Post 5 of 6. Previous: Incremental moments, full solves. I assume the 2025 post Introducing skfolio new online convex optimization module for mirror-map intuition. Next: What a research PR to skfolio 1.0.0 actually contains.

The branch at ~/workspace/skfolio on universal_portfolio is an Online Convex Optimization engine for portfolio selection (Hazan, 2016; Orabona, 2019; Li & Hoi, 2014), wired through BaseOptimization. It is not Cover’s mixture over CRPs (Cover, 1991); that module was deleted. Public imports today:

from skfolio.optimization.online import (
    FollowTheWinner, FollowTheLoser, BCRP, UCRP, CRP, BestStock, regret, RegretType,
)

optimization/__init__.py still does not re-export them. FirstOrderOCO, mirror maps, and projectors are internal. That split is the one I would keep for 1.0.

The engine

FirstOrderOCO in _foco.py is a COMID step (Duchi et al., 2010; McMahan, 2011; Xiao, 2010). Given a mirror map $\psi$, a projector onto $K$, a step-size schedule $\eta_t$, optional gradient predictor, and a mode:

OMD

\[x_{t+1} = \arg\min_{x\in K} \bigl\{ \eta_t\langle g_t, x\rangle + D_\psi(x, x_t) \bigr\}.\]

FTRL (as implemented: $\eta$ on the linear term)

\[x_{t+1} = \arg\min_{x\in K} \bigl\{ \eta_t\sum_{s=1}^t \langle g_s, x\rangle + \psi(x) \bigr\}.\]

The helper _composite_update solves the dual form $z=-\eta g + \nabla\psi(x_{\mathrm{center}})$ (centre absent in FTRL), then $x=\nabla\psi^*(z)$, then project_geom, then the external projector. Dynamic maps (AdaGrad diagonal) use grad_psi_after / grad_psi_star_after so the metric $H_{t+1}$ is the one Orabona’s notes want. Burg/PROD bypasses $\nabla\psi^*$ and solves a scalar dual for the simplex constraint (constrained_inverse).

Optimistic updates follow the Rakhlin–Sridharan effective gradient $g_{\mathrm{eff}}=g_t-\tilde g_t+\tilde g_{t+1}$. In OMD the full $g_{\mathrm{eff}}$ enters the proximal step. In FTRL the prediction is added to the cumulative sum. discount implements $G_t=\gamma G_{t-1}+g_t$. SwordMeta is exponential weights over expert engines; it is research code and should not be public in the first PR.

Default log-wealth loss on net returns $r_t$, relatives $x_t=\max(1+r_t,10^{-12})$:

\[\ell_t(w)=-\log(w^\top x_t)=-\log(1+w^\top r_t), \qquad \nabla\ell_t(w)=\frac{-r_t}{1+w^\top r_t}.\]

The analytic gradient lives in _autograd_objectives.py. use_autograd=True is a debug switch. Shipping autograd>=1.6 as a hard 1.0 dependency for a path the default does not use is how you pick a fight with the PR template.

Geometries the first PR can defend

`strategy`	$\psi$	Default mode	Closed form
`eg`	$\sum w_i\log w_i$	FTRL	softmax / multiplicative (Helmbold et al., 1998)
`ogd`	$\tfrac12\|w\|_2^2$	FTRL	Euclidean step + project
`prod`	$-\sum\log w_i$	Burg path	Soft-Bayes (Orseau et al., 2017)

learning_rate="auto" for EG is $\eta_t=\sqrt{\log n/(t+1)}$ (theory; moderate and empirical multiply by constants). OGD uses a diameter/gradient-bound schedule. These are the identities tests should lock: EG on the simplex is multiplicative, OGD is a projected gradient step, FTRL with entropy reproduces EG.

McMahan’s FTRL–OMD equivalence (McMahan, 2011) is the reason one engine can expose mode="omd" and mode="ftrl". The tests on the branch already check that entropy FTRL matches EG’s multiplicative form on a short synthetic path. That test is the reviewable heart of _foco.py. If it fails after the rebase onto 1.0, the PR is not ready, no matter how pretty the gallery looks.

AdaGrad, SWORD, Ada-BARRONS are implemented. They are also the part of the 2025 post that a maintainer cannot review without a paper. Out of the first diff. Tsallis maps exist and are not wired to any FTWStrategy. Deprecated AdaptiveLogBarrierMap should stay unpublished.

