow\\hat{\\mathbf{P}}(\\alpha^{*},\\beta^{*})=\\mathbf{D}\\big{(}\\exp\\big{(}-\\frac{1}{\\gamma}\\alpha^{*}\\big{)}\\big{)}\\mathbf{K}\\mathbf{D}\\big{(}\\exp(-\\frac{1}{\\gamma}\\beta^{*})\\big{)}$ \\\\ **11** & **return** $\\hat{\\mathbf{P}}^{*}$, $\\alpha^{*}$, $\\beta^{*}$ \\\\ \\hline \\end{tabular} \nhave been proposed to maintain the computational benefits of entropic regularization while yielding solutions closer to the unregularized problem (Schmitzer, 2019; Xie et al., 2020).\n\nIn this paper, we follow the work by Xie et al. (2020) and adapt an inexact Bregman proximal gradient for the negative entropy function (Teboulle, 1992) to the partial optimal transport case. The Bregman proximal gradient approach uses a proximal operator where the usual Euclidean distance(Parikh & Boyd, 2014) is replaced with the Bregman divergence associated with a continuously differentiable and strictly convex function (Beck, 2017). In the case of negative entropy, the Bregman divergence is the Kullback–Leibler divergence. Let ϕ(P ) = ⟨P , log(P ) − 1m×n⟩ denote the negative entropy of a non-negative matrix P , then given a non-negative matrix P\n′∈ R\nm×n\n+ , the Bregman divergence is\n\n$${\\mathcal{B}}_{\\phi}(\\mathbf{P}||\\mathbf{P}^{'}):=\\langle\\mathbf{P},\\mathbf{\\log}(\\mathbf{P}\\oslash\\mathbf{P}^{'})\\rangle-\\langle\\mathbf{P},\\mathbf{1}_{m\\times n}\\rangle+\\langle\\mathbf{P}^{'},\\mathbf{1}_{m\\times n}\\rangle.$$\n′, 1m×n⟩. (23)\nFor the subset selection problem 6, the Bregman proximal point evaluated at P\n(t), is\n\n$$\\begin{array}{r l}{{\\mathrm{\\bf~Breg\\mbox{-}prox}_{\\phi}(P^{(t)})=\\operatorname*{arg\\,min}_{P\\succcurlyeq0}}}&{{\\langle P,M\\rangle+\\lambda\\mathcal{B}_{\\phi}(P||P^{(t)})}}\\\\ {{\\mathrm{\\mathrm{\\mathrm{s.t.}}}\\quad P{\\bf1}_{n}=\\mu,\\ P^{\\top}{\\bf1}_{m}\\preccurlyeq\\zeta,}}\\end{array}$$\n$$(23)$$\n\n$$(24)$$\n\nwhere λ is positive scaling factor. By substituting Bϕ(P ||P\n(t)) from equation 23 into equation 24 and ignoring the constant term ⟨P\n(t), 1m×n⟩ we obtain\n\n$$\\begin{array}{r l}{{\\mathrm{Breg\\mbox{-}prox}_{\\phi}(\\mathbf{P}^{(t)})=\\operatorname*{arg\\,min}_{\\mathbf{P}\\geqslant0}}}&{{\\langle\\mathbf{P},\\mathbf{M}-\\mathbf{log}(\\mathbf{P}^{(t)})\\rangle+\\lambda\\langle\\mathbf{P},\\mathbf{log}(\\mathbf{P})-\\mathbf{1}_{m\\times n}\\rangle}}\\\\ {{\\mathrm{s.t.}\\quad\\mathbf{P}\\mathbf{1}_{n}=\\mu,\\ \\mathbf{P}^{\\top}\\mathbf{1}_{m}\\leqslant\\zeta,}}\\end{array}$$\n$$(25)$$\n\nwhich corresponds to the entropically regularized subset selection problem 13 with parameters γ and M\nin 13 replaced by λ and M − log(P\n(t)), respectively. Thus, solving the entropy-regularized support subset selection problem is required to solve an iteration of the proximal-step evaluation problem in equation 25. It has been shown in the work by Xie et al. (2020) that as t → ∞, the iterations P\n(t+1) = **Breg-prox**ϕ\n(P\n(t))\nconverge to an optimal solution of the original unregularized problem. Therefore, to solve 6 we can iteratively invoke SS-Entropic to obtain P\n(t+1) = **Breg-prox**ϕ\n(P\n(t)), while replacing γ and M in problem 13 by λ and M − λP\n(t)in problem 25, respectively.\n\nAlgorithm 2 (SS-Bregman) outlines the steps to solve the subset support selection using the Bregman proximal-point method, where the inner loop is solved by SS-Entropic. The nested loops of the exact proximal point algorithm can result in high computational costs, but this can be circumvented by choosing a lower number of iterations for the inner loop—stopping before its convergence. This is justified by the observation that the majority of the progress towards optimal solutions by gradient based methods is achieved during the first few iterations. Recently, an inertial variant of inexact-Bregman proximal point method for the optimal transport has been proposed (Yang & Toh, 2022), which may further accelerate the Bregman proximal point method, but to the best of our knowledge there are no guarantees for accelerated convergence.\n\nAlgorithm 2: (SS-Bregman) Inexact-Bregman Proximal Point Algorithm to approximately solve 6 via 25 Inputs : Target distribution µ, mass assignment upper bounding vector ζ, cost matrix M,\nBregman scaling parameter λ, and initial dual variable, β\n(i) ∈ R\nn\n+, inner-iteration limit *max-inner-iter* and outer-iteration limit *max-outer-iter* Outputs : Pˆ∗\nInitialization: β\n(0) ← β\n(i), P\n(0) ← 1 mn 1m×n 1 for t ← 0 to *max-outer-iter* − 1 do\n// repeatedly invoke Entropic-SS\n2 P\n(t+1), α(t+1), β\n(t+1) ← EntropicSS µ, ζ, β\n(t)*, λ,*M − log(P\n(t)), *max-inner-iter* 3 end 4 Pˆ∗ = P\n(t+1)\nDue to early stopping, SS-Bregman can yield infeasible solutions that do not satisfy the marginal constraints.\n\nIn practice, the number of iterations depends on problem in the hand. For applications related to point cloud registration, color transfer and PU learning, where number of data points in a data-batch is small, the algorithm is allowed to run with a large number of iterations yielding a highly accurate and feasible solution. Whereas, for the applications related to neural network training where training efficiency is more important than the solution accuracy, the algorithm is allowed to run for a smaller number of iterations.\n\n## 2.5 Point Cloud Registration With Subset Selection\n\nPoint cloud registration is a well studied problem that tries to find a correspondence of points in one sample\n(cloud) to another sample (Zhang et al., 2021; Zang et al., 2019). Practical settings include points sampled from the boundaries of 3D images such as captured by LIDAR or the points on the edges of objects in 2D\nimages (Xu et al., 2023). More generally, points correspond to data points in two or more samples. In both of these cases it is useful to consider the case that the two samples exist in different coordinate frames such that there is an affine transformation needed to align the samples before finding the correspondence.\n\nPartial optimal transport for point cloud registration is motivated by cases of occlusion in 2D or 3D imagery.\n\nIn the case of data, it could be that one sample has dropped modes either by the nature of the data gathering or generating process. Our proposed subset selection algorithms are applicable to cases where the source is assumed to have a complete or overcomplete representation of the target, i.e., only a subset of the target is available and all target points should be maintained.\n\nWe propose to use support subset selection as a loss function for optimizing affine transformations in partial point cloud registration. This can be posed as a bi-level optimization problem\n\n$$\\langle P,{\\hat{M}}(\\Theta)\\rangle\\quad\\mathrm{s.}$$\n\nmin Θmin P ≽0⟨P ,Mˆ (Θ)⟩ s.t. P 1n = µ, P\n⊤1m ≼ cν, (26)\nwhere Θ = [A, b] are the parameters of the affine transform, the entries of the cost matrix Mˆ (Θ) are\n[Mˆ (Θ)]ij = ∥xi−yˆ\nΘ\nj\n∥\n2 2i ∈ [m], j ∈ [n] for fixed target {xi}\nm i=1 and transformed source {yˆ\nΘ\nj = Ayj +b}\nn j=1.\n\n$\\mathbf{n}=\\mu,\\;\\;P$\n$$(26)$$\nThe standard approach to solve bi-level optimization problems in point cloud registration discussed in the works by Arun et al. (1987) and Myronenko & Song (2010), is an iterative alternating algorithm with two steps, where the sub-problem for the affine transform is solved exactly via ordinary least squares. If the coupling matrix during an iteration is given by P\n∗, the next subproblem is to find the affine transformation parameters Θ = [A, b] that minimize the weighted squared errors Pi,j [P\n∗]ij∥xi −(Ayj +b)∥\n2 2\n. The solution can be found analytically in terms of the source mass vector ν\n∗ = P\n∗⊤1m, weighted means of the target point cloud X = [x1*, . . . ,* xm]\n⊤ and the source point cloud Y = [y1*, . . . ,* yn]\n⊤ as x¯ = X⊤µ and y¯ = Y\n⊤ν\n∗,\nand centered point clouds X˜ = X − 1mx¯\n⊤ and Y˜ = Y − 1ny¯\n⊤, as\n\n$$\\begin{array}{l}{{A=\\left(\\tilde{X}^{\\top}P^{*}\\tilde{Y}\\right)\\left(\\tilde{Y}^{\\top}D(\\nu^{*})\\tilde{Y}\\right)^{\\dagger},}}\\\\ {{b=\\bar{x}-A\\bar{y},}}\\end{array}$$\n\n$$(27)$$\n\nwhere (·)\n†indicates the Moore-Penrose pseudo-inverse.\n\nTo find a solution to equation 26, we also use an iterative alternating algorithm with two steps. In contrast to Myronenko & Song (2010), instead of using complete source and target point clouds to obtain affine transformations, during every iteration we draw batches from both source and target point clouds to obtain the coupling matrix P\n∗ and update the affine transformation parameters Θ via a gradient update. The advantage of this mini-batch based approach is an implicit regularization and faster updates for affine transformation parameters. In the first step, given the affine transformation we obtain an approximate solution Pˆ∗ to the subset selection problem 6 via SS-Bregman. In the second step, we used automatic differentiation of the cost ⟨Pˆ∗,Mˆ (Θ)⟩ and perform gradient based update for the parameters Θ = [A, b]. It is important to mention that we follow the approach adopted by Xie et al. (2020) for gradient evaluation.\n\nTherefore, during an iteration, once the subset set selection map Pˆ∗ is obtained, it is deemed constant for the iteration in consideration, therefore the gradient is: ∇Θ⟨Pˆ∗,Mˆ (Θ)⟩ =Pi,j [Pˆ∗]ij∇Θ[Mˆ (Θ)]ij . More specifically, we used PyTorch based automatic differentiation for gradient evaluation (Paszke et al., 2017) and the Adam optimizer (Kingma & Ba, 2014) with learning rate of 0.5 for gradient based updates of parameters Θ. To initialize the affine mapping parameters we simply set A and b to the identity matrix and zero vector, respectively. However, since the bi-level optimization problem is not convex, even though the subset selection problem at each iteration is convex, in practice the algorithm could be allowed to run with multiple initialization to obtain the best fit.\n\n## 3 Experimental Results And Discussion\n\nIn this section we discuss the application of subset selection. Subsections 3.1 and 3.2 discuss the application of subset selection in toy data sets: point-clouds in 2D and 3D with and without affine transformations and color transfer, respectively. Subsection 3.3 discusses subset selection for positive-unlabeled learning tasks.\n\nSubsection 3.4 discusses the application of subset selection for semi-supervised training of neural networks.\n\nAll the experiments done in this paper use p = 2 and the Euclidean distance to define the cost matrix. Unless stated otherwise, experiments use µ =\n1 m 1m and ζ =\nc n 1n, where c ≥ 1 is the scaling factor.\n\n## 3.1 Subset Selection On Point Clouds\n\nCircle and Square: In order to demonstrate the proposed algorithms and highlight the difference between regular optimal transport and subset selection, we consider a target sample of points from a circle centered at the origin and a source sample of points from a 2D uniform distribution also centered at the origin. We allow the scaling parameter c to vary between 1 and 100, obtain the optimal transport plans P\n∗ using both SS-Entropic and SS-Bregman, and evaluate the cost values ⟨P\n∗,M⟩. Results for this toy case are shown in Figure 2. It can be observed that as c is increased the transport cost decreases until it saturates to the cost of the greedy solution Pi∈[m]\n1 m minj∈[n][M]ij , which corresponds to c = c\n∗ where the transport map could be found by greedily choosing nearest source point for each target point as in equation 4. Figure 2 also illustrates the transport couplings for c ∈ {1, 1.25, 1.5, 1.75, 2, 4, 8, 16}. A key observation is that transport maps obtained with SS-Bregman are sparser as compared to the denser maps obtained using SS-Entropic.\n\nAdditionally, they achieve smaller values of transport cost. Therefore in the subsequent sections, we focus\n\n\n\non results from SS-Bregman in the main body; results for Algorithm SS-Entropic are in the Appendix B\n\nFigure 2: (a) The toy data generated by uniformly sampling m = 100 points from a circle centered at the origin with unit diameter as the target. The source contains n = 80 points generated by sampling uniformly from [-\n1 2\n,\n1 2\n]×[-\n1 2\n,\n1 2\n]. (b) The optimal costs ⟨P\n∗,M⟩ obtained using SS-Entropic and SS-Bregman versus c for c ∈ [1, 100]. SS-Entropic is ran for 10,000 iterations with γ = 0.1. SS-Bregman is ran for *max-outer-iter* =\n100, *max-inner-iter* = 100 and λ = 0.1. (c) and (d) Support subset selection results obtained for c ∈\n{1, 1.25, 1.5, 1.75, 2, 4, 8, 16} using the Algorithms SS-Entropic and SS-Bregman, respectively.\nFragmented Hypercube with Mode Dropping: We demonstrate the utility of the support subset selection algorithm for partial point cloud registration on a toy case with one dropped mode and an affine transformation between the source and the target. Specifically, we consider data sampled from a uniform distribution over a hypercube (specifically, a square in 2D or a cube in 3D), which is then fragmented, where the target has one less fragment than the source. To generate the source we sample n points {vi}\nn i=1 from the uniform distribution over a unit hypercube centered at the origin [−\n1 2\n,\n1 2\n]\nd, d ∈ {2, 3}. These points are then fragmented into 2 dfragments according to their quadrant y˜i = vi + (d − 1) sign(vi) and then offset to obtain the source points as yi = y˜i + 5(d − 1) for i ∈ [n]. The target data is generated similarly: a sample of *m > m* ˆ points {zi}\nmˆ\ni=1 is obtained from [−\n1 2\n,\n1 2\n]\nd, then points with all negative coordinates are discarded, leaving m points, which are fragmented into 2 d − 1 fragments to obtain the target set {xi}\nm i=1 via xi = zi + (d − 1) sign(zi) for i ∈ [m]. Examples of the data for 2D and 3D are shown in Figure 3(a) and Figure 4(a), respectively.\n\n\n\nFigure 3: Results for affine transformation optimization with subset selection for partial optimal transport.\n\nTarget points X are sampled from a 2D fragmented hypercube centered at the origin with negative coordinates removed, whereas source points Y are sampled from a translated fragmented hypercube. (a) Source and target sample points. (b) Loss function plotted against Θ = [A, b] updates for scaling parameter c ∈\n{1, 1.25, 1.5, 1.75, 2, 4, 8, 16}. (c) Target and transformed source points after application of optimized affine transformation. Subset selection problems are solved using the SS-Bregman with λ = 0.1, *max-outer-iter* =\n100 and *max-inner-iter* = 500.\nDue to the translation by 5(d − 1) of the source point coordinates, direct application of the transport map will not yield a meaningful registration. Instead we use the bi-level optimization algorithm described in Section 2.5. The target and the transformed source after applying the affine transformation obtained using SS-Bregman for c ∈ {1, 1.25, 1.5, 1.75, 2, 4, 8, 16} are displayed in Figure 3(c) and Figure 4(c), respectively.\n\nClearly, the c = 1 case corresponding to the complete optimal transport fails to identify a meaningful affine transformation, instead skewing and rotating the source fragments to minimize the Wasserstein distance to the target. The figures also display the cost ⟨P\n∗,Mˆ ⟩ across iterations. It can be observed that, like the previous toy examples as the value of scaling factor c is increased, initially the value of the optimal loss ⟨P\n∗,Mˆ ⟩ decreases but after certain values of c, it saturates and stops decreasing and stays constant afterwards. Partial Point Cloud for 3D Shapes: We further apply this form of subset selection based point cloud registration to point clouds for 3D objects when the target points are only taken from a portion of the entire 3D point cloud. Results for the Stanford bunny and armadillo (Turk & Levoy, 1994; Krishnamurthy & Levoy, 1996) are shown in Figure 5. It can be observed that for the case c = 1, which corresponds to complete optimal transport, the entire set of source points are coupled to the target point cloud which results in a distorted affine transform. For c ∈ {2, 5, 10, 20}, subset selection allows an appropriate subset of the source points to be well-fit by an affine transform to the target point cloud.\n\n\n\nFigure 4: Results for affine transformation optimization with subset selection for partial optimal transport. Target points X are sampled from a 3D fragmented hypercube centered at the origin with negative co-ordinates removed, whereas source points Y are sampled from a translated fragmented hypercube. (a)\nSource and target sample points. (b) Loss function plotted against 0 = [4, b] updates for scaling parameter c E {1, 1.25, 1.5, 1.75, 2, 4, 8, 16}. (c) Target and transformed source points after application of the optimized affine transformation. Subset selection problems are solved using the SS-Bregman with \\ = 0.1, max-outer-iter = 200 and max-inner-iter = 500.\n\n## 3.2 Color Transfer\n\nColor transfer is the problem of finding a correspondence in the colors of pixels (represented as points in a 3D color space) between two images and then using this map to assign the colors of the source image to the target image (Reinhard et al., 2001). Color transfer is essentially an optimal transport problem in the color space, but with the added context that the pixels have their image coordinates, which are not used by the algorithm. For practical application to high resolution images, the pixel colors are first quantized using k-means clustering, as using partial optimal transport on the full set of pixel colors is computationally demanding. While in standard optimal transport the relative mass of each color cluster has to be preserved, here we exploit our formulation of partial optimal transport as support subset selection to allow a subset of colors to be used at a higher proportion than in the original source and allow a subset of colors to be completely discarded. For example, if a color cluster represents 1% of the original source's pixels, then it could represent up to c% of the target's pixels.\n\nWe apply k-means clustering to the set of vectors in RGB color space representing the source's M pixels and the target's N pixels separately to obtain m < M color centroids {z;},22 C R2 for the target image and n < N color centroids {y}};_1 C R2 for the source image, with u E Am and u E An being the vectors of proportion of colors in the target and source image color clusters, respectively. After that, we define the\n\n\n\nFigure 5: Affine transformation optimization for partial alignment of point clouds where a subset of the source point-cloud Y can be perfectly aligned (after rotation and scaling) with the target point cloud X.\n\nWe use the optimization algorithm described in Section 2.5, where SS-Bregman is employed to obtain the coupling P\n∗ given the affine transformation parameters A and b, which are updated using equation 27. (a)\nStanford bunny point cloud. (b) Stanford armadillo point cloud.\ncost matrix between the color centroids as Mij = ∥xi −yj∥\n2 2\n, ∀i ∈ [m], j ∈ [n] and obtain the support subset selection map P\n∗ ∈ R\nm×n\n+ using SS-Bregman, such that P\n∗1n = µ and P\n∗⊤1m ≼ cν. The support subset selection is then used to obtain the barycenter projections by solving (Blondel et al., 2018)\n\n$$\\hat{\\mathbf{x}}_{i}=\\operatorname*{arg\\,min}_{\\mathbf{x}\\in\\mathbb{R}^{3}}\\sum_{j=1}^{n}P_{ij}^{*}\\|\\mathbf{x}-\\mathbf{y}_{j}\\|_{2}^{2},\\quad\\forall i\\in[m].\\tag{28}$$\n\nThe analytic solution of the barycenter projections can be compactly written as\n\n$$\\mathrm{n~as}$$\n$${\\hat{\\mathbf{X}}}=(\\mathbf{P}^{*}\\oslash(\\mathbf{\\mu1}_{n}^{\\top}))\\mathbf{Y}\\in\\mathbb{R}^{m\\times3},$$\n$$(29)$$\nm×3, (29)\nwhere X = [x1, x2*, . . . ,* xm]\n⊤ ∈ R\nm×3 and Y = [y1, y2*, . . . ,* ym]\n⊤ ∈ R\nn×3 are matrices of the color centroids.\n\nEach pixel in the target image is assigned the corresponding barycenter projection xˆπ(i), where π(i) ∈ [m]\nis the cluster assignment for the ith pixel of target image, i ∈ [M].\n\nWe apply this color transfer scheme to images freely available though a Creative Commons licence, the\n\"Louisiana Nature Scene Barataria Preserve\" by Neil O as target and \"Autumn in Toronto\" by Bahman AMahmoodi as source. The color transfer results with m = n = 128 and SS-Bregman with λ = 0.1 are shown in Figure 6. It can be observed that the results for larger values of c are smoother within objects or areas of similar color (e.g., the dark backdrop behind the peppers) and sharper in the color transitions between different objects (the colors in the orange versus red pepper on the right side of the photograph) as compared to the optimal transport case c = 1. This is due to the fact that larger values of c allow certain colors to be reused more than their prevalence in the source image and allow some colors to be discarded, which enables smoother transitions in colors for areas of the target images with smooth color gradients. Similar observations can be seen in Figure 7 which uses the same settings and MATLAB test images: \"peppers\" as target and \"corn\" as source.\n\n## 3.3 Subset Selection For Positive-Unlabeled Learning\n\nIn this section, we discuss the application of subset selection to the one-class semi-supervised classification scheme known as positive-unlabeled (PU) learning (Bekker & Davis, 2020). In PU learning, the training\n\n\n\nFigure 6: Color transfer results for c ∈ {1, 1.25, 1.5, 1.75, 2, 4, 8, 16} for \"Louisiana Nature Scene Barataria Preserve\" by Neil O as target and \"Autumn in Toronto\" as source by Bahman A-Mahmoodi as target. The value of c for each image is indicated at the top of image.\n\n\n\nFigure 7: Color transfer results for c ∈ {1, 1.25, 1.5, 1.75, 2, 4, 8, 16} for MATLAB image \"peppers\" as target and \"corn\" as source. The value of c for each image is indicated at the top of image.\nsample consists of purely positively labeled instances, and the unlabeled test sample consists of both positive and negative instances. Previous work often assumes that a prior on the probability of positive instances in unlabeled data is known (Kato et al., 2019; Hsieh et al., 2019; Chapel et al., 2020). Partial optimal transport is then used to find a subset with cardinality proportional to the prior of the test sample (source/unlabeled) that corresponds to all or a subset of the training sample (target/positive). We argue that all of the target mass should be preserved in cases of a relatively small and curated positive training sample. This motivates the application of our proposed subset selection approach to find the subset of the source that covers the positive target, compared to fully-relaxed approaches. To illustrate the difference between fully and semi-relaxed partial optimal transport for PU learning, we consider two-dimensional random variables for the positive and unlabeled points where the support of positive random variable is a subset of the support of unlabeled random variable and both have long-tailed distributions, as shown in Figure 8. The lower accuracy for the fully-relaxed solution (79.5% versus subset selection's 84.1%) can be accounted for by the increased distances between source and target points away from the origin, which causes the target points to be dropped, preventing true positives in the source from being selected. This result holds for more general settings discussed in Appendix E.\n\nWe applied subset selection to PU learning using the experimental settings adapted from the work of Chapel et al. (2020), who explored using the PU-Wasserstein (PUW) 9 and the partial Gromov-Wasserstein distance\n(PGW) on various UCI, MNIST, colored-MNIST, and Caltech-office data sets. For the UCI, MNIST, and\n\n\n\nFigure 8: Results of PU learning on toy data using subset selection (accuracy 84.1%) and fully-relaxed partial optimal transport (accuracy 79.5%). Data is generated as [r cos(ψ), r sin(ψ)] where the radius r is drawn from a truncated exponential distribution with density 2 exp(−r)ı[log 2,∞)(r) and ψ is uniform over a subset of angles [0, π]. The points belong to four classes corresponding to the angle falling in one of the intervals [0, π 4\n], (\nπ 4\n,\nπ 2\n], (\nπ 2\n,\n3π 4\n], or (\n3π 4\n, π]. The m = 100 target/positive points are all from the third class. The solutions are obtained using the known number of positives n+ out of the n = 400 source points, setting c =n n+\nfor our semi-relaxed approach and s =\nn+\nnfor the fully-relaxed partial optimal transport.\ncolored-MNIST data sets, we randomly draw m = 400 positive points and n = 800 unlabeled points. For Caltech-office data sets, we randomly sampled m = 100 positive points from the first domain and n = 100 unlabeled data points from the second domain. Following the experiments from the works by Chapel et al.\n\n(2020) and Kato et al. (2019), for multi-class data sets, we chose the data points from the class labeled 1 as positive and a random mixture of all classes as unlabeled, and the prior probability of positive class in the unlabeled set π+ is set to be exactly the proportion of positives in unlabeled sample π+ =\nn+\nn\n, where n+\nis the number of true positives. This informs the PU-Wasserstein, partial Gromov-Wasserstein, and fullyrelaxed partial Wasserstein optimal transport problems on the amount of mass to be transported as s = π+,\nwhereas for subset selection we set the scaling parameter to be c =n n+\n. Classification accuracy is evaluated by assigning positive predictions to the n+ largest source mass assignments and negative predictions to the remaining source points. We also compute the ROC curve by using the source mass assignment ν\n∗to rank the unlabeled source points. We ran the experiment 10 times and report the average of classification accuracy and the area under the ROC curve (ROC-AUC) in Table 1. It can be observed that the proposed subset-selection performs better than PU Wasserstein in terms of accuracy in 8 out of the 10 data sets with same domain (UCI, MNIST, and Caltech-office with same domains). Subset selection does best overall on 6 out of these 10, with fully-relaxed partial optimal transport performing better on 4 datasets. For the Caltech-office data sets with domain transfer, the partial Gromov-Wasserstein optimal transport does best.\n\nIn terms of ROC-AUC, subset selection does better than PU Wasserstein in 9 out of the 10 intra-domain data sets, which is not surprising since the relative ranking is more meaningful than when the mass assignments are restricted to be binary valued {0, 1 n+\n} as in the solutions from PU Wasserstein.\n\nWe attribute, higher AUC-ROC of subset selection as compared to fully-relaxed Wasserstein, to the fact that, subset selection problem has equality constraints on target mass which ensures coverage (lower false negatives). The differences between our solutions (SS) and those for the PU Wasserstein (PUW) are subtle, as both are semi-relaxed and maintain equality constraints on the target distribution. The small, but consistent, differences in the accuracy between our method and PUW may be due to the additional uniform mass assignment constraints in PUW, which is achieved through group-LASSO. The uniform mass assignment constraints in PUW may result into larger transport costs as compared to solutions obtained using SS with same cardinality. The largest mass assignments in the solution to SS might be more reliable than constrained solutions to PU Wasserstein.\n\n| Dataset | π+ | Accuracy | ROC-AUC | | | | | | |\n|-------------------|-------------------|-------------|-----------|--------|---------------------|--------|--------|--------|--------|\n| PUW PGW FR-PW | SS | PUW | PGW | FR-PW | SS | | | | |\n| mushrooms | 0.518 95.15 | 94.85 | 99.45 | 96.63 | 0.9657 | 0.3336 | 0.9948 | 0.9883 | |\n| shuttle | 0.786 95.13 | 93.63 | 96.35 | 96.20 | 0.9321 | 0.6215 | 0.9467 | 0.9718 | |\n| pageblocks | 0.898 91.90 | 90.35 | 91.88 | 92.40 | 0.8036 | 0.7197 | 0.7817 | 0.8513 | |\n| usps | 0.167 98.28 | 95.55 | 97.08 | 98.48 | 0.9815 | 0.5096 | 0.9476 | 0.9927 | |\n| connect-4 | 0.658 60.95 58.05 | 60.95 | 60.73 | 0.5692 | 0.5126 | 0.5666 | 0.5871 | | |\n| spambase | 0.394 78.80 | 68.40 | 69.08 | 79.28 | 0.7952 | 0.5834 | 0.6770 | 0.8369 | |\n| mnist | 0.1 | 99.08 | 98.23 | 99.15 | 99.18 | 0.9874 | 0.7638 | 0.9768 | 0.9971 |\n| mnist-colored | 0.1 | 91.58 | 96.78 | 97.66 | 91.88 | 0.8189 | 0.6619 | 0.9360 | 0.9521 |\n| surf C → surf C | 0.1 | 90.00 | 87.20 | 82.00 | 90.40 0.8576 0.4622 | 0.7469 | 0.7333 | | |\n| surf C → surf A | 0.1 | 81.60 86.80 | 81.40 | 81.60 | 0.4546 | 0.4764 | 0.5337 | 0.4889 | |\n| surf C → surf W | 0.1 | 82.20 86.40 | 81.20 | 82.20 | 0.4707 | 0.4807 | 0.4451 | 0.5056 | |\n| surf C → surf D | 0.1 | 80.00 87.00 | 80.00 | 80.00 | 0.3756 | 0.4328 | 0.4056 | 0.4444 | |\n| decaf C → decaf C | 0.1 | 94.00 | 86.20 | 82.00 | 94.40 | 0.9498 | 0.5713 | 0.7667 | 0.9682 |\n| decaf C → decaf A | 0.1 | 80.20 88.20 | 81.80 | 80.40 | 0.3986 | 0.5031 | 0.5349 | 0.4564 | |\n| decaf C → decaf W | 0.1 | 80.20 88.60 | 82.00 | 80.00 | 0.4299 | 0.5827 | 0.5611 | 0.5965 | |\n| decaf C → decaf D | 0.1 | 80.80 92.20 | 80.40 | 80.40 | 0.4546 0.5042 | 0.4617 | 0.4530 | | |\n\nTable 1: PU learning on data sets as in the work by (Chapel et al., 2020). For subset selection (SS)\nand fully-relaxed partial-Wasserstein (FR-PW), accuracy is evaluated by assigning label 1 to n+ largest mass assignments and label 0 to the remaining mass assignments. For PU Wasserstein (PUW), the mass assignment are constrained to be binary valued in the set {0, p}, the data points with mass assignments 0 are labeled 0 and the data points with mass p =1 nπ+\n=1 n+\nare labeled 1.\n\n## 3.3.1 Pu Learning On Mnist/Emnist\n\nTo further illustrate how SS-Bregman operates on PU learning, we apply it to the case where the positive training sample (target) consists of MNIST digit images and the unlabeled test sample contains 50% points (MNIST digits) and 50% negative points (alphabetic letters from EMNIST). When c = 1, which is equivalent to standard optimal transport, initially all the images in the unlabeled source sample are assigned uniform masses. As c is increased, we hypothesize that the true positive MNIST digits will been assigned larger mass and remain in the selected support, whereas the EMNIST letters will receive relatively lower or zero mass. Our hypothesis is confirmed by the results displayed in Figure 9(a), which displays the ROC curve across different choices of c, and in Figure 9(c), which displays the area under the ROC curve (AUC). As c is increased, source points with largest mass assignments are mostly MNIST digits. Likewise, Figure 9(e) shows the images with highest mass for different values of c which are mainly MNIST numbers or EMNIST letters with close resemblance to a number. Figure 9(b) visualizes the distribution of source point masses by graphing the sorted masses for different values of c. From these curves the cardinality of the subset is easily seen for different values of c. Notably, for values of c ≤ 4 there are exists a subset of the selected source points with uniform mass, but for larger values of c, the mass is non-uniform across all instances. These changes correspond to the change in slope of the entropy of the distribution for different values of c is displayed in Figure 9(d). We further compared our approach for PU learning with semi-relaxed optimal transport approaches using the the squared ℓ2-norm penalty Chapel et al. (2021) and total-variation (TV) divergence Séjourné et al. (2019).\n\nThe formulation of semi-relaxed problems is discussed in Appendix C. We used POT-toolbox Flamary et al.\n\n(2021) to solve the semi-relaxed optimal transport problems. For both squared ℓ2-norm and total-variation penalties, we varied regularization parameter ρ with 32 uniform steps on logarithmic scale between 10−3 and 103. For subset-selection, c parameter is varied uniformly on logarithmic scale with 32 steps between 1 and 32. In order to evaluate the performance of each method, we assigned label 1 to all the selected points and zero to all the remaining points. We also computed the cardinalities and entropies of mass-assignments ν\n∗. In Figure 10 we compare the effect of the cardinality of the support of the mass assignment vector ν\n∗ on the accuracy and entropy H(ν\n∗). We observe that for the mass assignments with same cardinality, mass assignments obtained through subset selection have higher entropy as compared to both TV and ℓ2 penalized semi-relaxed optimal transport. For each regularization discussed above, mass-assignment paths across scaling parameter c for SS-Bregman are discussed in the Appendix C.\n\nWe also adapt the approach proposed in the work by Phatak et al. (2023) in the context of fully-relaxed partial optimal transport to automatically select the proportion of mass to separate inliers and outliers to automatically find a choice of c for subset selection for PU learning. The approach finds the knee of the smoothed version of the first derivative of 1 c Sp(*µ, cν*) as a function of 1 c\n∈ (0, 1], using the kneedle method (Satopaa et al., 2011). The results in Figure 10(c) show that the automatically selected value of c is at the highest accuracy.\n\n\n\nFigure 9: Subset selection results obtained using SS-Bregman with parameters λ = 1, *max-outer-iter* = 250 and *max-inner-iter* = 20. Source sample n = 512 consists of 50% digit images from MNIST and 50% letter images from EMNIST. The target sample contains m = 512 digit images drawn from MNIST. (a) ROC\ncurves for different values for c ∈ {1, 1.25, 1.50, 1.75, 2, 4, 8, 16, 20, 24, 28, 32}. (b) Mass assignments to source images in descending order ν\n∗↓. (c) AUC of ROC versus c. (d) Entropy H(ν\n∗) of the mass assignments ν\n∗\nversus c.\n\n## 3.3.2 Pu Learning For Cifar-10 Neural-Network Representations\n\nWe now consider the proposed subset selection algorithm for PU learning on the CIFAR-10 data set, where a single class from the training set is treated as the positive target and a mixture of all classes from the test set is the unlabeled source. Fundamentally, the performance of optimal transport methods on PU learning depends on the distance metric defining the cost matrix. Thus, the method performs poorly if a Euclidean distance metric is applied to complex data such as natural images. Instead, a learning representation extracted from\n\n \n\nFigure 10: PU learning with semi-relaxed optimal transport approaches on MNIST/EMNIST. Solutions for different cardinalities are obtained by varying the regularization/constraint parameters across a grid, uniform on a logarithmic scale. (a) Accuracy versus cardinality of selected subset using SS-Bregman, semi-relaxed optimal transport with total variation (TV) and squared-ℓ2 penalties for PU learning on MNIST/EMNIST. (b)\nEntropy of mass assignments versus cardinality of selected subset using SS-Bregman, semi-relaxed optimal transport with total variation (TV) and squared ℓ2 penalties. (c) Accuracy of PU learning on MNIST/EMNIST using SS-Bregman versus scaling parameter c. Vertical lines indicate the location of the knee obtained using Phatak et al. (2023), the true peak accuracy value of c, and the break point c\n∗.\na pretrained neural network can be used. Here each image is represented as the vector of activations of the penultimate layer of the pre-trained ResNet-20 classifier (trained on CIFAR-10), and the Euclidean distance between the activation vectors defines the cost matrix for the transport problem.The results are given in Figure 11. The results are similar to the previous MNIST/EMNIST data set. Mass is uniformly distributed across a subset of images for values of 1 *< c <* 8. When the subset is greater than the proportion of positive instances in the unlabeled source, then the relative ranking of mass is not reliable: the top instances for c ∈ {1.5, 2, 4} are images from the target class, but c ∈ {1.25, 1.75} have images resembling it from other classes. As c is increased above c = 8 the mass assignment is non-uniform, but constant for further increment in c. Values of c greater than 2 have an AUC >90%. Similar to PU learning on MNIST/EMNIST, we also compared our approach for PU learning on neural network representations of CIFAR-10 with TV and ℓ2 penalized semi-relaxed optimal transport. For semirelaxed formulations (ℓ2 and TV penalties) we varied ρ on logarithmic scale between 10−4 and 103 with 32 steps. For subset selection we varied the scaling parameter c logarithmically between 1 and 60 in 32 steps. We adopted the same strategy as previous case (PU learning on MNIST/EMNIST) to evaluate accuracy, cardinality, and entropy of mass assignments ν\n∗ at different values regularization and scaling parameters.\n\nFigure 12 shows accuracy versus cardinality, and entropy versus cardinality, along with variation of PU learning accuracy across c and knee-based scaling parameter selection. Our observations in this case are similar to the case for EMNIST/EMNIST: subset selection has the highest entropy at a given cardinality, its accuracy is at or above the other solutions, and the knee method selects close to optimal value of c.\n\n## 3.4 Subset Selection For Semi-Supervised Learning\n\nWe consider the semi-supervised training of a classifier where the training set is divided into a reliably labeled (curated) target set and an unlabeled or noisily labeled source set. We apply our proposed subset selection algorithm to perform partial optimal transport of the unlabeled source to the labeled target. The transport plan is computed without knowledge of any labels but defines how the source points will be labeled, and subset selection removes points that cannot easily be aligned to labeled training points. Additionally, the new mass assignment source points may be relatively higher for unlabeled points relatively close to labeled training points and lower for unlabeled points from existing points. Used in this way, the optimal transport with subset selection automatically tunes how far to propagate labels in a manner that takes\n\n\n\nFigure 11: Subset selection results obtained using algorithm SS-Bregman with parameters λ = 1, max-outer-iter = 250 and *max-inner-iter* = 20. Target and source consist of ResNet-20 embeddings of m = 512 CIFAR-10 dog images and n = 512 randomly sampled CIFAR-10 images, respectively. (a) ROC\ncurves for c ∈ {1, 1.25, 1.50, 1.75, 2, 4, 8, 16, 20, 24, 28, 32}. (b) Mass assignments to source images in descending order ν\n∗↓. (c) AUC of the ROC versus c. (d) Entropy H(ν\n∗) of mass assignments ν\n∗ with versus c. (e)\nSource images with 10 largest mass assignments.\ninto consideration the geometry and distribution of the curated target data set rather than only the local distances. However, using the distances defined directly in the input space may not be suitable, and a pre-trained representation may not exist for various tasks. Instead, we propose to use the internal learning representation from the neural network classifier while it is being optimized with the semi-supervised loss function.\n\nLet S = {(xi, Li)}M\ni=1 denote the labeled portion of the training set with the input xi ∈ X and label encoded as a one-hot vector Li ∈ {0, 1}\nk ⊂ ∆k, ∥Li∥1 = 1 for i ∈ [M], and T = {yj}\nN\nj=1 denote the unlabeled portion, yj ∈ X for j ∈ [N]. We consider a neural-network classifier with soft-max activation f(· ; θ) : X → ∆k with parameters θ trained on data with k classes. The neural network's internal representation is a function g(· ; θ) : X → R\nd. The Euclidean distance between the internal representation of data points provides the distance function, dθ(xi, yj ) = ∥g(xi; θ) − g(yj ; θ)∥2, which is parameterized by the network's parameters.\n\nWe train the neural network using mini-batches and a semi-supervised cross-entropy loss. Equal-sized batches are drawn uniformly from the pooled training data set of size M + N. Let τ and σ denote the length-m and length-n vectors of indices of the labeled and unlabeled points in a given batch, respectively, where m + n is the constant batch size. The m-by-n ground cost matrix M(θ) is defined using the squared distances among the batch's latent representations, Mij (θ) = d2θ\n(xτi\n, yσj\n) = ∥g(xτi\n; θ) − g(yσj\n; θ)∥\n2 2\n.\n\n \n\nFigure 12: PU learning with semi-relaxed optimal transport approaches on CIFAR-10. Solutions for different cardinalities are obtained by varying the regularization/constraint parameters across a grid, uniform on a logarithmic scale. (a) Accuracy versus cardinality of selected subset using SS-Bregman, semi-relaxed optimal transport with total variation (TV), and squared ℓ2 for PU learning on CIFAR-10. (b) Entropy of mass assignments versus cardinality. (c) Accuracy of PU learning on CIFAR-10 using SS-Bregman versus scaling parameter c. Vertical lines indicate knee obtained using Phatak et al. (2023), true peak accuracy value of c, and break point c\n∗.\nGiven the cost matrix and hyper-parameters (including c ≤ n), the subset selection transport plan P\n∗ ∈\n[0, 1]m×n is obtained using SS-Bregman. Given the matrix of one-hot encoded labels L = [Lτ1\n, . . . , Lτm]\n⊤ ∈\n{0, 1}\nm×k, the matrix of pseudo-labels assigned by the algorithm of the unlabeled mini-batch points is computed L˜ = nP\n∗⊤L ∈ [0, c]\nn×k, where [P\n∗⊤L]jl =\n1 n L˜jl ∈ [0, 1] is the estimate of the joint probability that mini-batch unlabeled instance j ∈ [n] belongs to class l ∈ [k].\n\n3 Given the pseudo-labels, the semisupervised cross-entropy loss function for a batch is\n\n$$\\text{loss}(\\mathbf{\\theta})=-\\bigg{[}\\sum_{i=1}^{m}\\sum_{l=1}^{k}\\frac{1}{m}\\mathbf{L}_{il}\\log(f_{l}(\\mathbf{x}_{\\tau_{i}};\\mathbf{\\theta}))+\\sum_{j=1}^{n}\\sum_{l=1}^{k}\\frac{1}{n}\\tilde{\\mathbf{L}}_{jl}\\log(f_{l}(\\mathbf{y}_{\\sigma_{j}};\\mathbf{\\theta}))\\bigg{]}.\\tag{30}$$\n\nOur approach is similar to other recent work (Damodaran et al., 2020) that also employs optimal transport using a learning representation. While we address the semi-supervised case, Damodaran et al. (2020) address supervised learning in the presence of label noise and perform self optimal transport within batches to correct for label noise. As baseline comparisons, we compare our semi-supervised approach to supervised training with either only the labeled portion or with noisy labels on the unlabeled portion. Due to the curated labeled set, the latter is not the typical label noise scenario; however, the division of a training set into a curated portion and a portion with label noise is relevant to practical scenarios. While our semi-supervised approach does not use noisy labels, future extensions could consider how to leverage the noisy labels too. In order to evaluate our approach we used MNIST, Fashion-MNIST (FMNIST), and CIFAR-10. We split training data sets into 80/20 proportions for training and validation. We further split the training part into a reliably labeled and unreliably labeled parts. Labels for the unreliably labeled part are generated by uniformly corrupting the true labels to other classes depending on the noise level. For each of our experiments, the subset selection transport underlying the loss done is found via SS-Bregman with λ = 0.1, max-outer-iter = *max-inner-iter* = 20 with a batch-size of 512. We used PyTorch framework for our experiments. A ResNet-18 model architecture is used on the CIFAR-10 data set. We trained the ResNet-18 for 180 epochs using Adam optimizer with an initial learning rate of 0.001, which is scheduled to be halved\n\n3It can be seen that the total sum of this joint is 1, 1\n\n$$1,\\underbrace{{\\mathbf1}_{n}^{\\top}{\\boldsymbol{P}}^{*}{}^{\\top}{\\boldsymbol{L}}{\\mathbf1}_{k}}_{\\mu^{\\top}}=1.$$\n\nafter every 60 epochs. The model architectures containing two convolutional layers for MNIST and FMNIST\nare given in Appendix F. The neural network classification models for MNIST are trained using stochastic gradient descent with a learning rate of 0.001, whereas models for FMNIST are trained using Adam with a learning rate of 0.001 and weight decay 1e-4. Model training for MNIST and Fashion-MNIST are done on a desktop system containing Intel Core-i7 9700 CPU, with 32 GB memory and NVIDIA GeForce RTX\n2070 GPU. ResNet-18 based models for CIFAR-10 are trained using Lambda-labs cloud resources with 30 vCPUs, 200 GB memory, and NVIDIA A10 GPUs.\n\n| Dataset | Architecture | stand. | Subset Selection | | | | | | | | |\n|-----------|------------------|----------------------------------------------------------------|--------------------|-------|-------|-------|-------|-------|-------|-------|-------|\n| | c = 1 | | c = 2 | c = 3 | c = 4 | c = 5 | c = 6 | c = 7 | c = 8 | c = 20 | | | | | | | | | |\n| MNIST | 2-layer conv-net | 96.57 | 97.15 | 97.39 | 97.33 | 97.40 | 97.43 | 97.32 | 97.38 | 97.44 | 97.35 |\n| FMNIST | 2-layer conv-net | 88.68 | 90.14 | 90.30 | 90.27 | 90.51 | 90.23 | 90.38 | 90.21 | 90.37 | 90.16 |\n| CIFAR-10 | ResNet-18 | 79.10 | 87.18 | 89.45 | 89.27 | 89.21 | 89.53 | 89.54 | 89.72 | 89.47 | 89.57 |\n\nTable 2: Validation accuracies for different values for neural network classification models trained with 50%\nreliably labeled points and 50% points with noisy labels (noise level 0.8). Standard training (stand.) treats them equally, but subset selection treats them as unlabeled and assign pseudo-labels. Subset selection is done using the SS-Bregman with λ = 0.1 and max-outer-iter = *max-inner-iter* = 20.\n\nIn the first step of experiments, we split the training set for each data set into 50/50 proportions for unreliably and unreliably labeled parts. Unreliably labeled data is generated by uniformly corrupting the labels with a 80% chance (noise level 0.8). Validation accuracies for each data set are displayed in the Table 2. Notably, the performance for c = 1 is higher than training with noisy labels, which shows that the semi-supervised training performs better than training with data with a high noise level. (Because the algorithm is not run to convergence, the mass assignments for unlabeled points may not be be exactly uniform in the c = 1 case.) The performance of subset selection is consistently higher for values of c > 1 compared to the c = 1, and the validation accuracies do not exhibit much change between c = 2 and c = 20. Therefore, we further evaluated our approach by varying both noise levels and clean and noisy proportions only for c = 2 and c = 20.\n\nProgress of validation accuracies on CIFAR-10 are displayed in Figure 13 for clean/noisy proportions in\n{20/80, 40/60, 60/40, 80/20} with noise levels {0.2, 0.4, 0.6, 0.8}. It can be observed that standard neural network training process with label noise can divided into three phases, first in which the validation accuracy increases until a peak. In the second phase, validation accuracy decreases, where the magnitude of the decrement depends on the noise level: it decreases less for low noise levels and more for higher noise levels. In the third phase, validation accuracy increases again and then oscillates around a constant value. This kind of phenomenon is more pronounced for larger noise levels (Zheng et al., 2020). In contrast, for the proposed subset selection based semi-supervised learning the validation accuracy does not go down after hitting its peak during the training process. This indicates that the transport map tend to assign correct pseudo-labels to the data points nearest the labeled data points and does not introduce label noise. The test set accuracies are displayed in Table 3 for supervised training on only the labeled data versus the semi-supervised training. The semi-supervised training with subset selection outperforms training performs better on 2 of 3 data sets under a 20/80 split of labeled and unlabeled, but does not outperform supervised learning for the 40/60 split. Thus, the semi-supervised loss function equation 30 is most beneficial when there is a higher ratio of unlabeled to labeled points.\n\n## 4 Discussion And Further Work\n\nIn this paper, we have focused on selecting a subset of one distribution's support as a special case of partial optimal transport Figalli (2010); Chapel et al. (2020). This is useful to find meaningful alignment when the support of the target distribution is assumed to be a subset of the source distribution. Results on the partial point cloud alignment, color transfer, PU learning, and semi-supervised learning all demonstrate the utility of this approach.\n\n\n\nFigure 13: Progress of validation accuracies while training ResNet-18 for CIFAR-10 classification. (a) Uniform noise levels are varied between 0.2,0.4,0.6,0.8 for standard training. (b) and (c) Training with the subset selection based semi-supervised loss at different values of c, does not use the unreliable labels, and outperforms standard training with noisy labels when either the proportion of reliably labeled data is 40%\nor the noise level is 0.4 or greater. Subset selection is done using theSS-Bregman with c = 20, A = 0.1, max-outer-iter = max-inner-iter = 20.\n\n| Architecture | Labeled/Unlabeled % | | | |\n|----------------|-----------------------|-----------------------|-----------------|-------|\n| Dataset | | Labeled only | Semi-supervised | |\n| | | with subset selection | | |\n| 2-layer | | 20/80 | 98.19 | 96.09 |\n| MNIST | conv-net | 40/60 | 98.30 | 97.14 |\n| 2-layer | | 20/80 | 88.35 | 88.42 |\n| F-MNIST | conv-net | 40/60 | 89.93 | 89.48 |\n| | | 20/80 | 81.62 | 82.37 |\n| ResNet-18 | | | | |\n| CIFAR-10 | | 40/60 | 87.77 | 86.74 |\n\nTable 3: Semi-supervised learning test accuracies on MINIST, Fashion-MNIST, and CIFAR-10.\n\nSubset selection is done using SS-Bregman with c = 20, ) = 0.1, max-outer-iter = max-inner-iter = 20.\nIn particular, the results from the PU learning show that the proposed subset selection is useful when there is known target distribution (an existing training or validation set) and an additional source distribution, which has additional diversity, but also outliers, compared to the target. One application of PU learning is to filter a source of new data for relevant examples for further modeling. Future work could explore subset selection approach for source distributions created from synthetic generation mechanisms. While not explored in a machine learning context, it is possible that the partial optimal transport with affine (or nonlinear) transformation can be applied to account for global covariate shift between the synthetic and real data. In this case, a user would want to balance the diversity (entropy) of the filtered source with its purity.\n\nInterestingly, the solution for the subset selection are similar but consistently outperform those from PU\nWasserstein Chapel et al. (2020), which by design of the constraints, have a maximum entropy, uniform distribution over the selected subset achieved through the group LASSO penalty. The results from subset selection show that it maintains close to maximum entropy amongst the selected support and is as accurate as other semi-relaxed penalty based approaches. Additionally, the manual choice of c controlling the constraint is more intuitive than selecting a penalty parameter. Finally, the automatic selection of c using the straightforward knee-based selection adapted from fully-relaxed partial optimal transport (Phatak et al.,\n2023) shows promising results to separate inliers and outliers.\n\nIn our experiments related to semi-supervised learning, we employed optimal transport between a labeled target and unlabeled source, to assign pseudo-labels to source points that cover the labeled data distribution, while ignoring ambiguous cases, during training. In future extension, we can consider how to use class information in the optimal transport planning, perhaps by using class conditional optimal transport, as currently the transport plan is not informed of the known target labels nor the classifier's boundaries.\n\nAnother line of exploration is how to use the support subset selection to correct noisily labeled source.\n\nAnother key contribution of this work is the proposed support subset selection algorithm using the inexact Bregman proximal point algorithm (SS-Bregman), which as shown in Appendix B yields a solution with a sparse source marginal similar to solutions to the original linear program 6—unlike the entropically regularized solution from SS-Entropic. We also demonstrate that the mass assignments of the linear program solution are piece-wise linear as a function of c. While not fully investigated here, this behavior could be exploited to find the sequence of breakpoints where points leave the support and where points leave the active set of constraints (indicated by being on the upper diagonal).\n\nRecently, Gromov-Wasserstein optimal transport has seen applications in graph-matching and generative modeling (Brogat-Motte et al., 2022; Li et al., 2023; Nekrashevich et al., 2023; Bunne et al., 2019; Mémoli, 2009). Due to inherent ability to match structural correspondences across spaces, partial GromovWasserstein optimal transport can be used to solved robust graph-alignment problems. Recently, efficient locally convergent solutions for a relaxed Gromov-Wasserstein distance have been proposed (Peyré et al.,\n2016; Li et al., 2023). Future work can explore the subset selection case of the partial Gromov-Wasserstein optimal transport, where one domain is expected to have a complete or overcomplete source distribution compared to the target. This may be useful in robust domain adaptation, semi-supervised domain adaptation, and metric alignment.\n\n## Author Contributions\n\nBilal Riaz and Austin Brockmeier worked on the formulation and design of experiments for subset-selection.\n\nBilal Riaz implemented the algorithms and results. Manuscript was written and edited by all authors.\n\n## Acknowledgments\n\nBilal Riaz is supported by Higher Education Commission of Pakistan and University of Delaware. We would like to express gratitude to Matthew S. Emigh for insightful discussions and the Office of Naval Research for funding this research. Research at the University of Delaware was sponsored by the Department of the Navy, Office of Naval Research under ONR award number N00014-21-1-2300.