Learning from many trajectories
Authors: Stephen Tu, Roy Frostig, Mahdi Soltanolkotabi
JMLR 2024 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We conduct a simple numerical simulation illustrating the benefits of multiple trajectories on learning. We construct a family of LDS-LS problem instances, parameterized by a scalar ρ (0, ) as follows. The covariate distribution Px is the linear dynamical system xt+1 = Axt + wt with: A = U diag( ρ, . . . , ρ | {z } n/2 times , ρ, . . . , ρ)UT, U Unif(O(n)), wt N(0, I/4). Here, O(n) denotes the set of n n orthonormal matrices. By construction, ρ is the spectral radius of A. The labels yt are set as yt = xt+1, so that the ground truth W Rn n is equal to A. We compare the risk of the OLS estimator (3.6) on the LDS-LS problem instance, compared with its risk on the corresponding Ind-LDS-LS baseline. Specifically, we plot the ratio between OLS excess risks E[L( ; T, Px)] on the two problem instances (Px), respectively. We fix the covariate dimension n = 5 and the trajectory horizon length T = 10n, and vary the number of trajectories m {1, . . . , 10}. Figure 4 shows the result of this experiment, where we also vary ρ {0.98, 0.99, 1.0, 1.01, 1.02}. The error bars are plotted over 1000 trials. All computations are implemented using jax (Bradbury et al., 2018), and run with float64 precision on a single machine. |
| Researcher Affiliation | Collaboration | Stephen Tu EMAIL University of Southern California Ming Hsieh Department of Electrical and Computer Engineering Los Angeles, CA 90089, USA Roy Frostig EMAIL Google Deep Mind San Francisco, CA 94105, USA Mahdi Soltanolkotabi EMAIL University of Southern California Ming Hsieh Department of Electrical and Computer Engineering Los Angeles, CA 90089, USA |
| Pseudocode | No | The paper does not contain any explicitly labeled pseudocode or algorithm blocks. The methods are described through mathematical formulations and textual explanations. |
| Open Source Code | No | The paper mentions "JAX: composable transformations of Python+Num Py programs, 2018. URL http://github.com/google/jax." However, this refers to a third-party library used, not the authors' own implementation code for the methodology described in the paper. There is no explicit statement about releasing their own code or a direct link to a repository containing their implementation. |
| Open Datasets | No | The paper describes generating its own data for the numerical simulations: "We construct a family of LDS-LS problem instances, parameterized by a scalar ρ (0, ) as follows. The covariate distribution Px is the linear dynamical system xt+1 = Axt + wt..." It does not mention using any existing publicly available datasets, nor does it provide access information (link, DOI, citation) for the data generated in its simulations. |
| Dataset Splits | No | The paper's numerical simulation involves constructing a generative model and running multiple trials ( |
| Hardware Specification | No | All computations are implemented using jax (Bradbury et al., 2018), and run with float64 precision on a single machine. The phrase "on a single machine" is too general and does not provide specific details about the CPU, GPU, or other hardware components. |
| Software Dependencies | No | All computations are implemented using jax (Bradbury et al., 2018), and run with float64 precision on a single machine. While "jax" is mentioned and cited, no specific version number for the JAX library itself is provided. The underlying languages/libraries (Python, NumPy) are also not specified with versions. |
| Experiment Setup | Yes | We construct a family of LDS-LS problem instances, parameterized by a scalar ρ (0, ) as follows. The covariate distribution Px is the linear dynamical system xt+1 = Axt + wt with: A = U diag( ρ, . . . , ρ | {z } n/2 times , ρ, . . . , ρ)UT, U Unif(O(n)), wt N(0, I/4). Here, O(n) denotes the set of n n orthonormal matrices. By construction, ρ is the spectral radius of A. The labels yt are set as yt = xt+1, so that the ground truth W Rn n is equal to A. ... We fix the covariate dimension n = 5 and the trajectory horizon length T = 10n, and vary the number of trajectories m {1, . . . , 10}. Figure 4 shows the result of this experiment, where we also vary ρ {0.98, 0.99, 1.0, 1.01, 1.02}. The error bars are plotted over 1000 trials. |