We propose the Hyperbolic Tangent Exponential Linear Unit (TeLU), a neural network hidden activation function defined as TeLU(x)=xtanh(exp(x)). TeLU's design is grounded in the core principles of key activation functions, achieving strong convergence by closely approximating the identity function in its active region while effectively mitigating the vanishing gradient problem in its saturating region. Its simple formulation enhances computational efficiency, leading to improvements in scalability and convergence speed. Unlike many modern activation functions, TeLU seamlessly combines the simplicity and effectiveness of ReLU with the smoothness and analytic properties essential for learning stability in deep neural networks. TeLU's ability to mimic the behavior and optimal hyperparameter settings of ReLU, while introducing the benefits of smoothness and curvature, makes it an ideal drop-in replacement. Its analytic nature positions TeLU as a powerful universal approximator, enhancing both robustness and generalization across a multitude of experiments. We rigorously validate these claims through theoretical analysis and experimental validation, demonstrating TeLU's performance across challenging benchmarks; including ResNet18 on ImageNet, Dynamic-Pooling Transformers on Text8, and Recurrent Neural Networks (RNNs) on the Penn TreeBank dataset. These results highlight TeLU's potential to set a new standard in activation functions, driving more efficient and stable learning in deep neural networks, thereby accelerating scientific discoveries across various fields.

Energy-based learning algorithms, such as predictive coding (PC), have garnered significant attention in the machine learning community due to their theoretical properties, such as local operations and biologically plausible mechanisms for error correction. In this work, we rigorously analyze the stability, robustness, and convergence of PC through the lens of dynamical systems theory. We show that, first, PC is Lyapunov stable under mild assumptions on its loss and residual energy functions, which implies intrinsic robustness to small random perturbations due to its well-defined energy-minimizing dynamics. Second, we formally establish that the PC updates approximate quasi-Newton methods by incorporating higher-order curvature information, which makes them more stable and able to converge with fewer iterations compared to models trained via backpropagation (BP). Furthermore, using this dynamical framework, we provide new theoretical bounds on the similarity between PC and other algorithms, i.e., BP and target propagation (TP), by precisely characterizing the role of higher-order derivatives. These bounds, derived through detailed analysis of the Hessian structures, show that PC is significantly closer to quasi-Newton updates than TP, providing a deeper understanding of the stability and efficiency of PC compared to conventional learning methods.

Recurrent Neural Networks (RNNs) have been foundational in sequence modeling due to their ability to capture temporal dependencies. Architectures such as Elman RNNs, Gated Recurrent Units (GRUs), and Long Short-Term Memory networks (LSTMs) [1] are widely used in applications like speech recognition, machine translation, and time-series analysis. However, these models are constrained by their fixed memory capacity, limiting them to recognizing regular languages when implemented with finite precision [2, 3]. To enhance the computational capabilities of RNNs, researchers have explored augmenting them with external memory structures like stacks [4, 5, 6, 7, 8, 9, 10]. This approach extends the expressivity of RNNs to context-free languages (CFLs) [11], which are crucial in applications like natural language processing (NLP) where hierarchical structures are prevalent. Memory-augmented models have demonstrated significant improvements in recognizing complex formal languages by simulating operations similar to Pushdown Automata (PDA).

This study investigates the learnability of Recurrent Neural Networks (RNNs) in classifying structured formal languages, focusing on counter and Dyck languages. Traditionally, both first-order (LSTM) and second-order (O2RNN) RNNs have been considered effective for such tasks, primarily based on their theoretical expressiveness within the Chomsky hierarchy. However, our research challenges this notion by demonstrating that RNNs primarily operate as state machines, where their linguistic capabilities are heavily influenced by the precision of their embeddings and the strategies used for sampling negative examples. Our experiments revealed that performance declines significantly as the structural similarity between positive and negative examples increases. Remarkably, even a basic single-layer classifier using RNN embeddings performed better than chance. To evaluate generalization, we trained models on strings up to a length of 40 and tested them on strings from lengths 41 to 500, using 10 unique seeds to ensure statistical robustness. Stability comparisons between LSTM and O2RNN models showed that O2RNNs generally offer greater stability across various scenarios. We further explore the impact of different initialization strategies revealing that our hypothesis is consistent with various RNNs. Overall, this research questions established beliefs about RNNs' computational capabilities, highlighting the importance of data structure and sampling techniques in assessing neural networks' potential for language classification tasks. It emphasizes that stronger constraints on expressivity are crucial for understanding true learnability, as mere expressivity does not capture the essence of learning.