Chronology, or why costs are not `previous_weights`

OnlinePortfolioSelection.partial_fit accepts one row. The user-guide chronology is the economic state machine:

Start period $t$ with drifted holdings $\tilde w_{t-1}$.
Rebalance into the already-chosen traded book $w_t$.
Charge costs on $\|w_t-\tilde w_{t-1}\|_1$.
Observe relatives $x_t$.
Update wealth $$W_{t+1}=W_t\cdot(w_t^\top x_t^{\mathrm{net}}-c^\top w_t-\tilde w_{t-1} )$$.
Drift $\tilde w_t \propto w_t\odot x_t$.
OCO (or PA) produces $w_{t+1}$.

all_weights_[t] stores the traded $w_t$. max_turnover constrains $\|w_{t+1}-w_t\|_1$ against the last trade. Costs use the drifted book. MeanRisk’s previous_weights (post 2) is a third object: a parameter on a batch program, defaulting to zeros if unset, so the first buy is costed from cash. OLPS, if previous_weights is omitted, treats the first trade as a starting book and does not invent an entry cost.¹

I would keep both conventions and write them in the user guide with the numerical example already in docs/user_guide/online.rst ($w=(0.5,0.5)$, $x=(1.1,1.0)$, drift to $(0.5238,0.4762)$). Reviewers who have only used MeanRisk will otherwise “fix” OLPS to cost from zero.

fit is a row loop of partial_fit. warm_start=False calls _reset_state_for_fit. predict is inherited from BaseOptimization and applies final weights_ to whatever X you pass: lookahead if that $X$ is the training stream. fit_predict is overridden and returns a MultiPeriodPortfolio of one-row Portfolios using all_weights_[t]. predict_online deep-copies, keeps warm_start, and fit_predicts. Gallery code must call fit_predict, not predict. I already made that mistake in an early draft of the 2025 post’s mental model; the code is the spec.

Follow-the-loser is a different proximal problem

Li–Hoi FTL methods (Li et al., 2012; Li et al., 2015; Li et al., 2013) are default passive-aggressive, update_mode="pa", not FirstOrderOCO. OLMAR wants a predicted relative $\phi$ (SMA or EWMA) to satisfy $\phi^\top w\ge\varepsilon$. PAMR wants $w^\top x\le\varepsilon$ (penalize yesterday’s winners). RMR is OLMAR with an L1-median predictor. CWMR puts a Gaussian around $w$ and enforces a probability margin.

The PA step is

\[w_{t+1} = \arg\min_{w\in K} \tfrac12\|w-w_t\|_2^2 \quad\text{s.t. a linear margin on }w,\]

which collapses to a closed-form step along a centred direction plus a projection. update_mode="md" routes FTL through FOCO with hinge / squared-hinge / softplus. The CWMR docstring calls the MD path experimental. First PR: PA only.

FTL is outside the $O(\sqrt{T\log n})$ log-loss guarantee. It is in the Li–Hoi survey as a family that works on daily equity mean reversion. That is a product reason to ship it, and a documentation reason not to claim sublinear regret for PAMR.

BCRP is the code bridge

BCRP subclasses MeanRisk. With objective_measure=LOG_WEALTH it overwrites expected return with $\sum_t\log((1+r_t)^\top w)$ and solves a convex program: Kelly / Cover CRP in hindsight. Constraints, solver, prior_estimator are the convex stack from post 2. UCRP is $1/n$. CRP is user weights. BestStock is one-hot on the best log-wealth asset.

Static regret (Cover, 1991; Hazan, 2016)

\[R_t = \sum_{s\le t}\bigl(\ell_s(w_s)-\ell_s(u)\bigr), \qquad \ell=-\log(w^\top x),\]

with $u$ the BCRP (or a user comparator). The solver inside _regret._solve_bcrp_constant is Cover-style multiplicative weights on prefixes, not the Dirichlet integral. RegretType.DYNAMIC_UNIVERSAL builds a $T\times n$ CVXPY program with a path-length budget; fine for a notebook, not for a live loop, out of the first PR.

This is the only place OLPS should call MeanRisk: as a hindsight comparator, with an information set no causal strategy is allowed. Tests can check $R_T\ge 0$ on log-wealth (up to numerics) and that EG’s average regret shrinks on a synthetic multiplicative-weights-friendly sequence. They should not check that EG Sharpe matches minimum-variance MeanRisk.

Projection

AutoProjector: box plus budget, optional L1 turnover, via bisection (project_box_and_sum, project_with_turnover). The bisection is on a Lagrange multiplier for the sum constraint with clipping to $[\ell_i,u_i]$; it is the same algorithm you write on a whiteboard for “simplex with bounds”. Groups, linear inequalities, tracking error, variance bounds fall through to a CVXPY project_convex. First PR can ship box+budget (+ turnover) with tests against a simplex projection identity, and leave the cvxpy projector as follow-up issue A. Shipping the full MeanRisk constraint language on day one duplicates _get_weight_constraints badly. A single random feasible point of MeanRisk, projected by OLPS, is a good future test; it is not a reason to block EG.

