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+# VARIATIONAL INFERENCE FOR SDES DRIVEN BY FRACTIONAL NOISE
+
+Rembert Daems  $^{1,2}$  Manfred Opper  $^{3,4,5}$  Guillaume Crevecoeur  $^{1,2}$  Tolga Birdal
+
+$^{1}$  D2LAB, Ghent University, Belgium  
+$^{2}$  MIRO core lab, Flanders Make@UGent, Belgium  
+$^{3}$  Dept. of Theor. Comp. Science, Technical University of Berlin, Germany  
+4 Inst. of Mathematics, University of Potsdam, Germany  
+<sup>5</sup> Centre for Systems Modelling and Quant. Biomed., University of Birmingham, UK  
+$^{6}$  Dept. of Computing, Imperial College London, UK
+
+# ABSTRACT
+
+We present a novel variational framework for performing inference in (neural) stochastic differential equations (SDEs) driven by Markov-approximate fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling real-world continuous-time dynamic systems with inherent noise and randomness. Combining SDEs with the powerful inference capabilities of variational methods, enables the learning of representative function distributions through stochastic gradient descent. However, conventional SDEs typically assume the underlying noise to follow a Brownian motion (BM), which hinders their ability to capture long-term dependencies. In contrast, fractional Brownian motion (fBM) extends BM to encompass non-Markovian dynamics, but existing methods for inferring fBM parameters are either computationally demanding or statistically inefficient. In this paper, building upon the Markov approximation of fBM, we derive the evidence lower bound essential for efficient variational inference of posterior path measures, drawing from the well-established field of stochastic analysis. Additionally, we provide a closed-form expression to determine optimal approximation coefficients. Furthermore, we propose the use of neural networks to learn the drift, diffusion and control terms within our variational posterior, leading to the variational training of neural-SDEs. In this framework, we also optimize the Hurst index, governing the nature of our fractional noise. Beyond validation on synthetic data, we contribute a novel architecture for variational latent video prediction—an approach that, to the best of our knowledge, enables the first variational neural-SDE application to video perception.
+
+# 1 INTRODUCTION
+
+Our surroundings constantly evolve over time, influenced by several dynamic factors, manifesting in various forms, from the weather patterns and the ebb & flow of financial markets to the movements of objects (Yu et al., 2023; Rempe et al., 2021) & observers, and the subtle deformations that reshape our environments (Gojcic et al., 2021). Stochastic differential equations (SDEs) provide a natural way to capture the randomness and continuous-time dynamics inherent in these real-world processes. To extract meaningful information about the underlying system, i.e. to infer the model parameters and to accurately predict the unobserved paths, variational inference (VI) (Bishop & Nasrabadi, 2006) is used as an efficient means, computing the posterior probability measure over paths (Opper, 2019; Li et al., 2020; Ryder et al., 2018) $^{1}$ .
+
+The traditional application of SDEs assumes that the underlying noise processes are generated by standard Brownian motion (BM) with independent increments. Unfortunately, for many practical scenarios, BM falls short of capturing the full complexity and richness of the observed real data,
+
+![](images/f2d2139111e9c4e599d331566da4e48b1bb55cc23168307d6b77ef152171f450.jpg)  
+Figure 1: We leverage the Markov approximation, where the non-Markovian fractional Brownian motion with Hurst index  $H$  is approximated by a linear combination of a finite number of Markov processes  $(Y_{1}(t),\ldots ,Y_{K}(t))$ , and propose a variational inference framework in which the posterior is steered by a control term  $u(t)$ . Note the long-term memory behaviour of the processes, where individual  $Y_{k}(t)$ s have varying transient effects, from  $Y_{1}(t)$  having the longest memory to  $Y_{7}(t)$  the shortest, and tend to forget the action of  $u(t)$  after a certain time frame.
+
+which often contains long-range dependencies, rare events, and intricate temporal structures that cannot be faithfully represented by a Markovian process. The non-Markovian fractional Brownian motion (fBM) (Mandelbrot & Van Ness, 1968) extends BM to stationary increments with a more complex dependence structure, i.e. long-range dependence vs. roughness/regularity controlled by its Hurst index (Gatheringal et al., 2018). Yet, despite its desirable properties, the computational challenges and intractability of analytically working with fBMs pose significant challenges for inference.
+
+In this paper, we begin by providing a tractable variational inference framework for SDEs driven by fractional Brownian motion (Types I & II). To this end, we benefit from the relatively under-explored Markov representation of fBM and path-wise approximate fBM through a linear combination of a finite number of Ornstein-Uhlenbeck (OU) processes driven by a common noise (Carmona & Coutin, 1998a;b; Harms & Stefanovits, 2019). We further introduce a differentiable method to optimise for the associated coefficients and conjecture (as well as empirically validate) that this strong approximation enjoys super-polynomial convergence rates, allowing us to use a handful of processes even in complex problems.
+
+Such Markov-aisation also allows us to inherit the well-established tools of traditional SDEs including Girsanov's change of measure theorem (Øksendal & Øksendal, 2003), which we use to derive and maximise the corresponding evidence lower bound (ELBO) to yield posterior path measures as well as maximum likelihood estimates as illustrated in Fig. 1. We then use our framework in conjunction with neural networks to devise VI for neural-SDEs (Liu et al., 2019; Li et al., 2020) driven by the said fractional diffusion. We deploy this model along with a novel neural architecture for the task of enhanced video prediction. To the best of our knowledge, this is the first time either fractional or variational neural-SDEs are used to model videos. Our contributions are:
+
+- We make accessible the relatively uncharted Markovian embedding of the fBM and its strong approximation, to the machine learning community. This allows us to employ the traditional machinery of SDEs in working with non-Markovian systems.  
+- We show how to balance the contribution of Markov processes by optimising for the combination coefficients in closed form. We further estimate the (time-dependent) Hurst index from data.  
+- We derive the evidence lower bound for SDEs driven by approximate fBM of both Types I and II.  
+- We model the drift, diffusion and control terms in our framework by neural networks, and propose a novel architecture for video prediction.
+
+We make our implementation publicly available under: github.com/VideoNeuralSDE/MAFBM.
+
+# 2 RELATED WORK
+
+Fractional noises and neural-SDEs. fBM (Mandelbrot & Van Ness, 1968) was originally used for the simulation of rough volatility in finance (Gatheral et al., 2018). Using the Lemarie-Meyer wavelet representation, Allouche et al. (2022) provided a large probability bound on the deep-feedforward RELU network approximation of fBM, where up to log terms, a uniform error of  $O(N^{-H})$  is achievable with  $\log(N)$  hidden layers and  $O(N)$  parameters. Tong et al. (2022) approximated the fBM (only Type II) with sparse Gaussian processes. Unfortunately, they are limited to Euler-integration and to the case of  $H > 1/3$ . Their model was also not applied to videos. Recently, Yang et al. (2023) applied Levy driven neural-SDEs to times series prediction and Hayashi & Nakagawa (2022) considered neural-SDEs driven by fractional noise. Neither of those introduce a variational framework. Both Liao et al. (2019); Morrill et al. (2021) worked with rough path theory
+
+to model long time series via rough neural-SDEs. To the best of our knowledge, we are the firsts to devise a VI framework for neural-SDEs driven by a path-wise (strong) approximation of fBM.
+
+SDEs and visual understanding. Apart from the recent video diffusion models (Luo et al., 2023; Yang et al., 2022; Ho et al., 2022), SDEs for spatiotemporal visual generation is relatively unexplored. Park et al. (2021); Ali et al. (2023) used neural-ODEs to generate and manipulate videos while (Rempe et al., 2020) used neural-ODEs for temporal 3D point cloud modeling. SDENet (Kong et al., 2020) and MDSDE-Net (Zhang et al., 2023) learned drift and diffusion networks for uncertainty estimation of images using out-of-distribution data. Tong et al. (2022) used approximateBFMs in score-based diffusion modeling for image generation. Gordon & Parde (2021) briefly evaluated different neural temporal models for video generation. While Babaeizadeh et al. (2018) used VI for video prediction, they did not employ SDEs. To the best of our knowledge, we are the firsts to use neural-SDEs in a variational framework for video understanding.
+
+# 3 BACKGROUND
+
+We first tailor and make accessible the fractional Brownian Motion (fBM) and its relatively less explored Markov approximations for the learning community. We then describe the SDEs driven by fBM and its approximation before delving into the inference. We leave the proofs to our appendix.
+
+# 3.1 FRACTIONAL BROWNIAN MOTION (FBM) & ITS MARKOV APPROXIMATION
+
+Definition 1 (Fractional Brownian Motion (Types I & II)).  $fBM$  is a self-similar, non-Markovian, non-martingale, zero-mean Gaussian process  $(B_H(t))_{t\in [0,T]}$  for  $T > 0$  with a covariance of either
+
+$$
+\mathbb {E} \left[ B _ {H} ^ {(I)} (t) B _ {H} ^ {(I)} (s) \right] = \frac {1}{2} \left(| t | ^ {2 H} + | s | ^ {2 H} - | t - s | ^ {2 H}\right) \tag {1}
+$$
+
+$$
+\mathbb {E} \left[ B _ {H} ^ {(I I)} (t) B _ {H} ^ {(I I)} (s) \right] = \frac {1}{\Gamma^ {2} (H + 1 / 2)} \int_ {0} ^ {s} ((t - u) (s - u)) ^ {H - 1 / 2} d u \quad (\text {T y p e I I}) \tag {2}
+$$
+
+where  $t > s$ ,  $0 < H < 1$  is the Hurst index, superscripts denote the types and  $\Gamma$  is the Gamma function.
+
+fBM recovers Brownian motion (BM) for  $H = 1/2$  (regular diffusion) and generalizes it for other choices. The increments are (i) positively correlated for  $H > 1/2$  (super-diffusion) where the tail behaviour is infinitely heavier than that of BM, and (ii) negatively correlated for  $H < 1/2$  (sub-diffusion), with variance  $\mathbb{E}\left(|B_H^{(I)}(t) - B_H^{(I)}(s)|^2\right) = |t - s|^{2H}$  for Type I. The Type II model implies nonstationary increments of which the marginal distributions are dependent on the time relative to the start of the observed sample, i.e., all realizations would have to be found very close to the unconditional mean (i.e., the origin) (Lim & Sithi, 1995; Davidson & Hashimzade, 2009).
+
+Definition 2 (Integral representations of fBM).  $B_H^{(I,II)}$  admits the following integral forms due to the Mandelbrot van-Ness and Weyl representations, respectively (Mandelbrot & Van Ness, 1968):
+
+$$
+\begin{array}{l} B _ {H} ^ {(I)} (t) = \frac {1}{\Gamma (H + 1 / 2)} \int_ {- \infty} ^ {t} \left[ K ^ {(I)} (t, s) := \left((t - s) ^ {H - 1 / 2} - (- s) _ {+} ^ {H - 1 / 2}\right) \right] d W (s) \tag {3} \\ = \frac {1}{\Gamma (H + 1 / 2)} \left(\int_ {- \infty} ^ {0} \left((t - s) ^ {H - 1 / 2} - (- s) ^ {H - 1 / 2}\right) d W (s) + \int_ {0} ^ {t} (t - s) ^ {H - 1 / 2} d W (s)\right) \\ \end{array}
+$$
+
+$$
+B _ {H} ^ {(I I)} (t) = \frac {1}{\Gamma (H + 1 / 2)} \int_ {0} ^ {t} \left[ K ^ {(I I)} (t, s) := (t - s) ^ {H - 1 / 2} \right] \mathrm {d} W (s) \tag {4}
+$$
+
+where  $K^{(I)}$  and  $K^{(II)}$  are the kernels corresponding to Types I and II, respectively.
+
+Proposition 1 (Markov representation of fBM (Harms & Stefanovits, 2019)). The long memory processes  $B_H^{(I,II)}(t)$  can be represented by an infinite linear combination of Markov processes, all driven by the same Wiener noise, but with different time scales, defined by speed of mean reversion  $\gamma$ . For both types we have representations of the form:
+
+$$
+B _ {H} (t) = \left\{ \begin{array}{l} \int_ {0} ^ {\infty} \left(Y _ {\gamma} (t) - Y _ {\gamma} (0)\right) \mu (\gamma) \mathrm {d} \gamma , \quad H <   1 / 2, \\ - \int_ {0} ^ {\infty} \partial_ {\gamma} \left(Y _ {\gamma} (t) - Y _ {\gamma} (0)\right) \nu (\gamma) \mathrm {d} \gamma , \quad H > 1 / 2 \end{array} , \right. \tag {5}
+$$
+
+where  $\mu (\gamma) = \gamma^{-(H + 1 / 2)} / (\Gamma (H + 1 / 2)\Gamma (1 / 2 - H))$  and  $\nu (\gamma) = \gamma^{-(H - 1 / 2)} / (\Gamma (H + 1 / 2)\Gamma (3 / 2 - H))$ . Note, these non-negative densities are not normalisable. To simplify notation, we will drop explicit dependency on the types  $(I,II)$  in what follows. For each  $\gamma \geq 0$ , and for both types  $I$  and  $II$ , the processes  $Y_{\gamma}(t)$  are OU processes which are solutions to the SDE  $dY_{\gamma}(t) = -\gamma Y_{\gamma}(t)\mathrm{d}t + \mathrm{d}W(t)$ . This SDE is solved by
+
+$$
+Y _ {\gamma} (t) = Y _ {\gamma} (0) e ^ {- \gamma t} + \int_ {0} ^ {t} e ^ {- \gamma (t - s)} d W (s). \tag {6}
+$$
+
+"Type I" and "Type II" differ in the initial conditions  $Y_{\gamma}(0)$ . One can show that:
+
+$$
+Y _ {\gamma} ^ {(I)} (0) = \int_ {- \infty} ^ {0} e ^ {\gamma s} d W (s) \quad \text {a n d} \quad Y _ {\gamma} ^ {(I I)} (0) = 0. \tag {7}
+$$
+
+Definition 3 (Markov approximation of fBM (MA-fBM)). Eq. (5) suggests that  $B_H(t)$  could be well approximated by a Markov process  $\hat{B}_H(t)$  by (i) truncating the integrals at finite  $\gamma$  values  $(\gamma_1 \dots \gamma_K)$  and (ii) approximating the integral by a numerical quadrature as a finite linear combination involving quadrature points and weights  $\{\omega_k\}$ . Changing the notation  $Y_{\gamma_k}(t) \to Y_k(t)$ :
+
+$$
+B _ {H} (t) \approx \hat {B} _ {H} (t) \equiv \sum_ {k = 1} ^ {K} \omega_ {k} \left(Y _ {k} (t) - Y _ {k} (0)\right), \tag {8}
+$$
+
+where for fixed  $\gamma_{k}$  the choice of  $\omega_{k}$  depends on  $H$  and the choice of "Type I" or "Type II". For "Type II", we set  $Y_{k}(0) = 0$ . Since  $Y_{k}(t)$  is normally distributed (Harms & Stefanovits, 2019, Thm. 2.16) and can be assumed stationary for "Type I", we can simply sample  $\left(Y_{1}^{(I)}(0), \ldots, Y_{K}^{(I)}(0)\right)$  from a normal distribution with mean 0 and covariance  $C_{i,j} = 1 / (\gamma_i + \gamma_j)$  (see Eq. (28)).
+
+This strong approximation provably bounds the sample paths:
+
+Theorem 1 (Alfonsi & Kebaier (2021)). For rough kernels  $(H < 1/2)$  and  $\{\omega_k\}$  following a Gaussian quadrature rule, there exists a constant  $c$  per every  $t \in (0,T)$  such that:
+
+$$
+\mathbb {E} \left| B _ {H} ^ {(I I)} (t) - \hat {B} _ {H} ^ {(I I)} (t) \right| \leq O \left(K ^ {- c H}\right), \quad \text {w h e r e} \quad 1 <   c \leq 2, \tag {9}
+$$
+
+as  $K\to \infty$  . Note that, in our setting,  $B_H^{(II)}(0) = \hat{B}_H^{(II)}(0) = 0.$
+
+In the literature, different choices of  $\gamma_{k}$  and  $\omega_{k}$  have been proposed (Harms & Stefanovits, 2019; Carmona & Coutin, 1998a; Carmona et al., 2000) and for certain choices, it is possible to obtain a superpolynomial rate, as shown by Bayer & Breneis (2023) for the Type II case. As we will show in Sec. 4.1, choosing  $\gamma_{k} = r^{k - n}, k = 1,\dots ,K$  with  $n = (K + 1) / 2$  (Carmona & Coutin, 1998a), we will optimise  $\{\omega_k\}_k$  for both types, to get optimal rates.
+
+# 3.2 SDEs DRIVEN BY (FRACTIONAL) BM
+
+Definition 4 (SDEs driven by BM (BMSDE)). A common generative model for stochastic dynamical systems considers a set of observational data  $\mathcal{D} = \{O_1,\dots ,O_N\}$ , where the  $O_{i}$  are generated (conditionally) independent at random at discrete times  $t_i$  with a likelihood  $p_{\theta}(O_i\mid X(t_i))$ . The prior information about the unobserved path  $\{X(t);t\in [0,T]\}$  of the latent process  $X(t)\in \mathbb{R}^{M}$  is given by the assumption that  $X(t)$  fulfils the SDE:
+
+$$
+\mathrm {d} X (t) = b _ {\theta} (X (t), t) \mathrm {d} t + \sigma_ {\theta} (X (t), t) \mathrm {d} W (t) \tag {10}
+$$
+
+The drift function  $b_{\theta}(X,t) \in \mathbb{R}^{D}$  models the deterministic part of the change  $\mathrm{d}X(t)$  of the state variable  $X(t)$  during the infinitesimal time interval  $\mathrm{dt}$ , whereas the diffusion matrix  $\sigma_{\theta}(X(t),t) \in \mathbb{R}^{D\times D}$  (assumed to be symmetric and non-singular, for simplicity) encodes the strength of the added Gaussian white noise process, where  $\mathrm{d}W(t) \in \mathbb{R}^{D}$  is the infinitesimal increment of a vector of independent Wiener processes during  $\mathrm{dt}$ .
+
+Definition 5 (SDEs driven by fBM (fBMSDE)). Dfn. 4 can be formally extended to the case of fractional Brownian motion replacing  $\mathrm{d}W(t)$  by  $\mathrm{d}B_H(t)$  (Guerra & Nualart, 2008):
+
+$$
+\mathrm {d} X (t) = b _ {\theta} (X (t), t) \mathrm {d} t + \sigma_ {\theta} (X (t), t) \mathrm {d} B _ {H} (t). \tag {11}
+$$
+
+Remark 1. Care must be taken in a proper definition of the diffusion part in the fBMSDE Eq. (11) and in developing appropriate numerical integrators for simulations, when the diffusion  $\sigma_{\theta}(X(t),t)$  explicitly depends on the state  $X(t)$ . Corresponding stochastic integrals of the Ito type cannot be applied when  $H < 1/2$  and other approaches (which are generalisations of the Stratonovich SDE for  $H = \frac{1}{2}$ ) are necessary (Lysy & Pillai, 2013).
+
+# 4 METHOD
+
+Our goal is to extend variational inference (VI) Bishop & Nasrabadi (2006) to the case where the Wiener process in Eq. (10) is replaced by an fBM as in Dfn. 5. Unfortunately, the processes defined by Eq. (11) are not Markovian preventing us from resorting to the standard Girsanov change of measure approach known for "ordinary" SDE to compute KL-divergences and ELBO functionals needed for VI (Opper, 2019). While Tong et al. (2022) leverage sparse approximations for Gaussian processes, this makes  $B_{H}$  conditioned on a finite but larger number of so-called inducing variables. We take a completely different and conceptually simple approach to VI for fBMSDE based on the exact representation of  $B_{H}(t)$  given in Prop. 1. To this end, we first show how the strong Markovapproximation in Dfn. 3 can be used to approximate an SDE driven by fBM, before delving into the VI for the Markov-Approximate fBMSDE.
+
+Definition 6 (Markov-Approximate fBMSDE (MA-fBMSDE)). Substituting the  $fBM$ ,  $B_H(t)$ , in Dfn. 5 by the finite linear combination of OU-processes  $\hat{B}_H(t)$ , we define MA-fBMSDE as:
+
+$$
+\mathrm {d} X (t) = b _ {\theta} (X (t), t) \mathrm {d} t + \sigma_ {\theta} (X (t), t) \mathrm {d} \hat {B} _ {H} (t), \tag {12}
+$$
+
+where  $\mathrm{d}\hat{B}_H(t) = \sum_{k=1}^{K} \omega_k \mathrm{d}Y_k(t)$  with  $\mathrm{d}Y_k(t) = -\gamma_k Y_k(t) \, \mathrm{d}t + \mathrm{d}W(t)$  (cf. Dfn. 3).
+
+Proposition 2.  $X(t)$  can be augmented by the finite number of Markov processes  $Y_{k}(t)$  (approximating  $B_{H}(t)$ ) to a higher dimensional state variable of the form  $Z(t) \doteq (X(t), Y_{1}(t), \ldots, Y_{K}(t)) \in \mathbb{R}^{D(K + 1)}$ , such that the joint process of the augmented system becomes Markovian and can be described by an 'ordinary' SDE:
+
+$$
+\mathrm {d} Z (t) = h _ {\theta} (Z (t), t) \mathrm {d} t + \Sigma_ {\theta} (Z (t), t) \mathrm {d} W (t), \tag {13}
+$$
+
+where the augmented drift vector  $h_\theta \in \mathbb{R}^{D \times (K + 1)}$  and the diffusion matrix  $\Sigma_\theta(Z, t) \in \mathbb{R}^{D(K + 1) \times D}$  are given by
+
+$$
+h _ {\theta} (Z, t) = \left( \begin{array}{c} b _ {\theta} (X, t) - \sigma_ {\theta} (X, t) \sum_ {k} \omega_ {k} \gamma_ {k} Y _ {k} \\ - \gamma_ {1} Y _ {1} \\ \dots \\ - \gamma_ {K} Y _ {K} \end{array} \right) \quad \Sigma_ {\theta} (Z, t) = \left( \begin{array}{c} \bar {\omega} \sigma_ {\theta} (X, t) \\ \vec {1} \\ \vdots \\ \vec {1} \end{array} \right), \tag {14}
+$$
+
+where  $\vec{1} = (1,1,\dots ,1)^{\top}\in \mathbb{R}^{D}$ . We will refer to Eq. (13) as the variational prior.
+
+Proof. Each of the  $D$  components of the vectors  $Y_{k}$  use the same scalar weights  $\omega_{k} \in \mathbb{R}$ . Also, note that each  $Y_{k}$  is driven by the same vector of Wiener processes. Hence, we obtain the system of SDEs given by
+
+$$
+\mathrm {d} X (t) = b _ {\theta} (X (t), t) \mathrm {d} t - \sigma_ {\theta} (X (t), t) \sum_ {k} \omega_ {k} \gamma_ {k} Y _ {k} (t) \mathrm {d} t + \bar {\omega} \sigma_ {\theta} (X (t), t) \mathrm {d} W (t) \tag {15}
+$$
+
+$$
+\mathrm {d} Y _ {k} (t) = - \gamma_ {k} Y _ {k} (t) \mathrm {d} t + \mathrm {d} W (t) \quad \text {f o r} \quad k = 1, \dots , K
+$$
+
+where  $\bar{\omega} \doteq \sum_{k} \omega_{k}$ . This system of equations can be collectively represented in terms of the augmented variable  $Z(t) \coloneqq (X(t), Y_{1}(t), \ldots, Y_{K}(t)) \in \mathbb{R}^{D(K + 1)}$  leading to a single SDE specified by Eqs. (13) and (14).
+
+Eq. (13) represents a standard SDE driven by Wiener noise allowing us to utilise the standard tools of stochastic analysis, such as the Girsanov change of measure theorem and derive the evidence lower bounds (ELBO) required for VI. This is what we will exactly do in the sequel.
+
+Proposition 3 (Controlled MA-fBMSDE). The paths of Eq. (13) can be steered by adding a control term  $u(X, Y_1, \ldots, Y_K, t) \in \mathbb{R}^D$  that depends on all variables to be optimised, to the drift  $h_\theta$  resulting in the transformed SDE, a.k.a. the variational posterior:
+
+$$
+\mathrm {d} \tilde {Z} (t) = \left(h _ {\theta} (\tilde {Z} (t), t) + \sigma_ {\theta} (\tilde {Z} (t), t) u (\tilde {Z} (t), t)\right) \mathrm {d} t + \Sigma_ {\theta} (\tilde {Z} (t), t) \mathrm {d} W (t) \tag {16}
+$$
+
+Sketch of the proof. Using the fact that the posterior probability measure over paths  $\tilde{Z}(t) \{\tilde{Z}(t); t \in [0, T]\}$  is absolutely continuous w.r.t. the prior process, we apply the Girsanov theorem (cf. App. B.1) on Eq. (13) to write the new drift, from which the posterior SDE in Eq. (16) is obtained.
+
+We will refer to Eq. (16) as the variational posterior. In what follows, we will assume a parametric form for the control function  $u(\tilde{Z}(t), t) \equiv u_{\phi}(\tilde{Z}(t), t)$  (as e.g. given by a neural network) and will devise a scheme for inferring the variational parameters  $(\theta, \phi)$ , i.e. variational inference.
+
+Proposition 4 (Variational Inference for MA-fBMSDE). The variational parameters  $\phi$  are optimised by minimising the KL-divergence between the posterior and the prior, where the corresponding evidence lower bound (ELBO) to be maximised is:
+
+$$
+\log p \left(O _ {1}, O _ {2}, \dots , O _ {N} \mid \theta\right) \geq \mathbb {E} _ {\tilde {Z} _ {u}} \left[ \sum_ {i = 1} ^ {N} \log p _ {\theta} \left(O _ {i} \mid \tilde {Z} (t _ {i})\right) - \int_ {0} ^ {T} \frac {1}{2} \left\| u _ {\phi} (\tilde {Z} (t), t) \right\| ^ {2} d t \right], \tag {17}
+$$
+
+where the observations  $\{O_i\}$  are included by likelihoods  $p_{\theta}\left(O_i \mid \tilde{Z}(t_i)\right)$  and the expectation is taken over random paths of the approximate posterior process defined by (Eq. (16)).
+
+Sketch of the proof. Since we can use Girsanov's theorem II (Øksendal & Øksendal, 2003), the variational bound derived in Li et al. (2020) (App. 9.6.1) directly applies.
+
+Remark 2. It is noteworthy that the measurements with their likelihoods  $p_{\theta}\left(O_i \mid \tilde{X}(t_i)\right)$  depend only on the component  $\tilde{X}(t)$  of the augmented state  $\tilde{Z}(t)$ . The additional variables  $Y_k(t)$  which are used to model the noise in the SDE are not directly observed. However, computation of the ELBO requires initial values for all state variables  $\tilde{Z}(0)$  (or their distribution). Hence, we sample  $Y_k(0)$  in accordance with Dfn. 3.
+
+# 4.1 OPTIMISING THE APPROXIMATION
+
+We now present the details of our novel method for optimising our approximation  $\hat{B}_H^{(I,II)}(t)$  for  $\omega_{k}$ . To this end, we first follow Carmona & Coutin (1998a) and choose a geometric sequence of  $\gamma_{k} = (r^{1 - n},r^{2 - n},\ldots ,r^{K - n}),n = \frac{K + 1}{2},r > 1$ . Rather than relying on methods of numerical quadrature, we consider a simple measure for the quality of the approximation over a fixed time interval  $[0,T]$  which can be optimised analytically for both types I and II.
+
+Proposition 5 (Optimal  $\omega \doteq [\omega_1,\dots ,\omega_K]$  for  $\hat{B}^{(I,II)}(t))$  . The  $L_{2}$  -error of our approximation
+
+$$
+\mathcal {E} ^ {(I, I I)} (\boldsymbol {\omega}) = \int_ {0} ^ {T} \mathbb {E} \left[ \left(\hat {B} _ {H} ^ {(I, I I)} (t) - B _ {H} ^ {(I, I I)} (t)\right) ^ {2} \right] d t \tag {18}
+$$
+
+is minimized at  $\mathbf{A}^{(I,II)}\pmb {\omega} = \pmb{b}^{(I,II)}$  , where
+
+$$
+\boldsymbol {A} _ {i, j} ^ {(I)} = \frac {2 T + \frac {e ^ {- \gamma_ {i} T} - 1}{\gamma_ {i}} + \frac {e ^ {- \gamma_ {j} T} - 1}{\gamma_ {j}}}{\gamma_ {i} + \gamma_ {j}}, \quad \boldsymbol {A} _ {i, j} ^ {(I I)} = \frac {T + \frac {e ^ {- (\gamma_ {i} + \gamma_ {j}) T} - 1}{\gamma_ {i} + \gamma_ {j}}}{\gamma_ {i} + \gamma_ {j}} \tag {19}
+$$
+
+$$
+\boldsymbol {b} _ {k} ^ {(I)} = \frac {2 T}{\gamma_ {k} ^ {H + 1 / 2}} - \frac {T ^ {H + 1 / 2}}{\gamma_ {k} \Gamma (H + 3 / 2)} + \frac {e ^ {- \gamma_ {k} T} - Q (H + 1 / 2 , \gamma_ {k} T) e ^ {\gamma_ {k} T}}{\gamma_ {k} ^ {H + 3 / 2}} \tag {20}
+$$
+
+$$
+\boldsymbol {b} _ {k} ^ {(I I)} = \frac {T}{\gamma_ {k} ^ {H + 1 / 2}} P (H + 1 / 2, \gamma_ {k} T) - \frac {H + 1 / 2}{\gamma_ {k} ^ {H + 3 / 2}} P (H + 3 / 2, \gamma_ {k} T). \tag {21}
+$$
+
+$P(z,x) = \frac{1}{\Gamma(z)}\int_0^x t^{z - 1}e^{-t}\mathrm{d}t$  is the regularized lower incomplete gamma function and  $Q(z,x) = \frac{1}{\Gamma(z)}\int_{x}^{\infty}t^{z - 1}e^{-t}\mathrm{d}t$  is the regularized upper incomplete gamma function.
+
+Sketch of the proof. By expanding the  $L_{2}$ -error we find a tractable quadratic form of the criterion:
+
+$$
+\begin{array}{l} \mathcal {E} ^ {(I, I I)} (\boldsymbol {\omega}) = \int_ {0} ^ {T} \mathbb {E} \left[ \left(\hat {B} _ {H} ^ {(I, I I)} (t) - B _ {H} ^ {(I, I I)} (t)\right) ^ {2} \right] \mathrm {d} t \tag {22} \\ = \int_ {0} ^ {T} \left(\mathbb {E} \left[ \hat {B} _ {H} ^ {(I, I I)} (t) ^ {2} \right] + \mathbb {E} \left[ B _ {H} ^ {(I, I I)} (t) ^ {2} \right] - 2 \mathbb {E} \left[ \hat {B} _ {H} ^ {(I, I I)} (t) B _ {H} ^ {(I, I I)} (t) \right]\right) d t \\ = \boldsymbol {\omega} ^ {T} \boldsymbol {A} ^ {(I, I I)} \boldsymbol {\omega} - 2 \boldsymbol {b} ^ {(I, I I) ^ {T}} \boldsymbol {\omega} + \text {c o n s t}, \\ \end{array}
+$$
+
+whose non-trivial minimum is attained as the solution to the system of equations  $\mathbf{A}^{(I,II)}\pmb {\omega} = \pmb{b}^{(I,II)}$ . We refer the reader to App. D.2 for the full proof and derivation.
