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Replicable Learning of Large-Margin Halfspaces (2402.13857v2)

Published 21 Feb 2024 in cs.LG

Abstract: We provide efficient replicable algorithms for the problem of learning large-margin halfspaces. Our results improve upon the algorithms provided by Impagliazzo, Lei, Pitassi, and Sorrell [STOC, 2022]. We design the first dimension-independent replicable algorithms for this task which runs in polynomial time, is proper, and has strictly improved sample complexity compared to the one achieved by Impagliazzo et al. [2022] with respect to all the relevant parameters. Moreover, our first algorithm has sample complexity that is optimal with respect to the accuracy parameter $\epsilon$. We also design an SGD-based replicable algorithm that, in some parameters' regimes, achieves better sample and time complexity than our first algorithm. Departing from the requirement of polynomial time algorithms, using the DP-to-Replicability reduction of Bun, Gaboardi, Hopkins, Impagliazzo, Lei, Pitassi, Sorrell, and Sivakumar [STOC, 2023], we show how to obtain a replicable algorithm for large-margin halfspaces with improved sample complexity with respect to the margin parameter $\tau$, but running time doubly exponential in $1/\tau2$ and worse sample complexity dependence on $\epsilon$ than one of our previous algorithms. We then design an improved algorithm with better sample complexity than all three of our previous algorithms and running time exponential in $1/\tau{2}$.

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Citations (4)
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Summary

  • The paper introduces replicable algorithms for learning large-margin halfspaces with dimension-independent sample complexity and proven improvements over previous methods.
  • It employs techniques such as surrogate convex loss and boosting to optimize sample and time complexity within the replicable learning framework.
  • The study leverages dimensionality reduction and rounding to design robust, resource-efficient algorithms, setting a new standard for replicability in machine learning.

Replicable Algorithms for Learning Large-Margin Halfspaces with Improved Sample Complexity

Efficient and Replicable Learning Algorithms for Large-Margin Halfspaces

We examine the problem of learning large-margin halfspaces under the stringent requirement of replicability, a concept that has recently garnered growing interest within the machine learning theory community. Our paper presents several algorithms that outperform existing approaches in the literature, particularly those presented by Impagliazzo et al. [2022]. We specifically address several key objectives: achieving dimension-independent sample complexity, ensuring the replicability of learning processes, and maintaining computational efficiency.

Main Results and Contributions

We design new replicable algorithms for the problem of learning large-margin halfspaces with significant improvements over previous work. Notably:

  1. Algorithm 1: We introduce an efficient replicable learning algorithm that boasts dimension-independent sample complexity while requiring polynomial runtime. The algorithm achieves an optimal dependency with respect to the accuracy parameter ϵ\epsilon, exhibiting a sample complexity of O~(ϵ−1τ−7ρ−2log⁡(1/δ))\widetilde{O}(\epsilon^{-1}\tau^{-7}\rho^{-2}\log(1/\delta)).
  2. Algorithm 2: Our second contribution is an SGD-based replicable algorithm that, in specific regimes of the problem parameters, delivers better sample and time complexity than our first algorithm. This approach leverages a surrogate convex loss and boosting technique to achieve a sample complexity of O~(ϵ−2τ−6ρ−2log⁡(1/δ))\widetilde{O}(\epsilon^{-2}\tau^{-6}\rho^{-2}\log(1/\delta)).
  3. Algorithm 3: Departing from the constraint of polynomial-time solutions, we construct an algorithm that exhibits an improved sample complexity of O~(ϵ−1τ−4ρ−2log⁡(1/δ))\widetilde{O}(\epsilon^{-1}\tau^{-4}\rho^{-2}\log(1/\delta)) at the cost of a running time that is exponential in 1/τ21/\tau^{2}. This algorithm employs dimensionality reduction and rounding mechanisms to efficiently harness a finite hypothesis space.

Theoretical Implications and Experimental Outlook

Our work not only addresses the open question of whether the bounds set by Impagliazzo et al. [2022] are tight but also conclusively demonstrates that substantial improvements are possible. Our findings have several implications:

  • The introduction of dimension-independent sample complexity contributes significantly to our understanding of replicable learning processes and opens new pathways for resource-efficient algorithm design.
  • The achievement of replicability without compromising on sample complexity or computational efficiency marks a significant step forward in the development of robust learning algorithms that can be reliably replicated across different executions.
  • The methodologies developed, particularly around dimensionality reduction and rounding within a shared randomness framework, could have broader applications beyond the specific problem tackled in this paper.

Conclusion and Future Directions

In conclusion, our work provides new pathways and methodologies for replicably learning large-margin halfspaces with improved sample complexity and computational efficiency. Looking ahead, there are several intriguing questions that merit further investigation. Among them are exploring more efficient algorithms with sample complexity approaching the theoretical lower bounds, extending our approaches to other learning paradigms, and investigating the implications of our findings on the broader replicability crisis in AI and machine learning. Our results represent a promising step toward more reliable and efficient learning algorithms that adhere to the critical principle of replicability.

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