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Title: Fluctuation-dissipation relations for stochastic gradient descent

Paper content:
The notion of the stationary equilibrium ensemble has played a central role in statistical mechanics.
In machine learning as well, training serves as generalized equilibration that drives the probability distribution of model parameters toward stationarity.
Here, we derive stationary fluctuation-dissipation relations that link measurable quantities and hyperparameters in the stochastic gradient descent algorithm.
These relations hold exactly for any stationary state and can in particular be used to adaptively set training schedule.
We can further use the relations to efficiently extract information pertaining to a loss-function landscape such as the magnitudes of its Hessian and anharmonicity.
Our claims are empirically verified.
Equilibration rules the long-term fate of many macroscopic dynamical systems.
For instance, as we pour water into a glass and let it be, the stationary state of tranquility is eventually attained.
Zooming into the tranquil water with a microscope would reveal, however, a turmoil of stochastic fluctuations that maintain the apparent stationarity in balance.
This is vividly exemplified by the Brownian motion BID3 : a pollen immersed in water is constantly bombarded by jittery molecular movements, resulting in the macroscopically observable diffusive motion of the solute.
Out of the effort in bridging microscopic and macroscopic realms through the Brownian movement came a prototype of fluctuation-dissipation relations BID6 BID37 .
These relations quantitatively link degrees of noisy microscopic fluctuations to smooth macroscopic dissipative phenomena and have since been codified in the linear response theory for physical systems BID28 BID9 BID16 , a cornerstone of statistical mechanics.Machine learning begets another form of equilibration.
As a model learns patterns in data, its performance first improves and then plateaus, again reaching apparent stationarity.
This dynamical process naturally comes equipped with stochastic fluctuations as well: often given data too gigantic to consume at once, training proceeds in small batches and random selections of these mini-batches consequently give rise to the noisy dynamical excursion of the model parameters in the loss-function landscape, reminiscent of the Brownian motion.
It is thus natural to wonder if there exist analogous fluctuation-dissipation relations that quantitatively link the noise in mini-batched data to the observable evolution of the model performance and that in turn facilitate the learning process.Here, we derive such fluctuation-dissipation relations for the stochastic gradient descent algorithm.
The only assumption made is stationarity of the probability distribution that governs the model parameters at sufficiently long time.
Our results thus apply to generic cases with non-Gaussian mini-batch noises and nonconvex loss-function landscapes.
Practically, the first relation (FDR1) offers the metric for assessing equilibration and yields an adaptive algorithm that sets learning-rate schedule on the fly.
The second relation (FDR2) further helps us determine the properties of the lossfunction landscape, including the strength of its Hessian and the degree of anharmonicity, i.e., the deviation from the idealized harmonic limit of a quadratic loss surface and a constant noise matrix.Our approach should be contrasted with recent attempts to import the machinery of stochastic differential calculus into the study of the stochastic gradient descent algorithm BID21 BID20 BID22 BID19 BID33 BID4 BID12 BID39 .
This line of work all assumes Gaussian noises and sometimes additionally employs the quadratic harmonic approximation for loss-function landscapes.
The more severe drawback, however, is the usage of the analogy with continuous-time stochastic differential equations, which is inconsistent in general (see Section 2.3.3).
Instead, the stochastic gradient descent algorithm can be properly treated within the framework of the KramersMoyal expansion BID36 BID7 BID30 BID29 BID18 .The
paper is organized as follows. In
Section 2, after setting up notations and deriving a stationary fluctuation-dissipation theorem (FDT), we derive two specific fluctuation-dissipation relations. The
first relation (FDR1) can be used to check stationarity and the second relation (FDR2) to delineate the shape of the loss-function landscape, as empirically borne out in Section 3. An
adaptive scheduling method is proposed and tested in Section 3.3. We
conclude in Section 4 with future outlooks.
In this paper, we have derived the fluctuation-dissipation relations with no assumptions other than stationarity of the probability distribution.
These relations hold exactly even when the noise is nonGaussian and the loss function is nonconvex.
The relations have been empirically verified and used to probe the properties of the loss-function landscapes for the simple models.
The relations further have resulted in the algorithm to adaptively set learning-rate schedule on the fly rather than presetting it in an ad hoc manner.
In addition to systematically testing the performance of this adaptive scheduling algorithm, it would be interesting to investigate non-Gaussianity and noncovexity in more details through higher-point observables, both analytically and numerically.
It would also be interesting to further elucidate the physics of machine learning by extending our formalism to incorporate nonstationary dynamics, linearly away from stationarity BID28 BID9 BID16 and beyond BID11 BID5 , so that it can in particular properly treat overfitting cascading dynamics and time-dependent sample distributions.

assistant: text: Paper's concepts work in the discrete-time formalism, use the master equation, and  remove reliance on a locally quadratic approximation of the loss function or on any Gaussian asumptions of the SGD noise.