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Prospective Learning: Learning for a Dynamic Future
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Develops prospective learning, a theoretical framework where learners output time-indexed sequences of predictors to minimize future risk, proposes prospective ERM with a sinusoidal time embedding, and validates it on synthetic, MNIST and CIFAR-10 non-stationary tasks against online continual learning baselines.
Problem
Standard PAC learning assumes IID data and a time-agnostic hypothesis, so it fails when data distributions and the optimal hypothesis change over time; existing online and continual learners minimize past or current error rather than future risk.
Approach
A prospective learner maps past data to an infinite sequence of predictors and minimizes expected cumulative future loss; prospective ERM incorporates time as an input (via sinusoidal embedding) so a single network can produce time-dependent predictions, and is shown to be a strong learner under consistency and uniform concentration conditions.
Key results
- On MNIST and CIFAR-10 prospective tasks, prospective ERM's prospective risk approaches Bayes risk while Follow-the-Leader, online SGD and Bayesian GD do not achieve low prospective risk (chance is 0.742).
- On synthetic tasks, prospective ERM's instantaneous and prospective risks eventually converge to zero, with minimal spikes at task switches unlike online baselines.
- For a period-2 Gaussian example, time-agnostic ERM converges to a risk above Bayes, whereas prospective ERM achieves Bayes risk Phi(-mu/sigma).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add time as an input to the predictor | Prospective ERM · Synthetic independent non-identically distributed tasks | prospective risk: prospective risk ... converge to zero | Helped |
| Add time as an input to the predictor | Prospective ERM · MNIST and CIFAR-10 tasks | prospective risk | Helped |
| Use prospective ERM instead of time-agnostic online learning baselines | Synthetic, MNIST, and CIFAR-10 independent non-identically distributed tasks · Independent non-identically distributed tasks | prospective risk | Helped |
| Concatenate the time embedding directly to input images instead of to a layer before softmax (ablation) | Prospective ERM CNN · CIFAR-10 tasks | prospective risk: worse by almost 0.2 | Hurt |
| Forecast future parameter estimates using the MAP estimate's rate of change | Prospective MAP · IID Bernoulli example | prospective risk convergence rate | Helped |
| Use a prior that is unfavorable to the true Bernoulli probability | MAP estimator · IID Bernoulli example | prospective risk convergence rate | Hurt |
Surprising findings
- Follow-the-Leader performs as well as or better than online SGD and Bayesian gradient descent on the prospective tasks.
- Prospection can help even in the IID setting, as shown by a forecasting-based MAP estimator with an unfavorable prior.
Limitations
- Experiments are limited to small-scale synthetic, MNIST and CIFAR-10 tasks; the authors leave more elaborate experiments to future work.
- Strong and weak prospective learnability equivalence is unknown.
- Theory relies on assumptions such as finite families of processes and countable hypothesis classes.
Experimental setup
Datasets: MNIST CIFAR-10 synthetic 1D and 2D binary classification
Benchmarks: MNIST prospective learning tasks CIFAR-10 prospective learning tasks synthetic prospective learning tasks
Metrics: prospective risk instantaneous risk zero-one error
Baselines: Follow-the-Leader Online SGD Bayesian gradient descent time-agnostic ERM
Builds on: empirical risk minimization PAC learning Bayesian gradient descent transformer positional encoding
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- Universal Rates of Empirical Risk Minimization (NeurIPS 2024 · similarity 0.85)
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