+
+![](images/b76160bd10dfce26088a239ae2aaf8599a906a9e5dcaacaa6a2b8daefb2362e3.jpg)  
+(a)  $H = 0.3,\theta = 0.0$
+
+![](images/ef93862fad7ab3136a1f4c145ce1e71b82e3f1eca14f71c663835e3a42d9c79b.jpg)  
+(b)  $H = 0.7,\theta = 0.0$  
+Figure 2: The true variance (blue) of a fOU bridge matches the empirical variance (dashed orange) of our trained models. The transparent black lines are the sampled approximate posterior paths used to calculate the empirical variance.
+
+![](images/798aabfec70f8ed458bb1953ad7229e2398b494ac028a2536e78acefa5b107f0.jpg)  
+(c)  $H = 0.6, \theta = 1.0$
+
+![](images/a16a6cd39e311bc1ab15fb7e4a9d87b9e7f3d9c3d4ce2b366b60a7f24b4fe64c.jpg)  
+(d)  $H = 0.8,\theta = 1.0$
+
+# 5 EXPERIMENTS
+
+We implemented our method in JAX (Bradbury et al., 2018), using Diffrax (Kidger, 2021) for SDE solvers, Optax (Babuschkin et al., 2020) for optimization, Diffrax (Babuschkin et al., 2020) for distributions and Flax (Heek et al., 2023) for neural networks. Unlike Tong et al. (2022) our approach is agnostic to discretization and the choice of the solver. Hence, in all experiments we can use the Stratonovich-Milstein solver, cf. App. E for more details.
+
+Recovering the fractional Ornstein-Uhlenbeck bridge. Applying our method on linear problems, allows comparing empirical results to analytical formulations derived e.g. using Gaussian process methodology Rasmussen et al. (2006). We begin by assessing the reconstruction capability of our method on a fractional Ornstein-Uhlenbeck (fOU) bridge, that is an OU-process driven by fBM:  $\mathrm{d}X(t) = -\theta X(t)\mathrm{d}t + \mathrm{d}B_H$ , starting at  $X(0) = 0$  and conditioned to end at  $X(T) = 0$ . Following the rules of Gaussian process regression (Rasmussen et al., 2006, Eq. 2.24), we have an analytical expression for the posterior covariance:
+
+$$
+\mathbb {E} \left[ \tilde {X} (t) ^ {2} \right] = K (t, t) - [ K (t, 0) \quad K (t, T) ] \left[ \begin{array}{c c} K (0, 0) & K (T, 0) \\ K (0, T) & K (T, T) + \sigma^ {2} \end{array} \right] ^ {- 1} \left[ \begin{array}{l} K (0, t) \\ K (T, t) \end{array} \right] \tag {23}
+$$
+
+where  $K(t,\tau)$  is the prior kernel and the observation noise is 0 for  $X(0)$  and  $\sigma$  for  $X(T)$ . If  $\theta = 0$ ,  $K(t,\tau) = \mathbb{E}\left[B_H(t)B_H(\tau)\right]$  (Eq. (1)) and if  $\theta > 0$  and  $H > 1/2$ , the kernel admits the following form (Lysy & Pillai, 2013, Appendix A):
+
+$$
+K (t, \tau) = \frac {(2 H ^ {2} - H)}{2 \theta} \left(e ^ {- \theta | t - \tau |} \left[ \frac {\Gamma (2 H - 1) + \Gamma (2 H - 1 , | t - \tau |)}{\theta^ {2 H - 1}} + \int_ {0} ^ {| t - \tau |} e ^ {\theta u} u ^ {2 H - 2} \mathrm {d} u \right]\right) \tag {24}
+$$
+
+where  $\Gamma(z, x) = \int_{x}^{\infty} t^{z-1} e - t \, \mathrm{d}t$  is the upper incomplete Gamma function. This allows us to compare the true posterior variance with the empirical variance of a model that is trained by maximizing the ELBO. for a data point  $X(T) = 0$ . As this is equivalent to the analytical result (Eq. (23)), we can compare the variances over time. As plotted in Fig. 2, for various  $H$  and  $\theta$  values, our VI can correctly recover the posterior variance, cf. App. F for additional results.
+
+Estimating time-dependent Hurst index. Since our method of optimizing  $\omega_{k}$  is tractable and differentiable, we can directly optimize a parameterized  $H$  by maximizing the ELBO. Also a time-dependent Hurst index  $H(t)$  can be modelled, leading to multifractional Brownian Motion (Peltier & Vehel, 1995). We directly compare with a toy problem presented in (Tong et al., 2022, Sec. 5.2). We use the same model for  $H(t)$ , a neural network with one hidden layer of 10 neurons and activation function tanh, and a final sigmoid activation, and the same input  $[\sin(t), \cos(t), t]$ . We use  $\hat{B}_{H}^{(II)}$  since their method is Type II. Fig. 3 shows a reasonable estimation of  $H(t)$ , which is more accurate than the result from Tong et al. (2022), cf.App. E for more details.
+
+![](images/20509f5037084527fbfb74a76c5f47a2c253cfc62c827bf0c11be01df1236b3d.jpg)  
+Figure 3: Estimating time-dependent  $H(t)$  from data.
+
+Latent video models To assess the video modelling capabilities of our framework, we train models on stochastic video datasets. The prior drift  $h_{\theta}$ , diffusion  $\sigma_{\theta}$  and control term  $u$  are parameterized
+
+![](images/0ec48e1b57400f1ef718130ca4a963145d63f672c75f39618c7aead1fc0f67e1.jpg)  
+Figure 4: Schematic of the latent SDE video model. Video frames  $\{o_i\}_i$  are encoded to vectors  $\{h_i\}_i$ . The static content vector  $w$ , that is free of the dynamic information, is inferred from  $\{h_i\}_i$ . The context model processes the information with temporal convolution layers, so that its outputs  $\{g_i\}_i$  contain information from neighbouring frames. A linear interpolation on  $\{g_i\}_i$  allows the posterior SDE model to receive time-appropriate information  $g(t)$ , at (intermediate) time-steps chosen by the SDE solver. Finally, the states  $\{x_i\}_i$  and static  $w$  are decoded to reconstruct frames  $\{o_i'\}_i$ .
+
+by neural networks. The prior model is used as a stochastic video predictor, where we condition on the first  $N$  frames to predict the next frames in the sequence. More intuitively, the posterior model reconstructs the given sequence of frames, while minimizing the control actions of  $u$ . This leads to a prior that will model the dataset, so that the posterior will be able to model the specific data sequence during training with minimal  $u$  input. It is paramount that the control function  $u$  receives relevant information during the SDE integration, so that it can steer the SDE in the right direction. See Fig. 4 for a schematic explanation of our model and App. E for a detailed explanation of submodel architectures and hyperparameters.
+
+We evaluate the stochastic video predictions by sampling 100 predictions and reporting the Peak Signal-to-Noise Ratio (PSNR) of the best sample, calculated frame-wise and averaged over time. This is the same approach as Franceschi et al. (2020) which allows a direct comparison. Furthermore, we report the ELBO on the test set, indicating how well the model has captured the data.
+
+We train models on Stochastic Moving MNIST (Denton & Fergus, 2018), a video dataset where two MNIST numbers move on a canvas and bounce off the edge with random velocity in a random direction. Our MA-fBM driven model is on par with closely related discrete-time methods such as SVG (Denton & Fergus, 2018) or SLRVP Franceschi et al. (2020), in terms of PSNR, and is better than the BM baseline in terms of PSNR and ELBO (Tab. 1).
+
+The Hurst index was optimized during training, and reached  $H = 0.90$  at convergence (long-term memory), indicating that MA-fBM is better suited to the data than BM.
+
+Table 1: Stochastic Moving MNIST results.  
+
+<table><tr><td>Model</td><td>ELBO</td><td>PSNR</td></tr><tr><td>SVG</td><td>N/A</td><td>14.50</td></tr><tr><td>SLRVP</td><td>N/A</td><td>16.93</td></tr><tr><td>BM</td><td>-913.60</td><td>14.90</td></tr><tr><td>MA-fBM</td><td>-608.00</td><td>15.30</td></tr></table>
+
+Table 2: Double pendulum.  
+
+<table><tr><td>Model</td><td>ELBO</td><td>PSNR</td></tr><tr><td>BM</td><td>-545.13</td><td>26.11</td></tr><tr><td>MA-fBM</td><td>-636.61</td><td>27.09</td></tr></table>
+
+We also report results on a real-world video dataset of a double pendulum (Asseman et al., 2018), where we investigate whether the chaotic behaviour can be modelled by an SDE driven by fBM. Our MA-fBM driven model is better than the BM baseline, both for the test set ELBO as for the PSNR metric (Tab. 2).
+
+The Hurst index reached a value of  $H = 0.93$  at convergence. See Fig. 5 for stochastic video predictions and App. F.3 for additional qualitative results.
+
+![](images/0ad7cb95c111e025ad25b02e70be7effcae5249864a1a27f8a8d4001f24a7019.jpg)  
+(a) BM
+
+![](images/ee0eae43e3fff23d0738b8c5fc75330b608bcfe622fffc8998133edd671cfd3c.jpg)  
+(b) MA-fBM  
+Figure 5: Stochastic video predictions using the trained prior of a model driven by BM (a) and a model driven by MA-fBM (b) trained on the double pendulum dataset. The initial state is conditioned on the same data for all samples. Two samples are shown for each model, and 7 evenly spaced frames from the total of 20 frames in the sequence are shown. The MA-fBM samples show a more diverse, chaotic behaviour, thus better capturing the dynamics in the data.
+
+# 5.1 ABLATIONS & FURTHER STUDIES
+
+Numerical study of the Markov approximation. By numerically evaluating the criterion  $\mathcal{E}^{(II)}$  we can investigate the effect of  $K$ , the number of OU-processes, on the quality of the approximation. Fig. 6 indicates that the approximation error diminishes by increasing  $K$ . However, after a certain threshold the criterion saturates, depending on  $H$ . Adding more processes, especially for low  $H$  brings diminishing returns. The rapid convergence evidenced in this empirical result well agrees with the theoretical findings of (Bayer & Breneis, 2023) especially for the rough processes where  $H < 1/2$ , as recalled in Thm. 1.
+
+MSE of the generated trajectories for MA-fBM and for varying  $K$ . On a more practical level, we take integration and numerical errors into account by simulating paths using MA-fBM and comparing to paths of the true integral driven by the same Wiener noise. This is only possible for Type II, as for Type I one would need to start the integration from  $-\infty$ . Paths are generated from  $t = 0$  to  $t = 10$ , with 4000 integration steps for the approximation and 40000 for the true integral. We generate the paths over a range of Hurst indices and different  $K$  values. For each setting, 16 paths are sampled. Our approach for optimising  $\omega_{k}$  values (Sec. 4.1) is compared to a baseline where  $\omega_{k}$  is derived by a piece-wise approximation of the Laplace integral (cf. App. D.1). Fig. 7 shows considerably better results in favor of our approach. Increasing  $K$  h accuracy of the approximation with diminishing returns, further cor in Sec. 3. We provide examples of individual trajectories generated
+
+Impact of  $K$  and the #parameters on inference time. We investigate the factors that influence the training time in Fig. 8, where  $K$  OU-processes are gradually included to systems with increasing number of network parameters. Note that, since our approximation is driven by 1 Wiener process, and the control function  $u(\tilde{Z}(t), t)$  is scalar, the impact on computational load of including more processes is limited and the run-time is still dominated by the size of the neural networks. This is good news as different applications might demand different number of OU-processes.
+
+![](images/090eeabc60815efbea5b8ec5b8fb6e2dfe9de5fce8e799f62ece5bac68137494.jpg)  
+Figure 6:  $\mathcal{E}^{(II)}$  vs.  $K$ .
+
+![](images/f436b07ec958ef1f167c2aa2449549ec47894129eafa57a78bed63e38766aaa3.jpg)  
+Figure 7: Mean square error (MSE) with  $95\%$  confidence intervals vs.  $H$  for varying  $K$ .
+
+is a rapid positive impact on the affirming our theoretical insights in this experiment in App. F.1.
+
+![](images/f501d39371294bab8616a5cde31d679013c216f1ad945b0df30e32476976a6ff.jpg)  
+Figure 8:  $K$  vs. the run-time.
+
+# 6 CONCLUSION
+
+In this paper, we have proposed a new approach for performing variational inference on stochastic differential equations driven by fractional Brownian motion (fBM). We began by uncovering the relatively unexplored Markov representation of fBM, allowing us to approximate non-Markovian paths using a linear combination of Wiener processes. This approximation enabled us to derive evidence lower bounds through Girsanov's change of measure, yielding posterior path measures as well as likelihood estimates. We also solved for optimal coefficients for combining these processes, in closed form. Our diverse experimental study, spanning fOU bridges and Hurst index estimation, have consistently validated the effectiveness of our approach. Moreover, our novel, continuous-time architecture, powered by Markov-approximate fBM driven neural-SDEs, has demonstrated improvements in video prediction, particularly when inferring the Hurst parameter during inference.
+
+Limitations and future work. In our experiments, we observed increased computational overhead for larger time horizons due to SDE integration, although the expansion of the number of processes incurred minimal runtime costs. We have also observed super-polynomial convergence empirically and recalled weaker polynomial rates in the literature. Our Markov approximation still lacks a tight convergence bound. Our future work will also extend our framework to (fractional) Levy processes, which offer enhanced capabilities for modeling heavy-tailed noise/data distributions.
+
+Acknowledgments. The authors thank Jonas Degrave and Tom Lefebvre for insightful discussions. This research received funding from the Flemish Government under the "Onderzoeksprogramma Artificielle Intelligentie (AI) Vlaanderen" programme. Furthermore it was supported by Flanders Make under the SBO project CADAIVISION. MO has been partially funded by Deutsche Forschungsgemeinschaft (DFG) - Project - ID 318763901 - SFB1294.
+
+# ETHICS STATEMENT
+
+Our work is driven by a dedication to the advancement of knowledge and the betterment of society. While being largely theoretical, similar to many works advancing artificial intelligence, our work deserves an ethical consideration, which we present below.
+
+All of our experiments were either run on publicly available datasets or on data that is synthetically generated. No human or animal subjects have been involved at any stage of this work. Our models are designed to enhance the understanding and prediction of real-world processes without causing harm or perpetuating unjust biases, unless provided in the datasets. While we do not foresee any issue with methodological bias, we have not analyzed the inherent biases of our algorithm and there might be implications in applications demanding utmost fairness.
+
+We aptly acknowledge the contributions of researchers whose work laid the foundation for our own. Proper citations and credit are given to previous studies and authors. All authors declare that there are no conflicts of interest that could compromise the impartiality and objectivity of this research. All authors have reviewed and approved the final manuscript before submission.
+
+# REPRODUCIBILITY STATEMENT
+
+We are committed to transparency in research and for this reason make our implementation publicly available under: github.com/VideoNeuralSDE/MAFBM. Considerable parts involve: (i) the Markov approximation and optimisation of the  $\omega_{k}$  coefficients; (ii) maximising ELBO to perform variational inference between the prior  $\mathrm{d}Z(t)$  and the posterior  $\mathrm{d}\hat{Z}(t)$  and (iii) the novel neural-SDE based video prediction architecture making use of all our contributions. Our code replicates some of our evaluations for both of the datasets involved.
+
+# REFERENCES
+
+Aurelien Alfonsi and Ahmed Kebaier. Approximation of stochastic volterra equations with kernels of completely monotone type. arXiv preprint arXiv:2102.13505, 2021.  
+Moayed Haji Ali, Andrew Bond, Tolga Birdal, Duygu Ceylan, Levent Karacan, Erkut Erdem, and Aykut Erdem. Vidstyleode: Disentangled video editing via stylegan and neuralodes. In International Conference on Computer Vision (ICCV), 2023.  
+Michaël Allouche, Stéphane Girard, and Emmanuel Gobet. A generative model for fbm with deep relu neural networks. Journal of Complexity, 73:101667, 2022.  
+Alexis Asseman, Tomasz Kornuta, and Ahmet Ozcan. Learning beyond simulated physics. In Modeling and Decision-making in the Spatiotemporal Domain Workshop, 2018. URL https://openreview.net/pdf?id=HylajWsRF7.  
+Mohammad Babaeizadeh, Chelsea Finn, Dumitru Erhan, Roy H Campbell, and Sergey Levine. Stochastic variational video prediction. In International Conference on Learning Representations, 2018.  
+Igor Babuschkin, Kate Baumli, Alison Bell, Surya Bhupatiraju, Jake Bruce, Peter Buchovsky, David Budden, Trevor Cai, Aidan Clark, Ivo Danihelka, Antoine Dedieu, Claudio Fantacci, Jonathan Godwin, Chris Jones, Ross Hemsley, Tom Hennigan, Matteo Hessel, Shaobo Hou, Steven Kapturowski, Thomas Keck, Iurii Kemaev, Michael King, Markus Kunesch, Lena Martens, Hamza Merzic, Vladimir Mikulik, Tamara Norman, George Papamakarios, John Quan, Roman Ring, Francisco Ruiz, Alvaro Sanchez, Laurent Sartran, Rosalia Schneider, Eren Sezener,
+
+Stephen Spencer, Srivatsan Srinivasan, Miloš Stanojevic, Wojciech Stokowiec, Luyu Wang, Guangyao Zhou, and Fabio Viola. The DeepMind JAX Ecosystem, 2020. URL http://github.com/deepmind.  
+Christian Bayer and Simon Breweis. Markovian approximations of stochastic volterra equations with the fractional kernel. Quantitative Finance, 23(1):53-70, 2023.  
+Christopher M Bishop and Nasser M Nasrabadi. Pattern recognition and machine learning, volume 4. Springer, 2006.  
+James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. JAX: composable transformations of Python+NumPy programs, 2018. URL http://github.com/google/jax.  
+Philippe Carmona and Laure Coutin. Fractional brownian motion and the markov property. Electronic Communications in Probability [electronic only], 3:95-107, 1998a.  
+Philippe Carmona and Laure Coutin. Simultaneous approximation of a family of (stochastic) differential equations. Unpublished, June, 915(10.1051), 1998b.  
+Philippe Carmona, Laure Coutin, and Gérard Montseny. Approximation of some gaussian processes. Statistical inference for stochastic processes, 3:161-171, 2000.  
+Annie AM Cuyt, Vigdis Petersen, Brigitte Verdonk, Haakon Waadeland, and William B Jones. Handbook of continued fractions for special functions. Springer Science & Business Media, 2008.  
+James Davidson and Nigar Hashimzade. Type i and type ii fractional brownian motions: A reconsideration. Computational Statistics & Data Analysis, 53(6):2089-2106, 2009.  
+Emily Denton and Rob Fergus. Stochastic video generation with a learned prior. In International conference on machine learning, pp. 1174-1183. PMLR, 2018.  
+Jean-Yves Franceschi, Edouard Delasalles, Mickaël Chen, Sylvain Lamprier, and Patrick Gallinari. Stochastic latent residual video prediction. In International Conference on Machine Learning, pp. 3233-3246. PMLR, 2020.  
+Crispin W Gardiner et al. Handbook of stochastic methods, volume 3. Springer Berlin, 1985.  
+Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum. Volatility is rough. Quantitative finance, 18(6):933-949, 2018.  
+Zan Gojcic, Or Litany, Andreas Wieser, Leonidas J Guibas, and Tolga Birdal. Weakly supervised learning of rigid 3d scene flow. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 5692-5703, 2021.  
+Cade Gordon and Natalie Parde. Latent neural differential equations for video generation. In NeurIPS 2020 Workshop on Pre-registration in Machine Learning, pp. 73-86. PMLR, 2021.  
+Joao Guerra and David Nualart. Stochastic differential equations driven by fractional brownian motion and standard brownian motion. Stochastic analysis and applications, 26:1053-1075, 2008.  
+Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete and Continuous Dynamical Systems-B, 26(10):5567-5579, 2020.  
+Philipp Harms and David Stefanovits. Affine representations of fractional processes with applications in mathematical finance. Stochastic Processes and their Applications, 129:1185-1228, 2019.  
+Kohei Hayashi and Kei Nakagawa. Fractional sde-net: Generation of time series data with long-term memory. In 2022 IEEE 9th International Conference on Data Science and Advanced Analytics (DSAA), pp. 1-10. IEEE, 2022.
+
+Jonathan Heek, Anselm Levskaya, Avital Oliver, Marvin Ritter, Bertrand Rondepierre, Andreas Steiner, and Marc van Zee. Flax: A neural network library and ecosystem for JAX, 2023. URL http://github.com/google/flax.  
+Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J. Fleet. Video diffusion models. arXiv:2204.03458, 2022.  
+Hilbert Johan Kappen and Hans Christian Ruiz. Adaptive importance sampling for control and inference. Journal of Statistical Physics, 162:1244-1266, 2016.  
+Patrick Kidger. On Neural Differential Equations. PhD thesis, University of Oxford, 2021.  
+Patrick Kidger, James Foster, Xuechen Chen Li, and Terry Lyons. Efficient and accurate gradients for neural sdes. Advances in Neural Information Processing Systems, 34:18747-18761, 2021.  
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.  
+Lingkai Kong, Jimeng Sun, and Chao Zhang. Sde-net: equipping deep neural networks with uncertainty estimates. In Proceedings of the 37th International Conference on Machine Learning, pp. 5405-5415, 2020.  
+Xuechen Li, Ting-Kam Leonard Wong, Ricky TQ Chen, and David K Duvenaud. Scalable gradients and variational inference for stochastic differential equations. In Symposium on Advances in Approximate Bayesian Inference, pp. 1-28. PMLR, 2020.  
+Shujian Liao, Terry Lyons, Weixin Yang, and Hao Ni. Learning stochastic differential equations using rn with log signature features. arXiv preprint arXiv:1908.08286, 2019.  
+SC Lim and VM Sithi. Asymptotic properties of the fractional brownian motion of riemann-liouville type. Physics Letters A, 206(5-6):311-317, 1995.  
+Xuanqing Liu, Tesi Xiao, Si Si, Qin Cao, Sanjiv Kumar, and Cho-Jui Hsieh. Neural sde: Stabilizing neural ode networks with stochastic noise. arXiv preprint arXiv:1906.02355, 2019.  
+Zhengxiong Luo, Dayou Chen, Yingya Zhang, Yan Huang, Liang Wang, Yujun Shen, Deli Zhao, Jingren Zhou, and Tieniu Tan. Videofusion: Decomposed diffusion models for high-quality video generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10209-10218, 2023.  
+Martin Lysy and Natesh S Pillai. Statistical inference for stochastic differential equations with memory. arXiv preprint arXiv:1307.1164, 2013.  
+Benoit B Mandelbrot and John W Van Ness. Fractional brownian motions, fractional noises and applications. SIAM review, 10(4):422-437, 1968.  
+Domenico Marinucci and Peter M Robinson. Alternative forms of fractional brownian motion. Journal of statistical planning and inference, 80(1-2):111-122, 1999.  
+James Morrill, Christopher Salvi, Patrick Kidger, and James Foster. Neural rough differential equations for long time series. In International Conference on Machine Learning, pp. 7829-7838. PMLR, 2021.  
+Bernt Øksendal and Bernt Øksendal. Stochastic differential equations. Springer, 2003.  
+Manfred Opper. Variational inference for stochastic differential equations. Annalen der Physik, 531 (3):1800233, 2019.  
+Sunghyun Park, Kangyeol Kim, Junsoo Lee, Jaegul Choo, Joonseok Lee, Sookyung Kim, and Edward Choi. Vid-ode: Continuous-time video generation with neural ordinary differential equation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, 2021.  
+Romain-François Peltier and Jacques Lévy Véhel. Multifractional Brownian motion: definition and preliminary results. PhD thesis, INRIA, 1995.
+
+Carl Edward Rasmussen, Christopher KI Williams, et al. Gaussian processes for machine learning, volume 1. Springer, 2006.  
+Davis Rempe, Tolga Birdal, Yongheng Zhao, Zan Gojcic, Srinath Sridhar, and Leonidas J. Guibas. Caspr: Learning canonical spatiotemporal point cloud representations. In Advances in Neural Information Processing Systems (NeurIPS), 2020.  
+Davis Rempe, Tolga Birdal, Aaron Hertzmann, Jimei Yang, Srinath Sridhar, and Leonidas J Guibas. Humor: 3d human motion model for robust pose estimation. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 11488-11499, 2021.  
+Tom Ryder, Andrew Golightly, A Stephen McGough, and Dennis Prangle. Black-box variational inference for stochastic differential equations. In International Conference on Machine Learning, pp. 4423-4432. PMLR, 2018.  
+Evangelos A Theodorou. Nonlinear stochastic control and information theoretic dualities: Connections, interdependencies and thermodynamic interpretations. Entropy, 17(5):3352-3375, 2015.  
+Anh Tong, Thanh Nguyen-Tang, Toan Tran, and Jaesik Choi. Learning fractional white noises in neural stochastic differential equations. In Advances in Neural Information Processing Systems, 2022.  
+Luxuan Yang, Ting Gao, Yubin Lu, Jinqiao Duan, and Tao Liu. Neural network stochastic differential equation models with applications to financial data forecasting. Applied Mathematical Modelling, 115:279-299, 2023.  
+Ruihan Yang, Prakhar Srivastava, and Stephan Mandt. Diffusion probabilistic modeling for video generation. arXiv preprint arXiv:2203.09481, 2022.  
+Zhengdi Yu, Shaoli Huang, Yongkang Cheng, and Tolga Birdal. Signavatars: A large-scale 3d sign language holistic motion dataset and benchmark. arXiv preprint arXiv:2310.20436, 2023.  
+Xiao Zhang, Wei Wei, Zhen Zhang, Lei Zhang, and Wei Li. Milstein-driven neural stochastic differential equation model with uncertainty estimates. Pattern Recognition Letters, 2023.
+
+# APPENDIX
+
+# A FURTHER DISCUSSIONS
+
+Difference between a Type I and a Type II fBM. Type I, also called Mandelbrot-Van Ness or 'standard' fBM is the most prevalent definition of fBM. Type II, also called Riemann-Liouville fBM, is historically most used in econometric literature. As can be seen in the integral definitions (Eqs. (3) and (4)), Type II omits the first integral from  $-\infty$  to 0 in the definition of Type I. So Type II is, in a sense, a simplification. Yet, to the best of our knowledge, its covariance (Eq. (2)) has no simple analytical expression. On the other hand, the Type I covariance is the straightforward, well known Eq. (1).
+
+As described by Lim & Sithi (1995); Marinucci & Robinson (1999) the main difference is that Type I has stationary increments, and Type II has non-stationary increments. This means that Type II has a larger emphasis on the origin  $t = 0$ , which might not be favourable for some applications. For example, if during training we sample a sequence from a video dataset at a random start point, this  $t = 0$  has no special or distinguished meaning and should not be treated differently by the driving fBM process. In other words, Type I ensures a shift in time has no effect on its increments. However, this is not the case for Type II. This difference is relevant for our framework, since the increments are driving the SDE.
+
+Optimal choices for  $\omega$  and  $\gamma$  values. Regarding the Type II case, there are different ways of determining  $\gamma_{k}$  and  $\omega_{k}$  in the literature (Carmona & Coutin, 1998a; Bayer & Breneis, 2023; Harms & Stefanovits, 2019) some of which can lead to super-polynomial convergence (Bayer & Breneis, 2023) under certain assumptions, while more general choices are still shown to converge, though with a weaker rate (Alfonsi & Kebaier, 2021) while still being strong (path-wise) and of arbitrarily high polynomial order (Harms, 2020). Some of these works state that such geometric choice of the quadrature intervals simplifies the proofs while being not optimal and smarter choices can exist (even with better rate of convergence). This is the reason why we believe that our computationally tractable, closed form expressions which optimally solve for these values lead to good, super-polynomial convergence both for types II and I (since the first type also admits a similar type of analysis).
+
+Practical considerations for choosing  $\gamma_{k}$ . Defining  $\gamma_{k}$  as  $(1 / \gamma_{\mathrm{max}},\dots,\gamma_{\mathrm{max}})$  is a convenient way to indicate some practical considerations for choosing  $\gamma_{k}$ . Carmona & Coutin (1998b) show that  $\gamma \mathrm{d}t > 1 / 2$  leads to unstable integration of the OU-process, where  $\mathrm{d}t$  is the integration step. Care should be taken that  $\gamma_{\mathrm{max}}\mathrm{d}t < 1 / 2$ , either by decreasing  $\gamma_{\mathrm{max}}$  or decreasing the integration step  $\mathrm{d}t$ . Additionally, choosing large values for  $\gamma$  is undesirable for numerical reasons. Especially when using lower precision, numerical overflow can be a problem. Since an OU-process reaches equilibrium after time  $1 / \gamma$ , a practical lower bound for  $\gamma_{\mathrm{max}}$  is the length of the modelled sequences. This ensures that memory of the MA-fBM process is modelled for at least the length of the sequence.
+
+Time horizon for optimising  $\omega_{k}$ . The closed form expressions for  $\omega_{k}$  are in function of  $H$  and the time horizon  $T$  (Prop. 5). Since the criterion is defined over the time interval  $[0, T]$ , it makes sense to choose  $T$  equal to the typical (or maximal) length of sequences in the modelled dataset. Specifically for "Type I", we advise to choose  $T$  at two or three times the modelled sequence length, as at  $t = 0$ , this process is already at equilibrium, and its 'history' should be accounted for in the criterion. We have observed better empirical results when choosing  $T$  at a multiple of the sequence length.
+
+Further clarification on the distinction with Tong et al. (2022). Our work mainly differs with Tong et al. (2022) in two ways: (i) fractional Brownian motion (fBM) is approximated as a Gaussian process (GP), (ii) only the Type II representation of fBM is used as an integral over increments of the Wiener process. Tong et al. (2022) perform a finite time discretization of the Type II integral to obtain a first approximation of the increments of fBM. In a second step, this approximate GP is further approximated using a sparse GP approach based on a smaller set of pseudo or inducing points which are distributed over time. Conditioned on the inducing points, samples from the sparse GP are independent random variables at each discrete time point. Finally, this (conditioned) white noise process is further interpreted in terms of the Euler discretization of an ordinary
+
+SDE leading to effective drift and diffusions. For the latter SDE, one can apply Girsanov's theorem and the corresponding ELBO (conditioned on the inducing points) to perform inference.
