The simple linear threshold units used in many artificial neural networks have a limited computational capacity. Famously, a single unit cannot handle nonlinearly separable problems like XOR. In contrast, real neurons exhibit complex morphologies as well as active dendritic integration, suggesting that their computational capacities outperform those of simple linear units. Considering specific families of Boolean functions, we empirically examine the computational limits of single units that incorporate more complex dendritic structures. For random Boolean functions, we show that there is a phase transition in learnability as a function of the input dimension, with most random functions below a certain critical dimension being learnable and those above not.

There is mounting evidence of emergent phenomena in the capabilities of deep learning methods as we scale up datasets, model sizes, and training times. While there are some accounts of how these resources modulate statistical capacity, far less is known about their effect on the computational problem of model training. This work conducts such an exploration through the lens of learning a $k$-sparse parity of $n$ bits, a canonical discrete search problem which is statistically easy but computationally hard. Empirically, we find that a variety of neural networks successfully learn sparse parities, with discontinuous phase transitions in the training curves. On small instances, learning abruptly occurs at approximately $n^{O(k)}$ iterations; this nearly matches SQ lower bounds, despite the apparent lack of a sparse prior. Our theoretical analysis shows that these observations are not explained by a Langevin-like mechanism, whereby SGD stumbles in the dark until it finds the hidden set of features (a natural algorithm which also runs in $n^{O(k)}$ time). Instead, we show that SGD gradually amplifies the sparse solution via a Fourier gap in the population gradient, making continual progress that is invisible to loss and error metrics.

There is mounting evidence of emergent phenomena in the capabilities of deep learning methods as we scale up datasets, model sizes, and training times. While there are some accounts of how these resources modulate statistical capacity, far less is known about their effect on the computational problem of model training. This work conducts such an exploration through the lens of learning a k -sparse parity of n bits, a canonical discrete search problem which is statistically easy but computationally hard. Empirically, we find that a variety of neural networks successfully learn sparse parities, with discontinuous phase transitions in the training curves. On small instances, learning abruptly occurs at approximately n {O(k)} iterations; this nearly matches SQ lower bounds, despite the apparent lack of a sparse prior.

We study the computational limits of Low-Rank Adaptation (LoRA) update for finetuning transformer-based models using fine-grained complexity theory. Our key observation is that the existence of low-rank decompositions within the gradient computation of LoRA adaptation leads to possible algorithmic speedup. This allows us to (i) identify a phase transition behavior and (ii) prove the existence of nearly linear algorithms by controlling the LoRA update computation term by term, assuming the Strong Exponential Time Hypothesis (SETH). For the former, we identify a sharp transition in the efficiency of all possible rank-$r$ LoRA update algorithms for transformers, based on specific norms resulting from the multiplications of the input sequence $\mathbf{X}$, pretrained weights $\mathbf{W^\star}$, and adapter matrices $\alpha \mathbf{B} \mathbf{A} / r$. Specifically, we derive a shared upper bound threshold for such norms and show that efficient (sub-quadratic) approximation algorithms of LoRA exist only below this threshold. For the latter, we prove the existence of nearly linear approximation algorithms for LoRA adaptation by utilizing the hierarchical low-rank structures of LoRA gradients and approximating the gradients with a series of chained low-rank approximations. To showcase our theory, we consider two practical scenarios: partial (e.g., only $\mathbf{W}_V$ and $\mathbf{W}_Q$) and full adaptations (e.g., $\mathbf{W}_Q$, $\mathbf{W}_V$, and $\mathbf{W}_K$) of weights in attention heads.

We investigate the computational limits of the memory retrieval dynamics of modern Hopfield models from the fine-grained complexity analysis. Our key contribution is the characterization of a phase transition behavior in the efficiency of all possible modern Hopfield models based on the norm of patterns. Specifically, we establish an upper bound criterion for the norm of input query patterns and memory patterns. Only below this criterion, sub-quadratic (efficient) variants of the modern Hopfield model exist, assuming the Strong Exponential Time Hypothesis (SETH). To showcase our theory, we provide a formal example of efficient constructions of modern Hopfield models using low-rank approximation when the efficient criterion holds. This includes a derivation of a lower bound on the computational time, scaling linearly with $\Max\{$# of stored memory patterns, length of input query sequence$\}$. In addition, we prove its memory retrieval error bound and exponential memory capacity.

