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Title: Stochastic Learning of Additive Second-Order Penalties with  Applications to Fairness

Paper content:
Many notions of fairness may be expressed as linear constraints, and the resulting constrained objective is often optimized by transforming the problem into its Lagrangian dual with additive linear penalties.
In non-convex settings, the resulting problem may be difficult to solve as the Lagrangian is not guaranteed to have a deterministic saddle-point equilibrium.
In this paper, we propose to modify the linear penalties to second-order ones, and we argue that this results in a more practical training procedure in non-convex, large-data settings.
For one, the use of second-order penalties allows training the penalized objective with a fixed value of the penalty coefficient, thus avoiding the instability and potential lack of convergence associated with two-player min-max games.
Secondly, we derive a method for efficiently computing the gradients associated with the second-order penalties in stochastic mini-batch settings.
Our resulting algorithm performs well empirically, learning an appropriately fair classifier on a number of standard benchmarks.
Machine learning systems are becoming increasingly prevalent in real-world applications, consequently affecting the decisions that determine a person's life and future, such as playing a role in parole conditions BID1 , loan applications BID12 , and airport screening BID13 .
Recent work has shown that such machine learning models often have biases which can unfairly disadvantage certain groups.
For example, learned word embeddings exhibit gender-specific biases in what should be gender neutral words BID4 .
In another case, a machine learning model's predictions regarding convict recidivism were found to be unfairly biased against African-Americans ( BID1 .
While it may seem at first that simply ignoring the features corresponding to these protected traits when training can alleviate this, previous work BID18 has shown that enforcing such blindness is largely ineffective due to redundant encodings in the data.
In other words, while the learning algorithm used may not be biased, the data can be inherently biased in complex ways, and this leads to models which perpetuate these undesirable biases.Research into the challenging problem of machine learning fairness is therefore of great interest.
To better specify this problem, previous work has elaborated on precise notions of fairness, such as demographic parity BID8 , equal opportunity BID14 , etc.
These notions can often be expressed mathematically as a linear constraint on the output of a machine learning model, taken in expectation over the entire data distribution.
Accordingly, a number of recent works have proposed to incorporate fairness during training by expressing the objective as a constrained optimization problem BID24 BID11 .
If the original objective is convex, the addition of linear constraints results in a problem which may be readily solved by Lagrangian methods.However, modern machine learning models are often not in a convex form.
Indeed, the success of deep neural networks over the past decade makes it clear that the most well-performing models are often highly non-convex and optimized via stochastic gradient methods over large amounts of data BID19 BID20 .
It is unfortunate that much of the existing work on fairness in machine learning has provided methods which are either focused on the convex, small data-set setting BID24 BID11 , or otherwise require sophisticated and complex training methods BID6 .In
this paper, we present a general method for imposing fairness conditions during training, such that it is practical in non-convex, large data settings. We
take inspiration from the standard Lagrangian method of augmenting the original loss with linear penalties. In
non-convex settings, this dual objective must be optimized with respect to both model parameters and penalty coefficients concurrently, and in general is not guaranteed to converge to a deterministic equilibrium.We propose to re-express the linear penalties associated with common fairness criteria as secondorder penalties. Second-order
penalties are especially beneficial in non-convex settings, as they may be optimized using a fixed non-negative value for the penalty coefficient λ. When λ → 0 the
optimization corresponds to an unconstrained objective, while as λ → ∞, the problem approaches that of a hard equality constraint. This allows us
to avoid sophisticated optimization methods for potentially non-convergent two-player games. Instead, we only
need to choose a fixed value for the penalty coefficient, which may be easily determined via standard hyperparameter optimization methods, such as cross-validation. As an additional
benefit, by choosing the penalty coefficient on a separate validation set, we can improve generalization performance.Second-order penalties, however, potentially introduce a new problem: By squaring an expectation over the entire data distribution, the resulting penalized loss is no longer an expectation of loss functions on individual data points sampled from the distribution, and therefore not readily approachable by stochastic gradient methods. We solve this by
presenting an equivalent form of the second-order penalty as an expectation of individual loss functions on pairs of independently sampled data points.Our resulting algorithm is thus not only more practical to optimize in non-convex settings, using a fixed value for the penalty coefficient, but is also easily optimized in large-data settings via standard stochastic gradient descent. We evaluate the
performance of our algorithm in a number of different settings. In each setting
, our algorithm is able to adequately optimize the desired constraints, such as encouraging feature orthonormality in deep image autoencoders and imposing predictive fairness across protected data groups.

assistant: text: We propose a method to stochastically optimize second-order penalties and show how this may apply to training fairness-aware classifiers.