\n\n## Code Availability\n\nCode for the paper is available at https://github.com/Bilal092/support-subset-selection.\n\n## References\n\nAkshay Agrawal, Robin Verschueren, Steven Diamond, and Stephen Boyd. 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In *International Conference on Machine Learning*, 2020.\n\n## A Appendix: Lipschitz Smoothness Of Dual\n\nThe dual form 16 considered in this paper does not explicitly enforce the primal problem's marginal simplex constraints on the transport plan. Consequently, the dual form is not necessarily Lipschitz smooth (Lin et al., 2019; Cuturi & Peyré, 2018; Lin et al., 2022). But in the proposed algorithm, we first update the dual variable α using the Sinkhorn-like update, which implicitly enforces the simplex constraint, making the semi-dual problem 1 γ\n-Lipschitz smooth with respect to ℓ1, ℓ2 and ℓ∞ norms. This justifies the use of η\n(k)\ns =\n1 γ in the accelerated proximal-gradient based approach to solve 16.\n\nRecall that the Lagrangian of 16 given in equation 14 is\n\n$${\\mathcal{L}}(P,\\alpha,\\beta)=\\langle P,\\ M+\\gamma(\\log(P)-{\\mathbf{1}_{m}}{\\mathbf{1}_{n}^{\\top}})+\\alpha{\\mathbf{1}_{n}^{\\top}}+{\\mathbf{1}_{m}}\\beta^{\\top}\\rangle-\\langle\\alpha,\\mu\\rangle-\\langle\\beta,\\zeta\\rangle.$$\n\nBy Slater's conditions, the problem 16 is strongly dual therefore\n\n$$(31)$$\n$$\\min_{P\\ni0}\\,\\max_{\\mathbf{\\alpha},\\mathbf{\\beta}}\\,{\\cal L}(\\mathbf{P},\\mathbf{\\alpha},\\mathbf{\\beta})=\\max_{\\mathbf{\\alpha},\\mathbf{\\beta}}\\,\\min_{P\\ni0}\\,{\\cal L}(\\mathbf{P},\\mathbf{\\alpha},\\mathbf{\\beta}).\\tag{1}$$\n$$(32)$$\n$$(33)$$\n\nIn order to find the minimum of the Lagrangian with respect to P , one takes its element-wise derivative with respect to P and obtains P˜(α, β) = Dexp (−\n1 γ α)exp (−\n1 γ M)Dexp (−\n1 γ β), which can then be substituted back into the Lagrangian 31 to obtain the problem\n\n$$\\operatorname*{max}_{\\alpha,\\beta}\\quad\\left\\{g(\\alpha,\\beta)=-\\gamma\\mathbf{1}_{m}^{\\top}{\\tilde{P}}(\\alpha,\\beta)\\mathbf{1}_{n}-\\langle\\alpha,\\mu\\rangle-\\langle\\beta,\\zeta\\rangle\\right\\},{\\mathrm{~s.t.~}}\\beta\\succcurlyeq0,$$\n\nwhich can be converted to the convex minimization problem 16 by defining f(α, β) = −g(α, β), as in min α,β f(α, β) = γ1\n⊤\nmP˜(α, β)1n + ⟨α, µ⟩ + ⟨β, ζ⟩\t, s.t. β ≽ 0, (34)\nThe partial gradients of f(α, β) with respect to α and β are\n\ny defining $f(\\pmb{\\alpha},\\pmb{\\beta})=-g(\\pmb{\\alpha},\\pmb{\\beta})$, as in \n$$\\iota)+\\langle\\beta,\\zeta\\rangle\\,\\}\\,,\\,\\,\\mathrm{s.t.}\\,\\,\\beta$$\n$$(34)$$\n$$\\nabla_{\\alpha}f(\\alpha,\\beta)=\\mu-\\exp\\big{(}-\\frac{\\alpha}{\\gamma}\\big{)}\\odot K\\exp\\big{(}-\\frac{\\beta}{\\gamma}\\big{)},$$ $$\\nabla_{\\beta}f(\\alpha,\\beta)=\\zeta-\\exp\\big{(}-\\frac{\\beta}{\\gamma}\\big{)}\\odot K^{\\top}\\exp\\big{(}-\\frac{\\alpha}{\\gamma}\\big{)}.$$\n$$(35\\mathrm{a})$$\n$$(35\\mathrm{b})$$\nFor twice continuously differentiable functions, the Lipschitz smoothness parameter is determined by the Hessian. The Hessian for f(α, β) is\n\n$$H_{f}(\\alpha,\\beta)=\\begin{bmatrix}\\nabla_{\\alpha}^{\\top}\\nabla_{\\alpha}f(\\alpha,\\beta)&\\nabla_{\\beta}^{\\top}\\nabla_{\\alpha}f(\\alpha,\\beta)\\\\ \\nabla_{\\alpha}^{\\top}\\nabla_{\\beta}f(\\alpha,\\beta)&\\nabla_{\\beta}^{\\top}\\nabla_{\\beta}f(\\alpha,\\beta)\\end{bmatrix},$$\n$$(36)$$\n\nwhere\n\n∇⊤ α∇αf(α, β) = 1 γ D exp − α γ ⊙ K exp (− β γ ) = 1 γ DP˜(α, β)1n ∇⊤ β ∇βf(α, β) = 1 γ D exp − β γ ⊙ K⊤ exp (− α γ ) = 1 γ DP˜(α, β) ⊤1m ∇⊤ β ∇αf(α, β) = 1 γ K ⊙ exp − α γ exp (− β ⊤ γ ) = 1 γ P˜(α, β), (37c) ∇⊤ α∇βf(α, β) = 1 γ K⊤ ⊙ exp − β γ exp (− α⊤ γ ) = 1 γ P˜(α, β)\n, (37a)\n, (37b)\n⊤. (37d)\nThe Sinkhorn update for α in equation 19 ensures that after each update of α, the transport plan lies on the probability simplex and matches the target marginal µ = P˜(α(k+1), β\n(k))1n. Defining ν˜ = P˜(α(k+1), β\n(k))\n⊤1m, the Hessian equation 36 at (α(k+1), β\n(k)) is compactly written as\n\n$${\\mathbf{H}}_{f}({\\mathbf{\\alpha}}^{(k+1)},{\\mathbf{\\beta}}^{(k)})={\\frac{1}{\\gamma}}\\begin{bmatrix}{\\mathbf{D}({\\mathbf{\\mu}})}&{\\tilde{\\mathbf{P}}({\\mathbf{\\alpha}}^{(k+1)},{\\mathbf{\\beta}}^{(k)})}\\\\ {\\tilde{\\mathbf{P}}({\\mathbf{\\alpha}}^{(k+1)},{\\mathbf{\\beta}}^{(k)})^{\\top}}&{\\mathbf{D}({\\tilde{\\mathbf{\\nu}}})}\\end{bmatrix}.$$\n (37a) $$\\left(37\\text{a}\\right)$$ (37b) $$\\left(37\\text{c}\\right)$$ (37d) $$\\left(37\\text{d}\\right)$$. \nWe use the induced-norms of the Hessian Hf (α(k+1), β\n(k)) to characterize the smoothness at (α(k+1), β\n(k)).\n\nFor a matrix A ∈ R\nm×n, the induced norm ∥ · ∥p,q is defined as\n\n$$\\|\\mathbf{A}\\|_{p,q}:=\\max_{\\mathbf{x}:\\|\\mathbf{x}\\|_{p}\\leq1}\\|\\mathbf{A}\\mathbf{x}\\|_{q}.\\tag{1}$$\n$$(38)$$\n$$(39)$$\n\nThe twice continuously differentiable function f(α(k+1), β\n(k)) is L-Lipschitz with respect to ℓp norm, if\n∥Hf (α(k+1), β\n(k))∥p,q ≤ L, where ℓq is the dual norm of the ℓp norm. Since the Hessian Hf (α(k+1), β\n(k)) is a non-negative matrix, one can observe that all its matrix entries are less than 1 γ max {µmax, ν˜max}, where µmax and ν˜max are maximum entries of µ and ν˜ respectively. Therefore for p = 1, if one can find the column index k corresponding to a matrix entry with value 1 γ max{µmax, ν˜max}, then x = ek is the vertex of the ℓ1 norm-ball where ∥Hf (α(k+1), β\n(k))x∥∞ =\n1 γ max{µmax, ν˜max}. Thus,\n\n$$\\|\\mathbf{H}_{f}(\\mathbf{\\alpha}^{(k+1)},\\mathbf{\\beta}^{(k)})\\|_{1,\\infty}={\\frac{1}{\\gamma}}\\operatorname*{max}\\,\\left\\{\\mu_{\\operatorname*{max}},{\\tilde{\\nu}}_{\\operatorname*{max}}\\right\\}\\leq{\\frac{1}{\\gamma}},$$\n\nwhich proves the function f(α(k+1), β\n(k)) is 1 γ\n-Lipschitz with respect to the ℓ1 norm. Since all the matrix entries of the Hessian Hf (α(k+1), β\n(k)) are less than 1 γ\n, its spectral radius is less than 1 γ\n(Horn & Johnson, 2012)(Theorem 8.1.18) and\n\n$$\\|\\mathbf{H}_{f}(\\mathbf{\\alpha}^{(k+1)},\\mathbf{\\beta}^{(k)})\\|_{2,2}=\\lambda_{\\mathrm{max}}(\\mathbf{H}_{f})\\leq\\frac{1}{\\gamma}.\\tag{1}$$\n$$(40)$$\n\nTherefore, the function f(α(k+1), β\n(k)) is 1 γ\n-Lipschitz with respect to the ℓ2 norm. For p = ∞, one can maximize the norm ∥Hf (α(k+1), β\n(k))x∥1, at the vertex of the ℓ∞ ball where all entries are unit magnitude, in particular x = 1m+n, which results into\n\n$${\\mathbf{H}}_{f}({\\boldsymbol{\\alpha}}^{(k+1)},{\\boldsymbol{\\beta}}^{(k)}){\\mathbf{1}}_{m+n}={\\frac{1}{\\gamma}}\\begin{bmatrix}{\\boldsymbol{\\mu}}+{\\dot{\\mathbf{P}}}({\\boldsymbol{\\alpha}}^{(k+1)},{\\boldsymbol{\\beta}}^{(k)}){\\mathbf{1}}_{n}\\\\ {\\dot{\\mathbf{P}}}({\\boldsymbol{\\alpha}}^{(k+1)},{\\boldsymbol{\\beta}}^{(k)})^{\\top}{\\mathbf{1}}_{m}+{\\dot{\\boldsymbol{\\nu}}}\\end{bmatrix}={\\frac{2}{\\gamma}}\\begin{bmatrix}{\\boldsymbol{\\mu}}\\\\ {\\dot{\\boldsymbol{\\nu}}}\\end{bmatrix}.$$\n\nTherefore,\n\n$$(41)$$\n$$\\|\\mathbf{H}_{f}(\\mathbf{\\alpha}^{(k+1)},\\mathbf{\\beta}^{(k)})\\|_{\\infty,1}=\\frac{2}{\\gamma}\\left\\|\\mathbf{\\mu}\\right\\|_{1}=\\frac{4}{\\gamma},\\tag{1}$$\n$$\\left(42\\right)$$\n\nand the function f(α(k+1), β\n(k)) is 4 γ\n-Lipschitz with respect to the ℓ∞ norm. In summary, the partial gradients are all 1 γ\n-Lipschitz smooth with respect to ℓ1, ℓ2 and ℓ∞ norms. Additionally, considering the dual variables α and β, seperately, one can see that\n\n$$\\|\\nabla_{\\mathbf{a}}^{\\top}\\nabla_{\\mathbf{a}}f(\\alpha^{(k+1)},\\beta^{(k)})\\|_{1,\\infty}=\\frac{\\mu_{\\max}}{\\gamma}\\leq\\frac{1}{\\gamma},\\ \\|\\nabla_{\\mathbf{a}}^{\\top}\\nabla_{\\mathbf{a}}f(\\alpha^{(k+1)},\\beta^{(k)})\\|_{2,2}\\leq\\frac{1}{\\gamma},\\quad\\|\\nabla_{\\mathbf{a}}^{\\top}\\nabla_{\\mathbf{a}}f(\\alpha^{(k+1)},\\beta^{(k)})\\|_{\\infty,1}=\\frac{1}{\\gamma},$$ and\nand\n\n$$\\|\\nabla_{\\beta}^{\\top}\\nabla_{\\beta}f(\\alpha^{(k+1)},\\beta^{(k)})\\|_{1,\\infty}=\\frac{\\rho_{\\max}}{\\gamma}\\leq\\frac{1}{\\gamma},\\ \\|\\nabla_{\\beta}^{\\top}\\nabla_{\\beta}f(\\alpha^{(k+1)},\\beta^{(k)})\\|_{2,2}\\leq\\frac{1}{\\gamma},\\quad\\|\\nabla_{\\beta}^{\\top}\\nabla_{\\beta}f(\\alpha^{(k+1)},\\beta^{(k)})\\|_{\\infty,1}=\\frac{1}{\\gamma}.$$\n\nTherefore, f(α(k+1), β\n(k)) is separately 1 γ\n-Lipschitz for both α and β with respect to ℓ1, ℓ2 and ℓ∞ norms.\n\n## B Appendix: Entropic Regularization Results\n\nIn this Appendix, results for point cloud registration and color transfer for SS-Entropic are displayed, which can be compared to the results for SS-Bregman in the main body.\n\nFragmented Hypercubes: Figure 14 shows the results for affine transformation optimization in 2D, which can be compared with the results in Figure 3(a). Visually it is clear that the alignment is much worse for values of c ∈ {1.25, 1.5}, but quantitatively it is worse for all values of c > 1 as SS-Bregman achieves cost below 10−2 at 500 iterations of Θ updates. Figure 15 shows the results in 3D using SS-Entropic, which can be compared with the results in Figure 4(a). In this case, results are quantitatively worse for all values of c > 1 as SS-Bregman achieves cost below 10−1 at 500 iterations Θ updates.\n\n\n\nFigure 14: Results for affine transformation optimization with subset selection for partial optimal transport. Target points X are sampled from a 2D fragmented hypercube centered at the origin with negative coordinates removed, whereas source points Y are sampled from a translated fragmented hypercube. (a)\nTarget and transformed source points after application of optimized affine transformation. Subset selection problems are solved using the SS-Entropic with y = 0.01 with max-iter = 4000.(b) Loss function curves for scaling parameter c E {1, 1.25, 1.5, 1.75, 2, 4, 8, 16}.\n\n\n\nFigure 15: Results for affine transformation optimization with subset selection for partial optimal transport. Target points X are sampled from a 3D fragmented hypercube centered at the origin with negative coordinates removed, whereas source points Y are sampled from a translated fragmented hypercube. (a)\nTarget and transformed source points after application of optimized affine transformation. Subset selection problems are solved using the SS-Entropic with y = 0.01 with max-iter = 4000.(b) Loss function curves for scaling parameter c E {1, 1.25, 1.5, 1.75, 2, 4, 8, 16}.\nPartial point cloud registration: The results for partial point cloud registration with entropically regularized subset selection (SS-Entropic) for the Stanford bunny and armadillo point clouds. It is clear that the entropically regularized form alone fails to find a meaningful correspondence, transforming the source such that is completely covered by the partial point cloud.\n\nThe results for color transfer with the entropically regularized subset selection Color Transfer:\nSS-Entropic are shown in Figure 17, which can be compared to results from SS-Bregman shown in Figure 6 and Figure 7. Namely, for the first image \"Louisiana Nature Scene Barataria Preserve\" the entropically\n\n\n\nFigure 16: Bunny and Armadillo partial point cloud registration using entropically regularized subset selection SS-Entropic y = 0.05, fails to find a accurate alignment of the source with the partially occluded target.\nregularized results appear more monochromatic with less distinct colors. In the second set of images, there is no visual difference between the outputs of SS-Entropic and SS-Bregman.\n\n\n\n(b)\nFigure 17: Color transfer results for c E {1, 1.25, 1.5, 1.75, 2, 4, 8, 16} using SS-Entropic.\n\n## C Appendix: Comparison With Semi-Relaxed Formulations\n\nWe analyze the variation of subset-selection mass assignments ν\n∗\nj =Pm i=1 P\n∗\nij , for j ∈ [n] as a function of c for source and target points (m = 100, n = 80) given in Figure 2. We use SS-Entropic and SS-Bregman and compare to solutions for the subset selection linear program using CVXPY (Diamond & Boyd, 2016; Agrawal et al., 2018). We also use CVXPY to solve following semi-relaxed unbalanced optimal transport problems of the form\n\n$$\\begin{array}{r l}{{\\operatorname*{min}_{\\mathbf{P}\\succcur\\succcur\\succcur}}}&{{\\langle\\mathbf{P},\\mathbf{M}\\rangle+{\\frac{1}{\\rho}}{\\mathcal{D}}_{\\varphi}\\!\\left(\\mathbf{P}^{\\top}\\mathbf{1}_{m}\\|\\mathbf{\\nu}\\right)}}\\\\ {{\\mathrm{s.t.}}}&{{\\mathbf{P}\\mathbf{1}_{n}=\\mathbf{\\mu},}}\\end{array}$$\n\nwhere Dφ ∈ {TV, ℓ2, ℓ∞} can be either TV distance (TV), squared Euclidean (ℓ2), or ℓ∞-norm based distance, and ρ > 0. It can be observed from Figure 18 that the sparsity patterns of mass assignments obtained using SS-Bregman closely match the linear program solutions across a range of c, whereas mass assignments obtained using SS-Entropic are more dense with less mass assignments equal to 0. The sparsity patterns for penalty-based relaxations differ from the linear program solutions. By comparing the mass assignments of different solutions in terms of number of non-zeros, i.e., the cardinality of the support of solution's marginal ν\n∗, denoted as (card), we note that the assignments obtained using subset selection have the largest entropy, which implies that subset selection tends to assign uniform masses to the selected subset of points. In comparison, the TV-based penalty, corresponding to the ℓ1-norm tends to have sparse deviations from uniform 1n with many points at exactly this value. Entropies of ν\n∗ are plotted in Figure 19.\n\nTo further elaborate our observations, we plot the mass assignments to unlabeled data points ν, across different values of scaling parameter c for MNIST/EMNIST PU learning in Figure 20 and Figure 21 for CIFAR-10 PU learning, as discussed in 3.3, and compared that with the mass assignments using semirelaxed formulations (ℓ2 and TV) with different values of regularization parameter ρ. We observe that paths taken by our formulation differs from both ℓ2 and TV . But the solutions correspond at the extreme limits:\nas ρ → 0 and c → 1 all solutions approach optimal transport solution, similarly as ρ → ∞ and ∀ c ≥ c\n∗\nmass assignments correspond to nearest neighbor solutions discussed in the Section 2.1. For PU learning experiments on MNIST/EMNIST and CIFAR10, we used the POT toolbox Flamary et al. (2021) function optim.semirelaxed_cg to obtain the solutions for semi-relaxed optimal transport problem with TV and squared-ℓ2 penalties via the Frank-Wolfe conditional gradient algorithm.\n\n## D Appendix: Relation Between Entropically Regularized Subset Selection And Unbalanced Optimal Transport\n\nUsing the notation from our paper, we have adapted the formulations for the unbalanced optimal transport with asymmetric penalties from the work by Séjourné et al. (2019). For φ1, φ2, the generating functions for the divergence penalty functions on the marginals, and entropic regularization parameter γ > 0, the unbalanced optimal transport for discrete probability measures µ, ν, and cost M is\n\nUOT (φ1,φ2) γ(µ, ν) = min P ∈R m×n ≥0 ⟨P ,M⟩ + Dφ1 (P 1∥µ) + Dφ2 (P ⊤1∥ν) + γKL(P ∥µν⊤) = min P ∈R m×n ≥0 ⟨P ,M⟩ + Xm i=1 µiφ1( Pj Pij µi) +Xn i=1 νiφ2( Pi Pij νi) + γ( X ij Pij (log(Pij ) − log(µiνj )) − X ij Pij + X ij µiνj ) = min P ∈R m×n ≥0 ⟨P ,M⟩ + Xm i=1 µiφ1( Pj Pij µi) +Xn i=1 νiφ2( Pi Pij νi) + γ ⟨P , log(P ) − 1m×n⟩ −X i µilog(µi) +X j X i Pij log(νi) + 1 .\n\n \nFigure 18: (a) The variation of mass assignment vector ν\n∗ with c for toy problem in Figure 2, using SS-Entropic, SS-Bregman, and the subset selection linear program 2 is solved using CVXPY Agrawal et al.\n\n(2018); Diamond & Boyd (2016). All divergence regularized semi-relaxed problems are also solved using CVXPY. (b) Entropy versus cardinality selected set of points. It can be observed that SS-Bregman and ℓ∞\nsolutions match each other until a support cardinality of around 40. This is due to fact that subset selection solution matches a unique ℓ∞ penalized solution for all values of c > 2.\n\n \nFigure 19: (a) Entropy of mass assignment vector ν\n∗ with c for toy problem in Figure 2. (b) Entropy as function of scaling parameter for penalized relaxed-OT problems c(ρ), obtained using formula c(ρ) =\nmax i ν\n∗\ni\n(ρ)\nνi, which implies that for sufficiently large values of ρ, c(ρ) = c\n∗.\n\n\n\n\n\n\n\n*j\n\n\n\n\n\n\n\n0.0202-penalty\nFigure 20: Mass assignments ν\n∗to unlabeled data points in PU learning on MNIST/EMNIST discussed in Section 3.3.1 across c for SS-Bregman and across ρ for TV and ℓ2 penalties.\n\n\n\nFigure 21: Mass assignments ν\n∗to unlabeled data points in PU learning on CIFAR-10 neural network representations, discussed in Section 3.3.2 across c for SS-Bregman and across ρ for TV and ℓ2 penalties.\nWhen $\\varphi_1(x)=\\iota_{\\{1\\}}(x)=\\begin{cases}0,&x=1\\\\ \\infty,&\\text{otherwise}\\end{cases}$ is the $\\varphi_2(x)=\\iota_{[0,c]}(x)=\\begin{cases}0,&x\\in[0,c]\\\\ \\infty,&\\text{otherwise}\\end{cases}$ is a range of marginal, then\nis the indicator function for the ratio of the target marginal and\nis a range constraint to the interval of [0, c] for the ratio of the source\n$${\\mathcal{U O T}}_{\\gamma}^{(\\varphi_{1},\\varphi_{2})}(\\mu,$$\nγ(*µ, ν*) = min\n$\\nu)=\\min_{\\mathbf{P}\\in\\mathbb{R}_{\\geq0}^{n\\times n}}\\langle\\mathbf{P},\\mathbf{M}\\rangle+\\gamma\\left(\\langle\\mathbf{P},\\mathbf{\\log(P)-1}_{m\\times n}\\rangle-\\langle\\mathbf{\\mu},\\mathbf{\\log(\\mu)}\\rangle+\\langle\\mathbf{P}^{\\top}\\mathbf{1}_{m},\\mathbf{\\log(\\nu)}\\rangle+1\\right)$ s.t. $\\mathbf{P1}_{n}=\\mathbf{\\mu},\\ \\mathbf{P}^{\\top}\\mathbf{1}_{m}\\preccurlyeq\\nu$. \nWith these choices of φ1, φ2 and a uniformly weighted source νi =\n1 n∀i ∈ [n], we can relate S\n(γ)\np (*µ, cν*)\nto UOT (φ1,φ2)\nγ(*µ, ν*), since ⟨P\n⊤1m, log(ν)⟩ =Pj Pi Pij log(νi) = − log(n), then UOT (φ1,φ2)\nγ(*µ, ν*) =\nS\n(γ)\np (*µ, cν*) + γ(H(µ) − log(n) + 1), where H(µ) = −⟨µ, log(µ)⟩ = −Pi µilog(µi). The uniformly weighted source is necessary to eliminate the bias towards ν for the marginal that the Kullback-Leilber divergence penalty induces compared to our entropy penalty, which only considers ν through the constraint.\n\n## E Pu Learning Toy Comparison Test\n\nIn order to show evidence that subset selection, as a semi-relaxed partial optimal transport, can consistently outperform fully-relaxed partial optimal transport on PU learning, we plot the mean accuracy for a range of examples using of the same design as those in Figure 22.\n\n\n\n, 2\n]\nFigure 22: PU learning accuracy on toy data using subset selection and fully-relaxed partial optimal transport. Subset set consistently outperforms fully-relaxed on this toy data. Data is generated as [r cos(ψ), r sin(ψ)] where the radius r is drawn from a truncated exponential distribution with density 2 exp(−r)ı[log 2,∞)(r) and ψ is uniform over the set of angles [−\nπ 2\n,\nπ 2\n] for the source and different subsets for the target as indicated in the legend. For each run m = 100 positive points are drawn and the size of the unlabeled sample is varied from n = 100 to n = 800 while are all from the third class. The solutions are obtained using the known number of positives n+ out of the n source points, setting c =n n+\nfor our semi-relaxed approach and s =\nn+\nnfor the fully-relaxed partial optimal transport. Confidence intervals show\n±1 standard deviations of the accuracy across 100 runs.\n\n## F Appendix: Neural Network Model Architectures\n\nFor semi-supervised learning, we used neural networks to both perform the classification and provide a\n\n\n\n\n\n\n\n\n\n\n\n learning representation space in which to perform the optimal transport to assign pseudo-labels to both unlabeled points. For the CIFAR-10 data set, we used the ResNet-18 (He et al., 2016) architecture. For MNIST and Fashion-MNIST, we used custom, but simple, model architectures. Both architectures contain two convolutional layers, followed by three fully-connected layers. The model used to train Fashion-MNIST\n(FMNIST) classifier contains additional batch-normalization layer between the convolutional layers. Optimization algorithms along with related hyper-parameters are in Section 3.4. The code below details the exact architectures along with types and shapes of all transformations.\n\n1 import torch 2 import torch . nn as nn 3 import torch . nn . functional as F\n4 5 class MNIST_classifier ( nn . Module ) :\n6 def __init__ ( self ): 7 super () . __init__ ()\n8 self . conv1 = nn . Conv2d ( in_channels =1 , out_channels =5 , kernel_size =(5 ,5) )\n9 self . conv2 = nn . Conv2d ( in_channels =5 , out_channels =1 , kernel_size =(5 ,5) )\n10 self . fc1 = nn . Linear (400 , 128)\n11 self . fc2 = nn . Linear (128 , 64) 12 self . fc3 = nn . Linear (64 , 10)\n13 14 def forward ( self , x): 15 x = F . relu ( self . conv1 (x) )\n16 x = F . relu ( self . conv2 (x) )\n17 x = torch . flatten (x , 1) 18 x = F . relu ( self . fc1 (x))\n19 x_rep = F. relu ( self . fc2 (x) )\n20 x = self . fc3 ( x_rep ) 21 return x , x_rep 22 23 class FMNIST_classifier ( nn . Module ): 24 def __init__ ( self ): 25 super () . __init__ () 26 self . conv1 = nn . Conv2d ( in_channels =1 , out_channels =32 , kernel_size =(5 , 5) )\n27 self . batchN1 = nn . BatchNorm2d ( num_features =32)\n28 self . conv2 = nn . Conv2d ( in_channels =32 , out_channels =64 , kernel_size =(5 , 5) ) 29 self . fc1 = nn . Linear ( in_features =64*4*4 , out_features =128)\n30 self . fc2 = nn . Linear ( in_features =128 , out_features =64)\n31 self . fc3 = nn . Linear ( in_features =64 , out_features =10)\n32 33 def forward ( self , x):\n34 x = self . conv1 (x) 35 x = F . relu (F. max_pool2d ( input =x , kernel_size =2 , stride =2) )\n36 x = self . batchN1 (x)\n37 x = self . conv2 (x) 38 x = F . relu (F. max_pool2d ( input =x , kernel_size =2 , stride =2) )\n39 x = torch . flatten (x , 1)\n40 x = F . relu ( self . fc1 (x))\n41 x_rep = self . fc2 ( x)\n42 x = self . fc3 ( x_rep ) 43 return x , x_rep Listing 1: Models used for training classifiers for MNIST and Fashion-MNIST\n\n\n\n\n\n\n\n\n\n\n\n\n\n"},"reviews":{"kind":"string","value":"Review 1:\nSummary: The article deals with partial optimal transport, that is, when the target and source measures do not maintain the entirety of their mass over the transport. In particular, the authors focus on a specific case where the measures are discrete and uniform, to propose a methodology for a particular case where it is only for the source that only part of the mass is transported. \n\nThe paper is well presented with sufficient experimental validation\n\nStrengths and Weaknesses: The paper is very well written; it is extensive, but that is because the authors give a great amount of detail, which I consider to be positive in this type of article, blending concepts with applications. \n\nThe conceptual part is very clearly presented, the authors re-derive the Sinkhron approach for the particular case of partial OT, and the procedure is presented in a detailed manner to the reader.\n\nThe experimental validation is superb. The treatment and explanation of the experimental aspects are very deep and informative. \n\nSome points can be improved in the paper, but they are mainly related to the presentation and not methodology (so listed in the next section of the review) **except the following ones**: \n\n- how can the scaling factor be chosen? The article provides multiple instances to compare how the methodology performs for different choices of $c$. However, in real applications, $c$ is not known and needs to be chosen. By arbitrarily choosing this factor, the method will leave parts of the support unmatched to the target (as shown in the first experiment), so a naive choice can lead to misleading conclusions. Can $c$ be chosen in a way that is more principled than just the ratio between the points one wants to match and all the points? (because one could not know how many points should be matched)\n\n- Also, why is only the uniform case considered? In the toy example, the application of the method to this kind of distribution appears to result in removing the parts of the support that are far away from the target, to then rescale the source and apply complete OT\n\nRequested Changes: i) Please refer to my comments above\n\nii) Sec 2.2: *Cuturi applied entropic regularization* please refer to works, not persons\n\niii) Secs 2.2, 2.3, 2.4 seem to propose a methodology and then the next section refutes them to propose something \"better\". In my opinion (if I am correct) this weakens the proposal of the paper. If possible, please reformulate. \n\niv) Possibly rename \"algorithm 1 / algorithm 2\" so that in the experiments they are not referred to as such. Idea: SS-entropic and SS-Bregman (or other appropriate names)\n\nv) Sec 3.2 reads *It can be observed that the results for larger values of c are smoother and sharper as compared to the optimal transport case c = 1.* Can something be **smoother and sharper** at the same time?\n\nBroader Impact Concerns: NA\n\n==================================================\n\nReview 2:\nSummary: This paper considers (a special case of) the _partial optimal transport_ (POT) problem, as introduced by Figalli (2010). Namely, in this work, authors consider two probability measures $\\mu,\\nu$ (typically discrete), \n\n$$ \\min_{P} \\braket{P,M}, $$\n\nfor some cost matrix $M$, provided that the first marginal $1^T P$ of $P$ should be $\\mu$ (so the total mass of $P$ must be $1$), but its second marginal is only required to satisfy $P 1 \\leq c \\nu$ for some prescribed $c \\geq 1$ (hyper-parameter). \nThe main idea is that the second marginal of the optimal solution to this linear program, denoted by $\\nu^*_c \\leq c \\nu$ is a probability measure that can be seen as a \"subset selection\" problem. \n\nEven though this linear program can be solved by standard (simplex, interior points) methods, better computational properties can be achieved by considering its entropic regularization, adding a term $+ \\gamma \\braket{P, \\log(P) - 1}$. Standard duality considerations yields a Sinkhorn-like algorithm with respect to two dual variables $(\\alpha,\\beta)$, where the standard log-sum-exp iteration holds for $\\alpha$ while for $\\beta$ a projection step on the non-negative ortant is required (as a consequence of the submarginal constraint). \n\nAs typical in entropic OT, the resulting (partial) transport plan has a density with respect to $\\mu \\otimes \\nu$, which is somewhat in contradiction with the \"subset selection\" motivation of the problem. \nTo mitigate this effect, authors follow the work of (Xie et al., 2020) and rely on an additional Bregman projection scheme , which can also be solved using a Sinkhorn like algorithm, yielding a sequence of transportation plans $P^{(t)}$ which are guaranteed to converge toward a solution of the _unregularized_ problem (with sparse support in general).\n\nAuthors support the feasibility and usefulness of their approach on an extensive set of experiments.\n\nStrengths and Weaknesses: # Strengths\n\n- Fairly well motivated problem / solution. \n- Theory and experiments seem sounded overall. \n- Large set of experiments, and performances seem convincing as far as I can tell. \n\n\n# Weaknesses\n\n- As far as I can tell, from a theoretical perspective, the paper only plugs together known tools (Partial OT from Figalli, entropic OT a la Cuturi, Bregman projection from Xie et al). In particular the first two points of the contributions (page 2) are disputable as, unless I am missing something, there is no real novel problem/method introduced in this paper from a theoretical perspective. \n- Related to the previous point, I would have appreciated more insights on the problem considered. Namely, the paper consider a subcase of Figalli's partial OT problem (transporting the whole $\\mu$ and considering discrete measures with mostly uniform mass), but as far as I can tell the specificity of this formulation is never used (aside from the experimental intuition); everything developed in this paper could be adapted to POT faithfully, isn't it? \n- A more detailed comparison with state-of-the-art theoretical and/or computational approach would be important. For instance, [this work](https://arxiv.org/pdf/1910.12958.pdf) by Séjourné et al., already propose a Sinkhorn scheme for Partial OT (which correspond to taking the total variation as marginal divergence) and I think that the dual problem / Sinkhorn algorithm derived is substantially equivalent to their algorithm. The work of Chapel et al. is not described either while being used as a major competitor in experiments. Right now, it is hard to assess how \"fair\" the comparison is. \n- The choice of the parameter $c$ is not discussed while it is showcased to play an important role in terms of experimental results. Intuitively, it interpolates between standard OT (possibly with entropic regularization) and \"nearest-neighbor Transport\", the latter being simpler to compute than the former. As such, I think that a natural question (that would be an interesting contribution as it would depend on the specific context of the paper) would be to know whether the computational complexity decreases as $c$ increases (similarly to the entropic regularization parameter, see (Feydy et al., 2019)). \n- Clarity can be improved in some places, see the \"Minor remarks\" below.\n\n\n# Minor remarks and suggestions\n\n- While the Remark in page 4 feels interesting, it is never leveraged later on as far as I can tell. If not, I think it could be deferred to the Appendix. If it plays an important role (e.g. as a way to somewhat bound the support of $\\nu^*_c$ or so), this should be highlighted. \n- [clarity] I'm a bit failing to see the point of introducing the two settings $P1 \\leq c \\nu$ and $P 1 \\leq \\zeta$, given that we could always write the later $P 1 \\leq \\|\\zeta\\| \\underbrace{\\frac{\\zeta}{\\|\\zeta\\|}}_{=: \\nu}$. I think that it would make things clearer to stick with one or another formulation, and only briefly mention the equivalence between these two settings. \n- [clarity] The variable $\\xi^{(k)}$ is only defined in the Algorithm 1; which makes equation (13) hard to parse at first glance. \n- A more extensive presentation of the work of (Xie et al., 2020) would be useful given the role it plays in the current work. \n- [clarity-ish] In Section 2.4, I think the presentation would be enhanced by immediately saying that we have access to a close form for $\\theta$, but that relying on mini-batch gradient descent has merits (I was first really confused about the need of running a gradient descent for a simple weighted linear regression). \n- [clarity] In the experiments (Figure 3 and 4), what are the 500 iterations on the $x$-axis? The caption only mentions 200 inter and outer iterations. From page 26, it seems to be a comparison point between Alg. 1 and 2. Please clarify this. \n- The overhead (if any) of using Algorithm 2 instead of Algorithm 1 is not discussed as far as I can tell (even in the supplementary material).\n\nRequested Changes: - [Critical] Having a better understanding of the benefits of the proposed setting (having one exact marginal, and one submarginal ; working with discrete measures). And if there is no such benefits, re-write the theoretical part of the paper in a more general setting. \n- [Critical] A better presentation of related works to make the contribution of the present paper more salient. \n- [Enhancement] Improving the clarity of the paper (see \"Minor remarks and suggestion\" above).\n\nBroader Impact Concerns: No broader impact concerns as far as I can tell.\n\n==================================================\n\nReview 3:\nSummary: The paper deals with the problem of solving an instance of a semi-relaxed optimal transport problem, which differs from the original OT problem by relaxing the constraint on the marginal of one of the distributions. Algorithms to solve the problem are described, and a set of experiments, ranging from toy examples that illustrate the behavior of the solution, to color transfer, PU learning, and semi-supervised learning, is provided.\n\nStrengths and Weaknesses: **Strengths of the paper**\n\nThe paper is mostly comprehensive in the formulation of the problem and on the description of the algorithms. The experiments provided showcase the behavior of the proposed method in a wide variety of contexts, from toy examples that illustrate the behavior of the solution (in particular, the impact of the parameter that tunes the \"partialness\" of the solution) to experiments in image and machine learning contexts, showing that the problem that is addressed is useful in a number of applications.\n\n\n**Weaknessess of the paper**\n\nWhile relaxing one of the two marginals' constraints has previously been proposed in existing literature, there is a lack of discussion regarding how the proposed approach positions itself in relation to the current state of the art. For instance, [1], p. 2, paragraph \"Relaxed Weighted OT,\" relaxes the matching constraint towards one distribution by introducing a set of relaxation parameters to increase (or decrease) the capacity value of each sample; [2], eq. (14), defines the \"semi-relaxed OT,\" in which one of the marginals is free.\n\nIn the context of support subset selection, one advantage of using partial OT is that it defines a set of active regions, that is to say a subset of the samples. As stated in section 2.3, the entropic formulation of the problem yields denser couplings, which may be detrimental to the identification of active regions. A Bregman Proximal-point method is provided in Algorithm 2; as mentioned in the paper, it can result in infeasible solutions and/or high computational time. One might then wonder why not utilize a different yet related problem for active region identification that would inherently leads to sparse solutions, such as unbalanced OT with an $\\ell_2$ penalization rather than an $\\ell_1$ one, as in [2]. \n\nFurthermore, the experimental section lacks comparison with competitors.\n\n\n[1] RABIN, Julien, FERRADANS, Sira, et PAPADAKIS, Nicolas. Adaptive color transfer with relaxed optimal transport. IEEE international conference on image processing (ICIP), 2014.\n\n[2] BLONDEL, Mathieu, SEGUY, Vivien, et ROLET, Antoine. Smooth and sparse optimal transport. International conference on artificial intelligence and statistics, 2018.\n\n\n*Other comments*\n- In the introduction, the second paragraph discussing partial OT might be misleading. When $\\mu$ is discrete, the TV norm is equivalent to the $\\ell_1$ norm of vectors, hence it comes down to a partial OT problem.\n- On page 4, the claims regarding extreme points and breakpoints lack supporting arguments.\n- Regarding the PU learning experiments, it remains unclear why the relative ranking provided by the entropic formulation outperforms the unconstrained one. Could you please elaborate on this point?\n\nRequested Changes: See weaknesses about the position of the paper with respect to the literature and justification of the formulation (entropic formulation of a partial OT problem).\n\nBroader Impact Concerns: There seems to be no ethical implications of the work.\n\n==================================================\n\nMetareview:\nRecommendation: Accept with minor revision\n\nComment: All reviewers agreed that while this work might not be totally original, it provides a new formulation for partial OT and interesting experiments, they all lean toward accept.\n\nSome minor comments from the final discussion with reviewers that should be taken into account in the camera ready:\n\n- p2 \"Recent works have proposed algorithms for computing the entire scaling or regularization path of solution [...] semi-relaxed unbalanced optimal transport (Chapel et al., 2021)\": the work of (Chapel et al., 2021) also computes the regularization path of the UOT problem, not only the semi-relaxed version.\n- in Fig. 2, I would recommend scaling the point size wrt the amount of mass points to better illustrate the creation of mass.\n- Fig 3 and 4 : the points are not visible (for all c>1 and theta opdate) in the pdf reader from google chrome (they are visible in local PDF reader) please solve this problem for final version.\n- It is claimed \"However, the main difference with this relaxation [(Chapel et al 2020)] as compared to our approach (or other semi-relaxed approaches) is that points in the target may lose mass or be completely dropped\". It is still unclear why the proposed method and the one of Chapel et al, 2020 should give different accuracy results in a PU learning context as \"accuracy is evaluated by assigning label 1 to the largest mass assignments to and label 0 to the remaining mass assignment\". Please discuss and give more details about this.\n- Fig. 9 (a): It is not clear why the-sampling is not the same.\n\n==================================================\n\n"}}},{"rowIdx":96,"cells":{"paper":{"kind":"string","value":"# C2P: Featuring Large Language Models With Causal Reasoning\n\nAnonymous authors Paper under double-blind review\n\n## Abstract\n\nCausal reasoning is the primary bottleneck that Large Language Models (LLMs) must overcome to attain human-level intelligence. To address this, we introduce the Causal Chain of Prompting (C2P) as the first reasoning framework that equips current LLMs with causal reasoning capabilities. C2P operates autonomously, avoiding reliance on external tools or modules during both the causal learning and reasoning phases, and can be seamlessly implemented during the training or fine-tuning of LLMs. Experimental results across various benchmark datasets demonstrate a significant improvement in causal learning and subsequent reasoning accuracy of LLMs. We illustrate how C2P enhances LLMs' ability to causally reason in real-world scenarios, addressing complex problems in fields such as healthcare, medicine, economics, education, social sciences, environmental science, and marketing.\n\nWith few-shot learning, GPT-4 Turbo using C2P with as few as six examples achieves significant performance improvements, boasting over a 33% increase in reasoning accuracy over the most state-of-the-art LLMs, which perform nearly randomly in similar circumstances.\n\nThis demonstrates the transformative potential of integrating C2P into LLM training or fine-tuning processes, thereby empowering these models with advanced causal reasoning capabilities.\n\n## 1 Introduction\n\nRecent advancements in Large Language Models (LLMs) have impacted existing AI paradigms and heightened expectations regarding AI's capabilities (Achiam et al., 2023; Brown et al., 2020). Despite significant architectural differences in LLMs, they generally produce outputs based on the most likely results learned from vast amounts of training data (Vaswani et al., 2017). This enables them to acquire extensive knowledge ranging from common sense to specialized domains such as mathematics and science (Jiralerspong et al.,\n2024). Despite this, the inefficiency of LLMs in addressing causal reasoning questions remains their primary bottleneck and simple tasks can completely break down reasoning in state-of-the-art LLMs (Nezhurina et al., 2024). Additionally, studies such as Kalai & Vempala (2023); Xu et al. (2024) have demonstrated that, despite the training data containing numerous examples of interventions, outcomes, and explanations, as well as similar tasks, hallucinatory responses persist and there is a lack of causal reasoning capability. As a result, while they may talk causality, they are not causal (Zečević et al., 2023). This deficiency represents a fundamental drawback of LLMs as AI systems compared to human intelligence, which goes beyond mere correlations and depends on causal relationships for decision-making (Penn & Povinelli, 2007; Anwar et al., 2024). Recently, answering cause and effect questions with LLMs has gained extensive interest (Shin et al., 2020; Jin et al., 2023b; Ashwani et al., 2024). To address this issue, it is important to note that LLMs have been utilized in conjunction with external tools to extract causal structures, as demonstrated in (Jiralerspong et al., 2024). However, their architectures lack specialized modules specifically designed to enhance the understanding of cause-and-effect relationships within their outputs (Wang et al., 2023; Imani et al., 2023).\n\nAside from studies that reason causally based on the knowledge that is already in their training data (Petroni et al., 2019; Jiang et al., 2020) and the ones that use LLMs in causality (Kıcıman et al., 2023; Zhang et al.,\n2023; Feder et al., 2024; Khatibi et al., 2024), chain-of-thought prompting is presented in (Wei et al., 2022) as\n\n \nFigure 1: Example of the standard prompting vs few-shot learned GPT-4 with C2P in open problem in astrophysics (Pasquato et al., 2023)\n.\none of the initial attempts in enhancing the reasoning in LLMs that shows improvement based on the data of the given query. However, LLMs still struggle with rigorous numerical and abstract reasoning among many other tasks (Xu et al., 2023). For example, a recent work, Causal Reasoning Assessment Benchmark (CRAB, Romanou et al., 2023), is designed to evaluate the causal understanding of events in real-world narratives. This study demonstrated that most systems perform poorly in identifying cause-and-effect tasks. Similarly, in (Jin et al., 2023b), the CORR2CAUSE dataset is introduced and demonstrated that current models often perform no better than random chance when tasked with causal questions. Following that, in (Jin et al.,\n2023a), the CLADDER dataset is introduced to asses Average Treatment Effects with LLMs and it is demonstrated that these models struggle with causal tasks. In their study, with the implementation of the proposed framework, CAUSALCoT, progress has been made in evaluating the average treatment effect where LLMs are provided with a collection of causal graphs and various types of queries (associational, interventional, and counterfactual), such as those included in the CLADDER dataset. More recently, in (Ashwani et al., 2024), a novel architecture called the Context-Aware Reasoning Enhancement with Counterfactual Analysis\n(CARE-CA) framework is presented to enhance causal reasoning and explainability. Their proposed framework incorporates an external explicit causal detection module with ConceptNet (Speer et al., 2017) and counterfactual statements, as well as implicit causal detection through LLMs, showing progress in causal reasoning in short and simple queries. Several other works at the intersection of causal inference and LLMs are discussed in an extensive survey by Liu et al. (2024).\n\nAs argued by Pearl (1995), causal Directed Acyclic Graphs (DAGs), along with d-separation, allow for the investigation of cause-and-effect relationships without relying on structural equation models in computational studies. Inspired by Pearl's foundational work, we propose a novel framework, named Causal Chain of Prompting (C2P), to address the inefficiencies of LLMs in handling causal queries (see Figure 1 as an example). We show that by identifying the adjacency matrix, equivalent to and instead of the causal DAG in Pearl's work, the causal relationships among the variables in the premise's cause-and-effect relationships can be effectively reasoned within the context of language models. Contrary to the existing research in addressing weaknesses of LLMs in causal reasoning, the C2P operates autonomously, avoiding reliance on external tools or modules during both the learning and reasoning phases to answer causal questions. Additionally, C2P can easily be implemented in the training or fine-tuning process of LLMs. C2P includes five simple main sub-tasks, as follows: (1) Prompting for extracting the random variables from the provided data. (2)\nPrompting for extracting all conditional and unconditional relations and cause and effect relations specifically mentioned among the random variables. (3) Prompting to create the initial adjacency matrix with values 1 for all elements except the diagonal elements and the ones corresponding to effect-cause relations (the cause and effects elements are also one). (4) Prompting of the conditional and unconditional independencies and identification of the colliders, step by step, to extract the causal adjacency matrix; and, (5) Prompting for reasoning questions or hypotheses. To evaluate the accuracy and reliability of implementing the C2P on LLMs, we initially assess it using publicly available synthetic datasets such as (Jin et al., 2023b). Subsequently, we evaluate it in more realistic and complex scenarios in real-world problems presented in (Pearl &\nMackenzie, 2018) and (Pasquato et al., 2023). Moreover, we present results on few-shot learning with C2P\non both synthetics and realistic scenarios. Contributions. In this work, we present several important contributions to facilitate causal reasoning in language models. Concretely, 1. We introduce the C2P framework as the first reasoning framework to equip LLMs with the ability of causal reasoning within real-world scenarios, without relying on external tools.\n\n2. Through extensive experiments with our framework, we demonstrate a significant improvement of LLMs in causal reasoning in various benchmarks. Additionally, we examine the performance of the C2P framework on more complex and real-world scenarios in various domains.\n\n3. We perform few-shot learning experiments in section 4 with GPT-4 Turbo using our C2P framework and show how integrating C2P during the training or fine-tuning of LLMs can revolutionize existing models. This approach equips them with causal reasoning capabilities akin to the transformative impact of 'Chain of Tought' (Wei et al., 2022) on LLMs, as discussed by Chung et al. (2024).\n\nThe codes for sections 3 and 4 are publicly available at https://anonymous.4open.science/r/C2P-5C2A/.\n\n## 2 Preliminaries On Causal Learning And Reasoning\n\nTo lay the groundwork for our framework, understanding the fundamental concept of cause and effect is paramount. A foundational and widely acknowledged principle in causality is the one proposed by Reichenbach, which posits the following:\nPrinciple 1 (Common cause principle (Reichenbach, 1991)) If two random variables X1 and X2 are statistically dependent, i.e., X1 ̸⊥⊥ X2, then there exists a third variable X3 that causally influences both. (As a special case, X3 may coincide with either X1 or X2.) Furthermore, this variable X3 screens X1 and X2 from each other in the sense that given X3*, they become independent,* X1 ⊥⊥ X2|X3.\n\nMarkov Property: The Markov property in a DAG G states that each node Xiis conditionally independent of its non-descendants given its parents, namely Xi ⊥⊥ NonDe(Xi)|Pa(Xi), where NonDe(Xi) denotes the non-descendants of Xi excluding itself, and Pa(Xi) denotes the parents of Xi. Using the Markov property, we can factorize the joint distribution of all the nodes in the graph into P(X1*, . . . , X*N ) = QN\ni=1 P(Xi|Pa(Xi)).\n\nTo infer the causal graph from probability distributions, a common assumption is faithfulness, namely the validity to infer all the d-separation sets in the graph from the independence relations in the probability distribution. In our work, we also take this broadly taken assumption which holds for most real-world scenarios. Directed Acyclic graph: A graph G is called a partially directed acyclic graph (PDAG) if there is no directed cycle, that is, if there is no pair (*j, k*) with directed paths from j to k and from k to j. G is called a directed acyclic graph (DAG) if it is a PDAG and all edges are directed.\n\nMarkov Equivalence of Graphs: We denote two DAGs as Markov equivalent if they induce the same joint distribution P(X). The set of DAGs that are Markov equivalent to each other is called a Markov equivalence class (MEC). Causal graphs in the same MEC can be easily identified since they have the same skeleton (i.e., undirected edges) and V-structures (i.e., structures in the form of X1 → X2 ← X3 where X1 and X3 are not connected).\n\nThe be able to perform any level of causal reasoning, the Ladder of Causation is proposed with three main rungs, \"seeing\", \"doing\", and \"imagining\", in (Pearl & Mackenzie, 2018). To fulfill all rungs of the ladder, more information on the causal structure is needed, which can be mostly provided with structural causal models (SCM) (Assuming all the assumptions are satisfied) (Bareinboim et al., 2022). The current causal discovery methods are divided into two groups, constraint-based algorithm such as the PC algorithm, which has a PDAG as an output, that represents the MEC of the true underlying graph and is the best result such methods can extract. Second is score-based methods such as NOTEARS (Zheng et al., 2018), GOLEM (Ng et al., 2020), and DAGMA (Bello et al., 2022), among many others, that extract a DAG that mostly fits the data and demonstrated very high accuracy in extracting Bayesian networks. Score-based methods require solving an optimization problem based on the numerical values of variables, rendering them impractical for implementation in LLM models. PC algorithm: Following these materials, the PC algorithm is developed based on Reichenbach's common cause principle and Markov property as follows for a causal mechanism with p random variables (Spirtes et al., 2001);\n(i) Form a complete undirected graph\n(ii) Eliminate edges between variables that are unconditionally independent\n(iii) For each pair of variables (X1, X2) having an edge between them, and for each variable, X3 with an edge connected to either of them, eliminate the edge between X1 and X2 if X1 ⊥⊥ X2|X3\n(iv) For each pair of variables X1, X2 having an edge between them, and for each pair of variables\n{X3, X4} with edges both connected to X1 or both connected to B, eliminate the edge between X1 and X2 if X1 ⊥⊥ X2|{X3, X4}.\n\n(v) For each triple of variables (X1, X2, X3) such that X1 and X2 are adjacent, X2 and X3 are adjacent, and A and C are not adjacent, orient the edges X1 − X2 − X3 as X1 → X2 ← X3, if X2 was not in the set conditioning on which X1 and X3 became independent and the edge between them was accordingly eliminated. We call such a triple of variables a V-structure.\n\n(vi) For each triple of variables such that X1 → X2 − X3, and X1 and X2 are not adjacent, orient the edge X2 − X3 as X2 → X3. This is called orientation propagation.\n\nd-separation and do calculus (Pearl, 1995): do-calculus, developed by Judea Pearl and others, is a set of rules used to transform and manipulate causal expressions within causal diagrams (or graphical models).\n\nIt is a generalization of do-calculus, which is itself a formal tool used to reason in causal relationships from a mixture of experimental and observational data. do-calculus consists of three main rules that enable one to rewrite expressions involving interventions (typically represented as do(x), indicating an intervention to set variable X to value x):\n1. **Insertion/deletion of observations**: If a variable X2 is independent of X1 given a set of other variables X3, then the conditional distribution of X2 given do(X1) and X3 is the same as the conditional distribution of X2 given X3 alone. This rule can be applied when X1 and X2 are d-separated by X3 in the graph modified by removing all arrows pointing into X1.\n\n2. **Action/observation exchange**: If X2 is independent of X1 given X3 and another set W where X1 is intervened, then the graph can be modified to reflect this exchange.\n\n3. **Insertion/deletion of actions**: This rule states that if intervening on X1 does not affect X2 after accounting for X3, the intervention can be ignored in the expression.\n\nThese rules are crucial for determining the identifiability of causal effects from data, allowing researchers to reason in causal relationships from a combination of experimental and observational data.\n\nd-Separation is a criterion used in the analysis of Bayesian networks to determine whether a set of variables X1 is independent of another set of variables X2, given a third set of variables X3. This concept is foundational in understanding the flow of causal influence in graphical models and helps in deciding whether a path between two variables is \"blocked\" or not by conditioning on other variables. The basic rule of d-separation is: A path between two variables is blocked if it includes an intermediate variable that is a collider and is not conditioned on or a non-collider that is conditioned on. Here, a **collider** is a variable that has inward arrows from two other nodes (i.e., X1 → X2 ← X3), whereas a **non-collider** does not meet this criterion.\n\n## 3 Developing Causal Chain Of Prompting\n\nTo develop C2P framework, we aim to extract the adjacency matrix of variables in a given premise. The adjacency matrix acts as equivalent to and instead of the causal DAG in Pearl's work leading to answering the causal reasoning question. In developing the Causal Chain of Prompting, we use a few steps similar to the PC algorithm for the given premise.\n\n## 3.1 Causal Chain Of Prompting (C2P)\n\nThe C2P frameworks consist of 5 main sub-tasks as follows, for learning and reasoning cause and effect relation for the given premise; 1. Prompting to extract the random variables in the provided data. 2. Prompting to extract all the cause and effect relations along with all conditional and unconditional relations among the random variables specifically mentioned in the given premise.\n\n3. Prompting to create an initial adjacency matrix where all elements are 1, except for the diagonal elements and the elements corresponding to the cause-and-effect relationships specifically mentioned in the given premise (extracted in sub-task 2).\n\n4. Prompting of the conditional and unconditional independency valuation and identifying the colliders, step by step, to extract the causal PDAG.\n\n5. Prompting for cause-and-effect questions or hypotheses.\n\nEach sub-task in C2P includes multiple steps and overall, to execute the frameworks, 9 main prompts have to be completed. The exact prompts for all the steps are provided in the Appendix, A.1.