Prompting techniques have significantly enhanced the capabilities of Large Language Models (LLMs) across various complex tasks, including reasoning, planning, and solving math word problems. However, most research has predominantly focused on language-based reasoning and word problems, often overlooking the potential of LLMs in handling symbol-based calculations and reasoning. This study aims to bridge this gap by rigorously evaluating LLMs on a series of symbolic tasks, such as addition, multiplication, modulus arithmetic, numerical precision, and symbolic counting. Our analysis encompasses eight LLMs, including four enterprise-grade and four open-source models, of which three have been pre-trained on mathematical tasks. The assessment framework is anchored in Chomsky's Hierarchy, providing a robust measure of the computational abilities of these models. The evaluation employs minimally explained prompts alongside the zero-shot Chain of Thoughts technique, allowing models to navigate the solution process autonomously. The findings reveal a significant decline in LLMs' performance on context-free and context-sensitive symbolic tasks as the complexity, represented by the number of symbols, increases. Notably, even the fine-tuned GPT3.5 exhibits only marginal improvements, mirroring the performance trends observed in other models. Across the board, all models demonstrated a limited generalization ability on these symbol-intensive tasks. This research underscores LLMs' challenges with increasing symbolic complexity and highlights the need for specialized training, memory and architectural adjustments to enhance their proficiency in symbol-based reasoning tasks.

An intelligent system capable of continual learning is one that can process and extract knowledge from potentially infinitely long streams of pattern vectors. The major challenge that makes crafting such a system difficult is known as catastrophic forgetting - an agent, such as one based on artificial neural networks (ANNs), struggles to retain previously acquired knowledge when learning from new samples. Furthermore, ensuring that knowledge is preserved for previous tasks becomes more challenging when input is not supplemented with task boundary information. Although forgetting in the context of ANNs has been studied extensively, there still exists far less work investigating it in terms of unsupervised architectures such as the venerable self-organizing map (SOM), a neural model often used in clustering and dimensionality reduction. While the internal mechanisms of SOMs could, in principle, yield sparse representations that improve memory retention, we observe that, when a fixed-size SOM processes continuous data streams, it experiences concept drift. In light of this, we propose a generalization of the SOM, the continual SOM (CSOM), which is capable of online unsupervised learning under a low memory budget. Our results, on benchmarks including MNIST, Kuzushiji-MNIST, and Fashion-MNIST, show almost a two times increase in accuracy, and CIFAR-10 demonstrates a state-of-the-art result when tested on (online) unsupervised class incremental learning setting.

One major criticism of deep learning centers around the biological implausibility of the credit assignment schema used for learning -- backpropagation of errors. This implausibility translates into practical limitations, spanning scientific fields, including incompatibility with hardware and non-differentiable implementations, thus leading to expensive energy requirements. In contrast, biologically plausible credit assignment is compatible with practically any learning condition and is energy-efficient. As a result, it accommodates hardware and scientific modeling, e.g. learning with physical systems and non-differentiable behavior. Furthermore, it can lead to the development of real-time, adaptive neuromorphic processing systems. In addressing this problem, an interdisciplinary branch of artificial intelligence research that lies at the intersection of neuroscience, cognitive science, and machine learning has emerged. In this paper, we survey several vital algorithms that model bio-plausible rules of credit assignment in artificial neural networks, discussing the solutions they provide for different scientific fields as well as their advantages on CPUs, GPUs, and novel implementations of neuromorphic hardware. We conclude by discussing the future challenges that will need to be addressed in order to make such algorithms more useful in practical applications.