+
+Note, that their current derivation of effective drift and diffusion relies on the Euler discretization of SDE. For higher order SDE solvers, the approximation has to be adapted, which requires new derivations. As a main difference, in our paper, the approximation is not based on the discretization in the time domain but of the discretization of an integral representation (Prop. 1) over a spectrum of decay constants of Ornstein-Uhlenbeck (OU) processes (driven by the same Wiener noise). Since each OU process already represents a noise process with temporal correlations, we can expect that a linear combination of a small number of such processes can yield a good approximation of the covariance of fBM over some given time interval. Our approximation leads to a system of SDEs (without conditioning) for which the ELBO can be easily obtained. Since the time discretization of the resulting SDE is performed after the OU approximation, any SDE solver can be directly applied. With this flexibility, in our paper, we have chosen the second order Stratonovich-Milstein solver.
+
+State dependent diffusions. For the case, where the diffusion  $\sigma(X,t)$  explicitly depends on the state variable  $X$ , our Markovian approximation results in a 'standard' white noise SDE for the augmented system. As such, it does not suffer from problems with proper definitions of stochastic integrals as compared to the original SDE driven by fBM for such cases. Hence, a straightforward Itô-interpretation of our augmented SDE is, in principle, possible. This might indicate, at first glance, that simple numerical solvers such as Euler's method could be sufficient for simulating the augmented SDE required for computing posterior expectations for the ELBO. While this point needs further theoretical investigation, preliminary simulations for simple models with state-dependent diffusions indicate that an Euler approximation (in accordance with known results for direct simulations of SDE driven by fBM (Lysy & Pillai, 2013)) quickly lead to deviations from known analytical results. Hence, for state dependent diffusions, we resort to the Stratonovich interpretation of the augmented system and use corresponding higher order solvers Kidger (2021)<sup>2</sup>. This approach yields excellent (pathwise) agreements with exact analytical results as we show in Sec. 5. Although the ELBO for SDE is derived from Girsanov's change of measure theorem for Itô-SDE, by the known correspondence (resulting in a change of drift functions, when diffusions are state dependent) (Gardiner et al., 1985) between Itô and Stratonovich SDE we conclude that within this approach, optimisation of the ELBO with respect to model parameters will also yield the corresponding estimates for the Stratonovich interpretation.
+
+On initial values for "Type I". The initial values for "Type I" can be understood as resulting from an OU-process which was started at some negative time  $t \to -\infty$  so that
+
+$$
+Y _ {k} ^ {(I)} (0) = \int_ {- \infty} ^ {0} e ^ {\gamma_ {k} s} \mathrm {d} W (s) \tag {25}
+$$
+
+and  $Y_{k}^{(I)}(0)$  can be considered as samples from the joint stationary distribution. Because the stationary distribution is normal (Harms & Stefanovits, 2019, Theorem 2.16) we can simply sample initial states of the  $Y_{k}(t)$  processes for Type I with covariance  $\mathbb{E}\left[Y_i(0)Y_j(0)\right]$ . Using Itô isometry (Øksendal & Øksendal, 2003):
+
+$$
+\begin{array}{l} \mathbb {E} \left[ Y _ {i} (0) Y _ {j} (0) \right] = \mathbb {E} \left[ \int_ {- \infty} ^ {0} e ^ {\gamma_ {i} s} d W (s) \int_ {- \infty} ^ {0} e ^ {\gamma_ {j} s} d W (s) \right] (26) \\ = \int_ {- \infty} ^ {0} e ^ {(\gamma_ {i} + \gamma_ {j}) s} d s (27) \\ = \frac {1}{\gamma_ {i} + \gamma_ {j}}. (28) \\ \end{array}
+$$
+
+# B PROOFS AND FURTHER THEORETICAL DETAILS
+
+# B.1 THE GIRSANOV THEOREM II AND THE KL DIVERGENCE OF MEASURES
+
+We now state the variation II of the Girsanov theorem (Øksendal & Øksendal, 2003) in our notation. Let  $X(t) \in \mathbb{R}^n$  be an Itô process w.r.t. measure  $P$  of the form:
+
+$$
+\mathrm {d} X (t) = b _ {\theta} (X (t), t) \mathrm {d} t + \sigma_ {\theta} (X (t), t) \mathrm {d} W (t), \tag {29}
+$$
+
+where  $0 \leq t \leq T$ ,  $W(t) \in \mathbb{R}^m$ ,  $b_{\theta}(X(t),t) \in \mathbb{R}^n$  and  $\sigma_{\theta}(X(t),t) \in \mathbb{R}^{n \times m}$ . Define a measure  $Q$  via:
+
+$$
+\frac {d Q}{d P} = M _ {T} := \exp \left[ - \int_ {0} ^ {T} u (X (t), t) d W (t) - \frac {1}{2} \int_ {0} ^ {T} u ^ {2} (X (t), t) d t \right]. \tag {30}
+$$
+
+Then
+
+$$
+W ^ {\prime} (t) := \int_ {0} ^ {T} u (X (t), t) \mathrm {d} t + W (T) \tag {31}
+$$
+
+is a Brownian motion w.r.t.  $Q$  and the process  $X(t)$  has the following representation in terms of  $B'(t)$ :
+
+$$
+\mathrm {d} X (t) = \alpha_ {\theta} (X (t), t) \mathrm {d} t + \sigma_ {\theta} (X (t), t) \mathrm {d} W ^ {\prime} (t), \tag {32}
+$$
+
+where the new drift is:
+
+$$
+\alpha_ {\theta} (X (t), t) = b _ {\theta} (X (t), t) - \sigma_ {\theta} (X (t), t) u (X (t), t). \tag {33}
+$$
+
+We can also rewrite the Radon-Nykodim derivative in Eq. (30) as
+
+$$
+\begin{array}{l} \frac {d Q}{d P} = \exp \left[ \int_ {0} ^ {T} u (X (t), t) \mathrm {d} W (t) - \frac {1}{2} \int_ {0} ^ {T} u ^ {2} (X (t), t) \mathrm {d} t \right] (34) \\ = \exp \left[ \int_ {0} ^ {T} u (X (t), t) (\mathrm {d} W ^ {\prime} (t) + u (X (t), t) \mathrm {d} t) - \frac {1}{2} \int_ {0} ^ {T} u (X (t), t) \mathrm {d} t \right] (35) \\ = \exp \left[ \int_ {0} ^ {T} u (X (t), t) \mathrm {d} W ^ {\prime} (t) + \frac {1}{2} \int_ {0} ^ {T} u (X (t), t) \mathrm {d} t \right]. (36) \\ \end{array}
+$$
+
+Thus, similar to Li et al. (2020), we get the KL divergence
+
+$$
+E _ {Q} \left[ \ln \frac {d Q}{d P} \right] = \frac {1}{2} \int_ {0} ^ {T} E _ {Q} \left[ u ^ {2} (X (t), t) \right] d t. \tag {37}
+$$
+
+# C COVARIANCES
+
+The full derivation of covariances between some processes relevant to this work are described here.
+
+Fractional Brownian motion (Type II). Using Itô isometry (Øksendal & Øksendal, 2003) we know that for  $t > s$
+
+$$
+\mathbb {E} \left[ \int_ {0} ^ {t} (t - u) ^ {H - 1 / 2} d W _ {u} \int_ {0} ^ {s} (s - u) ^ {H - 1 / 2} d W _ {u} \right] = \int_ {0} ^ {s} ((t - u) (s - u)) ^ {H - 1 / 2} \mathrm {d} u \tag {38}
+$$
+
+Thus
+
+$$
+\mathbb {E} \left[ B _ {H} ^ {(I I)} (t) B _ {H} ^ {(I I)} (s) \right] = \frac {1}{\Gamma^ {2} (H + 1 / 2)} \int_ {0} ^ {s} ((t - u) (s - u)) ^ {H - 1 / 2} d u \tag {39}
+$$
+
+OU-processes driven by the same Wiener process. Observe two Ornstein-Uhlenbeck processes driven by the same Wiener process:
+
+$$
+\begin{array}{l} \left\{ \begin{array}{l} d Y _ {i} (t) = - \gamma_ {i} Y _ {i} (t) \mathrm {d} t + d W (t) \\ d W (t) = \gamma_ {i} Y _ {i} (t) \mathrm {d} t + W (t) \end{array} \right. \tag {40} \\ \left\lfloor d Y _ {j} (t) = - \gamma_ {j} Y _ {j} (t) \mathrm {d} t + d W (t) \right. \\ \end{array}
+$$
+
+Their covariance can be written as:
+
+$$
+\begin{array}{l} \operatorname {C o v} \left(Y _ {i} (t), Y _ {j} (t)\right) = \mathbb {E} \left[ \left(Y _ {i} (t) - \mathbb {E} \left[ Y _ {i} (t) \right]\right) \left(Y _ {j} (t) - \mathbb {E} \left[ Y _ {j} (t) \right]\right) \right] (41) \\ = \mathbb {E} \left[ Y _ {i} (t) Y _ {j} (t) \right] (42) \\ = \mathbb {E} \left[ \int_ {0} ^ {t} e ^ {- \gamma_ {i} (t - s)} d W (s) \int_ {0} ^ {t} e ^ {- \gamma_ {j} (t - s)} d W (s) \right] (43) \\ = \int_ {0} ^ {t} e ^ {- (\gamma_ {i} + \gamma_ {j}) (t - s)} d s (44) \\ = \frac {1}{\gamma_ {i} + \gamma_ {j}} - \frac {e ^ {- (\gamma_ {i} + \gamma_ {j}) t}}{\gamma_ {i} + \gamma_ {j}} (45) \\ \end{array}
+$$
+
+where Eq. (44) is obtained following the Itô isometry (Øksendal & Øksendal, 2003).
+
+Markov approximated fractional Brownian motion (Type I). Recall that (Dfn. 3)
+
+$$
+\hat {B} _ {H} ^ {(I)} (t) = \sum_ {k} \omega_ {k} \left(Y _ {k} (t) - Y _ {k} (0)\right)
+$$
+
+where (Eq. (6))
+
+$$
+Y _ {k} (t) - Y _ {k} (0) = Y _ {k} (0) \left(e ^ {- \gamma_ {k} t} - 1\right) + \int_ {0} ^ {t} e ^ {- \gamma_ {k} (t - s)} d W (s)
+$$
+
+and  $\mathbb{E}[Y_i(0)Y_j(0)] = \frac{1}{\gamma_i + \gamma_j}$  (Eq. (28)). For  $t > \tau$ :
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \hat {B} _ {H} ^ {(I)} (t) \hat {B} _ {H} ^ {(I)} (\tau) \right] = \mathbb {E} \left[ \left(\sum_ {k} \omega_ {k} \left(Y _ {k} (t) - Y _ {k} (0)\right)\right) \left(\sum_ {k} \omega_ {k} \left(Y _ {k} (\tau) - Y _ {k} (0)\right)\right) \right] (46) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \mathbb {E} \left[ \left(Y _ {i} (t) - Y _ {i} (0)\right) \left(Y _ {j} (\tau) - Y _ {j} (0)\right) \right] (47) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \mathbb {E} \left[ \left(Y _ {i} (0) \left(e ^ {- \gamma_ {i} t} - 1\right) + \int_ {0} ^ {t} e ^ {- \gamma_ {i} (t - s)} \mathrm {d} W (s)\right) \right. (48) \\ \left. \cdot \left(Y _ {j} (0) \left(e ^ {- \gamma_ {j} \tau} - 1\right) + \int_ {0} ^ {\tau} e ^ {- \gamma_ {j} (\tau - s)} \mathrm {d} W (s)\right) \right] \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \left(\mathbb {E} \left[ Y _ {i} (0) Y _ {j} (0) \right] \left(e ^ {- \gamma_ {i} t} - 1\right) \left(e ^ {- \gamma_ {j} \tau} - 1\right) \right. (49) \\ + \int_ {0} ^ {\tau} \left(e ^ {- \gamma_ {i} (t - s)} e ^ {- \gamma_ {j} (\tau - s)} d s\right) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \frac {1 - e ^ {- \gamma_ {i} t} - e ^ {- \gamma_ {j} \tau} + e ^ {- \gamma_ {i} (t - \tau)}}{\gamma_ {i} + \gamma_ {j}} (50) \\ \end{array}
+$$
+
+Markov approximated fractional Brownian motion (Type II). Recall that (Dfn. 3)
+
+$$
+\hat {B} _ {H} ^ {(I I)} (t) = \sum_ {k} \omega_ {k} Y _ {k} (t), \qquad Y _ {k} (0) = 0, \quad k = 1, \dots , K
+$$
+
+and for  $t > \tau$ :
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \hat {B} _ {H} ^ {(I I)} (t) \hat {B} _ {H} ^ {(I I)} (\tau) \right] = \mathbb {E} \left[ \left(\sum_ {k} \omega_ {k} Y _ {k} (t)\right) \left(\sum_ {k = 1} ^ {K} \omega_ {k} Y _ {k} (\tau)\right) \right] (51) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \mathbb {E} \left[ Y _ {i} (t) Y _ {j} (\tau) \right] (52) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \mathbb {E} \left[ \int_ {0} ^ {t} e ^ {- \gamma_ {i} (t - s)} \mathrm {d} W (s) \int_ {0} ^ {\tau} e ^ {- \gamma_ {j} (\tau - s)} \mathrm {d} W (s) \right] (53) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \int_ {0} ^ {\tau} e ^ {- \gamma_ {i} (t - s) - \gamma_ {j} (\tau - s)} d s (54) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \left(\frac {e ^ {- \gamma_ {i} (t - \tau)}}{\gamma_ {i} + \gamma_ {j}} - \frac {e ^ {- \gamma_ {i} t - \gamma_ {j} \tau}}{\gamma_ {i} + \gamma_ {j}}\right) (55) \\ \end{array}
+$$
+
+fBM and MA-fBM (Type I). Since (Dfn. 3)
+
+$$
+\hat {B} _ {H} ^ {(I)} (t) = \sum_ {k} \omega_ {k} \left(Y _ {k} (t) - Y _ {k} (0)\right)
+$$
+
+where (Eq. (6))
+
+$$
+Y _ {k} (t) - Y _ {k} (0) = Y _ {k} (0) \left(e ^ {- \gamma_ {k} t} - 1\right) + \int_ {0} ^ {t} e ^ {- \gamma_ {k} (t - s)} \mathrm {d} W (s)
+$$
+
+and (Eq. (25))
+
+$$
+Y _ {k} (0) = \int_ {- \infty} ^ {0} e ^ {\gamma_ {k} s} d W (s).
+$$
+
+we can write
+
+$$
+\hat {B} _ {H} ^ {(I)} (t) = \sum_ {k} \omega_ {k} \left(\left(e ^ {- \gamma_ {k} t} - 1\right) \int_ {- \infty} ^ {0} e ^ {\gamma_ {k} s} \mathrm {d} W (s) + \int_ {0} ^ {t} e ^ {- \gamma_ {k} (t - s)} \mathrm {d} W (s)\right). \tag {56}
+$$
+
+This leads to the following derivation (using Itô isometry (Øksendal & Øksendal, 2003)):
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \hat {B} _ {H} ^ {(I)} (t) B _ {H} ^ {(I)} (t) \right] = \frac {1}{\Gamma (H + 1 / 2)} \sum_ {k} \omega_ {k} \mathbb {E} \left[ \left(\int_ {- \infty} ^ {0} \left((t - s) ^ {H - 1 / 2} - (- s) ^ {H - 1 / 2}\right) \mathrm {d} W (s) \right. \right. \\ \left. + \int_ {0} ^ {t} (t - s) ^ {H - 1 / 2} \mathrm {d} W (s)\right) \\ \left. \cdot \left(\left(e ^ {- \gamma_ {k} t} - 1\right) \int_ {- \infty} ^ {0} e ^ {\gamma_ {k} s} \mathrm {d} W (s) + \int_ {0} ^ {t} e ^ {- \gamma_ {k} (t - s)} \mathrm {d} W (s)\right) \right] (57) \\ = \frac {1}{\Gamma (H + 1 / 2)} \sum_ {k} \omega_ {k} \left(\left(e ^ {- \gamma_ {k} t} - 1\right) \int_ {- \infty} ^ {0} \left((t - s) ^ {H - 1 / 2} - (- s) ^ {H - 1 / 2}\right) e ^ {\gamma_ {k} s} d s \right. \\ \left. + \int_ {0} ^ {t} (t - s) ^ {H - 1 / 2} e ^ {- \gamma_ {k} (t - s)} \mathrm {d} s\right) (58) \\ = \sum_ {k} \omega_ {k} \frac {2 - e ^ {- \gamma_ {k} t} - Q (H + 1 / 2 , \gamma_ {k} t) e ^ {\gamma_ {k} t}}{\gamma_ {k} ^ {H + 1 / 2}} (59) \\ \end{array}
+$$
+
+where  $Q(z,x) = \frac{1}{\Gamma(z)}\int_{x}^{\infty}t^{z - 1}e^{-t}\mathrm{d}t$  is the regularized upper incomplete gamma function.
+
+fBM and MA-fBM (Type II).
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \hat {B} _ {H} ^ {(I I)} (t) B _ {H} ^ {(I I)} (t) \right] = \frac {1}{\Gamma (H + 1 / 2)} \sum_ {k} \omega_ {k} \mathbb {E} \left[ \int_ {0} ^ {t} e ^ {- \gamma_ {k} (t - s)} \mathrm {d} W (s) \int_ {0} ^ {t} (t - s) ^ {H - 1 / 2} \mathrm {d} s \right] (60) \\ = \frac {1}{\Gamma (H + 1 / 2)} \sum_ {k} \omega_ {k} \int_ {0} ^ {t} e ^ {- \gamma_ {k} (t - s)} (t - s) ^ {H - 1 / 2} d s (61) \\ = \sum_ {k} \omega_ {k} \frac {P (H + 1 / 2 , \gamma_ {k} t)}{\gamma_ {k} ^ {H + 1 / 2}} (62) \\ \end{array}
+$$
+
+where  $P(z,x) = \frac{1}{\Gamma(z)}\int_0^x t^{z - 1}e^{-t}\mathrm{d}t$  is the regularized lower incomplete gamma function.
+
+# D CHOOSING  $\omega_{k}$  VALUES
+
+# D.1 BASELINE
+
+To approximate the integral in equation (8) for  $H < 1/2$  we do a piece-wise linear approximation of the integral between the known  $Y_{k}(t)$  values:
+
+$$
+\sum_ {k = 1} ^ {K} \omega_ {k} Y _ {k} (t) = \sum_ {k = 1} ^ {K - 1} \int_ {\gamma_ {k}} ^ {\gamma_ {k + 1}} \left(\frac {\gamma_ {k + 1} - \gamma}{\gamma_ {k + 1} - \gamma_ {k}} Y _ {k} (t) + \frac {\gamma - \gamma_ {k}}{\gamma_ {k + 1} - \gamma_ {k}} Y _ {k + 1} (t)\right) \mu (\gamma) d \gamma \tag {63}
+$$
+
+For  $H > 1 / 2$  we approximate  $\partial_{\gamma}Y_{\gamma}(t)$  with finite differences:
+
+$$
+\sum_ {k = 1} ^ {K} \omega_ {k} Y _ {k} (t) = \sum_ {k = 1} ^ {K - 1} - \frac {Y _ {k + 1} (t) - Y _ {k} (t)}{\gamma_ {k + 1} - \gamma_ {k}} \int_ {\gamma_ {k}} ^ {\gamma_ {k + 1}} \nu (\gamma) \mathrm {d} \gamma \tag {64}
+$$
+
+This leads to the following proposal for  $\omega_{k}$ :
+
+$$
+\omega_ {k} = \left\{ \begin{array}{l} \frac {1}{\Gamma (\alpha) \Gamma (1 - \alpha)} \left(\mathbf {1} _ {\mathrm {k} > 1} \frac {\frac {\gamma_ {k} ^ {2 - \alpha} - \gamma_ {k - 1} ^ {2 - \alpha}}{2 - \alpha} - \gamma_ {k - 1} \frac {\gamma_ {k} ^ {1 - \alpha} - \gamma_ {k - 1} ^ {1 - \alpha}}{1 - \alpha}}{\gamma_ {k} - \gamma_ {k - 1}} + \mathbf {1} _ {\mathrm {k} <   \mathrm {K}} \frac {\gamma_ {k + 1} \frac {\gamma_ {k + 1} ^ {1 - \alpha} - \gamma_ {k} ^ {1 - \alpha}}{1 - \alpha} - \frac {\gamma_ {k + 1} ^ {2 - \alpha} - \gamma_ {k} ^ {2 - \alpha}}{2 - \alpha}}{\gamma_ {k + 1} - \gamma_ {k}}\right), H <   1 / 2 \\ \frac {- 1}{(2 - \alpha) \Gamma (\alpha) \Gamma (2 - \alpha)} \left(\mathbf {1} _ {\mathrm {k} > 1} \frac {\gamma_ {k} ^ {2 - \alpha} - \gamma_ {k - 1} ^ {2 - \alpha}}{\gamma_ {k} - \gamma_ {k - 1}} - \mathbf {1} _ {\mathrm {k} <   \mathrm {K}} \frac {\gamma_ {k + 1} ^ {2 - \alpha} - \gamma_ {k} ^ {2 - \alpha}}{\gamma_ {k + 1} - \gamma_ {k}}\right), H > 1 / 2 \end{array} \right. \tag {65}
+$$
+
+where  $\alpha = H + 1 / 2$
+
+# D.2 A PROOF FOR THE OPTIMIZED  $\omega_{k}$  VALUES
+
+To optimize  $\omega_{k}$  values, we first provide a closed form expression for the approximation error and then show how we can solve for the  $\omega_{k}$  that minimize this error.
+
+Type I. We will start by optimizing  $\omega_{k}$  for Type I. Consider the error:
+
+$$
+\begin{array}{l} \mathcal {E} ^ {(I)} (\boldsymbol {\omega}) = \int_ {0} ^ {T} \mathbb {E} \left[ \left(\hat {B} _ {H} ^ {(I)} (t) - B _ {H} ^ {(I)} (t)\right) ^ {2} \right] \mathrm {d} t (66) \\ = \int_ {0} ^ {T} \left(\mathbb {E} \left[ \hat {B} _ {H} ^ {(I)} (t) ^ {2} \right] + \mathbb {E} \left[ B _ {H} ^ {(I)} (t) ^ {2} \right] - 2 \mathbb {E} \left[ \hat {B} _ {H} ^ {(I)} (t) B _ {H} ^ {(I)} (t) \right]\right) d t (67) \\ \end{array}
+$$
+
+Using Eqs. (1), (50) and (59)
+
+$$
+\begin{array}{l} \mathcal {E} ^ {(I)} (\omega) = \int_ {0} ^ {T} \left(\sum_ {i, j} \omega_ {i} \omega_ {j} \frac {2 - e ^ {- \gamma_ {i} t} - e ^ {- \gamma_ {j} t}}{\gamma_ {i} + \gamma_ {j}} + t ^ {2 H} \right. \\ \left. - 2 \sum_ {k} \omega_ {k} \frac {2 - e ^ {- \gamma_ {k} t} - Q (H + 1 / 2 , \gamma_ {k} t) e ^ {\gamma_ {k} t}}{\gamma_ {k} ^ {H + 1 / 2}}\right) d t (68) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \frac {2 T + \frac {e ^ {- \gamma_ {i} T} - 1}{\gamma_ {i}} + \frac {e ^ {- \gamma_ {j} T} - 1}{\gamma_ {j}}}{\gamma_ {i} + \gamma_ {j}} + \frac {T ^ {2 H + 1}}{2 H + 1} \\ - 2 \sum_ {k} \omega_ {k} \left(\frac {2 T}{\gamma_ {k} ^ {H + 1 / 2}} - \frac {T ^ {H + 1 / 2}}{\gamma_ {k} \Gamma (H + 3 / 2)} + \frac {e ^ {- \gamma_ {k} T} - Q (H + 1 / 2 , \gamma_ {k} T) e ^ {\gamma_ {k} T}}{\gamma_ {k} ^ {H + 3 / 2}}\right) (69) \\ \end{array}
+$$
+
+This leads to the quadratic form  $\mathcal{E}^{(I)}(\pmb {\omega}) = \pmb{\omega}^T\pmb {A}^{(I)}\pmb {\omega} - 2\pmb{b}^{(I)^T}\pmb {\omega} + c^{(I)}$  with
+
+$$
+\boldsymbol {A} _ {i, j} ^ {(I)} = \frac {2 T + \frac {e ^ {- \gamma_ {i} T} - 1}{\gamma_ {i}} + \frac {e ^ {- \gamma_ {j} T} - 1}{\gamma_ {j}}}{\gamma_ {i} + \gamma_ {j}} \tag {70}
+$$
+
+$$
+\boldsymbol {b} _ {k} ^ {(I)} = \frac {2 T}{\gamma_ {k} ^ {H + 1 / 2}} - \frac {T ^ {H + 1 / 2}}{\gamma_ {k} \Gamma (H + 3 / 2)} + \frac {e ^ {- \gamma_ {k} T} - Q (H + 1 / 2 , \gamma_ {k} T) e ^ {\gamma_ {k} T}}{\gamma_ {k} ^ {H + 3 / 2}} \tag {71}
+$$
+
+$$
+c ^ {(I)} = \frac {T ^ {2 H + 1}}{2 H + 1}. \tag {72}
+$$
+
+Type II. We now repeat a similar procedure for the Type II.
+
+$$
+\begin{array}{l} \mathcal {E} ^ {(I I)} (\boldsymbol {\omega}) = \int_ {0} ^ {T} \mathbb {E} \left[ \left(\hat {B} _ {H} ^ {(I I)} (t) - B _ {H} ^ {(I I)} (t)\right) ^ {2} \right] \mathrm {d} t (73) \\ = \int_ {0} ^ {T} \left(\mathbb {E} \left[ \hat {B} _ {H} ^ {(I I)} (t) ^ {2} \right] + \mathbb {E} \left[ B _ {H} ^ {(I I)} (t) ^ {2} \right] - 2 \mathbb {E} \left[ \hat {B} _ {H} ^ {(I I)} (t) B _ {H} ^ {(I I)} (t) \right]\right) d t (74) \\ \end{array}
+$$
+
+Using Eqs. (2), (55) and (62)
+
+$$
+\begin{array}{l} \mathcal {E} ^ {(I I)} (\omega) = \int_ {0} ^ {T} \sum_ {i, j} \omega_ {i} \omega_ {j} \frac {1 - e ^ {- (\gamma_ {i} + \gamma_ {j}) t}}{\gamma_ {i} + \gamma_ {j}} + \frac {t ^ {2 H}}{2 H \Gamma (H + 1 / 2) ^ {2}} - 2 \sum_ {k} \omega_ {k} \frac {P (H + 1 / 2 , \gamma_ {k} t)}{\gamma_ {k} ^ {H + 1 / 2}} d t (75) \\ = \sum_ {i, j} \omega_ {i} \omega_ {j} \frac {T + \frac {e ^ {- (\gamma_ {i} + \gamma_ {j}) T} - 1}{\gamma_ {i} + \gamma_ {j}}}{\gamma_ {i} + \gamma_ {j}} + \frac {T ^ {2 H + 1}}{2 H (2 H + 1) \Gamma (H + 1 / 2) ^ {2}} (76) \\ - 2 \sum_ {k} \omega_ {k} \left(\frac {T}{\gamma_ {k} ^ {H + 1 / 2}} P (H + 1 / 2, \gamma_ {k} T) - \frac {H + 1 / 2}{\gamma_ {k} ^ {H + 3 / 2}} P (H + 3 / 2, \gamma_ {k} T)\right) (77) \\ \end{array}
+$$
+
+This leads to the quadratic form  $\mathcal{E}^{(II)}(\omega) = \omega^T\pmb{A}^{(II)}\pmb{\omega} - 2\pmb{b}^{(II)^T}\pmb{\omega} + c^{(II)}$  with
+
+$$
+\boldsymbol {A} _ {i, j} ^ {(I I)} = \frac {T + \frac {e ^ {- (\gamma_ {i} + \gamma_ {j}) T} - 1}{\gamma_ {i} + \gamma_ {j}}}{\gamma_ {i} + \gamma_ {j}} \tag {78}
+$$
+
+$$
+\boldsymbol {b} _ {k} ^ {(I I)} = \frac {T}{\gamma_ {k} ^ {H + 1 / 2}} P (H + 1 / 2, \gamma_ {k} T) - \frac {H + 1 / 2}{\gamma_ {k} ^ {H + 3 / 2}} P (H + 3 / 2, \gamma_ {k} T) \tag {79}
+$$
+
+$$
+c ^ {(I I)} = \frac {T ^ {2 H + 1}}{2 H (2 H + 1) \Gamma (H + 1 / 2) ^ {2}}. \tag {80}
+$$
+
+Exactly one solution for  $\omega$ . There is exactly one solution if  $A^{(I,II)}$  is positive definite, which is defined as
+
+$$
+\boldsymbol {\omega} ^ {T} \boldsymbol {A} ^ {(I, I I)} \boldsymbol {\omega} > 0 \text {f o r a l l} \boldsymbol {\omega} \in \mathbb {R} ^ {K} \backslash \{\mathbf {0} \}. \tag {81}
+$$
+
+Recall that
+
+$$
+\boldsymbol {\omega} ^ {T} \boldsymbol {A} ^ {(I, I I)} \boldsymbol {\omega} = \int_ {0} ^ {T} \mathbb {E} \left[ \hat {B} _ {H} ^ {(I, I I)} (t) ^ {2} \right] \mathrm {d} t \tag {82}
+$$
+
+thus there is exactly one solution if
+
+$$
+\int_ {0} ^ {T} \mathbb {E} \left[ \hat {B} _ {H} ^ {(I, I I)} (t) ^ {2} \right] \mathrm {d} t > 0 \text {f o r a l l} \boldsymbol {\omega} \in \mathbb {R} ^ {K} \backslash \{\mathbf {0} \}. \tag {83}
+$$
+
+Recall that  $\hat{B}_H^{(I,II)}(t)$  is a linear combination of  $K$  Ornstein-Uhlenbeck processes with speed of mean reversion  $\gamma_{k}$  driven by the same Brownian motion. Under the trivial conditions that  $\gamma_{i} \neq \gamma_{j}$  (so they can not cancel out) and  $\gamma_{k} < \infty$ , this will never be 0. Hence the last inequality holds unless  $\hat{B}_H^{(I,II)}(t) = 0$ . This concludes the proof.