\n\n## 3.2 Few-Shot Learning Agenda\n\nLLMs have powerful zero-shot capabilities, yet they might struggle with complex tasks with insufficient examples in the training process. In these situations, few-shot learning is a versatile and efficient technique for in-context learning that can be used to quickly adapt LLMs to new tasks and significantly enhance their performance (Min et al., 2022; Touvron et al., 2023). This approach involves providing several examples with desired answers, which condition the LLMs to produce the correct response for new instances with similar patterns. The few-shot learning process of C2P is similar to the previous subsection, with the prompts being based on the more abstract prompts presented there (Due to the token limitation). The prompts and an example of the given story are included in the Appendix, A.2. Depending on the token limitation of employed LLMs, the number of given examples (shots) varies. Comparing the token limitation of GPTs, we use GPT-4 Turbo, which has a 30000 token limit. Steps from other constraint-based methods can be used instead or along with some steps of the PC algorithm in C2P methods, comprehensively discussed in (Glymour et al., 2019). For instance, steps in the FCI method can be used in cases where the causal sufficiency assumption is violated, i.e., latent variables and selection bias may be present (Spirtes et al., 1995). Additionally, it is important that the classifiers proposed by Ceraolo et al. (2024) that formalize the definition of causal questions and establish a taxonomy for finergrained classification can be used before our framework to identify the causal question as a prerequisite of our method.\n\n## 4 Results Of The Experiment For Featuring Llms With Causal Reasoning 4.1 Experimental Setup\n\nThe experimental setup is divided into two data groups: synthetic dataset and realistic scenarios. In a first synthetic dataset, and similar to the experimental setup of (Jin et al., 2023b), we set up the following list of LLMs for the experiments on the CORR2CAUSE dataset. Various types of LLMs are tested in (Jin et al., 2023b) and we only include powerful LLMs in our experiment to avoid reproducing the same results. To test existing LLMs, we first include two BERT-based NLI models in the transformers library (Wolf et al., 2020): BART (Lewis et al., 2019), DistilBART (Shleifer & Rush, 2020). We also evaluate the recent, more efficient models, LLaMa3-8B (Touvron et al., 2023) and LLaMa3-70B (Taori et al., 2023).\n\nWe assess the GPT-3.5 (i.e., ChatGPT), and the latest GPT-4 (OpenAI, 2023) by April 2023, using the OpenAI API (https://openai.com/api/) with temperature 0. We employed the most recent GPTO as well.\n\nThen, we assess C2P on a proposed synthetic dataset, similar to the one in (Jin et al., 2023a), but more complex and with less information provided on the premises. Then we test the reliability of C2P in different numbers of nodes. Due to the limitation of tokens for different versions of GPTs, we employed GP4-Turbo which has a 30000 maximum token limit. We selected 50 samples at random from the corr2cause dataset for our experiment, ensuring a balanced distribution with 50 \"Yes\" and 50 \"No\" answers. In the corr2cause dataset, the majority of responses are \"No\", as a result, they used the F1 score as the primary metric for accuracy. Our experiment design enhances both the realism and comprehensibility of the results. These samples explore reasoning within three causal scenarios: direct cause-and-effect relationships (Fig. 2 i),\nindirect cause-and-effect relationships (Fig. 2 ii), and the presence of an effect due to two causes (Fig. 2 iii).\n\n\n\nFigure 2: All possible in cause and effect relations. i. X1 directly causes X2 (X2 is directly effect of X1). ii. X1 indirectly causes X3 (X3 is indirectly effect of X1). iii. X1 and X3 are causes of X2 (X2 is common effect of X1 and X3)\nTo evaluate the proposed framework on natural stories, we utilized the state-of-the-art large language model, GPT-4, which excels at story generation. We crafted detailed instructions in the prompts and generated approximately 20 stories for our case study in fields such as healthcare and medicine, economics, education, social sciences, environmental science, and marketing, all emphasizing the importance of causality. This process is similar to the one presented in (Jin et al., 2023b), but our generated stories are more complex.\n\nOur approach can be tested using the examples of the Simpson's Paradox discussed in (Pearl & Mackenzie, 2018). However, since these examples are already included in the training data of current LLMs, the models simply repeat the correct answers based on that training data, similar to the parrot study in (Zečević et al.,\n2023). Consequently, our generated natural stories replicate these examples in a manner that the LLM cannot address within its existing training data. Additionally, we assess the coevolution of supermassive black holes (SMBHs) and their host galaxies using our proposed framework, replicating the results provided in (Pasquato et al., 2023).\n\n## 4.2 Evaluation Of The C2P On Synthetic Dataset\n\nResults of the C2P on CORR2CAUSE dataset: In Table 1, we show the performance of the more powerful LLMs in the cause-effect task reported in (Jin et al., 2023b) compared to the results of the C2P framework. According to Table 1, performing pure causal reasoning tasks presents a significant challenge for all existing LLMs, even when it is asked to think step by step similar to (Wei et al., 2022). Notably, several models perform worse than a random guess, indicating a complete failure in this specific task of pure causal reason. Nevertheless, applying our proposed framework to GPT-4 Turbo has boosted at least 33%\nin the model's ability to respond to causal inquiries, suggesting a substantial improvement over the existing\n\n| Table 1: Main results: Comparison of using C2P frameworks compared to the exist | | ing LLMs | | |\n|-----------------------------------------------------------------------------------|-------|------------|--------|----------|\n| Models | F1 | Precision | Recall | Accuracy |\n| Random Baselines | | | | |\n| Random (Proportional) | 13.5 | 12.53 | 14.62 | 71.46 |\n| Random (Uniform) | 20.38 | 15.11 | 31.29 | 62.78 |\n| BERT-Based Models | | | | |\n| DistilBART MNLI | 26.74 | 15.92 | 83.63 | 30.23 |\n| BART MNLI | 33.38 | 31.59 | 35.38 | 78.50 |\n| LLaMa-Based Models | | | | |\n| LLaMa3-8B | 48.97 | 51.06 | 47.05 | 51 |\n| LLaMa3-70B | 70.8 | 60.7 | 56.6 | 60 |\n| GPT-Based Models | | | | |\n| GPT-3.5 | 48.8 | 55 | 44 | 54 |\n| GPT-4 | 48.4 | 53.3 | 48 | 53 |\n| GPT-4O | 57.7 | 59.5 | 56 | 59 |\n| C2P on GPT-4 Turbo | 92.78 | 95.74 | 90 | 93 |\n| Few-shot learned GPT-4 Turbo | 81.06 | 82.97 | 79.24 | 81 |\n\nTable 1: Main results: Comparison of using C2P frameworks compared to the existing LLMs\ncapabilities of LLMs. It is important to note that the primary factor contributing to the discrepancy between the LLaMa and GPT results and those reported in (Jin et al., 2023b) stems from the distribution of the provided premise. This table indicates that when reasoning questions are posed to LLMs, the responses slightly tend towards the \"No\" answer. More interestingly, in cases where existing models without C2P\nresponded correctly, their explanations often revealed flawed logic. This highlights the randomness of their responses more clearly.\n\n| Table 2: Accuracy by Number of Variables and sub-tasks | | | | | |\n|----------------------------------------------------------|----------|----------|----------|----------------|-------|\n| Number of variables | Accuracy | | | | |\n| First sub-task | Second | Third | Fourth | Fifth sub-task | |\n| | sub-task | sub-task | sub-task | | |\n| 3 variables | 100% | 100% | 100% | 100% | 99% |\n| 4 variables | 100% | 100% | 100% | 98% | 83.2% |\n| 5 variables | 100% | 100% | 100% | 88.3% | 75.2% |\n| 6 variables | 100% | 100% | 100% | 79.1% | 70.1% |\n\nRobustness analysis: To assess the robustness of the proposed prompts, Table 2 shows the computed accuracy computed in each sub-task of implementing C2P on GPT-4 for different numbers of variables in the given premise.\n\n## 4.3 Evaluation Of The C2P On Natural Stories\n\nTo evaluate the proposed framework on natural stories, we utilize the same process presented in (Jin et al.,\n2023b) and generate premises. The result of applying the C2P framework, both step-by-step and few-shot learned, is provided in Table 3 and compared to the result of GPTs. This table clearly demonstrates that the existing method responds randomly to reasoning prompts. It is important to note that even when GPTs provide the correct final answer, the reasoning process leading to that answer is still incorrect. In contrast, GPT-4 Turbo, enhanced with step-by-step prompting of the C2P framework and few-shot learned C2P,\nboosts the accuracy of these models in similar scenarios involving natural stories.\n\nTable 3: Result of applying the C2P framework and GPTs, TP: True Positives, FP: False Positives, TN: true negatives, FN: False Negatives\n\nModels TP FP TN FN\n\nGPTs\n\nGPT-4.5 2 2 3 3 GPT-4 2 2 3 3 GPT-4o 3 2 3 3\n\nFeatured GPT\n\nC2P on GPT-4 Turbo 5 0 5 0\n\nFew-shot learned GPT-4 Turbo 4 0 5 1\n\nAdditionally, Table 4 provides an example of such natural stories and demonstrates the result of applying a few-shot learned GP4-Turbo with C2P to it. This showcases how GPT models (including versions 3.5, 4, 4 Turbo, and 4o) fail to respond correctly. It is important to note that the few-shot learning process of GPT-4 Turbo was performed using only six examples from the CORR2CAUSE dataset, without including any natural story examples. Note that there are no natural stories employed in the few-shot learning process.\n\n| Table 4: Example of a natural story and the result of few-shot learned GPT-4-Turbo with C2P | |\n|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| Example of true story | Results of few-shot learned GPT-4 Turbo with C2P |\n| Premise: eating junk food (A), obesity (C), and watching television (B) have relations with each other. There is a correlation between eating junk food and obesity, and between watching television and obesity. However, eating junk food and watching television are independent from each other. Hypothesis: Eating junk food directly affects obesity | Step 1: Number of random variable:3 Names of random variable: A, B, C Step 2: All of Statistical Relations:{ Dependencies: [[A, C], [B, C]], Unconditional Independencies: [[A, B]], Conditional Independencies: []} Step 3: A: {A: 0, B: 1, C: 1}, B: {A: 1, B: 0, C: 1}, C: {A: 1, B: 1, C: 0} Step 4: A: {A: 0, B: 0, C: 1}, B: {A: 0, B: 0, C: 1}, C: {A: 1, B: 1, C: 0} Step 5: A: {A: 0, B: 0, C: 1}, B: {A: 0, B: 0, C: 1}, C: {A: 1, B: 1, C: 0} Step 6: C: [A, B] Step 7: C: [A, B] Step 8: A: {A: 0, B: 0, C: 1}, B: {A: 0, B: 0, C: 1}, C: {A: 0, B: 0, C: 0} Step 9: Checking matrix[A][D] = 1 and matrix[D][A] = 0. According to rule 2, this suggests A is a direct cause of D, or D is a direct effect of A. |\n\nTable 4: Example of a natural story and the result of few-shot learned GPT-4-Turbo with C2P\nExample of true story Results of few-shot learned GPT-4 Turbo with C2P\n\n## 4.4 Evaluation Of The C2P Framework On The (Co)Evolution Of Supermassive Black Holes And Their Host Galaxies\n\nIn astrophysics, experiments are impossible. Thus, causal reasoning must rely exclusively on observational data. Similar to the study in (Pasquato et al., 2023), we (co)evolute supermassive black holes (SMBHs) and their host galaxies. We implement C2P on GPT-4 Turbo to a comprehensive catalog of galaxy properties to obtain a PDAG and then reason the causal questions.\n\nFig. 3 shows the results of each sub-tasks of implementing C2P According to the extracted PDAG, many\n\n\n\nFigure 3: Prompts (Q) and results (A) of sub-tasks application of the C2P framework to real-world complex scenarios and steps of sub-task 3 for the given premise.\nquestions such as the causal given can be answered which is too complicated for the existing LLMs to answer. It is important to note that by implementing C2P, not only many causal questions can be answered with this approach, but it is easy to show which causal questions can not be answered based on the given premise and which more information has to be given to be able to causally reason. More importantly, it is worth noting that this is not the limitation of the C2P framework, any rational agent needs such information to answer such causal questions.\n\n## 5 Discussion, Limitations And Future Work\n\nCausal reasoning and identifiability: Having a PDAG of an SCM of a causal mechanism is the most effective way to achieve this with observational data. Based on this, two main questions arise: Given a PDAG,\nunder what conditions can we make causal reasoning? This involves determining the necessary assumptions and data required to address a causal question—this issue is known as identification. It is generally possible to answer some cause-effect questions based on the PDAG, represented by a causal graph, and lower-level data. These questions are mainly the ones that are related to the part where we have extracted a collider, or if the question is just about the existence of a cause-effect relation; in some cases, these questions can be answered as well. However, it is almost impossible to answer all the causal questions based on the given information, as discussed in (Bareinboim et al., 2022; Pearl & Bareinboim, 2022). As a result, the second question is, what else is needed to answer all the causal questions? In such cases, interventional data is necessary to explicitly extract the direction of causal relations in a mechanism. To do so, while a naive baseline approach would require O(n 2) interventions, various methods have been proposed, such as those in\n(Kocaoglu et al., 2017; Choo & Shiragur, 2023; von Kügelgen et al., 2024; Squires & Uhler, 2023), for cases in different situations. However, one clear thing is that it is not possible to reason all the causal questions.\n\nFor instance,\n- Directions cannot be discovered: PDAGs typically include both directed and undirected edges. The undirected edges indicate uncertainty about the direction of causality. It is important to note that without further data or assumptions, it is not possible to definitively determine the causal direction for these relationships with any method.\n\n- Full Causal Path Analysis: While PDAGs can indicate possible paths between variables, they may not fully reveal which paths are indeed causal and which are due to confounding or indirect effects.\n\nQuestions about specific causal pathways can thus be hard to answer definitively.\n\n- Predictions Under Interventions: Questions about the outcomes of hypothetical or actual interventions on one or more variables (do-calculus questions) often require a fully specified causal model.\n\nPDAGs, with their partial specification, might not support detailed predictions under interventions without resolving the ambiguities in causal direction.\n\nCausal reasoning is a broad and complex task in AI and LLMs. While current machine learning methods struggle with extracting causal structures and subsequently reasoning causally, this problem is even more intricate in the context of LLMs. The primary reason for this complexity is the inherent structure of LLMs and their reliance on the concept of attention. To enable LLMs to reason causally, the first step is to identify the causal question. Recently, a comprehensive database was compiled, and an AI model was trained to identify causal questions (Ceraolo et al., 2024). This model can be used prior to applying our framework to determine whether a causal reasoning process has to be activated or not. Additionally, more comprehensive examples and scenarios need to be generated to aid in the learning process of an LLM. Due to the vast variety of causal scenarios and the token limitations of current models, our approach must be integrated into the learning or fine-tuning process of an LLM. This ensures that the model can effectively learn and apply causal reasoning within its capabilities.\n\nScaling Instruction-Finetuned Language Models: LLMs primarily function through pattern recognition without the capability to recognize causal relationships, as indicated by prior research (). This limitation highlights the importance of implementing these models with causal analysis techniques to enhance their causal reasoning ability. Previous attempts to boost LLMs' causal reasoning involved using external tools or manually crafted prompts, which, although insightful, are impractical for easy integration. Our study introduced an algorithm that leverages LLMs' inherent abilities to foster causal reasoning autonomously. Our findings demonstrate a marked increase in the accuracy of LLMs' causal understanding, even in scenarios involving complex causal interactions. Additionally, incorporating a chain of thought into LLMs' training has been proven to enhance performance across various tasks (Chung et al., 2024). Based on the extracted results on the benchmark and provided datasets in our study, and the results of the few-shot learning process, we strongly hypothesize that our approach could similarly benefit the training of language models, particularly in boosting their causal reasoning, even in smaller models that would lead to a revelation in LLMs. This enhancement could be realized using a carefully selected set of examples or synthetic samples created by LLMs, following the guidelines outlined in our algorithm.\n\nFuture works As demonstrated in the results section, C2P has significantly enhanced the reasoning capabilities of LLMs. We have shown that extracting the adjacency matrix of the variables in a given premise can function as the DAG in Pearl's works, leading to improved reasoning ability. Our experimental results of few-shot learned LLMs indicate a significant increase in accuracy for tasks where users inquire about direct cause-and-effect relationships or common cause questions. However, the models still struggle with indirect cause-and-effect questions. Pearl demonstrated that the concepts of a DAG and d-separation complement each other in causal reasoning tasks. To address the low accuracy in answering indirect causal reasoning questions, more sub-tasks can be introduced to LLMs to perform d-separation in a more structured manner.\n\nAs mentioned, due to the token limitation of LLMs, we were able to use only six or even fewer examples in the few-shot process of GPT-4-Turbo. As a result, our proposed framework can be seamlessly integrated into the fine-tuning or learning process of an LLM by introducing more extensive examples with more complex scenarios. 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Dags with no tears: Continuous optimization for structure learning. *Advances in neural information processing systems*, 31, 2018.\n\n## A Appendix A.1 Prompts For C2P Frameworks\n\nThe proposed prompts for step-by-step proceeding with C2P are as follows:\n1. **Prompting for extracting the random variables from the provided data.**\n(i) Please give me the number of random variables in the given premise\n(ii) Write the names of all random variables.\n\n2. Prompting to extract all the cause and effect relations along with all conditional and unconditional relations among the random variables specifically mentioned in the given premise\n- If 2 random variables, for instance, R1 and R2, are independent, write it in this form: \"R1 is independent of R2\". If there exist 2 random variables, for instance, R1 and R2, are conditionally independent given a third random variable, for instance, R3, write it in this form: \"R1 and R2 are independent given R3\". If two random variables, for instance, R1 and R2, are specially mentioned to have cause and effect relation, write it in this form: \"R1 is the cause of R2\".\n\n3. **Prompting to create an adjacency matrix where all elements are 1, except for the**\ndiagonal elements and the elements corresponding to the cause-and-effect relationships specifically mentioned in the given premise.\n\n- In this phase, each random variable is treated as a node within a fully connected undirected graph. Then, for each pair, for instance, R1 and R2, presented in the form: \"R1 is the cause of R2\", set the element in [\"R2\", \"R1\"] in the adjacency matrix to 0.\n\n4. **Prompting of the conditional and unconditional independency valuation and identifying**\nthe colliders, step by step, to extract the causal PDAG.\n\n(i) Update the adjacency matrix based on the specified unconditional independencies between random variables. Each pair of variables that is declared independent should have their corresponding value set to zero in the adjacency matrix. The initial adjacency matrix and the list of independencies are provided below. Please ensure all independencies are correctly reflected in the updated matrix. Instructions: - For each pair of variables listed as independent, set their corresponding entries in the adjacency matrix to 0.\n\n(ii) Update the adjacency matrix based on the specified conditional independencies between random variables. Each pair of variables that is declared independent should have their corresponding value set to zero in the adjacency matrix. The initial adjacency matrix and the list of independencies are provided below. Please ensure all independencies are correctly reflected in the updated matrix. Instructions: - For each pair of variables listed as independent given other variable(s), set their corresponding entries in the adjacency matrix to 0.\n\n(iii) Task: Given an initial adjacency matrix, follow these steps: Step 1: Identify all rows (key values) in the matrix where there are two or more than two columns with the value \"1\" in them.\n\nFor each identified row, find all pairs of different columns where the values are \"1\".Ensure to exclude rows that do not contain any pairs from the results. Step 2: Display these pairs, \"All Pairs\", where each row name is key, and the value is a list of column names that are identified in Step 1.\n\n(iv) Given the \"All Pairs\" and the list of independencies, follow these instructions step by step: Instruction: For each key in \"All Pairs\", delete all the pairs that are not mentioned as independent in the \"independencies\" list and return other with all their values. The \"All Pairs\" contains pairs of elements associated with each key. The goal is to update this by removing pairs that are not mentioned as independent. The list of independencies provides information about which pairs are independent of each other.\n\n(v) Given the initial adjacency matrix represented and the \"All Pairs\" list, for each key-value pair\n(\"R\") in \"All Pairs\", modify the initial adjacency matrix as follows: 1- Set the value in the \"C1\" row and \"R\" column to 0: (\"C1\", \"R\") = 0. 2- Set the value in the \"C2\" row and \"R\" column to 0: (\"C2\", \"R\") = 0.\n\n## 5. Prompting For Cause-And-Effect Questions Or Hypotheses\n\n- To extract and understand causal relations in the adjacency matrix: For each specified variable\n\"R\" and \"C\", for instance, that are listed in the adjacency matrix: - If matrix entry at [\"R\", \"C\"] = 1 and [\"C\", \"R\"] = 1, then the causal direction between \"R\" and \"C\" is undetermined. - If matrix entry at [\"R\", \"C\"] = 1 and [\"C\", \"R\"] = 0, then \"R\" is a direct cause of \"C\" or \"C\" is a direct effect of \"R\". - If matrix entry at [\"R\", \"C\"] = 0 and [\"C\", \"R\"] = 1, then \"C\" is a direct cause of \"R\" or \"R\" is a direct effect of \"C\". If two variables directly affect a third variable, the first two variables are common causes, and the third variable is a collider. Evaluate the hypothesis based on the given partially directed acyclic graph presented as an adjacency matrix with the given Instruction.\n\n## A.2 Prompts For Few-Shot Learning Of C2P\n\nAn example of given prompts for the few-shot learning process of C2P is as follows: Premise: Suppose there is a closed system of 5 variables, A, B, C, D, and E. All the statistical relations among these 5 variables are as follows: A correlates with C. A correlates with D. A correlates with E.\n\nB correlates with D. B correlates with E. C correlates with D. C correlates with E. D correlates with E.\n\nHowever, A is independent of B. A and B are independent given C. B is independent of C. B and C are independent given A. C and E are independent given A, B, and D.\n\nHypothesis: There exists at least one collider (i.e., common effect) of A and B.\n\n- **\"Subtask 1\"**- The number of random variables and their names in the given premise in JSON\nformat:\nOutput: \"number of random variables: 5, \"names of random variables\": [\"A\", \"B\", \"C\", \"D\", \"E\"]\n- **\"Subtask 2\"**- All the dependencies, conditional and unconditional independencies between all random variables extracted in \"subtask 1\":\nOutput: {\"All of Statistical Relations\": {\"Dependencies\": [[\"A\", \"C\"], [\"A\", \"D\"], [\"A\", \"E\"], [\"B\",\n\"D\"], [\"B\", \"E\"], [\"C\", \"D\"], [\"C\", \"E\"], [\"D\", \"E\"]], \"Unconditional Independencies\": [[\"A\", \"B\"], [\"B\",\n\"C\"]], \"Conditional Independencies\": [[\"A\", \"B\"], [\"B\", \"C\"], [\"C\", \"E\"]]}}\n- **\"Subtask 3\"**- The adjacency matrix of all random variables extracted in \"subtask 1\" where each random variable is treated as a node within a fully connected undirected graph: Output: {\"A\": {\"A\": 0, \"B\": 1, \"C\": 1, \"D\": 1, \"E\": 1}, \"B\": {\"A\": 1, \"B\": 0, \"C\": 1, \"D\": 1, \"E\": 1}, \"C\": {\"A\": 1, \"B\": 1, \"C\": 0, \"D\": 1, \"E\": 1}, \"D\": {\"A\": 1, \"B\": 1, \"C\": 1, \"D\": 0, \"E\": 1}, \"E\": {\"A\": 1, \"B\": 1, \"C\": 1, \"D\": 1, \"E\": 0}}\n- **\"Subtask 4\"**- Update the adjacency matrix extracted in the output of \"subtask 3\" based on the specified unconditional independencies between random variables. Each pair of variables that are declared independent should have their corresponding value set to zero in the adjacency matrix. - For each pair of variables listed as unconditional independent in \"subtask 2\", we set their corresponding entries in the adjacency matrix to 0. - We do not change any other entries except those specified by the independence. Output: {\"A\": {\"A\": 0, \"B\": 0, \"C\": 1, \"D\": 1, \"E\": 1}, \"B\": {\"A\": 0, \"B\": 0, \"C\": 0, \"D\": 1, \"E\": 1}, \"C\": {\"A\": 1, \"B\": 0, \"C\": 0, \"D\": 1, \"E\": 1}, \"D\": {\"A\": 1, \"B\": 1, \"C\": 1, \"D\": 0, \"E\": 1}, \"E\": {\"A\": 1, \"B\": 1, \"C\": 1, \"D\": 1, \"E\": 0}}\n- **\"Subtask 5\"**- Update the adjacency matrix in the output of \"Subtask 4\" based on the specified conditional independencies between random variables extracted in \"subtask 2\". Each pair of variables that are declared conditional independent should have their corresponding value set to zero in the adjacency matrix. - For each pair of variables listed as conditionally independent given other variable(s), we set their corresponding entries in the adjacency matrix to 0. - We do not change any other entries except those specified by the conditional independence.\n\nOutput: {\"A\": {\"A\": 0, \"B\": 0, \"C\": 1, \"D\": 1, \"E\": 1}, \"B\": {\"A\": 0, \"B\": 0, \"C\": 0, \"D\": 1, \"E\":\n1}, \"C\": {\"A\": 1, \"B\": 0, \"C\": 0, \"D\": 1, \"E\": 0}, \"D\": {\"A\": 1, \"B\": 1, \"C\": 1, \"D\": 0, \"E\": 1}, \"E\":\n{\"A\": 1, \"B\": 1, \"C\": 0, \"D\": 1, \"E\": 0}}\n- **\"Subtask 6\"**: In this subtask, for each key value in the \"Adjacency Matrix\" mentioned in the output of \"Subtask 5\", the task is to extract only pairs of columns with values \"1\" step by step according to the provided steps and put it in \"Candidates\" dictionary: follow these steps: Step 1: Identify all rows (key values) in the matrix where there are at least two or more columns with the value \"1\". For each identified row, find all pairs of different columns where the values are \"1\". Ensure to exclude rows that do not contain any pairs from the results. Make sure to check each row individually and include all valid pairs for every row. Step 2: Display these pairs in a simplified JSON format, where each row name is a key, and the pair values are a list of column names that are identified in step 1. If there are no such rows, show an empty JSON object. Step 3: Do not include any rows with one or zero columns with \"1\" values in the output. Example output format: Candidates: { \"row1 name\": [[\"1value column1 name\", \"1value column2 name\"], [\"1value column1 name\", \"1value column3 name\"], [\"1value column2 name\", \"1value column3 name\"]], \"row2 name\": [[\"1value column5 name\", \"1value column6 name\"]], ... } The values in each row have to be pairs and the output cannot be as follows: { \"row1 name\": [[\"1value column1 name\"], [\"1value column1 name\"], [\"1value column2 name\", \"1value column3 name\"]], \"row2 name\": [[\"1value column5 name\"]], ... } Please provide only the desired output formatted exactly as shown in the example without any further explanation.\n\nOutput: {\"A\": [[\"C\", \"D\"], [\"C\", \"E\"], [\"D\", \"E\"]], \"B\": [[\"D\", \"E\"]], \"C\": [[\"A\", \"D\"]], \"D\": [[\"A\", \"B\"],\n[\"A\", \"C\"], [\"A\", \"E\"], [\"B\", \"C\"], [\"B\", \"E\"], [\"C\", \"E\"]], \"E\": [[\"A\", \"B\"], [\"A\", \"D\"], [\"B\", \"D\"]]}\n- **\"Subtask 7\"**- In this subtask, given the output of \"Subtask 6\", \"Candidates\", and \"Unconditional Independencies\" in \"Subtask 2\", the task is to identify and extract all the pairs in the \"Candidates\" that are also present in the \"Unconditional Independencies\" step by step according to the provided steps. follow these steps: 1- For each pair in the \"Candidates\" list, check if it is present in the \"Unconditional Independencies\" list. 3-Only keep all the pairs from \"Candidates\" that are also present in\n\"Unconditional Independencies\". If a pair in \"Candidates\" is found in \"Conditional Independencies\",\nkeep it. 4-Remove any pairs in \"Candidates\" that are not found in \"Conditional Independencies\".\n\nIf a pair in \"Candidates\" is not found in \"Unconditional Independencies\", remove it. 5-Output the result as the modified \"Candidates\" dictionary without any additional text or explanation. Only the updated \"Candidates\" dictionary and nothing else.\n\nOutput: {\"D\": [[\"A\", \"B\"], [\"B\", \"C\"]], \"E\": [[\"A\", \"B\"]]}\n- **\"Subtask 8\"**- Given the adjacency matrix in the output of \"Subtask 5\" and the \"All Pairs\" list in the output of \"Subtask 7\", for each key-value pair (\"R\") in \"All Pairs\", we modify the adjacency matrix as follows: -For each key \"R\" and pair (\"C1\", \"C2\") in the candidates, change the values in key \"R\" and pairs (\"C1\", \"C2\") to zero. -Ensure that only the specified modifications are made, and all other entries in the adjacency matrix remain unchanged.\n\nOutput: {\"A\": {\"A\": 0, \"B\": 0, \"C\": 1, \"D\": 1, \"E\": 1}, \"B\": {\"A\": 0, \"B\": 0, \"C\": 0, \"D\": 1, \"E\":\n1}, \"C\": {\"A\": 1, \"B\": 0, \"C\": 0, \"D\": 1, \"E\": 0}, \"D\": {\"A\": 0, \"B\": 0, \"C\": 0, \"D\": 0, \"E\": 1}, \"E\":\n{\"A\": 0, \"B\": 0, \"C\": 0, \"D\": 1, \"E\": 0}}\n- **\"Subtask 9\"**- In this subtask, given the \"Adjacency Matrix\" mentioned in the output of \"Subtask 8\", evaluate the correctness of the hypothesis step by step according to the provided rules. Rules: 1- If matrix[\"R\"][\"C\"] = 1 and matrix[\"C\"][\"R\"] = 1, then the causal direction between \"R\" and \"C\" is undetermined. 2- If matrix[\"R\"][\"C\"] = 1 and matrix[\"C\"][\"R\"] = 0, then \"R\" is a direct cause of \"C\", or \"C\" is a direct effect of \"R\". 3- If matrix[\"R\"][\"C\"] = 0 and matrix[\"C\"][\"R\"] = 1, then \"C\" is a direct cause of \"R\", or \"R\" is a direct effect of \"C\". 4- Test each variable in the matrix, for instance,\n\"R\", \"R\" is a collider if the matrix entries in [\"R\", \"C1\"] = 0, [\"C1\", \"R\"] = 1, [\"R\", \"C2\"] = 0, and\n[\"C2\", \"R\"] = 1 Perform it step by step and provide the final \"Yes\" or \"No\" answer.\n\nOutput: Step 1: According to the rules, a node is a collider (common effect) if the matrix entries in [\"R\", \"C1\"] = 0, [\"R\",\"C2\"] = 0, [\"C1\", \"R\"] = 1 and [\"C2\", \"R\"] = 1.\n\nStep 2: According to rule 4, \"R\" is a collider if the matrix entries in [\"R\", \"D\"] = 0, [\"R\", \"E\"] = 0,\n[\"D\", \"R\"] = 1 and [\"E\", \"R\"] = 1.\n\nStep 3: Checking each variable in the matrix Node C: [\"C\", \"A\"] = 1 and [\"A\", \"C\"] = 1, so C is not a common effect of A and B.\n\nNode D: [\"D\", \"A\"] = 0 and [\"A\", \"D\"] = 1, and [\"D\", \"B\"] = 0 and [\"B\", \"D\"] = 1, so D is a common effect of A and B.\n\nNode E: [\"E\",\"A\"] = 0 and [\"A\",\"E\"] = 1, and [\"E\",\"B\"] = 0 and [\"B\", \"E\"] = 1, so E is a common effect of A and B.\n\nStep 4: So, the hypothesis that there exists at least one collider (i.e., common effect) of A and B is correct. The answer is \"Yes\"."},"reviews":{"kind":"string","value":""}}},{"rowIdx":97,"cells":{"paper":{"kind":"string","value":"# Population Priors For Matrix Factorization\n\nAnonymous authors Paper under double-blind review\n\n## Abstract\n\nWe develop an empirical Bayes prior for probabilistic matrix factorization. Matrix factorization models each cell of a matrix with two latent variables, one associated with the cell's row and one associated with the cell's column. How to set the priors of these two latent variables? Drawing from empirical Bayes principles, we consider estimating the priors from data, to find those that best match the populations of row and column latent vectors. Thus we develop the twin population prior. We develop a variational inference algorithm to simultaneously learn the empirical priors and approximate the corresponding posterior. We evaluate this approach with both synthetic and real-world data on diverse applications: movie ratings, book ratings, single-cell gene expression data, and musical preferences. Without needing to tune Bayesian hyperparameters, we find that the twin population prior leads to high-quality predictions, outperforming manually tuned priors.\n\n## 1 Introduction\n\nThis paper is about empirical Bayes methods for setting the priors in Bayesian matrix factorization (Mnih & Salakhutdinov, 2007; Gopalan et al., 2015). Matrix factorization models each cell of a matrix with two latent variables, one associated with its row and one associated with its column. Matrix factorization has found broad applications across many fields, including studying consumer behavior, understanding legislative patterns, assessing pharmaceutical impacts, and exploring social networks (Gopalan et al., 2015; Koren et al., 2009; Gerrish & Blei, 2011; Jamali & Ester, 2010).\n\nSuppose Xi,j is the observed entry in row i and cell j, such as user i's rating of movie j. As a hierarchical model, a matrix factorization generates the data from the following process:\n\n$$\\begin{array}{l}{{U_{i}\\sim P_{\\theta^{r}}(U)}}\\\\ {{V_{j}\\sim P_{\\theta^{c}}(V)}}\\\\ {{X_{i,j}\\sim P(X_{i,j}\\mid U_{i},V_{j}).}}\\end{array}$$\n$\\left(1\\right)$. \nr (U) (1)\n$\\left(3\\right)$. \nc (V ) (2)\nXi,j ∼ P(Xi,j | Ui,Vj ). (3)\nHere Ui and Vj are per row and per column specific latent vectors, and Pθ r and Pθ c are the priors, each with hyperparameters θ r and θ c.\n\nThis formulation encompasses many factorization models. In Gaussian matrix factorization (Mnih &\nSalakhutdinov, 2007), the priors are Gaussians and Xi,j is drawn from a Gaussian with mean U >\ni Vj and variance σ 2. In Poisson matrix factorization (Canny, 2004; Dunson & Herring, 2005; Gopalan et al., 2015),\nthe priors are over the positive reals and Xi,j is drawn from a Poisson with rate U >\ni Vj .\n\nAn observed matrix of data X then defines a posterior distribution P(U,V | X) over the row variables and the column variables. The posterior can provide interpretations of the data and an avenue to form predictions about missing entries, for example, for a recommendation system.\n\nThe prior distributions on the row variables and column variables significantly impact the quality of the model posterior. How should we set them? Practitioners typically assume a simple parametric family for the priors, such as a Gaussian or a Gamma, and then find the prior hyperparameters best suited to the data, e.g., with cross-validation (Salakhutdinov & Mnih, 2008; Schmidt et al., 2009). This approach can be effective, but it is expensive and only allows for priors from a simple parametric family.\n\nIn this paper, we develop an empirical Bayes (EB) methodology for setting the priors (Robbins, 1992; Efron, 2012), learning them from data. The EB idea is to set the priors using the data, for example by finding the one that maximizes the marginal log-likelihood of the data. It is applicable for priors over variables that have repeated draws from them, such as the row and column variables in matrix factorization.\n\nEB has found applications in applied sciences in diverse fields such as astronomy (Bovy et al., 2011),\nactuarial sciences (Bühlmann & Gisler, 2005), genomics (Smyth, 2005; Love et al., 2014), economics (Frost\n& Savarino, 1986; Angrist et al., 2017), and survey sampling (Rao & Molina, 2015). EB priors have been successfully employed in simple hierarchical models, such as in variational autoencoders (Kingma & Welling, 2013; Tomczak & Welling, 2018; Kim & Mnih, 2018). Matrix factorization, however, provides a different type of application of EB. In matrix factorization, there are two priors to set, one for the row variables and one for the column variables, and the data that informs us about them is overlapping. Thus, we will find empirical Bayes priors for both row variables and column variables. The result is the *twin empirical Bayes prior* (TwinEB), a practical EB method for matrix factorization. Other methods for EB on matrix factorization include Wang & Stephens (2021) and da Silva et al.\n\n(2023); we discuss these below. Specifically, we model the two priors with mixture distributions, one for each prior. We use mixtures since they are a flexible family of distributions that can approximate a wide range of distributions (Titterington et al., 1985; Nguyen et al., 2020). We use variational inference (Blei et al., 2017; Wainwright & Jordan, 2008) to simultaneously estimate the priors and approximate the corresponding posterior. We verify the efficiency and the robustness of this approach with real-world data about recommendation systems and computational biology, and for both Poisson matrix factorization and Gaussian matrix factorization. We summarize the contributions as follows:\n1. We develop the *twin population prior*, an EB prior for Bayesian matrix factorization. 2. We derive a variational inference algorithm to approximate the matrix factors' posteriors and learn the twin population prior simultaneously.\n\n3. We study the twin population prior on both synthetic and real data and with two types of factorization. The automatically learned EB prior performs as well as the best prior chosen retrospectively.\n\n## 2 Related Work\n\nOne approach to setting the hyperparameters of priors involves using hierarchical Bayesian models (Gelman, 2006). In this line of work, the prior's hyperparameters are treated as unknown and assigned a prior with hyperpriors to be determined. Gopalan et al. (2015) employs this method for introducing hierarchical Poisson factorization, and Levitin et al. (2019) applies it to gene signature discovery from scRNAseq data. Our work involves more expressive priors (mixtures), which leads to a more flexible class of models while keeping a simpler model as no extra variables for the hyperparameters are introduced.\n\nA fruitful direction of research for setting priors for matrix factorization has also been the use of empirical Bayes, a methodology by which one tries to find the prior that matches, in some sense, the population distribution of the data. In Wang & Stephens (2021), the authors formulate prior elicitation for matrix factorization (MF) as steps in a variational expectation maximization algorithm, and in that context, find that it is equivalent to solving the EB normal means (EBNM) problem (Jiang & Zhang, 2009). Our proposed approach is closely related to Wang & Stephens (2021), but we propose to update both priors simultaneously which bypasses potentially costly numerical integration required in the EBNM step. A related line of research makes the connection to EBNM via random-matrix theory, focusing specifically on denoising principal components (Zhong et al., 2022). Closely related to TwinEB, da Silva et al. (2023) optimizes hyperparameters to match the prior predictive distribution to statistics computed from the data, and derives closed-form solutions for PMF and its hierarchical extensions. In Section 4, we compare our method to Wang & Stephens (2021) and da Silva et al. (2023).\n\nEmpirical Bayes (EB) priors have been explored in the context of variational autoencoders (VAEs) (Kingma\n& Welling, 2013) under the name aggregated posterior, average encoding distribution (Hoffman & Johnson, 2016) or VampPrior (Tomczak & Welling, 2018). The VampPrior learns an amortized posterior and a prior over the latent variables using a shared neural network. It models a prior on the rows only and was used to address posterior collapse (Tomczak & Welling, 2018) or to learn disentangled representations (Kim &\nMnih, 2018). Our method focuses on matrix factorization, and we derive EB priors for both row and column latent variables.\n\n## 3 Empirical Bayes Priors For Probabilistic Matrix Factorization\n\nOur goal is to develop empirical Bayes (EB) priors for Bayesian matrix factorization models. We will focus here on Poisson matrix factorization (PMF). In the supplement, we derive EB priors for Gaussian matrix factorization (GMF). With matrix factorization, the presence of repeated and identically distributed latent variables for each row and each column provides the opportunity to learn their prior distribution from data. This is a form of empirical Bayes (Robbins, 1992; Efron, 2012) that prescribes a *population prior* (see Section 3.1).\n\nThis population prior aims to align the model's marginal distribution of observations with the observed population distribution. In the special case of matrix factorization, there are two distinct populations: the populations of row vectors, and the population of column vectors. With TwinEB, we learn one prior for each population. This is a form of hierarchical modeling without introducing an extra layer of latent variables.\n\nNotations. Define [N] := 1*, . . . , N*. Let X = [Xi,j ] ∈ N\nN×D represent the measured feature j in individual i, where i ∈ [N] and j ∈ [D]. The dimension of the latent variables (the number of factors) is denoted as L.\n\n## 3.1 Background: Population Priors For Simple Hierarchical Models\n\nA crucial step in Bayesian statistics is the choice of the prior distribution; if done arbitrarily, it can lead to suboptimal posterior inference (Wang et al., 2021). We choose to follow an empirical Bayes principle that prescribes a *population prior* (Hoffman & Johnson, 2016; Tomczak & Welling, 2018). This prior, by design, aligns the model's marginal distribution of observations with the population distribution.\n\nWe first focus on a family of latent variable models called simple hierarchical models (Agrawal & Domke, 2021). The joint distribution factorizes as follows:\n\n$$P(\\mathbf{Z},\\mathbf{X})=\\prod_{i=1}^{N}P_{\\pi}(\\mathbf{z}_{i})P_{\\theta}(\\mathbf{x}_{i}\\mid\\mathbf{z}_{i}),$$\n\n$$\\left(4\\right)$$\nPπ(zi)Pθ(xi| zi), (4)\nwhere θ and Z = [zi] are global and local latent variables respectively and observations X = [xi] are exchangeable conditioned on the latent variables. The variable π indexes the prior distribution of zi. We assume that the global latent variables are fixed (e.g., they are learned through MAP estimation). We use a subscript notation to indicate fixed variables. To further simplify notations, we focus on the marginal likelihood of a single observation, x, and its corresponding local latent variable z.\n\nLet P\n?(x) be the true (unobserved) distribution of observations. An empirical Bayes criterion is that the marginal distribution of observations under the model, Pπ,θ,(x), should align with their true population distribution (Ignatiadis & Wager, 2022), that is:\n\n$$\\begin{array}{r l}{P^{\\star}(\\mathbf{x})=P_{\\pi,\\theta}(\\mathbf{x})}\\\\ {=\\int P_{\\pi}(\\mathbf{z})P_{\\theta}(\\mathbf{x}\\mid\\mathbf{z})\\ d\\mathbf{z}.}\\end{array}$$\nZPπ(z)Pθ(x | z) dz.(5)\n$\\left(5\\right)$. \nOur goal is to set π such that, for a fixed θ, Equation (5) holds. For a fixed likelihood function Pθ(x | z)\n(i.e., θ is fixed), the expression for the prior Pπ? that satisfies this conditions is:\n\n$$P_{\\pi^{\\star}}(\\mathbf{z})\\approx\\int P_{\\pi,\\theta}(\\mathbf{z}\\mid\\mathbf{x})P^{\\star}(\\mathbf{x})d\\mathbf{x}$$ $$=\\mathbb{E}_{P^{\\star}(\\mathbf{x})}[P_{\\pi,\\theta}(\\mathbf{z}\\mid\\mathbf{x})],$$ is all the time of the last two-sided version with the above theorem. \n$$(6)$$\n\nwhere Pπ,θ(z | x) is the (local) posterior distribution of the latent variable z given the observation x under the model. The definition presents two issues: the unknown true marginal distribution P\n?(x), and the fact that the target prior Pπ? (z) is on both sides of Equation (6), explicitly on the left and implicitly via the posterior on the right. The research literature has approximated Equation (6) with Monte Carlo estimates of P\n?(x) and variational inference of P(z | x) (Hoffman & Johnson, 2016; Tomczak & Welling, 2018).\n\n## 3.2 Population Priors For Probabilistic Matrix Factorization\n\nOur goal is to develop population priors for Bayesian matrix factorization models. The challenge is that unlike simple hierarchical models, there is no distinction between local and global random variables, rather latent variables denote row- and column-specific random variables.\n\n## 3.2.1 Twin Population Priors\n\nWe establish population priors for two latent variables, one for the rows PV ? (Ui) and one for the columns PU? (Vj ). These priors will match two different populations, one of the row vectors and one of the column vectors:\n\n$P^{\\star}_{\\rm row}(\\mathbf{X}_{i,:}):=$ Population distribution of row vectors $P^{\\star}_{\\rm column}(\\mathbf{X}_{:,j}):=$ Population distribution of column vectors\nWe begin with population priors for row latent variables. As in section 3.1, we specify the prior based on an empirical Bayes principle such that the true marginal distribution of the rows P\n?\n\nrow(Xi,:) is aligned with the distribution of the rows under the model PV (Xi,:), that is:\n\n$$P_{\\rm row}^{\\star}(\\mathbf{X}_{i,:})=P_{\\mathbf{V}}(\\mathbf{X}_{i,:})$$ $$=\\int P(\\mathbf{U}_{i})\\prod_{j=1}^{D}P(X_{i,j}\\mid\\mathbf{U}_{i},\\mathbf{V}_{j})d\\mathbf{U}_{i},$$ $\\mathbf{U}_{i}$\\(\\mathbf{U}_\n$$\\left(7\\right)$$\n\nfor a fixed set of column variables V . For Equation (7) to hold, a population prior should be used:\n\n$$P_{V^{\\star}}(U_{i})=\\mathbb{E}_{P_{\\mathrm{ref}}^{\\star}}$$\n\nrow(Xi,:)[PV (Ui| Xi,:)]. (8)\nSimilarly for columns, for a fixed set of row latent variables U, the empirical Bayes criterion is:\n\n$$P_{\\mathrm{column}}^{\\star}(X_{:,j})=P_{U}(X_{i,:})$$\n$\\mathbf{\\hat{\\varepsilon}}_{\\text{row}}(\\mathbf{\\hat{x}})$\n$P_{U}(\\mathbf{X}_{i,:})$ $$=\\int P(\\mathbf{V}_{i})\\prod_{i=1}^{N}P_{U_{j}}(X_{i,j}\\mid\\mathbf{V}_{i})d\\mathbf{V}_{i}.\\tag{1}$$\n$$({\\boldsymbol{\\delta}})$$\n$$\\mathbf{\\tau}\\cdot(U_{i}\\ |\\ \\mathbf{X}_{i,:})].$$\n$$(10)$$\n$$(9)$$\n\nThe prior that satisfies this criterion is the column population prior:\n\n$P_{U}\\cdot(V_{j})=\\mathbb{E}_{P_{\\rm column}^{*}}(X_{:,j})[P_{U}(V_{j}\\mid X_{:,j})]$. \nSince there are two populations in need of prior specification, we call Equations (8 and 10) the *twin* population priors. We have established the form of population priors in probabilistic matrix factorization. Next we focus on how to estimate the twin population priors and how to approximate posterior inference under them. In the remainder of the paper, we focus on the Poisson matrix factorization (PMF). We derive the priors for Gaussian matrix factorization (GMF) in the supplement.\n\n## 3.3 Twin Eb Prior For Poisson Matrix Factorization\n\nIn Poisson matrix factorization (PMF), the row and column latent variables Ui and Vi are non-negative L-vectors and the likelihood in Equation (3) is Poisson:\n\n$$X_{i,j}\\mid{\\boldsymbol{U}}_{i},{\\boldsymbol{V}}_{j}\\sim\\mathrm{Poisson}\\left({\\boldsymbol{U}}_{i}^{T}{\\boldsymbol{V}}_{j}\\right),$$\n, (11)\n$${\\mathrm{for~}}i\\in[N],j\\in[D].$$\n\nThe log-likelihood of the data is:\n\n$$\\log P(\\mathbf{X}\\mid\\mathbf{U},\\mathbf{V})=\\log\\prod_{i=1}^{N}\\prod_{j=1}^{D}P(X_{i,j}\\mid\\mathbf{U}_{i},\\mathbf{V}_{j})\\tag{1}$$ $$=\\sum_{i,j}^{N,D}\\log\\text{Poisson}\\left(X_{i,j};\\mathbf{U}_{i}^{T}\\mathbf{V}_{j}\\right).$$\n$$(11)$$\n$$\\left(12\\right)$$\n\nSome methods place Gamma priors on Vj and Ui (Gopalan et al., 2015). Note that this is a Bayesian formulation of non-negative matrix factorization (Cemgil, 2009).\n\nTo compute the population prior for the rows, PV (Ui) = EP ?\n\nrow(Xi,:)[PV (Ui| Xi,:)], we face two problems.\n\nNamely, we do not know the true population distribution of the rows P\n?\n\nrow(Xi,:), and the population prior PV (Ui) appears on both sides of the equality.\n\nTo find the population prior, we first notice that a Monte Carlo estimate of Equation (8) writes as:\n\n$$P_{\\mathbf{V}}(\\mathbf{U}_{i})\\approx\\frac{1}{N}\\sum_{i^{\\prime}=1}^{N}P_{\\mathbf{V}}(\\mathbf{U}_{i}\\mid\\mathbf{X}_{i^{\\prime},:}).\\tag{13}$$\n\nWhen the prior satisfies Equation (13), this property is called self-consistency (Laird, 1978).\n\nThe structure of Equations (13) suggests to use families of mixtures of parametric distributions to approximate the row and column population priors (Tomczak & Welling, 2018). Mixtures can approximate complex distributions when their number of components increases while having the convenience of remaining parametric (Titterington et al., 1985; Nguyen et al., 2020). We choose to model the priors as,\n\n$$P_{\\mathbf{V}}(\\mathbf{U}_{i}):=P_{\\theta^{r}}(\\mathbf{U}_{i})=\\sum_{k=1}^{K_{r}}\\pi_{k}P_{\\mathbf{\\mu}_{k},\\mathbf{\\sigma}_{k}}(\\mathbf{U}_{i}),$$ $$P_{\\mathbf{U}}(\\mathbf{V}_{j}):=P_{\\theta^{c}}(\\mathbf{V}_{j})=\\sum_{k=1}^{K_{c}}\\rho_{k}P_{\\mathbf{\\nu}_{k},\\mathbf{\\eta}_{k}}(\\mathbf{V}_{j}),$$\n$$\\left(14\\right)$$\n$$\\left(15\\right)$$\n$$(16)$$\n\nwhere Kr and Kc are the number of components in the mixtures, θ r = {µ,σ,π} and θ c = {ν, η, ρ}. The locations µ ∈ R\nKr×L and ν ∈ R\nKc×L, the scales σ ∈ R\nKr×L and η ∈ R\nKc×L, and the mixture weights π ∈ ∆Kr and ρ ∈ ∆Kc are the parameters of the mixtures priors. As Kr and Kc increase, the priors become more and more expressive (see Figure 4). Figure 1 shows a graphical model representation of matrix factorization with EB priors.\n\nIn a classical empirical Bayes setup, the idea is to set the priors that maximize the marginal likelihood of the data:\n\n$$\\hat{\\theta}^{r},\\hat{\\theta}^{c}=\\operatorname*{arg\\,max}_{\\theta^{r},\\theta^{c}}\\log P(X;\\theta^{r},\\theta^{c}),$$\n\nwhere\n\n$$\\log P(\\mathbf{X};\\theta^{r},\\theta^{c})=\\log\\int P_{\\theta^{r}}(\\mathbf{U})P_{\\theta^{c}}(\\mathbf{V})P\\left(\\mathbf{X}\\mid\\mathbf{U},\\mathbf{V}\\right)d\\mathbf{U}d\\mathbf{V}.$$\n\nIn Section 3.4, we find θ r, θc at the same time as we approximate the model posterior P(U,V | X; θ r, θc).\n\n$$(17)$$\n\n\n\nFigure 1: Twin population priors for Poisson matrix factorization model. Shaded nodes are observed while other nodes represent latent random variables. The empty squares indicate that we will fit these priors to the data.\n\n## 3.4 Posterior Inference In Pmf With Twin Eb Priors\n\nGiven data X, our goal is to calculate the posterior P(U,V | X; θ r, θc), which also depends on our choice of priors θ r, θc. The challenges are that this posterior is intractable (for any prior) and we simultaneously want to fit the priors to satisfy the EB criterion in Equation (16).\n\nOur strategy will be as follows. We will use variational inference (VI) (Blei et al., 2017) to approximate the posterior, taking gradient steps in the variational objective with respect to the posterior approximation (the variational family). At the same time, however, the variational objective of VI is an approximation (lower bound) of the log-marginal from Equation (16). So we also take gradient steps with respect to the EB priors to maximize it. The result is an algorithm that simultaneously approximates the posterior and learns the EB prior.\n\nThe variational posterior. Consider a parameterized mean-field variational family,\n\n$$q_{\\bf A}(U,V\\mid X)=\\prod_{i,l}q_{\\lambda_{i,l}^{c}}(U_{i,l})\\prod_{j,l}q_{\\lambda_{j,l}^{c}}(V_{j,l}),\\tag{1}$$\n\n```\nThis family has parameters for each row's latent vector and each column's latent vector, λ\n r\n i\n and λ\n c\n i\n respec-\n\n```\n\ntively. We further define Λr:= [λ r i,l], and Λc:= [λ c j,l]. The full set of variational parameters is Λ = {Λr, Λc}.\n\nFrom the perspective of posterior inference, our goal is to set qΛ to minimize the KL divergence to the exact posterior:\n\n$$\\hat{\\mathbf{A}}=\\operatorname*{arg\\,min}_{\\mathbf{A}}\\operatorname{KL}(q_{\\mathbf{A}};P(\\mathbf{U},\\mathbf{V}\\mid\\mathbf{X};\\theta^{r},\\theta^{c})).$$\nr, θc)). (19)\nIn detail, the variational family is a bank of Log-Normals:\n\n$$(18)$$\n$$(19)$$\n\n$$\\lambda^{r}_{i,l}:=(a^{\\prime}_{i,l},b^{\\prime}_{i,l}),$$ $$\\lambda^{c}_{j,l}:=(a_{j,l},b_{j,l}),$$ $$q\\chi^{r}_{i,l}(U_{i,l}):=\\mathcal{L}\\mathcal{N}(a^{\\prime}_{i,l},b^{\\prime}_{i,l}),$$ $$q\\chi^{c}_{j,l}(V_{j,l}):=\\mathcal{L}\\mathcal{N}(a_{j,l},b_{j,l}).