This section includes all proofs referenced in the main part of the paper, along with the associated theorems and lemmas for completeness. For the second part, note that it also holds trivially for τ = 1. The case, τ = 1, is easy to verify, and we assume the claim holds for τ = t. Hence, the convergence is of linear rate as claimed. In the following we show empirical results of performing a simple self-distillation procedure with deep neural networks with varying choices of α to investigate the large scale effects.

Distributionally robust reinforcement learning (DRRL), often framed as a robust Markov decision process (RMDP), seeks to find a robust policy that achieves good performance under the worst-case scenario among all environments within a prespecified uncertainty set centered around the training environment. Unlike previous work, which relies on a generative model or a pre-collected offline dataset enjoying good coverage of the deployment environment, we tackle robust RL via interactive data collection, where the learner interacts with the training environment only and refines the policy through trial and error. In this robust RL paradigm, two main challenges emerge: managing the distributional robustness while striking a balance between exploration and exploitation during data collection. Initially, we establish that sample-efficient learning without additional assumptions is unattainable owing to the curse of support shift; i.e., the potential disjointedness of the distributional supports between training and testing environments. To circumvent such a hardness result, we introduce the vanishing minimal value assumption to RMDPs with a total-variation distance robust set, postulating that the minimal value of the optimal robust value function is zero. Such an assumption effectively eliminates the support shift issue for RMDPs with a TV distance robust set, and we present an algorithm with a provable sample complexity guarantee. Our work makes the initial step to uncovering the inherent difficulty of robust RL via interactive data collection and sufficient conditions for sample-efficient algorithms with sharp sample complexity.

Graph diffusion, which iteratively propagates real-valued substances among the graph, is used in numerous graph/network-involved applications. However, releasing diffusion vectors may reveal sensitive linking information in the data such as transaction information in financial network data. Protecting the privacy of graph data is challenging due to its interconnected nature. This work proposes a novel graph diffusion framework with edge-level differential privacy guarantees by using noisy diffusion iterates. The algorithm injects Laplace noise per diffusion iteration and adopts a degree-based thresholding function to mitigate the high sensitivity induced by low-degree nodes. Our privacy loss analysis is based on Privacy Amplification by Iteration (PABI), which to our best knowledge, is the first effort that analyzes PABI with Laplace noise and provides relevant applications. We also introduce a novel -Wasserstein distance tracking method, which tightens the analysis of privacy leakage and makes PABI practically applicable. We evaluate this framework by applying it to Personalized Pagerank computation for ranking tasks. Experiments on real-world network data demonstrate the superiority of our method under stringent privacy conditions.

Through representative empirical studies, we validate the superior performance of SAMoSSA compared to existing baselines.

Automated scientific discovery promises to accelerate progress across scientific domains. However, developing and evaluating an AI agent's capacity for endto-end scientific reasoning is challenging as running real-world experiments is often prohibitively expensive or infeasible.

We thank Jiacheng Liu for his work on collecting chess-related data and chess book list. B.1 Universal Chess Interface (UCI) The UCI format is widely used for communication between chess engines and user interfaces. It represents chess moves by combining the starting and ending squares of a piece, such as "e2e4" to indicate moving the pawn from e2 to e4. SAN (Standard Algebraic Notation) is a widely used notation system in the game of chess for recording and describing moves. It provides a standardized and concise representation of moves that is easily understood by chess players and enthusiasts. In SAN, each move is represented by two components: the piece abbreviation and the destination square. The piece abbreviation is a letter that represents the type of piece making the move, such as "K" for king, "Q" for queen, "R" for rook, "B" for bishop, "N" for knight, and no abbreviation for pawns. The destination square is denoted by a combination of a letter (a-h) representing the column and a number (1-8) representing the row on the chessboard. Additional symbols may be used to indicate specific move types. The symbol "+" is used to indicate a check, while "#" denotes a checkmate. Castling moves are represented by "O-O" for kingside castling and "O-O-O" for queenside castling. PGN is a widely adopted format for recording chess games. It includes not only the SAN moves but also additional information like player names, event details, and game results. PGN files are human-readable and can be easily shared and analyzed. FEN is a notation system used to describe the state of a chess game. It represents the positions of pieces on the chessboard, active color, castling rights, en passant targets, and the half-move and full-move counters. The active color is represented by "w" for white or "b" for black.

Neural algorithmic reasoning is an emerging area of machine learning that focuses on building neural networks capable of solving complex algorithmic tasks. Recent advancements predominantly follow the standard supervised learning paradigm - feeding an individual problem instance into the network each time and training it to approximate the execution steps of a classical algorithm. We challenge this mode and propose a novel open-book learning framework. In this framework, whether during training or testing, the network can access and utilize all instances in the training dataset when reasoning for a given instance. Empirical evaluation is conducted on the challenging CLRS Algorithmic Reasoning Benchmark, which consists of 30 diverse algorithmic tasks. Our open-book learning framework exhibits a significant enhancement in neural reasoning capabilities. Further, we notice that there is recent literature suggesting that multi-task training on CLRS can improve the reasoning accuracy of certain tasks, implying intrinsic connections between different algorithmic tasks.