+
+# D.3 NUMERICALLY STABLE IMPLEMENTATION OF  $Q(z, x)e^{x}$
+
+The term  $Q(H + 1/2, \gamma_k T) e^{\gamma_k T}$  in Prop. 5 leads to numerical instability, since  $\gamma_k T$  is typically a high number (for the highest  $\gamma_k$ ). On the other hand,  $Q(H + 1/2, \gamma_k T)$  is a low number for high  $\gamma_k T$ . Our stable implementation makes use of a continued fraction (Cuyt et al., 2008, eq. (12.6.17)), using the 'Kettenbruch' notation (Cuyt et al., 2008, sec. 1.1) for continued fractions:
+
+$$
+\begin{array}{l} Q (H + 1 / 2, \gamma_ {k} T) e ^ {\gamma_ {k} T} = \frac {\Gamma (H + 1 / 2 , \gamma_ {k} T)}{\Gamma (H + 1 / 2)} e ^ {\gamma_ {k} T} (84) \\ = \frac {1}{\Gamma (H + 1 / 2) (\gamma_ {k} T) ^ {H + 1 / 2}} \underset {m = 1} {\overset {\infty} {\operatorname {K}}} \left(\frac {a _ {m} (H + 1 / 2) / (\gamma_ {k} T)}{1}\right) (85) \\ \end{array}
+$$
+
+where  $a_{m}(a)$  is given by
+
+$$
+a _ {1} (a) = 1, \quad a _ {2 j} (a) = j - a, \quad a _ {2 j + 1} (a) = j, \quad j \geq 1 \tag {86}
+$$
+
+In practice we observe better accuracy with the original equation for  $\gamma_k T < 10$ , where it is still stable, and only need 5 fractions to approximate the equation for  $\gamma_k T > 10$ .
+
+# E DETAILS ON MODEL ARCHITECTURES & HYPERPARAMETERS
+
+# E.1 FOU BRIDGE
+
+For all experiments,  $K = 5$  and  $\gamma_{k} = (\frac{1}{20},\dots ,20)$ . We use "Type I" and the optimal definitions for  $\omega_{k}$ , with a time horizon  $T = 6$ . The control function is a neural network with two hidden layers of each 1000 neurons, with tanh activation function. Its input is represented as  $[\sin t,\cos t,X(t),Y_1(t),\ldots ,Y_K(t)]$ . The control function is initialized so that its output is 0 at the start of training. Models are trained for 2000 training steps with a batch size of 32. We use the Adam (Kingma & Ba, 2014) optimizer with fixed learning rate  $10^{-3}$ . We use the Stratonovich-Milstein SDE solver (Kidger, 2021) with an integration step of 0.01. The length of the bridge  $T = 2$  and observation noise  $\sigma = 0.1$ .
+
+# E.2 TIME DEPENDENT HURST INDEX
+
+We directly compare our method with the data and estimate found in the published codebase of Tong et al. (2022) $^3$ . We choose  $K = 5$  and  $\gamma_{k} = \left(\frac{1}{20}, \dots, 20\right)$  and use "Type II" (to match the data and noise type in Tong et al. (2022)). The optimal definitions for  $\omega_{k}$ , with time horizon  $T = 2$  are used. The control function is a neural network with two hidden layers of each 1000 neurons, with tanh activation function. Its input is represented as  $[\sin t, \cos t, \sin 2t, \cos 2t, \dots, \sin 5t, \cos 5t, X(t), Y_1(t), \dots, Y_K(t)]$ . The model is trained for 1000 training steps with a batch size of 4. We use the Adam (Kingma & Ba, 2014) optimizer with a learning rate  $3 \times 10^{-3}$ , scheduled with cosine decay to  $3 \times 10^{-4}$  by the end of training. We use the Stratonovich-Milstein SDE solver (Kidger, 2021). The integration step is 0.005 and observation noise  $\sigma = 0.025$  (both identical to Tong et al. (2022)).
+
+# E.3 LATENTVIDEOMODEL
+
+Stochastic moving MNIST. For the MA-fBM model,  $K = 5$  and  $\gamma_{k} = (\frac{1}{20},\dots ,20)$ . We use "Type I" and the corresponding definitions for  $\omega_{k}$ , with a time horizon  $T = 2.4$ . For the BM model,  $K = 1$ ,  $\gamma_{1} = 0$  and  $\omega = 1$ , which naturally corresponds to white Brownian motion. The number of latent dimensions  $D = 6$ .
+
+The encoder model consists of four blocks, containing a convolution layer, maxpool, groupnorm and SiLU activation. Each block reduces spatial dimension by 2, and the number of features in each block is (64, 128, 256, 256). The last output is flattened and is the input of a dense layer, with  $h$  as output with 64 features.
+
+The median over the time axis of  $h$  is fed into a two layers neural network to produce the static content vector  $w$ . Since the median is permutation invariant,  $w$  contains no dynamic information, only static information.  $w$  also has 64 features.
+
+The context model consists of two subsequent  $1 - D$  convolutions in the temporal dimension. Thus, information is shared over different frames, which is necessary for inference. The output of this model is  $g$ .
+
+To start the SDE integration, we need an initial state that is conditioned on the data. We define a three layer neural network model that receives  $(g_{1},h_{1},h_{2},h_{3})$  and outputs the parameters of the posterior distribution  $q_{x_1}$  of the initial state of the SDE.  $x_{1}$  is sampled from  $q_{x_1}$ , which we model as a diagonal Normal distribution. The parameters of a prior model  $p_{x_1}$  are also optimized, and the Kullback-Leibler divergence  $D_{\mathrm{KL}}(p_{x_1},q_{x_1})$  is added to the loss function. This approach for training neural SDEs is similar to others in literature (Li et al., 2020).
+
+The prior drift  $b_{\theta}(X,t)$  and the control function  $u(Z(t),t)$  have the same architecture, a neural network with two hidden layers of each 200 neurons, with tanh activation functions. The shared diffusion  $\sigma_{\theta}(X,t)$  is implemented so that the noise is commutative to allow Milstein solvers (Li et al., 2020; Kidger et al., 2021), i.e.  $\sigma_{\theta}(X,t)$  is diagonal and the  $i$ -th component on the diagonal only receives  $X_{i}(t)$  as input, where we have defined  $D$  separate neural networks for each component. Each neural network has two layers with 200 neurons and tanh activations.
+
+$b_{\theta}$  and  $\sigma_{\theta}$  receive  $X(t)$  as input. The control function a concatenated vector of  $(X(t),Y_1(t),\ldots ,Y_K(t),g(t))$ $g(t)$  is a linear interpolation of  $g$  at time  $t$  . This enables the control function to use appropriate information to be able to steer the process correctly.
+
+The resulting states  $x$  after integration of the SDE are fed, together with the static content vector  $w$  in the decoder model. The decoder model has first a dense layer. The outputs of this first layer are shaped in a  $4 \times 4$  spatial grad. Subsequently, four blocks with a convolution layer, groupnorm, a spatial nearest neighbour upsampling layer and a SiLU activation. Thus, the model reaches the correct resolution of  $64 \times 64$ . Two additional convolution layers with SiLU activation and a final sigmoid activation complete the decoder model.
+
+We train on sequences of 25 frames, with a time length of 2.4 (0.1 per frame). The frames have resolution  $64 \times 64$  and 1 color channel. Each model was trained for 187500 training steps with a batch size of 32. We use the Adam (Kingma & Ba, 2014) optimizer with fixed learning rate  $3 \times 10^{-4}$ . We use the Stratonovich-Milstein SDE solver (Kidger, 2021) with an integration step of 0.033 (3 integration steps per data frame). Models were trained on a single NVIDIA GeForce RTX 4090, which takes around 39 hours for one model.
+
+Double pendulum. We use the train-test split from the original dataset (Asseman et al., 2018). The videos are recorded with a high speed camera, we used every 10th frame to increase the challenge of the dataset. We resized the frames to a resolution of  $128 \times 128$  resolution. Therefore, we added one block to the encoder and decoder model to achieve this resolution, compared to the model for stochastic moving MNIST. We did not use the static content vector  $w$ , since there is minimal static information in this dataset, and used  $D = 8$  latent dimensions. The models were trained for 124916 training steps, and we trained around 32 hours for one model. Beyond these outlined differences, all other details are equal to the stochastic moving MNIST model.
+
+# F ADDITIONAL EXPERIMENTAL RESULTS
+
+# F.1 GENERATED TRAJECTORIES OF MA-FBM FOR VARYING  $K$
+
+Included here are some of the trajectories used to calculate the MSE of the generated trajectories for MA-fBM for varying  $K$  (Fig. 7). We show trajectories of MA-fBM with our approach (Sec. 4.1) and the baseline method (cf. App. D.1) for choosing  $\omega_{k}$ . True paths are plotted in black, the approximations with varying  $K$  in a color-scale as indicated in the legends, see Figs. 9 to 12 and 14 to 16. Our method quickly converges to the true path for increasing  $K$ , while much slower for the baseline method.
+
+![](images/18af6c98d504f9eec91c1705a41f96a9aa190e0b16239f4ab120b8036126a153.jpg)  
+(a) Baseline
+
+![](images/a5222a97c399d0071349bd63f60332eb6a00e3b37d10907b8db03649282afeed.jpg)  
+(b) Ours
+
+![](images/d86152d55f1683b337cbe98b44f59f813f8cad63710bdb2cb24efa5987b82177.jpg)  
+Figure 9: Generated trajectories of (a) the baseline method summarized in App. D.1 and (b) our method, MA-fBM (Sec. 4.1), for varying  $K$  and  $H = 0.1$ .  
+(a) Baseline
+
+![](images/25753629d0f67879e1255eabb014115d5d5e524880f2ce7ccb5b2471b6a3c969.jpg)  
+(b) Ours  
+Figure 10: Generated trajectories of (a) the baseline method summarized in App. D.1 and (b) our method, MA-fBM (Sec. 4.1), for varying  $K$  and  $H = 0.2$ .
+
+# F.2 FOU BRIDGE
+
+Fig. 17 shows additional results of the fractional Ornstein-Uhlenbeck bridge. The variances are calculated with Eq. (23), and Eq. (24) for  $\theta > 0$  and  $H > 1/2$  or Eq. (1) for  $\theta = 0$ . Note that we do not have a useful covariance equation for  $\theta > 0$  and  $H < 1/2$  (Lysy & Pillai, 2013), so this setting is not included in the experiments.
+
+![](images/f1ef7f553a1e675b10b30dd91afac432ed2eff44e4be1042c8d4b0dadcb191a7.jpg)  
+(a) Baseline
+
+![](images/832d6b831ffbc2b3850523f277dacff25e0f5d053ac30d66475f81a287124251.jpg)  
+(b) Ours  
+Figure 11: Generated trajectories of (a) the baseline method summarized in App. D.1 and (b) our method, MA-fBM (Sec. 4.1), for varying  $K$  and  $H = 0.3$ .
+
+![](images/b964b0d182092d44f9b2fee549a0f4eae4fa53819a11ecd8a0e0973a66b4ea2c.jpg)  
+(a) Baseline
+
+![](images/53fa8aeca197c05d8e42035df4f83c9af6ec51d5097cf2ce6919bb7380d3d1b1.jpg)  
+(b) Ours
+
+![](images/f015f75f1f1ef7cd0588d39f0dd1ea00c8ed84b81812dd951533d89ff9bdb370.jpg)  
+Figure 12: Generated trajectories of (a) the baseline method summarized in App. D.1 and (b) our method, MA-fBM (Sec. 4.1), for varying  $K$  and  $H = 0.4$ .  
+(a) Baseline  
+Figure 13: Generated trajectories of (a) the baseline method summarized in App. D.1 and (b) our method, MA-fBM (Sec. 4.1), for varying  $K$  and  $H = 0.5$ .
+
+![](images/f9b522a9fbd19c61c595a0ede43869b25a7abb5a0a181792520caee66acc82ce.jpg)  
+(b) Ours
+
+![](images/1e8068c8e571f7c290400ade3585db5248d4c6b8436dd64539415d6327ccccd9.jpg)  
+(a) Baseline
+
+![](images/cd1f2b64cdc8cef3da7fda1c13671dc32cfd3f6df247f0a16d954957c37b30a3.jpg)  
+(b) Ours  
+Figure 14: Generated trajectories of (a) the baseline method summarized in App. D.1 and (b) our method, MA-fBM (Sec. 4.1), for varying  $K$  and  $H = 0.6$ .
+
+![](images/34b2f9e4b53c08da3a2485de05c5f134d0208437ef1f056697c0e808945495b6.jpg)  
+(a) Baseline
+
+![](images/cd5ea0b0c846849e0c26bbd44677913f558ea971abbb67608184cc450e2be360.jpg)  
+(b) Ours
+
+![](images/423af7c996b372d2e4e00994d399a3ce34cef5ac779e959ba66312734bddfd70.jpg)  
+Figure 15: Generated trajectories of (a) the baseline method summarized in App. D.1 and (b) our method, MA-fBM (Sec. 4.1), for varying  $K$  and  $H = 0.8$ .  
+(a) Baseline  
+Figure 16: Generated trajectories of (a) the baseline method summarized in App. D.1 and (b) our method, MA-fBM (Sec. 4.1), for varying  $K$  and  $H = 0.9$ .
+
+![](images/4d4a85003b56f18c03639dc42ed145b6c43ae26c65f6ead74b8917822c9dc1c1.jpg)  
+(b) Ours
+
+![](images/5ae68c97d04c631a6fe6093ff04b5fad8b2d0d1b1e70350328445f47ca81528a.jpg)
+
+![](images/fe7bb73a0d408fcbfb5f6ffe4d0913cf78ab4d7b454d4193d41cc29f5a2221c4.jpg)
+
+![](images/b6777277811f774ce441a493ef4bbdcf2c15ba15c3aed57c94d1337b86ec4724.jpg)
+
+![](images/3879ac6ff5a179b510cc9dbc57d9303cb2e099dfe8b0d563020a7ea86d6c2445.jpg)  
+(a)  $H = 0.1, \theta = 0.0$
+
+![](images/4d1f3f8b6e023e28e4e8fbcc6f12e0074d9ac13b20ce15b637b5a476eccd9d4e.jpg)  
+(b)  $H = 0.2,\theta = 0.0$
+
+![](images/86fd4de4e28d4b294fa56e1e97765ac74b7bd870c69795754521d2a7a75f31f6.jpg)  
+(c)  $H = 0.3,\theta = 0.0$
+
+![](images/b3da696c863cacc5f9c021edb42b2177dda3f9363558330e085810b1811aa06c.jpg)  
+(d)  $H = 0.4,\theta = 0.0$
+
+![](images/114c3ca8c102f300d7471e310b8ef81d325bac58f945d0f40d42551d92356bda.jpg)  
+(e)  $H = 0.5,\theta = 0.0$
+
+![](images/5013a36efd8f46c9a9c9fa4b40a13dd2969f6561337a17c51a47d775d414422c.jpg)  
+(f)  $H = 0.6$ $\theta = 0.0$
+
+![](images/c06ac66942aaa4a744cf6d8f34188045c4e66f27bd8fdfb2c5bb0e3988e9b3c6.jpg)  
+(g)  $H = 0.7, \theta = 0.0$  
+(j)  $H = 0.6, \theta = 1.0$
+
+![](images/ec30cd961b3df2f8596f8256495ae1590eb0491e8101251f9804e0aca03d5915.jpg)  
+(h)  $H = 0.8,\theta = 0.0$
+
+![](images/a08b74f472507de8906d303edd67ea9694d50f4c623cc213ee13cf5fa1fc9dc9.jpg)  
+(i)  $H = 0.9, \theta = 0.0$  
+(1)  $H = 0.8,\theta = 1.0$
+
+![](images/1861276c5c16f7b399430d9f43f054f54453792846dc31e16dfa0e36c5de22cf.jpg)  
+(m)  $H = 0.9,\theta = 1.0$  
+Figure 17: The true variance (blue) of a fOU bridge matches the empirical variance (dashed orange) of our trained models. The transparent black lines are the sampled approximate posterior paths used to calculate the empirical variance.
+
+# F.3 VIDEO MODELS
+
+On the choice of video datasets. We conducted experiments on two video datasets: Stochastic Moving MNIST (SM-MNIST) (Denton & Fergus, 2018) and the real video dataset of a chaotic double pendulum (Asseman et al., 2018).
+
+SM-MNIST and the double pendulum dataset contain different forms of nuisances and present different challenges to our stochastic model. First, SM-MNIST digits move with a constant velocity along a trajectory until they hit at wall at which point they bounce off with a random speed and direction. This sudden event intersperses the deterministic motion with moments of uncertainty, i.e. each time a digit hits a wall. This is the reason why a stochastic model fits better than an ODE and unlike BM, our noise can model the smooth and correlated trajectory simply by raising the Hurst index.
+
+On the other hand, the double pendulum dataset is actually governed by a set of coupled ordinary differential equations. However, despite being a simple physical system, it exhibits a rich dynamic behavior with a strong sensitivity to initial conditions and noises in the environment (motion of the air in the room, sound vibrations, vibration of the table due to coupling with the pendulum etc.). Combined with the chaotic nature of the system, this creates a major challenge for any model based upon smooth ODEs. Our model on the other hand heavy lifts this difficulty onto the (fractional) stochastic noise, leading to a more appropriate model. As shown in Tab. 2, our model outperforms the BM baseline also in this dataset.
+
+Figs. 18 and 19 show the posterior reconstructions of models trained on the Stochastic Moving MNIST and the double pendulum dataset respectively. Fig. 20 shows stochastic video prediction samples of Stochastic Moving MNIST.
+
+![](images/0f47e815ca253bec1d4de2057e1bba03c7744f24c1a2befaa1ccf9816d937005.jpg)  
+Figure 18: Posterior reconstructions of a model driven by BM and a model driven by MA-fBM, conditioned on the same data ('Ground truth').
+
+![](images/f7c02562f887968b636571ce0b3daa0951614f7673a3cdc11ed932d0687318f8.jpg)  
+Figure 19: Posterior reconstructions of a model driven by BM and a model driven by MA-fBM, trained on the double pendulum dataset. Both are conditioned on the same data ('Ground truth'). We show 7 evenly spaced frames of the total 20 frames.
+
+<table><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>8</td><td>9</td><td>9</td><td>9</td><td>9</td><td>9</td><td>9</td><td>9</td><td>9</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>91</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>0</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>91</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>0</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>91</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>0</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr><tr><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td><td>94</td></tr></table>
+
+Figure 20: Stochastic predictions using the trained prior of a model driven by BM and a model driven by MA-fBM, where the initial state is conditioned on the same data. Four samples are shown for each model. The MA-fBM samples show more diverse movements, thus better capturing the dynamics in the data. The BM samples are more similar, indicating a less powerful prior was learned.
+
+BM (1)
+
+BM (2)
+
+BM (3)
+
+BM (4)
+
+MA-fBM (1)
+
+MA-fBM (2)
+
+MA-fBM (3)
+
+MA-fBM (4)

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+# VIEWS CAN BE DECEIVING: IMPROVED SSL THROUGH FEATURE SPACE AUGMENTATION
+
+Kimia Hamidieh<sup>1</sup> Haoran Zhang<sup>1</sup> Swami Sankaranarayanan<sup>2</sup> Marzyeh Ghassemi<sup>1</sup>
+
+$^{1}$ MIT,  $^{2}$ Sony AI
+
+{hamidieh,haoranz,swamiviv,mghassem}@mit.edu
+
+# ABSTRACT
+
+Supervised learning methods have been found to exhibit inductive biases favoring simpler features. When such features are spuriously correlated with the label, this can result in suboptimal performance on minority subgroups. Despite the growing popularity of methods which learn from unlabeled data, the extent to which these representations encode spurious features is unclear. In this work, we explore the impact of spurious features on Self-Supervised Learning (SSL) for visual representation learning. We first empirically show that commonly used augmentations in SSL can cause undesired invariances in the image space, and illustrate this with a simple example. We further show that classical approaches in combating spurious correlations, such as dataset re-sampling during SSL, do not consistently lead to invariant representations. Motivated by these findings, we propose LATETVG to remove spurious information from these representations during pretraining, by regularizing later layers of the encoder via pruning. We find that our method produces representations which outperform the baselines on several benchmarks, without the need for group or label information during SSL.
+
+# 1 INTRODUCTION
+
+Standard supervised machine learning models exhibit high overall performance but often perform poorly on minority subgroups (Shah et al., 2020; McCoy et al., 2019; Gururangan et al., 2018). One potential cause is the presence of spurious correlations, which are features that are only correlated with the label for specific subsets of data. For instance, a machine learning model tasked with predicting bird species from images across different habitats may use the background the bird commonly appears in as a "shortcut", instead of core features specific to the bird such as the shape of their beak or plumage. This results in poor performance on bird groups that appear in unexpected environments (Sagawa et al., 2020a). Identifying spurious correlations in the supervised learning setting has been well studied, where empirical risk minimization has been shown to exploit spurious correlations and result in poor performance for minority subgroups (Hashimoto et al., 2018). As downstream tasks are explicitly defined, the label can be used to distinguish between core and spurious features (Liu et al., 2021a; Zhang et al., 2022). Recent work has proposed various methods to identify and mitigate the effects of spurious features, such as learning multiple prediction heads (Lee et al., 2022b), causal inference (Creager et al., 2021), data augmentation (Gao et al., 2023) and targeted strategies such as importance weighting (Lahoti et al., 2020), re-sampling Idrissi et al. (2021); Tu et al. (2020), or approaches based on group distributionally robust optimization (Sagawa et al., 2020a; Duchi et al., 2019).
+
+More recently, self-supervised learning (SSL) has emerged as a common form of pre-training for task-agnostic learning with large, unlabeled datasets (Chen et al., 2020a; He et al., 2019; Grill et al., 2020; Chen & He, 2020; Caron et al., 2020; Zbontar et al., 2021; Chen et al., 2020b). SSL methods learn representations from unlabeled datasets by solving an auxiliary pretext task (Doersch et al., 2015), such as inducing invariance between the representations of two augmented views of the same image (He et al., 2019; Chen et al., 2020a). These methods have shown impressive results for a wide range of downstream tasks and datasets (Liu et al., 2021b; Jaiswal et al., 2020; Tamkin et al., 2021).
+
+Capturing core features – rather than spurious features – is essential for learning effective representations that can be used in downstream tasks, but is particularly difficult in the case of SSL due to the absence of labeled data during the pre-training process. Given only unlabeled data, we define spurious features as those that strongly correlate with core features for most examples in the training set, but
+
+are not useful for downstream tasks. For example, when training an SSL model on multi-object images, larger objects may interfere with the learning of smaller objects (Chen et al., 2021). If the downstream task involves only the prediction of smaller objects, the larger (spurious) object may suppress the smaller (core) object from being learned. Large-scale unlabeled datasets that are commonly used in machine learning are inevitably imbalanced (Van Horn et al., 2021), have been found to be biased towards spuriously correlated sensitive attributes (Calude & Longo, 2017) such as gender or race (Agarwal et al., 2021), and can also include label-irrelevant features (Torralba & Efros, 2011; Fan et al., 2014).
+
+In this paper, we investigate the impact of spurious correlations on SSL pre-training. We first show theoretically that image augmentations used in SSL pre-training can lead to spurious connectivity when learning representations, causing the model to fail to predict the label using core features in downstream tasks. We empirically evaluate spurious connectivity, and then show that existing methods for utilizing group information in ERM based approaches do not provide an analogous improvement in SSL pre-training. We then propose Late-layer Transformation-based View Generation or LATETVG - a method that induces invariance to spurious features in the representation space by regularizing final layers of the featurizer via pruning. Importantly, since our approach addresses SSL pre-training, we do not assume that model developers know apriori the identity or values of the spurious features that exist in the data. We first evaluate LATETVG on several popular benchmarks for spurious feature learning, and then connect our method to the theoretical analysis by showing that LATETVG models empirically exhibit lower spurious connectivity. Our method demonstrates improved discriminative ability, especially over minority subgroups, for downstream predictive tasks, without access to group or label information. We make the following contributions:
+
+- We provide theoretical arguments (Sec 3.3) that illustrate how common augmentations used in SSL pre-training affect the model's ability to rely on spurious features, for downstream linear classifiers.  
+- We explore the extent of spurious learning in self-supervised representations through the lens of downstream worst-group performance. We empirically show that known techniques for avoiding spurious correlations, such as re-sampling of the training set given group information, do not consistently improve core feature representations (Sec 4.4).  
+- We propose LATEGV - an approach that corrects for the biases caused by augmentations, by modifying views of samples in the representation space (Sec 5.1). We find that LATEGV effectively improves worst-group performance in downstream tasks on four datasets by enforcing core feature learning (Sec 5.2).
+
+# 2 RELATED WORK
+
+Spurious Correlations. Spurious correlations arise in supervised learning models Koh et al. (2021); Joshi et al. (2023); Singla & Feizi (2021) in a variety of domains, from medical imaging (Zech et al., 2018; DeGrave et al., 2021) to natural language processing (Tu et al., 2020; Wang & Culotta, 2020). A variety of approaches have been proposed to learn classifiers which do not make use of spurious information. Methods like GroupDRO (Sagawa et al., 2020a) and DFR (Kirichenko et al., 2022) require group information during training, while methods like JTT (Liu et al., 2021a), LfF (Nam et al., 2020), CVaR DRO (Duchi et al., 2019), and CnC (Zhang et al., 2022) do not. However, all methods require group information for model selection.
+
+Self-supervised Representation Learning. Self-supervised learning methods learn representations from large-scale unlabeled datasets where annotations are scarce. In vision applications, the pretext task is typically to maximize similarity between two augmented views of the same image (Jing & Tian, 2020). This can be done in a contrastive fashion using the InfoNCE loss (Oord et al., 2018), such as in Chen et al. (2020a) and Chen et al. (2020b), or without the need for negative samples at all, as in Grill et al. (2020); Caron et al. (2020); Chen & He (2020); Caron et al. (2021); Oquab et al. (2023); Zbontar et al. (2021). Prior work has shown that SSL models may learn to spuriously associate certain foreground items with certain backgrounds (Meehan et al., 2023). In this work, we explore one potential mechanism for this phenomenon, both theoretically and empirically.
+
+Representation Learning under Dataset Imbalance and Shortcuts. Self-supervised models have demonstrated increased robustness to dataset imbalance (Liu et al., 2021b; Jiang et al., 2021b:a), and the dominance of easier or larger features suppressing the learning of other features (Chen et al., 2021). Some prior work has addressed shortcut learning in contrastive learning through adversarial feature modification without group labels (Robinson et al., 2021). However, other approaches to
+
+group robustness or fairness in self-supervised learning require group information or labels (Tsai et al., 2020; Song et al., 2019; Wang et al., 2021; Bordes et al., 2023; Scalbert et al., 2023). This paper focuses on learning representations from an unlabeled dataset with spurious correlations, encompassing both dataset imbalance and features of varying difficulty.
+
+Regularization in Self-supervised Learning. The concept of regularizing a specific subset of the network is relatively unexplored in self-supervised learning but finds motivation in recent findings from supervised settings, such as addressing minority examples (Hooker et al., 2019), out-of-distribution generalization (Zhang et al., 2021), late-layer regularizations through head weight-decay (Abnar et al., 2021), and initialization (Zhou et al., 2022). Additionally, Lee et al. (2022a) propose surgically fine-tuning specific layers of the network to handle distribution shifts in particular categories. These studies provide support for the approach of targeting a specific component of the network in self-supervised learning.
+
+# 3 SPURIOUS CONNECTIVITY INDUCES DOWNSSTREAM FAILURES
+
+In this section, we introduce a toy setting to demonstrate that common augmentations used in SSL pre-training affect a model's ability to rely on spurious features for downstream linear classifiers. We consider a binary classification problem with a binary spurious attribute, with an equal number of samples per group (Section 3.2). We show that augmentations applied during SSL pre-training can introduce undesired invariances in the representation space learned by a contrastive objective, making the downstream linear classifier trained on representations more reliant on the spurious feature (Section 3.3).
+
+# 3.1 BACKGROUND AND SETUP
+
+Setup. BWe consider learning representations from an unlabeled data space  $\mathcal{X}$  generated from an underlying latent feature space  $\mathcal{Z} \in \mathbb{R}^m \coloneqq \{z_{\mathrm{core}}, z_{\mathrm{spur}}, \ldots, z_m\}$ , where  $z_{\mathrm{core}}$  and  $z_{\mathrm{spur}}$  are correlated features. For a given downstream task with labeled samples, we assume that each  $x \in \mathcal{X}$  belongs to a class given by the ground-truth labeling function  $y: \mathcal{X} \to \mathcal{Y}$  where  $z_{\mathrm{core}}$  determines the labels for our downstream task of interest, while  $z_{\mathrm{spur}}$  determines the spurious attribute, which is easier to learn, and is not of interest for downstream tasks. We can define a deterministic attribute function  $a: \mathcal{X} \to \mathcal{S}$  where each  $x \in \mathcal{X}$  takes a value in  $\mathcal{S}$ . Let  $g = (y(x), a(x))$  denote the subgroup of a given sample  $x$ , where  $\mathcal{G} = \mathcal{Y} \times \mathcal{S}$  is the set of all possible subgroups. Figure 1 illustrates the subgroups on the Waterbirds dataset, where the background is a spurious feature that correlates with the bird species.
+
+Contrastive learning. We aim to learn representations by bringing together data-augmented views of the same input, which we refer to as positive pairs, using a contrastive objective. Let  $P_{+}$  be the distribution of positive pairs, which can be defined as the marginal probability of generating the augmented pair  $x$  and  $x'$  from the same image in the (natural) population data. Thus the distribution  $P_{+}$  relies both on original data distribution and the choice of SSL augmentations. To analyze the representation space learned in contrastive learning and core feature predictivity of the representations, consider a weighted graph with vertex set  $\mathcal{X}$  where the undirected edge  $(x, x')$  has weight  $w_{xx'} = P_{+}(x, x')$  similar to augmentation graph in HaoChen et al. (2021).
+
+Although the augmentation graph learns semantically similar structures that enables generalization to new domains (Shen et al., 2022), the inductive biases set by these augmentations is not well studied. In this work, we draw attention to cases where augmentations can create spurious connectivities within subgroups of the data, and when and why these connectivities can cause the downstream linear model to rely on the spurious feature.
+
+# 3.2 SPURIOUS CONNECTIVITY IN A TOY SETUP
+
+In this section, we introduce a setting in which contrastive objectives can learn representations that cause linear downstream models fail on downstream tasks. To start, we investigate how augmentations can transform the samples such that the subgroup assignment changes.
+
+Definition 3.1. Subgroup connectivity. Define the average subgroup connectivity given two disjoint subsets  $G_{1}, G_{2} \subseteq \mathcal{X}$  as  $w(G_{1}, G_{2}) = \frac{1}{|G_{1}| \cdot |G_{2}|} \sum_{x \in G_{1}, x' \in G_{2}} w_{xx'}$ . where  $w_{xx'}$  is the probability of generating the augmented pair  $x$  and  $x'$  from the same image in the natural population data.
+
+![](images/0c541442c54e638eb349baf70533b163ad2f2be79779275676a4c1f21c230352.jpg)  
+Figure 1: Analysing SSL augmentations. (a) Images generated from a latent space with correlating features. (b) If the connectivity induced by SSL augmentations between subgroups with the same spurious features is higher than the ones with the same invariant features, learned representations lead a downstream linear model to separate the data based on the spurious feature (red dashed line) instead of the invariant feature (green dashed line). Our empirical evaluation in Table 4 shows that this is indeed the case across different datasets considered in this work.