$$\n\nEach Log-Normal is parameterized by its natural parameters a and b:\n\n$${\\mathcal{L N}}(x;a,b)\\propto\\exp\\left(-{\\frac{a}{2b}}\\log(x)-{\\frac{(\\log x)^{2}}{2b}}\\right).$$\n\n$$(23)$$\n\nTo minimize the KL divergence in Equation (19), VI optimizes the variational parameters Λ to, equivalently, maximize the evidence lower bound (ELBO) (Blei et al., 2017):\nL(X; Λ, θr, θc) =\nEqΛ(U,V | X)[log P(X | U,V )] + EqΛ\n[log P(U; θ r, θc) + log P(V ; θ r, θc)] − EqΛ\n[log qΛ(U,V | X)]. (24)\nHere we use gradient ascent to maximize L(X; θ, Λ) with respect to Λ (Ranganath et al., 2014). We further use stochastic reparameterization gradients to take such steps (Kingma & Welling, 2013; Rezende et al., 2014). Maximum marginal likelihood. At the same time, we would like to set the prior parameters to maximize the marginal likelihood of the data (Equation (16)). The variational objective in Equation (24) conveniently also provides a lower-bound on the marginal likelihood (Blei et al., 2017):\nlog P(X; θ r, θc) ≥ L(X; θ, Λ). (25)\nSo, we will also follow stochastic gradients of the ELBO with respect to the prior parameters θ r, θcto maximize L(X; θ, Λ) with respect to θ r, θc. This strategy has been used in the context of linear regression (Mukherjee et al., 2023). Twin EB. Putting these two pieces together, our algorithm is a stochastic gradient ascent of the ELBO with respect to two sets of parameters. In optimizing with respect to Λ, we minimize the KL divergence between qΛ and the posterior; in optimizing with respect to θ r, θc, we maximize the (approximate) marginal likelihood of the data. We use the Adam algorithm for stochastic optimization (Kingma & Ba, 2014) with a batch size of 128, and using ten particles to obtain unbiased noisy estimates of the gradient of the ELBO via the reparameterization trick (the particles are samples from qΛ to estimate the expectation EqΛ\nof the ELBO with Monte-Carlo).\n\nThe details of the algorithm are in Algorithm 1. Our implementation is available at git@github.com:\nxxx/xxx.git.\n\nAlgorithm 1 Variational inference for Poisson matrix factorization with twin EB priors Input: Data X, number of particles S, learning rate ζ, number of iterations T, number of components Kr, Kc, number of latent dimensions L.\n\nOutput: Variational posterior parameters Λ∗, prior parameters θ r∗, θc∗.\n\nInitialize: Λ(0), θ r(0), θc(0).\n\nfor t = 1 to T do for i = 1 to N, l = 1 to L do for s = 1 to S do Sample \n(s)\ni,l ∼ N (0, 1).\n\nCompute U\n(s)\ni,l = exp(a 0\n(t−1)\ni,l + b 0\n(t−1)\ni,l \n(s)\ni,l ).\n\nend for end for for j = 1 to D, l = 1 to L do for s = 1 to S do Sample φ\n(s)\nj,l ∼ N (0, 1).\n\nCompute V\n(s)\nj,l = exp(a\n(t−1)\nj,l + b\n(t−1)\nj,l φ\n(s)\nj,l ).\n\nend for end for Estimate L(X; θ rt−1, θc(t−1), Λ(t−1)) using Monte-Carlo in Equation (24) with samples U(s), V\n(s)in place of EqΛ\n.\n\nΛ(t), θr(t), θc(t) ← Adam(∇(θ r,θc,Λ)L, ζ).\n\nend for return Λ(T), θr(T), θc(T).\n\n\n\nFigure 2: Twin population priors induce robustness to prior selection The held-out likelihood is sensitive to the choice of the prior hyper-parameters. GMF endowed with population priors on both row and column latent variables, TwinEB (GMF), achieves comparable or better results than other methods. Each sub-panel displays held-out log-likelihood from adjusting the column prior variance with a fixed row prior, while the right sub-panel does the opposite, varying the row prior variance with a constant column prior.\n\nWe demonstrate four datasets, from the top to bottom, MovieLens 1M, Ru1322b-scRNAseq, UserArtists, and GoodBooks. In all datasets, we set L = 15. Similar results hold for other values of L (see supplement).\n\n## 4 Experiments\n\nWe studied Algorithm 1 in several real-world matrix factorization settings: book ratings, movie ratings, artist preferences, and single-cell RNA sequence gene counts. For all datasets, we studied both Gaussian and Poisson matrix factorization. In all datasets, we set L = 15. We found that TwinEB performs as well or better than manually searching for a parameterized prior, and performs as well or better than setting a simple parameterized prior by empirical Bayes.\n\n## 4.1 Datasets\n\nIn this section we first review four real-world datasets, ranging from user-preferences to genomics, and then explore the impact of twin population priors on the performance of Gaussian and Poisson matrix factorization.\n\nIn our experiments, we fix the number of row and column mixtures to K .- 70 and Kc = 100 respectively.\n\nMovieLens 1M. This dataset comprises 1 million ratings from 6,000 users (rows) on 4,000 movies\n(columns) (Harper & Konstan, 2015). The ratings are on a scale of 1 to 5. Sparsity of this dataset, defined as the number of nonzero elements divided by the total number of entries, is 0.04. We can use matrix factorization to capture different aspects of user preferences and movie characteristics. Specifically, Ui,l may signify user i's affinity for aspect l (e.g., genre), while Vj,l may represent the degree to which movie j exhibits aspect l.\n\nRu1322b. We analyze single cell gene expression data from a patient with small cell lung cancer (Chan et al., 2021). This dataset comprises 4,000 highly variable genes (columns) across 5,308 cells (rows). For the GMF family, we applied a two-step transformation. First, we performed a log transformation on the counts after adding a pseudo-count of one. Then, we standardized the non-zero elements. Sparsity of this dataset is 0.16. Each entry of the matrix denotes the number of transcripts of gene j in cell i. We explain the gene expression matrix via L gene-modules, with Ui,l as the activity of module l in cell i, and Vj,l as gene j's contribution to module l. Matrix factorization can be used as exploratory data analysis (finding gene modules associated with malignancy in cancer) or as a component in a more complex analysis (causal inference (Wang & Blei, 2019)). See the supplement for details on preprocessing of the sequencing data. UserArtists We use the data introduced in Cantador et al. (2011), comprising 92,834 user-listened artist relations, across 1,892 users and 17,632 artists, with a maximum value of 352, 698. Similar to MovieLens 1M, we use matrix factorization to explain different aspects of user preferences and artist groupings. Sparsity of this dataset is 0.003. We note that this dataset was analyzed in da Silva et al. (2023).\n\nGoodBooks Contains 6 million ratings across 51,288 users and 10,000 books1. Ratings range from 1 to 5.\n\nSparsity of this dataset is 0.01.\n\n## 4.2 Evaluation Metric And Baselines\n\nWe evaluate each model based on the likelihood of held-out data. For a given row-wise data split into held-in and held-out rows, let Xout = {Xout i} denote the set of Nout masked entries of the held-out rows. We estimate the held-out log-likelihood as follows:\n\n$$\\begin{array}{l}{{\\frac{1}{N^{\\mathrm{out}}}\\sum_{i=1}^{N^{\\mathrm{out}}}\\log P(X_{i}^{\\mathrm{out}}\\mid X^{\\mathrm{in}})\\approx}}\\\\ {{\\frac{1}{N^{\\mathrm{out}}}\\sum_{i=1}^{N^{\\mathrm{out}}}\\log\\frac{1}{M}\\sum_{m=1}^{M}P(X_{i}^{\\mathrm{out}}\\mid\\mathbf{U}^{(m)},\\mathbf{V}^{(m)}),}}\\end{array}$$\n$$(26)$$\n$$(27)$$\n$\\boldsymbol{\\mathit{T}}$, $\\boldsymbol{\\mathit{V}}$). \nwhere M is the number of Monte Carlo samples from qΛ(U,V ).\n\nWe randomly assign 20% of the rows as the test set, and the rest as training data. We then mask 20% of the entries at random, and train the model on this train set using ten random restarts. We use the 20% masked entries as a *validation set*. At test time, we put aside 30% of the entries of the test rows at random - these entries constitute the *test set* - then we train the model on 40% of the rest of the entries. This procedure measures *strong generalization* (Steck, 2019).\n\nTo compute the held-out likelihood, after training the model on the held-in data, we fix its column parameters and then learn row parameters of the held-out rows. We then report the likelihood of the masked entries in the test set. For each model, we report the test HOLL of the random restart that achieved the best validation HOLL. In our experiments, we set M = 500. See the supplement for the derivation of Equation (27) and more details on the experiments. Baseline. We evaluate the performance of the PMF (GMF) model with TwinEB against multiple baselines, namely (i) TwinEB-Single, (ii) PMF (GMF). The former is a simple form of TwinEB where all latent dimensions have an identical prior, which we learn. The latter is a prior of the same family as the TwinEB, but with fixed hyperparameters. We compare TwinEB against a large choice of fixed parameters, akin to hyperparameter selection.\n\n1Accessed at https://github.com/zygmuntz/goodbooks-10k\n\n## 4.3 Results: Gaussian Matrix Factorization\n\nWe preprocess the data as follows. We standardize each column by subtracting the mean from non-zero entries and dividing the result by their standard deviation. We study two scenarios: (i) maintaining a fixed prior on row-wise variables while varying the prior on column-wise variables, and (ii) holding the prior on column-wise variables constant and adjusting the prior on row-wise variables. We set the variational family as well as the mixture components for the population priors to be Gaussian. We set the fixed prior to N (1.0, 1.0) and vary the variance of the non-fixed one over {0.001, 0.01, 0.1, 1.0, 10}.\n\nWe compared the GMF model to TwinEB (GMF) and TwinEB-Single (GMF). We treat zero entries as missing values.\n\nFigure 2 displays the outcomes for four real-world datasets. In GoodBooks and MovieLens-1M datasets, TwinEB achieves the highest test HOLL. For the UserArtists dataset, TwinEB does better than the fixed prior. Similar results were obtained by varying L to other values and are reported in Figures 7, 8 in Appendix D.\n\nWe also compare to the method of Wang & Stephens (2021), we used the corresponding package flashr, that currently supports GMF. We designed an imputation experiment where we held-out 10% of entries of a standardized matrix, and compare reconstruction accuracy on non-zero enteries. We applied flashr to the above four real-world datasets. flashr crashes on all but the least sparse dataset, Ru1322b. For the Ru1322b dataset it achieves a mean absolute error of 0.64 vs 0.51 for the TwinEB. We note that it is not straightforward to compare our method to that of Jiang & Zhang (2009); the software implementation does not immediately support missing data and imputation.\n\n## 4.4 Results: Poisson Matrix Factorization\n\nFor the MovieLens 1M and GoodBooks datasets, we binarize the matrix, setting entries to 'one' if a user has rated a movie and to 'zero' otherwise. For the Ru1322b dataset, we normalize the rows such that the sum of all rows are equal; we then round each value to the nearest integer. This is to account for the effect of library size (Heumos et al., 2023). We treat zeros as missing data. In all models, we set the variational family to be Log-Normal, and the prior (or the mixture components for the population priors) to be Gamma. Similar to GMF, we vary the row or column prior parameters, while keeping the other one fixed. We parameterize the Gamma prior by its mean and variance Gamma(*µ, σ*2), where µ = α/β and σ 2 = α/β 2. In each scenario, we set the fixed prior to Gamma(1, 10). For the varying prior, we set its mean to µ = 1 and change its variance along {0.01, 0.1, 0.25, 1.0, 10.0, 100.0}. Figure 3 shows the results.\n\nIn datasets except for GoodBooks, TwinEB outperforms TwinEBSingle. In the UserArtists dataset, TwinEB\noutperforms other methods. TwinEB, on average, takes about 1.5X the runtime of PMF (ranging form 1.1X\nto 2.1X). We point out that finding a good prior using grid-search incurs the sum of the cost of evaluations of the individual points in the grid, over 6.5X the running time of the PMF with TwinEB prior. We note that using a grid to find fixed priors (or alternatively, learning the scalar prior) does not guarantee reaching a best test HOLL. Indeed in the UserArtists dataset, TwinEB yields the best test HOLL. We also compared TwinEB to the method of da Silva et al. (2023); given the data, it estimates values for the shape and rate parameters of the Gamma prior for both the row and column r.v.s. Table 1 displays the estimated hyperparameters and the resulting test HOLL.\n\nPMF equipped with fixed priors values calculated from this methods yields Test HOLL that are on par or slightly worse than the TwinEB method. This method yielded negative hyperparameter estimates for the MovieLens-1M and GoodBooks datasets. These parameters are effectively not usable, since the parameters of a Gamma must be positive. Hence, we emphasize that unlike ours, this methods cannot be used on all datasets.\n\n\n\nFigure 3: Twin population priors induce robustness to prior selection. As in Figure 2, but for PMF.\nTable 1: Test Heldout Loglikelihood from running PMF with prior hyperparameters sets using the method of da Silva et al. (2023).\n\nFor MovieLens-1M and GoodBooks datasets, da Silva et al. (2023) yielded inadmissible hyperparameters.\n\n| | test HOLL | |\n|--------------|------------------------|--------|\n| Dataset | da Silva et al. (2023) | TwinEB |\n| Ru1322b | -9.53 | -9.46 |\n| UserArtists | -1,583 | -1,481 |\n| MovieLens-1M | NA | -3.04 |\n| GoodBooks | NA | -3.95 |\n\n## 4.5 Simulation: Complexity Of The Prior\n\nHere we examine the performance of the population priors as the number of mixture components are varied. To this end, we simulate a 1,000 by 1,500 dataset, with L = 64 dimensional row and column-wise r.v.s. We sample the row- and column-wise r.v.s from a mixture of 15 and 20 Gamma distributions respectively, the\n\nSimulated Data - Study of K\n\n\n\nFigure 4: Increasing the number of mixtures improves the model performance. Each line shows the average test-HOLL over 10 seeds for a fixed value for Kc, the number of column mixtures, and varying values of Kr, the number of row mixtures. Here we set L = 64 to its true simulated value. The results are similar for L = 32 (see supplement).\n\nrate and shape parameters of which are sampled from a Gamma(1, 1). The mixture weights are sampled from a Dirichlet(e0*, . . . , e*0) where the concentration parameter is e0 = 10.\n\nPerformance Increase with Number of Mixture Components. The more mixture components, the more flexible the EB prior is. If learning a EB prior is beneficial, then we expect the performance to increase with the number of mixture components. We run PMF with population priors with varying number of row and column mixtures, setting the dimension of the latent variable to its true value, and, to avoid model misspecification, set the mixture prior family to Gamma. We then report the average of the test HOLL across ten different seeds. Figure 4, the log-likelihood of test held-out data increases with the number of mixture components in the row prior.\n\n## 5 Discussion\n\nWe introduced the twin population priors for probabilistic matrix factorization. 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Martin J Wainwright and Michael I Jordan. *Graphical models, exponential families, and variational inference*.\n\nNow Publishers, Inc., 2008.\n\nWei Wang and Matthew Stephens. Empirical Bayes matrix factorization. *JMLR*, 22(1):5332–5371, 2021.\n\nYixin Wang and David M. Blei. The Blessings of multiple causes. *JASA*, 114(528):1574–1596, 2019. Publisher:\nTaylor & Francis.\n\nYixin Wang, David Blei, and John P Cunningham. Posterior collapse and latent variable non-identifiability.\n\nNeurIPS, 34:5443–5455, 2021.\n\nF Alexander Wolf, Philipp Angerer, and Fabian J Theis. SCANPY: large-scale single-cell gene expression data analysis. *Genome biology*, 19:1–5, 2018.\n\nXinyi Zhong, Chang Su, and Zhou Fan. Empirical Bayes PCA in high dimensions. *J. R. Stat. Soc.*, 84(3):\n853–878, 2022.\n\n## A Introduction\n\nThis is the supplement for the manuscript titled \"Population priors for matrix factorization\". The figures in this document supplement Figures 2, 3, and 4 in the main text. We provide derivations for twin population priors for Gaussian matrix factorization, as well as computing the likelihood of the held-out data in section B.\n\nWe then give more details about our experimental setup, including the parameters used in training (e.g., batch-size) in section C. Finally, we provide results for additional experiments in section D: for additional values of the latent dimension L for the datasets studied in the main text.\n\nNote that this document is accompanied by an archive file code.zip, that contains the source code, instructions to install and run the code, and scripts to recreate the experiments and plot the figures in the manuscript. All scripts have been de-identified.\n\n## B Derivations\n\nIn this section, we give more details and derivations for the quantities defined in the main text. Specifically, in section B.1 we derive the twin population priors for the Gaussian matrix factorization (GMF). In section B.2, we derive the variational approximation to the twin population priors in GMF. Finally, in section B.3, we derive the expression for held-out likelihood in matrix factorization.\n\n## B.1 Twin Population Priors For Gaussian Matrix Factorization\n\nIn this section we derive population priors for Gaussian matrix factorization (GMF). In the classical GMF, the likelihood and priors on the latent variables are Gaussian (Mnih & Salakhutdinov, 2007). The generative model is as follows:\n\n$$W_{j,l}\\sim{\\cal N}(0,\\sigma_{W}^{2}),\\quad j=1\\ldots D,l=1\\ldots L,$$ $$Z_{i,l}\\sim{\\cal N}(0,\\sigma_{Z}^{2}),\\quad i=1\\ldots N,l=1\\ldots L,\\tag{1}$$ $$X_{i,j}\\mid\\mathbf{U}_{i},\\mathbf{V}_{j}\\sim{\\cal N}(\\sum_{l=1}^{L}Z_{i,l}W_{jl},\\sigma^{2}),\\quad i=1\\ldots N,j=1\\ldots D,$$\n$$(28)$$\n\nwhere U = [Zi,l] ∈ R\nN×L and [Wj,l] ∈ R\nD×L, with Ui and Vj and are L-vectors representing the row and column wise latent variables, and σ, σW and σZ are constant.\n\nFor TwinEB, our goal is to learn the prior for the row and column latent variables, so, as in the main text, we construct mixture priors (Equations 14 and 15), where Pµk,σk and Pνk,ηk are Gaussian parameterized by their mean and scale parameters. That is,\n\n$$P_{\\theta^{r}}(\\mathbf{U}_{i})=\\sum_{k=1}^{K_{r}}\\pi_{k}\\mathcal{N}(\\mathbf{U}_{i};\\mathbf{\\mu}_{k},\\mathbf{\\sigma}_{k}),$$ $$P_{\\theta^{c}}(\\mathbf{V}_{j})=\\sum_{k=1}^{K_{c}}\\rho_{k}\\mathcal{N}(\\mathbf{V}_{j};\\mathbf{\\nu}_{k},\\mathbf{\\eta}_{k}),$$\n$$(29)$$\n$$(30)$$\n$$(31)$$\n\nwhere Kr and Kc are the number of components in the mixtures, θ r = {µ,σ,π} and θ c = {ν, η, ρ}. The locations µ ∈ R\nKr×L and ν ∈ R\nKc×L, the scales σ ∈ R\nKr×L and η ∈ R\nKc×L, and the mixture weights π ∈ ∆Kr and ρ ∈ ∆Kc are the parameters of the mixtures priors.\n\n## B.2 Posterior Inference In Gmf With Twin Population Priors\n\nOur goal is to calculate the posterior P(U,V | X.θr, θc), given data X, which also depends on our choice of priors θ r, θc. Our strategy is the same as in the main text, namely, to simultaneously optimize for the parameters of the variational distribution and the TwinEB prior. We substitute the variational family in Equations (19) and (19) with Gaussian, as follows:\n\n$$\\begin{array}{l}{{\\cal X}_{i,l}^{r}:=(a_{i,l}^{\\prime},b_{i,l}^{\\prime}),}}\\\\ {{\\cal X}_{j,l}^{c}:=(a_{j,l},b_{j,l}),}}\\\\ {{q_{{\\cal X}_{i,l}^{r}}(U_{i,l}):={\\cal N}(a_{i,l}^{\\prime},b_{i,l}^{\\prime}),}}\\\\ {{q_{{\\cal X}_{j,l}^{c}}(V_{j,l}):={\\cal N}(a_{j,l},b_{j,l}).}}\\end{array}$$\n\nEach Gaussian is parameterized by its natural parameters a and b:\nN (x; *a, b*) ∝ exp ax + bx2 The log-likelihood of the data is:\n\n$$\\log P(\\mathbf{X}\\mid\\mathbf{Z},\\mathbf{W})=\\log\\prod_{i=1}^{N}\\prod_{j=1}^{D}P_{\\sigma}(X_{i,j}\\mid\\mathbf{Z}_{i},\\mathbf{W}_{j})\\tag{36}$$ $$=\\sum_{i=1}^{N}\\sum_{j=1}^{D}\\log\\mathcal{N}\\left(X_{i,j};\\sum_{l=1}^{L}Z_{i,l}W_{j,l},\\sigma^{2}\\right)$$ $$=-\\sum_{i=1}^{N}\\sum_{j=1}^{D}\\left[\\log\\sigma\\sqrt{2\\pi}+\\frac{1}{2}\\left(\\frac{X_{i,j}-\\sum_{l=1}^{L}Z_{i,l}W_{j,l}}{\\sigma}\\right)^{2}\\right].$$\n\n$$(37)$$\n$$(38)$$\n\n## B.3 Held-Out Likelihood For Matrix Factorization\n\nWe use the log likelihood of held-out data as the score for each model. Let Xout = {Xi,j} denote Nout entries of the held-out rows that were masked. We compute the likelihood of the masked entries via the posterior predictive distribution:\n\n$$\\log P(X^{\\text{out}}\\mid X^{\\text{in}})=\\log\\prod_{i=1}^{N^{\\text{out}}}P(X_{i}^{\\text{out}}\\mid X^{\\text{in}})$$ $$=\\sum_{i=1}^{N^{\\text{out}}}\\log P(X_{i}^{\\text{out}}\\mid X^{\\text{in}}).$$\n (39) $\\binom{40}{41}$ (41) ... \nFor Bayesian matrix factorization we expand the summand in Equation (38) as follows:\n\n$$\\log P(X_{i}^{\\rm out}\\mid X^{\\rm in})\\approx\\log\\iint P(X_{i}^{\\rm out}\\mid\\mathbf{U},\\mathbf{V})P(\\mathbf{U},\\mathbf{V}\\mid X^{\\rm in})d\\mathbf{U}d\\mathbf{V}$$ $$\\approx\\log\\iint P(X_{i}^{\\rm out}\\mid\\mathbf{U},\\mathbf{V})q_{\\mathbf{A}}(\\mathbf{U},\\mathbf{V}\\mid X^{\\rm in})d\\mathbf{U}d\\mathbf{V}$$ $$=\\log_{\\mathbf{U},\\mathbf{V}\\sim q_{\\mathbf{A}}(\\cdot)}[P(X_{i}^{\\rm out}\\mid\\mathbf{U},\\mathbf{V})]$$ $$\\approx\\log\\frac{1}{M}\\sum_{m}^{M}P(X_{i}^{\\rm out}\\mid\\mathbf{U}^{(M)},\\mathbf{V}^{(M)}),$$\n\nwhere M is the number of Monte Carlo samples from qΛ(U,V | Xin). In Equation (40) we approximate the true posterior P(U,V | Xin) with its variational counterpart qΛ. The log-likelihood score for the entire held-out data is then:\n\n$$\\sum_{i}^{N^{\\mathrm{out}}}\\log P(X_{i}^{\\mathrm{out}}\\mid X^{\\mathrm{in}})\\approx\\sum_{i}^{N^{\\mathrm{out}}}\\log\\frac{1}{M}\\sum_{m}^{M}P(X_{i}^{\\mathrm{out}}\\mid\\mathbf{U}^{(M)},\\mathbf{V}^{(M)}).\\tag{1}$$\n$$\\quad(42)$$\n$$(43)$$\n\n17\n\n## C Experimental Details\n\nIn this section we give more details on our experimental studies in the main manuscript. In section C.1 we describe preprocessing for the gene expression in the Ru1322-scRNAseq dataset. In section C.2, we specify the parameters used during training. In section C.3 we give a brief description of the artifacts that accompany this supplementary material.\n\n## C.1 Preprocessing Of The Ru1322-Scrnaseq Gene Expression Dataset\n\nWe use CellRanger version 6.0.1 to process the FASTQ and generate the unique molecular identifier (UMI)\ncount matrices. We use scanpy to preprocess the data, and the seuratv3 algorithm to select highly variable genes Wolf et al. (2018).\n\n## C.2 Training Details\n\nWe set a batch size of 128 in all our experiments. We ran Poisson and Gaussian matrix factorization experiments for a maximum of 20, 000 iterations. By this step, all runs had converged.\n\nWe initialized the learning rate for the row and column variables, rlr and clr separately. We fix the initial learning rate rlr ∈ {0.01} and clr ∈ {0.01}. In the experiments in the main text, we use 10 Monte Carlo samples to approximate the ELBO, while in the supplemental experiments, we use a single particle.\n\nFor the PMF experiments in the supplement, we subsample zeros as is standard (Gopalan et al., 2015).\n\nWe uniformly randomly subsample the same number of zeros as non-zero values to estimate the likelihood.\n\nConcretely, let L(Xout) denote the log-likelihood of the masked entries of the held-out rows, Xout = {Xi,j}. Then L(Xout) can be decomposed as the sum of non-zero and zero Xi,j :\n\n$${\\mathcal{L}}(X^{\\mathrm{out}})={\\mathcal{L}}(X_{x_{i,j}\\neq0}^{\\mathrm{out}})+{\\mathcal{L}}(X_{x_{i,j}=0}^{\\mathrm{out}}),$$\n$$(44)$$\nxi,j=0), (44)\nwhere Xout xi,j 6=0 and Xout xi,j=0 have cardinalities Nout non-zero and Nout zero respectively.\n\nWe approximate Equation (44) by subsampling Nsub = min(N\nout zero, N\nout non-zero) of the zeros. Let Xzero Nsub denote a multiset of zeros of cardinality Nsub, then:\n\n$$\\hat{\\mathcal{L}}(X^{\\rm out})\\approx\\mathcal{L}(X^{\\rm out}_{x_{i,j}\\neq0})+\\frac{\\mathrm{N}^{\\rm out}_{\\mathrm{zero}}}{\\mathrm{N}_{\\mathrm{sub}}}\\,\\mathbf{X}^{\\rm zero}_{\\mathrm{sub}}.\\tag{45}$$\n\nWe ran our experiment on a machine equipped with an NVIDIA A100 GPU with 80GB memory. We implemented all methods in pytorch (Paszke et al., 2019).\n\n## C.3 Implementation\n\nPlease refer to the code directory for the source code and instructions on how to run the model. The file README.md contains instructions for installing the software, and running it. Under the data directory, we included preprocessed data for the MovieLens-100K dataset, a smaller version of the MovieLens-1M studied in the main text. In the notebooks directory, we put notebooks used for preprocessing the data, and plotting the figures in the manuscript. Finally, in the pipelines directory, we put nextflow scripts that recreate experiments that we have run for the manuscript.\n\n## D Additional Experiments\n\nIn this section we present additional experimental results. We show results for additional values of the latent dimension L. Concretely, section D.1 studies the effects of the twin population priors on Poisson and Gaussian matrix factorization, on three real world datasets, while section D.2 examines the sensitivity of the twin population priors to the choice of its hyper-parameters. We find that these results corroborate those that were presented in the main manuscript, that is, matrix factorization with traditional priors is sensitive to the choice of the hyper-parameters of the prior, and twin population priors is a robust way to set the prior in this family of models.\n\n## D.1 Additional Values Of Latent Dimensions\n\nIn the main text, we study the effects of twin population priors on the Poisson and Gaussian matrix factorization over four real datasets, with the dimension of the latent variables set to L = 15. Here we present results for additional values of L. Figures 5, 6, 7, 8, and 9 show the results. However, the twin population priors find a comparable or better prior, in the sense of higher held-out likelihood.\n\n\n\n## D.2 Performance Increase With Number Of Mixture Components\n\nWe show the performance of TwinEB on simulated data as we vary the number of row and column mixture components . Here we add results for additional values of L, and also study the MovieLens 100K dataset. The result in Figures 9 suggests that as the number of mixture components increases, the performance of TwinEB improves. The gain is apparent when the number of row components K, increases. This is expected, as our evaluation procedure, measures the generalizability for held-out rows, but not held-out columns.\n\n\n\nFigure 6: As in Figure 5, but with additional values of latent dimension L.\n\n\n\nFigure 7: As in Figure 3, but with additional values of latent dimension L.\n\n\n\nFigure 8: As in Figure 7, but with additional values of latent dimension L.\nL = 32\n\n\n\n\n\n.....\n\n.....\n\n\\#Column components: 1\n\\#Column components: 2\n\\#Column components: 4\n\\#Column components: 8\n\\#Column components: 16\n\\#Column components: 20\n\\#Column components: 32\n\nFigure 9: As in Figure 4, but with additional column components as well as latent dimension L = 32.\n\n## L = 64\n"},"reviews":{"kind":"string","value":""}}},{"rowIdx":98,"cells":{"paper":{"kind":"string","value":"# Generalization Error Bounds For Learning Under Censored Feedback\n\nAnonymous authors Paper under double-blind review\n\n## Abstract\n\nGeneralization error bounds from learning theory provide statistical guarantees on how well an algorithm will perform on previously unseen data. In this paper, we characterize the impacts of data non-IIDness due to censored feedback (a.k.a. selective labeling bias) on such bounds. We first derive an extension of the well-known Dvoretzky-Kiefer-Wolfowitz (DKW) inequality, which characterizes the gap between empirical and theoretical CDFs given IID data, to problems with *non-IID data due to censored feedback*. We then use this CDF error bound to provide a bound on the generalization error guarantees of a classifier trained on such non-IID data. We show that existing generalization error bounds (which do not account for censored feedback) fail to correctly capture the model's generalization guarantees, verifying the need for our bounds. We further analyze the eectiveness of (pure and bounded) exploration techniques, proposed by recent literature as a way to alleviate censored feedback, on improving our error bounds. Together, our findings illustrate how a decision maker should account for the trade-o between strengthening the generalization guarantees of an algorithm and the costs incurred in data collection when future data availability is limited by censored feedback.\n\n## 1 Introduction\n\nGeneralization error bounds are a fundamental concept in machine learning, which provide (statistical) guarantees on how a machine learning algorithm trained on some given dataset will perform on new, unseen data. However, many implicit or explicit assumptions about training data are often made when training ML models and deriving theoretical guarantees for their performance. These assumptions include access to independent and identically distributed (IID) training data, the availability of correct labels, and static underlying data distributions (Bartlett & Mendelson, 2002; Bousquet & Elissee, 2002; Cortes et al., 2019; 2020). Some studies in this area, e.g. Cheng et al. (2018); Kuznetsov & Mohri (2017); Mohri & Rostamizadeh\n(2007; 2008), have provided bounds when these assumptions are removed. In this paper, we are similarly interested in the impact of non-IID training data, specifically due to *censored feedback*, on the learned algorithm's generalization error guarantees.\n\nCensored feedback, also known as selective labeling bias, arises in many applications wherein human or algorithmic decision-makers set certain thresholds or criteria for favorably classifying individuals, and subsequently only observe the true label of individuals who pass these requirements. For example, schools may require a minimum GPA or standardized exam score for admission; yet, graduation rates are only observed for admitted students. Financial institutions may set limits on the minimum credit score required for loan approval; yet, loan return rates are only observed for approved applicants. In these types of classification tasks, the algorithm's training dataset grows over time (as students are admitted, loans are granted); however, the new data is selected in a non-IID manner from the underlying domain, due to the unobservability of the true label of rejected data. This type of bias also arises when determining recidivism in courts, evaluating the eectiveness of medical treatments, flagging fraudulent online credit card transactions, etc. Despite this ubiquity, to the best of our knowledge, generalization error bounds given non-IID training data due to censored feedback remain unexplored. We close this gap by providing such bounds in this work, show the need for them, and formally establish the extent to which censored feedback hinders generalization. One of the commonly proposed methods to alleviate the impacts of censored feedback is to *explore* the data domain, and admit (some of) the data points that would otherwise be rejected, with the goal of expanding the training data. Existing approaches to exploration can be categorized into *pure exploration* (Bechavod et al., 2019; Kazerouni et al., 2020; Kilbertus et al., 2020; Nie et al., 2018), where any individual in the exploration range may be admitted (with some probability '), and *bounded exploration* (Balcan et al., 2007; Lee et al., 2023; Wei, 2021; Yang et al., 2022), in which the exploration range is further limited based on cost or informativeness of the new samples. The additional data samples collected through (pure or bounded)\nexploration may not only help improve the accuracy of the learned model when evaluated on a given test data (as shown by these prior works), but may also help tighten the generalization error guarantees of the learned model; we formalize the latter improvement, and show how the frequency and range of exploration can be adjusted accordingly. We note that censored feedback may or may not be avoidable depending on the application (given, e.g.,\nthe costs or legal implications of exploration). We therefore present generalization error bounds both with and without exploration, establishing the extent to which the decision maker should be concerned about censored feedback's impact on the learned model's guarantees, and how well they might be able to alleviate it if exploration is feasible. Our approach. We characterize the generalization error bounds as a function of the gap between the empirically estimated cumulative distribution function (CDF) obtained from the training data, and the ground truth underlying distribution of data. At the core of our approach is noting that although censored feedback leads to training data being sampled in a non-IID fashion from the true underlying distribution, this non-IID data can be split into IID subdomains. Existing error bounds for IID data, notably the DvoretzkyKiefer-Wolfowitz (DKW) inequality (Dvoretzky et al., 1956; Massart, 1990), can provide bounds on the deviation of the empirical and theoretical subdomain CDFs, as a function of the number of available data samples in each subdomain. The challenge, however, lies in reassembling such subdomain bounds into an error bound on the full domain CDFs. Specifically, this will require us to shift and/or scale the subdomain CDFs, with shifting and scaling factors that are themselves empirically estimated from the underlying data, and can be potentially re-estimated as more data is collected. Our analysis identifies these factors, and highlights the impacts of each on the error bounds.\n\n## Summary Of Findings And Contributions:\n\n1. We generalize the well-known Dvoretzky-Kiefer-Wolfowitz (DKW) inequality, which characterizes the gap between empirical and theoretical CDFs given IID data, to problems with non-IID data due to censored feedback without exploration (Theorem 2) and with exploration (Theorem 3), and formally show the extent to which censored feedback hinders generalization.\n\n2. We characterize the change in these error bounds as a function of the severity of censored feedback\n(Proposition 1) and the exploration frequency (Proposition 2). We further show (Section 3.3) that a minimum level of exploration is needed to tighten the error bound.\n\n3. We derive a generalization error bound (Theorem 4) for a classification model learned in the presence of censored feedback using the CDF error bounds in Theorems 2 and 3.\n\n4. We numerically illustrate our findings (Section 5). We show that existing generalization error bounds\n(which do not account for censored feedback) fail to correctly capture the generalization error guarantees of the learned models. We also illustrate how a decision maker should account for the trade-o between strengthening the generalization guarantees of an algorithm and the costs incurred in data collection for reaching enhanced learning guarantees.\n\nRelated works. Although existing literature has studied generalization error bounds for learning from non-IID data, non-IIDness raised by censored feedback has been overlooked. We provide a detailed review of related work in Appendix A. Here, we discuss works most closely related to ours.\n\nFirst, our work is closely related to generalization theory in the PAC learning framework in non-IID settings, including (Mohri & Rostamizadeh, 2007; 2008; Yu, 1994) and (Kuznetsov & Mohri, 2017); these works consider dependent samples generated through a stationary, and non-stationary —-mixing sequence, respectively, where the dependence between samples weakens over time. To address the vanishing dependence issue, these works consider building blocks within which the data can be viewed as IID. The study of Yu (1994) is based on the VC-dimension, while Mohri & Rostamizadeh (2008) and Mohri & Rostamizadeh (2007) focus on the Rademacher complexity and algorithm stability, respectively. Our work is similar in that we also consider building IID blocks to circumvent data non-IIDness. However, we dier in our reassembly method, in the source of non-IID data, and in our study of the impacts of exploration.\n\nOur work is also closely related to partitioned active learning, including Cortes et al. (2019; 2020); Lee et al.\n\n(2023); Zheng et al. (2019). Cortes et al. (2019) partition the entire domain to find the best hypothesis for each subdomain, and a PAC-style generalization bound is derived compared to the best hypothesis over the entire domain. This idea is further extended to adaptive partitioning in Cortes et al. (2020). In Lee et al.\n\n(2023), the domain is partitioned into a fixed number of subdomains, and the most uncertain subdomain is explored to improve the mean-squared error. The work of Zheng et al. (2019) considers a special data nonIIDness where the data-generating process depends on the task property, partitions the domain according to the task types, and analyzes each subdomain separately. Our work is similar to these studies in that we also consider (active) exploration techniques, and partition the data domain to build IID blocks. However, we dier in problem setup and analysis approach, and in accounting for the cost of exploration when we consider bounded exploration techniques. Lastly, the technique of identifying IID-blocks within non-IID datasets has also been used in other contexts to address the challenge of generalization guarantees given non-IID data. For instance, Wang et al. (2023)\ninvestigate generalization performance with covariate shift and spatial autocorrelation in geostatistical learning. They address the non-IIDness issue by removing samples from the buer zone to construct spatially independent folds. Similarly, Tang et al. (2021) study generalization performance within the Federated Learning paradigm with non-IID data. They employ clustering techniques to partition clients into distinct clusters based on statistical characteristics, thus treating samples from clients within each cluster as IID\nand analyzing each cluster separately. We similarly explore generalization performance with non-IID data samples and employ the technique of identifying IID subdomains/blocks. However, we dier in the reason for the occurrence of non-IIDness, the setup of the problem, and our analytical approaches.\n\n## 2 Problem Setting\n\nConsider a decision maker (equivalently, the algorithm), and new agents (equivalently, data points) arriving sequentially. The algorithm is a binary classifier, used to make positive/negative decisions (e.g.,\naccept/reject) on each new data point. We use a bank granting loans as a running example.\n\nThe agents. Each agent/data point has a feature x and a true label y. The feature x œ X ™ R is the characteristic used for decision-making (e.g., a credit score). The true label y œ Y = {0, 1} reflects a qualification state, with y = 1 meaning the data point is qualified to receive a favorable decision (e.g., the applicant will return a loan if granted). We will use X and Y to denote the corresponding random variables, and x and y to denote realizations. Denote the proportion of qualified (unqualified) samples in the population by p1 (p0).\n\nThe algorithm. The decision maker begins with a fixed initial/historical training dataset containing ny IID samples 1 {xy1*,...,x*yny } for each label y (e.g., data on past loan repayments with n0 IID samples of individuals with credit scores {x01*,...,x*0n0 } who defaulted on their loans, and n1 IID samples of individuals with credit scores {x11*,...,x*1n1 } who paid o their loans on time). Based on these, the decision maker selects a threshold-based binary classifier f◊(x) : X æ {0, 1} to decide whether to admit or reject incoming agents\n(equivalently, assign labels 1 or 0). Specifically, f◊(x) = (x Ø ◊), with ◊ denoting the decision threshold\n(e.g., ◊ could be the minimum credit score to be approved for a loan).2\n\n1That is, we assume that any non-IIDness is introduced due to censored feedback impacting subsequent data collection.\n\nExtension to initially biased training data is also possible, but at the expense of additional notation. 2The single-dimensional features and threshold classifier assumptions are not too restrictive: Corbett-Davies et al. (2017, Thm 3.2) and Raab & Liu (2021) have shown that threshold classifiers can be optimal if multi-dimensional features can be appropriately converted into a one-dimensional scalar (e.g., with a neural network).\nThe decision threshold ◊ divides the data domain into two regions: the upper, *disclosed* region, where the true label of future admitted agents will become known to the decision maker, and the lower, *censored* region, where true labels are no longer observed. As new agents arrive, due to this censored feedback, additional data is only collected from the disclosed region of the data domain (e.g., we only find out if an agent repays the loan if it is granted the loan in the first place). This is what causes the non-IIDness of the (expanded)\ndataset: after new agents arrive, the training dataset consists of ny historical IID samples from both censored and disclosed regions on each label y, and an additional ky samples collected afterwards from each label y, but only from the disclosed region, making the entire ny + ky samples a non-IID subset of the respective label y's data domain.\n\nFormally, let Fy(x) denote the theoretical (ground truth) feature distribution for label y agents. Let –y :=\nFy(◊) be the theoretical fraction of the agents in the censored region, and my be the number of the initial ny training samples from label y agents located in the censored region. It is worth noting that my ny can provide an empirical estimate of –y, but the two are in general not equal. Let ky denote the number of additional training samples on label y that have been collected, all from the disclosed region, after new agents arrive. The decision maker will now have access to ny + ky total samples from label y agents, which are not identically distributed (my in the censored region, and ny ≠ my + ky in the disclosed region). Let Fyny+ky (x) denote the empirical CDF of the feature distribution for label y agents based on these ny + ky training data points. Our first goal is to provide an error bound, similar to the DKW inequality, of the discrepancy between Fyn+k and the ground truth CDF Fy, for each label y. We will then use these to bound the generalization error guarantees of the learned model from the (non-IID) {ny + ky}yœ{0,1} data points. Remark 1. *Note that we assume the decision maker starts with one* fixed realization of {ny}yœ{0,1} *data* points, and therefore the decision threshold ◊ *and the number of initial samples in the censored region*\n{my}yœ{0,1} *are (non-random) fixed values. For instance, the fixed training dataset and decision threshold can be likened to a financial institution with an existing loan repayment history dataset consisting of a* set number of defaults and on-time payo*s, and its initial decision threshold selected for approving future* applications. However, there are two possible ways to interpret the additional samples {ky}yœ{0,1}: a posteriori (i.e., outcomes after collecting exactly ky *new samples in the disclosed region), or* a priori (i.e., possible values once a total of T *new agents arrive, only some of which will fall in the disclosed region). The former* is a reasonable assumption if a decision maker has already collected samples under censored feedback, or alternatively, is willing to wait to collect the exact required number of samples until it can achieve a desired error bound. The latter is from the viewpoint of a decision maker contemplating potential outcomes if it waits for a total of T *new agents to arrive. We will present our new error bound under both interpretations.*\nWe summarize our algorithm dynamics and main notation below:\nStage I: initial data. The decision maker starts with ny fixed data points {xy1*,...,x*yny } from each label y œ {0, 1}, drawn IID from the respective true underlying distribution Fy(x). Accordingly, it selects a fixed decision threshold ◊. Given ◊, the ny samples from label y agents can be divided into my samples below ◊ (my = |{i : xyi < ◊}|; refered to as the censored region) and ny ≠ my samples above ◊ (referred to as the disclosed region).\n\nStage II: arrival of new agents. At each time t, a new agent arrives. Its true label is yˆ = y with probability py, and its feature xˆ is drawn uniformly at random from the corresponding Fyˆ(x).\n\nThe agent's feature xˆ is observed, and the agent is admitted if and only if xˆ Ø ◊. Due to censored feedback, yˆ will only be observed if the agent is admitted. When an agent is admitted, its data is used to expand the corresponding dataset of y = ˆy samples to {xy1,...,xyny , xyny+1*,...,x*yny+kty≠1, xˆ}.\n\nStage III: updating empirical distribution estimates. After T time steps (which can be fixed in advance, or denote the time at which a certain number of samples have been collected), the decision maker has access to ky new samples of label y agents, having expanded its trainig data on label y agents to (the non-IID collection) {xy1,...,xyny , xyny+1,...,xyny+ky≠1, xyny+ky }. Accordingly, it will find Fyny+ky (x), the empirical CDF of the feature distribution for label y agents based on the total of ny + ky data points on those agents.\n\n## 3 Error Bounds On Cumulative Distribution Function Estimates\n\nRecall that our first goal is to provide an error bound, similar to the DKW inequality, of the discrepancy between the empirical CDF of feature distribution Fyny+ky (x) and the ground truth CDF Fy(x), for each label y. Note that the empirical CDF is found for each label y separately based on its own data samples. Therefore, we drop the label y from our notation throughout this section for simplicity. Further, we first derive the *a posteriori* bounds for given realizations of ky, and develop the *a priori* version of the bound accordingly in Corollary 1.\n\nWe first state the Dvoretzky-Kiefer-Wolfowitz inequality (an extension of the Vapnik–Chervonenkis (VC)\ninequality for real-valued data) which provides a CDF error bound given IID data. Theorem 1 (The Dvoretzky-Kiefer-Wolfowitz (DKW) inequality (Dvoretzky et al., 1956; Massart, 1990)).\n\nLet Z1,...,Zn *be IID real-valued random variables with cumulative distribution function* F(z) = P(Z1 Æ z).\n\nLet the empirical distribution function be Fn(z) = 1n qni=1 (Zi Æ z). Then, for every n and ÷> 0,\n\n$$\\mathbb{P}\\bigg(\\operatorname*{sup}_{z\\in\\mathbb{R}}\\left|F(z)-F_{n}(z)\\right|\\geq\\eta\\bigg)\\leq2\\exp\\left(-2n\\eta^{2}\\right)\\;.$$\n\nIn words, the DKW inequality shows how the likelihood that the maximum discrepancy between the empirical and true CDFs exceeds a tolerance level ÷ decreases in the number of (IID) samples n.\n\nWe now extend the DKW inequality to the case of non-IID data due to censored feedback. We do so by first splitting the data domain into blocks containing IID data, to which the DKW inequality is applicable.\n\nSpecifically, although the expanded training dataset is non-IID, the decision maker has access to m IID\nsamples in the censored region, and n ≠ m + k IID samples in the disclosed region. Let Gm and Kn≠m+k denote the corresponding empirical feature distribution CDFs. The DKW inequality can be applied to bound the dierence between these empirical CDFs and the corresponding ground truth CDFs G and K.\n\nIt remains to identify a connection between the full CDF F, and G (the censored CDF) and K (the disclosed\n\n\n\nCDF), to reach a DKW-type error bound on the full CDF estimate (see Figure 1 for an illustration). This reassembly from the bounds on the IID blocks into the full data domain is however more involved, as it requires us to consider a set of scaling and shifting factors, which are themselves empirically estimated and dierent from the ground truth values. We will account for these dierences when deriving our generalization of the DKW inequality, as detailed in the remainder of this section. All proofs are given in the Appendix.\n\nFigure 1: The empirical CDFs Fn+k (Full domain), Gm (Censored region), and Kn≠m+k (Disclosed region),\nand the theoretical CDFs of F, G, and K. Experiments based on randomly drawn samples from Gaussian data N(7, 1), ◊ = 7, n = 50, m = 24, and k = 0.\n\n## 3.1 Cdf Bounds Under Censored Feedback\n\nWe first present two lemmas that establish how the deviation of Gm and Kn≠m+k from their corresponding theoretical values relate to the deviation of the full empirical CDF Fn+k from its theoretical value F.\n\nLemma 1 (Censored Region). Let Z = {Xi|Xi Æ ◊} denote the m out of n + k samples that are in the censored region. Let G and Gm be the theoretical and empirical CDFs of Z*, respectively. Then,*\n\n$$\\operatorname*{sup}_{x\\in(-\\infty,\\theta)}|F(x)-F_{n+k}(x)|\\leq\\operatorname*{sup}_{x\\in(-\\infty,\\theta)}\\underbrace{\\left|\\operatorname*{min}\\left(\\alpha,{\\frac{m}{n}}\\right)(G(x)-G_{m}(x))\\right|}_{(s e a l e d)\\ c e n s o r e d\\ s u b o m a n i e r o r}+\\underbrace{\\left|\\alpha-{\\frac{m}{n}}\\right|}_{s e a l i n g\\ e r o r}.$$\n\nThe (partial) error bound in this lemma shows the maximum dierence between the true F and the empirical Fn+k in the censored region (i.e., for x œ (≠Œ, ◊)) can be bounded by the maximum dierence between G\nand Gm, modulated by the *scaling* (min(–, mn )) that is required to map from partial CDFs to full CDFs.\n\nSpecifically, to match the partial and full CDFs, we need to consider the dierent endpoints of the censored region's CDF and the full CDF at ◊, which are Gm(◊) = G(◊)=1, F(◊) = –, and Fn+k(◊) = mn , respectively.\n\nThe first term in the bound above accounts for this by scaling the deviation between the true and empirical partial CDF accordingly. The second term accounts for the error in this scaling since the empirical estimate m n is generally not equal to the true endpoint –.\n\nThe following is a similar result in the disclosed region.\n\nLemma 2 (Disclosed Region). Let Z = {Xi|Xi Ø ◊} denote the n ≠ m + k out of the n + k samples in the disclosed region. Let K and Kn≠m+k be the theoretical and empirical CDFs of Z*, respectively. Then,*\n\n$$\\sup_{x\\in(\\theta,\\infty)}|F(x)-F_{n+k}(x)|\\leq\\sup_{x\\in(\\theta,\\infty)}\\underbrace{\\left|\\min(1-\\alpha,1-\\frac{m}{n})(K(x)-K_{n-m+k}(x))\\right|}_{(\\text{coulomb,disoboundal domain error}}+\\underbrace{2\\left|\\alpha-\\frac{m}{n}\\right|}_{\\text{substituting and scaling errors}}$$\n\nSimilar to Lemma 1, we observe the need for a scaling factor. However, in contrast to Lemma 1, this lemma introduces an additional *shifting error*, resulting in a factor of two in the last term |– ≠ mn |. In particular, we need to consider the dierent starting points of the disclosed region's CDF and full CDF at ◊, which are Km(◊) = K(◊)=0, F(◊) = –, and Fn+k(◊) = mn , respectively, when mapping between the CDFs; one of the |– ≠ mn | captures the error of shifting the starting point of the partial CDF to match that of the full CDF.\n\nWe can now state our main theorem, which generalizes the well-known DKW inequality to problems with censored feedback.\n\nTheorem 2. Let x1, x2,...,xn *be fixed initial data samples, drawn IID from a distribution with CDF* F(x).\n\nLet ◊ partition the data domain into two regions, such that - = F(◊), and m of the initial n samples are located to the left of ◊. Assume we have collected k additional samples above the threshold ◊*, and let* Fn+k denote the empirical CDF estimated from these n + k (non-IID) data. Then, *for every* ÷ > 0,\n\n$$\\mathbb{P}\\bigg{[}\\sup_{x\\in\\mathbb{R}}\\big{|}F(x)-F_{n+h}(x)\\big{|}\\geq\\eta\\bigg{]}\\leq\\underbrace{2\\exp\\left(\\frac{-2m(\\eta-|\\alpha|-\\frac{\\eta}{2}|)^{2}}{\\min\\big{(}|\\alpha|\\big{)}}\\right)}_{\\text{measured region error(constant)}}+\\underbrace{2\\exp\\left(\\frac{-2(\\eta-m+h)(\\eta-|\\alpha|-\\frac{\\eta}{2}|)^{2}}{\\min\\big{(}1-\\alpha,\\frac{|\\alpha|-\\eta}{2}|\\big{)}}\\right)}_{\\text{dihedral region error(decreasing with additional data)}}$$\n\nThe proof proceeds by applying the DKW inequality to each subdomain, and combining the results using a union bound on the results of Lemmas 1 and 2.\n\nThe expression above shows that as the number of samples collected under censored feedback increases\n(k æ Œ), the disclosed region's error decreases exponentially (similar to the DKW bound). However, unlike the DKW bound, this error bound does not go to zero due to a constant error term from the censored region of the data domain (the first term in the error bound). This means that unless exploration strategies are adopted, we can not guarantee arbitrarily good generalization in censored feedback tasks. Finally, we note that the DKW inequality can be recovered as the special case of our Theorem 2 by letting ◊ *æ ≠Œ* (which makes - ¥ 0, m ¥ 0).\n\nFinally, recall from Remark 1 that, instead of considering an exact realization of k new samples in the disclosed region, a decision maker may want to know the error bound after waiting for T agents to arrive\n(only some of which will fall in the disclosed region). The following corollary provides an error bound that can be leveraged under this viewpoint.\n\nCorollary 1. Let x1, x2,...,xn *be fixed initial data samples, drawn IID from a distribution with CDF* F(x).\n\nLet ◊ partition the data domain into two regions, such that - = F(◊), and m of the initial n samples are located to the left of ◊. Assume we have waited for T additional samples to arrive, and let Fn+T denote the empirical CDF estimated accordingly. Then, for every ÷ > 0,\n\nP 5sup xœR ---F(x) ≠ Fn+T (x) --- Ø ÷ 6 Æ 2 exp 1 ≠2m(÷≠|–≠ mn |)2 min !