We introduce SIRI, Scaling Iterative Reinforcement Learning with Interleaved Compression, a simple yet effective RL approach for Large Reasoning Models (LRMs) that enables more efficient and accurate reasoning. Existing studies have observed repetitive thinking patterns in LRMs, and attempts to reduce them often come at the cost of performance. In this paper, we show that this trade-off can be overcome through a training regime that iteratively alternates between compressing and expanding the reasoning budget, by dynamically adjusting the maximum rollout length during training. The compression phase cuts the rollout length, forcing the model to make precise and valuable decisions within a limited context, which effectively reduces redundant tokens and increases reasoning density. The expansion phase then relaxes the length limit, providing space for the model to explore and plan in long-horizon settings. Remarkably, we find that after each compression-expansion cycle, the model's performance improves even as its output length decreases, steadily pushing it closer to the Pareto frontier in the performance-efficiency trade-off. Training on DeepSeek-R1-Distill-Qwen-1.5B, SIRI-low improves performance on AIME24 by 43.2% while reducing token usage by 46.9% after three iterations, and SIRI-high achieves the highest accuracy compared to all other methods (Figure 1). Our findings shed light on the potential of periodically oscillating the LRM's output truncation length during training to dynamically balance exploration and efficiency in reasoning, converging towards an optimal "sweet spot" between the two. Our models are publicly available.

Take Fig. S.1 as an example.

Using the information theory, this study provides insights into how the construction of latent space of autoencoder (AE) using deep neural network (DNN) training finds a smooth low-dimensional manifold in the stiff dynamical system. Our recent study [1] reported that an autoencoder (AE) combined with neural ODE (NODE) as a surrogate reduced order model (ROM) for the integration of stiff chemically reacting systems led to a significant reduction in the temporal stiffness, and the behavior was attributed to the identification of a slow invariant manifold by the nonlinear projection of the AE. The present work offers fundamental understanding of the mechanism by employing concepts from information theory and better mixing. The learning mechanism of both the encoder and decoder are explained by plotting the evolution of mutual information and identifying two different phases. Subsequently, the density distribution is plotted for the physical and latent variables, which shows the transformation of the \emph{rare event} in the physical space to a \emph{highly likely} (more probable) event in the latent space provided by the nonlinear autoencoder. Finally, the nonlinear transformation leading to density redistribution is explained using concepts from information theory and probability.

Visual question answering aims to provide responses to natural language questions given visual input. Recently, visual programmatic models (VPMs), which generate executable programs to answer questions through large language models (LLMs), have attracted research interest. However, they often require long input prompts to provide the LLM with sufficient API usage details to generate relevant code. To address this limitation, we propose AdaCoder, an adaptive prompt compression framework for VPMs. AdaCoder operates in two phases: a compression phase and an inference phase. In the compression phase, given a preprompt that describes all API definitions in the Python language with example snippets of code, a set of compressed preprompts is generated, each depending on a specific question type. In the inference phase, given an input question, AdaCoder predicts the question type and chooses the appropriate corresponding compressed preprompt to generate code to answer the question. Notably, AdaCoder employs a single frozen LLM and pre-defined prompts, negating the necessity of additional training and maintaining adaptability across different powerful black-box LLMs such as GPT and Claude. In experiments, we apply AdaCoder to ViperGPT and demonstrate that it reduces token length by 71.1%, while maintaining or even improving the performance of visual question answering.

This study comes as a timely response to mounting criticism of the information bottleneck (IB) theory, injecting fresh perspectives to rectify misconceptions and reaffirm its validity. Firstly, we introduce an auxiliary function to reinterpret the maximal coding rate reduction method as a special yet local optimal case of IB theory. Through this auxiliary function, we clarify the paradox of decreasing mutual information during the application of ReLU activation in deep learning (DL) networks. Secondly, we challenge the doubts about IB theory's applicability by demonstrating its capacity to explain the absence of a compression phase with linear activation functions in hidden layers, when viewed through the lens of the auxiliary function. Lastly, by taking a novel theoretical stance, we provide a new way to interpret the inner organizations of DL networks by using IB theory, aligning them with recent experimental evidence. Thus, this paper serves as an act of justice for IB theory, potentially reinvigorating its standing and application in DL and other fields such as communications and biomedical research.

Recent year, many researches attempt to open the black box of deep neural networks and propose a various of theories to understand it. Among them, information bottleneck theory (IB) claims that there are two distinct phases consisting of fitting phase and compression phase in the course of training. This statement attracts many attentions since its success in explaining the inner behavior of feedforward neural networks. In this paper, we employ IB theory to understand the dynamic behavior of convolutional neural networks (CNNs) and investigate how the fundamental features have impact on the performance of CNNs. In particular, through a series of experimental analysis on benchmark of MNIST and Fashion-MNIST, we demonstrate that the compression phase is not observed in all these cases. This show us the CNNs have a rather complicated behavior than feedforward neural networks.

Analyzing deep neural networks (DNNs) via information plane (IP) theory has gained tremendous attention recently as a tool to gain insight into, among others, their generalization ability. However, it is by no means obvious how to estimate mutual information (MI) between each hidden layer and the input/desired output, to construct the IP. For instance, hidden layers with many neurons require MI estimators with robustness towards the high dimensionality associated with such layers. MI estimators should also be able to naturally handle convolutional layers, while at the same time being computationally tractable to scale to large networks. None of the existing IP methods to date have been able to study truly deep Convolutional Neural Networks (CNNs), such as the e.g.\ VGG-16. In this paper, we propose an IP analysis using the new matrix--based R\'enyi's entropy coupled with tensor kernels over convolutional layers, leveraging the power of kernel methods to represent properties of the probability distribution independently of the dimensionality of the data. The obtained results shed new light on the previous literature concerning small-scale DNNs, however using a completely new approach. Importantly, the new framework enables us to provide the first comprehensive IP analysis of contemporary large-scale DNNs and CNNs, investigating the different training phases and providing new insights into the training dynamics of large-scale neural networks.

Why deep neural networks (DNNs) capable of overfitting often generalize well in practice is a mystery in deep learning. Existing works indicate that this observation holds for both complicated real datasets and simple datasets of one-dimensional (1-d) functions. In this work, for general low-frequency dominant 1-d functions, we find that a DNN with common settings first quickly captures the dominant low-frequency components, and then relatively slowly captures high-frequency ones. We call this phenomenon Frequency Principle (F-Principle). F-Principle can be observed over various DNN setups of different activation functions, layer structures and training algorithms in our experiments. F-Principle can be used to understand (i) the behavior of DNN training in the information plane and (ii) why DNNs often generalize well albeit its ability of overfitting. This F-Principle potentially can provide insights into understanding the general principle underlying DNN optimization and generalization for real datasets.