+
+Intuitively, this subgroup connectivity is the average weight of edges connecting  $G_{1}$  to  $G_{2}$ , and is proportional to the probability of a sample  $x \in G_{1}$  being transformed to a sample  $x' \in G_{2}$  via augmentations. See Appendix C for further details.
+
+We specifically define the following terms to be the expected value of  $w(G_1, G_2)$  from Definition 3.1 when subgroups  $G_1$  and  $G_2$  have the following properties:
+
+- Spurious connectivity  $(\alpha)$ :  $G_{1}$  and  $G_{2}$  share the same spurious attribute but differ in class  
+- Invariant connectivity  $(\beta)$ :  $G_{1}$  and  $G_{2}$  share the same class but differ in spurious attribute  
+- Opposite connectivity  $(\gamma)$ :  $G_{1}$  and  $G_{2}$  differ both in the spurious attribute and the label
+
+Where  $\alpha, \beta, \gamma$  are average values estimated across a dataset consisting of subgroups.
+
+Toy Setup. We consider a downstream classification problem where a spurious attribute is present, and both the input and the spurious attribute take binary values. We define the probability of sampling a positive pair  $(x,x^{\prime})$  based on the expected connectivity terms  $\alpha_{\mathrm{toy}}$ ,  $\beta_{\mathrm{toy}}$ ,  $\gamma_{\mathrm{toy}}$ , and  $\rho_{\mathrm{toy}}$  as follows:
+
+$$
+P _ {+} (x, x ^ {\prime}) = \left\{ \begin{array}{l l} \alpha_ {\mathrm {t o y}}, & \text {i f} a (x) \neq a (x ^ {\prime}) \text {a n d} y (x) = y (x ^ {\prime}) \\ \beta_ {\mathrm {t o y}}, & \text {i f} a (x) = a (x ^ {\prime}) \text {a n d} y (x) \neq y (x ^ {\prime}) \\ \gamma_ {\mathrm {t o y}}, & \text {i f} a (x) \neq a (x ^ {\prime}) \text {a n d} y (x) \neq y (x ^ {\prime}) \\ \rho_ {\mathrm {t o y}}, & \text {i f} a (x) = a (x ^ {\prime}) \text {a n d} y (x) = y (x ^ {\prime}) \end{array} \right.
+$$
+
+Note that the average subgroup connectivity for this setup, would be exactly the same as the corresponding connectivity variable. Thus in our running example we have  $\alpha = \alpha_{\mathrm{toy}}$ ,  $\beta = \beta_{\mathrm{toy}}$ ,  $\gamma = \gamma_{\mathrm{toy}}$  and we can use them interchangeably. For this simplified augmentation graph, the expected connectivity terms between groups are a property of the graph, and independent of the model or architecture we use for learning representations. Combined with a contrastive objective, the expected connectivity can be a proxy for how close different subgroups are going to be in the representation space.
+
+# 3.3 ANALYSIS OF THE TOY SETTING
+
+In Section 4.2, we empirically show that common augmentations used in contrastive learning can be detrimental to learning invariant representations, as they implicitly encourage samples to cluster primarily based on the spurious feature. Based on this observation, we make the following assumption.
+
+Assumption 3.2. Given a spurious attribute function  $a: \mathcal{X} \to |G|$  which is defined for all  $x \in \mathcal{X}$ , we assume that for a data point  $x \in \mathcal{X}$ , the probability of distorting the labeling of the augmented images sampled from the augmentation distribution  $\mathcal{A}(\cdot |\bar{x})$ , is greater than the probability of distorting the attribute. More formally,
+
+$$
+\operatorname * {P r} _ {\tilde {x} \sim \mathcal {A} (\cdot | x)} (y (\tilde {x}) \neq y (x), a (\tilde {x}) = a (x)) \geq \operatorname * {P r} _ {\tilde {x} \sim \mathcal {A} (\cdot | x)} (y (\tilde {x}) = y (x), a (\tilde {x}) \neq a (x))
+$$
+
+Lemma 3.3. Consider the set of (unlabeled) population data  $\mathcal{X}$  in a binary-class setting where the spurious attribute takes binary values, consisting of  $|\mathcal{G}| = 4$  groups, with the same number of
+
+examples per group. Consider a simplified augmentation graph with parameters  $\alpha, \beta, \rho, \gamma$  defined as in [3.2] and assume that augmentations are more likely to change either class or attribute, than to change neither of the two ( $\alpha > \gamma, \beta > \gamma$ ), and that augmentations are less likely to change both at the same time ( $\rho > \alpha, \rho > \beta$ ).
+
+Under these conditions, the spectral contrastive loss recovers both invariant and spurious features, and for each sample in the population data, the spurious feature is bounded by constant  $B_{sp} = \sqrt{\beta - \alpha - \gamma + \rho}$ , while the invariant feature is bounded by  $B_{inv} = \sqrt{\alpha - \beta - \gamma + \rho}$  in the representation space. Proof in Appendix C.
+
+Corollary 3.4. Given Assumption 3.2 where  $\alpha >\beta$  in the simplified augmentation graph, the margin of the spurious classifier is  $B_{sp}$ , and is less than the margin of the invariant classifier  $B_{inv}$ , and the max-margin classifier trained on representations given by spectral clustering converges to the spurious classifier.
+
+This suggests that even with the same number of samples across different groups during pre-training, downstream linear classifiers can rely on the spurious feature to make predictions, where the representations are determined by the simplified augmentation graph and the spectral contrastive loss.
+
+# 4 EXPLORING SPURIOUS LEARNING IN REPRESENTATIONS
+
+In this section, we investigate the performance of downstream linear models trained on self-supervised representations, empirically verify our assumption regarding spurious and invariant connectivity, and show that in practice - similar to our toy analysis - having the same number of examples across groups in the presence of spurious connectivity does not lead to performance gains.
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets We evaluate methods on five commonly used benchmarks in spurious correlations – CelebA (Liu et al., 2015), CMNIST (Arjovsky et al., 2019), MetaShift (Liang & Zou, 2022), Spurious CIFAR-10 (Nagarajan et al., 2020), and Waterbirds (Wah et al., 2011) (See Appendix D.1 for dataset descriptions). For each dataset, we train an encoder with an SSL-based pre-training step followed by a supervised training of a linear model that probes the representations learned using SSL for the downstream task.
+
+SSL Pre-training For the SSL pre-training, we train SimSiam (Chen & He 2020) models with a ResNet backbone throughout the paper. The training split used during the pre-training stage are unbalanced and contain spuriously correlated data. The group/label counts for each dataset and split is shown in Appendix D.1. The backbone network used for most of our experiments are initialized with random weights, unless specified otherwise. We additionally report results for SimCLR (Chen et al. 2020a) models in Section 5.2.1
+
+Downstream Task For downstream task prediction, we train a linear layer using logistic regression on top of the pretrained embeddings. Note that the backbone is frozen during this finetuning phase and only the linear layer is updated. We use a balanced dataset for training where the spurious correlation does not hold. To create this downstream training dataset, we subsample majority groups (Sagawa et al., 2020b; Idrissi et al., 2021), to avoid the geometrical skews (Nagarajan et al., 2020) of the linear classifier on representations. Then, we evaluate the learned representations on the standard test split of each dataset, where group information is given. For each run, we report the average and worst-group accuracy.
+
+Empirical Evaluation of Spurious Connectivity To evaluate the connectivity term for each pair of subgroups in datasets exhibiting spurious correlations, we conduct an empirical analysis similar to Shen et al. (2022). Specifically, we train a classifier to distinguish between each pair of subgroups and evaluate its performance on a subset of the data that has been augmented with SSL augmentations. The error of the classifier represents the probability that the augmentation module alters the subgroup assignment for each example between the two subgroups, making them indistinguishable. Figure I illustrates this procedure.
+
+The Role of Initialization In representation learning, encoders are not typically trained from scratch but initialized from a model pretrained on larger datasets, such as ImageNet (Deng et al., 2009). Recent work in transfer learning (Geirhos et al., 2018; Salman et al., 2022) has questioned this assumption and pointed out that biases in pretrained models linger even after finetuning on downstream target tasks. In this section and more broadly in our work, we focus on performing SSL pre-training from randomly initialized weights. In addition, since the datasets considered in this work
+
+are similar to ImageNet, the performance of off-the-shelf ImageNet pretrained models is expected to be higher. For completeness, we have added these results to Appendix G.2.
+
+# 4.2 HIGH LEVELS OF SPURIOUS CONNECTIVITY IN PRACTICE
+
+We measure connectivity across four datasets in Table 4 and on all of them, we find that the average spurious connectivity is higher than invariant connectivity. We also confirm that both these values are higher than the probability of simultaneously changing both spurious attributes and invariant attributes. This means that the samples within the training set are more likely to be connected to each other through the spurious attribute, rather than the core feature. This suggests that the contrastive loss prefers alignment based on the spurious attribute instead of the class.
+
+Table 1: We report the error of classifiers trained to distinguish between two subgroups as a proxy for the probability of augmentations flipping group assignments between each two groups in the dataset, or the connectivity of two subgroups in the image space.  
+
+<table><tr><td>Dataset</td><td>Spurious Connectivity</td><td>Invariant Connectivity</td><td>Opposite Connectivity</td></tr><tr><td>celebA</td><td>10.4</td><td>3.7</td><td>2.8</td></tr><tr><td>cmnist</td><td>31.6</td><td>8.3</td><td>6.8</td></tr><tr><td>metashift</td><td>16.3</td><td>13.6</td><td>5.0</td></tr><tr><td>waterbirds</td><td>25.3</td><td>11.2</td><td>7.8</td></tr></table>
+
+We compute the connectivity terms by training classifiers to distinguish augmented data from each combination of the two groups in the dataset and reporting their error rates.
+
+The details of the choice of augmentations and training for this step can be found in Appendix E.
+
+# 4.3 SSL MODELS LEARN SPURIOUS FEATURES
+
+To measure the reliance of downstream models to spurious correlations, we measure the accuracy of the downstream model on each group in the test set, and use the worst-performing group accuracy as a lens to reason about spurious correlations. We find across all datasets, SSL models exhibit gaps between worst-group and average accuracy when predicting the core feature (Table 5 in Appendix D.3).
+
+These results indicate, that unlike supervised learning (Menon et al., 2021; Kirichenko et al., 2022; Rosenfeld et al., 2022), training of the final layer on a balanced set where the spurious correlation does not hold is not sufficient for improving worst-group accuracy when predicting the core attribute.
+
+# 4.4 RESAMPLING DURING SSL DOES NOT IMPROVE DOWNSTREAM PERFORMANCE
+
+To probe the effect of availability of such group information during the SSL pre-training stage, we examine whether classical approaches for combating spurious correlations, such as re-sampling training examples (Idrissi et al., 2021), are effective in removing spurious information during SSL pre-training.
+
+Assuming that group information is available, we train SimSiam on datasets re-sampled using the following strategies: (i) Balancing groups by resampling training examples to match the downstream validation distribution. (ii) Downsampling examples in majority groups to have the same number of examples in all groups. (iii) Upsampling minority examples to have the same number of examples in all groups.
+
+Table 2: Worst-group accuracy difference (%) between each balancing strategy and the original training set. Original training performance are shown in parentheses below each dataset. Full results can be found in Appendix Table 10.  
+
+<table><tr><td>Sampling Strategy</td><td>celebA(77.5)</td><td>cmmist(75.4)</td><td>metashift(42.3)</td><td>spurcifar10(43.4)</td><td>waterbirds(48.3)</td></tr><tr><td>Balancing</td><td>-1.7</td><td>-8.7</td><td>-3.8</td><td>-8.3</td><td>+3.0</td></tr><tr><td>Downsampling</td><td>+0.3</td><td>-10.6</td><td>+3.9</td><td>-14.4</td><td>+0.5</td></tr><tr><td>Upsampling</td><td>+4.1</td><td>-5.3</td><td>+2.7</td><td>-19.4</td><td>-0.3</td></tr></table>
+
+We find that re-sampling during self-supervised pre-training does not improve downstream worst-group accuracy in a consistent manner as in Table 2. We do see minor improvements for metashift
+
+![](images/354b61eb8f4de8ef73515ba2e37560120aa7bc32633e4d0f2b130f556eaf8d58.jpg)  
+Figure 2: We use model transformation modules to create new views of training examples in the representation space. The introduced set of transformations removes the features learned in the final few layers, and provides final representations invariant to such transformations.
+
+and celebA, but contrast this with large drops for spurcifar10 and cnnist. Given that the downstream linear model is trained on a downsampled dataset where such correlations do not exist, this means that re-sampling during self-supervised training does not necessarily improve the linear separability of representations with respect to the core feature, even given a balanced finetuning dataset. This is analogous to our findings in the toy setting in Section 3.3
+
+# 5 CREATING ROBUST REPRESENTATIONS VIA FEATURE SPACE AUGMENTATIONS
+
+In the previous sections, we showed that augmentation mechanisms used in SSL result in poor performance under spuriously correlated features in the training set. Instead of curating specific image augmentations that correct for these biases in the image space, we propose an approach to target spurious connectivity in the representation space by modifying positive pairs. In this section, we describe our approach, LATEVG that improves the performance of SSL models by introducing pruning based regularization to the later layers of the encoder.
+
+# 5.1 LATE-LAYER TRANSFORMATION-BASED VIEW GENERATION
+
+Motivated by improved SSL model invariance when trained with augmentations in image space (Chen et al., 2020a), we propose a model transformation module that specifically targets augmentations that modify the spurious feature in representation space. We propose Late-layer Transformation-based View Generation - LATETVG, which uses feature space transformations to mitigate spurious learning in SSL models and improve learning of the core feature.
+
+Formally, we propose using a model transformation module  $\mathcal{U}$ , that transforms any given model  $f_{\theta}$  parameterized by  $\theta = \{W_1,\dots ,W_n\}$  to  $f_{\tilde{\theta}}$ . At each step, we draw a transformation  $\phi_{M,\theta} \sim \mathcal{U}$  to obtain the transformed encoder. Each model transformation can be defined with a mask  $M \in \{0,1\}^{|\theta|}$ , where we transform the unmasked weights  $(1 - M) \odot \theta$  by  $\phi$ , and keep the rest of the weights  $M \odot \theta$  the same to obtain  $\tilde{\theta}$ . Here, we propose a specific transformation module  $\mathcal{U}$ .
+
+Transformations. For mitigating spurious connectivity, we choose a simple transformation targeted towards regularizing the final layers of the encoder. In our experiments, we consider a threshold pruning transformation module, which uses magnitude pruning on  $a\%$  of the weights in all layers deeper than  $L$ . More specifically, we propose a model transformation module  $\mathcal{U}_{\mathrm{Prune,L,a}}$  with  $\phi (\theta) = 0$ ,  $M\coloneqq M_{L,a} = \{M_L^l\odot \operatorname {Top}_a(W_l)\mid l\in [n]\}$  and  $\mathrm{Top}_a(W_l)_{i,j} = \mathbb{I}(|W_{l_{(i,j)}}|)$  in top  $a\%$  of  $\theta$ . Note that in this specific setting, the module transformation is deterministic (i.e.  $|\mathcal{U}| = 1$ ), though our formalization also allows for random transformations such as randomized pruning or re-initialization.
+
+To learn these representations, given two random augmentations  $t, t' \sim \mathcal{T}$  from the augmentation module  $\mathcal{T}$ , two views  $x_1 = t(x)$  and  $x_2 = t'(x)$  are generated from an input image  $x$ . At each step, given a feature encoder  $f$ , and an augmentation module  $\mathcal{U}$ , we obtain a transformed model  $\tilde{f} = \phi(f)$  with  $\phi \sim \mathcal{U}$ . During training, examples  $x_1$  and  $x_2$  are respectively passed through the normal encoder  $v_1 = f(x_1)$ , and the transformed encoder  $\tilde{v}_2 = \tilde{f}(x_2)$ . Encoded feature  $\tilde{v}_2$  is now a positive example that should be close to  $v_1$  in the representation space. An algorithmic representation of the method can be found in Appendix B.
+
+Table 3: Worst-group accuracy (%) of SSL-Base and LATETVG for SimSiam and SimCLR pretraining. Results for average accuracy can be found in Table 8.  
+
+<table><tr><td></td><td colspan="2">SimSiam</td><td colspan="2">SimCLR</td></tr><tr><td></td><td>SSL-BASE</td><td>SSL-LATE-TVG</td><td>SSL-BASE</td><td>SSL-LATE-TVG</td></tr><tr><td>celebA</td><td>77.5</td><td>83.1</td><td>76.7</td><td>82.2</td></tr><tr><td>cmnist</td><td>80.7</td><td>83.1</td><td>81.7</td><td>83.8</td></tr><tr><td>metashift</td><td>42.3</td><td>79.6</td><td>45.5</td><td>59.3</td></tr><tr><td>spurcifar10</td><td>43.4</td><td>61.4</td><td>36.5</td><td>40.4</td></tr><tr><td>waterbirds</td><td>48.3</td><td>56.3</td><td>43.8</td><td>55.4</td></tr></table>
+
+Intuition for LateTVG. When learning a discriminative process that maps data to a separable space, the variance among different subpopulations is stored in distinct regions of the network (Lee et al., 2022a). As a result, both spurious and core features, which describe the high-level data distribution, tend to reside at the end of a neural network. Thus, in LATEVG, we aim to encourage the final layers to learn more difficult features, by applying a model transformation that targets these layers, and causing the model to be invariant to final layer transformations. As pruning in supervised models have been shown to affect minority examples more than majority ones (Hooker et al., 2019), we hypothesize that our transformation can be considered as a curated view generating operation for the minority groups. In particular, pruning would contribute to "forgetting" the minority examples from the network, resulting in upweighting the loss for these examples.
+
+# 5.2 EXPERIMENTS
+
+In this section, we demonstrate the efficacy of LATETVG in mitigating the dependence on spurious correlations. We use the same experimental setup as described in Section 4.1. For evaluation of LATETVG, we use our SSL-LATETVG approach during the pre-training stage. We compare this performance to SSL models pre-trained with the standard SSL-base trained with either SimSiam or SimCLR.
+
+# 5.2.1 LATEVG IMPROVES SSL WORST-GROUP PERFORMANCE
+
+The goal of this experiment is to understand how LATEGV affects worst-group performance in downstream tasks that use SSL representations. We compare the worst group accuracy of two approaches, SSL-Base and SSL-LATEGV on 5 different datasets. Both models used similar hyper-parameter grids and model selection criteria as noted previously. The results are presented in Table 3. We show the performance of the best hyperparameter combination here, and have provided figures of performance gains for all hyperparameters in Appendix D.2. It can be clearly observed that SSL-LATEGV outperforms the base model by large margins across most datasets and for both SimSiam and SimCLR. On cnnist, our performance is very close to the baseline model and we do not see significant improvement. We hypothesize that this is due to the fact that the base encoder on the easier cnnist dataset is already quite perform-. mant. On datasets where the base encoder performs poorly such as metashift and spurcifar10, our approach improves the performance by at least  $10\%$  over base SimSiam. On a dataset of a larger scale like celebA, LATEGV still improves upon a strong encoder baseline.
+
+![](images/2c10c58ebd8cc63aeb8786d6254fd24c7b47fbd720daebff7c3d6e97bf6ae4b9.jpg)  
+Figure 3: Downstream worst-group accuracy of SSL-Late-TVG on the metashift dataset as we vary the percentage of minority group in the downstream training set. For all cases except for extreme minority decrement, SSL-Late-TVG outperforms the baseline.
+
+Further, we find that LATETVG closes the gap in performance to supervised pretraining (Table 8). We emphasize that this is an unfair comparison to begin with, since supervised pretraining requires labeled data whereas SSL does not, hence reducing the annotation budget drastically. Regardless, we find that LATETVG narrows the gap between the SSL baseline and the ERM model significantly
+
+-  $17\%$  relative improvement for cmnist to  $50\%$  in the case of spurcifar10. In the case of celebA, we even outperform the ERM baseline.
+
+# 5.2.2 SSL DOWNSTREAM LINEAR PERFORMANCE IS LESS RELIANT ON A BALANCED DOWNSTREAM DATASET
+
+Traditional approaches that mitigate spurious correlations in ERM-based settings assume that the downstream training set is balanced (Kirichenko et al., 2022). However, this still requires knowledge of the spurious feature, which we may not always have in practice. In this experiment, we challenge this assumption and analyze how SSL models behave when the downstream training set is imbalanced.
+
+We vary the proportion of minority groups in the downstream training set, by first downsampling the training set to have the same number of samples across groups, and second randomly sampling minority groups with weight  $\lambda$  (x-axis in Figure 3) and majority groups with weights  $1 - \lambda$ . We measure the worst group accuracy of the trained linear models for each dataset. We show the results on metashift in Figure 3, comparing the performance of SSL-Base and SSL-LATETVG. We can observe that LATESVG outperforms the baseline across a range of minority weights – implying that LATESVG is more robust to imbalances in downstream training data. This is a crucial aspect where LATESVG differs from other approaches in the supervised pretraining literature, such as DFR (Kirichenko et al., 2022), which requires a balanced training set for the reweighting strategy to be successful. Similar results for other datasets and linear models are provided in in Appendix F.5.
+
+# 5.2.3 LATESVG REDUCES SPURIOUS CONNECTIVITY IN THE REPRESENTATION SPACE
+
+Finally, we relate our method back to the theoretical analysis presented in Section 3, by computing the connectivity of the representation space learned by the SSL models, using the procedure outlined in Section 4. In Table 4, we find that LATETVG empirically reduces the spurious connectivity, while increasing the invariant connectivity, for all datasets. Thus, we have shown that LATETVG successfully augments the representation space to induce desired invariances.
+
+Table 4: We report the error of classifiers trained to distinguish between the representations of two subgroups as a proxy for connectivity terms. We find that LATEVG decreases spurious connectivity while increasing invariant connectivity in comparison to the baseline.  
+
+<table><tr><td>Dataset</td><td>Representation Space</td><td>Spurious Connectivity</td><td>Invariant Connectivity</td><td>Opposite Connectivity</td></tr><tr><td rowspan="2">celebA</td><td>SSL-BASE</td><td>18.9</td><td>15.7</td><td>8.3</td></tr><tr><td>SSL-LATE-TVG</td><td>15.8</td><td>17.9</td><td>8.0</td></tr><tr><td rowspan="2">cmnist</td><td>SSL-BASE</td><td>37.3</td><td>3.2</td><td>2.7</td></tr><tr><td>SSL-LATE-TVG</td><td>34.8</td><td>3.8</td><td>3.0</td></tr><tr><td rowspan="2">metashift</td><td>SSL-BASE</td><td>28.6</td><td>21.4</td><td>21.8</td></tr><tr><td>SSL-LATE-TVG</td><td>27.3</td><td>27.3</td><td>21.3</td></tr><tr><td rowspan="2">waterbirds</td><td>SSL-BASE</td><td>44.9</td><td>9.4</td><td>8.4</td></tr><tr><td>SSL-LATE-TVG</td><td>44.6</td><td>13.5</td><td>12.8</td></tr></table>
+
+# 6 CONCLUSION
+
+In this paper, we have investigated the impact of spurious correlations on self-supervised learning (SSL) pre-training and proposed a new approach, called LATETVG to address the issue. Our experiments demonstrated that spurious correlations caused by data augmentation can lead to spurious connectivity and hinder the model's ability to learn core features, which ultimately impacts downstream task performance. We have shown that traditional debiasing techniques, such as re-sampling, are not effective in mitigating the impact of spurious correlations in SSL pre-training. In contrast, LATETVG effectively improves the worst-group performance in downstream tasks by inducing invariance to spurious features in the representation space throughout training. Our approach does not require access to group or label information during training and can be applied to large-scale, imbalanced datasets with spurious correlations. We believe our work will help advance the field of SSL pre-training and encourage future research in developing methods that are robust to spurious correlations.
+
+# REFERENCES
+
+Samira Abnar, Mostafa Dehghani, Behnam Neyshabur, and Hanie Sedghi. Exploring the limits of large scale pre-training. arXiv preprint arXiv:2110.02095, 2021.  
+Sandhini Agarwal, Gretchen Krueger, Jack Clark, Alec Radford, Jong Wook Kim, and Miles Brundage. Evaluating clip: towards characterization of broader capabilities and downstream implications. arXiv preprint arXiv:2108.02818, 2021.  
+Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019.  
+Florian Bordes, Randall Balestriero, Quentin Garrido, Adrien Bardes, and Pascal Vincent. Guillotine regularization: Why removing layers is needed to improve generalization in self-supervised learning. Transactions on Machine Learning Research, 2023.  
+Cristian S Calude and Giuseppe Longo. The deluge of spurious correlations in big data. Foundations of science, 22:595-612, 2017.  
+Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. arXiv preprint arXiv:2006.09882, 2020.  
+Mathilde Caron, Hugo Touvron, Ishan Misra, Hervé Jégou, Julien Mairal, Piotr Bojanowski, and Armand Joulin. Emerging properties in self-supervised vision transformers. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 9650-9660, 2021.  
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In International conference on machine learning, pp. 1597-1607. PMLR, 2020a.  
+Ting Chen, Calvin Luo, and Lala Li. Intriguing properties of contrastive losses. Advances in Neural Information Processing Systems, 34:11834-11845, 2021.  
+Xinlei Chen and Kaiming He. Exploring Simple Siamese Representation Learning. arXiv e-prints, art. arXiv:2011.10566, November 2020.  
+Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020b.  
+Elliot Creager, Jorn-Henrik Jacobsen, and Richard Zemel. Environment inference for invariant learning. In International Conference on Machine Learning, pp. 2189-2200. PMLR, 2021.  
+Alex J DeGrave, Joseph D Janizek, and Su-In Lee. Ai for radiographic Covid-19 detection selects shortcuts over signal. Nature Machine Intelligence, 3(7):610-619, 2021.  
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248-255. IEEE, 2009.  
+Carl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by context prediction. In Proceedings of the IEEE international conference on computer vision, pp. 1422-1430, 2015.  
+John C Duchi, Tatsunori Hashimoto, and Hongseok Namkoong. Distributionally robust losses against mixture covariate shifts. Under review, 2, 2019.  
+Jianqing Fan, Fang Han, and Han Liu. Challenges of big data analysis. National science review, 1(2): 293-314, 2014.  
+Irena Gao, Shiori Sagawa, Pang Wei Koh, Tatsunori Hashimoto, and Percy Liang. Out-of-domain robustness via targeted augmentations. arXiv preprint arXiv:2302.11861, 2023.  
+Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A Wichmann, and Wieland Brendel. Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. arXiv preprint arXiv:1811.12231, 2018.
+
+Jean-Bastien Grill, Florian Strub, Florent Alché, Corentin Tallec, Pierre H. Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, Bilal Piot, Koray Kavukcuoglu, Rémi Munos, and Michal Valko. Bootstrap your own latent: A new approach to self-supervised Learning. arXiv e-prints, art. arXiv:2006.07733, June 2020.  
+Suchin Gururangan, Swabha Swayamdipta, Omer Levy, Roy Schwartz, Samuel R Bowman, and Noah A Smith. Annotation artifacts in natural language inference data. arXiv preprint arXiv:1803.02324, 2018.  
+Jeff Z HaoChen, Colin Wei, Adrien Gaidon, and Tengyu Ma. Provable guarantees for self-supervised deep learning with spectral contrastive loss. Advances in Neural Information Processing Systems, 34:5000-5011, 2021.  
+Tatsunori Hashimoto, Megha Srivastava, Hongseok Namkoong, and Percy Liang. Fairness without demographics in repeated loss minimization. In International Conference on Machine Learning, pp. 1929-1938. PMLR, 2018.  
+Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. arXiv:1911.05722, 2019.  
+Sara Hooker, Aaron Courville, Gregory Clark, Yann Dauphin, and Andrea Frome. What do compressed deep neural networks forget? arXiv preprint arXiv:1911.05248, 2019.  
+Badr Youbi Idrissi, Martin Arjovsky, Mohammad Pezeshki, and David Lopez-Paz. Simple data balancing achieves competitive worst-group-accuracy. arXiv preprint arXiv:2110.14503, 2021.  
+Ashish Jaiswal, Ashwin Ramesh Babu, Mohammad Zaki Zadeh, Debapriya Banerjee, and Fillia Makedon. A survey on contrastive self-supervised learning. Technologies, 9(1):2, 2020.  
+Ziyu Jiang, Tianlong Chen, Ting Chen, and Zhangyang Wang. Improving contrastive learning on imbalanced data via open-world sampling. Advances in Neural Information Processing Systems, 34, 2021a.  
+Ziyu Jiang, Tianlong Chen, Bobak J Mortazavi, and Zhangyang Wang. Self-damaging contrastive learning. In International Conference on Machine Learning, pp. 4927-4939. PMLR, 2021b.  
+Longlong Jing and Yingli Tian. Self-supervised visual feature learning with deep neural networks: A survey. IEEE transactions on pattern analysis and machine intelligence, 43(11):4037-4058, 2020.  
+Siddharth Joshi, Yu Yang, Yihao Xue, Wenhan Yang, and Baharan Mirzasoleiman. Towards mitigating spurious correlations in the wild: A benchmark & a more realistic dataset. arXiv preprint arXiv:2306.11957, 2023.  
+Polina Kirichenko, Pavel Izmailov, and Andrew Gordon Wilson. Last layer re-training is sufficient for robustness to spurious correlations. arXiv preprint arXiv:2204.02937, 2022.  
+Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Bal-subramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al. Wilds: A benchmark of in-the-wild distribution shifts. In International Conference on Machine Learning, pp. 5637-5664. PMLR, 2021.  
+Preethi Lahoti, Alex Beutel, Jilin Chen, Kang Lee, Flavien Prost, Nithum Thain, Xuezhi Wang, and Ed H. Chi. Fairness without demographics through adversarially reweighted learning, 2020.  
+Yoonho Lee, Annie S Chen, Fahim Tajwar, Ananya Kumar, Huaxiu Yao, Percy Liang, and Chelsea Finn. Surgical fine-tuning improves adaptation to distribution shifts. arXiv preprint arXiv:2210.11466, 2022a.  
+Yoonho Lee, Huaxiu Yao, and Chelsea Finn. Diversify and disambiguate: Learning from underspecified data. arXiv preprint arXiv:2202.03418, 2022b.  
+Weixin Liang and James Zou. Metashift: A dataset of datasets for evaluating contextual distribution shifts and training conflicts. arXiv preprint arXiv:2202.06523, 2022.
+
+Evan Z Liu, Behzad Haghloo, Annie S Chen, Aditi Raghunathan, Pang Wei Koh, Shiori Sagawa, Percy Liang, and Chelsea Finn. Just train twice: Improving group robustness without training group information. In International Conference on Machine Learning, pp. 6781-6792. PMLR, 2021a.  