–, mn \"2 2 ¸ ˚˙ ˝ censored region error (constant) +ÿ T k=0 2 3T k 4(1 ≠ –) k–T ≠k exp 1 ≠2(n≠m+k)(÷≠2|–≠ mn |)2 min !1≠–, n≠m n\"2 ¸ ˚˙ ˝ expected disclosed region error (decreasing with wait time T)\n2\nThe proof is straightforward, and follows from writing the law of total probability for the left-hand side of the inequality by conditioning on the realization k of the samples in the disclosed region. We first note that the constant error term, as expected, is unaected by the wait time T. The second term is the expected value of the disclosed region error from Theorem 2; it is decreasing with T as the exponential error terms decrease with k, and higher k's are more likely at higher T.\n\n## 3.2 Censored Feedback And Exploration\n\nA commonly proposed method to alleviate censored feedback, as noted in Section 1, is to introduce exploration in the data domain. From the perspective of the generalization error bound, exploration has the advantage of reducing the constant error term in Theorem 2, by collecting more data samples from the censored region. Formally, we consider (bounded) exploration in the range x œ (LB, ◊), where samples in this range are admitted with an exploration *frequency* '. When LB *æ ≠Œ*, this is a pure exploration strategy.\n\nNow, the lowerbound LB and the decision threshold ◊ partition the data domain into three IID subdomains\n(see Figure 2 for an illustration). However, the introduction of the additional exploration region (LB, ◊)\nwill enlarge the CDF bounds, as it introduces new scaling and shifting errors when reassembling subdomain bounds into full domain bounds.\n\nSpecifically, of the n initial data, let l, m ≠ l, and n ≠ m of them be in the censored (below LB), exploration\n(between LB and ◊), and disclosed (above ◊) regions, respectively. Let - = F(LB) and - = F(◊), with initial empirical estimates ln and mn , respectively.\n\nAs new agents arrive, let k1 and k2 denote the additional samples collected in the exploration range and disclosed range, respectively. One main dierence of this setting with that of Section 3.1 is that as additional samples are collected, the empirical estimate of - can be re-estimated. Accordingly, we present a lemma similar to Lemmas 1 and 2 for the exploration region.\n\nLemma 3 (Exploration Region). Let Z = {Xi|LB Æ Xi Æ ◊} denote the m ≠ l + k1 *samples out of the* n + k1 + k2 samples that are in the exploration range. Let E and Em≠l+k1 *be the theoretical and empirical* CDFs of Z*, respectively. Then,*\n\n$$\\sup_{x\\in(L,B,\\theta)}|F(x)-F_{n+k_{1}+k_{2}}(x)|\\leq\\underbrace{\\left|\\beta-\\frac{l}{n}\\right|}_{subflip\\text{error}}+\\underbrace{\\left|\\alpha-\\beta-\\frac{n-l}{n}\\frac{m-l+k_{1}}{n-l+k_{1}+\\epsilon k_{2}}\\right|}_{\\text{re-rational scaling error}}$$ $$+\\underbrace{\\sup_{x\\in(L,B,\\theta)}\\left|\\min\\left(\\alpha-\\beta,\\frac{n-l}{n}\\frac{m-l+k_{1}+k_{2}}{n-l+k_{1}+\\epsilon k_{2}}\\right)(E(x)-E_{m-l+k_{1}}(x))\\right|\\right|}_{\\text{real depletion subsystem error}}$$\n\n\n\nFigure 2: The empirical CDFs Fn+k1+k2 (Full domain), Gl (Censored region), Em≠l+k1 (Explored region),\nand Kn≠m+k2 (Disclosed region), and the theoretical CDFs of *F, G, E,* and K. Experiments based on randomly drawn samples from Gaussian data N(7, 1), ◊ = 7 LB = 6, n = 50, l = 7, m = 27, and k1 = k2 = 0.\nObserve that here, we need both scaling and shifting factors to relate the partial and full CDF bounds, as in Lemma 2, but with an evolving scaling error as more data is collected. In particular, the initial empirical estimate mn is updated to ln + n≠l n m≠l+k1 n≠l+k1+'k2 after the observation of the additional k1 and k2 samples.\n\nWe now extend the DKW inequality when data is collected under censored feedback and with exploration.\n\nTheorem 3. Let x1, x2,...,xn *be fixed initial data samples, drawn IID from a distribution with CDF* F(x).\n\nLet LB and ◊ partition the domain into three regions, such that - = F(LB) and - = F(◊), with l and m of the initial n samples located to the left of LB and ◊, respectively. Assume we have collected *an additional* k1 samples between LB and ◊, under an exploration probability ', and an additional number of k2 *samples* above ◊. Let Fn+k1+k2 denote the empirical CDF estimated from these n + k1 + k2 *non-IID samples. Then,*\nfor every ÷ > 0,\n\nP 5sup xœR ---F(x) ≠ Fn+k1+k2 (x) --- Ø ÷ 6 Æ 2 exp 1 ≠2l(÷≠|—≠ ln |)2 min !—, ln \"2 2 + 2 exp 1 ≠2(m≠l+k1)!÷≠|—≠ ln |≠ --–≠—≠ n≠l nm≠l+k1 n≠l+k1+'k2 --\"2 min !–≠—, n≠l nm≠l+k1 n≠l+k1+'k2 \"2 2+ 2 exp 1 ≠2(n≠m+k2)!÷≠2 --–≠ ln ≠ n≠l nm≠l+k1 n≠l+k1+'k2 --\"2 min !1≠–, n≠l nn≠m+'k2 n≠l+k1+'k2 \"2 2.\nComparing this expression with Theorem 2, we first note that the last terms corresponding to the disclosed region are similar when setting k = k2, with the dierence being in the impact of re-estimating –.\n\nThe key dierence between the two error bounds is in the censored region, in that the first term in Theorem 2 is now broken into two parts: (still) censored region (≠Œ*, LB*), and the exploration region (LB, ◊). We can see that although there can still be a non-vanishing error term in the (still) censored region, as we collect more samples (k1 æ Œ) in the exploration region, the error from the exploration region will decrease to zero. Further, if we adopt pure exploration (LB *æ ≠Œ*, which makes - ¥ 0, l ¥ 0), the first term will vanish as well (however, note that pure exploration may not be a feasible option if exploration is highly costly). Lastly, we note that an *a priori* version of this bound can be derived using similar techniques to that of Corollary 1.\n\n## 3.3 When Will Exploration Improve Generalization Guarantees?\n\nIt might seem at first sight that the new vanishing error term in the exploration range of Theorem 3 necessarily translates into a tighter error bound than that of Theorem 2 when exploration is introduced. Nonetheless, the shifting and scaling factors, as well as the introduction of an additional union bound, enlarge the CDF error bound. Therefore, in this section, we elaborate on the trade-o between these factors, and evaluate when the benefits of exploration outweigh its drawbacks in providing error bounds on the data CDF estimates.\n\nWe begin by presenting two propositions that assess the change in the bounds of Theorems 2 and 3 as a function of the severity of censored feedback (as measured by ◊) and the exploration frequency '.\n\nProposition 1. Let B(◊) *denote the error bound in Theorem 2, and assume the conditions of that theorem* hold. Assume also that we can collect an additional k = O(n) *samples above the threshold. Then,* B(◊) is increasing in ◊.\n\nProposition 2. Let Be(LB, ◊, ') denote the error bound in Theorem 3, and assume the conditions of that theorem hold. Then, Be(LB, ◊, ') *is decreasing in* '.\n\nIn words, as intuitively expected, these propositions state that the generalization bounds worsen (i.e., are less tight) when the censored feedback region is larger, and that they can be improved (i.e., made more tight) as the frequency of exploration increases. Numerical illustration. We also conduct a numerical experiment to illustrate the bounds derived in Theorems 2 and 3. We proceed as follows: 8000 random samples are drawn from a Gaussian distribution with mean µ = 7 and standard deviation ‡ = 3, with an additional 40000 samples arriving subsequently, randomly sampled from across the entire data domain. We set ÷ = 0.015, the threshold\n◊ = 8, and the lower bound LB = 6. We run the experiment 5 times and report the error bounds accordingly.\n\nIn Figure 3, the \"original\" (blue) line represents the DKW\nCDF bound of the initial samples without additional data.\n\nThe \"B(◊)\" (orange) line and \"B(LB)\" (green) line represent the CDF bound in Theorem 2 without exploration, where the decision threshold is at ◊ and LB, respectively. The \"Be(LB, ◊, ')\" (red) line represents the bound in Theorem 3 with exploration probability '.\n\nFigure 3: A minimum exploration frequency is\n\n needed to tighten the CDF error bound.\n\nFrom Figure 3, we first observe that the green line (B(LB), which observes new samples with x Ø LB = 6) provides a tighter bound than the orange line (B(◊), which observes new samples with x Ø ◊ = 8), with both providing tighter bounds than the blue line (original DKW bound, before any new samples are observed).\n\nThis is because collecting more samples from the disclosed region results in a decrease in the CDF error bound, as noted by Proposition 1. Additionally, we can observe from the trajectory of the red line (Be(LB, ◊, '),\nwhich observes a fraction ' of new samples from (LB, ◊), and all new samples above ◊) that introducing exploration enlarges the CDF error bound due to the additional union bound, but it also enables the collection of more samples, leading to a decrease in the CDF error bound as ' increases; note that this observation aligns with Proposition 2.\n\nNotably, we see that a minimum level of exploration probability ' (accepting around 10% of the samples in the exploration range) is needed to improve the CDF bounds over no exploration. Note that this may or may not be feasible for a decision maker depending on the costs of exploration (see also Section 3.4). However, if exploration is feasible, we also see that accepting around 20% of the samples in the exploration range (when the red line is close to the green line) can be sucient to provide bounds nearly as tight as observing all samples in the exploration range.\n\n## 3.4 How To Choose An Exploration Strategy?\n\nWe close this section by discussing potential considerations in the choice of an exploration strategy in light of our findings. Specifically, a decision maker can account for a tradeo between *the costs of exploration* and the improvement in the generalization error bound when choosing its exploration strategy. Recall that the exploration strategy consists of selecting an exploration lowerbound/range LB and an exploration probability\n'. Formally, the decision maker can solve the following optimization problem to choose these parameters:\n\n$$\\operatorname*{max}_{\\epsilon\\in[0,1],L B\\in[0,\\theta]}\\quad\\left(B(\\theta)-B^{\\epsilon}(L B,\\theta,\\epsilon)\\right)-C(L B,\\theta,\\epsilon)\\ ,$$\n\"≠ C(LB, ◊, ') , (1)\nwhere B(◊) and Be(LB, ◊, ') denote the error bounds in Theorems 2 and 3, respectively, and C(LB, ◊, ') is an exploration cost which is non-increasing in (◊ ≠LB) (reducing the exploration range will weakly decrease the costs) and non-decreasing in ' (exploring more samples will weakly increase the cost). As an example, the cost function C(LB, ◊, ') can be given by\n\n$$C(LB,\\theta,\\epsilon)=\\epsilon\\int_{LB}^{\\theta}e^{\\frac{\\theta-x}{\\epsilon}}f^{0}(x)\\mathrm{d}x.\\tag{2}$$\n$$\\left(1\\right)$$\n\nIn words, unqualified (costly) samples at x have a density f 0(x), and when selected (as captured by the '\nmultiplier), they incur a cost e\n◊≠x c , where c > 0 is a constant. Notably, observe that the cost is increasing as the sample x gets further away from the threshold ◊. For instance, in the bank loan example, this could capture the assumption that individuals with lower credit scores default on a larger portion of their loans.\n\nAs noted in Proposition 2, Be(LB, ◊, ') is decreasing in '; coupled with any cost function C(LB, ◊, ') that is (weakly) increasing in ', this means that the decision maker's objective function in equation 1 captures a tradeo between reducing generalization errors and modulating exploration costs.\n\nThe optimization problem in equation 1 can be solved (numerically) by plugging in for the error bounds from Theorems 2 and 3 and an appropriate cost function (e.g., equation 2). For instance, in the case of the numerical example of Fig. 3, under the cost function of equation 2 with c = 5, and fixing LB = 6, the decision maker should select ' = 11.75%.\n\nAnother potential solution for modulating exploration costs is to use multiple exploration subdomains, each characterized by an exploration range [LBi*, LB*i≠1), and with a higher exploration probability 'i assigned to the subdomains closer to the decision boundary (which are less likely to contain high cost samples). For instance, with the choice of b subdomains, the cost function of equation 2 would change to (the lower) cost:\n\n$$C(\\{LB_{i}\\}_{i=1}^{b},\\theta,\\{\\epsilon_{i}\\}_{i=1}^{b})=\\sum_{i=1}^{b}\\epsilon_{i}\\int_{LB_{i}}^{LB_{i-1}}e^{\\frac{\\theta-\\epsilon}{\\epsilon}}f^{0}(x)\\mathrm{d}x.\\tag{3}$$\n\nIt is worth noting that while this approach can reduce the costs of exploration, it will also weaken generalization guarantees when we reassemble the b exploration subdomains' bounds back into an error bound of the full domain (similar to what was observed in Fig. 3 for b = 1). This again highlights a tradeo between improving learning error bounds and restricting the costs of data collection.\n\n## 4 Generalization Error Bounds Under Censored Feedback\n\nIn this section, we use the CDF error bounds from Section 3 to characterize the generalization error of a classification model that has been learned from data collected under censored feedback. Specifically, we will first establish a connection between the generalization error of a classifier (the quality of its learning) and the CDF error bounds on its training dataset (the quality of its data). With this relation in hand, we can then use any of the CDF error bounds from Theorems 1-3 to bound how well algorithms learned on data suering from censored feedback (and without or with exploration) can generalize to future unseen data.\n\nFormally, we consider a 0-1 learning loss function L : Y ◊ Y æ {0, 1}. Denote R(◊) = EXY L(f◊(X), Y ) as the expected risk incurred by an algorithm with a decision threshold ◊. Similarly, we define the empirical risk as Remp(◊). The *generalization error bound* is an upper bound to the error |R(ˆ◊) ≠ Remp(ˆ◊)|, where ˆ◊ is the minimizer of the empirical loss, i.e., ˆ◊ := arg min◊ Remp(◊). In words, the bound provides a (statistical)\nguarantee on the performance R(ˆ◊), when using the learned ˆ◊ on unseen data, relative to the performance Remp(ˆ◊) assessed on the training data. Our objective is to characterize this bound under censored feedback, and to evaluate how utilizing (pure or bounded) exploration can improve the bound.\n\nRecall that the decision maker starts with a training data containing ny IID samples from each label y, drawn from an underlying distribution with CDF Fy(x). Let n = n0 + n1 denote the size of the initial training data. Then, the expected loss of a binary classifier with decision threshold ◊ is given by,\n\n$$R(\\theta)=\\mathbb{E}_{X Y}\\mathcal{L}(f(X),Y)=p_{1}F^{1}(\\theta)+p_{0}(1-F^{0}(\\theta))\\ ,$$\nwhile the empirical loss Remp(◊) is given by,\n$$R_{emp}(\\theta)=\\frac{n_{1}}{n}\\frac{1}{n_{1}}\\sum_{(x_{i},y_{i})}\\mathds{1}\\{x_{i}\\leq\\theta,y_{i}=1\\}+\\frac{n_{0}}{n}\\Big{(}1-\\frac{1}{n_{0}}\\sum_{(x_{i},y_{i})}\\mathds{1}\\{x_{i}\\leq\\theta,y_{i}=0\\}\\Big{)}.$$\n\nSimilarly, if the decision maker can collect an additional ky samples of agents with features above the threshold ◊, the above empirical risk expression can be updated accordingly, by considering the ny + ky samples available from each label y.\n\nWe detail the derivations of these expressions in Appendix I. Using these expressions of the expected and empirical risks, the following theorem provides an upper bound on the generalization error |R(ˆ◊) ≠ Remp(ˆ◊)| as a function of the CDF error bound, where ˆ◊ denotes the minimizer of the empirical loss, i.e., ˆ◊ :=\narg min◊ Remp(◊).\n\nTheorem 4. *Consider a threshold-based classifier* f◊\nˆ(x) : X æ {0, 1}*, determined from a dataset containing* ny initial IID training samples from each label y, with n = n0 + n1, under a 0-1 loss function. Let py denote the proportion of agents from label y*. Subsequently, due to the censored feedback, the algorithm collects* ky additional samples from each label y. Let Fy and Fym denote the CDFs and empirical CDFs, respectively, given m samples from label y agents. Then, with probability at least 1 ≠ 2\", ---R(ˆ◊) ≠ Remp(ˆ◊)\n\n$$\\left|R(\\hat{\\theta})-R_{e m p}(\\hat{\\theta})\\right|\\leq3\\Big{|}p_{0}-\\frac{n_{0}}{n}\\Big{|}+\\sum_{y\\in\\{0,1\\}}\\operatorname*{min}\\Big{(}p_{y},\\frac{n_{y}}{n}\\Big{)}\\operatorname*{sup}_{\\theta}\\Big{|}F^{y}(\\theta)-F^{y}_{n_{y}+k_{y}}(\\theta)\\Big{|}\\.$$\n\nThe proof is given in Appendix H. First, we note that tightening the CDF error bounds leads to tightening the generalization error guarantees. More specifically, using this theorem together with Theorems 1, 2, and 3, we can provide a generalization error guarantee for an algorithm in terms of the number of available data samples in its training data from each label and in dierent parts of the data domain, particularly when future data availability is non-IID due to censored feedback.\n\nFor instance, the DKW inequality can be alternatively expressed as follows: given ny IID samples from a label y, with probability at least 1 ≠ \", the following inequality holds:\n\n$$\\operatorname*{sup}_{z}\\left|F(z)-F_{n_{y}}^{y}(z)\\right|\\leq{\\sqrt{\\frac{\\log{\\frac{2}{\\delta}}}{2n_{y}}}}\\ .$$\n\nUsing this expression in Theorem 4, we conclude that (without censored feedback, or with pure exploration with ' = 1) with probability at least 1 ≠ 2\",\n\n$$\\left|R({\\hat{\\theta}})-R_{e m p}({\\hat{\\theta}})\\right|\\leq3\\Big|p_{0}-{\\frac{n_{0}}{n}}\\Big|+\\sum_{y\\in\\{0,1\\}}\\operatorname*{min}\\Big(p_{y},{\\frac{n_{y}}{n}}\\Big){\\sqrt{\\frac{\\log{\\frac{2}{\\theta}}}{2n_{y}}}}.$$\n\nWe can similarly specialize Theorem 4 to tasks with censored feedback by linking it with Theorems 2 and 3. Given the complexity of the CDF error bounds under censored feedback, while we cannot derive a closedform expression for the bound as done for the DKW inequality, we can compute the bounds numerically, as shown in the next section.\n\n## 5 Numerical Experiments 5.1 Cdf Error Bounds\n\nWe first illustrate our derived bounds (with \" = 0.015) on the empirical CDF. We start with 50 random samples from a Gaussian distribution N(7,1). Next, 200 new samples are drawn from the same distribution, with all samples with features x Ø ◊ = 7 accepted, and samples with features LB = 6 Æ x Æ ◊ accepted\n\n with a probability ' œ {0, 0.5, 1}; higher values of ' the represent less censored feedback (' = 1 means no censored feedback).\n\nFigure 4: CDF error bounds when dierent levels of exploration (') are used to alleviate censored feedback.\n\nAs ' increases: (a) the empirical CDF estimates become more accurate, and (b) our CDF error bounds improve (i.e., more tightly enclose the true CDF).\nFrom Figure 4, we first note that our bounds (the dotted lines) eectively enclose the true distribution. We also note the distinction between empirical CDFs in the disclosed region (x Ø 7) and the censored region\n(x Æ 7): as intuitively expected, empirical CDFs (solid lines) in the disclosed region are \"smoother\" compared to those in the censored region. Furthermore, as ' (exploration) increases, we overcome censored feedback in the exploration region, resulting in more accurate empirical estimates. Additionally, as ' increases, our error bounds improve (i.e., more tightly enclose the true CDF).\n\n## 5.2 Model Generalization Error Bounds: Real-World Data And Adaptively Updated Algorithm\n\nWe now illustrate the ability of our generalization error bounds (derived in Theorem 4) in providing guarantees on the error of the learned models from data aected by censored feedback, using experiments on a real-world dataset. In addition, while our bounds are derived for a fixed model ˆ◊, the model can be updated as new samples are collected. Therefore, in these experiments we also assess the performance of our bounds based on whether we adaptively update the decision threshold ˆ◊ with new samples.\n\nWe conduct these experiments on the real-world *Adult* census dataset (Dua & Gra, 2017). The objective is to predict whether an individual earns more than $50k/year, based on a multi-dimensional feature set. We employ a logistic regression algorithm and 0-1 loss for the classification task, and compare the generalization error across dierent exploration probabilities (' = {0.5, 1}). We start with a 1000 sample training dataset.\n\nA total of 45000 new samples arrive throughout the experiment; in addition to accepting all samples with feature x Ø ˆ◊, the algorithm also accepts some samples that fall below ˆ◊. The decision threshold is updated periodically based on new data (after each 5000 batch of new samples arrives). The decision threshold is retrained using (most recent) training data. We report our experiment results for an average of 5 runs, where the randomness comes from the order of samples arrived and the exploration.\n\nFrom Figure 5, we observe that as the decision threshold ˆ◊ is adaptively updated when more samples are collected, it has even better generalization performance compared to a non-adaptive decision threshold.\n\nThis is expected as a refined decision threshold yields better performance on unseen data. Further, for the generalization error bounds (dotted lines in the right panel), we see that our bounds eectively contain the true generalization errors of the model (for both the fixed model and adaptively updated model cases).\n\nNotably, in the presence of censored feedback, we observe that the generalization error bound with adaptive updating is tighter than the non-adaptive one, pointing to a potential future research direction for further improving our bounds.\n\n\n\n\n\nFigure 5: Generalization error with(out) an adaptively updated model (0) and varying exploration (e).\n\n## 5.3\n\nComparison with existing generalization error bounds\n\n\n\nFigure 6: Existing bounds fail to capture generalization when there is censored feedback.\nWe now compare the performance of our bounds with a number of existing generalization error bounds, and show that by failing to account for censored feedback, prior works fail to correctly capture how well a model learned on data suffering from censored feedback generalizes to unseen data. We consider the following four benchmarks: The 'Hoeffding + Azuma' bounds represent those derived from Hoeffding and Azuma inequalities (Hoeffding, 1994; Azuma, 1967). The 'VC + binomial' bounds are VC generalization bounds (Vapnik & Chervonenkis, 2015; Abu-Mostafa et al., 2012, Thm 2.5) where the shatter coefficient is bounded through the binomial theorem. The 'VC + poly' bounds represent VC generalization bounds (Vapnik &\nChervonenkis, 2015; Devroye et al., 2013, Thm 13.11) applicable to any linear classifier whose empirical error is minimal, where the shatter coefficient is bounded by a polynomial function. Lastly, the 'GC' bounds (Glivenko, 1933; Cantelli, 1933) are derived based on the Glivenko-Cantelli Theorem for a threshold classifier and 0-1 loss.\n\nWe conduct this experiment on synthetic data. We start with 50 initial training samples for each label y E {0,1} randomly drawn from Gaussian distributions N(9,1) and N(10,1), respectively.\n\nThe decision threshold 0 is selected to be the one minimizing the misclassification error on the training data. Then, a total of 50000 new samples arrive throughout the experiment. They will be accepted if the feature x > 0, otherwise, they are rejected. We run the experiments 5 times and report the average results with corresponding error bars. From Figure 6(a), we can clearly see that the 'Hoeding-Azuma' (red), 'VC+binomial' (blue), and\n'GC' (purple) bounds are inadequate for accurately estimating the true generalization error guarantees of the model. For the 'VC+poly' (gray) bound, for the given number of new samples, it provides a very loose bound, even compared with our bounds. However, as the number of arrived samples increases, it will exhibit similar behaviors to the other three benchmarks, in that it will go lower than the true generalization error\n(black line/shades).\n\n## 6 Conclusion And Future Work\n\nWe studied generalization error bounds for classification models learned from non-IID data collected under censored feedback. We presented two generalizations of the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality, which characterizes the gap between empirical and theoretical CDFs given IID data, to problems with *nonIID* data due to censored feedback without exploration (Theorem 2) and with exploration (Theorem 3), and connected these bounds to generalization error guarantees of the learned model (Theorem 4). Our findings establish the extent to which a decision maker should be concerned about censored feedback's impact on the learned model's performance guarantees, and show that a minimum level of exploration is needed to alleviate it.\n\nFor future work, we are interested in strengthening our bounds by allowing the model (◊) to be adaptively updated as new samples are collected; as noted in Section 5, this could help further strengthen our error bounds. Generalization error bounds under a combination of censored feedback and domain adaptation are also worth exploring, wherein the initial training data distribution diers from the target domain distribution.\n\nFinally, we have provided extensions of the DKW inequality, which strengthens the VC inequality when data is real-valued, under censored feedback; providing similar extensions of the VC inequality for *multidimensional data* could be an interesting direction of future work. We discuss some initial findings and potential challenges of this extension below. Bounds for higher dimensional data. When assessing generalization error under censored feedback in higher dimensional data, one approach could be to first reduce the dimensionality, enabling direct application of our findings. For instance, we have performed a mapping of multi-dimensional features to a single-dimensional representation in our experiments on the real-world *Adult* census dataset. However, this reduction may lead to some loss of information, potentially impacting algorithm performance. An alternative would be to follow our approach of identifying IID subspaces in the higher-dimensional data space, apply a multivariate DKW inequality (e.g., (Naaman, 2021)) in these subspaces, and then identify the appropriate error coecients to re-assemble the subdomain bounds and find a CDF error bound for the entire data domain. We provide an analysis for 2D spaces based on this approach in Appendix J. 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The authors demonstrate that existing generalization error bounds fail to account for censored feedback. To overcome this challenge, they extend the Dvoretzky-Kiefer-Wolfowitz (DKW) inequality to non-IID data, providing new CDF error bounds and then using them to generalization error bounds for classifiers trained on such non-IID data. These error bounds depend on the amount of available censored feedback (get tighter as more information about censored feedback is given). Therefore, this paper also considers the effectiveness of pure and bounded exploration techniques in mitigating the effects of censored feedback. Finally, the authors show the various behaviors of the proposed approach (DKW inequality and generalization error bounds with censored feedback) via experiments.\n\nStrengths and Weaknesses: #### **Strengths of the paper:**\n1. The authors generalize the DKW inequality for CDF error bounds for non-IID data due to censored feedback (Theorem 3), which is a function of the availability of censored feedback. The key idea is to divide data into different regions (disclosed and censored regions), where data can be treated as IID and then adapt current DKW inequality with suitable scaling and shifting. \n2. Since the bound depends on the amount of data available from the censored region, a new region from censored feedback is selected, called the exploration region, from where the samples can be collected to further improve the bounds under certain conditions (Theorem 3 and Section 3.3). Findings show that exploration can significantly improve generalization guarantees, helping decision-makers balance data collection costs and learning performance.\n3. Using CDF error bounds with censored data, the generalization error bounds for classification models trained on non-IID data due to censored feedback (Theorem 4). \n4. Finally, the authors have considered different scenarios to show the various behaviors of the proposed approach (DKW inequality and generalization error bounds with censored feedback).\n\n\n#### **Weaknesses of the paper:**\nOverall, I do not find any major weakness in the paper. I have only passed through the proofs in the supplementary material, which seems correct.\n\nRequested Changes: **Requested changes:** \n\n1. Discuss whether there is a way to select exploration probabilities based on exploration cost. For example, when collecting data is expensive, such as giving loans to individuals with low credit scores, they are more likely to default.\n\n2. Include figures to illustrate different regions so readers can understand easily and quickly.\n\n\n**Minor change:** Typo in Line 2 of Section 2: classier -> classifier\n\nBroader Impact Concerns: Since it is a theoretical paper, I do not have any ethical concerns.\n\n==================================================\n\nReview 2:\nSummary: The paper studies generalization error bounds when the data is not i.i.d. but is collected under censored feedback. This is a well-motivated problem, given that settings such as college admissions, lending etc all involve a selection mechanism that implies that only a truncated version of the original distribution will be observed. The key technical tool is an extension of the DKW inequality to non-i.i.d. data, in particular under censored feedback.\n\nStrengths and Weaknesses: I like the basic problem studied in this paper, and the paper is quite well-written. However, I don’t think the current results can be formally correct, even though I do believe there is a version of the results that can be made formally correct.\n\nFirst of all, the problem setting is ambiguous. “The decision maker begins with an initial/historical training dataset containing n IID samples1 {X_1^y , X_2^y , . . . , X_n^y } of label y agents.” For which y? Both y=0 and y=1? So we have 2n points? This is not an i.i.d. sample from the underlying (X,Y) distribution if we are sampling the same number (n) draws from X|Y=0 and X|Y=1. Another reason this is strange is that in Section 4 it says “Recall that the decision maker starts with a training data containing n_y IID samples from each label y…” This contradicts the setup introduced at the beginning. And again, in Theorem 2 it says “Let X_1,X_2,…,X_n be initial/historical IID data samples…” So now it seems that we in fact observe n i.i.d. points.\n\nThe most obvious reason why the results can’t be formally correct is that m, the number of censored training data points, must be a random variable for any reasonable problem setup (which, again, is still ambiguous to me). In other words, the number of data points that fall below theta cannot be fixed, since (under any reasonable setting) every data point should have a positive probability of landing on either side of theta. But Theorem 2 is said to hold “for every m.” That doesn’t make mathematical sense.\n\nOverall, I like the problem studied in this paper, and I do believe that there is a formally justified version of these results. I just believe that the current statements are not mathematically mature. I would encourage the authors to revise the problem setting, statements, and proofs in order to ensure correctness, and I believe that, if correct, these results would make a nice contribution to TMLR.\n\n\nMinor:\n- Please attribute the DKW theorem appropriately. This is a very classical result (from 1956) and the reference Devroye et al. (2013) is not the original result.\n\nRequested Changes: Please see above. The mathematical setting and statements need to be polished.\n\nBroader Impact Concerns: No concerns.\n\n==================================================\n\nReview 3:\nSummary: From the perspective of machine learning tasks, the problem of interest in this paper is binary classification under a data distribution which has \"censored feedback,\" which here is formulated by the data coming in two stages, where the data-based decision in stage 1 impacts which data is actually available in stage 2. From a technical standpoint, when the data in stage 2 passes through a filter depending on all stage 1 data, we naturally do not have IID data any longer, and the authors are interested in providing formal guarantees (in terms of bounds on the true probability of error at test time) for classifiers in such a setting (or so I have interpreted the core motivations).\n\nTowards this goal, comparing sample/population CDFs and seeking uniform bounds on their absolute difference is one standard approach, but traditional concentration inequalities may fail without independence; the authors show how setting a censor threshold $\\\\theta$ (and the resulting censored/disclosed set sizes) impacts bounds on the (uniform) empirical CDF estimation error (a generalized DKW inequality), when the empirical CDF is derived from all stage 1 data and the $\\\\theta$-filtered stage 2 data. Building upon this, they analyze if/when \"exploration\" strategies that consider data below a given threshold can be sufficient to improve error bounds, and consider bounds on the expected 0-1 test loss of an stage 1 driven ERM style classifier (their Thm 4), based on the machinery afforded by the aforementioned generalized CDF error bounds.\n\nStrengths and Weaknesses: __Strengths:__\n\nThe introduction and literature overview is well-written as easy to follow. The basic motivations are mostly quite clear, and the overall structure of the paper is natural.\n\n\n__Weaknesses:__\n\nI personally found certain parts of the technical exposition very hard to parse. I feel like the formulation is rather lacking, and to most readers of the main paper it may not even be clear what \"non-IID\" actually entails. Furthermore, I found the performance guarantee for classification under censorship (Theorem 4) quite puzzling; I feel like the statements being made do not really align with the main claims of the authors.\n\nI will discuss this in more detail in the next section.\n\nRequested Changes: __Nature of data is unclear__\n\nIn what I call \"stage 1\" where we get $n$ IID data points, what exactly is being observed? The dataset notation with superscript $y$ makes things confusing. To start, two very basic points are unclear.\n\n- How many data points do we actually have? On one hand, the authors say we have $n$ points in stage 1, and yet they also say that we get $\\\\{X\\_{1}^{y},\\\\ldots,X\\_{n}^{y}\\\\}$ for *\"label $y$ agents\"*, and since $y \\\\in \\\\{0,1\\\\}$, this suggests both $\\\\{X\\_{1}^{0},\\\\ldots,X\\_{n}^{0}\\\\}$ and $\\\\{X\\_{1}^{1},\\\\ldots,X\\_{n}^{1}\\\\}$ are present. This notation is confusing.\n- Does the learning algorithm actually get *labels* in stage 1? One would think so given the definition of $R\\_{emp}$ way down in section 4, but this is not apparent in section 2. Anyways, if this is the case, why not just say that the decision maker begins with an $n$-point IID dataset of $\\\\{(X\\_{1},Y\\_{1}),\\\\ldots,(X\\_{n},Y\\_{n})\\\\}$, as is done in basically every binary classification paper ever written?\n\nNext, the formulation of the stage 2 data is even less clear, since nothing is made explicit. Here is how I understand it (after seriously reading between the lines):\n\n- In stage 1, we have $n$ data points that I will call $\\\\mathcal{D}\\_{1} = \\\\{(X\\_{1},Y\\_{1}),\\\\ldots,(X\\_{n},Y\\_{n})\\\\}$, based on which we can determine a threshold $\\\\widehat{\\\\theta}\\_{n}$.\n- In stage 2...\n - First we get some more data, let's denote them by $\\\\{(X\\_{1}^{\\\\prime},Y\\_{1}^{\\\\prime}), (X\\_{2}^{\\\\prime},Y\\_{2}^{\\\\prime}), \\\\ldots\\\\}$ since the actual number is not specified; maybe this isn't important.\n - Then we apply the $\\\\widehat{\\\\theta}\\_{n}$-threshold filter to get the censored feedback dataset, which I denote by $\\\\mathcal{D}\\_{2} = \\\\{(X\\_{i}^{\\\\prime},Y\\_{i}^{\\\\prime}) : X\\_{i}^{\\\\prime} \\\\geq \\\\widehat{\\\\theta}\\_{n}\\\\}$. My understanding is that the critical quantity $k$ used by the authors is the number of points in this filtered dataset, namely $k = \\\\vert \\\\mathcal{D} \\\\vert$. Similarly, I take $m$ to be the number of points in $\\\\{(X\\_{i},Y\\_{i}) : X\\_{i} < \\\\widehat{\\\\theta}\\_{n}\\\\}$.\n- Taking these two stages together, I understand the full \"censored feedback\" dataset to be all the random pairs in $\\\\mathcal{D}\\_{1}$ and $\\\\mathcal{D}\\_{2}$.\n\nI certainly hope that this understanding is correct; it is the best I could do with what is actually written in the paper. I really think the formulation needs some work.\n\n__Nature of non-IID data is unclear__\n\nNow, if the data is as I described above, with $\\\\mathcal{D}\\_{2}$ determined based on a $\\\\mathcal{D}\\_{1}$-dependent threshold $\\\\widehat{\\\\theta}\\_{n}$, then I completely get how independence breaks down. However, in section 3.1, $\\\\theta$ is fixed throughout. As I result, I would expect that data in $\\\\mathcal{D}\\_{2}$ (based on fixed $\\\\theta$) would have a different distribution from that of the data in $\\\\mathcal{D}\\_{1}$, but isn't the data in $\\\\mathcal{D}\\_{1}$ and $\\\\mathcal{D}\\_{2}$ still all *independent*? When $\\\\theta$ is fixed, I can see no reason why the stage 2 filtered data should be considered dependent on the stage 1 data. Perhaps the authors do not say the data is dependent anywhere, but they bundle it all up into the term *\"non-IID\"* which is way too broad. If we replace $\\\\theta$ in 3.1 (and beyond) with a $\\\\mathcal{D}\\_{1}$-dependent $\\\\widehat{\\\\theta}\\_{n}$, does everything still go through? Does *anything*?\n\n\n__CDF bounds are mysterious__\n\nIs everything in section 3 supposed to be *conditional* on the stage 1 data $\\\\mathcal{D}\\_{1}$? I cannot find such a statement anywhere, but the authors in Theorem 2 (and elsewhere) treat $k$ and $m$ like they are *constants*, but if $\\\\mathcal{D}\\_{1}$ is not conditioned upon, $k$ and $m$ are *random* (even for fixed $\\\\theta$), depending on the draw of $\\\\mathcal{D}\\_{1}$. As such, I find it strange when the upper bound in Theorem 2 is said to hold *\"for every\"* choice of $m$ and $k$. Presumably the upper bound holds with probability 1 over the draw of the random data? Again, unclear points.\n\n\n__ERM classifier bounds are mysterious__\n\nI apologize if I have misunderstood something, I have no idea what the point of Theorem 4 is. Based on what is written in the paper, $\\\\widehat{\\\\theta}$ in Theorem 4 is defined completely by $R\\_{emp}$, which in turn is completely determined by the stage 1 data $\\\\mathcal{D}\\_{1}$ only. That's fine, but then why is the stage 2 data $\\\\mathcal{D}\\_{2}$ used to construct an error bound in Thm 4? At face value, it seems like the stage 2 data is totally irrelevant to both the estimator $R\\_{emp}$ and its minimizer $\\\\widehat{\\\\theta}$; am I missing something? Furthermore, since everything empirical on the left-hand side of the Thm 4 inequality is based on $\\\\mathcal{D}\\_{1}$, standard IID ERM analysis can be done, i.e., VC analysis for half-line classifiers. If there is supposed to be an advantage to the proposed approach as compared with typical IID ERM analysis, this needs to be made much more clear.\n\nAnother point is that the authors say this inequality (Thm 4) holds with probability at least $1-2\\\\delta$, but $\\\\delta$ doesn't appear anywhere in the assumptions nor the upper bound.\n\n\n__Story is not clear__\n\nIn the motivating exposition of sections 1 and 2, the authors seem to suggest that they want to analyze learning algorithms (here, classification procedures) that depend on *both* $\\\\mathcal{D}\\_{1}$ and $\\\\mathcal{D}\\_{2}$. This is precisely what the \"bank loan\" example suggests, if the bank trains a classifer once to set the initial threshold, then trains another classifer based on the original data *and* censored feedback. This is perfectly natural, but this is *not* what is being studied in Theorem 4, since the \"classifier\" of interest ($\\\\widehat{\\\\theta}$ in the authors' notation) is based completely on $\\\\mathcal{D}\\_{1}$. I feel like the work that is done is not in line with the goals/claims made. In particular, the claim in section 1 that *\"We derive a generalization error bound for a classification model learned in the presence of censored feedback...\"* does not appear valid to me.\n\n\nI could continue with other related points, but I think the message is clear. TMLR requires that the exposition be clear, and that the main claims be solid. At present, I think this paper needs quite a bit of work to satisfy this.\n\nBroader Impact Concerns: Not applicable.\n\n==================================================\n\nMetareview:\nRecommendation: Reject\n\nComment: As the correctness concerns of the reviewers still persist after the rebuttal, we decide to recommend rejection this time. We encourage the authors to fix the proofs in the next interation.\n\n==================================================\n\n"}}},{"rowIdx":99,"cells":{"paper":{"kind":"string","value":"# A Unified View Of Differentially Private Deep Generative Modeling\n\nDingfan Chen∗dingfan.chen@cispa.de CISPA Helmholtz Center for Information Security Raouf Kerkouche *raouf.kerkouche@cispa.de* CISPA Helmholtz Center for Information Security Mario Fritz fritz@cispa.de CISPA Helmholtz Center for Information Security Reviewed on OpenReview: *https: // openreview. net/ forum? id= YgmBD2c9qX*\n\n## Abstract\n\nThe availability of rich and vast data sources has greatly advanced machine learning applications in various domains. However, data with privacy concerns comes with stringent regulations that frequently prohibit data access and data sharing. Overcoming these obstacles in compliance with privacy considerations is key for technological progress in many real-world application scenarios that involve sensitive data. Differentially private (DP) data publishing provides a compelling solution, where only a sanitized form of the data is publicly released, enabling privacy-preserving downstream analysis and reproducible research in sensitive domains. In recent years, various approaches have been proposed for achieving privacy-preserving high-dimensional data generation by private training on top of deep neural networks. In this paper, we present a novel unified view that systematizes these approaches. Our view provides a joint design space for systematically deriving methods that cater to different use cases. We then discuss the strengths, limitations, and inherent correlations between different approaches, aiming to shed light on crucial aspects and inspire future research. We conclude by presenting potential paths forward for the field of DP data generation, with the aim of steering the community toward making the next important steps in advancing privacy-preserving learning.\n\n## 1 Introduction\n\nData sharing is crucial for the growth of machine learning applications across various domains. However, in many application scenarios, data sharing is prohibited due to the private nature of data (e.g., individual data from mobile devices, medical treatments, and banking records) and associated stringent regulations, such as the General Data Protection Regulation (GDPR) and the American Data Privacy Protection Act (ADPPA),\nwhich largely hinders technological progress in sensitive areas. Fortunately, differentially private (DP) data publishing (Dwork, 2008; Dwork et al., 2009; Fung et al., 2010) provides a compelling solution, where only a sanitized form of the data, with rigorous privacy guarantees, is made publicly available. Such sanitized synthetic data can be leveraged as a surrogate for real data, enabling downstream statistical analysis using established analytic tools, and can be shared openly with the research community, promoting reproducible research and technological advancement in sensitive domains.\n\nTraditionally, the sanitization algorithms are designed for capturing low-dimensional statistical characteristics and target at specific downstream tasks (e.g., answering linear queries (Roth & Roughgarden, 2010; Hardt &\nRothblum, 2010; Blum et al., 2013; Vietri et al., 2020)), which are hardly generalizable to unanticipated tasks involving high-dimensional data with complex distributions. On the other hand, the latest research, inspired by the recent successes of deep generative models in learning high-dimensional representations, applies deep\n∗Correspondence to: Dingfan Chen (dingfan.chen@cispa.de)\ngenerative models as the foundation of the generation algorithm. This line of approaches, as demonstrated in recent studies (Cao et al., 2021; Chen et al., 2020a; Xie et al., 2018; Yoon et al., 2019; Beaulieu-Jones et al.,\n2017; Ghalebikesabi et al., 2023), have shown promising results in sanitizing high-dimensional samples for general purposes.\n\nTowards designing models that are better compatible with the privacy target, recent research typically customizes the training objective for privacy-centric scenarios (Cao et al., 2021; Harder et al., 2021; Chen et al., 2020a; Long et al., 2021), all building on top of a foundational generic generator framework. However, research is fragmented as contributions have been made in different domains, different modeling paradigms, different metric and discriminator choices, and different data modalities. So far, a unified view of private generative models is notably missing in the literature, despite its potential to consolidate the design space for systematic exploration of innovative architectures and leveraging strengths across diverse modeling frameworks.\n\nIn this paper, we pioneer in providing a comprehensive framework and a unified perspective on existing approaches for differentially private deep generative modeling. Our innovative framework, complemented by an insightful taxonomy, effectively encapsulates approaches from existing literature, categorizing them according to the intrinsic differences in their underlying privacy barriers. We thoroughly assess each category's characteristics, emphasizing crucial points relevant for privacy analysis, and discuss their inherent strengths and weaknesses, with the aim of laying a foundation that supports seamless transition into potential future research.\n\nMoreover, we present a thorough introduction to the key concepts of DP and generative modeling. We highlight the key considerations that should be accounted for when developing DP generative models to ensure results comparable, error-free results. Furthermore, we introduce a taxonomy of existing representative types of deep generative models, classifying them based on the distinctive privacy challenges present during DP training. This introduction aims to equip researchers and practitioners with a systematic approach for the design and implementation of future privacy-preserving data generation techniques.\n\nLastly, we discuss open issues and potential future directions in the broader field of developing DP generation methods. Our objective is not limited to reviewing existing techniques, but also aims to equip readers with a systematic perspective for devising new approaches or refining existing ones. This work is thoughtfully written to serve diverse audiences, with an effort of providing practitioners with a comprehensive overview of the recent advancements, while aiding experts in reassessing existing strategies and designing innovative solutions for privacy-preserving generative modeling.\n\n## 2 Preliminaries Of Differential Privacy\n\nSetting. In this paper, we focus on the standard *central* model of DP, which is commonly agreed upon by all the approaches referenced herein. In this model, a trusted party or server is responsible for managing all data points, executing DP algorithms, and producing sanitized data that conforms to privacy constraints. This sanitized data, generated from the implemented DP algorithms, can be later shared with untrusted parties or released to the public while ensuring strict privacy guarantees. It is noteworthy that although approaches based on local DP may seem to generate a form of synthetic data—where users typically modify their own data due to distrust in the central server and a desire to conceal private information—these methods are fundamentally distinct from the ones explored in this work due to differing threat models and the resulting privacy implications.\n\nDefinition 2.1 ((*ε, δ*)-DP (Dwork, 2008)). A randomized mechanism M with range R is (*ε, δ*)-DP, if Pr[M(D) ∈ O] ≤ e ε· Pr[M(D\n′) ∈ O] + δ holds for any subset of outputs *O ⊆ R* and for any adjacent datasets D and D′, where D and D′ differ from each other with only one training example. ε is the upper bound of privacy loss, and δ is the probability of breaching DP constraints. Smaller values of both ε and δ translate to stronger DP guarantees and better privacy protection. Typically, M refers to the training algorithm of a generative model. DP ensures that inferring the presence of an individual in the private dataset—by observing the trained generative models M(D)—is challenging, with D being the original private dataset. This same level of guarantee also holds when the attacker observes the samples generated by the trained generative models (i.e., the sanitized dataset)\ndue to the post-processing theorem (Theorem 2.1). Privacy notion. There are two widely used definitions for adjacent datasets in existing works of DP data generation, which result in different DP notions: the \"replace-one\" and the \"add-or-remove one\" notions:\n- **Replace-one**: adjacent datasets are formed by *replacing* one data sample, i.e., D′ ∪ {x\n′} = *D ∪ {*x} for some x and x\n′. This is sometimes referred to *bounded-DP* in literature.\n\n- **Add-or-remove-one**: adjacent datasets are constructed by adding or *removing* one data sample, i.e.,\nD′ = *D ∪ {*x} for some x (or vice versa).\n\nIt is crucial to understand that different notions of DP may not provide equivalent privacy guarantees even under identical (*ε, δ*) values, potentially leading to slight differences in comparisons when algorithms are developed under varying privacy notions, a sentiment also noted in Ponomareva et al. (2023). Specifically, the \"replacement\" operation in the bounded-DP notion can be understood as executing two edits: removing one data point x and adding another x\n′. This suggests that the *replace-one* notion may be nested within the *add-or-remove-one* notion, and a naive transformation would result in a (2ε, δ)-DP algorithm under the replace-one notion from an algorithm that was (ε, δ)-DP under the *add-or-remove-one* notion. To minimize potential confusion and promote fair comparisons, we emphasize that future researchers should clearly specify the chosen notion in their work. Moreover, we encourage future research to include a privacy analysis for both notions, if technically feasible.\n\nPrivacy-preserving data generation is building on top of the closedness of DP under post-processing: if a generative model is trained under a (*ε, δ*)-DP mechanism, releasing a sanitized dataset generated by the model (for conducting downstream analysis tasks) will also be privacy-preserving, with the privacy cost bounded by ε (and δ).\n\nTheorem 2.1 (Post-processing (Dwork et al., 2014)). If M satisfies (*ε, δ*)-DP, F ◦ M will satisfy (*ε, δ*)-DP\nfor any data-independent function F with ◦ denoting the composition operator.\n\nWhile (*ε, δ*)-DP provides an intuitive understanding of the mechanism's overall privacy guarantee, dealing with composition is more convenient under the notion of Rényi Differential Privacy (RDP). Existing approaches typically use RDP to aggregate privacy costs across a series of mechanisms (such as multiple DP gradient descent steps during generative model training) and then convert to the (*ε, δ*)-DP notion at the end (See Appendix D). The formal definitions and the corresponding theorems are listed below.\n\nDefinition 2.2 (Rényi Differential Privacy (RDP) (Mironov, 2017)). A randomized mechanism M is\n(*α, ρ*)-RDP with order α, if\n\n$$D_{\\alpha}({\\mathcal{M}}({\\mathcal{D}})\\|{\\mathcal{M}}({\\mathcal{D}}^{\\prime}))={\\frac{1}{\\alpha-1}}\\log\\mathbb{E}_{t\\sim{\\mathcal{M}}({\\mathcal{D}})}\\left[\\left({\\frac{\\operatorname*{Pr}[{\\mathcal{M}}({\\mathcal{D}})=t]}{\\operatorname*{Pr}[{\\mathcal{M}}({\\mathcal{D}}^{\\prime})=t]}}\\right)^{\\alpha}\\right]\\leq\\rho$$\n\nholds for any adjacent datasets D and D′, where Dα(P∥Q) = 1 α−1 log Et∼Q[(P(t)/Q(t))α] denotes the Rényi divergence.\n\nTheorem 2.2 (Composition (Mironov, 2017)). For a sequence of mechanisms M1*, ...,*Mk s.t. Miis\n(*α, ρ*i)-RDP ∀i, the composition M1 ◦ ... ◦ Mk is (α,Pi ρi)-RDP.\n\nTheorem 2.3 (From RDP to (*ε, δ*)-DP (Balle et al., 2020)). If a randomized mechanism M is (*α, ρ*)-RDP,\nthen M is also ρ + log((α − 1)/α) − (log δ + log α)/(α − 1), δ-DP for any 0 *< δ <* 1.\n\nIn literature, achieving DP typically involves adding calibrated random noise, with scale proportional to the\nsensitivity value (Definition 2.3), to the private dataset's associated quantity to conceal individual influence.\nA notable instance of this practice can be formularized as the Gaussian Mechanism, as defined below.\nDefinition 2.3 (Sensitivity). The (global) ℓp-sensitivity for a function f : X → R\ndthat outputs d-dimensional\nvectors is defined as:\n$$\\Delta_{f}^{p}=\\operatorname*{max}_{{\\mathcal{D}},{\\mathcal{D}}^{\\prime}}\\|f({\\mathcal{D}})-f({\\mathcal{D}}^{\\prime})\\|_{p}$$\n′)∥p (1)\nover all adjacent datasets D and D′. The sensitivity characterizes the maximum influence (measured by ℓp norm) of one individual datapoint on the function's output. When dealing with matrix and tensor outputs, the ℓp norm is computed over the vectors that result from flattening the matrices and tensors into vectors.