There is mounting evidence of emergent phenomena in the capabilities of deep learning methods as we scale up datasets, model sizes, and training times. While there are some accounts of how these resources modulate statistical capacity, far less is known about their effect on the computational problem of model training. This work conducts such an exploration through the lens of learning a $k$-sparse parity of $n$ bits, a canonical discrete search problem which is statistically easy but computationally hard. Empirically, we find that a variety of neural networks successfully learn sparse parities, with discontinuous phase transitions in the training curves. On small instances, learning abruptly occurs at approximately $n^{O(k)}$ iterations; this nearly matches SQ lower bounds, despite the apparent lack of a sparse prior. Our theoretical analysis shows that these observations are not explained by a Langevin-like mechanism, whereby SGD "stumbles in the dark" until it finds the hidden set of features (a natural algorithm which also runs in $n^{O(k)}$ time). Instead, we show that SGD gradually amplifies the sparse solution via a Fourier gap in the population gradient, making continual progress that is invisible to loss and error metrics.

Hardware is the foundation that deep learning is based on, providing its capabilities and readiness to help people categorize objects, improve speech recognition, understand visualizations, or any other purpose motivating people to use deep learning. When analyzing deep learning computational needs, remembering acronyms is the best way to spell out the hardware requirements for deep learning. GPU, TPU, FPGA, and ASICs are all key hardware components necessary for making deep learning work, especially amid concerns in recent that its progress has stunted. These types of hardware consume a lot of power and facilitate large deep learning models that CPUs and regular laptops can't manage. How does each of these hardware types facilitate these needs while addressing the computational limits restricting deep learning from achieving maximum potential?

Last week's announcement of AlphaCode, DeepMind's source code–generating deep learning system, created a lot of excitement--some of it unwarranted--surrounding advances in artificial intelligence. As I've mentioned in my deep dive on AlphaCode, DeepMind's researchers have done a great job in bringing together the right technology and practices to create a machine learning model that can find solutions to very complex problems. However, the sometimes-bloated coverage of AlphaCode by the media highlights the endemic problems with framing the growing capabilities of artificial intelligence in the context of competitions meant for humans. Here's what the experts say For decades, AI researchers and scientists have been searching for tests that can measure progress toward artificial general intelligence. And having envisioned AI in the image of the human mind, they have turned to benchmarks for human intelligence.

Last week's announcement of AlphaCode, DeepMind's source code–generating deep learning system, created a lot of excitement--some of it unwarranted--surrounding advances in artificial intelligence. As I've mentioned in my deep dive on AlphaCode, DeepMind's researchers have done a great job in bringing together the right technology and practices to create a machine learning model that can find solutions to very complex problems. However, the sometimes-bloated coverage of AlphaCode by the media highlights the endemic problems with framing the growing capabilities of artificial intelligence in the context of competitions meant for humans. For decades, AI researchers and scientists have been searching for tests that can measure progress toward artificial general intelligence. And having envisioned AI in the image of the human mind, they have turned to benchmarks for human intelligence.

Large-scale ride-sharing systems combine real-time dispatching and routing optimization over a rolling time horizon with a model predictive control(MPC) component that relocates idle vehicles to anticipate the demand. The MPC optimization operates over a longer time horizon to compensate for the inherent myopic nature of the real-time dispatching. These longer time horizons are beneficial for the quality of the decisions but increase computational complexity. To address this computational challenge, this paper proposes a hybrid approach that combines machine learning and optimization. The machine-learning component learns the optimal solution to the MPC optimization on the aggregated level to overcome the sparsity and high-dimensionality of the MPC solutions. The optimization component transforms the machine-learning predictions back to the original granularity via a tractable transportation model. As a consequence, the original NP-hard MPC problem is reduced to a polynomial time prediction and optimization. Experimental results show that the hybrid approach achieves 27% further reduction in rider waiting time than the MPC optimization, thanks to its ability to model a longer time horizon within the computational limits.