+Hong Liu, Jeff Z HaoChen, Adrien Gaidon, and Tengyu Ma. Self-supervised learning is more robust to dataset imbalance. arXiv preprint arXiv:2110.05025, 2021b.  
+Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaou Tang. Deep learning face attributes in the wild. In Proceedings of the IEEE international conference on computer vision, pp. 3730-3738, 2015.  
+R Thomas McCoy, Ellie Pavlick, and Tal Linzen. Right for the wrong reasons: Diagnosing syntactic heuristics in natural language inference. arXiv preprint arXiv:1902.01007, 2019.  
+Casey Meehan, Florian Bordes, Pascal Vincent, Kamalika Chaudhuri, and Chuan Guo. Do ssl models have déjà vu? a case of unintended memorization in self-supervised learning, 2023.  
+Aditya Krishna Menon, Ankit Singh Rawat, and Sanjiv Kumar. Overparameterisation and worst-case generalisation: friend or foe? In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=jphnJNOwe36.  
+Mazda Moayeri, Sahil Singla, and Soheil Feizi. Hard imagenet: Segmentations for objects with strong spurious cues. Advances in Neural Information Processing Systems, 35:10068-10077, 2022.  
+Vaishnavh Nagarajan, Anders Andreassen, and Behnam Neyshabur. Understanding the failure modes of out-of-distribution generalization. arXiv preprint arXiv:2010.15775, 2020.  
+Junhyun Nam, Hyuntak Cha, Sungsoo Ahn, Jaeho Lee, and Jinwoo Shin. Learning from failure: De-biasing classifier from biased classifier. Advances in Neural Information Processing Systems, 33:20673-20684, 2020.  
+Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.  
+Maxime Oquab, Timothee Darcet, Theo Moutakanni, Huy Vo, Marc Szafraniec, Vasil Khalidov, Pierre Fernandez, Daniel Haziza, Francisco Massa, Alaaeldin El-Nouby, et al. Dinov2: Learning robust visual features without supervision. arXiv preprint arXiv:2304.07193, 2023.  
+Joshua Robinson, Li Sun, Ke Yu, Kayhan Batmanghelich, Stefanie Jegelka, and Suvrit Sra. Can contrastive learning avoid shortcut solutions? arXiv preprint arXiv:2106.11230, 2021.  
+Elan Rosenfeld, Pradeep Ravikumar, and Andrej Risteski. Domain-adjusted regression or: Erm may already learn features sufficient for out-of-distribution generalization. arXiv preprint arXiv:2202.06856, 2022.  
+Shiori Sagawa, Pang Wei Koh, Tatsunori B. Hashimoto, and Percy Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization, 2020a.  
+Shiori Sagawa, Aditi Raghunathan, Pang Wei Koh, and Percy Liang. An investigation of why overparameterization exacerbates spurious correlations. In International Conference on Machine Learning, pp. 8346-8356. PMLR, 2020b.  
+Hadi Salman, Saachi Jain, Andrew Ilyas, Logan Engstrom, Eric Wong, and Aleksander Madry. When does bias transfer in transfer learning? arXiv preprint arXiv:2207.02842, 2022.  
+Marin Scalbert, Maria Vakalopoulou, and Florent Couzinié-Devy. Improving domain-invariance in self-supervised learning via batch styles standardization. arXiv preprint arXiv:2303.06088, 2023.  
+Ramprasaath R Selvaraju, Abhishek Das, Ramakrishna Vedantam, Michael Cogswell, Devi Parikh, and Dhruv Batra. Grad-cam: Why did you say that? arXiv preprint arXiv:1611.07450, 2016.  
+Harshay Shah, Kaustav Tamuly, Aditi Raghunathan, Prateek Jain, and Praneeth Netrapalli. The pitfalls of simplicity bias in neural networks. Advances in Neural Information Processing Systems, 33:9573-9585, 2020.
+
+Kendrick Shen, Robbie M Jones, Ananya Kumar, Sang Michael Xie, Jeff Z HaoChen, Tengyu Ma, and Percy Liang. Connect, not collapse: Explaining contrastive learning for unsupervised domain adaptation. In International Conference on Machine Learning, pp. 19847-19878. PMLR, 2022.  
+Sahil Singla and Soheil Feizi. Salient imagenet: How to discover spurious features in deep learning? arXiv preprint arXiv:2110.04301, 2021.  
+Jiaming Song, Pratyusha Kalluri, Aditya Grover, Shengjia Zhao, and Stefano Ermon. Learning controllable fair representations. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 2164-2173. PMLR, 2019.  
+Alex Tamkin, Vincent Liu, Rongfei Lu, Daniel Fein, Colin Schultz, and Noah Goodman. Dabs: A domain-agnostic benchmark for self-supervised learning. arXiv preprint arXiv:2111.12062, 2021.  
+Antonio Torralba and Alexei A Efros. Unbiased look at dataset bias. In CVPR 2011, pp. 1521-1528. IEEE, 2011.  
+Yao-Hung Hubert Tsai, Yue Wu, Ruslan Salakhutdinov, and Louis-Philippe Morency. Demystifying self-supervised learning: An information-theoretical framework. arXiv e-prints, pp. arXiv-2006, 2020.  
+Lifu Tu, Garima Lalwani, Spandana Gella, and He He. An empirical study on robustness to spurious correlations using pre-trained language models. Transactions of the Association for Computational Linguistics, 8:621-633, 2020.  
+Grant Van Horn, Elijah Cole, Sara Beery, Kimberly Wilber, Serge Belongie, and Oisin Mac Aodha. Benchmarking representation learning for natural world image collections. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 12884-12893, 2021.  
+Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The caltech-ucsd birds-200-2011 dataset. 2011.  
+Tan Wang, Zhongqi Yue, Jianqiang Huang, Qianru Sun, and Hanwang Zhang. Self-supervised learning disentangled group representation as feature. Advances in Neural Information Processing Systems, 34:18225-18240, 2021.  
+Zhao Wang and Aron Culotta. Identifying spurious correlations for robust text classification. arXiv preprint arXiv:2010.02458, 2020.  
+Yuzhe Yang, Haoran Zhang, Dina Katabi, and Marzyeh Ghassemi. Change is hard: A closer look at subpopulation shift. In International Conference on Machine Learning, 2023.  
+Jure Zbontar, Li Jing, Ishan Misra, Yann LeCun, and Stéphane Deny. Barlow twins: Self-supervised learning via redundancy reduction. In International Conference on Machine Learning, pp. 12310-12320. PMLR, 2021.  
+John R Zech, Marcus A Badgeley, Manway Liu, Anthony B Costa, Joseph J Titano, and Eric Karl Oermann. Variable generalization performance of a deep learning model to detect pneumonia in chest radiographs: a cross-sectional study. PLoS medicine, 15(11):e1002683, 2018.  
+Dinghuai Zhang, Kartik Ahuja, Yilun Xu, Yisen Wang, and Aaron Courville. Can subnetwork structure be the key to out-of-distribution generalization? In International Conference on Machine Learning, pp. 12356-12367. PMLR, 2021.  
+Michael Zhang, Nimit S Sohoni, Hongyang R Zhang, Chelsea Finn, and Christopher Ré. Correct-n-contrast: A contrastive approach for improving robustness to spurious correlations. arXiv preprint arXiv:2203.01517, 2022.  
+Hattie Zhou, Ankit Vani, Hugo Larochelle, and Aaron Courville. Fortuitous forgetting in connectionist networks. arXiv preprint arXiv:2202.00155, 2022.

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+# VISION-LANGUAGE FOUNDATION MODELS AS EFFECTIVE ROBOT IMITATORS
+
+Xinghang  $\mathbf{L}\mathbf{i}^{1,2,\dagger}$ , Minghuan  $\mathbf{L}\mathbf{u}^{2,3,\dagger}$ , Hanbo Zhang $^{2}$ , Cunjun  $\mathbf{Y}\mathbf{u}^{4}$ , Jie  $\mathbf{X}\mathbf{u}^{2}$ , Hongtao  $\mathbf{W}\mathbf{u}^{2}$ , Chilam Cheang $^{2}$ , Ya Jing $^{2}$ , Weinan Zhang $^{3}$ , Huaping  $\mathbf{L}\mathbf{u}^{1,\boxtimes}$ , Hang  $\mathbf{L}\mathbf{i}^{2}$ , Tao Kong $^{2,\boxtimes}$
+
+$^{1}$ Tsinghua University,  $^{2}$ ByteDance Research,
+
+$^{3}$ Shanghai Jiao Tong University,  $^{4}$ National University of Singapore
+
+lixingha23@mails.tsinghua.edu.cn, hpliu@tsinghua.edu.cn,
+
+{minghuanliu, wnzhang}@sjtu.edu.cn, kongtao@bytedance.com
+
+# ABSTRACT
+
+Recent progress in vision language foundation models has shown their ability to understand multimodal data and resolve complicated vision language tasks, including robotics manipulation. We seek a way of making use of existing vision-language models (VLMs) with fine-tuning on robotics data. To this end, we derive a simple and novel vision-language manipulation framework, dubbed RoboFlamingo, built upon the open-source VLMs, OpenFlamingo. Unlike prior works, RoboFlamingo utilizes pre-trained VLMs for single-step vision-language comprehension, models sequential history information with an explicit policy head, and is slightly finetuned by imitation learning only on language-conditioned manipulation datasets. Such a decomposition provides RoboFlamingo the flexibility for open-loop control and deployment on low-performance platforms. By surpassing the state-of-the-art performance on the benchmark by a significant margin, we demonstrate that RoboFlamingo presents itself as an effective and competitive alternative for adapting VLMs to robot control. Our extensive experimental results also reveal several interesting conclusions regarding the behavior of different pre-trained VLMs on manipulation tasks. RoboFlamingo can be trained or evaluated on a single GPU server, and we believe it has the potential to be a cost-effective and easy-to-use solution for robotics manipulation, empowering everyone with the ability to fine-tune their own robotics policy. Codes and models will be public.
+
+# 1 INTRODUCTION
+
+Recent progress in vision-language foundation models (VLM) has presented their exhilarating ability in modeling and aligning the representation of images and words, and the unlimited potential to resolve a wide range of downstream tasks with multi-modality data, for instance, visual question-answering (Li et al., 2023; Zhou et al., 2022), image captioning (Zeng et al., 2022; Wang et al., 2022; Li et al., 2021), human-agent interactions (Liu et al., 2022b; Oertel et al., 2020; Seaborn et al., 2021). These successes, undeniably, encourage people to imagine a generalist robot equipped with such a vision-language comprehension ability to interact naturally with humans and perform complex manipulation tasks.
+
+Therefore, we aim to explore integrating vision-language foundation models to serve as robot manipulation policies. While there have been some previous studies that incorporated large language models (LLMs) and vision-language models (VLMs) into robot systems as high-level planners (Ahn et al., 2022; Driess et al., 2023), making use of them directly for low-level control still poses challenges. Most VLMs are trained on static image-language pairs, whereas robotics tasks require video comprehension for closed-loop control. Additionally, VLM outputs primarily consist of language tokens, which significantly differ in representation compared to robot actions. A recent work (Brohan et al., 2023), namely Robotics Transformer 2 (RT-2), has demonstrated a possible solution for adapting VLMs to low-level robot control. However, democratizing such an expensive framework for all robotics practitioners proves difficult as it utilizes private models and necessitates
+
+![](images/bf9f972f399f759ccd186956067ce1c34ce4b256d81f7e27314324882b9a0413.jpg)
+
+![](images/c984f2c23a6703d74a706767f9c2434d6fa88f59370160ffb3fb990920114b4a.jpg)
+
+![](images/67b7101b1a5dfe86f315577ddf5a88d0aab7fe5a6ec8fc84f49d758ddb21153f.jpg)
+
+![](images/ff6fa7433410ef1ffb3c3889396a279b9d79a4aa8c9661f8c86d38b8b03d3bfe.jpg)  
+Figure 1: Comparison among RoboFlamingo and existing vision-language manipulation solutions.
+
+![](images/ac33638bfc5d47f5da77dd5b354c6f17fde105e907623ddf5166e58793a3da4a.jpg)
+
+co-fine-tuning on extensive vision-language data to fully showcase its effectiveness. Consequently, there is an urgent need for robot communities to have a low-cost alternative solution that effectively enables a robot manipulation policy with VLMs.
+
+To this end, we introduce RoboFlamingo, a novel vision-language manipulation framework that leverages publicly accessible pre-trained VLMs to effectively construct manipulation policies for robotics. Specifically, RoboFlamingo is grounded upon the open-source VLM, OpenFlamingo (Awadalla et al., 2023), and resolves the challenge by decoupling visual-language understanding and decision-making. Unlike previous works, RoboFlamingo takes advantage of pre-trained VLMs mainly for understanding vision observations and language instructions at every decision step, models the historical features with an explicit policy head, and is fine-tuned solely on language-conditioned manipulation datasets using imitation learning. With such a decomposition, we only need to combine a small amount of robotics demonstration to adapt the model to downstream manipulation tasks, and RoboFlamingo also offers flexibility for open-loop control and deployment on low-performance platforms. Moreover, benefiting from the pre-training on extensive vision-language tasks, RoboFlamingo achieves state-of-the-art performance with a large margin over previous works, and generalizes well to zero-shot settings and environments. It is worth noting that RoboFlamingo can be trained or evaluated on a single GPU server. As a result, we believe RoboFlamingo can be a cost-effective yet high-performance solution for robot manipulation, empowering everyone with the ability to fine-tune their own robots with VLMs.
+
+Through extensive experiments, we demonstrate that RoboFlamingo outperforms existing methods by a clear margin. Specifically, we evaluate its performance using the Composing Actions from Language and Vision benchmark (CALVIN) (Mees et al., 2022b), a widely-recognized simulation benchmark for long-horizon language-conditioned tasks. Our findings indicate that RoboFlamingo is an effective and competitive alternative for adapting VLMs to robot control, achieving a performance improvement that is two times greater compared to the previous state-of-the-art method. Our comprehensive results also yield valuable insights into the use of pre-trained VLMs for robot manipulation tasks, offering potential directions for further research and development.
+
+# 2 RELATED WORK
+
+Language can be the most intuitive and pivotal interface for human-robot interaction, enabling non-expert humans to seamlessly convey their instructions to robots for achieving diverse tasks. Consequently, the realm of language-conditioned multi-task manipulation has garnered substantial attention in recent years. Intuitively, such tasks require robots to have a good understanding of not only the visual captures of the outside world, but also the instructions represented by words. With the strong representation ability of pre-trained vision and language models, a lot of previous works have incorporated pre-trained models into the learning framework. Among them, we roughly classify them into the following three categories, which is also illustratively compared in Fig. 1.
+
+Fine-tuning. While some early works such as Jang et al. (2022); Lynch & Sermanet (2020) trained a vision encoder and a language encoder to learn representations for the input language and vision data from manipulation tasks, some recent work directly takes pre-trained models to obtain great representations, then trains the policy model beyond them from scratch or fine-tuning the whole model. For instance, Jiang et al. (2023) utilizes a pre-trained T5 (Raffel et al., 2020) model to encode the multi-modal prompts, and learn the actions by fine-tuning the T5 model and additionally training an object encoder and attention layers. HULC (Mees et al., 2022a) utilizes the vision encoder of Lynch & Sermanet (2020) trained on the CALVIN dataset (Mees et al., 2022b) and some pre-trained language encoder models such as sentence transformer (Reimers & Gurevych, 2019), and their HULC++ (Mees et al., 2023) also fine-tunes these encoders. Besides, Brohan et al. (2022) proposed RT-1, i.e., robotics transformers, a 35M vision-language-action model (VLA) which tokenizes the action and aligns the vision, language, and action in the token space and is trained on a large amount of real-world manipulation dataset, using the Universal Sentence Encoder (Cer et al., 2018) to obtain the language embedding and the pre-trained EfficientNet-B3 (Tan & Le, 2019) as the vision tokenizer.
+
+LLM planning. Some approaches have exploited large language models (LLMs) as a powerful zero-shot planner, e.g., SayCan Ahn et al. (2022), to generate step-by-step pre-defined plans with human-interactive prompts on given tasks, subsequently instructing different pre-trained low-level skill policies to execute those plans and finish multiple tasks. Compared to other works, the controlling policies do not require any ability to understand instructions, but rely on the pre-trained frozen LLM to select necessary skills.
+
+Co-Fine-Tuning. Driess et al. (2023) proposed 540B PaLM-E model, showing a different way of utilizing the pre-trained vision and language model. Specifically, they choose different pre-trained models to encode the input scene, and the PaLM (Chowdhery et al., 2022) model as the base model, train the model to generate pre-defined multi-step plans described by language by co-fine-tuning the whole VLM end-to-end using both mobile manipulation question-answering data and auxiliary vision-language training data such as image captioning and visual question answering data collected from the web. Similar to SayCan (Ahn et al., 2022), they require low-level control policies to execute the generated plans. Motivated by PaLM-E, Brohan et al. (2023) further introduced RT-2, which is based on RT-1 but is adapted to use large vision-language backbones like PaLI-X (Chen et al., 2023) and PaLM-E (Driess et al., 2023), training the policy utilizing both robot manipulation data and web data. Their method reveals that VLMs have the potential to be adapted into robot manipulation, yet their key co-fine-tuning training strategy requires a large amount of both web-scale data vision-language data and low-level robot actions. Additionally, the VLMs and the data they use are private, making it hard for every robotics practitioner to play on such a solution for their own.
+
+Although these previous models somehow bridge the gap between vision and language on robot manipulation tasks, they either reply on low-level skill policies, like SayCan and PaLM-E; or train a whole large model, such as RT-1; or require a huge amount of vision-language data and computational resources to ensure the model learns the manipulation policy without forgetting the great alignment of vision and language. Compared with these works, our proposed RoboFlamingo is a simple and intuitive solution to easily adapt existing VLMs (OpenFlamingo (Alayrac et al., 2022; Awadalla et al., 2023) used in this paper), only requiring fine-tuning on a small number of manipulation demonstrations. We hope RoboFlamingo provides a different perspective on fully leveraging the ability of VLMs, while requiring less data collection costs and computing consumption to make it an open and easy-to-use solution for everyone.
+
+# 3 BACKGROUND
+
+Robot manipulation. In this paper, we mainly consider robot manipulation tasks, where the agent (robot) does not have access to the ground-truth state of the environment, but visual observations from different cameras and its own proprioception states. As for the action space, it often includes the relative target pose and open/closed state of the gripper. For instance, in the testbed of CALVIN (Mees et al., 2022b), the observations consist of simulated camera captures from two different views, and the action is a 7-DoF control of a Franka Emika Panda robot arm with a parallel gripper, and the instructions are reaching goals, i.e., the after-the-fact descriptions.
+
+![](images/249a394a7f85292a8b057969ec63ce5bfd6fc3e11c7e25cf0cb512a4c887c1cb.jpg)  
+Figure 2: The illustration of the proposed RoboFlamingo framework. The Flamingo backbone models single-step observations, and the temporal features are modeled by the policy head.
+
+Imitation learning. Imitation learning (Pomerleau, 1988; Zhang et al., 2018; Liu et al., 2020; Jang et al., 2022) allows the agent to mimic the manipulation plans from instruction-labeled expert play data  $\mathcal{D} = \{(\tau ,l)_i\}_{i = 0}^D$  where  $D$  is the number of trajectories,  $l$  is the language instruction, and  $\tau = \{(o_t,a_t)\}$  contains preceding states and actions to reach the goal described by the given instruction. The learning objective can be simply concluded as a maximum likelihood goal-conditioned imitation objective to learn the policy  $\pi_{\theta}$ :
+
+$$
+\ell = \mathbb {E} _ {(\tau , l) _ {i} \sim \mathcal {D}} \left[ \sum_ {t = 0} ^ {| \tau |} \log \pi_ {\theta} \left(a _ {t} \mid o _ {t}, l\right) \right]. \tag {1}
+$$
+
+# 4 ROBOFLAMINGO
+
+RoboFlamingo, a generalized robotics agent, excels in resolving language-conditioned manipulation tasks. The key idea is to draw help from pre-trained vision-language models (VLMs) and adapt them to manipulation policies, acquiring the ability of object grounding, language comprehension, vision-language alignment, and long-horizon planning. Particularly, RoboFlamingo looks into one of the popular VLMs, Flamingo (Alayrac et al., 2022), and takes its open-source model OpenFlamingo (Awadalla et al., 2023) as the backbone. The overview of RoboFlamingo is shown in Fig. 2. To adapt large-scale vision-language models to robotic manipulation, RoboFlamingo simply adds a policy head for end-to-end finetuning. It addresses three main challenges: 1) it adapts vision-language models with static image inputs to video observations; 2) it generates robot control signals instead of text-only outputs; 3) it requires a limited amount of downstream robotic manipulation data to achieve high performance and generality with billions of trainable parameters. We will elaborate on the design of RoboFlamingo in this section.
+
+# 4.1 LANGUAGE-CONDITIONED ROBOT CONTROL
+
+The problem of language-conditioned robot control can be modeled as a goal-conditioned partially observable Markov decision process (GC-POMDP) (Liu et al., 2022a):  $\mathcal{M} = \langle S, \mathcal{O}, \mathcal{A}, \mathcal{T}, \rho_0, \mathcal{L}, \phi, f \rangle$ , where  $\mathcal{S}$  and  $\mathcal{O}$  are the set of states and observations separately,  $\mathcal{A}$  is the action space,  $\mathcal{T}: \mathcal{S} \times \mathcal{A} \to \mathcal{S}$  is the environment dynamics function,  $\rho_0: \mathcal{S} \to [0,1]$  is the initial state distribution,  $\phi(s)$  indicate if the task is successful, and  $f(o|s): \mathcal{S} \to \mathcal{O}$  is the observation function. Specifically, for each controlling episode, the robot is given a goal, represented by a length- $M$  free-form language instruction  $l \in \mathcal{L}$  at every time step  $t$ , and the observations  $o_t$  are typically two images  $I_t$ ,  $G_t$  from a third-perspective camera and a gripper camera. The controlling policy can be modeled as a goal-conditioned policy  $\pi(a|o,l): \mathcal{S} \times \mathcal{L} \to \mathcal{A}$  and the action  $a$  is typically the desired relative position and pose of the gripper, along with its open/close status.
+
+In our RoboFlamingo, the policy  $\pi_{\theta}(a|o,l)$  is parameterized by  $\theta$ . It consists of a backbone based on Flamingo  $f_{\theta}$  and a policy head  $p_{\theta}$ . The backbone takes visual observations and language-represented goals as the input and provides a latent fused representation at each time step for the policy head:  $X_{t} = f_{\theta}(o_{t},l)$ . Then the policy head further predicts the action to fulfill the specified goal for the robot:  $a_{t} = p_{\theta}(X_{t},h_{t - 1})$ , where  $h_{t - 1}$  is the hidden state from the last step that encodes the history information for decision-making. We will introduce each module in detail in the following sections.
+
+# 4.2 THE FLAMINGO BACKBONE
+
+We adopt the Flamingo backbone  $f_{\theta}$  for understanding the vision and language inputs at every decision step. Overall, Flamingo encodes the vision observations to the latent tokens by a vision encoder; and then fuses them with language goals through the feature fusion decoder. We explain these parts in detail below.
+
+# 4.2.1 VISION ENCODER
+
+The vision encoder consists of a vision transformer (ViT) (Yuan et al., 2021) and a perceiver resampler (Alayrac et al., 2022). At every time step  $t$ , the two-view camera images  $I_{t}$ ,  $G_{t}$  are encoded to  $\hat{X}_{t}$ , consisting of a visual token sequence, through the ViT module:
+
+$$
+\hat {X} _ {t} ^ {v} = \operatorname {V i T} \left(I _ {t}, G _ {t}\right), \tag {2}
+$$
+
+where  $\hat{X}_t^v = (\hat{x}_{t1}^v,\dots ,\hat{x}_{tN}^v)$  represents the visual token sequence at  $t$ ,  $N$  represents the token number of the encoded output. After encoding, RoboFlamingo utilizes a perceiver resampler to compress the number of visual tokens from  $N$  to  $N_r$ . In detail, the resampler maintains a set of learnable parameters and utilizes the attention mechanism to reduce the number of token sequences to  $K$ . Formally, the resampler is formulated as:
+
+$$
+K _ {R} = \hat {X} _ {t} ^ {v} W _ {K} ^ {R}, V _ {R} = \hat {X} _ {t} ^ {v} W _ {V} ^ {R}, X _ {t} ^ {v} = \operatorname {s o f t m a x} \left(\frac {Q _ {R} K _ {R} ^ {T}}{\sqrt {d}}\right) V _ {R}, \tag {3}
+$$
+
+where  $Q_{R} \in \mathbb{R}^{N_{r} \times d}$  corresponds to the learnable parameters of the resampler and serves as the query vector,  $d$  is the hidden dimension size,  $W_{K}^{R}, W_{V}^{R} \in \mathbb{R}^{d_{v} \times d}$  represents the linear transformation matrix of key and value,  $d_{v}$  is the feature dimension of the visual token,  $K_{R}$  and  $V_{R}$  are the transformed key and value vector of vision input  $V$ .
+
+# 4.2.2 FEATURE FUSION DECODER
+
+The compressed visual tokens output from the resampler  $X_{t}^{v} \in \mathbb{R}^{N_{r} \times d}$  are further passed to the feature fusion decoder, which is designed to generate the vision-language joint embedding by fusing the language instruction with the encoded vision feature  $X_{t}^{v}$ . In RoboFlamingo, we utilize the pre-trained decoder from OpenFlamingo (Awadalla et al., 2023) and fine-tune the decoder module following the way as in Awadalla et al. (2023). Specifically, the decoder consists of  $L$  layers, each of which involves a transformer decoder layer and a cross-attention layer. The transformer layers are directly copied from a pre-trained language model (such as LlaMA (Touvron et al., 2023), GPT-Neox (Black et al., 2022) and MPT (Team et al., 2023)) and are frozen during the whole training process; the cross-attention layer takes the language token as query, and the encoded visual token as key and value, which is fine-tuned by imitation learning objectives on manipulation data (see following sub-sections). Formally, if we denote  $x_{i} \in \mathbb{R}^{d}$  the  $i$ -th embedded token of the instruction,  $M$  the instruction length, and  $X \in \mathbb{R}^{M \times d}$  is the embedded matrix of the instruction, then the embedded natural language instruction should be  $X = (x_{1}, x_{2}, \dots, x_{M})$  and output  $X_{t}^{l + 1}$  of the  $l$ -th decoder layer given the input  $X_{t}^{l}$  is computed by:
+
+$$
+\begin{array}{l} \hat {X} _ {t} ^ {l} = \mathrm {T a n h} (\alpha) \cdot \mathbf {M L P} \big (A \big (X _ {t} ^ {l} W _ {Q} ^ {C}, X _ {t} ^ {v} W _ {K} ^ {C}, X _ {t} ^ {v} W _ {V} ^ {C} \big) \big) + X _ {t} ^ {l}, \\ X _ {t} ^ {l + 1} = \operatorname {M L P} \left(A \left(\hat {X} _ {t} ^ {l} W _ {Q} ^ {S}, \hat {X} _ {t} ^ {l} W _ {K} ^ {S}, \hat {X} _ {t} ^ {l} W _ {V} ^ {S}\right)\right) + \hat {X} _ {t} ^ {l}, \\ \end{array}
+$$
+
+where  $X_{t}^{1} = X$ ,  $\hat{X}_{t}^{l}$  corresponds to the output of the gated cross-attention layer at time instant  $t$ ,  $W_{Q}^{C}, W_{K}^{C}, W_{V}^{C} \in \mathbb{R}^{d \times d}$  represents the learnable parameters of the cross-attention layer.  $\alpha \in \mathbb{R}$  is a learnable gate parameter to control the mixing weights for stability.  $W_{Q}^{S}, W_{K}^{S}, W_{V}^{S} \in \mathbb{R}^{d \times d}$  represents the parameters of the self-attention layer and MLP represents a multi-layer perceptron network. With the deep interaction of the vision and language token, we expect the output  $X_{t} = X_{t}^{L} = \{x_{t,1}^{L}, x_{t,2}^{L}, \dots, x_{t,M}^{L}\}$  at time step  $t$  to be an informative vision-language joint embedding for robot manipulation.
+
+# 4.3 POLICY HEAD
+
+The output  $X_{t}^{L}$  from the feature fusion decoder is trained as the representation of the vision observation and language instruction, which will be further translated into low-level control signals. To
+
+achieve this, we simply adopt an additional policy head  $p_{\theta}$  to predict the action, e.g., the 7 DoF end-effector pose and gripper status. We test various strategies to model the historical observation sequences and behave as the policy head, e.g., a long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997) network with an MLP for the final prediction; a decoder-only transformer (Brown et al., 2020) similarly with an MLP; or a single MLP that only models single-step information (see Section 5 for more details). Taking the LSTM version as an example, with the vision-language joint embedding sequence  $X_{t}^{L}$ , we obtain an aggregated embedding through a max-pooling operation over the token dimension and predict the action as:
+
+$$
+\tilde {X} _ {t} = \operatorname {M a x P o o l i n g} \left(X _ {t}\right); h _ {t} = \operatorname {L S T M} \left(\tilde {X} _ {t}, h _ {t - 1}\right); a _ {t} ^ {\text {p o s e}}, a _ {t} ^ {\text {g r i p p e r}} = \operatorname {M L P} \left(h _ {t}\right), \tag {5}
+$$
+
+where  $h_t$  represents the hidden state at  $t$ , and  $a_t^{pose}$ ,  $a_t^{gripper}$  are the predicted end-effector pose and gripper status.
+
+# 4.4 TRAINING OBJECTIVE
+
+We utilize maximum likelihood imitation learning objectives to fine-tune the proposed pre-trained backbone and the policy head. Concretely, the desired relative pose is optimized via regression loss (we use mean squared error (MSE) loss) and the gripper status uses classification loss (we use binary cross-entropy (BCE) loss):
+
+$$
+\ell = \sum_ {t} \operatorname {M S E} \left(a _ {t} ^ {\text {p o s e}}, \hat {a} _ {t} ^ {\text {p o s e}}\right) + \lambda_ {\text {g r i p p e r}} \operatorname {B C E} \left(a _ {t} ^ {\text {g r i p p e r}}, \hat {a} _ {t} ^ {\text {g r i p p e r}}\right), \tag {6}
+$$
+
+where  $\hat{a}_t^{\mathrm{pose}},\hat{a}_t^{\mathrm{gripper}}$  is the demonstration for end effector pose and gripper status at timestep  $t$ ,  $\lambda_{\mathrm{gripper}}$  corresponds to the weight of gripper loss.
+
+In the training procedure, we follow the fine-tuning paradigm of OpenFlamingo by only training the parameters of the resampler, the gated cross-attention module of each decoder layer, and the policy head while freezing all other parameters.
+
+# 5 EXPERIMENTS
+
+We conduct extensive experiments to examine the proposed RoboFlamingo solution, and answer how pre-trained VL models (VLMs) benefit language-conditioned robotic manipulation. In short, we investigate RoboFlamingo from the following perspectives:
+
+1. Effectiveness. We wonder the imitation learning performance of RoboFlamingo by training it on the given demonstration data.  
