Ehrenfeucht-Haussler Rank and Chain of Thought

• URL: http://arxiv.org/abs/2501.12997v1
• Date: Wed, 22 Jan 2025 16:30:58 GMT
• Title: Ehrenfeucht-Haussler Rank and Chain of Thought
• Authors: Pablo Barceló, Alexander Kozachinskiy, Tomasz Steifer,
• Abstract summary: We show that the rank of a function $f$ corresponds to the minimum number of Chain of Thought steps required by a single-layer transformer decoder. We also analyze the problem of identifying the position of the $k$-th occurrence of 1 in a Boolean sequence, proving that it requires $k$ CoT steps.
• Score: 51.33559894954108
• License:
• Abstract: The notion of rank of a Boolean function has been a cornerstone in the theory of PAC learning, enabling quasipolynomial-time learning algorithms for polynomial-size decision trees. We present a novel characterization of rank, grounded in the well-known Transformer architecture. We show that the rank of a function $f$ corresponds to the minimum number of Chain of Thought (CoT) steps required by a single-layer transformer decoder with hard attention to compute $f$. Based on this characterization we establish tight bounds on the number of CoT steps required for specific problems, showing that $\ell$-fold function composition necessitates exactly $\ell$ CoT steps. Furthermore, we analyze the problem of identifying the position of the $k$-th occurrence of 1 in a Boolean sequence, proving that it requires $k$ CoT steps.

Related papers

• Generalized Quantum Signal Processing [0.6768558752130311]
  We present a Generalized Quantum Signal Processing approach, employing general SU(2) rotations as our signal processing operators. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that $|P|leq 1$. In cases where only $P$ is known, we provide an efficient GPU optimization capable of identifying in under a minute of time, a corresponding $Q$ for degree on the order of $107$.
  arXiv  Detail & Related papers  (2023-08-03T01:51:52Z)
• Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials [50.90125395570797]
  We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $mathbbRd$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/epsilon)O(k)$, whereepsilon>0$ is the target accuracy.
  arXiv  Detail & Related papers  (2023-07-24T14:37:22Z)
• Multi-block-Single-probe Variance Reduced Estimator for Coupled Compositional Optimization [49.58290066287418]
  We propose a novel method named Multi-block-probe Variance Reduced (MSVR) to alleviate the complexity of compositional problems. Our results improve upon prior ones in several aspects, including the order of sample complexities and dependence on strongity.
  arXiv  Detail & Related papers  (2022-07-18T12:03:26Z)
• A Projection-free Algorithm for Constrained Stochastic Multi-level Composition Optimization [12.096252285460814]
  We propose a projection-free conditional gradient-type algorithm for composition optimization. We show that the number of oracles and the linear-minimization oracle required by the proposed algorithm, are of order $mathcalO_T(epsilon-2)$ and $mathcalO_T(epsilon-3)$ respectively.
  arXiv  Detail & Related papers  (2022-02-09T06:05:38Z)
• Nearly Optimal Regret for Learning Adversarial MDPs with Linear Function Approximation [92.3161051419884]
  We study the reinforcement learning for finite-horizon episodic Markov decision processes with adversarial reward and full information feedback. We show that it can achieve $tildeO(dHsqrtT)$ regret, where $H$ is the length of the episode. We also prove a matching lower bound of $tildeOmega(dHsqrtT)$ up to logarithmic factors.
  arXiv  Detail & Related papers  (2021-02-17T18:54:08Z)
• Algorithms and SQ Lower Bounds for PAC Learning One-Hidden-Layer ReLU Networks [48.32532049640782]
  We study the problem of learning one-hidden-layer ReLU networks with $k$ hidden units on $mathbbRd$ under Gaussian marginals. For the case of positive coefficients, we give the first-time algorithm for this learning problem for $k$ up to $tildeOOmega(sqrtlog d)$.
  arXiv  Detail & Related papers  (2020-06-22T17:53:54Z)
• On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems [75.58134963501094]
  This paper analyzes the trajectories of gradient descent (SGD) We show that SGD avoids saddle points/manifolds with $1$ for strict step-size policies.
  arXiv  Detail & Related papers  (2020-06-19T14:11:26Z)
• Sample Efficient Reinforcement Learning via Low-Rank Matrix Estimation [30.137884459159107]
  We consider the question of learning $Q$-function in a sample efficient manner for reinforcement learning with continuous state and action spaces. We develop a simple, iterative learning algorithm that finds $epsilon$-Schmidt $Q$-function with sample complexity of $widetildeO(frac1epsilonmax(d1), d_2)+2)$ when the optimal $Q$-function has low rank $r$ and the factor $$ is below a certain threshold.
  arXiv  Detail & Related papers  (2020-06-11T00:55:35Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.