In the rapidly evolving landscape of neural networks, the choice of activation function plays a pivotal role in model performance and stability. While the Rectified Linear Unit (ReLU) [6, 20] has long been the cornerstone of numerous deep learning architectures [25, 8, 26] due to its simplicity and effectiveness in mitigating the vanishing gradient problem [10, 11], it is not without limitations. Particularly, ReLU suffers from the "dying ReLU" issue [18], where neurons can become inactive and cease to contribute to the learning process, potentially leading to suboptimal models. Enter the Gaussian Error Linear Unit (GELU) [9] and Mish [19] activation functions, which have emerged as sophisticated alternatives, addressing some of ReLU's shortcomings. GELU, leveraging the properties of the Gaussian distribution, offers a smooth, non-linear transition in its activation, which can lead to improved learning dynamics [27, 4, 15]. Mish, further building on this concept, introduces a self-gating mechanism, enabling a smoother information flow.

This paper analyzes two competing rule extraction methodologies: quantization and equivalence query. We trained $3600$ RNN models, extracting $18000$ DFA with a quantization approach (k-means and SOM) and $3600$ DFA by equivalence query($L^{*}$) methods across $10$ initialization seeds. We sampled the datasets from $7$ Tomita and $4$ Dyck grammars and trained them on $4$ RNN cells: LSTM, GRU, O2RNN, and MIRNN. The observations from our experiments establish the superior performance of O2RNN and quantization-based rule extraction over others. $L^{*}$, primarily proposed for regular grammars, performs similarly to quantization methods for Tomita languages when neural networks are perfectly trained. However, for partially trained RNNs, $L^{*}$ shows instability in the number of states in DFA, e.g., for Tomita 5 and Tomita 6 languages, $L^{*}$ produced more than $100$ states. In contrast, quantization methods result in rules with number of states very close to ground truth DFA. Among RNN cells, O2RNN produces stable DFA consistently compared to other cells. For Dyck Languages, we observe that although GRU outperforms other RNNs in network performance, the DFA extracted by O2RNN has higher performance and better stability. The stability is computed as the standard deviation of accuracy on test sets on networks trained across $10$ seeds. On Dyck Languages, quantization methods outperformed $L^{*}$ with better stability in accuracy and the number of states. $L^{*}$ often showed instability in accuracy in the order of $16\% - 22\%$ for GRU and MIRNN while deviation for quantization methods varied in $5\% - 15\%$. In many instances with LSTM and GRU, DFA's extracted by $L^{*}$ even failed to beat chance accuracy ($50\%$), while those extracted by quantization method had standard deviation in the $7\%-17\%$ range. For O2RNN, both rule extraction methods had deviation in the $0.5\% - 3\%$ range.

While current deep learning algorithms have been successful for a wide variety of artificial intelligence (AI) tasks, including those involving structured image data, they present deep neurophysiological conceptual issues due to their reliance on the gradients that are computed by backpropagation of errors (backprop). Gradients are required to obtain synaptic weight adjustments but require knowledge of feed-forward activities in order to conduct backward propagation, a biologically implausible process. This is known as the "weight transport problem". Therefore, in this work, we present a more biologically plausible approach towards solving the weight transport problem for image data. This approach, which we name the error kernel driven activation alignment (EKDAA) algorithm, accomplishes through the introduction of locally derived error transmission kernels and error maps. Like standard deep learning networks, EKDAA performs the standard forward process via weights and activation functions; however, its backward error computation involves adaptive error kernels that propagate local error signals through the network. The efficacy of EKDAA is demonstrated by performing visual-recognition tasks on the Fashion MNIST, CIFAR-10 and SVHN benchmarks, along with demonstrating its ability to extract visual features from natural color images. Furthermore, in order to demonstrate its non-reliance on gradient computations, results are presented for an EKDAA trained CNN that employs a non-differentiable activation function.