\n\nDefinition 2.4 (Gaussian Mechanism (Dwork et al., 2014)). Let f : X → R\nd be an arbitrary d-dimensional function with ℓ2-sensitivity ∆2 f\n. The Gaussian Mechanism Mσ, parameterized by σ, adds noise into the output, i.e.,\nMσ(x) = f(x) + N (0, σ2I). (2)\n\n$$(2)$$\n$\\blacksquare$\nMσ is (*ε, δ*)-DP for σ ≥p2 ln (1.25/δ)∆2\n\n$$\\begin{array}{c}{{{\\mathcal{M}}_{\\sigma}(\\mathbf{x})=f(\\mathbf{x})+\\mathcal{N}(0,\\sigma^{2}\\mathbf{I}).}}\\\\ {{{}}}\\\\ {{\\overline{{{\\mathbf{I}}}})\\Delta_{f}^{2}/\\varepsilon\\mathrm{~and~}(\\alpha,\\frac{\\alpha(\\Delta_{f}^{2})^{2}}{2\\sigma^{2}})\\mathrm{-RDP}.}}\\end{array}$$\n\n## 2.1 Training Deep Learning Models With Dp\n\nAdditionally, we present the most prominent frameworks for training deep learning models with DP guarantees:\nDifferentially Private Stochastic Gradient Descent (DP-SGD) in Section 2.1.1 and Private Aggregation of Teacher Ensembles (PATE) in Section 2.1.2.\n\n## 2.1.1 Differenetially Private Stochastic Gradient Descent (Dp-Sgd)\n\nDP-SGD (Abadi et al., 2016) is an adaptation of the standard SGD algorithm that injects calibrated random Gaussian noise into the gradients during the optimization process, which ensures DP due to the Gaussian mechanism. The algorithm consists of the following steps:\n1. Compute the per-example gradients for a mini-batch of training examples.\n\n2. Clip the gradients to bound their ℓ2-norm (i.e., ℓ2-sensitivity) to ensure that the influence of any individual training example is limited.\n\n3. Add Gaussian noise to the aggregated clipped gradients to introduce the required randomness for DP\nguarantees.\n\n4. Update the model parameters using the noisy gradients.\n\nThe privacy guarantees provided by DP-SGD are determined by the choice of noise multiplier (which defines the standard deviation of the Gaussian noise by multiplying it with the sensitivity), the mini-batch sampling ratio, and the total number of optimization steps. The overall privacy guarantee can be calculated using the composition rule, which accounts for the cumulative privacy loss over multiple iterations of the algorithm. By default, DP-SGD adopts the *add-or-remove-one* notion, leading to a sensitivity value equal to the gradient clipping bound (see Appendix C).\n\n## 2.1.2 Private Aggregation Of Teacher Ensembles (Pate)\n\nThe PATE framework (Papernot et al., 2017; 2018) consists of two main components: an ensemble of teacher models and a student model. The training process begins with the partitioning of sensitive data into multiple disjoint subsets. Each subset is then used to train a teacher model independently (and non-privately), limiting the effect of each individual training sample to influence only one teacher model. To train a DP student model, a public dataset with similar characteristics to the sensitive data is used. During the training process, the student model queries the ensemble of teacher models for predictions on the public dataset. The teacher models' predictions are then aggregated using a DP voting mechanism, which adds noise to the aggregated votes to ensure privacy. The student model subsequently learns from the noisy aggregated predictions, leveraging the collective knowledge of the teacher models while preserving the privacy of the original training data.\n\nThe sensitivity of PATE is measured as the maximum change in label counts for teacher models' predictions between neighboring datasets. Given m teacher models, c label classes, the counts for class j is defined by the number of teachers that assign class j to a query input x¯, i.e., nj (x¯) = |i : i ∈ [m], fi(x¯) = j| for j ∈ [c],\nwhere fi denotes the i-th teacher model. Changing a single data point (whether by replacing, adding, or removing) will at most affect one data partition and, consequently, the prediction for one teacher trained on the altered partition, increasing the counts by 1 for one class and decreasing the counts by 1 for another class.\n\nThis results in a global sensitivity equal to ∆2\n(n1*,...,n*c) =\n√2 for both the *replace-one* and *add-or-remove-one* notion (see Appendix C). To reduce privacy consumption, PATE is associated with a data-dependent privacy accountant method to exploit the fact that when teachers have a large agreement, the privacy cost is usually much smaller than the data-independent bound would suggest. Moreover, Papernot et al. (2018) suggest private threshold checking for queries to only use teacher predictions with high consensus for training the student model. Notably, to obtain comparable results to approaches with data-independent privacy costs, extra sanitization via smooth sensitivity analysis is required.\n\n## 2.2 Important Notes For Deploying Dp Models\n\nThe development of DP models necessitate a thorough examination to ensure their correctness for providing a fair comparison of research progress and maintaining public trust in DP methodologies. We present below a series of critical questions that serve as fundamental sanity checks when developing DP models. This enables researchers to rapidly identify and rule out approaches that are incompatible with DP, thereby optimizing their research efforts towards innovation in this domain.\n\n- **What will be released to the public and accessible to potential adversaries?** The most critical question is to determine which components (e.g., model modules, data statistics, intermediate results, etc.)\nwill be made public and, as a result, could be accessible to potential adversaries. This corresponds to the assumed *threat model* and establishes the essential concept of a *privacy barrier*, which separates components accessible to potential attackers from those that are not.\n\nAll components within the attacker-accessible domain must be provided with DP guarantees. One common oversight is neglecting certain data-related intermediate statistics utilized during the model's training phase. These statistics might constitute only a minor aspect of the entire process, or their existence might be implicit, given that they are incorporated into other quantities. Nevertheless, failing to implement DP sanitization for these aspects can undermine the intended DP protection for the outcomes, e.g., the trained model may no longer adhere to DP standards.\n\nFor instance, when pre-processing is required for the usage of a DP model, an additional privacy budget should be allocated for exposing related statistics such as the dataset's mean and standard deviation (Tramer\n& Boneh, 2021). From a research standpoint, innovations may involve carefully designing DP mechanisms that apply DP constraints only to components accessible by attackers, while other components can be trained or computed non-privately to maintain high utility. A concrete example includes training a discriminator non-privately and withholding it by the model owner in deploying DP generative adversarial networks (see Section 4.3) while only privatizing the generator's training and releasing it to the public with a dedicated DP mechanism.\n\n- **What is the adopted privacy notion and granularity?** While DP asserts that an algorithm's output remains largely unchanged when a single database entry is modified, the definition of a \"single entry\" can vary considerably (reflecting the concept of *granularity*), and the way to modify the single entry can also be different (embodying the *privacy notion*). Thus, the claims of DP necessitate an unambiguous declaration of the sense and level at which privacy is being promised. As discussed in the previous section, the distinction in privacy notion is universally crucial in the design of DP mechanisms. On the other hand, the granularity becomes particularly relevant when handling data modalities that exhibit relatively less structural representations, such as graphs and text. For instance, training DP (generative) language models that provide guarantees at different levels (tokens, sentences, or documents) will lead to substantial differences in the complexity and the application scenarios.\n\n- **What constitutes the sensitivity analysis?** Sensitivity analysis demands rigorous attention, focusing on two primary aspects. The first consideration calls for a clear statement of the *sensitivity type* in use, e.g., global, local, and smooth sensitivity. Notably, techniques predicated on local and smooth sensitivity are generally not directly comparable to those depending on the global sensitivity. Second, determining the sensitivity bound during the training of a generative model that consists of more than one trainable module may be challenging, as discussed in Section 4.4, which necessitates a meticulous analysis to ensure the correctness of the privacy cost computation.\n\n\n\nFigure 1: Overview of training pipeline of generative models. (**Blue arrow**: forward pass; **Red arrow**:\nbackward pass; Dashed arrows indicate optional processes that may not be present in all generative models.)\n\n## 3 Preliminaries Of Generative Models\n\nIn this section, we present a comprehensive overview of representative generative models, with the aim to develop a clear understanding of the essential operations required to achieve DP across different types of generative models, as well as to demonstrate the fundamental differences in their compatibility with private training.\n\n## 3.1 Overview & Taxonomy\n\nGiven real data samples x from a dataset of interest, the goal of a generative model is to learn and capture the characteristics of its true underlying distribution p(x) and subsequently allows the model to generate new samples from the learned distribution. At a high-level of abstraction, the training pipeline of generative models can be depicted as the diagram in Figure 1. The \"Measurement\" block in the diagram summarizes the general process of comparing the synthetic and real data distributions using a \"critic\", which yields a loss term L that quantifies the similarity between the two. This loss term then acts as the training objective for the generator, with the update signal computed and then backpropagated to adjust the generator's parameters and improve its ability to generate realistic samples. Furthermore, the diagram outlines two optional processes (indicated by dashed arrows), that are involved in some generative models but not all. The first optional process involves guiding the training of the generator by feeding (quantities derived from) real data as inputs, which enables the explicit maximum likelihood computation and categorizes the models into two types: *implicit density* and *explicit density*. The second optional process involves updating the critic to better capture the underlying structure of the data and more accurately reflect the similarity between the distributions. This distinction highlights the usage of either static (data-independent) or *learnable (data-dependent)* features for the critic function within implicit density models.\n\nWe present a taxonomy of existing representative types of generative models whose private training has been realized in literature in Figure 2. We examine the following tiers in the taxonomy trees that exert significant influence on the application scenarios and the design of corresponding private training algorithms:\n- *Explicit* vs. *Implicit* Density Models\n- *Learnable* vs. *Static* Critics\n- *Distribution-wise* vs. *Point-wise* Optimization\n- *Tractable* vs. *Approximate* Density Explicit vs. *Implicit* **Density Models.** Existing generative models can be divided into two main categories: *explicit density models* define an explicit density function pmodel(x; θ), while implicit density models learn a mapping that generates samples by transforming an easy-to-sample random variable, without explicitly defining a density function.\n\nThese distinctions in modeling design result in different paradigms during the training phase, particularly in how real data samples are used (or accessed) in the process. *Explicit* density models typically use real data samples as inputs to the generator and also for measurement (as demonstrated in Figure 1), thereby enabling the tractable computation or approximation of the data likelihood objective. In contrast, *implicit* density models necessitate real data samples solely for the purpose of distribution comparison measurements.\n\n\n\nFigure 2: Overview of different deep generative models.\n\nff\nThis distinction demarcates potential privacy barriers for these two types of models during DP model training. In the context of implicit models, it is sufficient to privatize the single access point to the real data\n(Section 4.1-4.3). However, when dealing with training private *explicit* density models, it becomes essential to apply DP mechanisms that take both access points into account.\n\nLearnable vs. *Static* **Critics.** The training of generative models necessitates a \"critic\" to assess the distance between the real and generated distributions, which then builds up the training objectives for optimizing the generator. Specifically for *implicit* density models, the use of different types of critics could potentially influence the placement of privacy barrier when training DP models (Section 4.1- 4.2).\n\nWithin this framework, the critics may exist in two primary forms, namely *learnable* and *static* (dataindependent) variants. The distinction between the two lies in whether the critic itself is a parameterized function that undergoes updates during the training of generative models (*learnable*), or a data-independent function that remains static during the training process (*static*).\n\nWe do not further differentiate for explicit density models as they typically employ simple, data-independent critic such as ℓ1 and ℓ2 losses. Meanwhile, in contrast to implicit models, varying the critics in explicit models typically does not alter the privacy barrier in DP training. This is due to the constraint imposed by multiple access to real data in training of explicit models, which restricts the flexibility in positioning the privacy barriers.\n\nDistribution-wise vs. *Point-wise* **Optimization.** Generative models are designed to be stochastic and capable of producing a distribution of data. This is achieved by supplying the generator with random inputs\n(i.e., latent variables), stochastically drawn from a simple distribution, such as the standard Gaussian. The optimization process generally proceeds through mini-batches, essentially serving as point-wise approximations.\n\nThrough substantial number of update steps that involve various random latent variable inputs, the model is trained to generalize over new random variables during the generation phase, enabling a smooth transition from a *point-wise* approximation to the *distribution-wise* objective.\n\nHowever, certain contexts may not necessitate the stochasticity nature in these models. Instead, there might be an intentional focus on generating a small set of representative samples, a notion that resonates with the \"coreset\" concept. This could involve optimizing the model over a limited, fixed set of random inputs rather than the entire domain. We label this as *point-wise optimization* to distinguish it from the default distribution-wise optimization used in training conventional generative models.\n\nRecent studies have revealed intriguing advantages of merging insights from both these strategies, particularly in the realm of private learning. For instance, the point-wise optimization method exhibits remarkable compatibility with private learning primarily arises from the fact that *point-wise* optimization is generally less challenging in comparison to the *distribution-wise* training that requires generalization, which generally improves model convergence, and consequently enhances privacy. However, this point-wise approach has its limitations. Unlike distribution-wise training, it does not inherently support generalization over new latent\n\n\n\nff**usion** \n\nFigure 3: Diagram illustrating training process in generative models. **Blue arrow**: forward pass; Red arrow: backward pass.\n\nff\ncode inputs. This may restrict the stochastic sampling of new synthetic samples during inference. As a result, there is a trade-off between the flexibility of use in downstream applications and improved privacy guarantees.\n\nWe do not expressly differentiate between potential optimization strategies for explicit density models within our taxonomy in Figure 2, as such distinction is not obvious in the context of explicit density models. In these models, the latent space is typically formulated through a transformation of the distribution within the data space. This transformation process in turn complicates the control of stochasticity throughout the training phase and diminishes the applicability of point-wise optimization.\n\nTractable vs. *Approximate* **Density.** For models defining explicit density, a key distinguishing factor that shows practical relevance pertains to whether they allow exact likelihood computations. These models can broadly be categorized into two types: *tractable* density and *approximate* density models. The classification primarily stems from the model structural designs, which either enable tractable density inference or fall within the realm of approximate density.\n\nExisting studies have demonstrated encouraging results when conducting DP training on both types of models.\n\nIntriguingly, the DP training mechanisms appear to exhibit minor distinctions when applied to these two different categories. On an optimistic note, such results implies that it might be feasible to attain tractable likelihood computations with a DP guarantee without considerable effort. However, it remains unclear as to whether the difference in model designs will systematically influence their compatibility with DP training.\n\n## 3.2 Representative Models\n\nWe provide an illustration of the operational flow of representative generative models in Figure 3. As demonstrated, existing representative generative models can be effectively encapsulated within our unified framework shown in Figure 1. We proceed to briefly discuss the key characteristics of each type of generative models and their relation to potential implementations for DP training in this subsection.\n\n## 3.2.1 Implicit Density Models\n\nAs a canonical example of implicit density model, *Generative Adversarial Network (GAN)* (Goodfellow et al., 2014) employs a generator, Gθ (parametrized by θ), to learn the data distribution with the aid of a discriminator Dϕ (parametrized by ϕ) trained jointly in an adversarial manner, obviating the need for explicit density definition. The generator's functionality is enabled by inputting random latent variables, z, drawn from simple distributions such as a standard Gaussian, and mapping these random inputs to the data space. Concurrently, the discriminator is provided with both synthetic and real samples and its training objective is to differentiate between the two. Throughout the training process, the generator and the discriminator compete and evolve, enabling the generator to create realistic samples that can deceive the discriminator, while the discriminator enhances its ability to distinguish between real and fake samples. The original GAN training objective can be interpreted as optimizing the generator to produce synthetic data that minimizes the Jensen-Shannon (JS) divergence between the synthetic and real data distributions. This idea has been expanded in various GAN training objective extensions explored in the literature. For instance, variants have been proposed based on generalizations to any f-divergence (Nowozin et al., 2016), Wasserstein distance (Arjovsky et al., 2017; Gulrajani et al., 2017), maximum mean discrepancy (MMD) (Bińkowski et al.,\n2018; Li et al., 2017), and Sinkhorn distance (Genevay et al., 2018).\n\nOf particular interest to DP training is the observation that many of these divergence metrics can be approximated without requiring the training of a discriminator network. This has led to recent research in private generative models, which use a static function as the critic instead of a discriminator network. While such approaches might fall short in standard (non-private) generative modeling due to a lower expressive power compared to using learnable critic (that is adaptable to large data with diverse properties), they are highly competitive in DP training, as a static critic can effectively speed up convergence, thereby improving privacy guarantees.\n\nIn the case of implicit density models, the generator's interaction with the private dataset is typically indirect\n(only via the backward pass), meaning that there exists no direct link between the data source, as illustrated in the accompanying diagrams (Figure 3). This configuration presents an opportunity to strategically position the privacy barrier anywhere along the backpropagation path where the generator retrieves signals from the real data, facilitating an improved signal-to-noise ratio or simplified implementation. A more comprehensive understanding is presented in Section 4.1-4.3.\n\n## 3.2.2 Explicit Density Models\n\nSeveral prominent explicit density models have been developed in literature, each with distinct characteristics: - The *Variational Autoencoder (VAE)* (Kingma & Welling, 2014) is trained to maximize the Evidence Lower Bound (ELBO), a lower bound of the log-likelihood, which typically simplifies to ℓ1/ℓ2 losses on the data sample and its reconstruction under standard Laplacian/Gaussian noise modeling assumptions. The model comprises trainable encoder and decoder modules. Encoding is conducted through the encoder qϕ, which maps observed data to its corresponding latent variables, denoted as x qϕ→ z. The dimensions of these latent variables are typically smaller than the data dimension d, embodying the concept of an information bottleneck (Tishby et al., 1999; Shwartz-Ziv & Tishby, 2017). The decoder module is responsible for data reconstruction or generation, i.e., z pθ→ x. Additionally, VAE imposes regularization on the latent distributions to match the pre-defined prior, thereby enabling the generation of valid novel samples during inference.\n\n- *Diffusion models* (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020) operate similarly to VAEs in terms of maximizing the ELBO. However, instead of using a trainable encoder to map data to latent variables, diffusion models transform the data iteratively through a linear Gaussian operation, represented as x q→ ...\n\nq→ xt−1 q→ xt q→ xT . This procedure causes the latent variables at the final step xT to form a standard Gaussian distribution and maintain the same dimensionality as the data. The generation process is executed by reversing the diffusion operation, which means iteratively applying pθ(xt−1|xt) for all time steps t ∈ [T]. The trainable component of diffusion models resides in the reverse diffusion process, while the forward process is pre-defined and does not require training.\n\n- *Flow-based* models (Rezende & Mohamed, 2015; Kingma & Dhariwal, 2018), in contrast, minimize the Negative Log-Likelihood (NLL) directly. Uniquely, flow-based models employ the same invertible model for both encoding (x fθ→ z) and generation (z f\n−1\n→θ x), by executing either the flow or its inverse. Due to the invertibility demanded by the model construction, the dimensions of the latent variables z are identical to those of the data.\n\n- *Autoregressive models* (Larochelle & Murray, 2011; Oord et al., 2016; Van Den Oord et al., 2016; Van den Oord et al., 2016), as another instance of model with tractable density, are also designed to minimize the NLL. Unlike some other models, they accomplish this without the need for explicit latent variables or an encoding mechanism. Instead, these models utilize partially observed data, denoted as x1:i−1, where each sample is regarded as a high-dimensional vector with observations up to the (i − 1)th element. The model is then trained to predict potential values for the subsequent element, xi. Data generation is conducted through an iterative autoregressive process, where elements of each data vector is predicted one-by-one, starting from initial seeds. This can be represented as x0 pθ→ ...\n\npθ→ x1:i−1 pθ→ x1:i pθ→ ...\n\npθ→ x1:d.\n\nThe component subject to training is the autoregressive model itself. Its parameters, denoted by θ, are optimized to best predict the next elements in the sequence based on previously observed values.\n\nAs illustrated in Figure 3, all these models require real data or derived quantities (such as latent variables)\nas inputs to the generator during the training phase. This necessitates a significant difference in the DP\ntraining of these models compared to implicit density models, which only need indirect data access through the backward pass. In the context of typical explicit density models, DP constraints must be accounted for, given the access to real data in both the forward and backward pass. This typically results in privacy barriers being directly integrated into the update process of the generator module, as further discussed in Section 4.4.\n\n## 3.2.3 Extensions\n\nOur diagram has been consciously designed to encompass future developments, including potential hybrid variants of generative models. It facilitates systematic analysis of the modifications required to transition the original training pipeline to a privacy preserving one. Specifically, to train a DP variant of such a model, one could follow the following steps: (1) Illustrate the model components and information flows using diagrams analogous to those shown in Figure 3. (2) Determine the component(s) that will be provided with DP\nguarantees, taking into account practical use requirements and a feasible privacy-utility trade-off. (3) Establish the privacy barrier to ensure the privacy of the targeted component, which will later be made accessible for potential threat exposure. This step should consider all access paths between the target component and the data source. (4) Calculate and bound the sensitivity. (5) Implement the DP mechanism and calculate the accumulated privacy cost of the entire training process.\n\n## 4 Taxonomy\n\nAccompanied by a comprehensive diagram encapsulating the complete spectrum of potential design choices for deep generative models, we put forth a classification system for current DP generative methods. This system is predicated on the positioning of the privacy barrier within the diagram (Figure 1). Specifically, for explanatory purposes, we consider the key components within our diagram (the Generator, **Synthetic data**, Measurement, and **Real data**), resulting in following options for positioning the privacy barrier:\n- B1: Between **Real data** and **Measurement** - B2: Within **Measurement** - B3: Between **Measurement** and **Synthetic data**\n- B4: Within **Generator**\nB1 through B4 are introduced sequentially, demonstrating the systematic transition of the privacy barrier from the real data source towards the generator end. The data-processing theorem 2.1 ensures that the DP\nguarantee is upheld as long as the data is \"sanitized\" through a DP mechanism prior to exposure to potential adversaries. In this context, if a DP training algorithm safeguards against threats introduced by B1, then it also provides the same protective guarantee against attackers defined by B2 through B4.\n\nThe generator end typically represents the smallest unit necessary for preserving the full functionality of the model, implying that the privacy barrier cannot be shifted further without compromising the operational capabilities of the generative model. Moreover, we reserve a more detailed discussion on the threat model\n(privacy barrier) integrated within the adopted DP mechanism (not specifically relevant to generative models)\nfor later sections, where individual approaches will be introduced.\n\n## 4.1 B1: Between Real Data And Measurement\n\nThreat Model. Establishing a privacy barrier between the **Real data** and the **Measurement** entails using a DP mechanism to directly sanitize the data (features), thereby obtaining statistics that characterize the real data distribution for subsequent operations like computing the loss L as a **Measurement** that serves as the training objective for the generator. This approach provides protection against attackers who might gain access to the sanitized data features or any resultant statistics derived from the sanitized features, such as the loss measured on the sanitized data, any gradient vectors for updating the generator, and the generator's model parameters. General Formulation. Methods within this category typically adopt the distribution matching framework\n(illustrated in Figure 3(b)), which aims to minimize the statistical distance between real and synthetic data distributions (Harder et al., 2021; Rakotomamonjy & Liva, 2021; Vinaroz et al., 2022). This distance is assessed with a static, unlearnable function, typically applying a data-independent feature extraction function ψ to project the data samples into a lower-dimensional embedding space and subsequently calculating the (Euclidean) distance between the resulting embeddings of real and synthetic data. The generator is optimized to reduce the disparity between the mean embeddings of synthetic and real data, which can be interpreted as minimizing the maximum mean discrepancy (MMD) between the real and synthetic data distributions (Bińkowski et al., 2018; Li et al., 2017).\n\nDuring DP training of these models, data points xi or feature vectors ψ(xi) are first clipped or normalized (by norm) to ensure bounded sensitivity. Subsequently, random noise is injected into the mean features derived from the real samples, e.g., via Gaussian mechanism (Definition 2.4). The objectives can be formulated as follows:\n\nNon-private: $$\\min_{\\mathbf{\\theta}}\\left\\|\\frac{1}{|\\mathcal{D}|}\\sum_{i=1}^{|\\mathcal{D}|}\\psi(\\mathbf{x}_{i})-\\frac{1}{|\\mathcal{S}|}\\sum_{i=1}^{|\\mathcal{S}|}\\psi(G_{\\mathbf{\\theta}}(\\mathbf{z}_{i}))\\right\\|_{2}^{2}=\\min_{\\mathbf{\\theta}}\\left\\|\\hat{\\mathbf{\\mu}}_{\\mathcal{D}}-\\widehat{\\mathbf{\\mu}}s\\right\\|_{2}^{2}$$ (3) $$\\text{DP:}\\min_{\\mathbf{\\theta}}\\left\\|\\hat{\\mathbf{\\mu}}_{\\mathcal{D}}-\\hat{\\mathbf{\\mu}}s\\right\\|_{2}^{2}\\quad\\text{with}\\quad\\hat{\\mathbf{\\mu}}_{\\mathcal{D}}=\\hat{\\mathbf{\\mu}}_{\\mathcal{D}}+\\mathcal{N}(0,\\Delta_{\\mathbf{\\mu}_{\\mathcal{D}}}^{2}\\sigma^{2}\\mathbf{I})$$ (4)\n$\\boldsymbol{\\Sigma}[S]$\nwith µbD =1 |D| P|D| i=1 ψ(xi) and µbS =1 |S| P|S| i=1 ψ(Gθ(zi)) representing the mean features of the real and synthetic data, respectively. Meanwhile, µeD denotes the DP-sanitized mean real data embedding with ∆2µbD\nbeing the sensitivity value that characterizes the influence of each real data point on the mean embedding. A\nvisual illustration can be found in Figure 4.\n\nRepresentative Methods. While all methods in this category adhere to the same general formulation, they primarily diverge in their construction of the feature extraction function ψ and the objective function that forms the training loss L for the generator. **DP-Merf** (Harder et al., 2021) employs the MMD minimization approach, optimizing a generator to minimize the difference between synthetic and real data embeddings, using random Fourier features (Rahimi & Recht, 2007) for the embedding function ψ. **DP-SWD** (Rakotomamonjy\n& Liva, 2021) instead employs random projections u ∈ S\nd−1for feature extraction. Specifically, DP-SWD\nuniformly samples k random directions for data projection, thereby enabling tractable computation of one-dimensional Wasserstein distances along each projection direction. The Sliced Wasserstein Distance\n(SWD) (Rabin et al., 2012; Bonneel et al., 2015), which is determined as the mean of one-dimensional Wasserstein distances over DP-sanitized projections, serves as the training objective for the generator. Similar\n\n\n\nFigure 4: Diagram illustrating the general training procedure of methods under B1.\nto DP-Merf, **PEARL** (Liew et al., 2021) employs the Fourier transform as the feature extraction function while offering an alternative interpretation of describing the data distribution using the characteristic function with the characteristic function distance as the objective. Furthermore, PEARL proposes learning a reweighting function for the embedding features, placing greater emphasis on the discriminative features, in order to enhance the expressiveness of the plain Fourier features employed in the DP-Merf approach. Recent research efforts have primarily focused on identifying informative features that can efficiently capture the underlying characteristics of the data distribution. Specifically, **DP-HP** (Vinaroz et al., 2022) employs Hermite polynomials as the feature embedding function. This choice of embedding function reduces the required feature dimension, which consequently decreases the effective sensitivity of the data mean embedding and leads to an improved signal-to-noise ratio in the DP training. **Harder et al. (2022)** further propose utilizing feature extraction layers from pre-trained classification networks that capture general concepts learned on large-scale public datasets. Additionally, **DP-NTK** (Yang et al., 2023) introduces the use of the Neural Tangent Kernel (NTK) to represent data, resulting in the gradient of the neural network function serving as the feature map, i.e., ψ(x) = ∇θf(x; θ).\n\nPrivacy Analysis. The privacy analysis for methods in this category involves computing the sensitivity and applying the privacy analysis of associated noise mechanisms, such as the Gaussian mechanism (Definition 2.4).\n\nThe sensitivity represents the maximum effect of an individual data point on the mean embedding:\n\n$$\\Delta^{2}=\\max_{\\mathcal{D},\\mathcal{D}^{\\prime}}\\|\\widehat{\\mathbf{\\mu}}_{\\mathcal{D}}-\\widehat{\\mathbf{\\mu}}_{\\mathcal{D}^{\\prime}}\\|_{2}=\\Big{\\|}\\frac{1}{|\\mathcal{D}|}\\sum_{i=1}^{|\\mathcal{D}|}\\psi(\\mathbf{x}_{i})-\\frac{1}{|\\mathcal{D}^{\\prime}|}\\sum_{i=1}^{|\\mathcal{D}^{\\prime}|}\\psi(\\mathbf{x}_{i}^{\\prime})\\Big{\\|}_{2}\\tag{5}$$\n\nIn existing literature, the replace-one privacy notion is commonly used to compute the sensitivity value\n∆2, resulting in an upper bound of 2 |D| when the feature vector by construction has a norm equal to 1 or is normalized with a maximum norm of 1, i.e., ∥ψ(x)∥2 ≤ 1. Deriving the sensitivity value for the add-or-remove-one notion is slightly more technically involved, but applying existing techniques used for the replace-one notion leads to a conservative bound of 2 |D|+1 (See Appendix). This implies two things:\nfirst, the sensitivity value decreases inversely proportional to the size of the dataset, showing the beneficial effect of the \"mean\" operation over large datasets, which smooths out individual effects through population aggregation. Second, there is a minor difference in the computed sensitivity between the two privacy notions:\n2 |D|+1 versus 2 |D| . This means that the current comparison results hold with negligible effect when the dataset size is sufficiently large. While achieving a tighter bound for the sensitivity value is possible with the add-or-remove-one privacy notion, it may require additional assumptions. In contrast to other studies that compute the (worst-case) global sensitivity (Definition 2.3), the sensitivity in DP-SWD represents a form of expected value, accompanied by a sufficiently small failure probability.\n\nThis efficiently harnesses the characteristics of random projections to achieve a tight sensitivity bound, but requires careful comparison to other methods. When combining this sensitivity definition with mechanisms that offer (*ε, δ*)-DP (i.e., the relaxed DP notion), the final privacy guarantee will be weaker than (*ε, δ*), due to the additional failure probability derived from the sensitivity itself.\n\n\n\nFigure 5: Diagram illustrating the training pipeline of **DP-GAN** with a (vertical) privacy barrier of type B2 as shown.\n\n\n\nFigure 6: Diagram illustrating the training pipeline of **PATE-GAN** with a (horizontal) privacy barrier of type B2 as shown. Lstu and Ltea denote the student and teacher training losses respectively, while Dstu is the student discriminator and D1*, ..., D*k represent the teacher discriminators.\n\nAnalysis, Insights, Implications. Methods under this category present several strengths. Firstly, the\n\"mean\" operation adopted during the extraction of descriptive feature embeddings significantly reduces the impact of each individual. This leads to a lower sensitivity value that scales in inverse proportion to the number of data points being aggregated through the \"mean\" operation. As a result, a strong privacy guarantee can be ensured with less randomness required from the DP mechanism. Moreover, they are straightforward to implement, typically necessitating just one instance of sanitization on the computed mean feature (known as \"one-shot sanitization\") throughout the training process, which further saves the privacy consumption in comparison to iterative methods. These methods also converge quickly and can yield acceptable results even under a low privacy budget, given the ease of fitting the static target, i.e., the noisy mean.\n\nNevertheless, they come with certain drawbacks. The static feature might not be sufficiently discriminative or informative, lacking the expressiveness found in methods that employ trainable models as critics. Furthermore, the \"mean\" operation could potentially induce unintended mode collapse in the generated distributions, trading off generation diversity for privacy protection. This situation warrants attention in future works, particularly in optimizing the trade-off between the expressiveness of the feature extraction method in the critic and the privacy cost of achieving such expressiveness. A promising direction could be to exploit knowledge from public non-sensitive data and/or pre-trained models that better describe data without compromising the privacy of the sensitive data.\n\n## 4.2 B2: Within Measurement\n\nThreat Model. The previous category focuses on a static, sanitized statistical summary, derived from a data-independent function, as a replacement for real data when training generative models. However, learnable functions that are able to adapt to diverse data distribution may offer superior expressive power. In this regard, a logical strategy is to incorporate DP into the measurement process, particularly by training a DP critic. This privacy barrier sits \"within **Measurement**\" and safeguards against adversaries with access to the critic and subsequent quantities, including information flows to the generator. If gradient sanitization techniques like DP-SGD are employed for updating the critic, the DP mechanism further protects against attacks targeting all intermediate gradients w.r.t. the critic's parameters during the training phase.\n\nGeneral Formulation. Methods in this category follow two main principles: Firstly, they use a learnable critic (feature extraction function) that dynamically adapts to the private dataset, necessitating a boundary on the potential privacy leakage of such critic. Secondly, the generator is prohibited from accessing private real data directly, its access limited to indirect interaction through the backward pass. This ensures the generator's update signals are fully derived from the learnable critic. As such, developing a DP critic is sufficient to assure DP for the generator module (and the entire model) for privacy-preserving generation.\n\nGAN models (depicted in Figure 3(a)) meet these criteria and serve as a foundational framework that most existing DP methods in this category generally conform to.\n\nRepresentative Methods. The implementation of the privacy barrier within the **Measurement** block is exemplified in **DP-GAN** (Zhang et al., 2018; Xie et al., 2018) and concurrent studies (Beaulieu-Jones et al.,\n2017; Triastcyn & Faltings, 2018; Alzantot & Srivastava, 2019; Xu et al., 2019; Torkzadehmahani et al., 2019; Frigerio et al., 2019). In this context, the discriminator, acting as the learnable critic model, is trained via DP-SGD (Section 2.1.1). The privacy of the generator is ensured by the post-processing theorem. As per the public timestamp of paper releases, this approach can be traced back to Beaulieu-Jones et al. (2017),\nwho proposed training an ACGAN (Auxiliary Classifier GAN) (Odena et al., 2017) in a DP manner to conditionally generate samples for downstream analysis tasks on medical data. The training pipeline can be formalized as follows, with the illustration shown in Figure 5:\n\ng (t) D = ∇ϕL(Gθ, Dϕ) (Discriminator gradient) (6) g (t) G = ∇θL(Gθ, Dϕ) (Generator gradient) (7) ge (t) D = Mσ,C (g (t) D ) = clip(g (t) D , C) + N (0, σ2C 2I) (Apply DP sanitization) (8) ϕ (t+1) = ϕ (t) − ηD · ge (t) D (Discriminator update) (9) θ (t+1) = θ (t) − ηG · g (t) G (Generator update) (10)\n$$(6)$$\n$$\\left(7\\right)$$\n$$({\\boldsymbol{\\delta}})$$\n$\\left(\\mathfrak{g}\\right)$. \n$\\left(10\\right)^3$\nThe generator Gθ and discriminator Dϕ are parameterized by θ and ϕ, respectively, with ηG and ηD denoting their learning rates. Mσ,C refers to the Gaussian mechanism in DP-SGD, with σ representing the noise scale and C indicating the gradient clipping bound. Although we have omitted the sample index in the above equations for the sake of brevity, it should be noted that the clipping function in Equation 8 is expected to take per-example gradients as inputs, adhering to the standard procedure of DP-SGD (Section 2.1.1).\n\nSpecifically, it suffices to apply the sanitization only to the gradients that depend on the real data samples, including indirect usage of real samples, such as through gradient penalty terms (Gulrajani et al., 2017).\n\nUnlike DP-GAN that employs DP-SGD for training the DP discriminator, **PATE-GAN** (Yoon et al., 2019)\nleverages the PATE framework (Section 2.1.2) to train its DP (student) discriminator. PATE-GAN comprises three main components that are jointly trained throughout the process: multiple (non-private) teacher discriminators, a DP student discriminator, and a DP generator. Similar to the original PATE framework, PATE-GAN starts by partitioning the real dataset into disjoint subsets, which subsequently serve to train the teacher discriminators independently. In each training iteration, PATE-GAN follows a sequence of steps:\n(1) independently updating the teacher discriminators using mini-batch samples from real data partitions and synthetic samples drawn from the generator; (2) querying the teacher discriminators with a set of synthetic samples; (3) the teacher discriminators then engage in a voting process on the real/fake predictions for the synthetic samples they have received, and apply DP noise to the results of the vote; (4) training the student discriminator with the query synthetic samples as input and the DP aggregation of teacher predictions as the label; (5) finally, jointly updating the generator and the student discriminator, with the generator querying the student discriminator with new synthetic samples and obtaining update gradient signals from the DP\nstudent discriminator. A visual illustration is presented in Figure 6.\n\nWhile the discriminator in the GAN framework aims to distinguish between two distributions, recent research uncovered intriguing results when the learnable critic is designed to target specific downstream tasks, such as classification. Specifically, **Private-Set** (Chen et al., 2022a) employs a classification network as a learnable feature extractor, which is trained with DP-SGD. This learnable feature extractor, combined with the alignment in the gradients serving as the critic, encourages the synthetic data to emulate the training trajectories of the real data during the training process within a classification network, making the synthetic data useful for training downstream classifiers and safe for public release due to the DP guarantees embedded within the measurement process.\n\nPrivacy Analysis. Methods in this category inherit the privacy notion and sensitivity computation from their respective framework for training the DP critic (See Section 2.1.1-Section 2.1.2), while also inheriting the need for careful consideration regarding the application of data-dependent privacy analysis or adherence to privacy notion constraints to ensure comparable results. For methods grounded by DP-SGD, this results in a noticeable disparity between the replace-one and add-or-remove-one DP notions, as illustrated by the doubled sensitivity value when transitioning from the default add-or-remove-one to the replace-one notion, i.e., C\nversus 2C with C denoting the gradient clipping bound. Consequently, a doubled noise scale is required to achieve an ostensibly identical privacy guarantee, inevitably resulting in utility degradation and unfavorable comparison outcomes under the replace-one notion.\n\nAnalysis, Insights, Implications. While this training paradigm enjoys several advantages, such as ease of implementation and representative features for characterizing the difference between distributions, several challenges persist when applying such a paradigm in practice. Firstly, the joint training of a generator alongside a critic, which typically necessitates an adversarial approach, is inherently unstable due to the difficulty in maintaining equilibrium between these two components. This instability can be further amplified by the incorporation of gradient clipping and noise addition operations introduced by DP-SGD, or the additional fitting process involved in transferring knowledge from the teacher discriminators to the student one through the PATE framework. Moreover, the DP training of the critic often impedes its convergence, resulting in a sub-optimal critic that may not effectively guide the generator.\n\nRecent studies have investigated various strategies to alleviate these challenges, particularly in the context of GANs. These include warm-starting the GAN discriminator by pre-training on public data (Zhang et al.,\n2018), dynamically adjusting the gradient clipping bounds during the training process (Zhang et al., 2018),\nre-balancing the discriminator and generator updates to restore parity to a discriminator weakened by DP noise (Bie et al., 2023), and exploiting public pre-trained GANs while restricting private modeling to the latent space (Chen et al., 2021). In the Private-Set (Chen et al., 2022a) framework that optimizes for downstream classification task, it is reported that optimizing the generator in a point-wise manner (as discussed in Section 3) or directly optimizing the synthetic set instead of the generator model can empirically lead to faster convergence and preferable when strong privacy guarantee is required. In this regard, we anticipate promising outcomes from the future development of new variants of DP-compliant training pipelines and objectives that offer improved convergence and, consequently, enhanced privacy guarantees.\n\n## 4.3 B3: Between Measurement And Synthetic Data\n\nThreat Model. In response to challenges associated with training the DP critic (Section 4.2), recent studies have proposed shifting the privacy focus from the **Measurement** to the sanitization of the intermediate signal that backpropagates to update the generator, i.e., between **Measurement** and **Synthetic data**. The goal is to preserve the critic's training stability and its utility for accurately comparing synthetic and real data, thereby guiding the generator's training effectively. This strategy ensures privacy when revealing sanitized intermediate gradients exchanged between the generator and the critic during the backward pass, as well as guarantees DP for the generator, which is updated with sanitized gradients. However, this scheme does not provide privacy guarantees for the release of the critics, since their training is conducted non-privately. General Formulation. Similar to the case outlined in Section 4.2, the backbone generative models for this category are typically implicit density models. This restriction is in place as these models do not invoke direct interaction between the real data and the generator during the forward pass, which means that sanitizing the intermediate signals transmitted between the **Measurement** and **Synthetic data** is sufficient for ensuring\n\n\n\nFigure 7: The diagram illustrates the general training process of methods incorporating the privacy barrier B3. In the figure, g up G and ge up G denote the upstream gradient (referenced as g upstream G in Equation 12) and its sanitized variant, respectively. Note that variations exist in the formulation of critics and their corresponding training paradigms.\nprivacy protection. Methods in this category adhere to the gradient sanitization scheme, which introduces a DP perturbation into the gradients communicated between the critic and generator during the backward pass. This can be formulated as follows:\n\n$$\\begin{array}{c}{{\\mathbf{g}_{G}^{(t)}=\\nabla_{\\theta}\\mathcal{L}_{G}(\\mathbf{\\theta}^{(t)})=\\nabla_{G_{\\theta}(z)}\\mathcal{L}_{G}(\\mathbf{\\theta}^{(t)})\\cdot\\mathbf{J}_{\\theta}G_{\\theta}(\\mathbf{z})}}\\\\ {{\\hat{\\mathbf{g}}_{G}^{(t)}=\\mathcal{M}\\big(\\underbrace{\\nabla_{G_{\\theta}(\\mathbf{z})}\\mathcal{L}_{G}(\\mathbf{\\theta}^{(t)})}_{\\mathbf{g}_{G}^{(t)\\mathrm{transform}}}\\big)\\cdot\\underbrace{\\mathbf{J}_{\\theta}G_{\\theta}(\\mathbf{z})}_{J_{G}^{\\mathrm{transform}}}}\\end{array}$$\n(12)\nHere, LG represents the generator's loss (originating from a critic), and M denotes a potential DP sanitization mechanism on g upstream G —the gradient information backpropagating from the critic to the generator. This can be considered as the gradient of the objective with respect to the current synthetic samples. It is important to note that the second term (J\nlocal G ), i.e., the local generator Jacobian, is computed independently of training data and thus does not require sanitization. The generator is subsequently updated with the DP sanitized gradient, i.e., θ\n(t+1) = θ\n(t) − ηG · ge\n(t)\nG . Meanwhile, the critic, if learnable, is updated normally (non-privately).\n\nA visual illustration is presented in Figure 7.\n\nRepresentative Methods. Existing methods explored various choices for the critic and different DP\nmechanisms to sanitize the upstream gradients g upstream G . **GS-WGAN** (Chen et al., 2020a) adopts the Gaussian mechanism for sanitization and capitalizes on the inherent bounding of the gradient norm. This follows from the Lipschitz property when employing the Wasserstein distance with gradient penalty (Arjovsky et al., 2017; Gulrajani et al., 2017) as the objective when training a GAN. In contrast, **G-PATE** (Long et al., 2021) incorporates the PATE framework as its sanitization mechanism. This approach discretizes the gradients and allows multiple teacher discriminator models to vote on these discretized gradient values. The DP noisy argmax is then transferred to the generator. **DataLens** (Wang et al., 2021) further improves the signal-to-noise ratio in the PATE sanitization by employing top-K dimension compression.\n\nIn a different vein, **DP-Sinkhorn** (Cao et al., 2021) presents compelling results using a nonparametric critic. Specifically, DP-Sinkhorn estimates the Sinkhorn divergence grounded on ℓ1 and ℓ2 losses in the data space, adhering to the distribution matching generative framework as depicted in Figure 3(b). This use of a data-independent critic contributes stability to the training process and capitalizes on the privacy enhancement brought by subsampling.\n\nPrivacy Analysis. The privacy analysis for this method category largely aligns with the established unit sanitization mechanisms, denoted as M, which function on upstream gradients g upstream G . Nevertheless, specific attention is necessary given that these intermediate gradients do not directly originate from real data samples. This scenario noticeably influences the sensitivity computation, defined formally by:\n\n$$\\Delta^{2}=\\operatorname*{max}_{{\\mathcal{D}},{\\mathcal{D}}^{\\prime}}\\|f(\\mathbf{g}_{G}^{\\mathrm{upstream}})-f(\\mathbf{g}_{G}^{\\prime}{}^{\\mathrm{upstream}})\\|_{2}$$\nupstream)∥2 (13)\n$$(13)^{\\frac{1}{2}}$$\n\nIn this equation, f encapsulates the operations required to set bounds on the sensitivity and to render the associated sanitization mechanism applicable. gG\nupstream and g\n′\nG\nupstream symbolize the intermediate upstream gradients originating from neighboring datasets D and D′respectively. Specifically, f performs distinct roles according to the method employed: For GS-WGAN and DP-Sinkhorn, f signifies the operation of norm clipping; In G-PATE, f encompasses the processes of dimension reduction and gradient discretization, and the computation of teacher voting histograms based on these discretized gradients; In the context of DataLens, rather than employing random projection and discretization as in G-PATE, f adopts a top-k stochastic sign quantization of the gradients. Subsequent to this operation, the teacher voting histograms are also calculated.\n\nA direct application of the triangle inequality reveals that ∆2equals 2C (with C representing the gradient clipping bound) in both GS-WGAN and DP-Sinkhorn for both the replace-one and add-or-remove-one notions, while C is further guaranteed to be 1 in GS-WGAN by the nature of the adopted Wasserstein objective.\n\nThis is notably different from the substantial disparity between the two privacy notions in the standard DP-SGD framework. In G-PATE, the voting histogram diverges by a maximum of 2 entries for each gradient dimension, which are processed independently via DP aggregation. As for the DataLens approch, the change of one data point will at most reverse all the signs of the top-k elements of gradients originated from one teacher model, leading to ∆2 = 2√k (See Appendix for details).