+2. Zero-shot Generalization. We focus on generalization on unseen tasks. In other words, we study how the model will behave given unseen vision contexts like different objects, even with unseen instructions.  
+3. Ablation Studies. We further explore the essential factors that matter in adapting VLMs to robot control policy in the framework of RoboFlamingo.
+
+# 5.1 BENCHMARK AND BASELINES
+
+We choose CALVIN (Mees et al., 2022b), an open-source simulated benchmark to learn long-horizon language-conditioned tasks, as our testbed, and the corresponding datasets as our imitation learning demonstration data. CALVIN encompasses a total of 34 distinct tasks and evaluates 1000 unique instruction chains for sequential tasks. In each experiment, the robot is required to successfully complete sequences of up to five language instructions consecutively. The policy for each consecutive task is dependent on a goal instruction, and the agent advances to the subsequent goal only if it successfully accomplishes the current task. The dataset contains four splits for environments A, B, C, and D. Each consists of 6 hours of human-teleoperated recording data (more than 2 million steps) that might contain sub-optimal behavior, and only  $1\%$  of that data is annotated with language instructions ( $\sim$ 24 thousand steps). See Fig. 4 in Appendix A.1 for a more detailed description and visualized examples of the benchmark.
+
+Table 1: The imitation performance on various settings, all results are reported using the best-behaved model checkpoints. Full and Lang denote if the model is trained using unpaired vision data (i.e., vision data without language pairs); Freeze-emb refers to freezing the embedding layer of the fusion decoder; Enriched denote using GPT-4 enriched instructions. The gray rows denote numerical results evaluated by our re-trained model. We re-implement RT-1 and take the original code of HULC provided by Mees et al. (2022a). All other results are reported by Mees et al. (2022a).  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Training Data</td><td rowspan="2">Test Split</td><td colspan="6">Task Completed in a Sequence (Success Rate)</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>Avg Len</td></tr><tr><td>MCIL</td><td>ABCD (Full)</td><td>D</td><td>0.373</td><td>0.027</td><td>0.002</td><td>0.000</td><td>0.000</td><td>0.40</td></tr><tr><td>HULC</td><td>ABCD (Full)</td><td>D</td><td>0.889</td><td>0.733</td><td>0.587</td><td>0.475</td><td>0.383</td><td>3.06</td></tr><tr><td>HULC</td><td>ABCD (Lang)</td><td>D</td><td>0.892</td><td>0.701</td><td>0.548</td><td>0.420</td><td>0.335</td><td>2.90</td></tr><tr><td>RT-1</td><td>ABCD (Lang)</td><td>D</td><td>0.844</td><td>0.617</td><td>0.438</td><td>0.323</td><td>0.227</td><td>2.45</td></tr><tr><td>RoboFlamingo (Ours)</td><td>ABCD (Lang)</td><td>D</td><td>0.964</td><td>0.896</td><td>0.824</td><td>0.740</td><td>0.66</td><td>4.09</td></tr><tr><td>MCIL</td><td>ABC (Full)</td><td>D</td><td>0.304</td><td>0.013</td><td>0.002</td><td>0.000</td><td>0.000</td><td>0.31</td></tr><tr><td>HULC</td><td>ABC (Full)</td><td>D</td><td>0.418</td><td>0.165</td><td>0.057</td><td>0.019</td><td>0.011</td><td>0.67</td></tr><tr><td>RT-1</td><td>ABC (Lang)</td><td>D</td><td>0.533</td><td>0.222</td><td>0.094</td><td>0.038</td><td>0.013</td><td>0.90</td></tr><tr><td>RoboFlamingo (Ours)</td><td>ABC (Lang)</td><td>D</td><td>0.824</td><td>0.619</td><td>0.466</td><td>0.331</td><td>0.235</td><td>2.48</td></tr><tr><td>HULC</td><td>ABCD (Full)</td><td>D (Enriched)</td><td>0.715</td><td>0.470</td><td>0.308</td><td>0.199</td><td>0.130</td><td>1.82</td></tr><tr><td>RT-1</td><td>ABCD (Lang)</td><td>D (Enriched)</td><td>0.494</td><td>0.222</td><td>0.086</td><td>0.036</td><td>0.017</td><td>0.86</td></tr><tr><td>Ours</td><td>ABCD (Lang)</td><td>D (Enriched)</td><td>0.720</td><td>0.480</td><td>0.299</td><td>0.211</td><td>0.144</td><td>1.85</td></tr><tr><td>Ours (freeze-emb)</td><td>ABCD (Lang)</td><td>D (Enriched)</td><td>0.737</td><td>0.530</td><td>0.385</td><td>0.275</td><td>0.192</td><td>2.12</td></tr></table>
+
+We compare a set of well-performed baselines in CALVIN: (1) MCIL (Lynch & Sermanet, 2020): a scalable framework combining multitask imitation with free-form text conditioning, which learns language-conditioned visuomotor policies, and is capable of following multiple human instructions over a long horizon in a dynamically accurate 3D tabletop setting. (2) HULC (Mees et al., 2022a): a hierarchical method that combines different observation and action spaces, auxiliary losses, and latent representations, which achieved the SoTA performance on CALVIN. (3) RT-1 (Brohan et al., 2022): robotics transformer, which directly predicts the controlling actions by action tokens, as well as vision and language inputs. RT-2 (Brohan et al., 2023) is not experimentally compared since we have no access to their code, data, and model weights.
+
+# 5.2 IMITATION PERFORMANCE
+
+We train RoboFlamingo (with the M-3B-IFT backbone) using demonstrations only with language annotation from all 4 splits (A, B, C, and D), and evaluate the imitation performance on episodes sampled on split D ( $ABCD \to D$ ). The performance comparison is shown in Tab. 1. RoboFlamingo outperforms all baseline methods over all metrics by a large margin, even for those methods that are trained on the full set of data. This demonstrates the effectiveness of RoboFlamingo as the solution for robotics manipulation, enabling VLMs to become effective robot imitators.
+
+In addition, the success rate of the subsequent tasks can be regarded as a notion of the generalizability of the manipulation policies, since the initial state of a subsequent task highly relies on the ending state of its former task. The later a task is arranged in the task sequence, the more diverse its initial state is, which will need more powerful visual-language alignment abilities to successfully complete the task. Among all methods, RoboFlamingo achieves the highest success rate over the latter tasks. This demonstrates that RoboFlamingo is able to utilize the visual-language grounding ability of pre-trained VLMs. In the appendix, we further include the results of RoboFlamingo co-trained with COCO and VQA data (Appendix B.1) and compare with recent robotics representation works (Appendix B.2). Appendix B.1 also reveals how the original VL abilities change after fine-tuning.
+
+# 5.3 ZERO-SHOT GENERALIZATION
+
+To assess the zero-shot generalization ability, we evaluate RoboFlamingo in two aspects: vision and language. For vision generalization, we train models on splits A, B, and C and test on split D, which presents a different vision context. Our method significantly outperforms baselines in this vision generalization scenario  $(ABC\rightarrow D)$ , as shown in Tab. 1. Regarding language generalization, we enrich the language setting by generating 50 synonymous instructions for each task using GPT-4 (Achiam et al., 2023). We then randomly sample instructions during evaluation. Our method exhibits superior performance compared to all baselines in this language generalization setting.
+
+Table 2: Variants of VLMs tested. Pre-train denotes the original performance of VLM on the pre-training VL dataset, BestAvg. Len. denotes the best performance of the average success length of VLMs within 5 epochs, and MeanAvg. Len. denotes the mean performance of the average success length of VLMs of the last 3 epochs on CALVIN.  
+
+<table><tr><td rowspan="2">Backbone Name</td><td rowspan="2">LLM Arch</td><td rowspan="2">Total 
+Param</td><td rowspan="2">LLM 
+Param</td><td rowspan="2">Trainable 
+Param</td><td rowspan="2">Instr. 
+Tuning</td><td colspan="2">Pre-trained (Public, 4-shot)</td><td colspan="2">Avg. Len.</td></tr><tr><td>COCO (CIDEr)</td><td>VQAv2 (Acc)</td><td>Best</td><td>Mean</td></tr><tr><td>M-3B</td><td rowspan="2">MPT</td><td rowspan="2">3B</td><td rowspan="2">1B</td><td rowspan="2">1B</td><td>X</td><td>77.3</td><td>45.8</td><td>3.94</td><td>3.81</td></tr><tr><td>M-3B-IFT</td><td>✓</td><td>82.7</td><td>45.7</td><td>4.09</td><td>4.02</td></tr><tr><td>G-4B</td><td rowspan="2">GPT-Neox</td><td rowspan="2">4B</td><td rowspan="2">3B</td><td rowspan="2">1B</td><td>X</td><td>81.8</td><td>49.0</td><td>3.67</td><td>3.53</td></tr><tr><td>G-4B-IFT</td><td>✓</td><td>85.8</td><td>49.0</td><td>3.79</td><td>3.72</td></tr><tr><td>L-9B</td><td>LLaMA</td><td rowspan="2">9B</td><td rowspan="2">7B</td><td rowspan="2">1B</td><td>X</td><td>74.3</td><td>44.0</td><td>2.79</td><td>2.71</td></tr><tr><td>M-9B</td><td>MPT</td><td>X</td><td>89.0</td><td>54.8</td><td>3.97</td><td>3.87</td></tr></table>
+
+![](images/8d3fedacd8d47aeb81f38a7c292072c2de71dfec5ccb98aa3132dc8e3a743503.jpg)  
+(a) Various policy formulation.
+
+![](images/c722d7c2bbf5cce79cc84745eaff9a8866d3e3c87da486883322bbaaebc47d90.jpg)  
+(b) Different training paradigms.
+
+![](images/13d09828a76a3e8c9673903782892dabf6ca9d3d81d2b51413325e48359c7288.jpg)  
+(c) Open loop control.  
+Figure 3: Ablation studies on the  $ABCD \to D$  setting.
+
+Note that the success rate of RoboFlamingo on subsequent tasks dropped more than HULC does. This may be due to our approach directly using word tokens as input during training, which can result in larger variations for synonymous sentences compared to HULC using a frozen sentence model for embedding instructions. To address this, we freeze the embedding layer of the feature fusion decoder in our method, leading to improved generalization and reduced performance drop.
+
+# 5.4 ABLATION STUDIES
+
+In this section, we conduct ablation studies for RoboFlamingo to answer the following questions:
+
+1) How does RoboFlamingo perform with different policy heads/formulations?  
+2) Does vision-language (VL) pre-training improve downstream robotic tasks?  
+3) How do critical factors in VL pre-training affect robotic tasks?
+
+# 5.4.1 HOW DOES RoboFlamingo PERFORM WITH DIFFERENT POLICY FORMULATIONS?
+
+We test RoboFlamingo with different policy heads/formulations. In particular, we compare 4 different implementations: (a)  $MLP w / o$  hist takes only the current observation as input to predict actions, which ignores the observation history. (b)  $MLP w$  hist takes the history frames into the vision encoder with position embedding, and encodes the history information through the cross-attention layers in the feature fusion decoder. (c)  $GPT$  and (d)  $LSTM$  both utilize the VLM backbone to process single-frame observations and integrate the history with the policy head.  $GPT$  explicitly takes the visual history as input to predict the next action.  $LSTM$  implicitly maintains a hidden state to encode memory and predict the action. See Appendix C.1 for detailed illustration. We compare their best performance on the  $ABCD \rightarrow D$  setting in Fig. 3 (a).  $MLP w / o$  hist performs the worst, indicating the importance of the history information in the manipulation task.  $MLP w$  hist performs better than  $MLP w / o$  hist, but is still much worse than  $GPT$  and  $LSTM$ . We hypothesize that this may stem from the fact that the VLM (OpenFlamingo) has only seen image-text pairs during pre-training and cannot process consequent frames effectively. Further, the performance of  $GPT$  and  $LSTM$  are similar, we choose  $LSTM$  as the default choice due to its simplicity.
+
+# 5.4.2 DOES VL PRE-TRAINING IMPROVE DOWNSTREAM ROBOTIC TASKS?
+
+To verify the necessity of VL pre-training, we train the same model without loading the pre-trained parameters of the cross-attention layers and the resampler trained by OpenFlamingo models (denoted
+
+Table 3: The performance on  $10\%$  language annotated data on  $ABCD \to D$  setting. All variants are trained and evaluated for the same training epochs.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">Task Completed in a Sequence (Success Rate)</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>Avg Len</td></tr><tr><td>M-3B</td><td>0.047</td><td>0.003</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.05</td></tr><tr><td>M-3B-IFT</td><td>0.120</td><td>0.007</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.13</td></tr><tr><td>G-4B</td><td>0.420</td><td>0.054</td><td>0.003</td><td>0.000</td><td>0.000</td><td>0.48</td></tr><tr><td>G-4B-IFT</td><td>0.448</td><td>0.084</td><td>0.014</td><td>0.003</td><td>0.001</td><td>0.55</td></tr><tr><td>M-9B</td><td>0.547</td><td>0.190</td><td>0.067</td><td>0.020</td><td>0.003</td><td>0.83</td></tr></table>
+
+as No VL Pre-train). Besides, we also conduct an ablation study to freeze the pre-trained VLM and only train the policy head (denoted as No VL Finetune). As shown in Fig. 3 (b), we can see that vision-language pre-training crucially improves the downstream robotic manipulation by a large margin. Besides, tuning on the VL model itself on robotic tasks is indispensable due to the limited capacity of the policy head.
+
+# 5.4.3 HOW DO CRITICAL FACTORS IN VL PRE-TRAINING AFFECT ROBOTIC TASKS?
+
+Model size. A larger model usually results in better VL performance. Yet, with full training data in CALVIN, we find that the smaller model is competitive with the larger model (see the comparison in Tab. 2 and Appendix B.4). To further validate the impact of model size on downstream robotic tasks, we train different variants with  $10\%$  of language annotated data in CALVIN, which is only  $0.1\%$  of the full data. From Tab. 3 we can observe that with limited training data, the performance of VLMs is highly related to the model size. The larger model achieves much higher performance, indicating that a larger VLM can be more data-efficient.
+
+Instruction fine-tuning. Instruction-Finetuning is a specialized technique that utilizes a further pretraining enhancement on the LLM with the IFT dataset (Conover et al., 2023; Peng et al., 2023), which provides a rich repertoire of instruction-following behaviors that inform its capabilities in language-conditioned tasks. We find that LLMs with such a training stage can improve the performance of the policy in both seen and unseen scenarios, revealed by the performance improvements of M-3B-IFT against M-3B, and G-4B-IFT against G-4B shown in Tab. 2.
+
+# 5.5 FLEXIBILITY OF DEPLOYMENT
+
+Since our RoboFlamingo adopts a structure that separates the perception and policy module and leaves the main computation on the perception module, we could perform open loop control to accelerate the inference of RoboFlamingo. Instead of taking only the next action to execute and performing VLM inference every time for new observations to predict future actions, open-loop control can be achieved by predicting an action sequence (stacked actions) with only one inference given the current observation, therefore alleviating the delay and the test-time computing requirement. However, as indicated in Fig. 3 (c), directly implementing open loop control without re-training may lead to deteriorated performance, retraining the model with jump step demonstration could alleviate the performance drop.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+This paper explores the potential of pre-trained vision-language models in advancing language-conditioned robotic manipulation. Our proposed RoboFlamingo, based on the pre-trained OpenFlamingo model, showcases state-of-the-art performance on a benchmark dataset. Moreover, our experimental findings highlight the benefits of pre-trained models in terms of data efficiency and zero-shot generalization ability. This research contributes to the ongoing efforts to develop intelligent robotic systems that can seamlessly understand and respond to human language instructions, paving the way for more intuitive and efficient human-robot collaboration. Due to the lack of real-robot data, this paper does not deploy on real-world robotics. To our delight, recent progress on large-scale real robotics data (Padalkar et al., 2023) has shown the potential of fine-tuning large VLMs for real robots, and the most exciting future work is to see how RoboFlamingo will behave in real-world tasks combined with such amount of data.
+
+# ACKNOWLEDGEMENTS
+
+This work was supported by the National Natural Science Foundation of China under Grant 62025304. The Shanghai Jiao Tong University team is partially supported by National Key R&D Program of China (2022ZD0114804), Shanghai Municipal Science and Technology Major Project (2021SHZDZX0102) and National Natural Science Foundation of China (62322603, 62076161). The author Minghuan Liu is also supported by the ByteDance Scholarship and Wu Wen Jun Honorary Doctoral Scholarship.
+
+# REFERENCES
+
+Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report. arXiv, pp. 2303–08774, 2023.  
+Michael Ahn, Anthony Brohan, Noah Brown, Yevgen Chebotar, Omar Cortes, Byron David, Chelsea Finn, Chuyuan Fu, Keerthana Gopalakrishnan, Karol Hausman, et al. Do as i can, not as i say: Grounding language in robotic affordances. arXiv preprint arXiv:2204.01691, 2022.  
+Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katherine Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. Advances in Neural Information Processing Systems, 35:23716-23736, 2022.  
+Anas Awadalla, Irena Gao, Josh Gardner, Jack Hessel, Yusuf Hanafy, Wanrong Zhu, Kalyani Marathe, Yonatan Bitton, Samir Gadre, Shiori Sagawa, et al. Openflamingo: An open-source framework for training large autoregressive vision-language models. arXiv preprint arXiv:2308.01390, 2023.  
+Sid Black, Stella Biderman, Eric Hallahan, Quentin Anthony, Leo Gao, Laurence Golding, Horace He, Connor Leahy, Kyle McDonell, Jason Phang, Michael Pieler, USVSN Sai Prashanth, Shivanshu Purohit, Laria Reynolds, Jonathan Tow, Ben Wang, and Samuel Weinbach. GPT-NeoX-20B: An open-source autoregressive language model. In Proceedings of the ACL Workshop on Challenges & Perspectives in Creating Large Language Models, 2022. URL https://arxiv.org/abs/2204.06745.  
+Anthony Brohan, Noah Brown, Justice Carbajal, Yevgen Chebotar, Joseph Dabis, Chelsea Finn, Keerthana Gopalakrishnan, Karol Hausman, Alex Herzog, Jasmine Hsu, et al. Rt-1: Robotics transformer for real-world control at scale. arXiv preprint arXiv:2212.06817, 2022.  
+Anthony Brohan, Noah Brown, Justice Carbajal, Yevgen Chebotar, Xi Chen, Krzysztof Choromanski, Tianli Ding, Danny Driess, Avinava Dubey, Chelsea Finn, et al. Rt-2: Vision-language-action models transfer web knowledge to robotic control. arXiv preprint arXiv:2307.15818, 2023.  
+Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877-1901, 2020.  
+Daniel Cer, Yinfei Yang, Sheng-yi Kong, Nan Hua, Nicole Limtiaco, Rhomni St John, Noah Constant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar, et al. Universal sentence encoder. arXiv preprint arXiv:1803.11175, 2018.  
+Xi Chen, Josip Djolonga, Piotr Padlewski, Basil Mustafa, Soravit Changpinyo, Jialin Wu, Carlos Riquelme Ruiz, Sebastian Goodman, Xiao Wang, Yi Tay, et al. Pali-x: On scaling up a multilingual vision and language model. arXiv preprint arXiv:2305.18565, 2023.  
+Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles Sutton, Sebastian Gehrmann, et al. Palm: Scaling language modeling with pathways. arXiv preprint arXiv:2204.02311, 2022.  
+Mike Conover, Matt Hayes, Ankit Mathur, Jianwei Xie, Jun Wan, Sam Shah, Ali Ghodsi, Patrick Wendell, Matei Zaharia, and Reynold Xin. Free dolly: Introducing the world's first truly open instruction-tuned llm, 2023. URL https://www.databricks.com/blog/2023/04/12/dolly-first-open-commercially-viable-instruction-tuned-llm.
+
+Danny Driess, Fei Xia, Mehdi SM Sajjadi, Corey Lynch, Aakanksha Chowdhery, Brian Ichter, Ayzaan Wahid, Jonathan Thompson, Quan Vuong, Tianhe Yu, et al. Palm-e: An embodied multimodal language model. arXiv preprint arXiv:2303.03378, 2023.  
+Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735-1780, 1997.  
+Eric Jang, Alex Irpan, Mohi Khansari, Daniel Kappler, Frederik Ebert, Corey Lynch, Sergey Levine, and Chelsea Finn. Bc-z: Zero-shot task generalization with robotic imitation learning. In Conference on Robot Learning, pp. 991-1002. PMLR, 2022.  
+Yunfan Jiang, Agrim Gupta, Zichen Zhang, Guanzhi Wang, Yongqiang Dou, Yanjun Chen, Li Fei-Fei, Anima Anandkumar, Yuke Zhu, and Linxi Fan. Vima: General robot manipulation with multimodal prompts. In Fortieth International Conference on Machine Learning, 2023.  
+Siddharth Karamcheti, Suraj Nair, Annie S Chen, Thomas Kollar, Chelsea Finn, Dorsa Sadigh, and Percy Liang. Language-driven representation learning for robotics. arXiv preprint arXiv:2302.12766, 2023.  
+Lei Li, Yuwei Yin, Shicheng Li, Liang Chen, Peiyi Wang, Shuhuai Ren, Mukai Li, Yazheng Yang, Jingjing Xu, Xu Sun, et al.  $m^3$  it: A large-scale dataset towards multi-modal multilingual instruction tuning. arXiv preprint arXiv:2306.04387, 2023.  
+Xinghang Li, Di Guo, Huaping Liu, and Fuchun Sun. Robotic indoor scene captioning from streaming video. In 2021 IEEE International Conference on Robotics and Automation (ICRA), pp. 6109-6115. IEEE, 2021.  
+Minghuan Liu, Tairan He, Minkai Xu, and Weinan Zhang. Energy-based imitation learning. arXiv preprint arXiv:2004.09395, 2020.  
+Minghuan Liu, Menghui Zhu, and Weinan Zhang. Goal-conditioned reinforcement learning: Problems and solutions. arXiv preprint arXiv:2201.08299, 2022a.  
+Xinzhu Liu, Xinghang Li, Di Guo, Sinan Tan, Huaping Liu, and Fuchun Sun. Embodied multi-agent task planning from ambiguous instruction. Proceedings of robotics: science and systems, New York City, NY, USA, pp. 1-14, 2022b.  
+Corey Lynch and Pierre Sermanet. Language conditioned imitation learning over unstructured data. arXiv preprint arXiv:2005.07648, 2020.  
+Oier Mees, Lukas Hermann, and Wolfram Burgard. What matters in language conditioned robotic imitation learning over unstructured data. IEEE Robotics and Automation Letters (RA-L), 7(4): 11205-11212, 2022a.  
+Oier Mees, Lukas Hermann, Erick Rosete-Beas, and Wolfram Burgard. Calvin: A benchmark for language-conditioned policy learning for long-horizon robot manipulation tasks. IEEE Robotics and Automation Letters (RA-L), 7(3):7327-7334, 2022b.  
+Oier Mees, Jessica Borja-Diaz, and Wolfram Burgard. Grounding language with visual affordances over unstructured data. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 11576-11582. IEEE, 2023.  
+Suraj Nair, Aravind Rajeswaran, Vikash Kumar, Chelsea Finn, and Abhinav Gupta. R3m: A universal visual representation for robot manipulation. arXiv preprint arXiv:2203.12601, 2022.  
+Catharine Oertel, Ginevra Castellano, Mohamed Chetouani, Jauwairia Nasir, Mohammad Obaid, Catherine Pelachaud, and Christopher Peters. Engagement in human-agent interaction: An overview. Frontiers in Robotics and AI, 7:92, 2020.  
+Abhishek Padalkar, Acorn Pooley, Ajinkya Jain, Alex Bewley, Alex Herzog, Alex Irpan, Alexander Khazatsky, Anant Rai, Anikait Singh, Anthony Brohan, et al. Open x-embodiment: Robotic learning datasets and rt-x models. arXiv preprint arXiv:2310.08864, 2023.
+
+Baolin Peng, Chunyuan Li, Pengcheng He, Michel Galley, and Jianfeng Gao. Instruction tuning with gpt-4. arXiv preprint arXiv:2304.03277, 2023.  
+Dean A Pomerleau. Alvinn: An autonomous land vehicle in a neural network. Advances in neural information processing systems, 1, 1988.  
+Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. Journal of Machine Learning Research, 21(140):1-67, 2020. URL http://jmlr.org/papers/v21/20-074.html.  
+Nils Reimers and Iryna Gurevych. Sentence-bert: Sentence embeddings using siamese bert-networks. arXiv preprint arXiv:1908.10084, 2019.  
+Katie Seaborn, Norihisa P Miyake, Peter Pennefather, and Mihoko Otake-Matsuura. Voice in human-agent interaction: A survey. ACM Computing Surveys (CSUR), 54(4):1-43, 2021.  
+Mingxing Tan and Quoc Le. Efficientnet: Rethinking model scaling for convolutional neural networks. In International conference on machine learning, pp. 6105-6114. PMLR, 2019.  
+MosaicML NLP Team et al. Introducing mpt-7b: A new standard for open-source, commercially usable llms, 2023. URL www.mosaicml.com/blog/mpt-7b.Accessed, pp. 05-05, 2023.  
+Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al. Llama: Open and efficient foundation language models. arXiv preprint arXiv:2302.13971, 2023.  
+Tiannan Wang, Wangchunshu Zhou, Yan Zeng, and Xinsong Zhang. Efficientvlm: Fast and accurate vision-language models via knowledge distillation and modal-adaptive pruning. arXiv preprint arXiv:2210.07795, 2022.  
+Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zi-Hang Jiang, Francis EH Tay, Jiashi Feng, and Shuicheng Yan. Tokens-to-token vit: Training vision transformers from scratch on imagenet. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 558-567, 2021.  
+Yan Zeng, Xinsong Zhang, Hang Li, Jiawei Wang, Jipeng Zhang, and Wangchunshu Zhou.  $x^{2}$ -vlm: All-in-one pre-trained model for vision-language tasks. arXiv preprint arXiv:2211.12402, 2022.  
+Tianhao Zhang, Zoe McCarthy, Owen Jow, Dennis Lee, Xi Chen, Ken Goldberg, and Pieter Abbeel. Deep imitation learning for complex manipulation tasks from virtual reality teleoperation. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 5628-5635. IEEE, 2018.  
+Wangchunshu Zhou, Yan Zeng, Shizhe Diao, and Xinsong Zhang. Vblue: A multi-task multi-dimension benchmark for evaluating vision-language pre-training. In International Conference on Machine Learning, pp. 27395-27411. PMLR, 2022.
+
+# A ENVIRONMENTAL SETUPS
+
+# A.1 THE CALVIN BENCHMARK
+
+CALVIN (Mees et al., 2022b) is an open-source simulated benchmark for evaluating long-horizon language-conditioned tasks.
+
+As shown in Fig. 4, CALVIN includes four different environments A, B, C, and D, each of which consists of 6 hours of human-teleoperated recording data (more than 2 million trajectories) that might contain sub-optimal behavior, and only  $1\%$  of that data is annotated with language instructions (around 24 thousand trajectories). Each split is settled with different settings of objects and environments, aiming to validate the performance, robustness, and generality of policies trained with different data combinations.
+
+This benchmark requires a 7-DOF Franka Emika Panda robot arm with a parallel gripper, utilizing onboard sensors and images from two camera views to successfully complete sequences of up to five language instructions consecutively. This setup further challenges the robot's ability to transition between various goals. CALVIN encompasses a total of 34 distinct tasks and evaluates 1000 unique instruction chains for sequences. The robot is reset to a neutral position after each sequence to prevent any policy bias resulting from its initial pose. This neutral initialization eliminates any correlation between the initial state and the task, compelling the agent to rely solely on language cues to comprehend and solve the given task. The policy for each consecutive task is dependent on the instruction of the current goal, and the agent advances to the subsequent goal only if it successfully accomplishes the current task.
+
+![](images/639db90786990c57fe05bef21eb7a3a8750d34f4ab5458ea1086f80034d6a32f.jpg)  
+Figure 4: The visualization of the four splits (left) and a full task sequence demonstration in CALVIN (right). The ID in the blue circle represents the end of which task among the five tasks. When the task is finished, the instructions for the next task will be given. For instance, the second image in the first row of the right side denotes the end of task 1, and at the next step the instruction shown in the blue circle “2” will be given. Blue circle 0 is not an instruction.
+
+# A.2 EXAMPLES OF ENRICHED INSTRUCTIONS
+
+Table 4: Examples of original and enriched instructions in the CALVIN benchmark.  
+
+<table><tr><td>Task Type</td><td>rotate red block right</td><td>push pink block</td><td>move slider left</td><td>open drawer</td><td>lift blue block slider</td></tr><tr><td>CALVIN Instruction</td><td>Take the red block and rotate it to the right</td><td>Go push the pink block left</td><td>Push the sliding door to the left side</td><td>Pull the handle to open the drawer</td><td>Lift the blue block from the sliding</td></tr><tr><td rowspan="3">Enriched Instruction</td><td>Rotate the red item in a clockwise direction</td><td>Shift the pink block to the left</td><td>Push the sliding doorway to the left</td><td>Grasp the handle firmly and pull to dislodge the drawer</td><td>Carefully hist the blue marker out of the mobile drawer</td></tr><tr><td>Give a rightward spin to the red block</td><td>Roll the pink cube on the left</td><td>Use your arm to slide the door towards the left</td><td>Grip the handle exert force to unfold the drawer</td><td>Lift upward the blue block from the sliding closet</td></tr><tr><td>Change the position of the red block to the right</td><td>Dislocate the pink cube to your left</td><td>Guide sliding passageway to the left</td><td>Tug the handle to make the drawer slide out</td><td>Grasp and lift the blue box from the rolling drawer</td></tr></table>
+
+To validate the performance of the policies over diversified language expressions, we utilize GPT4 to augment the language instruction in CALVIN. We showcase the enriched language instructions in Tab. 4. We can see that the enriched instructions do have the same meaning as the original one, yet they are organized with different words. As shown in Table 1, RoboFlamingo can still achieve better performance compared to HULC.
+
+# A.3 COMPUTING RESOURCE
+
+All experiments involved in this paper are conducted on a single GPU server with 8 NVIDIA Tesla A100 GPUs, and the default batch size is 6 on each GPU. The MPT-3B model takes 13 hours of training per epoch and achieves the best performance at the 3rd epoch, while the MPT-9B model also takes 26 hours of training per epoch and achieves the best performance at the 4rd epoch.
