This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation. At each stage of the process, we employ a localized inexact proximal point iteration to solve a simplified linear system. We show that the time complexity of such localized methods is upper bounded by min{ O(R2/ϵ2), O(m)}to obtain an ϵ-approximation of the PPR vector, where m denotes the number of edges in the graph and R is a constant defined via nested evolving set processes. Furthermore, the algorithms induced by our framework require solving only O(1/ α) such linear systems, where α is the damping factor. When 1/ϵ2 m, this implies the existence of an algorithm that computes an ϵ-approximation of the PPR vector with an overall time complexity of O(R2/( αϵ2)), independent of the underlying graph size.

Deep State Space Models (SSMs) reignite physics-grounded compute paradigms, as RNNs could natively be embodied into dynamical systems. This calls for dedicated learning algorithms obeying to core physical principles, with efficient techniques to simulate these systems and guide their design. We propose Recurrent Hamiltonian Echo Learning (RHEL), an algorithm which provably computes loss gradients as finite differences of physical trajectories of non-dissipative, Hamiltonian systems. In ML terms, RHEL only requires three "forward passes" irrespective of model size, without explicit Jacobian computation, nor incurring any variance in the gradient estimation. Motivated by the potential to implement our algorithm in non-digital physical systems, we first introduce RHEL in continuous time and demonstrate its formal equivalence with the continuous adjoint state method.

We establish novel structural and statistical results for entropy-regularized min-max inverse reinforcement learning (Min-Max-IRL) with linear reward classes in finite-horizon MDPs with Borel state and action spaces. On the structural side, we show that maximum likelihood estimation (MLE) and Min-Max-IRL are equivalent at the population level, and at the empirical level under deterministic dynamics. On the statistical side, exploiting pseudo-self-concordance of the Min-Max-IRL loss, we prove that both the trajectory-level KL divergence and the squared parameter error in the Hessian norm decay at the fast rate $\mathcal{O}(n^{-1})$, where $n$ is the number of expert trajectories. Our guarantees apply under misspecification and require no exploration assumptions. We further extend reward-identifiability results to general Borel spaces and derive novel results on the derivatives of the soft-optimal value function with respect to reward parameters.

This appendix contains all proofs of the results mentioned in the main body of the paper, plus further results which have been omitted there due to space limits. We recall the following lemma which upper bounds the probability measure of the ball around a point x X that contains its kth nearest neighbors. The proof immediately follows from the multiplicative Chernoff bound (see, e.g., Lemma 3.2 in [28]). When combined with Assumption 5.1 we obtain the following corollary. Corollary A.2. Suppose that the measure-smoothness assumption (Assumption 5.1) holds with parameters λ, Cλ, k k.

Assumption 1.1 ensures that the gradient estimator

Let us defineK Rn n a random diagonal sampling matrix whereKj,j Bernoulli(pj) for 1 j n. Therefore, Bernoulli-CRS will perform on average the same amount of computations as in the fixed-rankCRS. This formulation immediately hints atthe possibility tosample over the input channeldimension, similarly to sampling column-row pairs in matrices. Let ` be a β-Lipschitz loss function, and let the network be trained with SGD using properly decreasing learning rate. Let us denote the weight, bias and activation gradients with respect to a loss function` by Wl, bl, al respectively.

We now prove the first bullet. This is a contradiction, so in fact c κ. The first claim of the second bullet is analogous. To do so, we note the following technical lemma (proof below). To prove (#1), we proceed by induction on t.