\n\nTypically, the total privacy cost is calculated based on the RDP accountant (Theorem 2.2). Notably, each synthetic sample in a mini-batch constitutes one execution of the sanitization mechanism for the DP-SGD\nframework, or one query in the PATE framework. In other words, performing an update step with a mini-batch of synthetic samples on the generator can be regarded as a composition of *batch size* times its unit sanitization mechanism.\n\nAnalysis, Insights, Implications. Compared to previous categories (Section 4.1-4.2), shifting the privacy barrier away from the **Measurement** process itself offers several benefits. These include: (1) the flexibility to employ a powerful critic, thereby effectively guiding the generator towards capturing the characteristics of the data distribution; (2) seamless support for different privacy notions (as discussed in privacy analysis above); (3) practically simpler to properly bound the sensitivity. This can be achieved by exploiting the intrinsic properties of the objective (Chen et al., 2020a), or through the usage of the PATE framework (Long et al., 2021; Wang et al., 2021). This is particularly beneficial when compared to the previous scenario of learnable critics that typically necessitate a laborious and fragile hyperparameter search for a reasonable gradient clipping bound. However, the increased expressive capacity comes with the trade-off of relatively high privacy consumption.\n\nThe accumulation of privacy cost across iterations is notably faster in this scenario than in standard DP-SGD\ntraining of a single model: each DP update on the generator in this category equates to a *batch size* number of calls to the Gaussian mechanism, possibly without the advantage of subsampling, as detailed in the preceding privacy analysis section. This markedly contrasts with the standard DP-SGD training on a single discriminator, as mentioned in the previous category (refer to Section 4.2), where each individual DP gradient update equates to a *single* execution of the (subsampled) Gaussian mechanism.\n\nFortunately, this drawback has been partially mitigated through the use of data-dependent privacy analysis\n(as demonstrated in PATE-based methods like G-PATE and DataLens) that provides analytically tighter results that lead to stronger DP guarantees, or a data-independent critic (as in DP-Sinkhorn) that offers smooth compatibility with subsampling and better convergence. Looking forward, we anticipate further developments from refining this training paradigm, particularly through the utilization of strong backbone discriminators (and generators) trained on external non-private data, thereby optimizing privacy consumption.\n\n## 4.4 B4: Within Generator\n\nThreat Model. DP can be directly integrated into the training or deployment of a generator, the minimal unit within the generative models pipeline essential for maintaining the full generation functionality for future use. Generally, the privacy barrier safeguards against attackers who have access to the trained generator model while a more fine-grained distinguishment between the type of access (e.g., white-box or black-box) may be required depending on the application scenarios and the adopted DP mechanism. If the gradient\n\n\n\nFigure 8: Diagram showcasing the DP training of a VAE (Section 3.2.2). This representation is also applicable to other DP model training scenarios conforming to privacy barrier B4, e.g., by replacing the trainable encoder with a non-trainable module.\nsanitization scheme is adopted, it can protect against adversaries who can access the white-box generator\n(and possibly other trainable components subject to DP sanitization) and the intermediate sanitized gradients during the whole training process.\n\nGeneral Formulation. In this context, the training pipeline can be generally simplified to the standard process of training DP classification models. This process, as exemplified by the commonly used DP-SGD\nframework, entails bounding sensitivity through gradient clipping and subsequently injecting randomness into the generator's gradients. In contrast to category B3, where the upstream gradient g upstream G undergoes sanitization, in this case, it is the final generator gradient g\n(t)\nG (refer to Equation 11) that is being sanitized.\n\nThis results in a difference equivalent to the multiplicator of the local generator Jacobian (refer to Equation 12).\n\nSpecial attention should be paid when implementing DP-SGD here, as additional model components (e.g.,\nthe encoder in a VAE) alongside the generator could compromise the transparency of the privacy analysis.\n\nIt is crucial to ensure that the gradient clipping operation is executed accurately to effectively limit each individual real sample's influence on the generator. The presence of an additional model component may disperse individual effects across multiple gradients within a mini-batch, rendering standard per-example gradient clipping inadequate (refer to the discussion in the privacy analysis below). Moreover, to optimize model utility, it is necessary to precisely define the scope of gradient clipping and perturbation to ensure that the implementation does not introduce unnecessary noise exceeding the desired privacy guarantee.\n\nRepresentative Methods. Existing works have realized such privacy barrier for various types of generative models, particularly those within the explicit density category. Examples include DP Normalizing Flow (Waites & Cummings, 2021; Jiang & Sun, 2023), **DP VAE** (Chen et al., 2018; Acs et al., 2018; Abay et al., 2019; Takagi et al., 2021; Pfitzner & Arnrich, 2022), **DP Diffusion models** (Dockhorn et al., 2022; Ghalebikesabi et al., 2023; Lyu et al., 2023), and DP training of **language models** (McMahan et al., 2018; Li et al., 2022; Mattern et al., 2022; Yue et al., 2023), which collectively illustrate the extensive potential of DP generators across numerous applications such as natural language generation, density estimation, high-quality image generation, training downstream models, and model selection. In particular, Ghalebikesabi et al. (2023)\nhighlighted that certain training techniques advantageous for DP classification models (De et al., 2022), such as pre-training, utilization of large batch sizes, and augmentation multiplicity (Fort et al., 2021; De et al.,\n2022), also show effectiveness when applied to training DP generators in diffusion models. Furthermore, the work by Jiang & Sun (2023) underscores the potential efficacy of training a DP Flow model within a compressed, lower-dimensional latent space. This strategy not only circumvents the substantial computational demands (Rombach et al., 2022), but also synergizes well with DP protocols, given the direct correlation between the DP noise-to-signal-ratio and the model's dimensionality. Privacy Analysis. The privacy analysis follows from the adopted DP mechanism for training the generators, similar to the standard case of training a DP classifier. A key consideration lies in the correct implementation and analysis of the privacy cost when the models comprise multiple trainable components, such as the encoder and decoder in the VAE. In such cases, simply incorporating the DP-SGD into the generator module and conducting a standard privacy accountant is inappropriate. This is due to the fact that each training example's influence is assimilated into the encoder's parameters. Consequently, every training example, even those absent from the current mini-batch, can affect all latent variables (which serve as inputs to the generator/decoder) in each iteration, rendering the per-example gradient clipping itself insufficient for bounding the sensitivity. A proper implementation would require either enforcing DP also on the encoder\n(i.e., applying DP-SGD on both the encoder and decoder) or factoring this into the privacy cost computation\n(i.e., the DP-SGD step on the decoder should be counted as full batch Gaussian mechanism instead of a subsampled one). Moreover, in situations where each sample in a mini-batch is used more than once, such as their use over multiple time steps when training diffusion models, the cost must be accounted for every such occurrence. To deal with this, one can refer to the *multiplicity* technique (Fort et al., 2021; De et al., 2022; Ghalebikesabi et al., 2023; Dockhorn et al., 2022), which averages all gradients resulting from each unique training sample before clipping them.\n\nAnalysis, Insights, Implications. Methods in this category are generally easy to implement, particularly for models with only a generator as the learnable component. This reduces training to the standard classification cases, demonstrating significant potential and achieving state-of-the-art generation quality when adapted to the latest generative modeling techniques (Dockhorn et al., 2022; Ghalebikesabi et al., 2023).\n\nHowever, this privacy barrier setting may not be fully compatible with models containing multiple trainable components. The reason for this lies in the potential integration of training samples' effects into the parameters of components other than the generator (e.g., the encoder in VAEs, the discriminator in GANs), which substantially complicates the implementation of DP mechanisms and may lead to unexpectedly high privacy consumption. Moreover, DP methods are bounded by the expressive capability of the underlying generative model. Particularly in this category, which predominantly relies on explicit density models, the usage of simple critics (like static ℓ1 or ℓ2 loss functions) tends to restrict the capture of fine details, often delivering less desirable outcomes compared to trainable critics. For instance, VAEs have commonly produced blurrier images, whereas GANs pioneered the production of high-resolution photorealistic generations. While recent advancements in explicit density models have significantly improved their capabilities, particularly through innovative designs that enable training on extensive datasets, there is a potential limitation concerning their practical utility. This limitation primarily arises from the substantial need for sensitive training data, which is essential to achieve a satisfactory performance level with the resulting DP model in real-world applications.\n\nLooking forward, we envision future advancement on balancing the data efficiency and generation performance could largely improve the practicability of the DP methods under this category.\n\n## 5 Discussion 5.1 Connection To Related Fields\n\nWhile the data generation methods investigated in this work are mostly designed to capture the entire data distribution for general purposes, intriguing results are observed when the generator is intentionally guided towards enhancing its downstream utility for specific target tasks such as training neural network classifiers (Chen et al., 2022a) and answering linear queries (Liu et al., 2021). This can be achieved by employing objectives tailored for downstream tasks, rather than relying solely on general distribution divergence measures. If downstream tasks can be executed on a specific set of samples and do not require a complete understanding of the distribution, problem complexity can be further reduced by directly optimizing the synthetic samples instead of the generative models. This strategy, which trade-off the generality of general-purpose generative modeling for downstream utility, might be particularly beneficial considering the high complexity inherent to DP generation. Moreover, such framework naturally aligns with broader fields such as coreset generation, private query release, private Bayesian inference. In these scenarios, a set of synthetic data can be optimized to resemble real data for specific tasks (Wang et al., 2018; Bachem et al.,\n2017), substitute real data for answering queries to conserve the privacy budget under DP (Hardt & Rothblum, 2010; Hardt et al., 2012), or support privacy-preserving computation of the posterior distribution (Manousakas et al., 2020; Savitsky et al., 2022).\n\n## 5.2 Relation To Other Summary Papers\n\nSeveral related summary papers complement our work by focusing on different aspects. For instance, Tao et al. (2021) benchmark multiple DP models for tabular data; Fan (2020) and Cai et al. (2021) discuss early DP GANs; Jordon et al. (2022) and De Cristofaro (2023) provide high-level overviews of DP synthetic data generation for non-expert audiences; Hu et al. (2023) covers broad classes of DP data generation methods without focusing on the technical part of deep generative modeling; Lastly, Ponomareva et al. (2023) offer a comprehensive summary of developing and deploying general DP ML models, supplementing our focus on the technical aspects of DP generative modeling.\n\n## 5.3 Challenges And Future Directions\n\nPublic Knowledge. A promising future direction which holds significant practical relevance is the exploitation of public data/knowledge in training DP generative models. Recent studies have demonstrated promising improvements in DP generation introduced by leveraging public data (Chen et al., 2021; Liu et al.,\n2021; Harder et al., 2022; Lyu et al., 2023) and reported high-quality generation (Ghalebikesabi et al., 2023; Lin et al., 2023) with the aid of such resources. A prevalent method for leveraging public knowledge involves utilizing large foundation models, initially pre-trained on public datasets, and subsequently fine-tuned to align with private data distributions for various applications. This approach is particularly relevant in the field of natural language processing (NLP), where the widespread availability of foundation models and the typically significant semantic overlap between public and private data renders DP fine-tuning relatively effective (Li et al., 2022). Additionally, the rapid growth of efficient fine-tuning techniques also show great potential for facilitating DP learning (Yu et al., 2021; Duan et al., 2023). While these advancements are particularly notable in the NLP domain, exploring the specific benefits and most effective strategies for applying these techniques to other data modalities is a topic that warrants further research. Furthermore, challenges that are generally associated with private learning on public data (Tramèr et al., 2022) call for further investigation.\n\nIn particular, the unique difficulties specific to generative modeling, such as a small tolerance for distribution shift between the public and private data distributions, warrant additional exploration.\n\nTask-specific Generation. There exists a principled trade-off between the flexibility offered by generalpurpose generative modeling and the utility of task-specific data generation. In particular, capturing a complete high-dimensional data distribution is a difficult task. This task becomes even harder when considering the privacy constraints, thus making the models highly data-demanding and almost impossible for DP model to achieve reasonable performance in practice. It has also been recently questioned to what extend a well-performing general-purpose DP generative model can be realized at all (Stadler et al., 2022; Stadler\n& Troncoso, 2022). While it is difficult to predict how these trade-off develop in the future, task-specific\n(or task-guided) data generation can greatly relax the objectives, leading to real-world useful DP synthetic data (see examples discussed in Section 5.1). On the other hand, such task-specific generation is particularly advantageous for scenarios where the synthetic data is intentionally designed to be useful only for specific\n(benign) tasks, thereby preventing potential unauthorized data misuse.\n\nConditional Generation. While the formulas presented throughout Section 4 are illustrated through unconditional generation for simplicity and clarity, in practice, DP generation is typically executed in a conditional manner, whereby samples are generated given specific input conditions. Although implementing conditional generation is technically straightforward for all generative network backbones (Mirza & Osindero, 2014; Sohn et al., 2015; Odena et al., 2017; Winkler et al., 2019), it might necessitate additional consideration with respect to the privacy analysis. For instance, when modeling the class-conditional data feature distribution, an additional privacy budget may be allocated to learn the class label occurrence ratio for addressing class imbalance (Harder et al., 2021), contrasting with other methodologies that typically employ a data-independent uniform class-label distribution. Moreover, certain situations necessitate meticulous investigation into privacy implications and performance. Firstly, when the training process employs conditional\n(e.g., per-label class) sampling, additional consideration for privacy cost is imperative, as this contradicts the requirements of random sub-sampling incorporated in standard privacy cost computations. Secondly, some generative modules may integrate such conditional information in non-trivial ways (e.g., being embedded into the module parameters beyond mere gradients (Karras et al., 2020)). This integration can mean that the conditional input might no longer be protected under DP guarantees via a vanilla DP sanitization scheme.\n\nThese scenarios necessitate further exploration to ensure the reliability of privacy protections and to facilitate the development of more effective utility-preserving DP generative models.\n\nFederated Learning. DP data generation models have also shown promising potential in applications related to federated training (Augenstein et al., 2019; Xin et al., 2020; Zhang et al., 2021; Triastcyn & Faltings, 2020), facilitating tasks such as privacy-preserving data inspection and debugging that were previously infeasible due to privacy constraints. Specifically, Augenstein et al. (2019) incorporated DP-SGD into the training of a GAN in a federated setting, where each client maintains a local GAN model and communicates the gradients to the server during each communication round, with the gradients being sanitized under DP\nnoise. Moreover, Chen et al. (2020a) illustrated that the privacy barrier B3 (Section 4.3) is seamlessly compatible with the federated training setting. In this context, only the upstream gradient (Equation 12)\nneeds to be communicated, offering additional benefits such as improved communication efficiency. More recently, task-specific DP generation has proven particularly advantageous in alleviating non-iid challenges and enhancing convergence speed for federated learning (Xiong et al., 2023; Wang et al., 2023). Although these approaches might still require a substantial amount of client local data and computational resources, the future development of efficient algorithms is anticipated to yield fruitful outcomes. Evaluation and Auditing. Evaluating generative models has historically posed a significant challenge (Lucic et al., 2018), and the same holds true for DP generation methods. While evaluating them based on specific downstream tasks has been a common approach in existing literature, it has become evident that relying solely on a single metric may be inadequate. This limitation arises from the general lack of alignment among various aspects, including downstream utility, statistical properties, and visual appearance (Alaa et al., 2022; Stadler & Troncoso, 2022; Chen et al., 2022a; Ganev et al., 2022). Consequently, there arises a need for future investigations into comprehensive metrics that consider mixed objectives to more effectively address a wide range of potential practical applications.\n\nFurthermore, assessing the privacy guarantees of DP generators against real-world attacks (i.e., \"auditing\" (Jagielski et al., 2020; Nasr et al., 2023)), and quantifying the privacy risk associated with synthetic data (Stadler & Troncoso, 2022; Houssiau et al., 2022), presents a particularly intricate challenge for generative models. This complexity primarily arises from two key factors. Firstly, the measurement of privacy risks often conflicts with the primary objective of maximum likelihood, which aims to precisely fit the training data.\n\nWhile achieving an exact alignment with the training data aligns with training objectives, it raises a debatable question about compromising privacy protection. Deciding whether an exact match should be regarded as a privacy breach in such cases remains a matter of debate. Secondly, generative models typically exhibit low sensitivity to privacy attacks (Hayes et al., 2017; Chen et al., 2020b), which diminishes the informativeness of computed auditing scores. These challenges highlight the need for dedicated design tailored to the auditing of DP generative models.\n\n## 6 Conclusion\n\nIn summary, we introduce a unified view coupled with a novel taxonomy that effectively characterizes existing approaches in DP deep generative modeling. Our taxonomy, which encompasses critical aspects such as threat models, general formulation, detailed descriptions, privacy analysis, as well as insights and broader implications, provides a consolidated platform for systematically exploring potential innovative methodologies while leveraging the strengths of existing techniques. Furthermore, we present a comprehensive introduction to the core principles of DP and generative modeling, accompanied by substantial insights and discussions regarding essential considerations for future research in this area.\n\n## Acknowledgements\n\nThis work is partially funded by the Helmholtz Association within the project \"Privatsphären-schützende Analytik in der Medizin\" (PrivateAIM, 01ZZ2316G), the BMBF with the project \"Reprasentative, synthetische Gesundheitsdaten mit starken Privatspharengarantien\" (PriSyn, 16KISAO29K) and ELSA - European Lighthouse on Secure and Safe AI, funded by the European Union under grant agreement No. 101070617.\n\nViews and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or European Commission. Neither the European Union nor the European Commission can be held responsible for them. Dingfan Chen acknowledges the funding from the Qualcomm Innovation Fellowship Europe. 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Differentially private releasing via deep generative model\n(technical report). *arXiv preprint arXiv:1801.01594*, 2018.\n\n# Appendix\n\n## A Summary Of Existing Works\n\n| Approach | Privacy | Privacy | Sensitivity | Generative | DP | Code |\n|------------------------------------------|-----------|-------------------|---------------|-----------------------|--------------|--------|\n| barrier | notion | type | framework | framework | | |\n| DP-Merf (Harder et al., 2021) | B1 | Replace-one | Global | Distribution matching | Gaussian | 1 |\n| DP-SWD (Rakotomamonjy & Liva, 2021) | B1 | Replace-one | Smooth | Distribution matching | Gaussian | 2 |\n| PEARL (Liew et al., 2021) | B1 | Replace-one | Global | Distribution matching | Gaussian | 3 |\n| DP-HP Vinaroz et al. (2022) | B1 | Replace-one | Global | Distribution matching | Gaussian | 4 |\n| DP-GEN (Chen et al., 2022b) | ∼B1 | - | - | Energy-based model | - | 5 |\n| Harder et al. (2022) | B1 | Replace-one | Global | Distribution matching | Gaussian | 6 |\n| DP-NTK (Yang et al., 2023) | B1 | Replace-one | Global | Distribution matching | Gaussian | 7 |\n| DPSDA (Lin et al., 2023) | B1 | Add-or-remove-one | Global | Diffusion | Gaussian | 8 |\n| SPRINT-gan (Beaulieu-Jones et al., 2017) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | 9 |\n| dp-GAN (Zhang et al., 2018) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | 10 |\n| DPGAN Xie et al. (2018) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | 11 |\n| Triastcyn & Faltings (2018) | B2 | Add-or-remove-one | - | GAN | empirical DP | - |\n| PATE-GAN (Yoon et al., 2019) | B2 | Both | Local | GAN | PATE | 12 |\n| Alzantot & Srivastava (2019) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | 13 |\n| Xu et al. (2019) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | - |\n| DP-CGAN (Torkzadehmahani et al., 2019) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | 14 |\n| Frigerio et al. (2019) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | 15 |\n| DPMI (Chen et al., 2021) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | - |\n| DPautoGAN (Tantipongpipat et al., 2021) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | 16 |\n| Private-Set (Chen et al., 2022a) | B2 | Add-or-remove-one | Global | Distribution matching | DP-SGD | 17 |\n| Bie et al. (2023) | B2 | Add-or-remove-one | Global | GAN | DP-SGD | - |\n| GS-WGAN (Chen et al., 2020a) | B3 | Both | Global | GAN | DP-SGD | 18 |\n| G-PATE (Long et al., 2021) | B3 | Both | Local | GAN | PATE | 19 |\n| DataLens (Wang et al., 2021) | B3 | Both | Global/Local | GAN | PATE | 20 |\n| DP-Sinkhorn (Cao et al., 2021) | B3 | Both | Global | Distribution matching | DP-SGD | 21 |\n| DP-GM (Acs et al., 2018) | B4 | Add-or-remove-one | Global | VAE | DP-SGD | - |\n| DP-VaeGM (Chen et al., 2018) | B4 | Add-or-remove-one | Global | VAE , AE+ | DP-SGD | - |\n| DP-SYN (Abay et al., 2019) | B4 | Add-or-remove-one | Global | AE+ | DP-SGD | - |\n| P3GM (Takagi et al., 2021) | B4 | Add-or-remove-one | Global | VAE | DP-SGD | 22 |\n| DP-NF (Waites & Cummings, 2021) | B4 | Add-or-remove-one | Global | Flow | DP-SGD | 23 |\n| DP2 -VAE (Jiang et al., 2022) | B4 | Add-or-remove-one | Global | VAE | DP-SGD | - |\n| DPDM (Dockhorn et al., 2022) | B4 | Add-or-remove-one | Global | Diffusion | DP-SGD | 24 |\n| Ghalebikesabi et al. (2023) | B4 | Add-or-remove-one | Global | Diffusion | DP-SGD | - |\n| DP-LDM (Lyu et al., 2023) | B4 | Add-or-remove-one | Global | Diffusion | DP-SGD | 25 |\n| DP-LFlow Jiang & Sun (2023) | B4 | Add-or-remove-one | Global | Flow | DP-SGD | 26 |\n\nTable 1: Table summary of existing works most relevant to our contributions. The shaded area corresponds to approaches that require public data features.\n\n1 https://github.com/ParkLabML/DP-MERF\n2 https://github.com/arakotom/dp_swd 3 https://github.com/spliew/pearl 4 https://github.com/parklabml/dp-hp 5 https://github.com/chiamuyu/DPGEN\n6 https://github.com/ParkLabML/DP-MERF 7 https://github.com/FreddieNeverLeft/DP-NTK 8 https://github.com/microsoft/DPSDA 9 https://github.com/greenelab/SPRINT_gan 10 https://github.com/alps-lab/dpgan 11 https://github.com/illidanlab/dpgan 12 https://github.com/vanderschaarlab/mlforhealthlabpub/tree/main/alg/pategan 13 https://github.com/nesl/nist_differential_privacy_synthetic_data_challenge/\n14 https://github.com/reihaneh-torkzadehmahani/DP-CGAN 15 https://github.com/SAP-samples/security-research-differentially-private-generative-models 16 https://github.com/DPautoGAN/DPautoGAN\n17 https://github.com/DingfanChen/Private-Set 18 https://github.com/DingfanChen/GS-WGAN\n19 https://github.com/AI-secure/G-PATE\n20 https://github.com/AI-secure/DataLens 21 https://github.com/nv-tlabs/DP-Sinkhorn_code 22 https://github.com/tkgsn/P3GM\n23 https://github.com/ChrisWaites/jax-flows/tree/master/research/dp-flows 24 https://github.com/nv-tlabs/DPDM 25 https://github.com/SaiyueLyu/DP-LDM 26 https://github.com/dihjiang/DP-LFlow Note that the DP training of language models typically falls within DP generative modeling and can be categorized as the \"**within generator**\" category in our taxonomy. However, the training strategies adopted are more akin to the DP training of general (discriminative) models, which is not the primary focus of our work. Hence, these approaches are not exhaustively mentioned in the table above.\n\n## B Additional Notes On Potential Methods With Privacy Barrier B1\n\nIn the DP deep generative modeling literature, existing approaches with privacy barrier between **Real data** and Measurement (Section 4.1) typically release sanitized features in a *condensed and aggregated* form. In this sense, recent approaches, which may deviate from the general \"mean embedding\" formulation (as shown in Equation 3-4), but still publish a sanitized statistical summary of the private dataset, such as **DPSDA** (Lin et al., 2023), fall into this category. Specifically, **DPSDA** sanitizes a count histogram that summarizes the distribution of real data and employs it as a measurement to refine the synthetic data distribution, thereby rendering it more similar to the real private data distribution.\n\nHowever, one might wonder if it is feasible to release a DP database in the *original* form of the real data, *prior* to the training of a generative model. A positive example of this idea can be found in the Small Database Mechanism (**SmallDB**) in the context of private query release, introduced in Section 4.1 of Dwork et al.\n\n(2014). This mechanism outputs a sanitized database in the same form as the original data, by selecting the database (from all possible sets of the data universe) via the exponential mechanism with a utility function of the negative error to the query release problem (difference in the query answer on the synthetic versus the real database). However, as the name suggests, the use of such an algorithm is largely limited to small (low-dimensional) datasets. This is mainly due to the exponential growth of the data universe with dimensionality, which drastically increases the computational burden and undermines the accuracy guarantees.\n\nWhile **DP-GEN** (Chen et al., 2022b) attempted to apply a similar idea to deep generative models, the output space of their generation method only supports (has non-zero probability) combinations of its input private dataset (See detailed proofs in Appendix B of Dockhorn et al. (2022)), instead of the entire data universe. This invalidates their claimed privacy guarantee, and the performance of a proper implementation of such a \"direct database release\" approach on high-dimensional data remains unclear.\n\n## C Additional Sensitivity Analysis C.1 Privacy Barrier B1\n\nSensitivity of DP-Merf (Harder et al., 2021) and the General Formulation in Section 4.1. It can be clearly seen that the ℓ2-sensitivity for the replace-one notion is 2m , where m = |D| represents the size of the private dataset, as demonstrated in the original paper. Subsequently, we proceed to derive a conservative bound for the sensitivity value in the DP-Merf method under the add-or-remove-one DP notion, which can be generalized to other approaches within the same category (Section 4.1), including Liew et al. (2021); Vinaroz et al. (2022); Harder et al. (2022); Yang et al. (2023). For the add-one case, we let m = |D| and assume, without loss of generality, that D′ = *D ∪ {*x\n′\nm+1} and x\n′\ni = xi for all i = 1*, ..., m*.\n\n∆2 = max D,D′ 1 m + 1 mX +1 i=1 ϕ(x ′ i ) − 1 m Xm i=1 ϕ(xi) 2 = max x′m+1,A 1 m + 1 (ϕ(x ′ m+1) + A) − 1 m A 2 = max x′m+1,A 1 (m + 1)m A −1 m + 1 ϕ(x ′ m+1) 2 ≤ max A 1 (m + 1)m A 2 + max x′m+1 1 m + 1 ϕ(x ′ m+1) 2 ≤1 (m + 1)m m +1 m + 1 =2 m + 1\nwhere A =Pm i=1 ϕ(xi) for brevity. The inequalities follow from the triangle inequality and the fact that\n∥ϕ(·)∥2 = 1 Similarly, for the remove-one case, we let m = |D|, D′ ∪ {xm} = D and x\n′\ni = xi for all i = 1*, ..., m* − 1.\n\n∆2 = max D,D′ 1 m − 1 mX−1 i=1 ϕ(x ′ i ) − 1 m Xm i=1 ϕ(xi) 2 = max xm,A 1 m − 1 A − 1 m (A + ϕ(xm)) 2 = max xm,A 1 (m − 1)m A − 1 m ϕ(xm) 2 ≤ max A 1 (m − 1)m A 2 + max xm 1 m ϕ(xm) 2 ≤1 (m − 1)m (m − 1) + 1m = 2 m\nwith A =Pm−1 i=1 ϕ(xi). The inequalities follow from the triangle inequality and the fact that ∥ϕ(·)∥2 = 1 Sensitivity of DP-SWD (Rakotomamonjy & Liva, 2021). The sensitivity is calculated as the maximum difference over two embeddings, determined after performing random projections on two neighboring datasets.\n\nThe \"replace-one\" notion is adopted to simplify the analysis. With a probability of at least 1 − δ, it can be shown that:\n∥XU − X′U∥\n2 F ≤ w(*k, δ*)\nwith w(*k, δ*) = kd +\n2 3 ln 1 δ +\n2 d qk d−1 d+2 ln 1 δ\n. Here X, X′ denote data matrices in R\n|D|×dfor neighboring datasets D, D′ under the bounded-DP notion, while U ∈ R\nd×krepresents the random projection matrix with each column independently drawn from S\nd−1. Additionally, it is ensured that ∥Xi,: − X′\ni,:\n∥2 ≤ 1 for all i by pre-processing the dataset, making each sample record have unit norm. To prove the desired result, the sensitivity is first transformed into a summation of k i.i.d random variables following the beta distribution B(1/2,(d − 1)/2), which then allows the application of Bernstein's inequality to establish concentration bounds for the summation. For a more detailed proof, please refer to Appendix 8.1-8.2 in Rakotomamonjy &\nLiva (2021).\n\nSensitivity of DPSDA (Lin et al., 2023). The core component of DPSDA is the method of constructing a nearest neighbors histogram that describes the real data distribution while providing DP guarantees (refer to Algorithm 2 in Lin et al. (2023)). Specifically, for every real sample xiin the private dataset D, the algorithm identifies its nearest synthetic counterparts and constructs a histogram. This histogram represents the frequency of each existing synthetic sample sk being the closest to the real samples. Given a synthetic dataset consisting of n samples {sk}\nn k=1 and let m = |D|:\n\n$$h_{j}=\\left|i:i\\in[m],j=\\operatorname*{arg\\,min}_{k\\in[n]}d(\\mathbf{x}_{i},\\mathbf{s}_{k})\\right|\\quad{\\mathrm{for}}\\,j=1,...,n$$\n\nwhere h = (h1*, ..., h*n) builds up the histogram with each hj reflecting the number of real samples for which the corresponding synthetic sample sj is the nearest neighbor, based on the distance metric d. Subsequently, DP Gaussian noise is added to the histogram for providing privacy guarantees: h = h + N (0, σI).\n\nFor the add-or-remove-one notion, we can assume that w.l.o.g. the neighboring datasets D, D′satisfy D′ ∪ {xm} = D (or D′ = *D ∪ {*xm}). Let sj be the closest synthetic sample to xm and h, h\n′represent the histograms on D and D′respectively. The ℓ2-sensitivity is then given by:\n\n$$\\begin{array}{l}{{\\Delta^{2}=\\operatorname*{max}_{\\mathcal{D},\\mathcal{D}^{\\prime}}\\|(h_{1},\\cdots,h_{n})-(h_{1}^{\\prime},\\cdots,h_{n}^{\\prime})\\|_{2}}}\\\\ {{\\quad\\quad=\\operatorname*{max}_{h_{j},h_{j}^{\\prime}}\\|(0,...,0,h_{j}-h_{j}^{\\prime},0,...,0)\\|_{2}}}\\\\ {{\\quad\\quad=1}}\\end{array}$$\n\nFor the replace-one notion, we define neighboring datasets D, D′to satisfy D′ ∪ {xm} = *D ∪ {*x\n′\nm} with xm ̸= x\n′\nm. The ℓ2-sensitivity is defined by:\n\n$$\\begin{array}{l}{{\\Delta^{2}=\\operatorname*{max}_{\\mathcal{D},\\mathcal{D}^{\\prime}}\\|(h_{1},\\cdots,h_{n})-(h_{1}^{\\prime},\\cdots,h_{n}^{\\prime})\\|_{2}}}\\\\ {{\\quad=\\operatorname*{max}_{h_{j},h_{j}^{\\prime},h_{k},h_{k}^{\\prime}}\\|(0,...,0,h_{j}-h_{j}^{\\prime},0,...,0,h_{k}-h_{k}^{\\prime},0,...,0)\\|_{2}}}\\\\ {{\\quad=\\sqrt{1^{2}+1^{2}}=\\sqrt{2}}}\\end{array}$$\n\nwhere sj and sk are the closet synthetic samples to xm and x\n′\nm respectively, while w.l.o.g. *j < k*.\n\n## C.2 Privacy Barrier B2\n\nThe sensitivity analysis for methods in this category inherits the approach used in the DP-SGD and the PATE framework, which is presented below.\n\nSensitivity of DP-SGD (Section 2.1.1). The main component of the DP-SGD algorithm can be formalized as follows:\n\n$$\\textrm{Clip:}\\bar{\\mathbf{g}_t}(\\mathbf{x}_i)\\gets\\mathbf{g}_t(\\mathbf{x}_i)/\\max\\left(1,\\frac{\\|\\mathbf{g}_t(\\mathbf{x}_i)\\|_2}{C}\\right)$$ $$\\textrm{Add noise:}\\tilde{\\mathbf{g}_t}\\gets\\frac{1}{B}\\Big(\\sum_i\\bar{\\mathbf{g}_t}(\\mathbf{x}_i)+\\mathcal{N}(0,\\sigma^2C^2\\mathbf{I})\\Big)$$\n\nwhere gt(xi) = ∇θtL(θt, xi) denotes the gradient on sample xi at iteration t, C represents the clipping bound, B is the batch size, σ is the noise scale, and the summation is taken over all samples in the batch.\n\nThe sensitivity in DP-SGD is computed as:\n\n$$\\Delta^{2}=\\operatorname*{max}_{\\mathcal{D},\\mathcal{D}^{\\prime}}\\|\\sum_{i}\\bar{g}_{t}(\\mathbf{x}_{i})-\\sum_{i}\\bar{g}_{t}(\\mathbf{x}_{i}^{\\prime})\\|_{2}$$\n\nFor the add-or-remove-one DP notion, let D′, D only differ in the existence of x\n′ i\n, i.e., D′ = *D ∪ {*x\n′ i\n}, it is easy to see that\n\n$$\\Delta^{2}=\\operatorname*{max}_{\\mathbf{x}_{i}^{\\prime}}\\|{\\bar{\\mathbf{g}}}_{t}(\\mathbf{x}_{i}^{\\prime})\\|_{2}\\leq C$$\n\nFor the replace-one DP notion, w.l.o.g. let D′ ∪ {x\n′ i\n} = *D ∪ {*xi}, thus\n\n$$\\Delta^{2}=\\operatorname*{max}_{\\mathbf{x}_{i}^{\\prime},\\mathbf{x}_{i}}\\|{\\bar{\\mathbf{g}}}_{t}(\\mathbf{x}_{i})-{\\bar{\\mathbf{g}}}_{t}(\\mathbf{x}_{i}^{\\prime})\\|_{2}\\leq2C$$\n\ndue to the triangle inequality.\n\nSensitivity of PATE (Section 2.1.2). Given m teachers, c possible label classes and an input vector x, the \"votes\" of teachers that assign class j to a query input x¯ is denoted as:\n\n$$n_{j}({\\bar{\\mathbf{x}}})=|i:i\\in[m],f_{i}({\\bar{\\mathbf{x}}})=j|\\quad{\\mathrm{for~}}j=1,...,c$$\n\nwith fi denotes the i-th teacher model. And the histogram of the teachers' vote histogram is:\n\n$${\\bar{n}}({\\bar{\\mathbf{x}}})=(n_{1},\\cdots,n_{c})\\in\\mathbb{N}^{c}$$\n\nAs each training data sample only influences a single teacher due to the disjoint partitioning, changing one data sample in the training dataset—whether it's removal, addition, or replacement—will at most alter the votes (by 1) for two classes, denoted here as classes i and j, on any possible query sample x¯. Let the vote histograms resulting from neighboring datasets D, D′ be (n1, · · · , nc) and (n\n′\n1\n, · · · , n′c\n) respectively, the global sensitivity can be represented as:\n\n∆1 = max D,D′ ∥(n1, · · · , nc) − (n ′ 1 , · · · , n′c )∥1 = max ni,n′i ,nj ,n′j ∥(0, ..., 0, ni − n ′ i , 0, ..., 0, nj − n ′ j , 0, ..., 0)∥1 = max ni,n′i |ni − n ′ i | + max nj ,n′j |nj − n ′ j | ≤ 2 ∆2 = max ni,n′i ,nj ,n′j ∥(0, ..., 0, ni − n ′ i , 0, ..., 0, nj − n ′ j , 0, ..., 0)∥2 = max ni,n′i ,nj ,n′j q(ni − n ′ i ) 2 + (nj − n ′ j ) 2 ≤ √2\nThis holds for all possible query samples x¯.\n\nThe ℓ1- and ℓ2-sensitivities calibrate the two variants of noise mechanisms used in PATE: the Gaussian NoisyMax (GNMax) and the max-of-Laplacian (LNMax). The GNMax is defined as:\n\n$$\\operatorname{PATE}_{\\sigma}({\\bar{\\mathbf{x}}})=\\operatorname*{arg\\,max}_{j\\in[c]}\\{n_{j}({\\bar{\\mathbf{x}}})+{\\mathcal{N}}(0,\\sigma^{2})\\}$$\nand the LNMax as:\n$$\\operatorname{PATE}_{\\gamma}({\\bar{\\mathbf{x}}})=\\operatorname*{arg\\,max}_{j\\in[c]}\\{n_{j}({\\bar{\\mathbf{x}}})+L a p(1/\\gamma)\\}$$\n\n## C.3 Privacy Barrier B3\n\nSensitivity of GS-WGAN (Chen et al., 2020a) and DP-Sinkhorn (Cao et al., 2021). The sensitivity for both GS-WGAN and DP-Sinkhorn can be derived via triangle inequality:\n\n$$\\begin{split}\\Delta^{2}&=\\max_{\\mathcal{D},\\mathcal{D}^{\\prime}}\\|f(\\boldsymbol{g}_{G}^{\\mathrm{upstream}})-f(\\boldsymbol{g}_{G}^{\\prime\\mathrm{upstream}})\\|_{2}\\\\ &\\leq\\max_{\\mathcal{D}}\\|f(\\boldsymbol{g}_{G}^{\\mathrm{upstream}})\\|_{2}+\\max_{\\mathcal{D}^{\\prime}}\\|f(\\boldsymbol{g}_{G}^{\\prime\\mathrm{upstream}})\\|_{2}\\\\ &\\leq2C\\end{split}$$\n\nwith f denoting the gradient clipping operation and C the clipping bound. Notably, no matter which privacy notion is used, both terms (maxD ∥f(g upstream G )∥2 and maxD′ ∥f(g\n′\nG\nupstream)∥2) are upper-bounded by the gradient clipping bound C.\n\nSensitivity of DataLens (Wang et al., 2021). Given m teachers, the d-dimensional gradients yielded\nfrom each teacher i after applying top-k sign quantization take the following form (refer to Algorithm 2 in\nWang et al. (2021)):\n$${\\hat{\\mathbf{g}}}_{i}\\in\\{0,1,-1\\}^{d}\\quad{\\mathrm{with}}\\quad\\|{\\hat{\\mathbf{g}}}_{i}\\|_{1}=k\\quad{\\mathrm{and}}$$\n√\n$$||{\\hat{g}}_{i}||_{2}={\\sqrt{k}}$$\nIn other words, gi contains exactly k non-zero elements, with the non-zero elements taking values of either 1 or −1, depending on the sign of the original upstream gradient.\n\nConsider gradient sets {gˆi}\nm i=1 and {gˆ\n′\ni\n}\nm i=1 which originate from neighboring datasets D and D′respectively.\n\nAs the influence of each data point is limited to a single teacher model, these gradient sets differ by at most one element. Without loss of generality, let's assume they diverge in the i-th element. The ℓ2-sensitivity is then computed as follows:\n\n$$\\begin{array}{r l}{\\Delta^{2}=\\operatorname*{max}_{\\mathcal{D},\\mathcal{D}^{\\prime}}\\left\\|\\sum_{i=1}^{m}{\\hat{g}}_{i}-\\sum_{i=1}^{m}{\\hat{g}}_{i}^{\\prime}\\right\\|_{2}}\\\\ {}&{{}=\\operatorname*{max}_{\\hat{g}_{i},\\hat{g}_{i}^{\\prime}}\\left\\|{\\hat{g}}_{i}-{\\hat{g}}_{i}^{\\prime}\\right\\|_{2}}\\\\ {}&{{}\\leq\\|{\\hat{g}}_{i}\\|_{2}+\\|{\\hat{g}}_{i}^{\\prime}\\|_{2}=2{\\sqrt{k}}}\\end{array}$$\n\n## C.4 Privacy Barrier B4\n\nThe sensitivity analysis for methods in this category adheres to the DP-SGD framework. While special considerations may be required to ensure the implementation correctly adheres to this framework, these considerations typically do not alter the sensitivity analysis itself.\n\n## D Additional Background On Privacy Cost Accumulation\n\nTheorem 2.2 (presented in Section 2) provides a straightforward method for calculating the aggregated privacy cost when composing multiple (potentially heterogeneous) DP mechanisms. In this section, we present more details regarding determining the accumulated privacy cost over multiple executions of *sampled Gaussian* mechanisms (Definition D.1).\n\nDefinition D.1 (Sampled Gaussian Mechanism (SGM) (Abadi et al., 2016; Mironov et al., 2019)). Let f be an arbitrary function mapping subsets of D to R\nd. The sampled Gaussian mechanism (SGM) parametrized with the sampling rate 0 < q ≤ 1 and the noise multiplier σ > 0 is defined as\n\n$\\text{SG}_{q,\\sigma}\\stackrel{\\Delta}{=}f\\left(\\{\\pmb{x}:\\pmb{x}\\in\\mathcal{D}\\text{is sampled with probability}q\\}\\right)+\\mathcal{N}(0,\\sigma^2\\pmb{I}_d)$. \nwhere each element of D is sampled independently at random with probability q without replacement.\n\nThe sampled Gaussian mechanism consists of adding i.i.d Gaussian noise with zero mean and variance σ 2 to each coordinate of the true output of f, i.e., SGq,σ injects random vectors from a multivariate isotropic Gaussian distribution N (0, σ2Id) and into the true output, where Id is written as I if unambiguous in the given context.\n\nTheorem D.1. (Mironov et al., 2019) Let SGq,σ be the sampled Gaussian mechanism for some function f with ∆2 f ≤ 1 for any adjacent D, D′ under the add-or-remove-one notion. Then SGq,σ satisfies (*α, ρ*)-RDP if\n\n$$\\rho\\leq D_{\\alpha}\\Big{(}{\\cal N}(0,\\sigma^{2})\\,\\big{\\|}\\,(1-q){\\cal N}(0,\\sigma^{2})+q{\\cal N}(1,\\sigma^{2})\\Big{)}$$ and $\\rho\\leq D_{\\alpha}\\Big{(}(1-q){\\cal N}(0,\\sigma^{2})+q{\\cal N}(1,\\sigma^{2})\\,\\big{\\|}\\,{\\cal N}(0,\\sigma^{2})\\Big{)}$\nTheorem D.1 reduce the problem of proving the RDP bound for SGq,σ to a simple special case of a mixture of one-dimensional Gaussians.\n\nTheorem D.2. (Mironov et al., 2019) Let SGq,σ be the sampled Gaussian mechanism for some function f and under the assumption ∆2 f ≤ 1 for any adjacent D, D′ under the add-or-remove-one notion. Let µ0 denote the pdf of N (0, σ2), µ1 denote the pdf of N (1, σ2), and let µ be the mixture of two Gaussians µ = (1 − q)µ0 + qµ1. Then SGq,σ satisfies (*α, ρ*)-RDP if\n\n$$\\rho\\leq{\\frac{1}{\\alpha-1}}\\log\\left(\\operatorname*{max}\\{A_{\\alpha},B_{\\alpha}\\}\\right)$$\n\nwhere\n\n$$\\begin{array}{l}{{A_{\\alpha}\\stackrel{\\Delta}{=}\\mathbb{E}_{z\\sim\\mu_{0}}[\\left(\\mu(z)/\\mu_{0}(z)\\right)^{\\alpha}]}}\\\\ {{B_{\\alpha}\\stackrel{\\Delta}{=}\\mathbb{E}_{z\\sim\\mu}[\\left(\\mu_{0}(z)/\\mu(z)\\right)^{\\alpha}]}}\\end{array}$$\n\nTheorem D.2 states that applying SGM to a function of sensitivity (Equation 2.3) at most 1 satisfies (*α, ρ*)-\nRDP if ρ ≤1 α−1 log(max{Aα, Bα}). Thus, analyzing RDP properties of SGM is equivalent to upper bounding Aα and Bα.\n\nCorollary D.1. (Mironov et al., 2019) Aα ≥ Bα for any α ≥ 1.\n\nThis allows reformulation of the RDP bound as\n\n$$\\rho\\leq{\\frac{1}{\\alpha-1}}\\log A_{\\alpha}$$\n\nThe Aα can be calculated for a range of α values using the numerically stable computation approach presented in Section 3.3 of Mironov et al. (2019), which is implemented in standard DP packages such as Opacus27 and Tensorflow-privacy28. Then, the smallest Aα (tightest bound) is used to upper bound ρ and later the RDP\nprivacy cost is converted to (*ε, δ*)-DP via Theorem 2.3. Notably, this approach generalizes previous results such as moment accountant (Abadi et al., 2016) (See Table 1 in Mironov et al. (2019) for a summary).\n\n27 https://opacus.ai/ 28 https://github.com/tensorflow/privacy"},"reviews":{"kind":"string","value":"Review 1:\nSummary: This paper provides a unified view of differential privacy for generative models. It is a synthesis paper, that aptly summarizes about 10 years of methods in the intersection of DP and Gen AI into a coherent mathematical framework.\n\nStrengths and Weaknesses: The exposition is clear and thorough. I consider this a strong review paper. The novelty is limited, but I agree that the proposed framework and taxonomy is a strong contribution.\n\nI recommend this paper for acceptance to TMLR. I would consider this paper an solid resource for both academic researchers, industry practioners, and grad students.\n\nRequested Changes: I would recommend that the authors add the following to items to improve this work: A discussion on the similarities and distinctions when applying DP as a in fine-tuning a generative model compared to in pre-training. With generative models now serving as foundation models, it would be helpful to understand the differences along this dimension.\n\nFurthermore, the authors should probably be a bit more thorough concerning DP fine-tuning of transformer-based LLMs. This is an extremely timely topic, as these models are becoming increasingly pervasive.\n\nI do not considering this \"critical\", rather, strongly recommended.\n\nBroader Impact Concerns: NA.\n\n==================================================\n\nReview 2:\nSummary: This paper provides a systemization of the knowledge surrounding differentially private data-synthesis. It starts off by providing primers on differential privacy and generative models, and then moves on to categorizing different DP-synthesis methods based on where the privacy barrier is placed. They provide four such groups: barrier between measure and real data, barrier within measure, barrier between measure and synthetic data and barrier within generator. They then discuss the work that falls under each category.\n\nStrengths and Weaknesses: Strengths:\n1. The visualizations, I think they are all very intuitive, stimulating and to the point. I loved them. They are also pretty accurate and thorough. \n2. The flow of the paper and the categorization: I think the way the paper is written will benefit both experts and non-expert, which is not always the case for survey papers! \n\n\nWeaknesses:\n1. Although the paper claims to unify all DP-synthesis approaches, and to work on different modalities, the work from the NLP literature is missing. See the requested changes for a full list. In general, I think a discussion of modalities and different challenges for each one is needed. \n\n2. The paper is more like a good tutorial, than an SoK paper, as in it doesn't really fully help the reader understand when to use what method. What is next in this field? \n\n3. I think the within metric categorization is redundant. See below.\n\nRequested Changes: 1. Missing all NLP related synthesis references: The paper is missing a discussion on synthesized data in language, in particular the following papers:\n\n[1] Mattern, Justus, et al. \"Differentially Private Language Models for Secure Data Sharing.\" Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing. 2022.\n[2] Xiang Yue, Huseyin Inan, Xuechen Li, Girish Kumar, Julia McAnallen, Hoda Shajari, Huan Sun, David Levitan, and Robert Sim. 2023. Synthetic Text Generation with Differential Privacy: A Simple and Practical Recipe. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)\n[3] Fatemehsadat Mireshghallah, Yu Su, Tatsunori Hashimoto, Jason Eisner, and Richard Shin. 2023. Privacy-Preserving Domain Adaptation of Semantic Parsers. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)\n\nI believe all of these methods would fall under the 'within generator' category. \n\n\n2. The within measure category seems a bit unintuitive for me. Barrier is usually not within a thing, it is between componenets. I think both methods (PATE-GAN and DP-GAN) would be between real-data and measure. Because for PATE, it's actually the noisy output that is fed to the measure, so the noise is not in the measure. Similar case for GAN.\n\nBroader Impact Concerns: NA\n\n==================================================\n\nReview 3:\nSummary: This paper provides a comprehensive review on existing approaches for differentially private deep generative modeling from a unified prespective. Different methods are systematically introduced, categorized, and analyzed on the strength and weakness. Open issues and potential future issues in the field of DP generation methods are also discussed.\n\nStrengths and Weaknesses: ## Strength\n\n1. This paper is a well-written review paper for existing techniques of DP generative models. All key concepts are well introduced with high-quality figure illustrations. The organization and presentation is friendly to broader audiences\n2. The paper provides systematic categorizations of existing techniques under a unified framework, clearly discussing the setting, strength, and weakness of different methods\n3. The paper clearly identifies open questions in the field for future developments\n\n## Weakness\n\nAs the motivation of DP generative model is to enable downstream statistical analysis by sharing a sanitized form of the data, the tradeoff between privacy and utility naturally come into play. Such tradeoff is an important factor for end users to select a proper DP method for their need. Though the paper made extensive discussion on the privacy guarantee of different methods, the implication on utility is not discussed. There also lacks a comprehensive discussion on potential downstream task that can be benefited from the DP generative models. Adding the discussion on downstream task and the utility tradeoff would be especially helpful for practitioners aiming to utilize the DP model.\n\nRequested Changes: See weakness part\n\nBroader Impact Concerns: NA\n\n==================================================\n\nMetareview:\nRecommendation: Accept as is\n\nComment: This submission is recommended for a survey certification on the basis of the content and recommendation of the reviewers.\n\n==================================================\n\n"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":2732,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sInJlcG9JZHMiOlsiNjZkMGZmYjk5NzRjNWM2OTAyNDlhNDczIl0sImlhdCI6MTc5MjIzNjQyMCwianRpIjoiMjZmODg1NGYtNjZlYy00ZGM3LWIzMzMtMTE3OTlkYTQ3ZDQ0Iiwic3ViIjoiL2RhdGFzZXRzL1JlZFRhY2h5b24vdG1sci1vcGVucmV2aWV3IiwiZXhwIjoxNzkyMjQwMDIwLCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.83wLqZRqVt0crgK6I9712zUprIqO67MyMHAcwWUpnhjq7IDZYYm1VONeDatSw_8CRqDm9Q2oO2eoxSImLozxAA","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":2732,"numBytesParquet":136915276}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
paper stringlengths 13.4k 699k | reviews stringlengths 0 38.9k |
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# On The Near-Optimality Of Local Policies In Large Cooperative Multi-Agent Reinforcement Learning
Washim Uddin Mondal *wmondal@purdue.edu* School of IE and CE, Purdue University Vaneet Aggarwal vaneet@purdue.edu School of IE and ECE, Purdue University Satish V. Ukkusuri *sukkusur@purdue.edu* Lyles School of Civil Eng... | Review 1:
Summary: This paper considers a multi-agent optimization problem in which the interactions between agents of of mean field type (this means that that the rewards and behavior of one agent depend on other agents only through their empirical distribution of states and actions). A controller wants to optimize th... |
# Iterative State Estimation In Non-Linear Dynamical Systems Using Approximate Expectation Propagation
Sanket Kamthe∗*sanketkamthe@gmail.com* Department of Computing, Imperial College London Chief Technology Office, JP Morgan Chase So Takao∗so.takao@ucl.ac.uk UCL Centre for Artificial Intelligence, University College ... | Review 1:
Summary: The paper presents an algorithm to approximate the smoothing distribution for a nonlinear stochastic dynamical system. The approximation is in terms of a Gaussian distribution. The algorithm approximates the smoothing distribution p(x_t|y_1,...y_T) with product of certain Gaussian factors, and these ... |
# Correcting Flaws In Common Disentanglement Metrics
Louis Mahon lmahon@ed.ac.uk School of Informatics University of Edinburgh Lei Sha Artificial Intelligence Institute Beihang University Thomas Lukasiewicz Institute of Logic and Computation Vienna University of Technology Reviewed on OpenReview: https://openreview.ne... | Review 1:
Summary: This paper proposes 3 new metrics for assessing the disentanglement of learned representations - one aiming to fix issues in prior metrics with alignment of single neurons, one aiming to fix issues in prior metrics which don't consider entanglement of groups of neurons, and one which leans on a "comp... |
# Dynamic Structure Estimation From Bandit Feedback Using Nonvanishing Exponential Sums
Motoya Ohnishi* *mohnishi@cs.washington.edu* Paul G. Allen School of Computer Science & Engineering University of Washington Isao Ishikawa* *ishikawa.isao.zx@ehime-u.ac.jp* Ehime University RIKEN Center for Advanced Intelligence Pr... | Review 1:
Summary: This paper considers the following dynamical system:
$r_t(x_t)= {f^t(\theta)}^\top x_t + \eta_t,$.
