Anthony M. Polloreno, Reuben R. W. Wang, Nikolas A. Tezak
In this note we extend the definition of the Information Processing Capacity (IPC) by Dambre et al [1] to include the effects of stochastic reservoir dynamics. We quantify the degradation of the IPC in the presence of this noise. [1] Dambre et al. Scientific Reports 2, 514, (2012)
Anthony M. Polloreno
Reservoir computation is a recurrent framework for learning and predicting time series data, that benefits from extremely simple training and interpretability, often as the the dynamics of a physical system. In this paper, we will study the impact of noise on the learning capabilities of analog reservoir computers. Recent work on reservoir computation has shown that the information processing capacity (IPC) is a useful metric for quantifying the degradation of the performance due to noise. We further this analysis and demonstrate that this degradation of the IPC limits the possible features that can be meaningfully constructed in an analog reservoir computing setting. We borrow a result from quantum complexity theory that relates the circuit model of computation to a continuous time model, and demonstrate an exponential reduction in the accessible volume of reservoir configurations. We conclude by relating this degradation in the IPC to the fat-shattering dimension of a family of functions describing the reservoir dynamics, which allows us to express our result in terms of a classification task. We conclude that any physical, analog reservoir computer that is exposed to noise can only be used to perform a polynomial amount of learning, despite the exponentially large latent space, even with an exponential amount of post-processing.
Anthony M. Polloreno
In noisy physical reservoirs, the classical information-processing capacity $C_{\mathrm{ip}}$ quantifies how well a linear readout can realize tasks measurable from the input history, yet $C_{\mathrm{ip}}$ can be far smaller than the observed rank of the readout covariance. We explain this ``missing capacity'' by introducing the innovation capacity $C_{\mathrm{i}}$, the total capacity allocated to readout components orthogonal to the input filtration (Doob innovations, including input-noise mixing). Using a basis-free Hilbert-space formulation of the predictable/innovation decomposition, we prove the conservation law $C_{\mathrm{ip}}+C_{\mathrm{i}}=\mathrm{rank}(Σ_{XX})\le d$, so predictable and innovation capacities exactly partition the rank of the observable readout dimension covariance $Σ_{XX}\in \mathbb{R}^{\rm d\times d}$. In linear-Gaussian Johnson-Nyquist regimes, $Σ_{XX}(T)=S+T N_0$, the split becomes a generalized-eigenvalue shrinkage rule and gives an explicit monotone tradeoff between temperature and predictable capacity. Geometrically, in whitened coordinates the predictable and innovation components correspond to complementary covariance ellipsoids, making $C_{\mathrm{i}}$ a trace-controlled innovation budget. A large $C_{\mathrm{i}}$ forces a high-dimensional innovation subspace with a variance floor and under mild mixing and anti-concentration assumptions this yields extensive innovation-block differential entropy and exponentially many distinguishable histories. Finally, we give an information-theoretic lower bound showing that learning the induced innovation-block law in total variation requires a number of samples that scales with the effective innovation dimension, supporting the generative utility of noisy physical reservoirs.
Essential AI, Ishaan Shah, Anthony M. Polloreno et al.
We demonstrate that Muon, the simplest instantiation of a second-order optimizer, explicitly expands the Pareto frontier over AdamW on the compute-time tradeoff. We find that Muon is more effective than AdamW in retaining data efficiency at large batch sizes, far beyond the so-called critical batch size, while remaining computationally efficient, thus enabling more economical training. We study the combination of Muon and the maximal update parameterization (muP) for efficient hyperparameter transfer and present a simple telescoping algorithm that accounts for all sources of error in muP while introducing only a modest overhead in resources. We validate our findings through extensive experiments with model sizes up to four billion parameters and ablations on the data distribution and architecture.