Appendix A provides derivations supporting Section 3 in the main paper. In Appendix B, we explain our experimental setup, including dataset preparation and model implementation, in more detail. Finally, Appendix C provides additional results supporting our claims regarding the scalability of our method, together with additional results from the experiments presented in Section 4. In this section we provide detailed derivations of the ST-DGMRF joint distribution, for both firstorder transition models (Section A.1) and higher-order transition models (Section A.2). A.1 Joint distribution The LDS (see Section 2.2 and 3.1 in the main paper) defines a joint distribution over system states First, note that Eq. (1) can be written as a set of linear equations Moving all xk-terms to the left-hand side, we can rewrite this as a matrix-vector multiplication I F1 I F2 I ...... FKI | {z} Empty positions in F represent zero-blocks. Now, we can express x as an affine transformation of ϵ x = F 1c+F 1ϵ, (3) where F 1 exists because det(F) = 1. Since ϵ is distributed as ϵ N(0,Q 1) with Q = diag(Q0,Q1,...,QK), and c is deterministic, we can use the affine property of Gaussian distributions to obtain the joint distribution This reduces both computations and memory requirements. In contrast, the information vector η = Ωµcan be expressed compactly as η = FTQFF 1c = FTQc, (8) which can be computed efficiently using sparse and parallel matrix-vector multiplications on a GPU.

We present a novel actor-critic algorithm for an environment with delayed feedback, which addresses the state-space explosion problem of conventional approaches. Conventional approaches use an augmented state constructed from the last observed state and actions executed since visiting the last observed state Using the augmented state space, the correct Markov decision process for delayed environments can be constructed; however, this causes the state space to explode as the number of delayed timesteps increases, leading to slow convergence. Our proposed algorithm, called Belief-Projection-Based Q-learning (BPQL), addresses the state-space explosion problem by evaluating the values of the critic for which the input state size is equal to the original state-space size rather than that of the augmented one. We compare BPQL to traditional approaches in continuous control tasks and demonstrate that it significantly outperforms other algorithms in terms of asymptotic performance and sample efficiency. We also show that BPQL solves long-delayed environments, which conventional approaches are unable to do.

Appendix A provides derivations supporting Section 3 in the main paper. In this section we provide detailed derivations of the ST -DGMRF joint distribution, for both first-order transition models (Section A.1) and higher-order transition models (Section A.2). A.1 Joint distribution The LDS (see Section 2.2 and 3.1 in the main paper) defines a joint distribution over system states First, note that Eq. (1) can be written as a set of linear equations x We make use of this property in the DGMRF formulation and in the conjugate gradient method. Eq. 11 is converted into a discrete-time dynamical system by approximating ρ We consider two ST -DGMRF variants that capture different amounts of prior knowledge. DGMRF transition matrices can be parameterized accordingly. The air quality dataset is based on hourly PM2.5 measurements obtained from [ The raw PM2.5 measurements are log-transformed and standardized to zero mean and unit Ca. 50% of the nodes are masked out (purple nodes within We use a simple MLP with one hidden layer of width 16 with ReLU activations and no output non-linearity. The DGMRF parameters are not shared across time, allowing for dynamically changing spatial covariance patterns.

The supplementary material is organized as follows. We give details of the definitions and notation in Section B.1 . Then, we provide the technical details of the lower bound (Lemma 3.3). In Section D.4 we provide insights into auto-labeling using This suggests, in these settings auto-labeling using active learning followed by selective classification is expected to work well. This idea is captured by the Chow's excess risk [ Nevertheless, it would be interesting future work to explore the connections between auto-labeling and active learning with abstention.

TBAL is emerging as a widely-used solution in practice.

I{ } is the indicator function. It's sufficient to prove that the denominator converges to that of softmax at each point We have shown that softmax is translational invariant w.r.t. Without the loss of generality, we use τ = 1 in the following proof. To begin with, we prove the first equation and then give the proof of the second part of Theorem 3.3. We introduce some extra notations that are used throughout the proof.

Did you specify all the training details (e.g., data splits, hyperparameters, how they