The goals of the paper are:
1. Estimate the length of the period from the collection of rewards with sample efficiency guarantee.
2. Estimate eigenvalues of a linear dynamical system with sample effic... |
# Tackling Visual Control Via Multi-View Exploration Maximization
Anonymous authors Paper under double-blind review
## Abstract
We present MEM: Multi-view Exploration Maximization for tackling complex visual control tasks. To the best of our knowledge, MEM is the first approach that combines multi-view representatio... | Review 1:
Summary: This paper presents a new intrinsic reward approach for deep reinforcement learning (RL). The authors propose to enhance traditional intrinsic reward-driven exploration in RL with multi-view representation learning, which involves representing the environment from multiple viewpoints. The proposed ap... |
# Language Models Can See: Plugging Visual Controls In Text Generation
Anonymous authors Paper under double-blind review
## Abstract
Generative language models (LMs) such as GPT-2/3 can be prompted to generate text with remarkable quality. While they are designed for text-prompted generation, it remains an open ques... | Review 1:
Summary: This paper proposed a training-free framework for image caption generation. More specifically, the authors proposed a new decoding scheme called magic score. Following prior work (Su et al. 2022), which uses contrastive objective as a degeneration penalty, the magic score uses a pre-trained CLIP mode... |
# Understanding Compositionality In Data Embeddings
Anonymous authors Paper under double-blind review
## Abstract
Embeddings are used in AI to represent symbolic structures such as knowledge graphs.
However, the representations obtained cannot be directly interpreted by humans, and may further contain unintended in... | Review 1:
Summary: This paper studied the compositionality of embeddings in word embeddings, sentence embeddings, and knowledge graph embeddings. Although compositionality had previously been studied, the paper argued that no method existed to provide quantitative, statistical tests. The paper proposed two compositiona... |
# Adawavenet: Adaptive Wavelet Network For Time Series Analysis
Anonymous authors Paper under double-blind review
## Abstract
Time series data analysis is a critical component in various domains such as finance, healthcare, and meteorology. Despite the progress in deep learning for time series analysis, there remain... | |
# [Re] Gnninterpreter: A Probabilistic Generative Model-Level Explanation For Graph Neural Networks
Ana-Maria Vasilcoiu *ana-maria.vasilcoiu@student.uva.nl* University of Amsterdam Thijs Stessen thijs.stessen@student.uva.nl University of Amsterdam Thies Kersten thies.kersten@student.uva.nl University of Amsterdam Batu... | Review 1:
Summary: The paper provides a detailed examination of the GNNInterpreter model through the replication of original experiments and the addition of new ones on an extra dataset. It focuses on four main claims about GNNInterpreter’s ability to generate faithful, realistic explanations across various data types,... |
# A Decoder-Only Foundation Model For Time-Series Forecasting
Anonymous authors Paper under double-blind review
## Abstract
Motivated by recent advances in large language models for Natural Language Processing
(NLP), we design a time-series foundation model for forecasting whose out-of-the-box zeroshot performance o... | |
# Interpretable Node Representation With Attribute Decoding
Xiaohui Chen∗xiaohui.chen@tufts.edu Department of Computer Science Tufts University Xi Chen∗xi.chen15@rutgers.edu Department of Computer Science Rutgers University Li-Ping Liu *liping.liu@tufts.edu* Department of Computer Science Tufts University Reviewed on ... | Review 1:
Summary: This work proposes a Probabilistic Matrix Factorization method that uses an inductive (equivariant) GNN as a proposal distribution for the node embedding averages and variances. The method is based on Bayesian hierarchical models with spike and slab priors, so that we node embeddings are separated fr... |
# Some Supervision Required: Incorporating Oracle Policies In Reinforcement Learning Via Epistemic Uncertainty Metrics
Anonymous authors Paper under double-blind review
## Abstract
An inherent problem of reinforcement learning is performing exploration of an environment through random actions, of which a large porti... | Review 1:
Summary: The paper focuses on learning to control in the presence of oracle policies, and how such oracle policies can be used to guide an RL agent's learning. The proposed method simply chooses the oracle policies' actions when the RL agent is uncertain, where uncertainty is w.r.t. the Q function of the agen... |
# Leoclr: Leveraging Original Images For Contrastive Learning Of Visual Representations
Anonymous authors Paper under double-blind review
## Abstract
Contrastive instance discrimination approaches outperform supervised learning in downstream tasks like image classification and object detection. However, these approa... | |
# Introspective Experience Replay: Look Back When Surprised
Ramnath Kumar *ramnathk@google.com* Google Research Dheeraj Nagaraj *dheerajnagaraj@google.com* Google Research Reviewed on OpenReview: *https: // openreview. net/ forum? id= vWTZO1RXZR*
## Abstract
In reinforcement learning (RL), experience replay-based sa... | Review 1:
Summary: This paper presents a novel method for sampling data in the replay buffer for faster convergence of reinforcement learning algorithms. This paper provide detailed survey of existing methods and leverage simple examples to illustrate the benefit of the proposed method. Experiments in simulated domains... |
# Fundamental Problems With Model Editing: How Should Rational Belief Revision Work In Llms?
Anonymous authors Paper under double-blind review
## Abstract
The model editing problem concerns how language models should learn new facts about the world over time. While empirical research on model editing has drawn wides... | |
# Approximations To The Fisher Information Metric Of Deep Generative Models For Out-Of-Distribution Detection
Anonymous authors Paper under double-blind review
## Abstract
Likelihood-based deep generative models such as score-based diffusion models and variational autoencoders are state-of-the-art machine learning m... | Review 1:
Summary: The authors propose a way to perform out-of-distribution (OOD) detection using a pre-trained generative model $p_\theta$. To assess if a point $x$ is OOD, the key idea is to use the magnitude of the gradient of the log-likelihood $\nabla_\theta \log p_\theta(x)$. Instead of using the standard L2 norm... |
# Vision-Language Instruction Tuning: A Review And Analysis
†*ARC Lab, Tencent PCG*
‡*Foundation Technology Center, Tencent PCG*
Reviewed on OpenReview: *https: // openreview. net/ forum? id= ul2tbUPtIQ*
Chen Li †*palchenli@tencent.com* Yixiao Ge †*yixiaoge@tencent.com* Dian Li ‡*goodli@tencent.com* Ying Shan †*yingss... | Review 1:
Summary: This paper aims at the literature review of multimodal large language models (MLLMs), specifically for vision-language instruction tuning (VLIT). They cover various VLIT settings as well as different constructions of VLIT datasets. Built upon this, they conduct experiments and demonstrate performance... |
# Identifiable Causal Inference With Noisy Treatment And No Side Information
Anonymous authors Paper under double-blind review
## Abstract
In some causal inference scenarios, the treatment variable is measured inaccurately, for instance in epidemiology or econometrics. Failure to correct for the effect of this measu... | Review 1:
Summary: This paper studies causal inference under inaccurate observation variables and without side information. To measure the error, a new causal model is proposed. The authors propose a method to train the model and analyze its identifiability. Experiments on synthetic and real-world datasets are conducte... |
# Variable Complexity Weighted-Tempered Gibbs Samplers For Bayesian Variable Selection
Anonymous authors Paper under double-blind review
## Abstract
A subset weighted-tempered Gibbs Sampler (subset-wTGS) has been recently introduced by Jankowiak to reduce the computation complexity per MCMC iteration in high-dimensi... | Review 1:
Summary: This paper considers a variable selection problem for an additive noise linear model.
Authors specifically consider a Bayesian approach where the inclusion of a covariate is expressed by a Berounlli random variable -- the goal is to compute the posterior probability of including a covariate given ob... |
# Fair Evaluation Of Graph Markov Neural Networks
Anonymous authors Paper under double-blind review
## Abstract
Graph Markov Neural Networks (GMNN) have recently been proposed to improve regular graph neural networks (GNN) by including label dependencies into the semi-supervised node classification task. GMNNs do th... | Review 1:
Summary: In this paper the authors release a new dataset named WikiVitals for node classification task. Graph Markov Neural Networks (GMNN) model is applied to the dataset and the authors conduct fair comparison procedure which involves multi-fold cross validation. Ablation study is conducted to evaluate the ... |
# Mdae : Modified Denoising Autoencoder For Missing Data Imputation
Anonymous authors Paper under double-blind review
## Abstract
This paper introduces a methodology based on Denoising AutoEncoder (DAE) for missing data imputation. The proposed methodology, called mDAE hereafter, results from a modification of the l... | Review 1:
Summary: The paper introduces mDAE for missing data imputation. mDAE is a modified version of DAE (Denoising Auto-Encoder) for overcoming the problem that the latter one has when (pre-)training on fully complete data for later making a robust imputation on the missing values of the dataset. The principal stra... |
# Mandera: Malicious Node Detection In Federated Learning Via Ranking
Anonymous authors Paper under double-blind review
## Abstract
Byzantine attacks aim to hinder the deployment of federated learning algorithms by sending malicious gradients to degrade the model. Although the benign gradients and Byzantine gradient... | Review 1:
Summary: This paper considers the problem of defending against byzantine attacks by converting the gradient matrix of the local clients into a ranked matrix. Then the malicious and benign clients can be well-separated using k-means clustering algorithm based on mean and variance of rank for each node. Empiric... |
# Defense Against Multi-Target Backdoor Attacks
Anonymous authors Paper under double-blind review
## Abstract
Neural Trojan/Backdoor attacks pose a significant threat to the current deep-learning-based systems and are hard to defend against due to the lack of knowledge about triggers. In this paper, we first introdu... | Review 1:
Summary: This paper proposes a new version of Badnet, where instead of a single trigger, there are multiple triggers whose locations are not fixed. The authors also propose a defense mechanism against these attacks, demonstrating that their method works effectively.
Strengths and Weaknesses: Weaknesses:
Clar... |
# Unveiling Transformers With Lego: A Synthetic Reasoning Task
Anonymous authors Paper under double-blind review
## Abstract
We propose a synthetic reasoning task, LEGO (Learning Equality and Group Operations),
that encapsulates the problem of following a chain of reasoning, and we study how the Transformer architec... | Review 1:
Summary: The paper introduces LEGO, a synthetic reasoning task requiring a chain of reasoning, and investigates how Transformer architectures learn it. They examine factors like pretraining, dataset composition, and architectural variations. They identify a novel attention pattern and propose that pretraining... |
# Bayesian Methods For Constraint Inference In Reinforcement Learning
Dimitris Papadimitriou *dimitri@berkeley.edu* UC Berkeley Usman Anwar ua237@cam.ac.uk University of Cambridge Daniel S. Brown *daniel.s.brown@utah.edu* University of Utah Reviewed on OpenReview: *https: // openreview. net/ forum? id= oRjk5V9eDp& ref... | Review 1:
Summary: This paper considers the problem of inferring unknown constraint states in a constrained MDP with known transition model, nominal reward function, and finite state and action spaces. The dual formulation of the constrained RL objective leads to an MCMC algorithm where both the constraint states and t... |
# Polter: Policy Trajectory Ensemble Regularization For Unsupervised Reinforcement Learning
Frederik Schubert *schubert@tnt.uni-hannover.de* Institute for Information Processing Leibniz University Hannover Carolin Benjamins *benjamins@ai.uni-hannover.de* Institute of Artificial Intelligence Leibniz University Hannover... | Review 1:
Summary: This paper proposes a regularization technique for Unsupervised Reinforcement Learning (URL) which involves minimizing the KL divergence to an ensemble of policies discovered during the pre-training process. This approach can be easily applied to large set of URL algorithms and is shown to improve th... |
# The Eigenlearning Framework: A Conservation Law Perspective On Kernel Regression And Wide Neural Networks
James B. Simon james.simon@berkeley.edu University of California, Berkeley and Generally Intelligent Madeline Dickens *dickens@berkeley.edu* University of California, Berkeley Dhruva Karkada dkarkada@berkeley.ed... | Review 1:
Summary: This paper develops a unified framework giving closed-form estimates for the test risk and certain generalization metrics of KRR. In particular, it shows that KRR can be interpreted as an explicit competition among kernel eigenmodes. The derived framework is finally used to offer theoretical explanat... |
# Goplan: Goal-Conditioned Offline Reinforcement Learning By Planning With Learned Models
Mianchu Wang∗ *Mianchu.Wang@warwick.ac.uk* University of Warwick Rui Yang∗ryangam@connect.ust.hk Hong Kong University of Science and Technology
Xi Chen *pcchenxi@tsinghua.edu.cn*
Tsinghua University
Hao Sun hs789@cam.ac.uk Uni... | Review 1:
Summary: This paper presents a new method for goal-conditioned offline reinforcement learning that combines a multi-modal policy distribution with an MCTS-based approach for fine-tuning. Experimental results measure the approach's performance on a variety of continuous control environments and illustrate that... |
# Tier: Text-Image Entropy Regularization For Clip-Style Models
Anil Palepu *apalepu@mit.edu* Harvard-MIT Health Sciences & Technology Andrew L. Beam andrew_beam@hms.harvard.edu Harvard University
## Abstract
In this paper, we study the effect of a novel regularization scheme on contrastive languageimage pre-trained... | Review 1:
Summary: This paper presents a new learning strategy to further improve the popular vision-language pre-trained model CLIP. In particular, authors propose to include two regularization terms on the text-patch embeddings, motivated by the fact that, for a given text prompt, only a few patches should be highly ... |
# A Novel Information Theoretic Objective To Disentangle Representations For Fair Textual Classification
Anonymous authors Paper under double-blind review
## Abstract
One of the pursued objectives of deep learning is to provide tools that learn abstract representations of reality from the observation of multiple con... | Review 1:
Summary: The paper proposes a new objective to learn the fair classifier. Specifically, the classic minimax objective is $I(Z,Y)-\lambda I(Z,S)$ where $Z$ is the representation $Y$ is the class distribution and $S$ is the sensitive attribute. The proposed objective is $I(Z,Y)-\lambda I(Z,S|Y)$.
Strengths and... |
# From Promise To Practice: A Study Of Common Pitfalls Behind The Generalization Gap In Machine Learning
Anonymous authors Paper under double-blind review
## Abstract
The world of Machine Learning (ML) offers great promise, but often there is a noticeable gap between claims made in research papers and the model's pr... | |
# Mmd-Regularized Unbalanced Optimal Transport
Piyushi Manupriya *cs18m20p100002@iith.ac.in* Department of Computer Science and Engineering, IIT Hyderabad, INDIA.
J. SakethaNath saketha@cse.iith.ac.in Department of Computer Science and Engineering, IIT Hyderabad, INDIA.
Pratik Jawanpuria *pratik.jawanpuria@microsoft... | Review 1:
Summary: The authors study unbalanced optimal transport as a tool to compare probability measures, which is a central task in machine learning. Unbalanced optimal transport (UOT) is a variant of the standard Kantorovich problem, wherein the hard constraints on the transport plans are replaced with soft regula... |
# Test-Time Recalibration Of Conformal Predictors Under Distribution Shift Based On Unlabeled Examples
Anonymous authors Paper under double-blind review
## Abstract
Modern image classifiers are very accurate, but the predictions come without uncertainty estimates. Conformal predictors provide uncertainty estimates b... | Review 1:
Summary: The paper proposes a simple strategy for calibrating the threshold of conformal predictors under distribution shift. The method relies on calculating the quantiles of an unlabeled set and is easy to compute. The results are compared to a regression-based baseline.
Strengths and Weaknesses: ### Stren... |
# Differential Equation Scaling Limits Of Shaped And Unshaped Neural Networks
Mufan (Bill) Li mufan.li@princeton.edu Princeton University Mihai Nica nicam@uoguelph.ca University of Guelph and Vector Institute Reviewed on OpenReview: *https: // openreview. net/ forum? id= iRDwUXYsSJ*
## Abstract
Recent analyses of ne... | Review 1:
Summary: This article explores the behavior of neural networks with unshaped activations, where the activation function remains constant regardless of network size. Two main findings are presented:
1. Convergence of Two Architectures: It demonstrates that two different network architectures converge to the s... |
# New Guarantees For **Learning Revenue Maximizing Menus** Of Lotteries And Two-Part Tariffs
Maria-Florina Balcan ninamf@cs.cmu.edu Carnegie Mellon University Hedyeh Beyhaghi hedyeh@cmu.edu Carnegie Mellon University Reviewed on OpenReview: *https: // openreview. net/ forum? id= mhawjZcmrJ*
## Abstract
We advance a ... | Review 1:
Summary: This work investigates the learnability of two classes of economic mechanisms, namely menus of lotteries and two-part tariffs, at the intersection of learning theory and computational economics. The authors concentrate on learning high-revenue mechanisms of this type from buyer valuation data in both... |
# Gradient Masked Averaging For Federated Learning
Irene Tenison∗,1,∇**, Sai Aravind Sreeramadas**2,3, Vaikkunth Mugunthan1**, Edouard Oyallon**4, Irina Rish2,3**, and Eugene Belilovsky**3,5 1Massachussetts Institute of Technology, MA, USA, 2Universite de Montreal, Quebec, Canada, 3Mila, Quebec AI Institute, Canada, 4... | Review 1:
Summary: This paper provides a new aggregation scheme that can be used by various federated learning algorithms. The proposed method hinges upon soft masking of the local gradients prior to updating the global model. The authors then conduct extensive experimentation showing the efficacy of their approach and... |
# Reproducibility Study On Adversarial Attacks Against Robust Transformer Trackers
Fatemeh Nourilenjan Nokabadi fatemeh.nourilenjan-nokabadi.1@ulaval.ca IID, Université Laval & *Mila* Jean-François Lalonde *jflalonde@gel.ulaval.ca* IID, Université Laval Christian Gagné christian.gagne@gel.ulaval.ca IID, Université Lav... | Review 1:
Summary: This paper presents a study on the reproducibility of existing attack approaches against transformer trackers, aiming to understand their behavior under adversarial conditions and how these attacks affect tracking accuracy. By conducting experiments across different perturbation levels on the VOT2022... |
# A Survey On Causal Discovery Methods For I.I.D. And Time Series Data
Uzma Hasan uzmahasan@umbc.edu Causal AI Lab Department of Information Systems University of Maryland, Baltimore County Baltimore, MD, USA
Emam Hossain *emamh1@umbc.edu* Causal AI Lab Department of Information Systems University of Maryland, Baltimo... | Review 1:
Summary: This survey paper focuses on reviewing causal discovery methods for temporal and non-temporal data. Overall, this paper well presents the basic concepts, terminologies, and assumptions for causal discoveries, and provides a good collection of existing methods with details for both i.i.d. and time ser... |
# Unifying The Perspectives Of Nlp And Software Engineering: A Survey On Language Models For Code
Anonymous authors Paper under double-blind review
## Abstract
In this work we systematically review the recent advancements in software engineering with language models, covering 70+ models, 40+ evaluation tasks, 180+ d... | Review 1:
Summary: The paper presents a comprehensive literature review about the current research on text-code-related problems (e.g. text-to-code generation, code retrieval). The review covers an impressive number of things: 50+ models, 30+ evaluation tasks, 170+ datasets, and 700+ related works. The paper introduces... |
# Symbolic Regression On Probabilistic Flows For Network System Modeling
Anonymous authors Paper under double-blind review
## Abstract
Real-world complex systems often miss high-fidelity physical descriptions and are typically subject to partial observability. Learning the dynamics of such systems is a challenging a... | |
# A Survey On Data Selection For Language Models
Alon Albalak,* *UC Santa Barbara, SynthLabs* Yanai Elazar, *Allen Institute for AI, University of Washington* Sang Michael Xie, *Stanford University* Shayne Longpre, *Massachusetts Institute of Technology* Nathan Lambert, *Allen Institute for AI*
Xinyi Wang, *UC Santa B... | Review 1:
Summary: This paper systematically studies the data selection mechanism in language models across different stages, from pre-training to fine-tuning. It gives a thorough review to understand the techniques of data selection, including data quality, selection mechanism and so on. Generally, it summarizes a lot... |
# Cheap And Deterministic Inference For Deep State-Space Models Of Interacting Dynamical Systems
Andreas Look andreas.look@bosch.com Bosch Center for Artificial Intelligence Melih Kandemir kandemir@imada.sdu.dk University of Southern Denmark Barbara Rakitsch *barbara.rakitsch@bosch.com* Bosch Center for Artificial Int... | Review 1:
Summary: The authors introduce Graph Deep State Space Models (GDSSM) to learn state space models over $M$ interacting agents. The contributions of this work are twofold: 1) the combination of classical deep-state space models with graph neural networks and 2) a deterministic training scheme based on moment ma... |
# Low-Regret Active Learning
Anonymous authors Paper under double-blind review
## Abstract
We develop an online learning algorithm for identifying unlabeled data points that are most informative for training (i.e., active learning). By formulating the active learning problem as the prediction with sleeping experts p... | Review 1:
Summary: This paper considers the problem of active learning: choosing informative points to label in a sequential fashion to learn a classifier that is as good as one trained on the full dataset. To do so, the authors rely on the scheme of information gain maximization, which greedily labels points based whi... |
# Hierarchical Vae With A Diffusion-Based Vampprior
Anonymous authors Paper under double-blind review
## Abstract
Deep hierarchical variational autoencoders (VAEs) are powerful latent variable generative models. In this paper, we introduce Hierachical VAE with Diffusion-based Variational Mixture of the Posterior Pri... | |
# Unsupervised Federated Domain Adaptation For Segmentation Of Mri Images
Anonymous authors Paper under double-blind review
## Abstract
Automatic semantic segmentation of magnetic resonance imaging (MRI) images using deep neural networks greatly assists in evaluating and planning treatments for various clinical appl... | Review 1:
Summary: The paper proposes an unsupervised domain adaptation algorithm to improve the lesion segmentation performance on the target dataset. The algorithm adds a regularization term of the distance between source and target embedding distributions. It also ensemble the outputs of models trained from multiple... |
# Isr: Invertible Symbolic Regression
Anonymous authors Paper under double-blind review
## Abstract
We introduce an Invertible Symbolic Regression (ISR) method, a machine learning technique that generates analytical relationships between inputs and outputs of a given dataset via invertible maps (or architectures). T... | Review 1:
Summary: The paper introduces Invertible Symbolic Regression (ISR), which merges concepts from Invertible Neural Networks (INNs) and Equation Learner (EQL) to tackle interpretability in inverse problems. ISR aims to derive interpretable mathematical relationships between system parameters and observable quant... |
# Object-Aware Cropping For Self-Supervised Learning
Shlok Mishra1, Anshul Shah2, Ankan Bansal1, Janit Anjaria1, Abhyuday Jagannatha3, Abhishek Sharma, David Jacobs1**, Dilip Krishnan**4 1**University of Maryland, College Park,**
2Johns Hopkins University, 3University of Massachusetts Amherst 4**Google Research**
{shl... | Review 1:
Summary: The paper proposes a self-supervised representation learning method to learn from complex (multi-object) datasets, like OpenImages and COCO, which are more representative of real-world uncurated data. The proposed method optimizes three terms simultaneously: 1) a contrastive term, 2) a rotation predi... |
# Weight-Balancing Fixes And Flows For Deep Learning
Lawrence K. Saul lsaul@flatironinstitute.org Flatiron Institute, Center for Computational Mathematics 162 Fifth Avenue, New York, NY 10010 Reviewed on OpenReview: *https: // openreview. net/ forum? id= uaHyXxyp2r*
## Abstract
Feedforward neural networks with homog... | Review 1:
Summary: This paper makes two contributions to our understanding of feedforward neural networks with homogeneous activation functions. The first is a procedure for "gauge-fixing" - computing multiplicative rescaling factors at each hidden unit that rebalance the weights of these networks without altering the ... |
# Autoencoding Reduced Order Models For Control Through The Lens Of Dynamic Mode Decomposition
Anonymous authors Paper under double-blind review
## Abstract
Modeling and control of high-dimensional dynamical systems often involve some dimensionality reduction techniques to construct a lower-order model that makes th... | Review 1:
Summary: The paper aims to solve the problem of controlling a dynamical system by projecting observations to low-dimensional latent state space, solving the control problem there, and projecting back to the observation space in an auto-encoder fashion.The paper adopts ideas from three well-established methods... |
# Your Classifier Can Be Secretly A Likelihood-Based Ood Detector
Anonymous authors Paper under double-blind review
## Abstract
The ability to detect out-of-distribution (OOD) inputs is critical to guarantee the reliability of classification models deployed in an open environment. A fundamental challenge in OOD
dete... | |
# Self-Supervised Learning For Segmentation And Quantification Of Dopamine Neurons In Parkinson'S Disease
Fatemeh Haghighi fhaghigh@asu.edu Arizona State University, Tempe, AZ 85281, USA
Department of Neuroscience, Genentech Inc., South San Francisco, CA 94080, USA
Soumitra Ghosh∗ghoshs29@gene.com Department of Neuros... | Review 1:
Summary: In this paper, the authors proposed a novel method based on self-supervised learning for segmentation and quantification of dopamine neurons in Parkinson’s disease. By leveraging info extracted from natural images and medical data, they use self-supervised pre-training to transfer these knowledge to ... |
# Offline Reinforcement Learning With Mixture Of Deterministic Policies
Takayuki Osa osa@mi.t.u-tokyo.ac.jp The University of Tokyo, RIKEN
Akinobu Hayashi *akinobu_hayashi@jp.honda* Honda R&D Co., Ltd.
Pranav Deo *pranav_deo@jp.honda* Honda R&D Co., Ltd.
Naoki Morihira naoki_morihira@jp.honda Honda R&D Co., Ltd.
Ta... | Review 1:
Summary: In the offline reinforcement learning (RL) setting, it has been observed that querying the value of out-of-distribution (OOD) actions may cause an accumulation of bootstrapping errors. To address this issue, existing works may penalize deviation from behavior policy or underestimate OOD actions.
Th... |
# Learning To Switch Among Agents In A Team Via 2**-Layer** Markov Decision Processes
Vahid Balazadeh *vahid@cs.toronto.edu* University of Toronto Abir De *abir@cse.iitb.ac.in* Indian Institute of Technology Bombay Adish Singla *adishs@mpi-sws.dot.org* Max Planck Institute for Software Systems Manuel Gomez Rodriguez *... | Review 1:
Summary: In this work, the authors proposed a MDP framework to study an RL problem that is less autonomous. In particular, they restrcit the problem formulation into one that allow allows the RL agent to take some of the actions and
leave the remaining ones to human agents, and argue that the resulting perfor... |
# Active Learning For Level Set Estimation Using Randomized Straddle Algorithms
Anonymous authors Paper under double-blind review
## Abstract
Level set estimation (LSE) the problem of identifying the set of input points where a function takes a value above (or below) a given threshold is important in practical appli... | |
# Noncommutative C ∗**-Algebra Net: Learning Neural Networks With Powerful Product Structure In** C ∗**-Algebra**
Anonymous authors Paper under double-blind review
## Abstract
We propose a new generalization of neural networks with noncommutative C
∗-algebra. An important feature of C
∗-algebras is their noncommutat... | Review 1:
Summary: This paper introduces neural network architectures whose parameters belong to non-commutative C* -algebras. The relevant feature of C* algebras for deep learning is their product structure, which permits potentially interesting interactions between input data. This paper proposes some examples of non... |
# Accelerated Quality-Diversity Through Massive Parallelism
Bryan Lim bryan.lim16@imperial.ac.uk Department of Computing Imperial College London Maxime Allard *m.allard20@imperial.ac.uk* Department of Computing Imperial College London Luca Grillotti luca.grillotti16@imperial.ac.uk Department of Computing Imperial Coll... | Review 1:
Summary: In this paper the authors introduce an study on a subclass of evolutionary algorithms, notably the Quality-Diversity optimizations. The study focuses on hardware acceleration, parallelization and quality related changes due to highly parallel executions. The paper is well written, with appropriate ba... |
# Vision-Language Dataset Distillation
Xindi Wu1 Byron Zhang1 Zhiwei Deng2 **Olga Russakovsky**1 1Princeton University 2Google DeepMind
{xindiw, zishuoz, olgarus}@princeton.edu, zhiweideng@google.com Reviewed on OpenReview: *https: // openreview. net/ forum? id= 6L6cD65pot* Website: *https: // princetonvisualai. githu... | Review 1:
Summary: The submission gives an overview of a method for distilling a dataset for vision-language pretraining. The method is evaluated on a number of vision-language tasks, and the results are compared with several core-set methods. The method is shown to outperform the core-set methods by a substantial marg... |
# Maximally Expressive Gnns For Outerplanar Graphs
Anonymous authors Paper under double-blind review
## Abstract
We propose a *linear time* graph transformation that enables the Weisfeiler-Leman (WL) algorithm and message passing graph neural networks (MPNNs) to be maximally expressive on *outerplanar* graphs. Our a... | |
# Exploring Efficient Few-Shot Adaptation For Vision Transformers
Chengming Xu *cmxu18@fudan.edu.cn* School of Data Science, Fudan University Siqian Yang *seasonsyang@tencent.com* Yabiao Wang caseywang@tencent.com Youtu Lab, Tencent Zhanxiong Wang *maxzxwang@tencent.com* Tencent Yanwei Fu∗*yanweifu@fudan.edu.cn* Xiang... | Review 1:
Summary: This paper proposed a finetuning method for FSL, namely eTT, with two key components APT and DRA. By using eTT, finetuned ViT achieved consistently good performances on meta datasets, compared to ResNet-18 and -34 models.
Strengths and Weaknesses: Strength:
1. This work proposed an APT module that ... |
# A New Stochastic Optimization Technique For Combating Data Poisoning Attacks
Anonymous authors Paper under double-blind review
## Abstract
In this paper, we present new techniques for building and analyzing robust stochastic optimization algorithms. To solve the given d-dimensional optimization problem, our techni... | Review 1:
Summary: The paper studies solving large-scale optimization problems with an optimization variable in $\mathbb{R}^d$ by iteratively solving the problem over random subsets in a smaller dimension $k$ (Algorithm 1). The paper has results on the convergence analysis of the proposed algorithm using an assumption ... |
# Holistic Evaluation Of Language Models
Percy Liang†, Rishi Bommasani†, Tony Lee†, Dimitris Tsipras‡, Dilara Soylu‡, Michihiro Yasunaga‡, Yian Zhang‡, Deepak Narayanan‡, Yuhuai Wu‡, Ananya Kumar, Benjamin Newman, Binhang Yuan, Bobby Yan, Ce Zhang, Christian Cosgrove, Christopher D. Manning, Christopher Ré, Diana Acos... | Review 1:
Summary: The authors present an innovative approach, Holistic Evaluation of Language Models (HELM), which aims to enhance the transparency and understanding of large language models through comprehensive evaluation across various scenarios and metrics. HELM's abstract taxonomy allows for core and targeted ass... |
# Max-Affine Spline Insights Into Deep Network Pruning
Haoran You†∗, Randall Balestriero‡∗, Zhihan Lu†, Yutong Kou§, **Huihong Shi**⋄,
Shunyao Zhang†, Shang Wu†, **Yingyan Lin**†, and **Richard Baraniuk**†
†Rice University ‡Meta AI Research
§Huazhong University of Science and Technology ⋄Nanjing University
†{haoran.yo... | Review 1:
Summary: **Summary**
This paper proposes a new perspective on understanding neural network pruning. The paper employs spline theory to analyze how the decision boundaries of neural networks evolve over training and how they are removed when we apply pruning. The paper exposes an early-bird phenomenon in whic... |
# Mixture Degree-Corrected Stochastic Block Model For Multigroup Community Detection In Multiplex Graphs
Anonymous authors Paper under double-blind review
## Abstract
Multiplex graphs have emerged as a powerful tool for modeling complex data structures due to their ability to handle multiple relational layers. Clust... | |
# Federated Minimax Optimization With Client Heterogeneity
Pranay Sharma pranaysh@andrew.cmu.edu Department of Electrical and Computer Engineering Carnegie Mellon University Rohan Panda rohanpan@andrew.cmu.edu Department of Electrical and Computer Engineering Carnegie Mellon University Gauri Joshi *gaurij@andrew.cmu.e... | Review 1:
Summary: This paper considers optimizing a nonconvex minimax objective in a federated setting where the data distributions and system capabilities vary among participating data holders. The authors propose a general communication efficient (via periodic averaging) method that facilitates an unequal number of... |
# A Simple Convergence Proof Of Adam And Adagrad
Alexandre Défossez defossez@meta.com Meta AI
Léon Bottou Meta AI
Francis Bach INRIA / PSL
Nicolas Usunier Meta AI
Reviewed on OpenReview: *https: // openreview. net/ forum? id= ZPQhzTSWA7*
## Abstract
We provide a simple proof of convergence covering both the Adam and... | Review 1:
Summary: The paper proposes a "simple" proof for Adam and Adagrad for smooth objective functions with bounded gradients. The link between Adam and Adagrad is shown via this proof, along with experimental results with a toy problem and CIFAR10 dataset. The paper has the potential to be impactful for researcher... |
# A Simple And Efficient Method For Random Fourier Features
Anonymous authors Paper under double-blind review
## Abstract
Random Fourier features and random projection involve matrix multiplication with a k × D
random matrix, the original dimensionality D and the dimensionality in the projected space k. Large values... | |
# Recognition Models To Learn Dynamics From Partial Observations With Neural Odes
Mona Buisson-Fenet mona.buisson@minesparis.psl.eu Ansys Research Team, Ansys France Centre Automatique et Systèmes, Mines Paris - PSL University Institute for Data Science in Mechanical Engineering - RWTH Aachen University Valery Morgent... | Review 1:
Summary: The paper proposes a model learning technique that can learn/estimate the initial hidden state. To estimate the initial state, the paper uses a KKL-based recognition model. In the experiments, the proposed method is applied to learn dynamics models on three different datasets, the earthquake model, t... |
# An Attribute-Based Method For Video Anomaly Detection
Anonymous authors Paper under double-blind review
## Abstract
Video anomaly detection (VAD) identifies suspicious events in videos, which is critical for crime prevention and homeland security. In this paper, we propose a simple but highly effective VAD method ... | |
# Omnipred: Towards Language Models As Universal Regressors
Anonymous authors Paper under double-blind review
## Abstract
Over the broad landscape of experimental design, regression has been a powerful tool to accurately predict the outcome metrics of a system or model given a set of parameters, but has been traditi... | |
# Understanding Linearity Of Cross-Lingual Word Embedding Mappings
Xutan Peng †*x.peng@shef.ac.uk*
Chenghua Lin †*c.lin@shef.ac.uk*
Mark Stevenson † *mark.stevenson@shef.ac.uk* Chen Li ‡*palchenli@tencent.com*
†*Department of Computer Science, The University of Sheffield*
‡*Applied Research Center, Tencent PCG*
Revi... | Review 1:
Summary: This paper is motivated by the gap in understanding the conditions under which the relation ship between word embedding models of different language can be captured by a linear mapping. The authors argue that the existence of the linear relationship is explained by the similarity of the analogy struc... |
# Variational Learning Ista
Fabio Valerio Massoli *fmassoli@qti.qualcomm.com* Qualcomm AI Research∗
Christos Louizos *clouizos@qti.qualcomm.com* Qualcomm AI Research∗
Arash Behboodi abehboodi@qti.qualcomm.com Qualcomm AI Research∗
Reviewed on OpenReview: *https: // openreview. net/ forum? id= AQk0UsituG*
## Abstract
... | Review 1:
Summary: The paper proposes two major algorithms: (1) Augmented Dictionary Learning ISTA (A-DLISTA), which uses an augmentation
module to adaptively learn the parameters according to the sensing matrix; (2) Variational Learning ISTA (VLISTA) which can learn dictionary distributions in the variational framewor... |
# Towards Better Generalization With Flexible Representation Of Multi-Module Graph Neural Networks
Hyungeun Lee didls1228@hanyang.ac.kr Department of Electronic Engineering Hanyang University Kijung Yoon kiyoon@hanyang.ac.kr Department of Electronic Engineering Hanyang University Reviewed on OpenReview: *https: // ope... | Review 1:
Summary: This paper is concerned with how the graph structure influences the generalization of GNNs. OOD of GNNs is considered and is verified to have high correlation with the average degree of graphs by the random graph generators.
Strengths and Weaknesses: The papers combine multiple update modules and m... |
# Interventions In Graph Neural Networks Lead To New Neural Causal Models
Anonymous authors Paper under double-blind review
## Abstract
Causality can be described in terms of a structural causal model (SCM) that carries information on the variables of interest and their mechanistic relations. For most processes of i... | Review 1:
Summary: Authors propose using GNNs in neural causal models. They provide a set of definitions to better connect SCMs with GNNs both in observational and interventional layers of the hierarchy.
Strengths and Weaknesses: Please see below for my detailed comments and concerns. In short, the paper contains a n... |
# Federated Learning With Heterogeneous Differential Privacy
Anonymous authors Paper under double-blind review
## Abstract
Federated learning (FL) takes a first step towards privacy-preserving machine learning by training models while keeping client data local. Models trained using FL may still **indirectly**
leak p... | Review 1:
Summary: The paper consider a modification of a DP Federated Averaging algorithm, which takes into account the scenario, where the clients may have varying levels of (eps,delta)-privacy guarantees, and where there may be a small fraction of clients without privacy guarantees at all. The idea is simple: in the... |
# Discffusion: Discriminative Diffusion Models As Few-Shot Vision And Language Learners
Anonymous authors Paper under double-blind review
## Abstract
Diffusion models, such as Stable Diffusion (Rombach et al., 2022a), have shown incredible performance on text-to-image generation. Since text-to-image generation often... | Review 1:
Summary: The paper introduces a novel method, Discriminative Stable Diffusion (Discffusion), which repurposes pre-trained text-to-image diffusion models for few-shot discriminative tasks, specifically image-text matching. The method leverages the cross-attention scores from pretrained diffusion models to eval... |
# Cr-Moe: Consistent Routed Mixture-Of-Experts For Scaling Contrastive Learning
Ziyu Jiang *jiangziyu@tamu.edu* Texas A&M University Guoqing Zheng *zheng@microsoft.com* Microsoft Research Yu Cheng chengyu@cse.cuhk.edu.hk The Chinese University of Hong Kong Ahmed Hassan Awadallah hassanam@microsoft.com Microsoft Resear... | Review 1:
Summary: This paper proposed Consistent Routed Mixture-of-Experts (CR-MoE) for self-supervised contrastive learning (CL). Based on the native combination of CL and MoE, this paper introduced consistent routing regularization, which encourages the corresponding tokens to be routed to the same set of experts.
... |
# Generative Models Are Self-Watermarked: Declaring Model Authentication Through Re-Generation
Anonymous authors Paper under double-blind review
## Abstract
As machine- and AI-generated content proliferates, protecting the intellectual property of generative models has become imperative, yet verifying data ownership... | Review 1:
Summary: This paper focuses on the problem of watermarking or identifying if the a given image was generated by a blackbox GenAI model. Specifically, it focuses on image generation, and three text generation tasks: machine translation, paraphrasing, and summarization. The task is formulated as being able to v... |
# Directional Privacy For Deep Learning
Anonymous authors Paper under double-blind review
## Abstract
Differentially Private Stochastic Gradient Descent (DP-SGD) is a key method for applying privacy in the training of deep learning models. This applies isotropic Gaussian noise to gradients during training, which can... | Review 1:
Summary: This paper applies the von Mises-Fisher (VMF) mechanism to stochastic gradient descent. The paper calls the resulting SGD method DirDP-SGD because the VMF mechanism generally can guarantee directional differential privacy (DirDP).
In order to compare the performance of the DirDP-SGD with DP-SGD (wh... |
# Towards General Purpose Vision Foundation Models For Medical Image Analysis: An Experimental Study Of Dinov2 On Radiology Benchmarks
Anonymous authors Paper under double-blind review
## Abstract
The integration of deep learning systems into healthcare has been hindered by the resourceintensive process of data anno... | Review 1:
Summary: The paper explores the applicability of DINOv2 for medical image analysis, specifically focusing on radiology benchmarks.
Strengths and Weaknesses: # Strengths
The paper proposes leveraging a general-purpose vision foundation model, DINOv2 and fine-tuning it for medical image analysis. The authors h... |
# Scaling Vision-And-Language Navigation With Offline Rl
Valay Bundele†*valaybundele@gmail.com* University of Tübingen Mahesh Bhupati mail2mahesh0537@gmail.com Indian Institute of Technology Bombay Biplab Banerjee *bbanerjee@iitb.ac.in* Indian Institute of Technology Bombay Aditya Grover adityag@cs.ucla.edu University... | Review 1:
Summary: The authors present VLN-ORL, a simple reward-conditioning objective for VLN that adds a dense L2-distance based reward signal to condition models on so that they can learn from suboptimal data. They also introduce a new dataset/benchmark to evaluate VLN agents in simulation with suboptimal datasets.
... |
# Atnas: Automatic Termination For Neural Architecture Search
Anonymous authors Paper under double-blind review
## Abstract
Neural architecture search (NAS) is a framework for automating the design process of a neural network structure. While the recent one-shot approaches have reduced the search cost, there still e... | Review 1:
Summary: The paper propose a new early stopping criterion for neural architecture search. This allow to reduce the computation cost of NAS but can also be used to reduce the overfitting of this approaches.
Strengths and Weaknesses: Strengths:
- The paper is well written and easy to follow.
- The idea is in... |
# Rignn: A Rationale Perspective For Semi-Supervised Openworld Graph Classification
Xiao Luo1*xiaoluo@cs.ucla.edu*
Yusheng Zhao2*yusheng.zhao@stu.pku.edu.cn*
Zhengyang Mao2*zhengyang.mao@stu.pku.edu.cn*
Yifang Qin2*qinyifang@pku.edu.cn*
Wei Ju2*juwei@pku.edu.cn* Ming Zhang2 *mzhang_cs@pku.edu.cn*
Yizhou Sun1*yzsu... | Review 1:
Summary: The paper tackles the open-world semi-supervised graph classification where unlabeled graphs could come from unseen classes. The authors propose CIGNN, which takes a causal perspective to detect components containing the most information related to the label space and classify unlabeled graphs into k... |
# Normalized/Clipped Sgd With Perturbation For Differentially Private Non-Convex Optimization
Anonymous authors Paper under double-blind review
## Abstract
By ensuring differential privacy in the learning algorithms, one can rigorously mitigate the risk of large models memorizing sensitive training data. In this pap... | Review 1:
Summary: This paper proposes and analyzes two algorithms, DP-SGD and DP-NSGD, for training deep learning models with differential privacy guarantees. Both algorithms clip or normalize the per-sample gradients to limit sensitivity before adding noise for privacy. The paper provides convergence analysis for non... |
# Deep Operator Learning Lessens The Curse Of Dimensionality For Pdes
Ke Chen kechen@umd.edu Department of Mathematics University of Maryland, College Park Chunmei Wang chunmei.wang@ufl.edu Department of Mathematics University of Florida Haizhao Yang hzyang@umd.edu Department of Mathematics University of Maryland, Col... | Review 1:
Summary: This paper presents an analysis to explain how using deep neural networks to learn a class of PDEs have demonstrated success in the field. in particular, the merits of DNN in reducing the sample/computational complexity for a class of PDEs whose operator functions are Lipschitz continuity and exhibit... |
# When Is Momentum Extragradient Optimal? A Polynomial-Based Analysis
Junhyung Lyle Kim⋆jlylekim@rice.edu Rice University, Department of Computer Science Gauthier Gidel gauthier.gidel@umontreal.ca Université de Montréal, Department of Computer Science and Operations Research Mila - Quebec AI Institute, Canada CIFAR AI... | Review 1:
Summary: This paper analyzes the momentum extragradient (MEG) method for differentiable games using residual polynomial.
It is shown that the convergence rate of the MAG can be characterized by the residual polynomial (Theorem 1),
under the assumption that the vector field is given by an affine function.
Base... |
# How To Train Your Vit? Data, Augmentation, And Regularization In Vision Transformers
Andreas Steiner∗*andstein@google.com* Alexander Kolesnikov∗*akolesnikov@google.com* Xiaohua Zhai∗*xzhai@google.com* Jakob Uszkoreit *usz@google.com* Google Research, Brain Team, Zürich
∗ Equal technical contribution, †*independent r... | Review 1:
Summary: The paper conducts extensive experiments, e.g., training 50,000 models under different settings, to investigate the effects of different training recipes and their combinations for training Vision Transformers. Specific aspects include regularization, augmentation, model size, and training data size... |
# Understanding Convolution On Graphs Via Energies
Francesco Di Giovanni∗*fd405@cam.ac.uk* University of Cambridge James Rowbottom∗*jr908@cam.ac.uk* University of Cambridge Benjamin P. Chamberlain Charm Therapeutics Thomas Markovich Cash App Michael M. Bronstein University of Oxford Reviewed on OpenReview: *https: // ... | Review 1:
Summary: This paper proposes an analysis and re-evaluation of basic MPNNs such as GCN via an energetic perspective. The paper shows that by setting the appropriate weights in models like GCN, they can be viewed as a gradient flow that is either smoothing (a commonly known observation), but also over-sharpenin... |
# Bayesian Optimization With Derivatives Acceleration
Guillaume Perrin *guillaume.perrin@univ-eiffel.fr* Université Gustave Eiffel, COSYS, 5 Bd Descartes 77454 Marne-La-Vallée, France Rodolphe Leriche **leriche@emse.fr**
LIMOS (CNRS, Mines Saint-Etienne, UCA) Saint-Etienne, France Reviewed on OpenReview: *https: // op... | Review 1:
Summary: This work proposes the use of derivative & Hessian estimate of the target function in its acquisition criterion for BO. It provides an approximation of such criterion, and show that it indeed helps finding better estimate quicker than the one without those information on toy examples.
Strengths and... |
# Large Language Models As Your Personal Data Scientist
Anonymous authors Paper under double-blind review
## Abstract
Large Language Models (LLMs) have contributed to massive performance improvements for various language understanding and generation tasks; however, their limits are yet to be fully explored for "ill-... | Review 1:
Summary: The paper presents a case study in developing an automatic conversational data science assistant through large language models. It presents a conversational AI agent called VIDS that helps users conduct data science tasks, describing the collection of micro-agents that constitute the full system. The... |
# Training-Free Graph Neural Networks And The Power Of Labels As Features
Ryoma Sato rsato@nii.ac.jp National Institute of Informatics Reviewed on OpenReview: *https: // openreview. net/ forum? id= 7DzU88VrNU*
## Abstract
We propose training-free graph neural networks (TFGNNs), which can be used without training and... | Review 1:
Summary: The authors propose the somewhat strange yet defensible idea of graph neural networks which work without any training. The idea is that in many situations, graph neural networks are used for transductive node classification, where the goal is to take a graph and impute information about neighboring n... |
# A Survey Of Lottery Ticket Hypothesis
Anonymous authors Paper under double-blind review
## Abstract
The Lottery Ticket Hypothesis (LTH) states that a dense neural network model contains a highly sparse subnetwork (i.e., winning tickets) that can achieve even better performance than the original model when trained ... | |
# Coin++: Neural Compression Across Modalities
Emilien Dupont* *dupont@stats.ox.ac.uk* Hrushikesh Loya* *loya@stats.ox.ac.uk* Milad Alizadeh *milad.alizadeh@cs.ox.ac.uk*
Adam Goliński *adamg@robots.ox.ac.uk*
Yee Whye Teh *y.w.teh@stats.ox.ac.uk*
Arnaud Doucet doucet@stats.ox.ac.uk
University of Oxford
Reviewed on... | Review 1:
Summary: The authors propose to tackle the problem of data compression with the goal of having one method that is works across moalidites. To do so, the authors build on the recent COIN method that shifts away from the traditional encoder/decoder for data compression. Instead, the idea is to train a network o... |
# Multi-Dimensional Concept Discovery (Mcd): A Unifying Framework With Completeness Guarantees
Johanna Vielhaben **johanna.vielhaben@hhi.fraunhofer.de**
Explainable Artificial Intelligence Group Fraunhofer Heinrich-Hertz-Institute Stefan Blüecher **bluecher@tu-berlin.de**
Machine Learning Group TU Berlin Nils Strodtho... | Review 1:
Summary: This paper proposes a framework for concept discovery based on hidden representation of neural networks. Different from existing approaches, authors proposed a multi-dimensional linear subspaces without restricting to single directions or enforcing orthogonality between subspaces with complete relat... |
# Taming Diffusion Times In Score-Based Generative Models: Trade-Offs And Solutions
Anonymous authors Paper under double-blind review
## Abstract
Score-based diffusion models are a class of generative models whose dynamics is described by stochastic differential equations that map noise into data. While recent works... | Review 1:
Summary: In this paper, the authors provide a theoretical analysis of the role of the forward time T in denoising diffusion models. The authors advocate that there is a trade-off between the time needed for the forward diffusion to converge for which a large time T is needed and the score approximation which ... |
# Black Box Uncertainty Analysis For Semantic Segmentation
Anonymous authors
Paper under double-blind review Figure 1: A couple of examples show how well the entropy of predicted outputs correlates with misclassified pixels. Green indicates misclassified pixels with high entropy, red ... | |
# Partial Optimal Transport For Support Subset Selection
Bilal Riaz bilalria@udel.edu Department of Electrical & Computer Engineering University of Delaware Yüksel Karahan ykarahan@udel.edu Department of Electrical & Computer Engineering University of Delaware Austin J. Brockmeier ajbrock@udel.edu Department of Electr... | Review 1:
Summary: The article deals with partial optimal transport, that is, when the target and source measures do not maintain the entirety of their mass over the transport. In particular, the authors focus on a specific case where the measures are discrete and uniform, to propose a methodology for a particular case... |
# C2P: Featuring Large Language Models With Causal Reasoning
Anonymous authors Paper under double-blind review
## Abstract
Causal reasoning is the primary bottleneck that Large Language Models (LLMs) must overcome to attain human-level intelligence. To address this, we introduce the Causal Chain of Prompting (C2P) a... | |
# Population Priors For Matrix Factorization
Anonymous authors Paper under double-blind review
## Abstract
We develop an empirical Bayes prior for probabilistic matrix factorization. Matrix factorization models each cell of a matrix with two latent variables, one associated with the cell's row and one associated wit... | |
# Generalization Error Bounds For Learning Under Censored Feedback
Anonymous authors Paper under double-blind review
## Abstract
Generalization error bounds from learning theory provide statistical guarantees on how well an algorithm will perform on previously unseen data. In this paper, we characterize the impacts ... | Review 1:
Summary: This paper studies the generalization error bounds for classification models when the collected data is non-IID due to censored feedback. Here, censored feedback is due to the selection of data (or data collection biases), e.g., giving a loan or not based on credit score. The authors demonstrate that... |
# A Unified View Of Differentially Private Deep Generative Modeling
Dingfan Chen∗dingfan.chen@cispa.de CISPA Helmholtz Center for Information Security Raouf Kerkouche *raouf.kerkouche@cispa.de* CISPA Helmholtz Center for Information Security Mario Fritz fritz@cispa.de CISPA Helmholtz Center for Information Security Re... | Review 1:
Summary: This paper provides a unified view of differential privacy for generative models. It is a synthesis paper, that aptly summarizes about 10 years of methods in the intersection of DP and Gen AI into a coherent mathematical framework.
Strengths and Weaknesses: The exposition is clear and thorough. I co... |
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