Empirical caution, not a PR claim

The draft study in paper/follow_the_winner_vs_follow_the_loser.md found that at daily frequency, FTW methods sit next to UCRP, while FTL closes a large fraction of the BCRP–UCRP log-wealth gap; at monthly rebalancing the roles reverse, with a $\sqrt{\Delta}$ scaling of the EG weight deviation.² I wrote about that in 2025. It is a reason to ship both families and to put rebalancing frequency in the gallery. It is not a theorem the PR must prove, and it is not a license to dump fourteen strategies into optimization/__init__.py.

Export cut (repeated in post 6)

In: FollowTheWinner(strategy="eg"|"ogd"), FollowTheLoser(..., update_mode="pa") for olmar/pamr/rmr/cwmr, UCRP/CRP/BCRP/BestStock, regret(..., STATIC), fit / partial_fit / fit_predict / predict_online, analytic log-wealth gradient, box+budget projector.

Internal: FirstOrderOCO, mirror maps, predictors, SwordMeta.

Out: SWORD, Ada-BARRONS, Tsallis, FTL MD, objective= path-risk on $T=1$, DYNAMIC_UNIVERSAL, Cover UP, CORN, autograd as required, public online_walk_forward_predict.

prod is a close call: small, tested, Burg is a genuine third geometry. I would still leave it to a second PR unless tests already make it free. Reviewers can only digest so many mirror maps.

If you do pass previous_weights into an OLPS estimator, it is interpreted as the book already held before the first rebalance, so first-period costs and turnover are measured against it. That matches a live handoff from a previous strategy.⏎
The draft is not a submission. Use it as a caution about daily EG versus UCRP, and as a reason the gallery should show more than one rebalancing frequency. Cross-sectional gradient demeaning on the simplex is a no-op for entropy and Euclidean geometries; do not advertise it as a feature.⏎

References

Hazan, E. (2016). Introduction to online convex optimization. Foundations and Trends in Optimization, 2(3-4), 157–325.
Orabona, F. (2019). A modern introduction to online learning. Arxiv:1912.13213.
Li, B., & Hoi, S. C. H. (2014). Online portfolio selection: A survey. ACM Computing Surveys (CSUR), 46(3), 1–36.
Cover, T. M. (1991). Universal portfolios. Mathematical Finance, 1(1), 1–29.
Duchi, J., Shalev-Shwartz, S., Singer, Y., & Tewari, A. (2010). Composite objective mirror descent. Proceedings of the 23rd Annual Conference on Learning Theory, 14–26.
McMahan, H. B. (2011). Follow-the-Regularized-Leader and Mirror Descent: Equivalence Theorems and L1 Regularization. Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, 525–533.
Xiao, L. (2010). Dual averaging methods for regularized stochastic learning and online optimization. Journal of Machine Learning Research, 11, 2543–2596.
Helmbold, D. P., Schapire, R. E., Singer, Y., & Warmuth, M. K. (1998). On-Line Portfolio Selection Using Multiplicative Updates. Proceedings of the 11th Annual Conference on Computational Learning Theory, 126–135.
Orseau, L., Lattimore, T., & Legg, S. (2017). Soft-bayes: Prod for mixtures of experts with log-loss. International Conference on Algorithmic Learning Theory, 372–399.
Li, B., Zhao, P., & Hoi, S. C. H. (2012). On-Line Portfolio Selection with Passive Aggressive Mean Reversion. Proceedings of the 29th International Conference on Machine Learning.
Li, B., Hoi, S. C. H., Sahoo, D., & Liu, Z.-Y. (2015). Moving average reversion strategy for on-line portfolio selection. Artificial Intelligence, 222, 104–123.
Li, B., Hoi, S. C. H., Zhao, P., & Gopalkrishnan, V. (2013). Confidence Weighted Mean Reversion Strategy for Online Portfolio Selection. ACM Transactions on Knowledge Discovery from Data, 7(1), 1–38.

`strategy`	\(\psi\)	Default mode	Closed form
`eg`	\(\sum w_i\log w_i\)	FTRL	softmax / multiplicative (Helmbold et al., 1998)
`ogd`	\(\tfrac12\|w\|_2^2\)	FTRL	Euclidean step + project
`prod`	\(-\sum\log w_i\)	Burg path	Soft-Bayes (Orseau et al., 2017)