+
+# B EXTENDED EXPERIMENTAL RESULTS
+
+# B.1 CO-TRAINING
+
+From Tab. 1, the Enriched setting, we have noticed some evidence that the model may lose some foundation capabilities as the performance loss, which indicates that there is over-fitting during the fine-tuning. To further understand the phenomenon, we conduct further experiments by testing the fine-tuned RoboFlamingo model (the M-3B-IFT variant) on the COCO image caption and VQAv2, which verify our conjecture (see Tab. 6). To prevent such problems, we choose to co-train RoboFlamingo (the M-3B-IFT variant) with VQA and COCO datasets during fine-tuning on the robotics dataset. We test the co-train model on CALVIN, and the COCO image caption, VQAv2 tasks as well, as shown in Tab. 6 and Tab. 5. This provides a solution for fine-tuning VLMs to robotics models while preserving the ability on vision-language tasks, even though it may slightly deteriorate the performance on robotic tasks. In our implementation, we ensure that the model equally incorporates batches of VL and robot data in each epoch. From Fig. 5 and Fig. 5 we could observe a similar performance curve of the Co-trained and Fine-tune version of our model, while Co-trained model achieves higher performance in the early epochs and Fine-tune model ends up higher in the later epochs. One interesting observation is that under the Enriched setting, the performance of the co-trained model also drops, this may indicate the difference between understanding different sentences and aligning vision-language representations (as the pre-trained tasks do).
+
+![](images/a50e5af50ae0fcebb6e374afd304226ca162daffa5e58568ff2d68b5415cb2ef.jpg)  
+Figure 5: The performance of Co-Trained and Fine-tune model of MPT-3B-IFT at each epoch on  $ABC \rightarrow D$  split.
+
+![](images/a4e816d223297b9ed7aa197ad3aecff6b95ec6f5805a89106af236803f26e332.jpg)  
+Figure 6: The performance of Co-Trained and Fine-tune model of MPT-3B-IFT at each epoch on  $ABCD \to D$  split.
+
+# B.2 COMPARISON WITH PRE-TRAINED ROBOTICS REPRESENTATION MODELS
+
+We consider comparing our RoboFlamingo with recent pre-trained robotics representation models, such as R3M (Nair et al., 2022) and Voltron (Karamcheti et al., 2023). We loaded the pre-train
+
+Table 5: Comparison of co-trained models and fine-tuned models on the CALVIN benchmark. All results are selected from the best of the last 5 epochs.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Training Data</td><td rowspan="2">Test Split</td><td rowspan="2">1</td><td colspan="5">Task Completed in a Sequence</td><td rowspan="2">Avg Len</td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td></td></tr><tr><td>Co-trained</td><td>ABC</td><td>D</td><td>0.829</td><td>0.636</td><td>0.453</td><td>0.321</td><td>0.234</td><td>2.47</td><td></td></tr><tr><td>Fine-tune</td><td>ABC</td><td>D</td><td>0.824</td><td>0.619</td><td>0.466</td><td>0.331</td><td>0.235</td><td>2.48</td><td></td></tr><tr><td>Co-trained</td><td>ABCD</td><td>D</td><td>0.957</td><td>0.858</td><td>0.737</td><td>0.645</td><td>0.561</td><td>3.76</td><td></td></tr><tr><td>Fine-tune</td><td>ABCD</td><td>D</td><td>0.964</td><td>0.896</td><td>0.824</td><td>0.740</td><td>0.66</td><td>4.09</td><td></td></tr><tr><td>Co-trained</td><td>ABCD</td><td>D (Enriched)</td><td>0.678</td><td>0.452</td><td>0.294</td><td>0.189</td><td>0.117</td><td>1.73</td><td></td></tr><tr><td>Fine-tune</td><td>ABCD</td><td>D (Enriched)</td><td>0.720</td><td>0.480</td><td>0.299</td><td>0.211</td><td>0.144</td><td>1.85</td><td></td></tr></table>
+
+Table 6: Comparison of co-trained models and fine-tuned models on the COCO image caption and VQAv2 evaluation dataset. All results are selected from the epoch as in Tab. 5.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="8">COCO</td><td rowspan="2">VQA Acc</td></tr><tr><td>BLEU-1</td><td>BLEU-2</td><td>BLEU-3</td><td>BLEU-4</td><td>METEOR</td><td>ROUGE_L</td><td>CIDEr</td><td>SPICE</td></tr><tr><td>Fine-tune (3B, zero-shot)</td><td>0.157</td><td>0.052</td><td>0.018</td><td>0.008</td><td>0.038</td><td>0.147</td><td>0.005</td><td>0.006</td><td>4.09</td></tr><tr><td>Fine-tune (3B, 4-shot)</td><td>0.168</td><td>0.057</td><td>0.020</td><td>0.008</td><td>0.043</td><td>0.161</td><td>0.005</td><td>0.007</td><td>3.87</td></tr><tr><td>OpenFlamingo (3B, zero-shot)</td><td>0.580</td><td>0.426</td><td>0.301</td><td>0.209</td><td>0.208</td><td>0.464</td><td>0.757</td><td>0.153</td><td>40.92</td></tr><tr><td>OpenFlamingo (3B, 4-shot)</td><td>0.612</td><td>0.461</td><td>0.332</td><td>0.234</td><td>0.220</td><td>0.491</td><td>0.822</td><td>0.162</td><td>43.86</td></tr><tr><td>Co-Train (3B, zero-shot)</td><td>0.223</td><td>0.157</td><td>0.106</td><td>0.071</td><td>0.124</td><td>0.334</td><td>0.346</td><td>0.084</td><td>36.37</td></tr><tr><td>Co-Train (3B, 4-shot)</td><td>0.284</td><td>0.204</td><td>0.142</td><td>0.098</td><td>0.142</td><td>0.364</td><td>0.426</td><td>0.100</td><td>38.73</td></tr><tr><td>Original Flamingo (80B, fine-tuned)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>1.381</td><td>-</td><td>82.0</td></tr></table>
+
+weights of R3M and Voltron and fine-tuned them on CALVIN data, only training the policy head while freezing their representation parameters. As for Voltron, we also include a version that fine-tunes the representation layers. The results are shown in Tab. 7, which reveals the clear advantage of fine-tuning pre-trained VLMs compared with these specific robotics representation models.
+
+Table 7: Comparative performance of various VL representation models on various settings, all results are selected from the best of the last 5 epochs.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Training Data</td><td rowspan="2">Test Split</td><td rowspan="2">1</td><td colspan="5">Task Completed in a Sequence</td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>Avg Len</td></tr><tr><td>Voltron (Frozen)</td><td>ABC</td><td>D</td><td>0.026</td><td>0.001</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.03</td></tr><tr><td>Voltron (Fine-tuned)</td><td>ABC</td><td>D</td><td>0.569</td><td>0.272</td><td>0.105</td><td>0.038</td><td>0.014</td><td>1.00</td></tr><tr><td>RoboFlamingo (Ours)</td><td>ABC</td><td>D</td><td>0.824</td><td>0.619</td><td>0.466</td><td>0.331</td><td>0.235</td><td>2.48</td></tr><tr><td>R3M (Frozen)</td><td>ABCD</td><td>D</td><td>0.085</td><td>0.005</td><td>0.001</td><td>0.000</td><td>0.000</td><td>0.10</td></tr><tr><td>Voltron (Frozen)</td><td>ABCD</td><td>D</td><td>0.101</td><td>0.003</td><td>0.001</td><td>0.000</td><td>0.000</td><td>0.11</td></tr><tr><td>Voltron (Fine-tuned)</td><td>ABCD</td><td>D</td><td>0.837</td><td>0.566</td><td>0.352</td><td>0.208</td><td>0.115</td><td>2.08</td></tr><tr><td>RoboFlamingo (Ours)</td><td>ABCD</td><td>D</td><td>0.964</td><td>0.896</td><td>0.824</td><td>0.740</td><td>0.662</td><td>4.09</td></tr></table>
+
+# B.3 FINE-TUNE THE FULL MODEL
+
+In the fine-tuning of RoboFlamingo, we follow the training of Flamingo (Alayrac et al., 2022; Awadalla et al., 2023) that only trains the parameters of the resampler, the gated cross-attention module of each decoder layer, and the policy head while freezing all other parameters. This leads RoboFlamingo to have 1B trainable parameters (as shown in Tab. 2). In this part, we show the results of training the full model (the MPT-3B-IFT variant), which has 3B trainable parameters in Tab. 8, revealing an obvious performance deterioration.
+
+# B.4 PERFORMANCE CURVES IN TRAINING OF DIFFERENT BACKBONES
+
+Fig. 8 and Fig. 9 show the performance of RoboFlamingo with different VLMs on both  $ABC \rightarrow D$  and  $ABCD \rightarrow D$  settings in 5 training epochs. It is noticed that most variants converge in 5-
+
+Table 8: Comparison between full-model fine-tuning (3B trainable parameters) and RoboFlamingo-style fine-tuning (1B trainable parameters)  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Training Data</td><td rowspan="2">Test Split</td><td colspan="6">Task Completed in a Sequence (Success Rate)</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>Avg Len</td></tr><tr><td rowspan="2">Full model fine-tuned RoboFlamingo</td><td rowspan="2">ABCD (Lang)</td><td rowspan="2">D</td><td>0.415</td><td>0.070</td><td>0.009</td><td>0.002</td><td>0.001</td><td>0.50</td></tr><tr><td>0.964</td><td>0.896</td><td>0.824</td><td>0.740</td><td>0.66</td><td>4.09</td></tr></table>
+
+epoch training and achieve the best performance, benefiting from the pre-training on extensive vision-language tasks.
+
+![](images/2c30409f10015c5426c38a668dd1bc2f83786deb59861127365a6d60a0906af7.jpg)  
+Figure 7: The visualization of RoboFlamingo and HULC executing the same task sequence in the  $ABC \rightarrow D$  split.
+
+![](images/ed24f6bc343347b2f28607d016b52d988ca1e2c24fc5db8c98301776382dabb5.jpg)  
+Figure 8: The performance of VLMs at each epoch on  $ABC \to D$  split.
+
+# B.5 QUALITATIVE EXAMPLES
+
+We visualize the task frames and analyze how RoboFlamingo achieve such a great performance. As the example shown in Fig. 7, where RoboFlamingo successfully finishes the entire task sequence, while HULC stucks at the third one. RoboFlamingo only takes a dozen steps to locate and move to the top of the drawer, and simultaneously releases the gripper to complete the task; while HULC keeps moving above the desktop for hundreds of steps and fails to locate the drawer. Furthermore, although both methods are successful for the first two tasks, RoboFlamingo uses significantly fewer steps. This representative episode vividly illustrates that our method is much more effective and efficient and could better generalize to unseen vision context.
+
+# B.6 DETAILED IMITATION PERFORMANCES ON EACH TASK
+
+We present the detailed imitation performances by tasks in Tab. 9. All model are reported by their best checkpoint.
+
+# B.7 ROLLOUT EXAMPLES
+
+We present some rollout examples of RoboFlamingo on the  $ABCD \to D$  split.
+
+![](images/f6021a6b672147a74f98ab2fc4e0bf6d2ea76c4d7a4b08909a9c432fb41e28b3.jpg)  
+Figure 9: The performance of VLMs at each epoch on  $ABCD \to D$  split.
+
+Table 9: Success rates by task of variants of RoboFlamingo. Each task is evaluated 100 times.  
+
+<table><tr><td>Task Name</td><td>M-3B</td><td>M-3B-IFT</td><td>G-4B</td><td>G-4B-IFT</td><td>L-9B</td><td>M-9B</td></tr><tr><td>rotate blue block right</td><td>0.947</td><td>0.893</td><td>0.729</td><td>0.770</td><td>0.493</td><td>0.882</td></tr><tr><td>move slider right</td><td>0.996</td><td>0.993</td><td>0.996</td><td>0.992</td><td>0.987</td><td>0.996</td></tr><tr><td>lift red block slider</td><td>0.890</td><td>0.970</td><td>0.967</td><td>0.858</td><td>0.856</td><td>0.927</td></tr><tr><td>place in slider</td><td>0.904</td><td>0.828</td><td>0.582</td><td>0.911</td><td>0.874</td><td>0.910</td></tr><tr><td>turn off lightbulb</td><td>0.972</td><td>1.000</td><td>0.992</td><td>0.956</td><td>0.927</td><td>0.964</td></tr><tr><td>turn off led</td><td>0.988</td><td>1.000</td><td>1.000</td><td>0.994</td><td>0.970</td><td>0.981</td></tr><tr><td>push into drawer</td><td>0.777</td><td>0.821</td><td>0.731</td><td>0.770</td><td>0.705</td><td>0.703</td></tr><tr><td>lift blue block drawer</td><td>1.000</td><td>0.950</td><td>1.000</td><td>1.000</td><td>0.917</td><td>0.737</td></tr><tr><td>close drawer</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>0.986</td><td>0.995</td></tr><tr><td>lift pink block slider</td><td>0.940</td><td>0.971</td><td>0.944</td><td>0.862</td><td>0.861</td><td>0.918</td></tr><tr><td>lift pink block table</td><td>0.859</td><td>0.851</td><td>0.905</td><td>0.892</td><td>0.543</td><td>0.899</td></tr><tr><td>move slider left</td><td>0.996</td><td>0.996</td><td>1.000</td><td>1.000</td><td>0.970</td><td>0.996</td></tr><tr><td>open drawer</td><td>0.976</td><td>0.997</td><td>0.997</td><td>0.982</td><td>0.980</td><td>0.997</td></tr><tr><td>turn on lightbulb</td><td>0.988</td><td>0.994</td><td>1.000</td><td>0.988</td><td>0.949</td><td>0.988</td></tr><tr><td>rotate blue block left</td><td>0.923</td><td>0.939</td><td>0.820</td><td>0.925</td><td>0.636</td><td>0.848</td></tr><tr><td>push blue block left</td><td>0.746</td><td>0.955</td><td>0.841</td><td>0.836</td><td>0.677</td><td>0.909</td></tr><tr><td>rotate red block right</td><td>0.926</td><td>0.972</td><td>0.853</td><td>0.905</td><td>0.591</td><td>0.959</td></tr><tr><td>turn on led</td><td>0.988</td><td>0.988</td><td>0.994</td><td>0.976</td><td>0.985</td><td>0.994</td></tr><tr><td>push pink block right</td><td>0.652</td><td>0.754</td><td>0.833</td><td>0.651</td><td>0.627</td><td>0.750</td></tr><tr><td>push red block left</td><td>0.949</td><td>0.920</td><td>0.849</td><td>0.949</td><td>0.562</td><td>0.908</td></tr><tr><td>lift blue block table</td><td>0.891</td><td>0.956</td><td>0.925</td><td>0.927</td><td>0.611</td><td>0.931</td></tr><tr><td>place in drawer</td><td>0.988</td><td>0.989</td><td>0.988</td><td>0.975</td><td>0.971</td><td>0.976</td></tr><tr><td>rotate red block left</td><td>0.970</td><td>0.908</td><td>0.950</td><td>0.953</td><td>0.677</td><td>0.952</td></tr><tr><td>push pink block left</td><td>0.947</td><td>0.920</td><td>0.915</td><td>0.973</td><td>0.747</td><td>0.933</td></tr><tr><td>stack block</td><td>0.612</td><td>0.641</td><td>0.608</td><td>0.595</td><td>0.569</td><td>0.604</td></tr><tr><td>lift blue block slider</td><td>0.847</td><td>0.963</td><td>0.908</td><td>0.826</td><td>0.769</td><td>0.869</td></tr><tr><td>push red block right</td><td>0.657</td><td>0.732</td><td>0.797</td><td>0.451</td><td>0.457</td><td>0.653</td></tr><tr><td>lift red block table</td><td>0.948</td><td>0.939</td><td>0.942</td><td>0.975</td><td>0.606</td><td>0.989</td></tr><tr><td>lift pink block drawer</td><td>0.857</td><td>0.800</td><td>0.929</td><td>0.714</td><td>0.778</td><td>0.923</td></tr><tr><td>rotate pink block right</td><td>0.917</td><td>0.896</td><td>0.714</td><td>0.794</td><td>0.478</td><td>0.789</td></tr><tr><td>unstack block</td><td>1.000</td><td>0.982</td><td>0.957</td><td>0.980</td><td>0.946</td><td>0.979</td></tr><tr><td>rotate pink block left</td><td>0.929</td><td>0.839</td><td>0.906</td><td>0.818</td><td>0.698</td><td>0.927</td></tr><tr><td>push blue block right</td><td>0.479</td><td>0.597</td><td>0.471</td><td>0.478</td><td>0.400</td><td>0.594</td></tr><tr><td>lift red block drawer</td><td>0.947</td><td>1.000</td><td>1.000</td><td>1.000</td><td>0.769</td><td>0.933</td></tr></table>
+
+ABCD→D
+
+go push the blue block left
+
+![](images/07689ccdf2b209532e57e42f269d978eef20d155a0609573c3ad9730e278e9c8.jpg)  
+1
+
+![](images/35c8e7bc5b4ffa8c96ffa6e474c07a9ac66fbf3597f7779de3028a2ec015df96.jpg)
+
+![](images/e331f64755df6e48544e6567114c1074044fad5fdd69f0fe79990d5b85155bc1.jpg)
+
+![](images/aa7d83ad2322dbb54114b55da39816d2f3bda81d55098b139f32eca7c4f933b8.jpg)
+
+![](images/705df3bc0a1a4fd28dc038e4c46bddc2eb75f35d1f0dc7cf2811d471468c0fa2.jpg)
+
+![](images/79c230c36dd4f4526846fabba5213f11a8ffb7452eacad55b8348aa7714ca118.jpg)
+
+![](images/eb749998ca9fcbcfda758c38009b83c5b79dc11ef12aa4a4fe32cf45ba56cf18.jpg)
+
+![](images/41006aad22a97105e5e36a265c8b4c05c49df509085b796e2cbe0ea96a54c566.jpg)
+
+![](images/f4d12001bd99412c80a4e5a64b385bf91fa7437a6e956ad05e31b1e572338e21.jpg)
+
+![](images/d65d10ec104bf4a57da90d09571bd4eebdbce3b023f1c7a00001c42feb29e278.jpg)
+
+pull the handle to open the drawer
+
+![](images/6ceb22a01fe22e21c785a3704295423ab0b8add85f5feda5a7c886c7d8dff428.jpg)  
+2
+
+![](images/f5f7bbe7b71dafb270790262346b6492330e3d42f6dda72f8b81580d97f87139.jpg)
+
+![](images/680f79851082f6b822737e2826d4e9eed214bcd9c60908a3c81552a6f23b4632.jpg)
+
+![](images/5c12eaa30571d2056c35c523e303ce7f3e312dcda2c3e2196420de2110224c3f.jpg)
+
+![](images/43e470f7fea439fbec57d91de196908f4d2a7d5a5615f0915db709eab36c9096.jpg)
+
+![](images/6d62886cff1a9d665abc0c287da75cf5d91b41823521e82994a386759d2bd416.jpg)
+
+![](images/4c041796af7ef4a830a93435798d08f54ac44c2d61326acb8e527e088b528f3c.jpg)
+
+![](images/1f402b1909a64b85d1f90c518cfbb409b4aca21e2560798d4dd854b81f494a60.jpg)
+
+![](images/bde6ce6b2e1b4097041aa7d5100ebf4891da45be85f5c617bd10489a2e89f869.jpg)
+
+![](images/7dfcc7fe02d6a0edd1492fc30b9c87cd547d30bce8efbcfa3b9eeac70683f284.jpg)
+
+grasp and lift the red block
+
+![](images/d1b78300fb4a6166dd7e6eb98c171a185cf0128184fe55262a214e81177fd0a0.jpg)  
+3
+
+![](images/9bcf487cda659e9c07686d5cdef03a1c4a097dde731e46878b5cf18481db21ea.jpg)
+
+![](images/305d58fc07201ac9b55ad014131e0b3bc92219d81c1dade68dfc56f4a1a7a2a3.jpg)
+
+![](images/269d9bf6bc22884788300e80f7f432fd740aa5edf065586da6451b3fb0be8d89.jpg)
+
+![](images/204ff5307564ae496625e48d737e566ad355203041b03e458ef34ae5930194fa.jpg)
+
+![](images/2a6707b74249a7375e39d0ac85bfbdfbe65603c315a9eabb0f35a5d04aa3e142.jpg)
+
+![](images/11b7534330becf0e2a38829ce7616d349b22632028ae4ab065bdc530ef0e47b0.jpg)
+
+![](images/9ead047f79503c29ec25a4b40348d35bde1da2e4039d959aedbbcc30affa27ff.jpg)
+
+![](images/e05056c8b01a25c754bd7e87d41aff84a8638a4233192b4b9ca579f4b5c69965.jpg)
+
+![](images/0b85f405b103b0252f38677dbf7d9acebc5c9270e0e9caa5117491b76ba3277b.jpg)
+
+![](images/c6e490f11fd78474d6799fc1ea577ea3a90ade5e0b036c75fecdc087cc5c1e0b.jpg)  
+4
+
+![](images/795f1030ab92ad6b0dfda4fac5486eb6c4129900e10d49a25407b2b34ebe4aa4.jpg)
+
+![](images/299d10d361df69d5420cdf3387152142b9fa070d4a0aed3a8e05db471e473659.jpg)
+
+![](images/b73f446123d6e18d5816a3394f787f858ca49e6c81f43a1df4fda526b292bffa.jpg)
+
+![](images/424858f4b6a9d0a0ec1ebe914c4c39b4dcc9c1198c899e11b3e964076de1f754.jpg)
+
+![](images/72269695824881b45187db83b23712f1a84694e458f9273d383e13d4437d717f.jpg)
+
+![](images/c4d85320c3068d251233ebf7719f7b3f19973d5e44f55c46625345ee3536599c.jpg)
+
+![](images/3f8c6b7f01ee7b8f2b77564f2cbb8ec7ab6df54efa502c49304cfe15a03b14d7.jpg)
+
+![](images/6144724f028d039628e9bdd3fec00f4f6fdc482cf954ff8c5d17ed3f219e7d6e.jpg)
+
+![](images/b6656021840d96d7da0866130b507e03ba4c03ab937e2a60993659267a37fd9a.jpg)
+
+![](images/dc15e65f06ccb952dbc56c91529fb2aa10b6827dd24c68155aa19e827ab39543.jpg)  
+5
+
+![](images/d64b7ec00ccaa202de8951062eeec2d58848482021475ea127c4a79412a388b8.jpg)
+
+![](images/8e932d72dd161dc0d6911c7e96cbb2e75fa5ab820ac84dc3e9c1bd802c82c962.jpg)
+
+![](images/c48c6c30bfe5d81227ffb62e6310b630743c651c709856cabb34b75642cb5b7b.jpg)
+
+![](images/edf69e067a33c7f60de80a1bfc7d7321ecde7741d88f088e268078da1f3fb605.jpg)
+
+![](images/d6e8dd83011d2baa407772acbf9d9e49de645bc53f0f5fdd0557c693fa45f0eb.jpg)
+
+![](images/36fed2b4ba49da41e0c578399d4c3eaf0fe11aa01751c4661c58392365bf12bc.jpg)
+
+![](images/ec949d81882bb88ff9eebd44c1589c1aa70f2600e5bb9641eca21f045a3ec260.jpg)
+
+![](images/7b7b2d404b2e79bdcc3b592476fbb404a9c0d3a8d0e80873c5811079f84a59c3.jpg)
+
+![](images/fd3adbe4d17bd0fe4e9b844b747f2b60859328b830847875461b6fc339330ee7.jpg)
+
+take the red block and rotate it to the right
+
+![](images/559240de4bbfa444711721c3c4e7c5f04e40971c1e0968785cc45d58aaf51240.jpg)  
+1
+
+![](images/9e2f74bd7b051e749af97fba3da12bbbcadcff18192bada2a7bf85430ec1a93f.jpg)
+
+![](images/f74fd6a72c9e4b7179e47d402a803d1bdb46888536c91825e190e3ab5c42eed8.jpg)
+
+![](images/42f4378e63672dbf22a816be60a5fad6c6ae2953e6ab249e6d3f70d8c0989595.jpg)
+
+![](images/d0ac9ad2e71b47cdb1bcb6f924f15f8ae358810604f6cd50068a122531298c18.jpg)
+
+![](images/66905c1410cda9a6f559c65c60589c1b18e7eecd78a9c3850dd6ddc00e4d995a.jpg)
+
+![](images/79d7a021c0128a53052c7c4416ded8a20fa8b003ddf97dbb87c11b50f09f0326.jpg)
+
+![](images/b3d5852ff3f31d396331e01b4f2e303dc0cd00caa3f3b347f445377b8bf8dab6.jpg)
+
+![](images/bc86d4b96ec3d8ef7ee1bf860a2b3afe57f7b9a45640df9a0bfd81975c4a14df.jpg)
+
+![](images/d236c66b9ecf953430ab328365dffd0ebd3feadb6680e0275e499f07c2974905.jpg)
+
+![](images/e17172e36ae24e4634381f7c3173a39bccf5d0c2b152d4acdaa57d6a327d6e82.jpg)  
+2
+
+![](images/037f81ee13a8cb03e5bfda60137b8d10109c5278ed12f75232cdaa65eab96072.jpg)
+
+![](images/f5ba8b9669de04b749c8cafb4c0276776a2c00eaa2845f9157612819990b07d1.jpg)
+
+![](images/ddbcaa97efea41c21ec0c3433330b08c0282d72924f0d6dcbcb4380ab50f0c31.jpg)
+
+![](images/f2331ae661d0f2d22d6e8d75deb5193c61910e15f01b89f0178f31dfa386382a.jpg)
+
+![](images/1510bdeff15deb6c8caf041c6c1aa1dcbda8e4d41e51438e62105407e5012e9d.jpg)
+
+![](images/b3ad562e38023604cbc9495fccadf8d95615827c7f2aa76d47104328f2d6302b.jpg)
+
+![](images/282a655ac9ad6eb4afd0bf513c49306b005e0a59e50876722142502a4725c626.jpg)
+
+![](images/7c50002e848effc8217bf8be5292417ade05fc7397e6a89b33747dd82d7e0924.jpg)
+
+![](images/4aa40e34e6a0d54f5e1d274cbac8540f87cb93ec71aae86c63515999c3adde3a.jpg)
+
+![](images/42aa46c8b71274d67724402de673d39f396e3cde406933c41e563bb815342351.jpg)  
+3
+
+![](images/9065432ac710f844d1aba3b5e83cac3b87c6aef7aba99aebcde205335a088f86.jpg)
+
+![](images/6cbb732f94111921d3e540581132c6326ed38fad265aee6b32968da189014e21.jpg)
+
+![](images/b1a852adb68351730e5684f9e7cc09b0485169b097ae9275a53870935e45425d.jpg)
+
+![](images/511da032c70032b7ef5aabc7de813cd8792f9e102d040adee0f504c57a94de56.jpg)
+
+![](images/6361fe3b3d2bbaa3087e38e154b329511100b28c071c6fdd559e6782ccb50b05.jpg)
+
+![](images/a41b32aad87d98f5cacd8d54e57c22d77bcc94328e62d2a5d04f019edb91afc7.jpg)
+
+![](images/580bed55897db7e059b76cc3e50f02d564ba58567471cc3e9f143e50ef5519d1.jpg)
+
+![](images/c26b84987b7fd0f93d8171dfd36d5ab452d16383340709bfc886d8800abb726f.jpg)
+
+![](images/278bcca2fdf3101eeca94cb89f5888529c0ec4a5f731492060c6b4eaa5371e3b.jpg)
+
+![](images/f91eee1d86e507f1e4161411e9aac434975eb19916903a8ae2ba2c9900a8e3cf.jpg)  
+4
+
+![](images/cabe0b053c86b13d772f55d5e06b46c8f1a17eaed9c7408670fa98c917788677.jpg)
+
+![](images/e0b65ad52e08730e0ed4aea32ed728d52c4393654b7f7199390bd4645b0c7fda.jpg)
+
+![](images/6d9e21f9eed0a84a5cdf4ce1c1ddf341e76fe560477d2f51a1a2b732523122c1.jpg)
+
+![](images/35aabf06027392e66e7b2bef8d1783effe64d63f86af048f4d9ba82c68929564.jpg)
+
+![](images/e20caf2bbef2cc60abd9b0ef8d0fc2d7320b99c8d3feb1a9e89d94bed0bc25f8.jpg)
+
+![](images/2a224b576ea6f17eea678d3acf05f748ecb390a49ab3ba5376fa7d9cc613b3f9.jpg)
+
+![](images/0e603f5fc81b10db81f8b6576d530539dced2a0bd8883b26e9a5ac9e62e6ed15.jpg)
+
+![](images/cdc718b5007ac70ceca255d9bc6756d09132d4f83fc22011aaf3c2cedaa85625.jpg)
+
+![](images/151d821e864e2ac97876135c88488ca1b43d3460ca07b6c99bcddbc9048281e9.jpg)
+
+![](images/772927ba98d6008726cb6e9c3c3aef465142c3fd3d2b96925c6273c15d0e243b.jpg)  
+5
+
+![](images/d75cee89b7ec6a3275eccfacd8a0517a6a5230e7903e6611f65bf98ccd38fd66.jpg)  
+Figure 10: Rollouts on the  $ABCD \to D$  split of the CALVIN benchmark.
+
+![](images/59936d43764b9eb2b3829edd80bb3ad24b123c2efd8fcc813f61d1cf5cef2f65.jpg)
+
+![](images/3515f462da4abdde398f4e095e7a5d268ed4b8af2f688fe6bed60f7980ccb4eb.jpg)
+
+![](images/75cddb16fb95e6279c9406d6f34c7ab83a0560b72dac7ce48ebefd893f124c82.jpg)
+
+![](images/9b16bc6de74ccb8587bcd3802038048c87551148612ac1428f401d2583f5c558.jpg)
+
+![](images/b321776f46fbf1d6db4b48a6e2ac37d36d56a08565a9a141f388ae7faf5d48f2.jpg)
+
+![](images/5c4ca99c26d68a1110b53a2951593785819811dd14631f5baa95f72a3215dc73.jpg)
+
+![](images/da630a904575afb29a85192f04ec9cc2685042e241d607d5a3dd5446e9143053.jpg)
+
+![](images/4dcbb4a7860c918dd6ad6270d6aad745fcad6e618e37727e651f68ebc818ebf6.jpg)
+
+# C ADDITIONAL DETAILS
+
+# C.1 ILLUSTRATION OF POLICY HEADS/FORMULATION
+
+We illustrate the details of the four policy heads/formulation mentioned in Section 5.4: (a)  $MLP w / o$  hist takes only the current observation as input to predict actions, which ignores the observation history. (b)  $MLP w$  hist takes the history frames into the vision encoder with position embedding, and encodes the history information through the cross-attention layers in the feature fusion decoder. (c)  $GPT$  and (d)  $LSTM$  both utilize the VLM backbone to process single-frame observations and integrate the history with the policy head.  $GPT$  explicitly takes the visual history as input to predict the next action.  $LSTM$  implicitly maintains a hidden state to encode memory and predict the action.
+
+![](images/c473bc228147644b72f5e1dd0b53d7b77c8bbefef9d3a90979892c546a5de5e6.jpg)  
+Figure 11: Implementation details of all policy heads/formulation involved.

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